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+# CROSS-SUPERVISED OBJECT DETECTION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+After learning a new object category from image-level annotations (with no object bounding boxes), humans are remarkably good at precisely localizing those objects. However, building good object localizers (i.e., detectors) currently requires expensive instance-level annotations. While some work has been done on learning detectors from weakly labeled samples (with only class labels), these detectors do poorly at localization. In this work, we show how to build better object detectors from weakly labeled images of new categories by leveraging knowledge learned from fully labeled base categories. We call this learning paradigm cross-supervised object detection. While earlier works investigated this paradigm, they did not apply it to realistic complex images (e.g., COCO), and their performance was poor. We propose a unified framework that combines a detection head trained from instance-level annotations and a recognition head learned from image-level annotations, together with a spatial correlation module that bridges the gap between detection and recognition. These contributions enable us to better detect novel objects with image-level annotations in complex multi-object scenes such as the COCO dataset.
+
+# 1 INTRODUCTION
+
+Deep architectures have achieved great success in many computer vision tasks including object recognition and the closely related problem of object detection. Modern detectors, such as the Faster RCNN (Ren et al., 2015), YOLO (Redmon et al., 2016), and RetinaNet (Lin et al., 2017), use the same network backbone as popular recognition models. However, even with the same backbone architectures, detection and recognition models require different types of supervision. A good detector relies heavily on precise bounding boxes and labels for each instance (we shall refer to these as instance-level annotations), whereas a recognition model needs only image-level labels. Needless to say, it is more time consuming and expensive to obtain high quality bounding box annotations than class labels. As a result, current detectors are limited to a small set of categories relative to their object recognition counterparts. To address this limitation, it is natural to ask, “Is it possible to learn detectors with only class labels?” This problem is commonly referred to as weakly supervised object detection (WSOD).
+
+Early WSOD work (Hoffman et al., 2014) showed fair performance by directly applying recognition networks to object detection. More recently, researchers have used multiple instance learning methods (Dietterich et al., 1997) to recast WSOD as a multi-label classification problem (Bilen & Vedaldi, 2016). However, these weakly supervised detectors perform poorly at localization. Most WSOD experiments have been conducted on the ILSVRC (Russakovsky et al., 2015) data set, in which images have only a single object, or on the PASCAL VOC (Everingham et al., 2010) data set, which has only 20 categories. The simplicity of these data sets limits the number and types of distractors in an image, making localization substantially easier. Learning from only class labels, it is challenging to detect objects at different scales in an image that contains many distractors. In particular, as shown in our experiments, weakly supervised object detectors do not work well in complex multi-object scenes, such as the COCO dataset (Lin et al., 2014).
+
+To address this challenge, we focus on a form of learning in which the localization of classes with only object labels (weakly labeled classes) can benefit from other classes that have ground truth bounding boxes (fully labeled classes). We refer to this interesting learning paradigm as crosssupervised object detection (CSOD). While several works (Hoffman et al., 2014; Tang et al., 2016;
+
+Yang et al., 2019a; Redmon & Farhadi, 2017) have explored this problem before, they still have the same limitation as the WSOD work we mentioned above. Those cross-supervised object detectors work under simplified scenarios (e.g., ILSVRC data set) where images contain single objects and are object-centered. They struggle to learn under more complex and realistic scenarios, where there are multiple objects from potentially very different classes, and objects could be small and appear anywhere in the images. In this work, we show that by doing multi-task learning on both weaklysupervised base classes and fully-supervised novel classes, our model is able to learn a good detector under the CSOD setting.
+
+More formally, we define CSOD as follows. At training time, we are given 1) images contain objects from both base and novel classes, 2) both class labels and ground truth bounding boxes for base objects, and 3) only class labels for novel objects. Our goal is to detect novel objects. In CSOD, base classes and novel classes are disjoint. Thus, it can be seen as performing fullysupervised detection on the base classes and weakly supervised detection on the novel classes. It has similarities to both transfer learning and semi-supervised learning, since it transfer knowledge from base class to novel class and have more information about some instances than other instances. However, CSOD represents a distinct and novel paradigm for learning.
+
+The current weakly-supervised method has several drawbacks to learn from a multi objects image. As shown in Fig. 1, a weakly supervised object detector tends to detect only the most discriminating part of novel objects instead of the whole object. Notice how only the head of the person, and not the whole body, is detected. Another issue is that the localizer for one object (e.g., the horse) may be confused by the occurrence of another object, such as the person on the horse. This example illustrates the gap between detection and recognition: without ground truth bounding boxes, the detector acts like a standard recognition model – focusing on discriminating rather than detecting.
+
+In this paper, we explore two major mechanisms for improving on this. Our first mechanism is unifying detection and recognition. Using the same network backbone architecture, recognition and detection can be seen as image-level classification and region-level classification respectively, suggesting a strong relation between them. In particular, it suggests a shared training framework in which the same backbone is used with different heads for detection and recognition. Thus, we combine a detection head learned from ground truth bounding boxes, and a recognition head learned in a weakly supervised fashion from class labels. Unlike a traditional recognition head, our recognition head produces a class score for multiple proposals and is capable of detecting objects. The second mechanism is learning a spatial correlation module to reduce the gap between detection and recognition. It takes several high-confidence bounding boxes produced by the recognition head as input, and learns to regress ground truth bounding boxes. By combining these mechanisms together, our model outperforms all previous models when all novel objects are weakly labeled.
+
+In summary, our contributions are three-fold. First, we define a new task—cross-supervised object detection, which enables us to leverage knowledge from fully labeled base categories to help learn a robust detector from novel object class labels only. Second, we propose a unified framework in which two heads are learned from class labels and detection labels respectively, along with a spatial correlation module bridging the gap between recognition and detection. Third, we significantly outperform existing methods (Zhang et al. (2018a); Tang et al. (2017; 2018)) on PASCAL VOC and COCO, suggesting that CSOD could be a promising approach for expanding object detection to a much larger number of categories.
+
+# 2 RELATED WORK
+
+Weakly supervised object detection. WSOD (Kosugi et al. (2019); Zeng et al. (2019); Yang et al. (2019b); Wan et al. (2019); Arun et al. (2019); Wan et al. (2018); Zhang et al. (2018b); Ren et al. (2020); Zhang et al. (2018c); Li et al. (2019); Gao et al. (2019b); Kosugi et al. (2019)) attempts to learn a detector with only image category labels. Most of these methods adopt the idea of Multiple Instance Learning (Dietterich et al. (1997)) to recast WSOD as a multi-label classification task. Bilen & Vedaldi (2016) propose an end-to-end network by modifying a classifier to operate at the level of image regions, serving as a region selector and a classifier simultaneously. Tang et al. (2017) and Tang et al. (2018) find that several iterations of online refinement based on the outputs of previous iterations boosts performance. Wei et al. (2018) and Diba et al. (2017) use semantic segmentation based on class activation maps (Zhou et al. (2016)) to help generate tight bounding boxes. However,
+
+![](images/a6525e87d58457804058030797ef06c2c5c8237d897d3edcf64e7b4ca07b3647.jpg)  
+Figure 1: A comparison between weakly supervised object detector and our detector. Weakly supervised object detector only detects the most discriminating part of an object, e.g., focus on head of a person when detecting a person; or being distracted by co-occurring instances, e.g., distracted by the person on the horse when detecting a horse. Our detector can address these issues.
+
+WSOD methods tend to focus on the most discriminating part of an object and are prone to distractions from co-occurring objects. Detecting a part of the object or distractors represents convergence to a local optimum. Thus, their performance depends heavily on initialization. In comparison, our proposed cross-supervised object detector alleviates the issue of getting trapped in a local optimum by leveraging knowledge learned from fully labeled base categories.
+
+Cross-supervised object detection. There are several previous works using both image-level and instance-level annotations. Kuen et al. (2019) learned a parameter transferring function between a classifier and a detector, enabling an image-based classification network to be adapted to a regionbased classification network. Hoffman et al. (2014) and Tang et al. (2016) propose methods of adaptation for knowledge transfer from classification features to detection features. Uijlings et al. (2018) use a proposal generator trained on base classes to transfer knowledge by leveraging a MIL framework, organized in a semantic hierarchy. Hoffman et al. (2015) design a three-step framework to learn a feature representation from weakly supervised classes and strongly supervised classes jointly. However, these methods can only perform object localization in single object scenes such as ILSVRC, whereas our method can perform object detection in complex multi-object scenes as well, e.g. COCO. Also, it is worth noting that we are doing multi-task learning, which means that we jointly learn from base and novel classes. In comparison, some works (Uijlings et al., 2018) are doing transfer learning. They first learn a model on base classes and then transfer and fine-tune the model on novel classes. Gao et al. (2019a) use a few instance-level labels and a large scale of image-level labels for each category in a training-mining framework, which is referred to as semisupervised detection. Zhang et al. (2018a) propose a framework named MSD that learn objectness on base categories and use it to reject distractors when learning novel objects. In comparison, our spatial correlation module not only learns objectness, but also refines coarse bounding boxes. Further, our model learns from both base and novel classes instead of only novel classes.
+
+# 3 CROSS-SUPERVISED OBJECT DETECTION
+
+CSOD requires us to learn from instance-level annotations (detection labels) and image-level annotations (recognition labels). In this section, we explain the unification of detection and recognition and introduce our framework. In the next section, we describe our novel spatial correlation module.
+
+# 3.1 UNIFYING DETECTION AND RECOGNITION
+
+How to learn a detector from both instance-level and image-level annotations? Since detection and recognition can be seen as region-level and image-level classification respectively, a natural choice is to design a unified framework that combines a detection head and a recognition head that can learn from image-level and instance-level annotations respectively. Here we exploit several baselines to unify the detection and recognition head. (1) Finetune. We first learn through the detection head on base classes with fully labeled samples. Then, we finetune our model using the recognition head on novel classes with only class labels. (2) Two Head. We simultaneously learn the detection and recognition head on base and novel classes, respectively. The weights of the backbones are updated using the loss backpropagated from both heads jointly. (3) Two head +. Instead of learning only on novel classes, we learn the recognition head from class labels of both base and novel classes whereas the recognition head remain the same. (4) Two Branch. Instead of having two shared fully-connected layers after RoI pooling layer (see Fig. 2), we make these two fully-connected layers seperated, allowing the detection and recognition head to have separate unshared pair of fully-connected layers each. Everything else is the same as the Two Head baseline. Experiments are conducted to compare these baselines in $\ S \ S . 1$ and $\ S \ S . 2$ . Our proposed model is based on Two Head. We will discuss the details in $\ S 3 . 2$ .
+
+![](images/41ec47eab0504bf1e8dab709b878701e7e9a92c5e76db3b8f8be856947fd4534.jpg)  
+Figure 2: Our Detection-Recognition Network (DRN) without the spatial correlation module. In this illustration, Person belongs to novel classes and Boat belongs to base classes. The recognition head learns from the class label Person and outputs the top-scoring bounding box to help the detection head learn to detect the person. The spatial correlation module, discussed in $\ S 4$ , can be added to further refine the top-scoring bounding boxes.
+
+The connection between the recognition and detection head. The baselines mentioned above only use the recognition head to detect novel objects, ignoring the fact that a detection head can play the same role even better. A majority of WSOD methods (Tang et al. (2017); Wan et al. (2019); Wei et al. (2018)) find that re-train a new detector taking the top-scoring bounding boxes from a weakly supervised object detector as ground truth marginally improve the performance. Even with coarse and noisy pseudo bounding boxes, a standard object detector produces better detection results than a weakly supervised object detector. Keeping this hypothesis in mind, we introduce a guidance from the recognition head to the detection head. For each of the novel categories existing in a training sample, the recognition head outputs the top-scoring bounding box, which are then used by the detection head as supervision in that sample.
+
+# 3.2 DETECTION-RECOGNITION NETWORK
+
+The structure of our Detection-Recognition Network (DRN) is shown in Fig. 2. Given an image, we first generate 2000 object proposals by Selective Search (Uijlings et al. (2013)) or RPN (Ren et al. (2015)) trained on base classes. The image and proposals are fed into several convolutional (conv) layers followed by a region-of-interest (RoI) pooling layer (Girshick (2015)) to output fixed-size feature maps. Then, these feature maps are fed into two fully connected (fc) layers to produce a collection of proposal features, which are further branched into the recognition and detection head.
+
+Recognition Head. We followed previous WSOD methods to design our recognition head. Since OICR (Tang et al. (2017)) is simple, neat, and commonly being used, we make our recognition head the same as OICR, but with fewer refinement branches to reduce the computation cost. However, our recognition head can be replaced by any WSOD structure as shown in $\ S 5 . 3$ .
+
+Within the recognition head as shown in Fig. 2, the proposal features are branched into three streams producing three matrices $\mathbf { x } ^ { c } , \mathbf { x } ^ { d } , \mathbf { x } ^ { e } \in \mathbb { R } ^ { C \times | R | }$ , where $C$ is the number of novel classes and $| R |$ is the number of proposals. Then the two matrices $\mathbf { x } ^ { c }$ and $\mathbf { x } ^ { d }$ are passed through a softmax function
+
+![](images/1450e19e8f29587235da2e261c12a62149ab30614da707a9c2c6b3b5fee699e1.jpg)  
+Figure 3: Our spatial correlation module (SCM). Our SCM learns to capture spatial correlation among high-confidence bounding boxes, generating a class-agnostic heatmap for the whole image. A heatmap detector is then trained to learn ground truth bounding boxes.
+
+over classes and proposals respectively: $\sigma ( \mathbf { x } ^ { c } )$ and $\sigma ( \mathbf { x } ^ { d } )$ . A proposal score $\mathbf { x } _ { c r } ^ { R }$ , indicating the score of $c ^ { t h }$ novel class for $r ^ { t h }$ proposal, corresponds to the respective element of the matrix $\bar { \mathbf { x } } ^ { R } =$ $\sigma ( \mathbf { x } ^ { c } ) \odot \sigma ( \mathbf { x } ^ { d } )$ , where $\odot$ refers to an element-wise product. Finally, we obtain the image score of $c ^ { t h }$ class $\phi _ { c }$ by summing over all proposals: $\begin{array} { r } { \phi _ { c } = \sum _ { r = 1 } ^ { | R | } x _ { c r } ^ { R } } \end{array}$ . Then we culate a standard multi-class cross-entropy loss as shown in the first term of Eq.1. Another matrix $\mathbf { x } ^ { e }$   
+function over classes, the result of which is expresses as a weighted multi-class cross entropy loss as shown in the second term of Eq.1. We set the pseudo label for each proposal $r$ based on its IoU (or overlap) with the top-scoring proposal of $c ^ { t h }$ class, $y _ { c r } = 1$ if $\mathrm { I o U } > 0 . 5$ and $y _ { c r } = 0$ otherwise. The weight $w _ { r }$ for each proposal $r$ is its IoU with the top-scoring proposal. The total loss for the recognition head is
+
+$$
+L _ { r e c } = [ - \sum _ { c = 1 } ^ { C } y _ { c } l o g \phi _ { c } + ( 1 - y _ { c } ) l o g ( 1 - \phi _ { c } ) ] + [ - \frac { 1 } { | { \cal R } | } \sum _ { r = 1 } ^ { | { \cal R } | } \sum _ { c = 1 } ^ { C + 1 } w _ { r } y _ { c r } l o g x _ { c r } ^ { e } ]
+$$
+
+Supervision from our recognition head. We use the matrix $x ^ { e }$ to propose pseudos bounding boxes to guide the detection head. Specifically, we select one top-scoring proposal for each object category that appears in the image as a pseudo bounding box, as done in OICR. We introduce the spatial correlation module in $\ S 4$ , to further refine this pseudo ground truth.
+
+Detection Head. Now that we have pseudo bounding boxes for novel objects and ground truth bounding boxes for base objects, we train our detection head like a standard detector. For simplicity and efficiency, our detection head use the same structure of Faster R-CNN (Ren et al. (2015)). At inference time, the detection head produces detection results for both base categories and novel categories.
+
+# 4 LEARNING TO MODEL SPATIAL CORRELATION
+
+Our intuition is that there exists spatial correlation among high-confidence bounding boxes, and such spatial correlation can be captured to predict ground truth bounding boxes. By representing the spatial correlation in a class-agnostic heatmap, we can easily learn a mapping from recognitionbased bounding boxes to ground truth bounding boxes for base categories, and then transfer this mapping to novel categories.
+
+Thus, we propose a spatial correlation module (SCM). SCM is used as a guidance refinement technique, taking sets of high-confidence bounding boxes from the recognition head, and correspondingly returning pseudo ground truth bounding boxes to the detection head. These pseudo ground truth boxes act as supervision while training on novel categories. The framework of SCM is showed in Fig. 3. Within this module, we first generate a class agnostic heatmap based on the high-confidence bounding boxes predicted by our recognition head, and then we perform detection on top of the heatmap.
+
+Heatmap synthesis. We want to capture the information about how the high-confidence bounding boxes interact amongst themselves. Here, we introduce a simple way of achieving this using a class-agnostic heatmap. For each category existing in the image $y _ { c } = 1 , c \in C$ , we first threshold and select high-confidence bounding boxes of class $c$ . Then we synthesize a corresponding classagnostic heatmap, which is essentially a two-channel feature map of the same size as the original image. The value at each pixel is the sum and the maximum of confidence over all selected bounding boxes covering that pixel.
+
+Table 1: Object Detection performance (mAP $\%$ ) on PASCAL VOC 2007 test set. ∗ indicates using the structure of OICR in the recognition head. ”MSD-Ens” is the ensemble of AlexNet and VGG16. ”MSD-Ens+FRCN” indicates using an ensemble model to predict pseudo ground truths and then learn a Fast-RCNN (Girshick (2015)) using VGG-16.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Base mean</td><td colspan="9">Novel</td><td rowspan="2"></td></tr><tr><td>table</td><td>dog</td><td>horse</td><td>mbike</td><td>person</td><td>plant sheep</td><td></td><td>sofa</td><td>train tv</td></tr><tr><td>OICR</td><td>42.1</td><td>33.4</td><td>29.3</td><td>56.3</td><td>64.6</td><td>8.0</td><td>23.5</td><td>47.2</td><td>47.2</td><td>48.3 61.7</td><td>mean 42.0</td></tr><tr><td>PCL</td><td>49.2</td><td>51.5</td><td>37.3</td><td>63.3</td><td>63.9</td><td>15.8 23.6</td><td>48.8</td><td>55.3</td><td>61.2</td><td>62.1</td><td>48.3</td></tr><tr><td>MSD-VGG16</td><td>50.6</td><td>14.3</td><td>69.3</td><td>65.4 69.6</td><td>2.4</td><td>20.5</td><td>54.6</td><td>34.3</td><td>58.3</td><td>54.6</td><td>44.3</td></tr><tr><td>MSD-Ens</td><td>53.4</td><td>18.3</td><td>70.6</td><td>66.7 69.8</td><td></td><td>3.7 24.7</td><td>55.0</td><td>37.4</td><td>58.3</td><td>57.3</td><td>46.1</td></tr><tr><td>MSD-Ens+FRCN</td><td>53.9</td><td>15.3</td><td>72.0</td><td>74.4</td><td>65.2</td><td>15.4</td><td>25.1 53.6</td><td>54.4</td><td>45.6</td><td>61.4</td><td>48.2</td></tr><tr><td>Weight Transfer</td><td>68.4</td><td>10.4</td><td>61.0</td><td>58.0</td><td>65.1</td><td>19.8</td><td>19.5</td><td>58.0</td><td>50.8 58.6</td><td>52.7</td><td>45.4</td></tr><tr><td>Finetune*</td><td>71.8</td><td>17.8</td><td>22.9</td><td>15.2</td><td>71.2</td><td>10.2</td><td>15.1</td><td>61.7 36.6</td><td>21.9</td><td>61.3</td><td>33.4</td></tr><tr><td>Two Head*</td><td>72.9</td><td>60.6</td><td>33.2</td><td>47.7</td><td>70.2</td><td>3.9</td><td>25.5</td><td>52.6</td><td>58.4 54.7</td><td>64.4</td><td>47.1</td></tr><tr><td>Two Head+*</td><td>72.4</td><td>44.5</td><td>29.5</td><td>52.4</td><td>68.4</td><td>5.1</td><td>22.6 53.0</td><td>55.5</td><td>58.6</td><td>64.8</td><td> 45.4</td></tr><tr><td>Two Branch*</td><td>72.7</td><td>57.3</td><td>30.2</td><td>44.2 68.1</td><td>3.0</td><td>21.4</td><td>52.2</td><td>53.5</td><td>51.2</td><td>59.7</td><td>44.1</td></tr><tr><td>Ours w/o SCM</td><td>71.6</td><td>62.3</td><td>41.9</td><td>38.2</td><td>73.0</td><td>11.3</td><td>26.0</td><td>60.6</td><td>63.8</td><td>70.5 65.3</td><td>51.3</td></tr><tr><td>Ours</td><td>72.9</td><td>61.0</td><td>57.1</td><td>63.5</td><td>72.0</td><td>19.5</td><td>24.2</td><td>60.9</td><td>58.6</td><td>68.5 65.5</td><td>55.1+3.8</td></tr><tr><td>Ours* w/o SCM</td><td>72.7</td><td>66.8</td><td>50.4</td><td>57.0 71.5</td><td></td><td>12.1 27.6</td><td>57.1</td><td>62.7</td><td>54.2</td><td>64.2</td><td>52.4</td></tr><tr><td>Ours*</td><td>72.7</td><td>60.9</td><td>59.4</td><td>70.5</td><td>71.0</td><td>17.5</td><td>24.1</td><td>62.0</td><td>60.5 62.4</td><td>69.1</td><td>55.7+8.3</td></tr></table>
+
+Heatmap detection. We consider each class-agnostic heatmap as a two-channel image, and perform detection on it. Specifically, we learn a class-agnostic detector on base classes, that we further use to produce pseudo ground truth bounding boxes for novel objects.
+
+For this task, we use a lightweight one-stage detector, consisting of only five convolutional layers. We follow the same network architecture and loss as FCOS (Tian et al. (2019)), replacing the backbone and feature pyramid network with five max pooling layers. In our experiments, we also compare this tiny detector to a baseline: using three fully-connected layers to regress the groundtruth location taking the coordinates of high-confidence bounding boxes as input.
+
+Loss of DRN. After introducing our SCM, we can formulate the full loss function for DRN. We use $L _ { r e c }$ , $L _ { d e t }$ , and $L _ { s c m }$ to indicate the losses from our recognition head, detection head, and spatial correlation module respectively. $\lambda _ { r e c } , \lambda _ { d e t }$ , and $\lambda _ { s c m }$ are the regularization hyperparameters used to balance the three separate loss functions. We train our DRN using the following loss:
+
+$$
+L = \lambda _ { r e c } L _ { r e c } + \lambda _ { d e t } L _ { d e t } + \lambda _ { s c m } L _ { s c m }
+$$
+
+# 5 EXPERIMENTS
+
+# 5.1 PASCAL VOC
+
+Setup. PASCAL VOC 2007 and 2012 datasets contain 9, 962 and 22, 531 images respectively for 20 object classes. They are divided into train, val, and test sets. Here we follow previous work (Tang et al. (2017)) to choose the trainval set (5, 011 images from 2007 and 11, 540 images from 2012). We divide the first 10 classes into base classes and the other 10 classes into novel classes. To evaluate our methods, we calculate mean of Average Precision (mAP) based on the PASCAL criteria, i.e., $\mathrm { I O U } { > } 0 . 5$ between predicted boxes and ground truths.
+
+Implementation details. All our baselines, competitors and our framework are based on VGG16 (Simonyan & Zisserman (2015)) followed most of weakly supervised object detection methods. We set $\lambda _ { r e c } = 1$ , $\lambda _ { d e t } = 1 0$ , and $\lambda _ { s c m } = 1 0$ . We train the whole framework for 20 epochs using SGD with a momentum of 0.9, a weight decay of 0.0005 and a learning rate of 0.001, which is reduced by a factor of 10 at $1 4 ^ { t h }$ epoch. For a stable learning process, we don’t provide supervision from recognition head to detection head in the first 9 epochs.
+
+Table 2: The results on COCO. We compare our method with several strong baselines in $\ S \ 3 . 1$ and competitors. Our method significantly outperforms these approaches, showing that our crosssupervised object detector is capable of detecting novel objects in complex multi-object scenes.   
+
+<table><tr><td rowspan="2">method</td><td colspan="6">non-voc -→ voc: test on B = {voc}</td><td colspan="6"> sixty→ twenty: test on B= {twenty}</td></tr><tr><td>AP</td><td>AP50</td><td>AP75</td><td>APs APM</td><td></td><td>APL</td><td>AP</td><td>AP50</td><td>AP75 APsAPm APL</td><td></td><td></td><td></td></tr><tr><td>Rec. Head</td><td>4.0</td><td>15.4</td><td>0.9</td><td>1.2</td><td>5.7</td><td>5.8</td><td>4.7</td><td>16.4</td><td>1.3</td><td>1.7</td><td>8.0</td><td>6.9</td></tr><tr><td>OICR</td><td>4.2</td><td>15.7</td><td>1.0</td><td>1.3</td><td>5.5</td><td>5.9</td><td>4.5</td><td>16.6</td><td>1.4</td><td>2.0</td><td>8.2</td><td>7.1</td></tr><tr><td>PCL</td><td>9.2</td><td>19.6</td><td>-</td><td>-</td><td>1</td><td>-</td><td>9.2</td><td>19.6</td><td>-</td><td>-</td><td>1</td><td>-</td></tr><tr><td>Weight T.</td><td>9.3</td><td>26.4</td><td>5.7</td><td>5.8</td><td>11.7</td><td>12.4</td><td>8.7</td><td>25.5</td><td>5.5</td><td>5.4</td><td>11.5</td><td>11.7</td></tr><tr><td>Finetune</td><td>2.3</td><td>7.4</td><td>0.3</td><td>0.7</td><td>3.1</td><td>3.3</td><td>2.4</td><td>7.7</td><td>0.2</td><td>0.5</td><td>2.8</td><td>3.0</td></tr><tr><td>Two Head</td><td>11.0</td><td>30.2</td><td>6.1</td><td>6.2</td><td>15.4</td><td>15.4</td><td>11.3</td><td>29.5</td><td>5.8</td><td>6.3</td><td>14.8</td><td>15.0</td></tr><tr><td>Two Head+</td><td>9.1</td><td>26.7</td><td>5.4</td><td>5.5</td><td>12.1</td><td>12.3</td><td>9.0</td><td>27.1</td><td>5.4</td><td>5.7</td><td>11.7</td><td>11.6</td></tr><tr><td>Two Branch</td><td>9.4</td><td>26.6</td><td>5.6</td><td>5.7</td><td>12.3</td><td>12.4</td><td>8.5</td><td>24.4</td><td>4.5</td><td>4.3</td><td>11.9</td><td>11.9</td></tr><tr><td>Ours w/o SCM</td><td>12.5</td><td>33.6</td><td>6.6</td><td>7.3</td><td>19.2</td><td>16.4</td><td>12.6</td><td>32.3</td><td>7.8</td><td>7.0</td><td>19.4</td><td>17.4</td></tr><tr><td> Ours</td><td>13.9+1.4</td><td> 36.2+2.6</td><td>7.7</td><td>6.9</td><td>18.8</td><td>19.9</td><td>14.0+1.4</td><td> 34.5+2.2</td><td>8.9</td><td>7.1</td><td>19.2</td><td>20.6</td></tr></table>
+
+Baselines and competitors. We compare against several baselines as mentioned in $\ S \ 3 . 1$ , two WSOD methods: OICR (Tang et al. (2017)) and PCL (Tang et al. (2018)), and two cross-supervised object detector: MSD (Zhang et al. (2018a)), weight transfer (Kuen et al. (2019)).
+
+Results. As shown in Table 1, our method outperforms all other approaches by a large margin (over $7 \%$ relative increase in mAP on novel classes). The results are consistent with our discussion in $\ S \ 3 . 1$ . We note that (1) sharing backbone for the recognition and detection head learns a more discriminative embedding for novel objects. In Table 1, Two Head∗ boosts the performance by 5 points as compared to only using the recognition head (OICR). (2) A supervision from recognition head to detection head exploits the full potential of a detection model. By adding the supervision (Ours∗ w/o SCM ), the result is improved by 5 points as compared to Two Head. (3) Our spatial correlation module successfully captures the spatial correlation between high-confidence proposals. It further boosts the performance by 3 points.
+
+<table><tr><td>method</td><td>non-voc-→voc AP50 on B</td><td>sixty-→twenty AP50 on B</td></tr><tr><td>2 layer Fc layer 3 layer</td><td>31.0 30.8</td><td>28.7 28.3</td></tr><tr><td>4 layer</td><td>30.5</td><td>28.5</td></tr><tr><td>R-50-FPN</td><td>36.4</td><td>34.8</td></tr><tr><td>FCOS 4 conv</td><td>35.8</td><td>33.8</td></tr><tr><td>5 conv</td><td>36.2</td><td>34.5</td></tr><tr><td>w/o SCM</td><td>33.6</td><td>32.3</td></tr></table>
+
+(a) Ablation on Heatmap synthesis. The result suggests using two-channel heatmap consists of maximum confidence and sum of confidence over proposals covering that position.
+
+(b) Ablation on the structure of SCM. FCOS with 5 conv layers has nearly the best performance and very few parameters compared to a ResNet-50 backbone.
+
+Table 3: Ablation study of our method.   
+
+<table><tr><td></td><td>non-voc-→voc AP50 on B</td><td>sixty→twenty AP50 on B</td></tr><tr><td>method max</td><td>35.5</td><td>33.8</td></tr><tr><td>sum</td><td>36.0</td><td>34.0</td></tr><tr><td>num</td><td>31.5</td><td>29.5</td></tr><tr><td>max+sum</td><td>36.2</td><td>34.5</td></tr><tr><td>max+num</td><td>35.7</td><td>34.1</td></tr><tr><td>sum+num</td><td>35.9</td><td>34.2</td></tr><tr><td>max+sum+num</td><td>36.1</td><td>34.2</td></tr></table>
+
+<table><tr><td>method</td><td>non-voc-→voc AP50 on B</td><td>sixty-→twenty AP50 on B</td></tr><tr><td>WSDDN</td><td>35.7</td><td>33.8</td></tr><tr><td>OICR</td><td>36.6</td><td>34.7</td></tr><tr><td>Ours</td><td>36.4</td><td>34.5</td></tr></table>
+
+<table><tr><td colspan="2"></td><td colspan="2">base-novel AP50 on A AP5o on B</td></tr><tr><td rowspan="2">dataset PASCAL VOC</td><td>method</td><td></td><td rowspan="2">46.1</td></tr><tr><td>RPN</td><td>76.2</td></tr><tr><td rowspan="2">non-voc-&gt;voc</td><td>SS</td><td>72.7</td><td>55.7</td></tr><tr><td>RPN SS</td><td>46.3 42.5</td><td>36.2 34.5</td></tr></table>
+
+(c) Ablation on the structure of the recognition head. OICR has more refinement branches so it behaves a little better than our recognition head but takes double the computation time.
+
+(d) Ablation on the proposal generator. On PASCAL VOC, there are not enough categories to learn a good RPN. So, we use selective search and RPN to generate proposals for PASCAL VOC and COCO respectively.
+
+![](images/519db407a3bc2c8ce24f79950a80c288e4bc2f54370abe1a7373dbc299411de4.jpg)  
+Figure 4: Detection results on novel objects. The results are from our proposed model but with different heads. The first row shows the results of the recognition head. The second row lists the results from SCM. The third row displays the results from the detection head.
+
+# 5.2 COCO
+
+Setup. We train on the COCO train2017 split and test on val2017 split. We simulate the crosssupervised object detection scenario on COCO by splitting the 80 classes into base and novel classes. We use a 20/60 split same as Hu et al. (2018), dividing the COCO categories into all the 20 classes contained in PASCAL VOC and the 60 that are not. We refer to these as the ‘voc’ and ‘non-voc’ category sets. ‘voc non-voc’ indicates that we take ‘voc’ as our base classes and ‘non-voc’ as our novel classes. Similarly, we split the first 20 classes into ‘twenty’ and the last 60 classes into ’sixty’.
+
+Implementation details. The implementation details are the same as $\ S 5 . 1$ by default. We train the whole framework for 13 epochs. There is no supervision from recognition head to detection head in the first 5 epochs. The learning rate is reduced by a factor of 10 at $\bar { 8 } ^ { t h }$ , and $1 2 ^ { t h }$ epochs.
+
+Baselines and competitors. Most baselines and competitors are the same as $\ S \ S . 1$ . ’Rec. Head’ represents only using our recognition head structure as a weakly supervised object detector.
+
+Results. The results on COCO still support our discussion in $\ S 5 . 1$ . Even in complex multi objects scenes, our DRN outperforms all baselines and competitors by a large margin.
+
+# 5.3 ABLATION EXPERIMENTS
+
+Heatmap synthesis. In Table 3a, we compare the different methods to synthesize the heatmaps in the spatial correlation module. For each position in the heatmap, we consider three kinds of values: the maximum of confidence, the sum of confidence, and the number of proposals covering the position. This result informs us to use max and sum to create a two-channel heatmap.
+
+Structure of SCM. In Table 3b, we compare different implementations of SCM. We compare the FCOS (Tian et al. (2019)) with 5 convolutional layers and the standard FCOS with a ResNet-50 (He et al. (2016)) backbone. We also compare to the regression baseline mentioned in $\ S 4$ . Considering the computation cost, we choose FCOS with 5 convolutional layer as our heatmap detector.
+
+Structure of the Recognition head. In Table 3c, we compare different structures for the recognition head. WSDDN (Bilen & Vedaldi (2016)) and OICR are compared to our structure. The results support that our model can benefit from a stronger recognition head.
+
+Different proposal generation methods. Table 3d shows the ablation of different ways to generate proposals. In PASCAL VOC with only 10 base classes, RPN performs worse than selective search. In COCO with 60 base classes, RPN performs better than selective search.
+
+Visualization. Fig. 4 shows detection results on novel objects. Images in the first row, the second row, and the third row are detected by our model from the recognition head, the SCM, and the detection head respectively. The images in the first row tend to focus on the discriminating parts of the objects, e.g. the first and the second images contain only a part of the person. It also tends to detect co-occurring objects, e.g. the fourth image not only detects horse but also a large part of the person. Our SCM alleviates these problems. It tends to focus on the whole object, e.g. the first and the third samples detect the whole person instead of only the head. Also, it can correct unsatisfactory bounding boxes distracted by co-occurring objects, e.g. SCM correctly localizes the horse instead of localizing both the person and the horse in the fourth example. Obviously, bounding boxes in the third row are the best, indicating the efficacy of our framework.
+
+# 6 CONCLUSION
+
+In this paper, we have focused on cross-supervised object detection in realistic settings with complex imagery. We explore two major ways to build a good cross-supervised object detector: sharing network backbone between a recognition head and a detection head, and learning a spatial correlation module to bridge the gap between recognition and detection. Significant improvement on PASCAL VOC and COCO suggests a novel and promising approach for expanding object detection to a much larger number of categories.
+
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+    {
+        "type": "text",
+        "text": "CROSS-SUPERVISED OBJECT DETECTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "After learning a new object category from image-level annotations (with no object bounding boxes), humans are remarkably good at precisely localizing those objects. However, building good object localizers (i.e., detectors) currently requires expensive instance-level annotations. While some work has been done on learning detectors from weakly labeled samples (with only class labels), these detectors do poorly at localization. In this work, we show how to build better object detectors from weakly labeled images of new categories by leveraging knowledge learned from fully labeled base categories. We call this learning paradigm cross-supervised object detection. While earlier works investigated this paradigm, they did not apply it to realistic complex images (e.g., COCO), and their performance was poor. We propose a unified framework that combines a detection head trained from instance-level annotations and a recognition head learned from image-level annotations, together with a spatial correlation module that bridges the gap between detection and recognition. These contributions enable us to better detect novel objects with image-level annotations in complex multi-object scenes such as the COCO dataset. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep architectures have achieved great success in many computer vision tasks including object recognition and the closely related problem of object detection. Modern detectors, such as the Faster RCNN (Ren et al., 2015), YOLO (Redmon et al., 2016), and RetinaNet (Lin et al., 2017), use the same network backbone as popular recognition models. However, even with the same backbone architectures, detection and recognition models require different types of supervision. A good detector relies heavily on precise bounding boxes and labels for each instance (we shall refer to these as instance-level annotations), whereas a recognition model needs only image-level labels. Needless to say, it is more time consuming and expensive to obtain high quality bounding box annotations than class labels. As a result, current detectors are limited to a small set of categories relative to their object recognition counterparts. To address this limitation, it is natural to ask, “Is it possible to learn detectors with only class labels?” This problem is commonly referred to as weakly supervised object detection (WSOD). ",
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+    {
+        "type": "text",
+        "text": "Early WSOD work (Hoffman et al., 2014) showed fair performance by directly applying recognition networks to object detection. More recently, researchers have used multiple instance learning methods (Dietterich et al., 1997) to recast WSOD as a multi-label classification problem (Bilen & Vedaldi, 2016). However, these weakly supervised detectors perform poorly at localization. Most WSOD experiments have been conducted on the ILSVRC (Russakovsky et al., 2015) data set, in which images have only a single object, or on the PASCAL VOC (Everingham et al., 2010) data set, which has only 20 categories. The simplicity of these data sets limits the number and types of distractors in an image, making localization substantially easier. Learning from only class labels, it is challenging to detect objects at different scales in an image that contains many distractors. In particular, as shown in our experiments, weakly supervised object detectors do not work well in complex multi-object scenes, such as the COCO dataset (Lin et al., 2014). ",
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+        "type": "text",
+        "text": "To address this challenge, we focus on a form of learning in which the localization of classes with only object labels (weakly labeled classes) can benefit from other classes that have ground truth bounding boxes (fully labeled classes). We refer to this interesting learning paradigm as crosssupervised object detection (CSOD). While several works (Hoffman et al., 2014; Tang et al., 2016; ",
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+    {
+        "type": "text",
+        "text": "Yang et al., 2019a; Redmon & Farhadi, 2017) have explored this problem before, they still have the same limitation as the WSOD work we mentioned above. Those cross-supervised object detectors work under simplified scenarios (e.g., ILSVRC data set) where images contain single objects and are object-centered. They struggle to learn under more complex and realistic scenarios, where there are multiple objects from potentially very different classes, and objects could be small and appear anywhere in the images. In this work, we show that by doing multi-task learning on both weaklysupervised base classes and fully-supervised novel classes, our model is able to learn a good detector under the CSOD setting. ",
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+        "type": "text",
+        "text": "More formally, we define CSOD as follows. At training time, we are given 1) images contain objects from both base and novel classes, 2) both class labels and ground truth bounding boxes for base objects, and 3) only class labels for novel objects. Our goal is to detect novel objects. In CSOD, base classes and novel classes are disjoint. Thus, it can be seen as performing fullysupervised detection on the base classes and weakly supervised detection on the novel classes. It has similarities to both transfer learning and semi-supervised learning, since it transfer knowledge from base class to novel class and have more information about some instances than other instances. However, CSOD represents a distinct and novel paradigm for learning. ",
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+        "text": "The current weakly-supervised method has several drawbacks to learn from a multi objects image. As shown in Fig. 1, a weakly supervised object detector tends to detect only the most discriminating part of novel objects instead of the whole object. Notice how only the head of the person, and not the whole body, is detected. Another issue is that the localizer for one object (e.g., the horse) may be confused by the occurrence of another object, such as the person on the horse. This example illustrates the gap between detection and recognition: without ground truth bounding boxes, the detector acts like a standard recognition model – focusing on discriminating rather than detecting. ",
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+    {
+        "type": "text",
+        "text": "In this paper, we explore two major mechanisms for improving on this. Our first mechanism is unifying detection and recognition. Using the same network backbone architecture, recognition and detection can be seen as image-level classification and region-level classification respectively, suggesting a strong relation between them. In particular, it suggests a shared training framework in which the same backbone is used with different heads for detection and recognition. Thus, we combine a detection head learned from ground truth bounding boxes, and a recognition head learned in a weakly supervised fashion from class labels. Unlike a traditional recognition head, our recognition head produces a class score for multiple proposals and is capable of detecting objects. The second mechanism is learning a spatial correlation module to reduce the gap between detection and recognition. It takes several high-confidence bounding boxes produced by the recognition head as input, and learns to regress ground truth bounding boxes. By combining these mechanisms together, our model outperforms all previous models when all novel objects are weakly labeled. ",
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+        "type": "text",
+        "text": "In summary, our contributions are three-fold. First, we define a new task—cross-supervised object detection, which enables us to leverage knowledge from fully labeled base categories to help learn a robust detector from novel object class labels only. Second, we propose a unified framework in which two heads are learned from class labels and detection labels respectively, along with a spatial correlation module bridging the gap between recognition and detection. Third, we significantly outperform existing methods (Zhang et al. (2018a); Tang et al. (2017; 2018)) on PASCAL VOC and COCO, suggesting that CSOD could be a promising approach for expanding object detection to a much larger number of categories. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Weakly supervised object detection. WSOD (Kosugi et al. (2019); Zeng et al. (2019); Yang et al. (2019b); Wan et al. (2019); Arun et al. (2019); Wan et al. (2018); Zhang et al. (2018b); Ren et al. (2020); Zhang et al. (2018c); Li et al. (2019); Gao et al. (2019b); Kosugi et al. (2019)) attempts to learn a detector with only image category labels. Most of these methods adopt the idea of Multiple Instance Learning (Dietterich et al. (1997)) to recast WSOD as a multi-label classification task. Bilen & Vedaldi (2016) propose an end-to-end network by modifying a classifier to operate at the level of image regions, serving as a region selector and a classifier simultaneously. Tang et al. (2017) and Tang et al. (2018) find that several iterations of online refinement based on the outputs of previous iterations boosts performance. Wei et al. (2018) and Diba et al. (2017) use semantic segmentation based on class activation maps (Zhou et al. (2016)) to help generate tight bounding boxes. However, ",
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+        "type": "image",
+        "img_path": "images/a6525e87d58457804058030797ef06c2c5c8237d897d3edcf64e7b4ca07b3647.jpg",
+        "image_caption": [
+            "Figure 1: A comparison between weakly supervised object detector and our detector. Weakly supervised object detector only detects the most discriminating part of an object, e.g., focus on head of a person when detecting a person; or being distracted by co-occurring instances, e.g., distracted by the person on the horse when detecting a horse. Our detector can address these issues. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "WSOD methods tend to focus on the most discriminating part of an object and are prone to distractions from co-occurring objects. Detecting a part of the object or distractors represents convergence to a local optimum. Thus, their performance depends heavily on initialization. In comparison, our proposed cross-supervised object detector alleviates the issue of getting trapped in a local optimum by leveraging knowledge learned from fully labeled base categories. ",
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+        "text": "Cross-supervised object detection. There are several previous works using both image-level and instance-level annotations. Kuen et al. (2019) learned a parameter transferring function between a classifier and a detector, enabling an image-based classification network to be adapted to a regionbased classification network. Hoffman et al. (2014) and Tang et al. (2016) propose methods of adaptation for knowledge transfer from classification features to detection features. Uijlings et al. (2018) use a proposal generator trained on base classes to transfer knowledge by leveraging a MIL framework, organized in a semantic hierarchy. Hoffman et al. (2015) design a three-step framework to learn a feature representation from weakly supervised classes and strongly supervised classes jointly. However, these methods can only perform object localization in single object scenes such as ILSVRC, whereas our method can perform object detection in complex multi-object scenes as well, e.g. COCO. Also, it is worth noting that we are doing multi-task learning, which means that we jointly learn from base and novel classes. In comparison, some works (Uijlings et al., 2018) are doing transfer learning. They first learn a model on base classes and then transfer and fine-tune the model on novel classes. Gao et al. (2019a) use a few instance-level labels and a large scale of image-level labels for each category in a training-mining framework, which is referred to as semisupervised detection. Zhang et al. (2018a) propose a framework named MSD that learn objectness on base categories and use it to reject distractors when learning novel objects. In comparison, our spatial correlation module not only learns objectness, but also refines coarse bounding boxes. Further, our model learns from both base and novel classes instead of only novel classes. ",
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+    {
+        "type": "text",
+        "text": "3 CROSS-SUPERVISED OBJECT DETECTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "CSOD requires us to learn from instance-level annotations (detection labels) and image-level annotations (recognition labels). In this section, we explain the unification of detection and recognition and introduce our framework. In the next section, we describe our novel spatial correlation module. ",
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+    {
+        "type": "text",
+        "text": "3.1 UNIFYING DETECTION AND RECOGNITION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "How to learn a detector from both instance-level and image-level annotations? Since detection and recognition can be seen as region-level and image-level classification respectively, a natural choice is to design a unified framework that combines a detection head and a recognition head that can learn from image-level and instance-level annotations respectively. Here we exploit several baselines to unify the detection and recognition head. (1) Finetune. We first learn through the detection head on base classes with fully labeled samples. Then, we finetune our model using the recognition head on novel classes with only class labels. (2) Two Head. We simultaneously learn the detection and recognition head on base and novel classes, respectively. The weights of the backbones are updated using the loss backpropagated from both heads jointly. (3) Two head +. Instead of learning only on novel classes, we learn the recognition head from class labels of both base and novel classes whereas the recognition head remain the same. (4) Two Branch. Instead of having two shared fully-connected layers after RoI pooling layer (see Fig. 2), we make these two fully-connected layers seperated, allowing the detection and recognition head to have separate unshared pair of fully-connected layers each. Everything else is the same as the Two Head baseline. Experiments are conducted to compare these baselines in $\\ S \\ S . 1$ and $\\ S \\ S . 2$ . Our proposed model is based on Two Head. We will discuss the details in $\\ S 3 . 2$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/41ec47eab0504bf1e8dab709b878701e7e9a92c5e76db3b8f8be856947fd4534.jpg",
+        "image_caption": [
+            "Figure 2: Our Detection-Recognition Network (DRN) without the spatial correlation module. In this illustration, Person belongs to novel classes and Boat belongs to base classes. The recognition head learns from the class label Person and outputs the top-scoring bounding box to help the detection head learn to detect the person. The spatial correlation module, discussed in $\\ S 4$ , can be added to further refine the top-scoring bounding boxes. "
+        ],
+        "image_footnote": [],
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+        "text": "",
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+        "type": "text",
+        "text": "The connection between the recognition and detection head. The baselines mentioned above only use the recognition head to detect novel objects, ignoring the fact that a detection head can play the same role even better. A majority of WSOD methods (Tang et al. (2017); Wan et al. (2019); Wei et al. (2018)) find that re-train a new detector taking the top-scoring bounding boxes from a weakly supervised object detector as ground truth marginally improve the performance. Even with coarse and noisy pseudo bounding boxes, a standard object detector produces better detection results than a weakly supervised object detector. Keeping this hypothesis in mind, we introduce a guidance from the recognition head to the detection head. For each of the novel categories existing in a training sample, the recognition head outputs the top-scoring bounding box, which are then used by the detection head as supervision in that sample. ",
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+        "text": "3.2 DETECTION-RECOGNITION NETWORK ",
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+        "type": "text",
+        "text": "The structure of our Detection-Recognition Network (DRN) is shown in Fig. 2. Given an image, we first generate 2000 object proposals by Selective Search (Uijlings et al. (2013)) or RPN (Ren et al. (2015)) trained on base classes. The image and proposals are fed into several convolutional (conv) layers followed by a region-of-interest (RoI) pooling layer (Girshick (2015)) to output fixed-size feature maps. Then, these feature maps are fed into two fully connected (fc) layers to produce a collection of proposal features, which are further branched into the recognition and detection head. ",
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+        "type": "text",
+        "text": "Recognition Head. We followed previous WSOD methods to design our recognition head. Since OICR (Tang et al. (2017)) is simple, neat, and commonly being used, we make our recognition head the same as OICR, but with fewer refinement branches to reduce the computation cost. However, our recognition head can be replaced by any WSOD structure as shown in $\\ S 5 . 3$ . ",
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+        "type": "text",
+        "text": "Within the recognition head as shown in Fig. 2, the proposal features are branched into three streams producing three matrices $\\mathbf { x } ^ { c } , \\mathbf { x } ^ { d } , \\mathbf { x } ^ { e } \\in \\mathbb { R } ^ { C \\times | R | }$ , where $C$ is the number of novel classes and $| R |$ is the number of proposals. Then the two matrices $\\mathbf { x } ^ { c }$ and $\\mathbf { x } ^ { d }$ are passed through a softmax function ",
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+        "type": "image",
+        "img_path": "images/1450e19e8f29587235da2e261c12a62149ab30614da707a9c2c6b3b5fee699e1.jpg",
+        "image_caption": [
+            "Figure 3: Our spatial correlation module (SCM). Our SCM learns to capture spatial correlation among high-confidence bounding boxes, generating a class-agnostic heatmap for the whole image. A heatmap detector is then trained to learn ground truth bounding boxes. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "over classes and proposals respectively: $\\sigma ( \\mathbf { x } ^ { c } )$ and $\\sigma ( \\mathbf { x } ^ { d } )$ . A proposal score $\\mathbf { x } _ { c r } ^ { R }$ , indicating the score of $c ^ { t h }$ novel class for $r ^ { t h }$ proposal, corresponds to the respective element of the matrix $\\bar { \\mathbf { x } } ^ { R } =$ $\\sigma ( \\mathbf { x } ^ { c } ) \\odot \\sigma ( \\mathbf { x } ^ { d } )$ , where $\\odot$ refers to an element-wise product. Finally, we obtain the image score of $c ^ { t h }$ class $\\phi _ { c }$ by summing over all proposals: $\\begin{array} { r } { \\phi _ { c } = \\sum _ { r = 1 } ^ { | R | } x _ { c r } ^ { R } } \\end{array}$ . Then we culate a standard multi-class cross-entropy loss as shown in the first term of Eq.1. Another matrix $\\mathbf { x } ^ { e }$   \nfunction over classes, the result of which is expresses as a weighted multi-class cross entropy loss as shown in the second term of Eq.1. We set the pseudo label for each proposal $r$ based on its IoU (or overlap) with the top-scoring proposal of $c ^ { t h }$ class, $y _ { c r } = 1$ if $\\mathrm { I o U } > 0 . 5$ and $y _ { c r } = 0$ otherwise. The weight $w _ { r }$ for each proposal $r$ is its IoU with the top-scoring proposal. The total loss for the recognition head is ",
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+        "img_path": "images/2d79b14df0ad6e4eb5ec1e8023e9ee8e0329dee8eab36b8da68f418dfa3857a6.jpg",
+        "text": "$$\nL _ { r e c } = [ - \\sum _ { c = 1 } ^ { C } y _ { c } l o g \\phi _ { c } + ( 1 - y _ { c } ) l o g ( 1 - \\phi _ { c } ) ] + [ - \\frac { 1 } { | { \\cal R } | } \\sum _ { r = 1 } ^ { | { \\cal R } | } \\sum _ { c = 1 } ^ { C + 1 } w _ { r } y _ { c r } l o g x _ { c r } ^ { e } ]\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Supervision from our recognition head. We use the matrix $x ^ { e }$ to propose pseudos bounding boxes to guide the detection head. Specifically, we select one top-scoring proposal for each object category that appears in the image as a pseudo bounding box, as done in OICR. We introduce the spatial correlation module in $\\ S 4$ , to further refine this pseudo ground truth. ",
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+        "type": "text",
+        "text": "Detection Head. Now that we have pseudo bounding boxes for novel objects and ground truth bounding boxes for base objects, we train our detection head like a standard detector. For simplicity and efficiency, our detection head use the same structure of Faster R-CNN (Ren et al. (2015)). At inference time, the detection head produces detection results for both base categories and novel categories. ",
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+        "type": "text",
+        "text": "4 LEARNING TO MODEL SPATIAL CORRELATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our intuition is that there exists spatial correlation among high-confidence bounding boxes, and such spatial correlation can be captured to predict ground truth bounding boxes. By representing the spatial correlation in a class-agnostic heatmap, we can easily learn a mapping from recognitionbased bounding boxes to ground truth bounding boxes for base categories, and then transfer this mapping to novel categories. ",
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+        "type": "text",
+        "text": "Thus, we propose a spatial correlation module (SCM). SCM is used as a guidance refinement technique, taking sets of high-confidence bounding boxes from the recognition head, and correspondingly returning pseudo ground truth bounding boxes to the detection head. These pseudo ground truth boxes act as supervision while training on novel categories. The framework of SCM is showed in Fig. 3. Within this module, we first generate a class agnostic heatmap based on the high-confidence bounding boxes predicted by our recognition head, and then we perform detection on top of the heatmap. ",
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+        "type": "text",
+        "text": "Heatmap synthesis. We want to capture the information about how the high-confidence bounding boxes interact amongst themselves. Here, we introduce a simple way of achieving this using a class-agnostic heatmap. For each category existing in the image $y _ { c } = 1 , c \\in C$ , we first threshold and select high-confidence bounding boxes of class $c$ . Then we synthesize a corresponding classagnostic heatmap, which is essentially a two-channel feature map of the same size as the original image. The value at each pixel is the sum and the maximum of confidence over all selected bounding boxes covering that pixel. ",
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+    {
+        "type": "table",
+        "img_path": "images/aebcae5fe48440380a5163b89ade4d051d19099429f71fd2e8936f7caa412fbd.jpg",
+        "table_caption": [
+            "Table 1: Object Detection performance (mAP $\\%$ ) on PASCAL VOC 2007 test set. ∗ indicates using the structure of OICR in the recognition head. ”MSD-Ens” is the ensemble of AlexNet and VGG16. ”MSD-Ens+FRCN” indicates using an ensemble model to predict pseudo ground truths and then learn a Fast-RCNN (Girshick (2015)) using VGG-16. "
+        ],
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+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Base mean</td><td colspan=\"9\">Novel</td><td rowspan=\"2\"></td></tr><tr><td>table</td><td>dog</td><td>horse</td><td>mbike</td><td>person</td><td>plant sheep</td><td></td><td>sofa</td><td>train tv</td></tr><tr><td>OICR</td><td>42.1</td><td>33.4</td><td>29.3</td><td>56.3</td><td>64.6</td><td>8.0</td><td>23.5</td><td>47.2</td><td>47.2</td><td>48.3 61.7</td><td>mean 42.0</td></tr><tr><td>PCL</td><td>49.2</td><td>51.5</td><td>37.3</td><td>63.3</td><td>63.9</td><td>15.8 23.6</td><td>48.8</td><td>55.3</td><td>61.2</td><td>62.1</td><td>48.3</td></tr><tr><td>MSD-VGG16</td><td>50.6</td><td>14.3</td><td>69.3</td><td>65.4 69.6</td><td>2.4</td><td>20.5</td><td>54.6</td><td>34.3</td><td>58.3</td><td>54.6</td><td>44.3</td></tr><tr><td>MSD-Ens</td><td>53.4</td><td>18.3</td><td>70.6</td><td>66.7 69.8</td><td></td><td>3.7 24.7</td><td>55.0</td><td>37.4</td><td>58.3</td><td>57.3</td><td>46.1</td></tr><tr><td>MSD-Ens+FRCN</td><td>53.9</td><td>15.3</td><td>72.0</td><td>74.4</td><td>65.2</td><td>15.4</td><td>25.1 53.6</td><td>54.4</td><td>45.6</td><td>61.4</td><td>48.2</td></tr><tr><td>Weight Transfer</td><td>68.4</td><td>10.4</td><td>61.0</td><td>58.0</td><td>65.1</td><td>19.8</td><td>19.5</td><td>58.0</td><td>50.8 58.6</td><td>52.7</td><td>45.4</td></tr><tr><td>Finetune*</td><td>71.8</td><td>17.8</td><td>22.9</td><td>15.2</td><td>71.2</td><td>10.2</td><td>15.1</td><td>61.7 36.6</td><td>21.9</td><td>61.3</td><td>33.4</td></tr><tr><td>Two Head*</td><td>72.9</td><td>60.6</td><td>33.2</td><td>47.7</td><td>70.2</td><td>3.9</td><td>25.5</td><td>52.6</td><td>58.4 54.7</td><td>64.4</td><td>47.1</td></tr><tr><td>Two Head+*</td><td>72.4</td><td>44.5</td><td>29.5</td><td>52.4</td><td>68.4</td><td>5.1</td><td>22.6 53.0</td><td>55.5</td><td>58.6</td><td>64.8</td><td> 45.4</td></tr><tr><td>Two Branch*</td><td>72.7</td><td>57.3</td><td>30.2</td><td>44.2 68.1</td><td>3.0</td><td>21.4</td><td>52.2</td><td>53.5</td><td>51.2</td><td>59.7</td><td>44.1</td></tr><tr><td>Ours w/o SCM</td><td>71.6</td><td>62.3</td><td>41.9</td><td>38.2</td><td>73.0</td><td>11.3</td><td>26.0</td><td>60.6</td><td>63.8</td><td>70.5 65.3</td><td>51.3</td></tr><tr><td>Ours</td><td>72.9</td><td>61.0</td><td>57.1</td><td>63.5</td><td>72.0</td><td>19.5</td><td>24.2</td><td>60.9</td><td>58.6</td><td>68.5 65.5</td><td>55.1+3.8</td></tr><tr><td>Ours* w/o SCM</td><td>72.7</td><td>66.8</td><td>50.4</td><td>57.0 71.5</td><td></td><td>12.1 27.6</td><td>57.1</td><td>62.7</td><td>54.2</td><td>64.2</td><td>52.4</td></tr><tr><td>Ours*</td><td>72.7</td><td>60.9</td><td>59.4</td><td>70.5</td><td>71.0</td><td>17.5</td><td>24.1</td><td>62.0</td><td>60.5 62.4</td><td>69.1</td><td>55.7+8.3</td></tr></table>",
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+        "type": "text",
+        "text": "Heatmap detection. We consider each class-agnostic heatmap as a two-channel image, and perform detection on it. Specifically, we learn a class-agnostic detector on base classes, that we further use to produce pseudo ground truth bounding boxes for novel objects. ",
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+        "text": "For this task, we use a lightweight one-stage detector, consisting of only five convolutional layers. We follow the same network architecture and loss as FCOS (Tian et al. (2019)), replacing the backbone and feature pyramid network with five max pooling layers. In our experiments, we also compare this tiny detector to a baseline: using three fully-connected layers to regress the groundtruth location taking the coordinates of high-confidence bounding boxes as input. ",
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+        "text": "Loss of DRN. After introducing our SCM, we can formulate the full loss function for DRN. We use $L _ { r e c }$ , $L _ { d e t }$ , and $L _ { s c m }$ to indicate the losses from our recognition head, detection head, and spatial correlation module respectively. $\\lambda _ { r e c } , \\lambda _ { d e t }$ , and $\\lambda _ { s c m }$ are the regularization hyperparameters used to balance the three separate loss functions. We train our DRN using the following loss: ",
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+        "img_path": "images/579f00d2f806a7f62ac4dedd95dd1bd906c1fee7f49092331d62c69bedccf6e4.jpg",
+        "text": "$$\nL = \\lambda _ { r e c } L _ { r e c } + \\lambda _ { d e t } L _ { d e t } + \\lambda _ { s c m } L _ { s c m }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 PASCAL VOC ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Setup. PASCAL VOC 2007 and 2012 datasets contain 9, 962 and 22, 531 images respectively for 20 object classes. They are divided into train, val, and test sets. Here we follow previous work (Tang et al. (2017)) to choose the trainval set (5, 011 images from 2007 and 11, 540 images from 2012). We divide the first 10 classes into base classes and the other 10 classes into novel classes. To evaluate our methods, we calculate mean of Average Precision (mAP) based on the PASCAL criteria, i.e., $\\mathrm { I O U } { > } 0 . 5$ between predicted boxes and ground truths. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Implementation details. All our baselines, competitors and our framework are based on VGG16 (Simonyan & Zisserman (2015)) followed most of weakly supervised object detection methods. We set $\\lambda _ { r e c } = 1$ , $\\lambda _ { d e t } = 1 0$ , and $\\lambda _ { s c m } = 1 0$ . We train the whole framework for 20 epochs using SGD with a momentum of 0.9, a weight decay of 0.0005 and a learning rate of 0.001, which is reduced by a factor of 10 at $1 4 ^ { t h }$ epoch. For a stable learning process, we don’t provide supervision from recognition head to detection head in the first 9 epochs. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2836e932d9bb7f0b5dbaa5d2077a341a47e0a738af996c13b936d919138721b6.jpg",
+        "table_caption": [
+            "Table 2: The results on COCO. We compare our method with several strong baselines in $\\ S \\ 3 . 1$ and competitors. Our method significantly outperforms these approaches, showing that our crosssupervised object detector is capable of detecting novel objects in complex multi-object scenes. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">method</td><td colspan=\"6\">non-voc -→ voc: test on B = {voc}</td><td colspan=\"6\"> sixty→ twenty: test on B= {twenty}</td></tr><tr><td>AP</td><td>AP50</td><td>AP75</td><td>APs APM</td><td></td><td>APL</td><td>AP</td><td>AP50</td><td>AP75 APsAPm APL</td><td></td><td></td><td></td></tr><tr><td>Rec. Head</td><td>4.0</td><td>15.4</td><td>0.9</td><td>1.2</td><td>5.7</td><td>5.8</td><td>4.7</td><td>16.4</td><td>1.3</td><td>1.7</td><td>8.0</td><td>6.9</td></tr><tr><td>OICR</td><td>4.2</td><td>15.7</td><td>1.0</td><td>1.3</td><td>5.5</td><td>5.9</td><td>4.5</td><td>16.6</td><td>1.4</td><td>2.0</td><td>8.2</td><td>7.1</td></tr><tr><td>PCL</td><td>9.2</td><td>19.6</td><td>-</td><td>-</td><td>1</td><td>-</td><td>9.2</td><td>19.6</td><td>-</td><td>-</td><td>1</td><td>-</td></tr><tr><td>Weight T.</td><td>9.3</td><td>26.4</td><td>5.7</td><td>5.8</td><td>11.7</td><td>12.4</td><td>8.7</td><td>25.5</td><td>5.5</td><td>5.4</td><td>11.5</td><td>11.7</td></tr><tr><td>Finetune</td><td>2.3</td><td>7.4</td><td>0.3</td><td>0.7</td><td>3.1</td><td>3.3</td><td>2.4</td><td>7.7</td><td>0.2</td><td>0.5</td><td>2.8</td><td>3.0</td></tr><tr><td>Two Head</td><td>11.0</td><td>30.2</td><td>6.1</td><td>6.2</td><td>15.4</td><td>15.4</td><td>11.3</td><td>29.5</td><td>5.8</td><td>6.3</td><td>14.8</td><td>15.0</td></tr><tr><td>Two Head+</td><td>9.1</td><td>26.7</td><td>5.4</td><td>5.5</td><td>12.1</td><td>12.3</td><td>9.0</td><td>27.1</td><td>5.4</td><td>5.7</td><td>11.7</td><td>11.6</td></tr><tr><td>Two Branch</td><td>9.4</td><td>26.6</td><td>5.6</td><td>5.7</td><td>12.3</td><td>12.4</td><td>8.5</td><td>24.4</td><td>4.5</td><td>4.3</td><td>11.9</td><td>11.9</td></tr><tr><td>Ours w/o SCM</td><td>12.5</td><td>33.6</td><td>6.6</td><td>7.3</td><td>19.2</td><td>16.4</td><td>12.6</td><td>32.3</td><td>7.8</td><td>7.0</td><td>19.4</td><td>17.4</td></tr><tr><td> Ours</td><td>13.9+1.4</td><td> 36.2+2.6</td><td>7.7</td><td>6.9</td><td>18.8</td><td>19.9</td><td>14.0+1.4</td><td> 34.5+2.2</td><td>8.9</td><td>7.1</td><td>19.2</td><td>20.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Baselines and competitors. We compare against several baselines as mentioned in $\\ S \\ 3 . 1$ , two WSOD methods: OICR (Tang et al. (2017)) and PCL (Tang et al. (2018)), and two cross-supervised object detector: MSD (Zhang et al. (2018a)), weight transfer (Kuen et al. (2019)). ",
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+        "type": "text",
+        "text": "Results. As shown in Table 1, our method outperforms all other approaches by a large margin (over $7 \\%$ relative increase in mAP on novel classes). The results are consistent with our discussion in $\\ S \\ 3 . 1$ . We note that (1) sharing backbone for the recognition and detection head learns a more discriminative embedding for novel objects. In Table 1, Two Head∗ boosts the performance by 5 points as compared to only using the recognition head (OICR). (2) A supervision from recognition head to detection head exploits the full potential of a detection model. By adding the supervision (Ours∗ w/o SCM ), the result is improved by 5 points as compared to Two Head. (3) Our spatial correlation module successfully captures the spatial correlation between high-confidence proposals. It further boosts the performance by 3 points. ",
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+    {
+        "type": "table",
+        "img_path": "images/0e6185d41a461a802efa1a43ebf3f024f96611211da4572306b2b91ffeb31b2f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>method</td><td>non-voc-→voc AP50 on B</td><td>sixty-→twenty AP50 on B</td></tr><tr><td>2 layer Fc layer 3 layer</td><td>31.0 30.8</td><td>28.7 28.3</td></tr><tr><td>4 layer</td><td>30.5</td><td>28.5</td></tr><tr><td>R-50-FPN</td><td>36.4</td><td>34.8</td></tr><tr><td>FCOS 4 conv</td><td>35.8</td><td>33.8</td></tr><tr><td>5 conv</td><td>36.2</td><td>34.5</td></tr><tr><td>w/o SCM</td><td>33.6</td><td>32.3</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "(a) Ablation on Heatmap synthesis. The result suggests using two-channel heatmap consists of maximum confidence and sum of confidence over proposals covering that position. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(b) Ablation on the structure of SCM. FCOS with 5 conv layers has nearly the best performance and very few parameters compared to a ResNet-50 backbone. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/4571cf8d4feadee3f2e1648b3687e56b47b9d4bfd89dc36ae6851355760b6514.jpg",
+        "table_caption": [
+            "Table 3: Ablation study of our method. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>non-voc-→voc AP50 on B</td><td>sixty→twenty AP50 on B</td></tr><tr><td>method max</td><td>35.5</td><td>33.8</td></tr><tr><td>sum</td><td>36.0</td><td>34.0</td></tr><tr><td>num</td><td>31.5</td><td>29.5</td></tr><tr><td>max+sum</td><td>36.2</td><td>34.5</td></tr><tr><td>max+num</td><td>35.7</td><td>34.1</td></tr><tr><td>sum+num</td><td>35.9</td><td>34.2</td></tr><tr><td>max+sum+num</td><td>36.1</td><td>34.2</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/efbb49b9ed92ec1af551979fa652cca039a15b9203cc5a1c16c997b7ba7f41ca.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>method</td><td>non-voc-→voc AP50 on B</td><td>sixty-→twenty AP50 on B</td></tr><tr><td>WSDDN</td><td>35.7</td><td>33.8</td></tr><tr><td>OICR</td><td>36.6</td><td>34.7</td></tr><tr><td>Ours</td><td>36.4</td><td>34.5</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/51109980375222f7fdb86fe1ae068bde84a89ece724e9d6e2857c1d7eb69d77e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"2\">base-novel AP50 on A AP5o on B</td></tr><tr><td rowspan=\"2\">dataset PASCAL VOC</td><td>method</td><td></td><td rowspan=\"2\">46.1</td></tr><tr><td>RPN</td><td>76.2</td></tr><tr><td rowspan=\"2\">non-voc-&gt;voc</td><td>SS</td><td>72.7</td><td>55.7</td></tr><tr><td>RPN SS</td><td>46.3 42.5</td><td>36.2 34.5</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(c) Ablation on the structure of the recognition head. OICR has more refinement branches so it behaves a little better than our recognition head but takes double the computation time. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(d) Ablation on the proposal generator. On PASCAL VOC, there are not enough categories to learn a good RPN. So, we use selective search and RPN to generate proposals for PASCAL VOC and COCO respectively. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/519db407a3bc2c8ce24f79950a80c288e4bc2f54370abe1a7373dbc299411de4.jpg",
+        "image_caption": [
+            "Figure 4: Detection results on novel objects. The results are from our proposed model but with different heads. The first row shows the results of the recognition head. The second row lists the results from SCM. The third row displays the results from the detection head. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 COCO ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Setup. We train on the COCO train2017 split and test on val2017 split. We simulate the crosssupervised object detection scenario on COCO by splitting the 80 classes into base and novel classes. We use a 20/60 split same as Hu et al. (2018), dividing the COCO categories into all the 20 classes contained in PASCAL VOC and the 60 that are not. We refer to these as the ‘voc’ and ‘non-voc’ category sets. ‘voc non-voc’ indicates that we take ‘voc’ as our base classes and ‘non-voc’ as our novel classes. Similarly, we split the first 20 classes into ‘twenty’ and the last 60 classes into ’sixty’. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation details. The implementation details are the same as $\\ S 5 . 1$ by default. We train the whole framework for 13 epochs. There is no supervision from recognition head to detection head in the first 5 epochs. The learning rate is reduced by a factor of 10 at $\\bar { 8 } ^ { t h }$ , and $1 2 ^ { t h }$ epochs. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Baselines and competitors. Most baselines and competitors are the same as $\\ S \\ S . 1$ . ’Rec. Head’ represents only using our recognition head structure as a weakly supervised object detector. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results. The results on COCO still support our discussion in $\\ S 5 . 1$ . Even in complex multi objects scenes, our DRN outperforms all baselines and competitors by a large margin. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 ABLATION EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Heatmap synthesis. In Table 3a, we compare the different methods to synthesize the heatmaps in the spatial correlation module. For each position in the heatmap, we consider three kinds of values: the maximum of confidence, the sum of confidence, and the number of proposals covering the position. This result informs us to use max and sum to create a two-channel heatmap. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Structure of SCM. In Table 3b, we compare different implementations of SCM. We compare the FCOS (Tian et al. (2019)) with 5 convolutional layers and the standard FCOS with a ResNet-50 (He et al. (2016)) backbone. We also compare to the regression baseline mentioned in $\\ S 4$ . Considering the computation cost, we choose FCOS with 5 convolutional layer as our heatmap detector. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Structure of the Recognition head. In Table 3c, we compare different structures for the recognition head. WSDDN (Bilen & Vedaldi (2016)) and OICR are compared to our structure. The results support that our model can benefit from a stronger recognition head. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Different proposal generation methods. Table 3d shows the ablation of different ways to generate proposals. In PASCAL VOC with only 10 base classes, RPN performs worse than selective search. In COCO with 60 base classes, RPN performs better than selective search. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Visualization. Fig. 4 shows detection results on novel objects. Images in the first row, the second row, and the third row are detected by our model from the recognition head, the SCM, and the detection head respectively. The images in the first row tend to focus on the discriminating parts of the objects, e.g. the first and the second images contain only a part of the person. It also tends to detect co-occurring objects, e.g. the fourth image not only detects horse but also a large part of the person. Our SCM alleviates these problems. It tends to focus on the whole object, e.g. the first and the third samples detect the whole person instead of only the head. Also, it can correct unsatisfactory bounding boxes distracted by co-occurring objects, e.g. SCM correctly localizes the horse instead of localizing both the person and the horse in the fourth example. Obviously, bounding boxes in the third row are the best, indicating the efficacy of our framework. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            262,
+            318,
+            277
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have focused on cross-supervised object detection in realistic settings with complex imagery. We explore two major ways to build a good cross-supervised object detector: sharing network backbone between a recognition head and a detection head, and learning a spatial correlation module to bridge the gap between recognition and detection. Significant improvement on PASCAL VOC and COCO suggests a novel and promising approach for expanding object detection to a much larger number of categories. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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parse/train/Bki4EfWCb/Bki4EfWCb.md

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+# INFERENCE SUBOPTIMALITY IN VARIATIONAL AUTOENCODERS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Amortized inference has led to efficient approximate inference for large datasets. The quality of posterior inference is largely determined by two factors: a) the ability of the variational distribution to model the true posterior and b) the capacity of the recognition network to generalize inference over all datapoints. We analyze approximate inference in variational autoencoders in terms of these factors. We find that suboptimal inference is often due to amortizing inference rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation.
+
+# 1 INTRODUCTION
+
+There has been significant work on improving inference in variational autoencoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014) through the development of expressive approximate posteriors (Rezende & Mohamed, 2015; Kingma et al., 2016; Ranganath et al., 2016; Tomczak & Welling, 2016; 2017). These works have shown that with more expressive approximate posteriors, the model learns a better distribution over the data.
+
+In this paper, we analyze inference suboptimality in VAEs: the mismatch between the true and approximate posterior. In other words, we are interested in understanding what factors cause the gap between the marginal log-likelihood and the evidence lower bound (ELBO). We refer to this as the inference gap. Moreover, we break down the inference gap into two components: the approximation gap and the amortization gap. The approximation gap comes from the inability of the approximate distribution family to exactly match the true posterior. The amortization gap refers to the difference caused by amortizing the variational parameters over the entire training set, instead of optimizing for each datapoint independently. We refer the reader to Table 1 for detailed definitions and Figure 1 for a simple illustration of the gaps. In Figure 1, ${ \mathcal { L } } [ q ]$ refers to the ELBO using an amortized distribution $q$ , whereas $q ^ { * }$ is the optimal $q$ within its variational family.
+
+Our experiments investigate how the choice of encoder, posterior approximation, decoder, and model optimization affect the approximation and amortization gaps. We train VAE models in a number of settings on the MNIST, Fashion-MNIST (Xiao et al., 2017), and CIFAR10 datasets.
+
+Our contributions are: a) we investigate inference suboptimality in terms of the approximation and amortization gaps, providing insight to guide future improvements in VAE inference, b) we quantitatively demonstrate that the learned true posterior accommodates the choice of approximation, and c) we demonstrate that using parameterized functions to improve the expressiveness of the approximation plays a large role in reducing error caused by amortization.
+
+![](images/7083885f712ccbaedc70c4ec67996ea743f40445ce818540472026cabebf20b2.jpg)  
+Figure 1: Gaps in Inference
+
+<table><tr><td>Term</td><td>Definition</td><td>VAE Formulation</td></tr><tr><td>Inference</td><td>logp(x)-L[q]</td><td>KL(q(z|x)llp(z|x))</td></tr><tr><td>Approximation</td><td>logp(x）-L[q*]</td><td>KL(q*（z|x)lp(z|x)）</td></tr><tr><td>Amortization</td><td>C-C[a]</td><td>KL(q(z|x)llp(z|x))-KL(q*(z|x)llp(z|x))</td></tr></table>
+
+Table 1: Summary of Gap Terms. The middle column refers to the general case where our variational objective is a lower bound on the marginal log-likelihood (not necessarily the ELBO). The right most column demonstrates the specific case in VAEs. $q ^ { * } ( z | x )$ refers to the optimal approximation within a family $\mathcal { Q }$ , i.e. $\begin{array} { r } { q ^ { * } ( z | x ) = \mathrm { \bar { a r g m i n } } _ { q \in \mathcal { Q } } \mathrm { K L } \left( q ( z | \bar { x } ) | | \dot { p ( z | x ) } \right) } \end{array}$ .
+
+# 2 BACKGROUND
+
+# 2.1 INFERENCE IN VARIATIONAL AUTOENCODERS
+
+Let $x$ be the observed variable, $z$ the latent variable, and $p ( x , z )$ be their joint distribution. Given a dataset $X = \{ x _ { 1 } , x _ { 2 } , . . . , x _ { N } \}$ , we would like to maximize the marginal log-likelihood:
+
+$$
+\log p ( X ) = \sum _ { i = 1 } ^ { N } \log p ( x _ { i } ) = \sum _ { i = 1 } ^ { N } \log \int p ( x _ { i } , z _ { i } ) d z _ { i } .
+$$
+
+In practice, the marginal log-likelihood is computationally intractable due to the integration over the latent variable $z$ . Instead, VAEs optimize the ELBO of the marginal log-likelihood (Kingma & Welling, 2014; Rezende et al., 2014):
+
+$$
+\begin{array} { r l } & { \log p ( x ) = \mathbb { E } _ { z \sim q ( z \mid x ) } \left[ \log \left( \displaystyle \frac { p ( x , z ) } { q ( z \mid x ) } \right) \right] + { \mathrm { K L } } \left( q ( z \mid x ) | | p ( z | x ) \right) } \\ & { \phantom { \exp x } \geq \mathbb { E } _ { z \sim q ( z \mid x ) } \left[ \log \left( \displaystyle \frac { p ( x , z ) } { q ( z \mid x ) } \right) \right] = \mathcal { L } _ { \mathrm { V A E } } [ q ] . } \end{array}
+$$
+
+From the above we can see that the lower bound is tight if $q ( z | x ) = p ( z | x )$ . The choice of $q ( z | x )$ is often a factorized Gaussian distribution for its simplicity and efficiency. VAEs perform amortized inference by utilizing a recognition network (encoder), resulting in efficient approximate inference for large datasets. The overall model is trained by stochastically optimizing the ELBO using the reparametrization trick (Kingma & Welling, 2014).
+
+# 2.2 EXPRESSIVE APPROXIMATE POSTERIORS
+
+There are a number of strategies for increasing the expressiveness of approximate posteriors, going beyond the original factorized-Gaussian. We briefly summarize normalizing flows and auxiliary variables.
+
+# 2.2.1 NORMALIZING FLOWS
+
+Normalizing flow (Rezende & Mohamed, 2015) is a change of variables procedure for constructing complex distributions by transforming probability densities through a series of invertible mappings. Specifically, if we transform a random variable $z _ { \mathrm { 0 } }$ with distribution $q _ { 0 } ( z )$ , the resulting random variable $z _ { T } = T ( z _ { 0 } )$ has a distribution:
+
+$$
+q _ { T } ( z _ { T } ) = q _ { 0 } ( z _ { 0 } ) \left| \mathrm { d e t } \frac { \partial z _ { T } } { \partial z _ { 0 } } \right| ^ { - 1 }
+$$
+
+By successively applying these transformations, we can build arbitrarily complex distributions. Stacking these transformations remains tractable due to the determinant being decomposable: $\operatorname* { d e t } ( A { \bar { B } } ) = \operatorname* { d e t } ( A ) \operatorname* { d e t } ( B )$ . An important property of these transformations is that we can take expectations with respect to the transformed density $q _ { T } ( z _ { T } )$ without explicitly knowing its formula known as the law of the unconscious statistician (LOTUS):
+
+$$
+\mathbb { E } _ { q _ { T } } [ h ( z _ { T } ) ] = \mathbb { E } _ { q _ { 0 } } [ h ( f _ { T } ( f _ { T - 1 } ( \dots f _ { 1 } ( z _ { 0 } ) ) ) ) ]
+$$
+
+Using the change of variable and LOTUS, the lower bound can be written as:
+
+$$
+\log p ( x ) \geq \mathbb { E } _ { z _ { 0 } \sim q _ { 0 } ( z | x ) } \left[ \log \left( \frac { p ( x , z _ { T } ) } { q _ { 0 } ( z _ { 0 } | x ) \prod _ { t = 1 } ^ { T } \left| \operatorname* { d e t } \frac { \partial z _ { t } } { \partial z _ { t - 1 } } \right| ^ { - 1 } } \right) \right] .
+$$
+
+The main constraint on these transformations is that the determinant of their Jacobian needs to be easily computable.
+
+# 2.2.2 AUXILIARY VARIABLES
+
+Deep generative models can be extended with auxiliary variables which leave the generative model unchanged but make the variational distribution more expressive. Just as hierarchical Bayesian models induce dependencies between data, hierarchical variational models can induce dependencies between latent variables. The addition of the auxiliary variable changes the lower bound to:
+
+$$
+\begin{array} { r l } & { \log p ( x ) \geq \mathbb { E } _ { z , v \sim q ( z , v \mid x ) } \left[ \log \left( \displaystyle \frac { p ( x , z ) r ( v \mid x , z ) } { q ( z , v \mid x ) } \right) \right] } \\ & { \qquad = \mathbb { E } _ { q ( z \mid x ) } \left[ \log \left( \displaystyle \frac { p ( x , z ) } { q ( z \mid x ) } \right) - { \mathrm { K L } \Big ( q ( v \mid z , x ) \| r ( v \mid x , z ) \Big ) } \right] } \end{array}
+$$
+
+where $r ( v | x , z )$ is called the reverse model. From Eqn. 8, we see that this bound is looser than the regular ELBO, however the extra flexibility provided by the auxiliary variable can result in a higher lower bound. This idea has been employed in works such as auxiliary deep generative models (ADGM, Maaløe et al. (2016)), hierarchical variational models (HVM, Ranganath et al. (2016)) and Hamiltonian variational inference (HVI, Salimans et al. (2015)).
+
+# 2.3 MARGINAL LOG-LIKELIHOOD ESTIMATION
+
+We use two bounds to estimate the marginal log-likelihood of a model: IWAE (Burda et al., 2016) and AIS (Neal, 2001). Here we describe the IWAE bound. See Section 6.5 in the appendix for a description of AIS.
+
+The IWAE bound is a tighter lower bound than the VAE bound. More specifically, if we take multiple samples from the $q$ distribution, we can compute a tighter lower bound on the marginal log-likelihood:
+
+$$
+\log p ( x ) \geq \mathbb { E } _ { z _ { 1 } . . . z _ { k } \sim q ( z | x ) } \left[ \log \left( \frac { 1 } { k } \sum _ { i = 1 } ^ { k } \frac { p ( x , z _ { i } ) } { q ( z _ { i } | x ) } \right) \right] = \mathcal { L } _ { \mathrm { I W A E } } [ q ] .
+$$
+
+As the number of importance samples approaches infinity, the bound approaches the marginal loglikelihood. This importance weighted bound was introduced along with the Importance Weighted Autoencoder (Burda et al., 2016), thus we refer to it as the IWAE bound. It is often used as an evaluation metric for generative models (Burda et al., 2016; Kingma et al., 2016). As shown by Bachman & Precup (2015) and Cremer et al. (2017), the IWAE bound can be seen as using the VAE bound but with an importance weighted $q$ distribution.
+
+# 3 METHODS
+
+# 3.1 APPROXIMATION AND AMORTIZATION GAPS
+
+The inference gap $\mathcal { G }$ is the difference between the marginal log-likelihood $\log p ( x )$ and a lower bound ${ \mathcal { L } } [ q ]$ . Given the distribution in the family that maximizes the bound, $q ^ { * } ( z | x ) \ =$ arg $\operatorname* { m a x } _ { q \in \mathcal { Q } } \mathcal { L } [ q ]$ , the inference gap decomposes as the sum of approximation and amortization gaps:
+
+$$
+\mathcal { G } = \log p ( x ) - \mathcal { L } [ q ] = \underbrace { \log p ( x ) - \mathcal { L } [ q ^ { * } ] } _ { \mathrm { A p p r o x i m a t i o n } } + \underbrace { \mathcal { L } [ q ^ { * } ] - \mathcal { L } [ q ] } _ { \mathrm { A m o r t i z a t i o n } } .
+$$
+
+For VAEs, we can translate the gaps to KL divergences by rearranging (2):
+
+$$
+\mathcal { G } _ { \mathrm { V A E } } = \mathrm { K L } \big ( q ^ { * } ( z | x ) | | p ( z | x ) \big ) + \mathrm { K L } \big ( q ( z | x ) | | p ( z | x ) \big ) - \mathrm { K L } \big ( q ^ { * } ( z | x ) | | p ( z | x ) \big ) .
+$$
+
+# 3.2 FLEXIBLE APPROXIMATE POSTERIOR
+
+Our experimentation compares two families of approximate posteriors: the fully-factorized Gaussian (FFG) and a flexible flow (Flow). Our choice of flow is a combination of the Real NVP (Dinh et al., 2017) and auxiliary variables (Ranganath et al., 2016; Maaløe et al., 2016). Our model also resembles leap-frog dynamics applied in Hamiltonian Monte Carlo (HMC, Neal et al. (2011)).
+
+Let $z \in \mathbb { R } ^ { n }$ be the variable of interest and $v \in \mathbb { R } ^ { n }$ the auxiliary variable. Each flow step involves:
+
+$$
+\begin{array} { l } { { v ^ { \prime } = v \circ \sigma _ { 1 } ( z ) + \mu _ { 1 } ( z ) } } \\ { { z ^ { \prime } = z \circ \sigma _ { 2 } ( v ^ { \prime } ) + \mu _ { 2 } ( v ^ { \prime } ) } } \end{array}
+$$
+
+where $\sigma _ { 1 } , \sigma _ { 2 } , \mu _ { 1 } , \mu _ { 2 } : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ are differentiable mappings parameterized by neural nets and $\circ$ takes the Hadamard or element-wise product. The determinant of the combined transformation’s Jacobian, $| \mathrm { d e t } ( D f ) |$ , can be easily evaluated. See section 6.2 in the Appendix for a detailed derivation.
+
+Thus, we can jointly train the generative and flow-based inference model by optimizing the bound:
+
+$$
+\log p ( x ) \geq \mathbb { E } _ { z , v \sim q ( z , v \mid x ) } \left[ \log \left( { \frac { p ( x , z ^ { \prime } ) r ( v ^ { \prime } | x , z ^ { \prime } ) } { q ( z , v | x ) \left| \operatorname* { d e t } ( D f ) \right| ^ { - 1 } } } \right) \right] = \mathcal { L } _ { \mathrm { f l o w } } [ q ] .
+$$
+
+Additionally, multiple such type of transformations can be stacked to improve expressiveness. We refer readers to section 6.1.2 in the Appendix for details of our flow configuration adopted in the experimentation.
+
+# 3.3 EVALUATION BOUNDS
+
+We use several bounds to compute the inference gaps. To estimate the marginal log-likelihood, $\log { \hat { p } } ( x )$ , we take the maximum of our tightest lower bounds, specifically the maximum between the IWAE and AIS bounds. To compute the AIS bound, we use 100 chains, each with 500 intermediate distributions, where each transition consists of one HMC trajectory with 10 leapfrog steps. The initial distribution for AIS is the prior, so that it is encoder-independent.
+
+For our experiments, we test two different variational distributions: the fully-factorized Gaussian $q _ { F F G }$ and the flexible approximation $q F l o w$ as described in section 3.2. When computing ${ \mathcal { L } } _ { \mathrm { V A E } } [ q ]$ and $\mathcal { L } _ { \mathrm { I W A E } } [ q ]$ , we use 5000 samples. To compute $\mathcal { L } _ { \mathrm { V A E } } [ q ^ { * } ]$ , we optimize the parameters of the variational distribution for every datapoint. See Section 6.4 for details of the local optimization and stopping criteria.
+
+# 4 RELATED WORK
+
+Much of the earlier work on variational inference focused on optimizing the variational parameters locally for each datapoint, e.g. the original Stochastic Variational Inference scheme (SVI, Hoffman et al. (2013)) specifies the variational parameters to be optimized locally in the inner loop. Salakhutdinov & Larochelle (2010) perform such local optimization when learning deep Boltzmann machines. More recent work has applied this idea to improve approximate inference in directed Belief networks (Hjelm et al., 2015).
+
+Most relevant to our work is the recent work of Krishnan et al. (2017). They explicitly remark on two sources of error in variational learning with inference networks, and propose to optimize approximate inference locally from an initialization output by the inference network. They show improved training on high-dimensional, sparse data with the hybrid method, claiming that local optimization reduces the negative effects of random initialization in the inference network early on in training. Yet, their work only dwells on reducing the amortization gap and does analyze the error arising from the use of limited approximating distributions.
+
+![](images/a1d655761a82d6b4bc4ede47b36a7fbb2611e59eeea14fc4cf9c5c21bd3472f8.jpg)  
+Figure 2: True Posterior and Approximate Distributions of a VAE with 2D latent space. Columns: 4 different datapoints. FFG: Fully-factorized Gaussian. Flow: Using a flexible approximate distribution. Amortized: Using amortized parameters. Optimal: Parameters optimized for individual datapoints. The green distributions are the true posterior distributions, highlighting the mismatch with the approximation.
+
+Even though it is clear that failed inference would lead to a failed generative model, little quantitative assessment has been done showing the effect of the approximate posterior on the true posterior. Burda et al. (2016) visually demonstrate that when trained with an importance-weighted approximate posterior, the resulting true posterior is more complex than those trained with fully-factorized Gaussian approximations. We extend this observation quantitatively in the setting of flow-based approximate inference.
+
+# 5 EXPERIMENTAL RESULTS
+
+# 5.1 INTUITION THROUGH VISUALIZATION
+
+To begin, we would like to gain some insight into the properties of inference in VAEs by visualizing different distributions in the latent space. To this end, we trained a VAE with a two-dimensional latent space on MNIST. We show contour plots of various distributions in the latent space in Fig. 2. The first row contains contour plots of the true posteriors $p ( z | x )$ for four different training datapoints (columns). We have selected these four examples to highlight different inference phenomena. The amortized FFG row refers to the output of the recognition net, in this case, a fully-factorized Gaussian (FFG) approximation. Optimal FFG is the FFG that best fits the posterior of the datapoint. Optimal Flow is the optimal fit of a flexible distribution to the same posterior, where the flexible distribution we use is described in Section 3.2.
+
+Posterior A is an example of a distribution where FFG can fit well. Posterior B is an example of dependence between dimensions, demonstrating the limitation of having a factorized approximation. Posterior C highlights a shortcoming of performing amortization with a limited-capacity recognition network, where the amortized FFG shares little support with the true posterior. Posterior $\mathbf { D }$ is a bimodal distribution which demonstrates the ability of the flexible approximation to fit to complex distributions, in contrast to the simple FFG approximation. These observations raise the following question: in more typical VAEs, is the amortization of inference the leading cause of the distribution mismatch, or is it the choice of approximation?
+
+<table><tr><td rowspan="2"></td><td colspan="2">MNIST</td><td colspan="2">Fashion-MNIST</td><td colspan="2">CIFAR-10</td></tr><tr><td>qFFG</td><td>qFlow</td><td>qFFG</td><td>qFlow</td><td>qFFG</td><td>qFlow</td></tr><tr><td>log p(x)</td><td>-89.80</td><td>-88.94</td><td>-97.47</td><td>-97.41</td><td>-14913.15</td><td>-14914.45</td></tr><tr><td>LVAE[qFlow]</td><td>-90.80</td><td>-90.38</td><td>-98.92</td><td>-99.10</td><td>-14914.22</td><td>-14915.57</td></tr><tr><td>LVAE[qFFG]</td><td>-91.23</td><td>-113.54</td><td>-100.53</td><td>-132.46</td><td>-14915.40</td><td>-14919.08</td></tr><tr><td>LVAE[q]</td><td>-92.57</td><td>-91.79</td><td>-104.75</td><td>-103.76</td><td>-14976.57</td><td>-14975.12</td></tr><tr><td>Approximation</td><td>1.43</td><td>1.44</td><td>3.06</td><td>1.69</td><td>2.25</td><td>1.12</td></tr><tr><td>Amortization</td><td>1.34</td><td>1.41</td><td>4.22</td><td>4.66</td><td>61.17</td><td>59.55</td></tr><tr><td>Inference</td><td>2.77</td><td>2.85</td><td>7.28</td><td>6.35</td><td>63.42</td><td>60.67</td></tr></table>
+
+Table 2: Inference Gaps. The columns $q _ { F F G }$ and $q _ { F l o w }$ refer to the variational distribution used for training the model. All numbers are in nats.
+
+# 5.2 AMORTIZATION VS APPROXIMATION GAP
+
+Here we will compare the influence that the approximation and amortization errors have on the total inference gap. Table 2 are results from training on MNIST, Fashion-MNIST and CIFAR-10. For each dataset, we trained two different approximate posterior distributions: a fully-factorized Gaussian, $q _ { F F G }$ , and a flexible distribution, $q _ { F l o w }$ . Due to the computational cost of optimizing the local parameters for each datapoint, our evaluation is performed on a subset of 1000 datapoints for MNIST and Fashion-MNIST and a subset of 100 datapoints for CIFAR-10.
+
+For MNIST, we see that the amortization and approximation gaps each account for nearly half of the inference gap. On Fashion-MNIST, which is a more difficult dataset to model, the amortization gap becomes larger than the approximation gap. Similarly for CIFAR-10, we see that the amortization gap is much more significant than the approximation gap. Thus, for the three datasets and model architectures that we tested, the amortization gap seems to be the prominent cause of inference suboptimality, especially when the difficulty of the dataset increases. This analysis indicates that improvements in inference will likely be a result of reducing amortization error, rather than approximation errors.
+
+With these results in mind, would simply increasing the capacity of the encoder improve the amortization gap? We examined this by training the MNIST and Fashion-MNIST models from above but with larger encoders. See Section 6.1.2 for implementation details. Table 3 are the results of this experiment. Comparing to Table 2, we see that for both datasets and both variational distributions, the inference gap decreases and the decrease is mainly due to a reduction in the amortization gap.
+
+<table><tr><td rowspan="7">logp(x) LVAE[qFlow] LVAEqFFG]</td><td colspan="2">MNIST</td><td colspan="2">Fashion-MNIST</td></tr><tr><td>qFFG</td><td>qFlow</td><td>qFFG</td><td>qFlow</td></tr><tr><td>-89.61</td><td>-88.99</td><td>-95.99</td><td>-96.18</td></tr><tr><td>-90.65</td><td>-90.44</td><td>-97.40</td><td>-97.91</td></tr><tr><td>-91.07</td><td>-108.71</td><td>-99.64</td><td>-129.7</td></tr><tr><td>-92.18</td><td>-91.19</td><td>-102.73</td><td>-101.67</td></tr><tr><td>LVAE[q] Approximation 1.46</td><td>1.45</td><td>3.65</td><td>1.73</td></tr><tr><td>Amortization</td><td>1.11</td><td>0.75</td><td>3.09</td><td>3.76</td></tr><tr><td>Inference</td><td>2.56</td><td>2.20</td><td>6.74</td><td>5.49</td></tr></table>
+
+Table 3: Larger Encoder. The columns $q _ { F F G }$ and $q F l o w$ refer to the variational distribution used for training the model. All numbers are in nats.
+
+# 5.2.1 INFLUENCE OF FLOWS ON AMORTIZATION GAP
+
+The common reasoning for increasing the expressiveness of the approximate posterior is to minimize the difference between the true and approximate, i.e. reduce the approximation gap. However, given that the expressive approximation is often accompanied by many additional parameters, we would like to know if it has an influence on the amortization error.
+
+To investigate this, we trained a VAE in the same manner as Section 5.2. After training, we kept the generator fixed and trained new encoders to fit to the fixed posterior. Specifically, we trained a small encoder with a factorized Gaussian $q$ distribution to obtain a large amortization gap. We then trained a small encoder with a flow distribution. See Section 6.2 for the details of the experiment. The results are shown in Table 4. As expected, we observe that the small encoder has a very large amortization gap. However, when we use $q _ { F l o w }$ as the approximate distribution, we see the approximation gap decrease, but more importantly, there is a significant decrease in the amortization gap. This indicates that the parameters used for increasing the complexity of the approximation also play a large role in diminishing the amortization error.
+
+Table 4: Influence of Flows on the Amortization Gap. The parameters used to increase the flexibility of the approximate distribution also reduce the amortization gap. See Section 5.2.1 for details of the experiment.   
+
+<table><tr><td>Variational Family</td><td>qFFG</td><td>qFlow</td></tr><tr><td>logp(x) LVAE[q*]</td><td>-84.70 -86.61</td><td>-84.70 -85.48</td></tr><tr><td>LVAE[q] Approximation</td><td>-129.83 1.91</td><td>-98.58 0.78</td></tr><tr><td>Amortization</td><td>43.22</td><td>13.10</td></tr><tr><td>Inference</td><td>45.13</td><td>13.88</td></tr></table>
+
+These results are expected given that the parameterization of the Flow distribution can be interpreted as an instance of the RevNet (Gomez et al., 2017) which has demonstrated that Real-NVP like transformations (Dinh et al., 2017) can model complex functions similar to typical MLPs. Thus the flow transformations we employ should also be expected to increase the expressiveness while also increasing the capacity of the encoder. The implication of this observation is that models which improve the flexibility of their variational approximation, and attribute their improved results to the increased expressiveness, may have actually been due to the reduction in amortization error.
+
+# 5.3 INFLUENCE OF APPROXIMATE POSTERIOR ON TRUE POSTERIOR
+
+We have seen that increasing the expressiveness of the approximation improves the marginal likelihood of the trained model, but to what amount does it alter the true posterior? Will a factorized Gaussian approximation cause the true posterior to be more like a factorized Gaussian or is the true posterior mostly fixed? Just as it is hard to evaluate a generative model by visually inspecting samples from the model, its hard to say how Gaussian the true posterior is by visual inspection. We can quantitatively determine how close the posterior is to a fully factorized Gaussian (FFG) distribution by comparing the marginal log-likelihood estimate, $\log { \dot { \hat { p } } } ( x )$ , and the Optimal FFG bound, $\mathcal { L } _ { \mathrm { V A E } } [ q _ { F F G } ^ { * } ]$ . In other words, we are estimating the KL divergence between the optimal Gaussian and the true posterior, $\mathrm { K L } \left( q ^ { * } ( z | x ) | | p ( z | x ) \right)$ .
+
+In Table 2 on MNIST, the Optimal Flow improves upon the Optimal FFG for the FFG trained model by 0.4 nats. In contrast, on the Flow trained model, the difference increases to 12.5 nats. This suggests that the true posterior of a FFG-trained model is closer to FFG than the true posterior of the Flow-trained model. The same observation can be made on the Fashion-MNIST dataset. This implies that the decoder can learn to have a true posterior that fits better to the approximation. Although the generative model can learn to have a posterior that fits to the approximation, it seems that not having this constraint, ie. using a flexible approximate, results in better generative models.
+
+We can use these observations to help justify our approximation and amortization gap results of Section 5.2. Those results showed that the amortization error is often the main cause of inference suboptimality. One reason for this is that the generator accommodates to the choice of approximation, as shown above, thus reducing the approximation error.
+
+Given that we have seen that the generator could accommodate to the choice of approximation, our next question is whether a generator with more capacity can accommodate more. To this end, we trained VAEs with decoders of different sizes and measured the approximation gaps. Specifically, we trained decoders with 0, 2, and 4 hidden layers on MNIST. See Table 5 for the results. We see that as the capacity of the decoder increases, the approximation gap decreases. This result implies that the more flexible the generator, the less flexible the approximate distribution needs to be.
+
+Table 5: Increased decoder capacity reduces approximation gap. All numbers are in nats.   
+
+<table><tr><td>Generator HiddenLayers</td><td>0</td><td>2</td><td>4</td></tr><tr><td>logp(x)</td><td>-100.52</td><td>-86.61</td><td>-83.82</td></tr><tr><td>LVAE[qFFG]</td><td>-104.42</td><td>-84.78</td><td>-82.19</td></tr><tr><td>Approximation Gap</td><td>3.90</td><td>1.83</td><td>1.63</td></tr></table>
+
+# 5.3.1 ANNEALING THE ENTROPY
+
+Typical warm-up (Bowman et al., 2015; Sønderby et al., 2016) refers to annealing $\mathrm { K L } \left( q ( \boldsymbol { z } | \boldsymbol { x } ) | | p ( \boldsymbol { z } ) \right)$ during training. This can also be interpreted as performing maximum likelihood estimation (MLE) early on during training. This optimization technique is known to help prevent the latent variable from degrading to the prior (Burda et al., 2016; Sønderby et al., 2016). We employ a similar annealing scheme during training. Rather than annealing the KL divergence, we anneal the entropy of the approximate distribution $q$ :
+
+$$
+\begin{array} { r } { \mathbb { E } _ { z \sim q ( z | x ) } \left[ \log p ( x , z ) - \lambda \log q ( z | x ) \right] , } \end{array}
+$$
+
+where $\lambda$ is annealed from 0 to 1 over training. This can be interpreted as maximum a posteriori (MAP) in the initial phase. Due to its similarity, we will also refer to this technique as warm-up.
+
+We find that warm-up techniques, such as annealing the entropy, are important for allowing the true posterior to be more complex. Table 6 are results from a model trained without the entropy annealing schedule. Comparing these results to Table 2, we observe that the difference between $\mathcal { L } _ { \mathrm { V A E } } [ q _ { F F G } ^ { * } ]$ and $\mathcal { L } _ { \mathrm { V A E } } [ q _ { F l o w } ^ { * } ]$ is significantly smaller without entropy annealing. This indicates that the true posterior is more Gaussian when entropy annealing is not used. This suggests that, in addition to preventing the latent variable from degrading to the prior, entropy annealing allows the true posterior to better utilize the flexibility of the expressive approximation, resulting in a better trained model.
+
+<table><tr><td rowspan="7">log p(x) LVAE[qFlow LVAE[qFFG]</td><td colspan="2">MNIST</td><td colspan="2">Fashion-MNIST</td></tr><tr><td>qFFG</td><td>qFlow</td><td>qFFG</td><td>qFlow</td></tr><tr><td>-89.82</td><td>-89.52</td><td>-102.56</td><td>-102.88</td></tr><tr><td>-90.96</td><td>-90.45</td><td>-103.73</td><td>-104.02</td></tr><tr><td>-90.84</td><td>-92.25</td><td>-103.85</td><td>-105.80</td></tr><tr><td>-92.33</td><td>-91.75</td><td>-106.90</td><td>-107.01</td></tr><tr><td>LvAE[q] Approximation 1.02</td><td>0.93</td><td>1.29</td><td>1.14</td></tr><tr><td>Amortization</td><td>1.49</td><td>1.30</td><td>3.05</td><td>2.29</td></tr><tr><td>Inference</td><td>2.51</td><td>2.23</td><td>4.34</td><td>4.13</td></tr></table>
+
+Table 6: Models trained without entropy annealing. The columns $q _ { F F G }$ and $q _ { F l o w }$ refer to the variational distribution used for training the model. All numbers are in nats.
+
+# 6 CONCLUSION
+
+In this paper, we investigated how encoder capacity, approximation choice, decoder capacity, and model optimization influence inference suboptimality in terms of the approximation and amortization gaps. We found that the amortization gap is often the leading source of inference suboptimality and that the generator reduces the approximation gap by learning a true posterior that fits to the choice of approximate distribution. We showed that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation. We confirmed that increasing the capacity of the encoder reduces the amortization error. We also showed that optimization techniques, such as entropy annealing, help the generative model to better utilize the flexibility of the expressive variational distribution. Computing these gaps can be useful for guiding improvements to inference in VAEs. Future work includes evaluating other types of expressive approximations and more complex likelihood functions.
+
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+
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+
+# APPENDIX
+
+# 6.1 MODEL ARCHITECTURES AND TRAINING HYPERPARAMETERS
+
+# 6.1.1 2D VISUALIZATION
+
+The VAE model of Fig. 2 uses a decoder $p ( x | z )$ with architecture: $2 - 1 0 0 - 7 8 4$ , and an encoder $q ( z | x )$ with architecture: $7 8 4 - 1 0 0 - 4$ . We use tanh activations and a batch size of 50. The model is trained for 3000 epochs with a learning rate of $1 0 ^ { - 4 }$ using the ADAM optimizer (Kingma & Ba, 2014).
+
+# 6.1.2 MNIST & FASHION-MNIST
+
+Both MNIST and Fashion-MNIST consist of a training and test set with 60k and 10k datapoints respectively, where each datapoint is a $2 8 \mathbf { x } 2 8$ grey-scale image. We rescale the original images so that pixel values are within the range [0, 1]. For MNIST, We use the statically binarized version described by Larochelle & Bengio (2008). We also binarize Fashion-MINST statically. For both datasets, we adopt the Bernoulli likelihood for the generator.
+
+The VAE models for MNIST and Fashion-MNIST experiments have the same architecture given in table 7. The flow configuration is given in table 8.
+
+Table 7: Neural net architecture for MNIST/Fashion-MNIST experiments.   
+
+<table><tr><td>Inference Network</td><td>Generator</td></tr><tr><td>Input ∈R784</td><td>Input ∈R50</td></tr><tr><td>FC.200-ELU-FC.200-ELU-FC.50+50</td><td>FC.200-ELU-FC.200-ELU-FC.784-Sigmoid</td></tr></table>
+
+In the large encoder setting, we change the number of hidden units for the inference network to be 500, instead of 200. The warm-up models are trained with a linear schedule over the first 400 epochs according to Section 5.3.1.
+
+The activation function is chosen to be the exponential linear unit (ELU, Clevert et al. (2015)), as we observe improved performance compared to tanh. We follow the same learning rate schedule and train for the same amount of epochs as described by Burda et al. (2016). All models are trained with the a batch-size of 100 with ADAM.
+
+# 6.1.3 CIFAR-10
+
+CIFAR-10 consists of a training and test dataset with $5 0 \mathrm { k }$ and $1 0 \mathrm { k }$ datapoints respectively, where each datapoint is a $3 2 \times 3 2$ color image. We rescale individual pixel values to be in the range [0, 1]. We follow the discretized logistic likelihood model adopted by Kingma et al. (2016), where each input channel has its own scale learned by an MLP. For the latent variable, we use a 32-dimensional factorized Gaussian for $q ( z | x )$ following Kingma et al. (2016). For all neural networks, ELU is chosen to be the activation function. The specific network architecture is shown in Table 9.
+
+We adopt a gradually decreasing learning rate with an initialize value of $1 0 ^ { - 3 }$ . Warm-up is applied with a linear schedule over the first 20 epochs. All models are trained with a batch-size of 100 with ADAM. Early-stopping is applied based on the performance on the held-out set.
+
+For the model with expressive inference, we use four flow steps as opposed to only two in MNIST/Fashion-MNIST experiments.
+
+# 6.2 INFLUENCE OF FLOWS ON AMORTIZATION GAP EXPERIMENT
+
+The aim of this experiment is to show that the parameters used for increasing the expressiveness of the approximation also contribute to reducing the amortization error. To show this, we train a VAE on MNIST, discard the encoder, then retrain two encoders on the fixed decoder: one with a factorized Gaussian distribution and the other with a parameterized ’flow’ distribution. We use fixed decoder so that the true posterior is constant for both encoders. See 5.2.1 for the results and below for the architecture details.
+
+The architecture of the decoder is: $D _ { Z } - 2 0 0 - 2 0 0 - D _ { X }$ . The architecture of the encoder used to train the decoder is $D _ { X } - 2 0 0 - 2 0 0 - 2 D _ { Z }$ . The approximate distribution $q ( z | x )$ is a factorized Gaussian.
+
+Next, we describe the encoders which were trained on the fixed trained decoder. In order to highlight a large amortization gap, we employed a very small encoder architecture: $D _ { X } - 2 D _ { Z }$ . This encoder has no hidden layers, which greatly impoverishes its ability and results in a large amortization gap.
+
+We compare two approximate distributions $q ( z | x )$ . Firstly, we experiment with the typical fully factorized Gaussian (FFG). The second is what we call a flow distribution. Specifically, we use the transformations of Dinh et al. (2017). We also include an auxiliary variable so we don’t need to select how to divide the latent space for the transformations. The approximate distribution over the latent $z$ and auxiliary variable $v$ factorizes as: $q ( z , v | x ) = q ( z | x ) \bar { q } ( v )$ . The $q ( v )$ distribution is simply a ${ \bf N } ( 0 , 1 )$ distribution. Since we’re using a auxiliary variable, we also require the $r ( v | z )$ distribution which we parameterize as $r ( v | z )$ : $[ D z ] - 5 0 - 5 0 - 2 D z$ . The flow transformation is the same as in Section 3.2, which we apply twice.
+
+<table><tr><td>q(uolz0)</td><td>r(Ur|zT)</td></tr><tr><td>Input ∈R50</td><td>Input ∈ R50</td></tr><tr><td>FC.100-ELU-FC.100-ELU-FC.50+50</td><td>FC.100-ELU-FC.100-ELU-FC.50+50</td></tr></table>
+
+<table><tr><td colspan="2">q(Ut+1, Zt+1lUt, zt)</td></tr><tr><td>01（）,02（）</td><td>μ1(.)，μ2(.)</td></tr><tr><td>Input ∈ R50</td><td>Input ∈ R50</td></tr><tr><td>FC.100-ELU-FC.100-ELU-FC.50</td><td>FC.100-ELU-FC.100-ELU-FC.50</td></tr></table>
+
+Table 8: Flow setting for MNIST/Fashion-MNIST experiments. $q ( v _ { T } , z _ { T } | v _ { 0 } , z _ { 0 } )$ consists of two normalizing flows given in the second tabular.
+
+<table><tr><td>Inference Network</td><td>Generator</td></tr><tr><td>Input 32 × 32 color image</td><td>Input ∈ R32</td></tr><tr><td>4 × 4 conv. 64 channels.stride 2.BN 4 × 4 conv.128 channels.stride 2.BN</td><td>FC.256×2×2ELU;FC.64-ELU-FC.32-ELU-FC.3 4 × 4 deconv. 128 channels. stride 2. BN</td></tr><tr><td>4 × 4 conv. 256 channels. stride 2.BN</td><td>4 × 4 deconv.64 channels.stride 2.BN</td></tr><tr><td>FC.32 + 32.output layer for mean and log-variance</td><td>4 × 4 deconv.3 channels.stride 2. Sigmoid</td></tr></table>
+
+Table 9: Network architecture for CIFAR-10 experiments. For the generator, one of the MLPs immediately after the input layer of the generator outputs channel-wise scales for the discretized logistic likelihood model. BN stands for batch-normalization.
+
+# 6.3 COMPUTATION OF THE DETERMINANT FOR FLOW
+
+The overall mapping $f$ that performs $( z , v ) \mapsto ( z ^ { \prime } , v ^ { \prime } )$ is the composition of two sheer mappings $f _ { 1 }$ and $f _ { 2 }$ that respectively perform $( z , v ) \mapsto ( z , v ^ { \prime } )$ and $( z , v ^ { \prime } ) \mapsto ( z ^ { \prime } , v ^ { \prime } )$ . Since the Jacobian of either one of the sheer mappings is diagonal, the determinant of the composed transformation’s Jacobian $D f$ can be easily computed:
+
+$$
+\operatorname * { d e t } ( D f ) = \operatorname * { d e t } ( D f _ { 1 } ) \mathrm { d e t } ( D f _ { 2 } ) = \Bigl ( \prod _ { i = 1 } ^ { n } \sigma _ { 1 } ( z ) _ { i } \Bigr ) \Bigl ( \prod _ { j = 1 } ^ { n } \sigma _ { 2 } ( v ^ { \prime } ) _ { j } \Bigr ) .
+$$
+
+# 6.4 LOCAL OPTIMIZATION OF APPROXIMATE DISTRIBUTION
+
+For the local FFG optimization, we initialize the mean and variance as the prior, i.e. $\mathcal { N } ( 0 , I )$ . We optimize the mean and variance using the Adam optimizer with a learning rate of $1 0 ^ { - 3 }$ . To determine convergence, after every 100 optimization steps, we compute the average of the previous 100 ELBO values and compare it to the best achieved average. If it does not improve for 10 consecutive iterations then the optimization is terminated. For the Flow model, the same process is used to optimize all of its parameters. All neural nets for the flow were initialized with a variant of the Xavier initilization (Glorot & Bengio, 2010). We use 100 Monte Carlo samples to compute the ELBO to reduce variance.
+
+# 6.5 ANNEALED IMPORTANCE SAMPLING
+
+Annealed importance sampling (AIS, Neal (2001); Jarzynski (1997)) is a means of computing a lower bound to the marginal log-likelihood. Similarly to the importance weighted bound, AIS must sample a proposal distribution $\bar { f } _ { 1 } ( z )$ and compute the density of these samples, however, AIS then transforms the samples through a sequence of reversible transitions $\mathcal { T } _ { t } ( z ^ { \prime } | z )$ . The transitions anneal the proposal distribution to the desired distribution $f _ { T } ( z )$ .
+
+Specifically, AIS samples an initial state $z _ { 1 } \sim f _ { 1 } ( z )$ and sets an initial weight $w _ { 1 } = 1$ . For the following annealing steps, $z _ { t }$ is sampled from $\mathcal { T } _ { t } { \left( z ^ { \prime } \right| } z )$ and the weight is updated according to:
+
+$$
+w _ { t } = w _ { t - 1 } \frac { f _ { t } ( z _ { t - 1 } ) } { f _ { t - 1 } ( z _ { t - 1 } ) } .
+$$
+
+This procedure produces weight $w _ { T }$ such that $\mathbb { E } \left[ w _ { T } \right] = \mathcal { Z } _ { T } / \mathcal { Z } _ { 1 }$ , where $Z _ { T }$ and $Z _ { 1 }$ are the normalizing constants of $f _ { T } ( z )$ and $f _ { 1 } ( z )$ respectively. This pertains to estimating the marginal likelihood when the target distribution is $p ( x , z )$ when we integrate with respect to $z$ .
+
+Typically, the intermediate distributions are simply defined to be geometric averages: $f _ { t } ( z ) ~ =$ $\dot { f _ { 1 } } \dot { ( z ) } ^ { 1 - \dot { \beta _ { t } } } f _ { T } ( z ) ^ { \beta _ { t } }$ , where $\beta _ { t }$ is monotonically increasing with $\beta _ { 1 } = 0$ and $\beta _ { T } = 1$ . When $f _ { 1 } ( z ) =$ $p ( z )$ and $f _ { T } ( z ) = p ( x , z )$ , the intermediate distributions are: $f _ { i } ( x ) = p ( z ) p ( x | z ) ^ { \beta _ { i } }$ .
+
+Model evaluation with AIS appears early on in the setting of deep belief networks (Salakhutdinov & Murray, 2008). AIS for decoder-based models was also used by $\mathrm { { W u } }$ et al. (2017). They validated the accuracy of the approach with Bidirectional Monte Carlo (BDMC, Grosse et al. (2015)) and demonstrated the advantage of using AIS over the IWAE bound for evaluation when the inference network overfits to the training data.
+
+# 6.6 THE INFERENCE GAP
+
+How well is inference done in VAEs during training? Are we close to doing the optimal or is there much room for improvement? To answer this question, we quantitatively measure the inference gap: the gap between the true marginal log-likelihood and the lower bound. This amounts to measuring how well inference is being done during training. Since we cannot compute the exact marginal log-likelihood, we estimate it using the maximum of any of its lower bounds, described in 3.3.
+
+Fig. 3a shows training curves for a FFG and Flow inference network as measured by the VAE, IWAE, and AIS bounds on the training and test set. The inference gap on the training set with the FFG model is 3.01 nats, whereas the Flow model is 2.71 nats. Accordingly, Fig. 3a shows that the training IWAE bound is slightly tighter for the Flow model compared to the FFG. Due to this lower inference gap during training, the Flow model achieves a higher AIS bound on the test set than the FFG model.
+
+To demonstrate that a very small inference gap can be achieved, even with a limited approximation such as a factorized Gaussian, we train the model on a small dataset. In this experiment, our training set consists of 1000 datapoints randomly chosen from the original MNIST training set. The training curves on this small datatset are show in Fig. 3b. Even with a factorized Gaussian distribution, the inference gap is very small: the AIS and IWAE bounds are overlapping and the VAE is just slightly below. Yet, the model is overfitting as seen by the decreasing test set bounds.
+
+![](images/27084004bf0862c2b0451b27bad5bd7bbe3e9917819b0dc3cd0ee5c7c6bd4713.jpg)  
+Figure 3: Training curves for a FFG and a Flow inference model on MNIST. AIS provides the tightest lower bound and is independent of encoder overfitting. There is little difference between FFG and Flow models trained on the 1000 datapoints since inference is nearly equivalent.
+
+# 6.6.1 ENCODER AND DECODER OVERFITTING
+
+We will begin by explaining how we separate encoder from decoder overfitting. Decoder overfitting is the same as in the regular supervised learning scenario, where we compare the train and test error. To measure decoder overfitting independently from encoder overfitting, we use the AIS bound since it is encoder-independent. Thus we can observe decoder overfitting through the AIS test training curve. In contrast, the encoder can only overfit in the sense that the recognition network becomes unsuitable for computing the marginal likelihood on the test set. Thus, encoder overfitting is computed by: $\mathcal { L } _ { \mathrm { A I S } } \ - \mathcal { L } _ { \mathrm { I W } }$ on the test set.
+
+For the small dataset of Fig. 3b, it clear that there is significant encoder and decoder overfitting. A model trained in this setting would benefit from regularization. For Fig. 3a, the model is not overfit and would benefit from more training. However, there is some encoder overfitting due to the gap between the AIS and IWAE bounds on the test set. Comparing the FFG and Flow models, it appears that the Flow does not have a large effect on encoder or decoder overfitting.
+
+# 6.7 GAUSSIAN LATENTS WITH FULL COVARIANCE
+
+The flexiblity of the Gaussian family with arbitrary covariance lies between that of FFG and Flow. With covariance, the Gaussian distribution can model interactions between different latent dimensions. Yet, compared to Flow, its expressiveness is limited due to its inability to model higher order interactions and its unimodal nature.
+
+To apply the reparameterization trick, we perform the Cholesky decomposition on the covariance matrix: $\overrightharpoon { \Sigma } = L \overrightharpoon { L } ^ { \top }$ , where $L$ is lower triangular. A sample from $\mathcal { N } ( \boldsymbol { \mu } , \boldsymbol { \Sigma } )$ could be obtained by first sampling from a unit Gaussian $\epsilon \sim \mathcal { N } ( 0 , \bar { I } )$ , then computing $z = \mu + L \epsilon$ .
+
+To analyze the capability of the Gaussian family, we train several VAEs on MNIST and FashionMNIST with the approximate posterior $q ( z | x )$ being a Gaussian with full covariance. To inspect how well inference is done, we perform the local optimizations described in Section 5.2 with FFG and Flow.
+
+Table 10: Gaussian latents trained with full covariance.   
+
+<table><tr><td></td><td>MNIST</td><td>Fashion-MNIST</td></tr><tr><td>logp(x)</td><td>-89.28</td><td>-96.46</td></tr><tr><td>LVAElqFlow]</td><td>-90.69</td><td>-98.19</td></tr><tr><td>LvAE[FFG]</td><td>-101.84</td><td>-107.89</td></tr><tr><td>LvAE[q]</td><td>-92.05</td><td>-102.93</td></tr></table>
+
+We can see from table 10 that local optimization with FFG on a model trained with full covariance inference produces a bad lower bound. This resonates with the argument that the approximation has a significant influence on the true posterior as described in section 5.3.
+
+Comparing to numbers in table 2, we can see that the full-covariance VAE trained on MNIST is nearly on par with that trained with Flow (-89.28 vs -88.94). For Fashion-MNIST, the fullcovariance VAE even performs better by a large margin in terms of the estimated log-likelihood (-96.46 vs -97.41).

parse/train/HJxYwiC5tm/HJxYwiC5tm.md

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+# WHY DO DEEP CONVOLUTIONAL NETWORKS GENERALIZE SO POORLY TO SMALL IMAGE TRANSFORMATIONS?
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Deep convolutional network architectures are often assumed to guarantee generalization for small image translations and deformations. In this paper we show that modern CNNs (VGG16, ResNet50, and InceptionResNetV2) can drastically change their output when an image is translated in the image plane by a few pixels, and that this failure of generalization also happens with other realistic small image transformations. Furthermore, we see these failures to generalize more frequently in more modern networks. We show that these failures are related to the fact that the architecture of modern CNNs ignores the classical sampling theorem so that generalization is not guaranteed. We also show that biases in the statistics of commonly used image datasets makes it unlikely that CNNs will learn to be invariant to these transformations. Taken together our results suggest that the performance of CNNs in object recognition falls far short of the generalization capabilities of humans.
+
+# 1 INTRODUCTION
+
+Deep convolutional neural networks (CNNs) have revolutionized computer vision. Perhaps the most dramatic success is in the area of object recognition, where performance is now described as "superhuman" (He et al., 2015). A key to the success of any machine learning method is the inductive bias of the method, and clearly the choice of architecture in a neural network significantly affects the inductive bias. In particular, the choice of convolution and pooling in CNNs is motivated by the desire to endow the networks with invariance to irrelevant cues such as image translations, scalings, and other small deformations (Fukushima & Miyake, 1982; Zeiler & Fergus, 2014). This motivation was made explicit in the 1980s by Fukushima in describing the "neocognitron" architecture, which served as inspiration for modern CNNs (LeCun et al., 1989), "After finishing the process of learning, pattern recognition is performed on the basis of similarity in shape between patterns, and is not affected by deformation, nor by changes in size, nor by shifts in the position of the input patterns." (Fukushima, 1988)
+
+Despite the excellent performance of CNNs on object recognition, the vulnerability to adversarial attacks suggests that superficial changes can result in highly non-human shifts in prediction (e.g. (Bhagoji et al., 2017; Su et al., 2017). In addition, filtering the image in the Fourier domain (in a way that does not change human prediction) also results in a substantial drop in prediction accuracy (Jo & Bengio, 2017). These and other results (Rodner et al., 2016) indicate that CNNs are not invariant to cues that are irrelevant to the object identity.
+
+An argument against adversarial attacks on CNNs is that they often involve highly unnatural transformations to the input images, hence in some sense we would not expect CNNs to be invariant to these transformations. When considering more natural transformations, there is preliminary evidence that AlexNet (Krizhevsky et al., 2012) is robust to some of them (Zeiler & Fergus, 2014). On the other hand, there is also preliminary evidence for lack of robustness in the more modern networks for object classification (Bunne et al., 2018) and detection (Rosenfeld et al., 2018) along with studies suggesting that with small CNNs and the MNIST data, data augmentation is the main feature affecting CNN invariance (Kauderer-Abrams, 2017). An indirect method to probe for invariances measures the linearity of the learned representations under natural transformations to the input image (Lenc & Vedaldi, 2015; Hénaff & Simoncelli, 2015; Fawzi & Frossard, 2015; Cohen & Welling, 2014). The recent work of (Engstrom et al., 2017) investigates adversarial attacks that use only rotations and translations. They find that "simple transformations, namely translations and rotations alone, are sufficient to fool neural network-based vision models on a significant fraction of inputs" and show that advanced data augmentation methods can make the networks more robust.
+
+![](images/92b13decc6d7a5ecd856ae52ff3d3939757f136ce0017965aa457404cf336bc7.jpg)  
+Figure 1: Examples of jagged predictions of modern deep convolutional neural networks. Top: A negligible vertical shift of the object (Kuvasz) results in an abrupt decrease in the network’s predicted score of the correct class. Middle: A tiny increase in the size of the object (Lotion) produces a dramatic decrease in the network’s predicted score of the correct class. Bottom: A very small change in the bear’s posture results in an abrupt decrease in the network’s predicted score of the correct class. Colored dots represent images chosen from interesting $\mathbf { X }$ -axis locations of the graphs on the right. These dots illustrate sensitivity of modern neural networks to small, insignificant (to a human), and realistic variations in the image.
+
+In this paper, we directly ask "why are modern CNNs not invariant to natural image transformations despite the architecture being explicitly designed to provide such invariances?". Specifically, we systematically examine the invariances of three modern deep CNNs: VGG-16 (Simonyan & Zisserman, 2014), ResNet-50 (He et al., 2016), and InceptionResNet-V2 (Szegedy et al., 2017). We find that modern deep CNNs are not invariant to translations, scalings and other realistic image transformations, and this lack of invariance is related to the subsampling operation and the biases contained in image datasets.
+
+# 2 FAILURES OF MODERN CNNS
+
+Figure 1 contains examples of abrupt failures following tiny realistic transformations for the InceptionResNet-V2 CNN. Shifting or scaling the object by just one pixel could result in a sharp change in prediction. In the top row, we embed the original image in a larger image and shift it in the image plane (while filling in the rest of the image with a simple inpainting procedure). In the middle row, we repeat this protocol with rescaling. In the bottom row, we show frames from a BBC film in which the ice bear moves almost imperceptibly between frames and the network’s output changes dramatically1.
+
+![](images/9846430a7dbf8d7be4b46ec6f4f2c1932f1d8ab18f7d55cf0e60e002d26e1274.jpg)  
+Figure 2: Modern deep convolutional neural networks are sensitive to small image translations. A) Comparison of three networks of various depths in the task of vertical image translation depicted in figure 1. Images (rows) are randomly chosen from the ImageNet dataset (Deng et al., 2009), and are sorted by the network’s prediction sum in a descending order. B) More modern networks have more jagged predictions.
+
+In order to measure how typical these failures are, we randomly chose images from the ImageNet validation set and measured the output of three modern CNNs as we embedded these images in a larger image and systematically varied the vertical translation. As was the case in figure 1, we used a simple inpainting procedure to fill in the rest of the image.
+
+Results are shown in figure 2. Each row corresponds to an image under different translations and the color denotes the network’s estimate of the probability of the correct class. Thus a row that is all light corresponds to a correct classification that is invariant to translation, while a row that is all dark corresponds to an incorrect classification that is invariant to translation. Surprisingly, many rows show abrupt transitions from light to dark, indicating that the classification changes abruptly as the object is translated. We quantify the lack of invariance by a measure we call "jaggedness": the number of times the network’s predictions had the correct class in its top-5 and after just one pixel shift it moved outside of the top-5 (and also the opposite transition from non-top-5 to top5). Using this measure, we find that for approximately $30 \%$ of the images, the output is "jagged", i.e the network changes its prediction by a shift of a single pixel. Also, as shown in the right of figure 2, jaggedness is greater for the modern, deeper, networks compared to the less modern VGG16 network. While the deeper networks have better test accuracy, they are also less invariant. In the appendix we also show an alternative to the "jaggedness" measure, which gives similar results.
+
+A natural criticism of these results is that they are somehow related to the image resizing and inpainting procedures that we used. To test this possibility, we repeated the experiment with a different protocol where we chose different crops of the original ImageNet image while making sure that the object bounding box remained within the crop. This protocol does not require any inpainting while still translating the object location within the new image. Results are shown in the appendix. We still have a large fraction of images for which the prediction is not invariant to translation. We also show in the appendix similar results for scaling rather than translation. Overall we find that regardless of the protocol used, modern CNNs often change their output significantly as a result of a small translation or scaling.
+
+# 3 IGNORING THE SAMPLING THEOREM
+
+The failure of CNNs to generalize to image translations is particularly puzzling. Intuitively, it would seem that if all layers in a network are convolutional then the representation should simply translate when an image is translated. If the final features for classification are obtained by a global pooling operation on the representation (as is done for example in ResNet50 and InceptionResNetV2) then these features should be invariant to translation. Where does this intuition fail?
+
+![](images/c2c6c1eeffb9c3ee10ebc723e3bca2e59d9e945c463eccdc0390f9338be39180.jpg)  
+Figure 3: The deeper the network, the less shiftable are the feature maps. A) A vertical shift of a "Kuvasz" dog in the image plane. B) Feature maps from three different network architectures in response to the translated Kuvasz image. Layer depth assignments reflect the number of trainable convolutional layers preceding the selected layer. The last layer is always the last convolutional layer in each network.
+
+This intuition ignores the subsampling operation which is prevalent in modern CNNs, also known as "stride". This failure of translation invariance in systems with subsampling was explicitly discussed in Simoncelli et al. (Simoncelli et al., 1992) who wrote "We cannot literally expect translation invariance in a system based on convolution and subsampling: translation of the input signal cannot produce simple translations of the transform coefficients, unless the translation is a multiple of each of the subsampling factors in the system". Since deep networks often contain many subsampling operations, the subsampling factor of the deep layers may be very large so that "literal" translation invariance only holds for very special translations. In InceptionResnetV2, for example, the subsampling factor is 60, so we expect exact translation invariance to hold only for $\scriptstyle { \frac { 1 } { 6 0 ^ { 2 } } }$ of possible translations.
+
+Simoncelli et al. also defined a weaker form of translation invariance, which they called "shiftability" and showed that it can hold for systems with subsampling (this is related to weak translation invariance as defined by (Lenc & Vedaldi, 2015), see also (Esteves et al., 2017; Cohen & Welling, 2014) for related ideas applied to neural networks). Here we extend the basic shiftability result to show that when shiftability holds, then global pooling will indeed yield invariant representations.
+
+We define $r ( x )$ as the response of a feature detector at location $x$ in the image plane. We say that this response is "convolutional" if translating the image by any translation $\delta$ yields a translation of the response by the same $\delta$ . This definition includes cases when the feature response is obtained by convolving the input image with a fixed filter, but also includes combinations of linear operations and nonlinear operations that do not include any subsampling.
+
+We start by a trivial observation:
+
+Observation: If $r ( x )$ is convolutional then global pooling $\begin{array} { r } { r = \sum _ { x } r ( x ) } \end{array}$ is translation invariant.
+
+Proof: This follows directly from the definition of a convolutional response. If $r ( x )$ is the feature response to one image and $r _ { 2 } ( x )$ is the feature response to the same image translated, then $\textstyle \sum _ { x } r ( x ) =$ $\textstyle \sum _ { x } ^ { * } r _ { 2 } ( x )$ since the two responses are shifts of each other.
+
+Definition: A feature detector $r ( x )$ with subsampling factor $s$ is called “shiftable” if for any $x$ the detector output at location $x$ can be linearly interpolated from the responses on the sampling grid:
+
+$$
+r ( x ) = \sum _ { i } B _ { s } ( x - x _ { i } ) r ( x _ { i } )
+$$
+
+where $x _ { i }$ are located on the sampling grid for subsampling factor $s$ and $B _ { s } ( x )$ is the basis function for reconstructing $r ( x )$ from the samples.
+
+The classic Shannon-Nyquist theorem tells us that $r ( x )$ will be shiftable if and only if the sampling frequency is at least twice the highest frequency in $r ( x )$ .
+
+Claim: If $r ( x )$ is shiftable then global pooling on the sampling grid $\begin{array} { r } { r = \sum _ { i } r ( x _ { i } ) } \end{array}$ is translation invariant.
+
+Proof: This follows from the fact that global pooling on the sampling grid is (up to a constant) the same as global pooling for all $x$ .
+
+$$
+\begin{array} { l c l } { \displaystyle \sum _ { x } r ( x ) } & { = } & { \displaystyle \sum _ { x } \sum _ { i } r ( x _ { i } ) B ( x - x _ { i } ) } \\ { \displaystyle } & { = } & { \displaystyle \sum _ { i } r ( x _ { i } ) \sum _ { x } B ( x - x _ { i } ) } \\ { \displaystyle } & { = } & { K \sum _ { i } r ( x _ { i } ) } \end{array}
+$$
+
+where $\begin{array} { r } { K = \sum _ { x } B ( x - x _ { i } ) } \end{array}$ and $K$ does not depend on $x _ { i }$
+
+While the claim focuses on a global translation, it can also be extended to piecewise constant transformations.
+
+Corollary: Consider a set of transformations $T$ that are constant on a set of given image subareas. If $r ( x )$ is shiftable and for a given image, the support of $r ( x )$ and its receptive field is contained in the same subregion for all transformations in $T$ , then global pooling on the sampling grid is invariant to any transformation in $T$ .
+
+Proof: This follows from the fact that applying any transformation in $T$ to an image has the same effect on the feature map $r ( x )$ as translating the image.
+
+To illustrate the importance of the sampling theorem in guaranteeing invariance in CNNs, consider a convolutional layer in a deep CNN where each unit acts as a localized "part detector" (this has been reported to be the case for many modern CNNs (Zeiler & Fergus, 2014; Zhou et al., 2014)). Each such part detector has a spatial tuning function and the degree of sharpness of this tuning function will determine whether the feature map can be subsampled while preserving shiftability or not. For example, consider a part detector that fires only when the part is exactly at the center of its receptive field. If there is no subsampling, then as we translate the input image, the feature map will translate as well, and the global sum of the feature map is invariant to translation. But if we subsample by two (or equivalently use a stride of two), then there will only be activity in the feature map when the feature is centered on an even pixel, but not when it is centered on an odd pixel. This means that the global sum of the feature map will not be invariant to translation.
+
+In the language of Fourier transforms, the problem with a part detector that fires only when the part is exactly at the center of the receptive field is that the feature map contains many high frequencies and hence it cannot be subsampled while preserving shiftability. On the other hand, if we have a part detector whose spatial tuning function is more broad, it can be shiftable and our claim (above) shows that the global sum of activities in a feature map will be preserved for all translations, even though the individual firing rates of units will still be different when the part is centered at an odd pixel or an even pixel. Our corollary (above), shows the importance of shiftability to other smooth transformations: in this case each "part detector" will translate with a different translation but it is still the case that nonshiftable representations will not preserve the global sum of activities as the image is transformed, while shiftable representations will.
+
+Figure 3 examines the extent to which the representations learned by modern CNNs are invariant or shiftable. The top row shows an image that is translated vertically, while the bottom three rows show the representations in different layers for the three CNNs we consider. For VGG16 the representation appears to shift along with the object, including the final layer where the blurred pattern of response is not a simple translation of the original response, but seems to preserve the global sum for this particular image. For the two more modern networks, the responses are sharper but lose their shiftability in the later layers. In particular, the final layers show approximate invariance to one special translation but no response at all to another translation, suggesting that the many layers of subsampling yield a final response that is not shiftable.
+
+We also performed a more quantitative measure of shiftability by counting for a given image the number of times the global sum of activities in each layer changes significantly (more than $20 \%$ of mean) as the input is shifted (for each image, we only considered feature maps where the maximum response was above a threshold). We call this measure "nonshiftability". According to the preceding analysis, in architectures that obey the sampling theorem, the global sum should be invariant to input translation so nonshiftability should be zero in all layers. We find that for all three networks, the initial layers have nonshiftability close to zero but as we go deeper and deeper nonshiftability increases. Furthermore, the deeper, more modern networks, exhibit larger nonshiftability in their deep layers compared to VGG16 (see appendix for graphs).
+
+How can we guarantee that representations in CNNs will be shiftable? As explained above, we need to make sure that any feature map that uses stride does not contain frequencies above the Nyquist frequency. If CNNs were purely linear, we could simply blur the input images so that they would not include any frequencies higher than the Nyquist limit determined by the final sampling factor of the network. But since CNNs also include nonlinearities, they can add high frequencies that were not present in the input.
+
+An important message of the sampling theorem is that you should always blur before subsampling. Translated to the language of neural networks this means that stride (i.e. subsampling) should always be combined with pooling (i.e. blurring) in the preceding layer. Indeed if we have an arbitrarily deep CNN where all the layers use stride $^ { = 1 }$ followed by one layer that has a stride greater than one, then by choosing the pooling window appropriately we can guarantee that the final layer will still be shiftable. If we do not use appropriate pooling then there is no guarantee that this layer will be shiftable. Even if we use appropriate pooling that ensures that a given layer is shiftable, the subsequent nonlinearities in a CNN may not preserve the shiftability, as the nonlinearities may again introduce high frequencies.
+
+To illustrate the effect of pooling on shiftability in modern CNNs we replaced the $2 \times 2$ max pooling layers of VGG16 with $6 \times 6$ average pooling. This has the effect of reducing low freqencies but given the nonlinearities, it does not guarantee shiftability. As shown in figure 4 this simple change makes the representations approximately shiftable and, as predicted by our theory, the global sum is now invariant to both translations and rescalings of the input. This invariance of course comes with a price: the feature maps now have less detail and in preliminary experiments we find that recognition performance decreases somewhat. But the sampling theorem tells us that if we want to use subsampling while avoiding aliasing, we need to ensure that no high frequencies (relative to the Nyquist frequency) are present in the feature maps.
+
+As an alternative to pooling, Ruderman et al. (Ruderman et al., 2018) have shown that networks may learn smooth filters that will lead to reduced sensitivity to transformations. Evidently, the filters learned in standard VGG16 are not smooth enough.
+
+# 4 WHY DON’T MODERN CNNS LEARN TO BE INVARIANT FROM DATA?
+
+While the preceding discussion suggests that the CNN architecture will not yield translation invariance "for free", there is still the possibility that the CNN will learn a translation invariant prediction from the training examples. This requires that the training set will actually be invariant to the irrelevant transformations. We examined the degree of invariance in the ImageNet training set by manually labeling the training images in five categories: Tibetan terrier, elephant, pineapple, bagel and rowing paddle.
+
+Consistent with previous results on "dataset bias" (Simon et al., 2007; Raguram & Lazebnik, 2008; Berg & Berg, 2009; Torralba & Efros, 2011; Weyand & Leibe, 2011; Mezuman & Weiss, 2012) we find that the ImageNet dataset is not invariant to translations and rescalings. Figure 5 shows the distribution of the distances between the eyes of a "Tibetan terrier" and the positions of the center point between the dog’s eyes. Notice that both distributions are far from uniform. Similar results are obtained for the other four categories.
+
+![](images/24fecbc7f848b672f4bee1c0d89154ae077eb2c305f5959d8742a51bfd5d9aca.jpg)  
+Figure 4: Average pooling makes VGG representations approximately shiftable. Plotted are summation of feature maps from the last layer of VGG (spatial dimension $7 { \bf x } 7 $ ) as an input image is vertically translated in the image plane (top), or rescaled (bottom). Left: the original VGG16 with its $2 \mathbf { x } 2$ max pooling layers. Right: VGG16 where every $2 \mathbf { x } 2$ max pooling layer was replaced by a 6x6 average pooling layer. More randomly selected images are shown in the appendix.
+
+![](images/086cee2ee3cec063d5efaeebe2f70147eff5ecac249bb8e40ca7d033388356ec.jpg)  
+Figure 5: Photographer’s biases in the ImageNet’s “Tibetan terrier” category. Left: Example of the hand-labeling procedure. Middle: Positions of the middle point between the dog’s eyes. Right: Histogram of distances between the dog’s eyes. Notice the bias in both the object’s position and scale.
+
+To be more quantitative, we used the available bounding-box labels, and extracted the center point of the bounding-box and its height as proxies for the object position and size respectively. We then applied a statistical significance test to ask whether object location and object sizes were uniform for that category. For more than 900 out of the 1000 categories we found that location and size were highly non uniform $( P < 1 0 ^ { - 1 0 }$ ). Given these strong biases, we cannot expect a learning system to learn to be invariant.
+
+Even if the training set is not invariant, we can make it invariant using data augmentation. Will this make the CNN learn an invariant prediction? First, we note that we used pretrained networks and according to the authors’ description of the training procedure, all three networks were trained using data augmentation. Obviously, not any data augmentation is sufficient for the networks to learn invariances. To understand the failure of data augmentation, it is again instructive to consider the subsampling factor. Since in modern networks the subsampling factor is approximately 60, then for a system to learn complete invariance to translation only, it would need to see $6 0 ^ { \mathrm { { 2 } } } = 3 6 0 0$ augmented versions of each training example, or it would need to have an inductive bias that allows it to generalize over transformations. If we also add invariance to rotations and scalings, the number grows exponentially with the number of irrelevant transformations. Engstrom et al. (Engstrom et al., 2017) suggest a sophisticated data augmentation method and show that it increases the invariance to translation and rotation. However, for challenging datasets such as ImageNet the lack of invariance largely persists.
+
+![](images/8687e7fea4e72e1813da5fc3562a8991c87990ba4dbd8ba66b1a09134f0d9887.jpg)  
+Figure 6: The performance of modern CNNs on test images from ImageNet that are embedded in a random location in a larger image is quite poor (less than $50 \%$ accuracy). Human performance is not affected. Right: An example of a full sized image and the same image resized to $1 0 0 \mathrm { x } 1 0 0$ .
+
+# 5 IMPLICATIONS FOR PRACTICAL SYSTEMS
+
+Although our results show that modern CNNs fail to generalize for small image transformations, their performance on the ImageNet test set is still amazingly good and far better than previous techniques. This is related to the fact that the ImageNet test set contains the same photographer’s biases as the training set, so generalization to very different sizes and locations is not required. To highlight this point, we created a new test set in which ImageNet images were embedded in a larger image in a random location (and the missing pixels were filled in using a simple inpainting algorithm). Figure 6 shows that human performance is not affected by the rescaling and random translations, while the performance of modern CNNs deteriorates dramatically. In fact, when images are scaled to half their original size and randomly translated, the accuracy of modern CNNs is less than $50 \%$ , typically considered poor performance.
+
+One way in which modern systems partially address the lack of invariance is using test time augmentation in which the system output on a given image is computed by a majority vote among many random crops of the image. Clearly this is wasteful in resources and still only provides partial invariance.
+
+# 6 DISCUSSION
+
+CNN architectures were designed based on an intuition that the convolutional structure and pooling operations will give invariance to translations and small image deformations "for free". In this paper we have shown that this intuition breaks down once subsampling, or "stride" is used and we have presented empirical evidence that modern CNNs do not display the desired invariances since the architecture ignores the classic sampling theorem. This still leaves open the possibility of a CNN learning invariance from the data but we have shown that the ImageNet training and testing examples include significant photographer’s bias so that it is unlikely that a system will learn invariance using these examples.
+
+In addition to pointing out these failures, the sampling theorem also suggests a way to impose translation invariance by ensuring that all representations are sufficiently blurred to overcome the subsampling. However, such blurred representations may lead to a decrease in performance, especially in datasets and benchmarks that contain photographer’s bias. Alternatively, one could use specially designed features in which invariance is hard coded or neural network architectures that explicitly enforce invariance (Sifre & Mallat, 2013; Gens & Domingos, 2014; Cheng et al., 2016a;b; Dieleman et al., 2016; 2015; Xu et al., 2014; Worrall et al., 2017; Cohen & Welling, 2016). Again, as long as the datasets contain significant photographer’s bias, such invariant approaches may lead to a decrease in performance.
+
+# REFERENCES
+
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+Gong Cheng, Peicheng Zhou, and Junwei Han. Learning rotation-invariant convolutional neural networks for object detection in vhr optical remote sensing images. IEEE Transactions on Geoscience and Remote Sensing, 54(12):7405–7415, 2016a.
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+Gong Cheng, Peicheng Zhou, and Junwei Han. Rifd-cnn: Rotation-invariant and fisher discriminative convolutional neural networks for object detection. In Computer Vision and Pattern Recognition (CVPR), 2016 IEEE Conference on, pp. 2884–2893. IEEE, 2016b.
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+Logan Engstrom, Dimitris Tsipras, Ludwig Schmidt, and Aleksander Madry. A rotation and a translation suffice: Fooling cnns with simple transformations. CoRR, abs/1712.02779, 2017. URL http://arxiv.org/abs/1712.02779.
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+Alhussein Fawzi and Pascal Frossard. Manitest: Are classifiers really invariant? arXiv preprint arXiv:1507.06535, 2015.
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+Robert Gens and Pedro M Domingos. Deep symmetry networks. In Advances in neural information processing systems, pp. 2537–2545, 2014.
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+Jason Jo and Yoshua Bengio. Measuring the tendency of cnns to learn surface statistical regularities. arXiv preprint arXiv:1711.11561, 2017.
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+Eric Kauderer-Abrams. Quantifying translation-invariance in convolutional neural networks. arXiv preprint arXiv:1801.01450, 2017.
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+Karel Lenc and Andrea Vedaldi. Understanding image representations by measuring their equivariance and equivalence. In Computer Vision and Pattern Recognition (CVPR), 2015 IEEE Conference on, pp. 991–999. IEEE, 2015.
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+Elad Mezuman and Yair Weiss. Learning about canonical views from internet image collections. In Advances in Neural Information Processing Systems, pp. 719–727, 2012.
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+Rahul Raguram and Svetlana Lazebnik. Computing iconic summaries of general visual concepts. In Computer Vision and Pattern Recognition Workshops, 2008. CVPRW’08. IEEE Computer Society Conference on, pp. 1–8. IEEE, 2008.
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+Erik Rodner, Marcel Simon, Robert B Fisher, and Joachim Denzler. Fine-grained recognition in the noisy wild: Sensitivity analysis of convolutional neural networks approaches. arXiv preprint arXiv:1610.06756, 2016.
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+Amir Rosenfeld, Richard Zemel, and John K Tsotsos. The elephant in the room. arXiv preprint arXiv:1808.03305, 2018.
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+Laurent Sifre and Stéphane Mallat. Rotation, scaling and deformation invariant scattering for texture discrimination. In Computer Vision and Pattern Recognition (CVPR), 2013 IEEE Conference on, pp. 1233–1240. IEEE, 2013.
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+Ian Simon, Noah Snavely, and Steven M Seitz. Scene summarization for online image collections. In Computer Vision, 2007. ICCV 2007. IEEE 11th International Conference on, pp. 1–8. IEEE, 2007.
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+Eero P Simoncelli, William T Freeman, Edward H Adelson, and David J Heeger. Shiftable multiscale transforms. IEEE transactions on Information Theory, 38(2):587–607, 1992.
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+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.
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+Jiawei Su, Danilo Vasconcellos Vargas, and Sakurai Kouichi. One pixel attack for fooling deep neural networks. arXiv preprint arXiv:1710.08864, 2017.
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+Christian Szegedy, Sergey Ioffe, Vincent Vanhoucke, and Alexander A Alemi. Inception-v4, inceptionresnet and the impact of residual connections on learning. In AAAI, volume 4, pp. 12, 2017.
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+Antonio Torralba and Alexei A Efros. Unbiased look at dataset bias. In Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, pp. 1521–1528. IEEE, 2011.
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+Tobias Weyand and Bastian Leibe. Discovering favorite views of popular places with iconoid shift. In Computer Vision (ICCV), 2011 IEEE International Conference on, pp. 1132–1139. IEEE, 2011.
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+Daniel E Worrall, Stephan J Garbin, Daniyar Turmukhambetov, and Gabriel J Brostow. Harmonic networks: Deep translation and rotation equivariance. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), volume 2, 2017.
+
+Yichong Xu, Tianjun Xiao, Jiaxing Zhang, Kuiyuan Yang, and Zheng Zhang. Scale-invariant convolutional neural networks. arXiv preprint arXiv:1411.6369, 2014.
+
+Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pp. 818–833. Springer, 2014.
+
+Bolei Zhou, Aditya Khosla, Àgata Lapedriza, Aude Oliva, and Antonio Torralba. Object detectors emerge in deep scene cnns. CoRR, abs/1412.6856, 2014. URL http://arxiv.org/abs/ 1412.6856.
+
+# APPENDIX
+
+# A PIPELINE FOR PRODUCING THE BOTTOM ROW OF FIGURE 1
+
+We download this video: https://www.youtube.com/watch?v=0mgnf6t9VEc using an online downloader. We load the video frames and resize them to 299 by 299 as used by the standard Keras applications framework (https://keras.io/applications/). We preprocess the frames using the standard Keras preprocessing function. Finally, we use the predictions of the InceptionV3 model to demonstrate the jagged behavior shown in figure 1.
+
+# B OTHER SUPPLEMENTARY MATERIAL
+
+<table><tr><td>Network</td><td>Top-1</td><td>Top-5</td><td>Parameters</td><td>Depth</td></tr><tr><td>VGG16 (Simonyan &amp; Zisserman, 2014)</td><td>0.715</td><td>0.901</td><td>138,357,544</td><td>16</td></tr><tr><td>ResNet50 (He et al., 2016)</td><td>0.759</td><td>0.929</td><td>25,636,712</td><td>50</td></tr><tr><td>InceptionResNetV2 (Szegedy et al., 2017)</td><td>0.804</td><td>0.953</td><td>55,873,736</td><td>134</td></tr></table>
+
+Table 1: The networks used (taken from (https://keras.io/applications/))
+
+![](images/a71df43d23932beae44db3fe1f86af62e842a039bccc1744ae6f6a5a13b43090.jpg)  
+Figure 7: We also measure jaggedness using the Mean Absolute Difference (MAD) in the probability of the correct response as the image is shifted by a single pixel. Results are similar to those using the jaggedness measure described in the text.
+
+![](images/b224b7ac821e3722354ddb36cd7f8e8fdcd75330482d741d76e42df5f1cbaba0.jpg)  
+Figure 8: Modern deep convolutional neural networks are sensitive to small image translations (Without image downscaling). A) Example of InceptionResNetV2 sensitivity to very small horizontal translations. B) Comparison of three networks of various depths (16, 50, and 134 for VGG16, ResNet50, and InceptionResNetV2 respectively) in the task of horizontal image translation. Rows are sorted by sum. B) Modern networks have more jagged predictions. Jaggedness is calculated by counting the number of times the network’s predictions had the correct class in its top-5 and after just one pixel shift it moved outside of the top-5 (and also the opposite transition from non-top-5 to top5). Similar results can be seen with the alternative measure of jaggedness: Mean Absolute Difference (MAD)
+
+![](images/461b9bb8f402b1fe1034878c41ff3c84da22c54d2181d9efe5c17603bf7490dd.jpg)  
+Figure 9: Modern deep convolutional neural networks are sensitive to small image rescalings A) Comparison of three networks of various depths (16, 50, and 134 for VGG16, ResNet50, and InceptionResNetV2 respectively) in the task of image rescaling. Rows are sorted by their sum. B) Quantification of the jagged behaviour of the deep neural networks.
+
+![](images/a18d4c5c49d12344dd249d611faf0645c92d07b90746efe496d666cacc637d8a.jpg)  
+Figure 10: Nonshiftability as a function of depth in the three networks. Nonshiftability is defined as the number of times the global sum of a feature map changes by more than $20 \%$ of the mean as the input is translated. We only consider feature maps where the maximum response was above a threshold. According to our analysis, this measure should be zero if the representation is shiftable. Each line shows the nonshiftability in different layers in response to a randomly selected image.
+
+![](images/736a357904deebb3c77412a16d918f4ebba396b7bf0dd53a75b4bcd7b1f06c5a.jpg)  
+Figure 11: Another randomly selected image. Plotted are summation of feature maps from the last layer of VGG (spatial dimension $7 \mathrm { x } 7$ ) as the input is vertically translated in the image plane (top), or rescaled (bottom). Left: the original VGG16 with its $2 \mathbf { x } 2$ max pooling layers. Right: VGG16 where every $2 \mathbf { x } 2$ max pooling layer was replaced by a 6x6 average pooling layer.
+
+![](images/d5f9473f2cfb1d47f8e298c4f5541896cfbe98ed884eb1d06c9c00cf27628e28.jpg)  
+Figure 12: Another randomly selected image. Plotted are summation of feature maps from the last layer of VGG (spatial dimension $7 \mathrm { x } 7$ ) as the input is vertically translated in the image plane (top), or rescaled (bottom). Left: the original VGG16 with its $2 \mathbf { x } 2$ max pooling layers. Right: VGG16 where every 2x2 max pooling layer was replaced by a 6x6 average pooling layer.
+
+![](images/d9682b73986d14dd062e66ccce5d52708494e1951d88ed769031a05502fe93d0.jpg)  
+Figure 13: Another randomly selected image. Plotted are summation of feature maps from the last layer of VGG (spatial dimension $7 \mathrm { x } 7$ ) as the input is vertically translated in the image plane (top), or rescaled (bottom). Left: the original VGG16 with its $2 \mathbf { x } 2$ max pooling layers. Right: VGG16 where every $2 \mathbf { x } 2$ max pooling layer was replaced by a 6x6 average pooling layer.

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+        "text": "WHY DO DEEP CONVOLUTIONAL NETWORKS GENERALIZE SO POORLY TO SMALL IMAGE TRANSFORMATIONS? ",
+        "text_level": 1,
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+    {
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        ],
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+    },
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+        "text": "Deep convolutional network architectures are often assumed to guarantee generalization for small image translations and deformations. In this paper we show that modern CNNs (VGG16, ResNet50, and InceptionResNetV2) can drastically change their output when an image is translated in the image plane by a few pixels, and that this failure of generalization also happens with other realistic small image transformations. Furthermore, we see these failures to generalize more frequently in more modern networks. We show that these failures are related to the fact that the architecture of modern CNNs ignores the classical sampling theorem so that generalization is not guaranteed. We also show that biases in the statistics of commonly used image datasets makes it unlikely that CNNs will learn to be invariant to these transformations. Taken together our results suggest that the performance of CNNs in object recognition falls far short of the generalization capabilities of humans. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            336,
+            517
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep convolutional neural networks (CNNs) have revolutionized computer vision. Perhaps the most dramatic success is in the area of object recognition, where performance is now described as \"superhuman\" (He et al., 2015). A key to the success of any machine learning method is the inductive bias of the method, and clearly the choice of architecture in a neural network significantly affects the inductive bias. In particular, the choice of convolution and pooling in CNNs is motivated by the desire to endow the networks with invariance to irrelevant cues such as image translations, scalings, and other small deformations (Fukushima & Miyake, 1982; Zeiler & Fergus, 2014). This motivation was made explicit in the 1980s by Fukushima in describing the \"neocognitron\" architecture, which served as inspiration for modern CNNs (LeCun et al., 1989), \"After finishing the process of learning, pattern recognition is performed on the basis of similarity in shape between patterns, and is not affected by deformation, nor by changes in size, nor by shifts in the position of the input patterns.\" (Fukushima, 1988) ",
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+    {
+        "type": "text",
+        "text": "Despite the excellent performance of CNNs on object recognition, the vulnerability to adversarial attacks suggests that superficial changes can result in highly non-human shifts in prediction (e.g. (Bhagoji et al., 2017; Su et al., 2017). In addition, filtering the image in the Fourier domain (in a way that does not change human prediction) also results in a substantial drop in prediction accuracy (Jo & Bengio, 2017). These and other results (Rodner et al., 2016) indicate that CNNs are not invariant to cues that are irrelevant to the object identity. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "An argument against adversarial attacks on CNNs is that they often involve highly unnatural transformations to the input images, hence in some sense we would not expect CNNs to be invariant to these transformations. When considering more natural transformations, there is preliminary evidence that AlexNet (Krizhevsky et al., 2012) is robust to some of them (Zeiler & Fergus, 2014). On the other hand, there is also preliminary evidence for lack of robustness in the more modern networks for object classification (Bunne et al., 2018) and detection (Rosenfeld et al., 2018) along with studies suggesting that with small CNNs and the MNIST data, data augmentation is the main feature affecting CNN invariance (Kauderer-Abrams, 2017). An indirect method to probe for invariances measures the linearity of the learned representations under natural transformations to the input image (Lenc & Vedaldi, 2015; Hénaff & Simoncelli, 2015; Fawzi & Frossard, 2015; Cohen & Welling, 2014). The recent work of (Engstrom et al., 2017) investigates adversarial attacks that use only rotations and translations. They find that \"simple transformations, namely translations and rotations alone, are sufficient to fool neural network-based vision models on a significant fraction of inputs\" and show that advanced data augmentation methods can make the networks more robust. ",
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+    {
+        "type": "image",
+        "img_path": "images/92b13decc6d7a5ecd856ae52ff3d3939757f136ce0017965aa457404cf336bc7.jpg",
+        "image_caption": [
+            "Figure 1: Examples of jagged predictions of modern deep convolutional neural networks. Top: A negligible vertical shift of the object (Kuvasz) results in an abrupt decrease in the network’s predicted score of the correct class. Middle: A tiny increase in the size of the object (Lotion) produces a dramatic decrease in the network’s predicted score of the correct class. Bottom: A very small change in the bear’s posture results in an abrupt decrease in the network’s predicted score of the correct class. Colored dots represent images chosen from interesting $\\mathbf { X }$ -axis locations of the graphs on the right. These dots illustrate sensitivity of modern neural networks to small, insignificant (to a human), and realistic variations in the image. "
+        ],
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+    {
+        "type": "text",
+        "text": "In this paper, we directly ask \"why are modern CNNs not invariant to natural image transformations despite the architecture being explicitly designed to provide such invariances?\". Specifically, we systematically examine the invariances of three modern deep CNNs: VGG-16 (Simonyan & Zisserman, 2014), ResNet-50 (He et al., 2016), and InceptionResNet-V2 (Szegedy et al., 2017). We find that modern deep CNNs are not invariant to translations, scalings and other realistic image transformations, and this lack of invariance is related to the subsampling operation and the biases contained in image datasets. ",
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+    {
+        "type": "text",
+        "text": "2 FAILURES OF MODERN CNNS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 1 contains examples of abrupt failures following tiny realistic transformations for the InceptionResNet-V2 CNN. Shifting or scaling the object by just one pixel could result in a sharp change in prediction. In the top row, we embed the original image in a larger image and shift it in the image plane (while filling in the rest of the image with a simple inpainting procedure). In the middle row, we repeat this protocol with rescaling. In the bottom row, we show frames from a BBC film in which the ice bear moves almost imperceptibly between frames and the network’s output changes dramatically1. ",
+        "bbox": [
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+    {
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+        "img_path": "images/9846430a7dbf8d7be4b46ec6f4f2c1932f1d8ab18f7d55cf0e60e002d26e1274.jpg",
+        "image_caption": [
+            "Figure 2: Modern deep convolutional neural networks are sensitive to small image translations. A) Comparison of three networks of various depths in the task of vertical image translation depicted in figure 1. Images (rows) are randomly chosen from the ImageNet dataset (Deng et al., 2009), and are sorted by the network’s prediction sum in a descending order. B) More modern networks have more jagged predictions. "
+        ],
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+    {
+        "type": "text",
+        "text": "In order to measure how typical these failures are, we randomly chose images from the ImageNet validation set and measured the output of three modern CNNs as we embedded these images in a larger image and systematically varied the vertical translation. As was the case in figure 1, we used a simple inpainting procedure to fill in the rest of the image. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Results are shown in figure 2. Each row corresponds to an image under different translations and the color denotes the network’s estimate of the probability of the correct class. Thus a row that is all light corresponds to a correct classification that is invariant to translation, while a row that is all dark corresponds to an incorrect classification that is invariant to translation. Surprisingly, many rows show abrupt transitions from light to dark, indicating that the classification changes abruptly as the object is translated. We quantify the lack of invariance by a measure we call \"jaggedness\": the number of times the network’s predictions had the correct class in its top-5 and after just one pixel shift it moved outside of the top-5 (and also the opposite transition from non-top-5 to top5). Using this measure, we find that for approximately $30 \\%$ of the images, the output is \"jagged\", i.e the network changes its prediction by a shift of a single pixel. Also, as shown in the right of figure 2, jaggedness is greater for the modern, deeper, networks compared to the less modern VGG16 network. While the deeper networks have better test accuracy, they are also less invariant. In the appendix we also show an alternative to the \"jaggedness\" measure, which gives similar results. ",
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+    {
+        "type": "text",
+        "text": "A natural criticism of these results is that they are somehow related to the image resizing and inpainting procedures that we used. To test this possibility, we repeated the experiment with a different protocol where we chose different crops of the original ImageNet image while making sure that the object bounding box remained within the crop. This protocol does not require any inpainting while still translating the object location within the new image. Results are shown in the appendix. We still have a large fraction of images for which the prediction is not invariant to translation. We also show in the appendix similar results for scaling rather than translation. Overall we find that regardless of the protocol used, modern CNNs often change their output significantly as a result of a small translation or scaling. ",
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+    {
+        "type": "text",
+        "text": "3 IGNORING THE SAMPLING THEOREM ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The failure of CNNs to generalize to image translations is particularly puzzling. Intuitively, it would seem that if all layers in a network are convolutional then the representation should simply translate when an image is translated. If the final features for classification are obtained by a global pooling operation on the representation (as is done for example in ResNet50 and InceptionResNetV2) then these features should be invariant to translation. Where does this intuition fail? ",
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+    {
+        "type": "image",
+        "img_path": "images/c2c6c1eeffb9c3ee10ebc723e3bca2e59d9e945c463eccdc0390f9338be39180.jpg",
+        "image_caption": [
+            "Figure 3: The deeper the network, the less shiftable are the feature maps. A) A vertical shift of a \"Kuvasz\" dog in the image plane. B) Feature maps from three different network architectures in response to the translated Kuvasz image. Layer depth assignments reflect the number of trainable convolutional layers preceding the selected layer. The last layer is always the last convolutional layer in each network. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This intuition ignores the subsampling operation which is prevalent in modern CNNs, also known as \"stride\". This failure of translation invariance in systems with subsampling was explicitly discussed in Simoncelli et al. (Simoncelli et al., 1992) who wrote \"We cannot literally expect translation invariance in a system based on convolution and subsampling: translation of the input signal cannot produce simple translations of the transform coefficients, unless the translation is a multiple of each of the subsampling factors in the system\". Since deep networks often contain many subsampling operations, the subsampling factor of the deep layers may be very large so that \"literal\" translation invariance only holds for very special translations. In InceptionResnetV2, for example, the subsampling factor is 60, so we expect exact translation invariance to hold only for $\\scriptstyle { \\frac { 1 } { 6 0 ^ { 2 } } }$ of possible translations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Simoncelli et al. also defined a weaker form of translation invariance, which they called \"shiftability\" and showed that it can hold for systems with subsampling (this is related to weak translation invariance as defined by (Lenc & Vedaldi, 2015), see also (Esteves et al., 2017; Cohen & Welling, 2014) for related ideas applied to neural networks). Here we extend the basic shiftability result to show that when shiftability holds, then global pooling will indeed yield invariant representations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We define $r ( x )$ as the response of a feature detector at location $x$ in the image plane. We say that this response is \"convolutional\" if translating the image by any translation $\\delta$ yields a translation of the response by the same $\\delta$ . This definition includes cases when the feature response is obtained by convolving the input image with a fixed filter, but also includes combinations of linear operations and nonlinear operations that do not include any subsampling. ",
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+    {
+        "type": "text",
+        "text": "We start by a trivial observation: ",
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+    {
+        "type": "text",
+        "text": "Observation: If $r ( x )$ is convolutional then global pooling $\\begin{array} { r } { r = \\sum _ { x } r ( x ) } \\end{array}$ is translation invariant. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Proof: This follows directly from the definition of a convolutional response. If $r ( x )$ is the feature response to one image and $r _ { 2 } ( x )$ is the feature response to the same image translated, then $\\textstyle \\sum _ { x } r ( x ) =$ $\\textstyle \\sum _ { x } ^ { * } r _ { 2 } ( x )$ since the two responses are shifts of each other. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition: A feature detector $r ( x )$ with subsampling factor $s$ is called “shiftable” if for any $x$ the detector output at location $x$ can be linearly interpolated from the responses on the sampling grid: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8eca73341a20c8fadf33b2a6e1d23a276f6623119228078446f818a058f05ab3.jpg",
+        "text": "$$\nr ( x ) = \\sum _ { i } B _ { s } ( x - x _ { i } ) r ( x _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $x _ { i }$ are located on the sampling grid for subsampling factor $s$ and $B _ { s } ( x )$ is the basis function for reconstructing $r ( x )$ from the samples. ",
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+    {
+        "type": "text",
+        "text": "The classic Shannon-Nyquist theorem tells us that $r ( x )$ will be shiftable if and only if the sampling frequency is at least twice the highest frequency in $r ( x )$ . ",
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+    {
+        "type": "text",
+        "text": "Claim: If $r ( x )$ is shiftable then global pooling on the sampling grid $\\begin{array} { r } { r = \\sum _ { i } r ( x _ { i } ) } \\end{array}$ is translation invariant. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Proof: This follows from the fact that global pooling on the sampling grid is (up to a constant) the same as global pooling for all $x$ . ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/a807cc9e5590322990f445043557245941d52bf288b16228e933d4a94f7c97bc.jpg",
+        "text": "$$\n\\begin{array} { l c l } { \\displaystyle \\sum _ { x } r ( x ) } & { = } & { \\displaystyle \\sum _ { x } \\sum _ { i } r ( x _ { i } ) B ( x - x _ { i } ) } \\\\ { \\displaystyle } & { = } & { \\displaystyle \\sum _ { i } r ( x _ { i } ) \\sum _ { x } B ( x - x _ { i } ) } \\\\ { \\displaystyle } & { = } & { K \\sum _ { i } r ( x _ { i } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { K = \\sum _ { x } B ( x - x _ { i } ) } \\end{array}$ and $K$ does not depend on $x _ { i }$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While the claim focuses on a global translation, it can also be extended to piecewise constant transformations. ",
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+    {
+        "type": "text",
+        "text": "Corollary: Consider a set of transformations $T$ that are constant on a set of given image subareas. If $r ( x )$ is shiftable and for a given image, the support of $r ( x )$ and its receptive field is contained in the same subregion for all transformations in $T$ , then global pooling on the sampling grid is invariant to any transformation in $T$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof: This follows from the fact that applying any transformation in $T$ to an image has the same effect on the feature map $r ( x )$ as translating the image. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To illustrate the importance of the sampling theorem in guaranteeing invariance in CNNs, consider a convolutional layer in a deep CNN where each unit acts as a localized \"part detector\" (this has been reported to be the case for many modern CNNs (Zeiler & Fergus, 2014; Zhou et al., 2014)). Each such part detector has a spatial tuning function and the degree of sharpness of this tuning function will determine whether the feature map can be subsampled while preserving shiftability or not. For example, consider a part detector that fires only when the part is exactly at the center of its receptive field. If there is no subsampling, then as we translate the input image, the feature map will translate as well, and the global sum of the feature map is invariant to translation. But if we subsample by two (or equivalently use a stride of two), then there will only be activity in the feature map when the feature is centered on an even pixel, but not when it is centered on an odd pixel. This means that the global sum of the feature map will not be invariant to translation. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "In the language of Fourier transforms, the problem with a part detector that fires only when the part is exactly at the center of the receptive field is that the feature map contains many high frequencies and hence it cannot be subsampled while preserving shiftability. On the other hand, if we have a part detector whose spatial tuning function is more broad, it can be shiftable and our claim (above) shows that the global sum of activities in a feature map will be preserved for all translations, even though the individual firing rates of units will still be different when the part is centered at an odd pixel or an even pixel. Our corollary (above), shows the importance of shiftability to other smooth transformations: in this case each \"part detector\" will translate with a different translation but it is still the case that nonshiftable representations will not preserve the global sum of activities as the image is transformed, while shiftable representations will. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Figure 3 examines the extent to which the representations learned by modern CNNs are invariant or shiftable. The top row shows an image that is translated vertically, while the bottom three rows show the representations in different layers for the three CNNs we consider. For VGG16 the representation appears to shift along with the object, including the final layer where the blurred pattern of response is not a simple translation of the original response, but seems to preserve the global sum for this particular image. For the two more modern networks, the responses are sharper but lose their shiftability in the later layers. In particular, the final layers show approximate invariance to one special translation but no response at all to another translation, suggesting that the many layers of subsampling yield a final response that is not shiftable. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We also performed a more quantitative measure of shiftability by counting for a given image the number of times the global sum of activities in each layer changes significantly (more than $20 \\%$ of mean) as the input is shifted (for each image, we only considered feature maps where the maximum response was above a threshold). We call this measure \"nonshiftability\". According to the preceding analysis, in architectures that obey the sampling theorem, the global sum should be invariant to input translation so nonshiftability should be zero in all layers. We find that for all three networks, the initial layers have nonshiftability close to zero but as we go deeper and deeper nonshiftability increases. Furthermore, the deeper, more modern networks, exhibit larger nonshiftability in their deep layers compared to VGG16 (see appendix for graphs). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "How can we guarantee that representations in CNNs will be shiftable? As explained above, we need to make sure that any feature map that uses stride does not contain frequencies above the Nyquist frequency. If CNNs were purely linear, we could simply blur the input images so that they would not include any frequencies higher than the Nyquist limit determined by the final sampling factor of the network. But since CNNs also include nonlinearities, they can add high frequencies that were not present in the input. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "An important message of the sampling theorem is that you should always blur before subsampling. Translated to the language of neural networks this means that stride (i.e. subsampling) should always be combined with pooling (i.e. blurring) in the preceding layer. Indeed if we have an arbitrarily deep CNN where all the layers use stride $^ { = 1 }$ followed by one layer that has a stride greater than one, then by choosing the pooling window appropriately we can guarantee that the final layer will still be shiftable. If we do not use appropriate pooling then there is no guarantee that this layer will be shiftable. Even if we use appropriate pooling that ensures that a given layer is shiftable, the subsequent nonlinearities in a CNN may not preserve the shiftability, as the nonlinearities may again introduce high frequencies. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To illustrate the effect of pooling on shiftability in modern CNNs we replaced the $2 \\times 2$ max pooling layers of VGG16 with $6 \\times 6$ average pooling. This has the effect of reducing low freqencies but given the nonlinearities, it does not guarantee shiftability. As shown in figure 4 this simple change makes the representations approximately shiftable and, as predicted by our theory, the global sum is now invariant to both translations and rescalings of the input. This invariance of course comes with a price: the feature maps now have less detail and in preliminary experiments we find that recognition performance decreases somewhat. But the sampling theorem tells us that if we want to use subsampling while avoiding aliasing, we need to ensure that no high frequencies (relative to the Nyquist frequency) are present in the feature maps. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As an alternative to pooling, Ruderman et al. (Ruderman et al., 2018) have shown that networks may learn smooth filters that will lead to reduced sensitivity to transformations. Evidently, the filters learned in standard VGG16 are not smooth enough. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 WHY DON’T MODERN CNNS LEARN TO BE INVARIANT FROM DATA? ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "While the preceding discussion suggests that the CNN architecture will not yield translation invariance \"for free\", there is still the possibility that the CNN will learn a translation invariant prediction from the training examples. This requires that the training set will actually be invariant to the irrelevant transformations. We examined the degree of invariance in the ImageNet training set by manually labeling the training images in five categories: Tibetan terrier, elephant, pineapple, bagel and rowing paddle. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Consistent with previous results on \"dataset bias\" (Simon et al., 2007; Raguram & Lazebnik, 2008; Berg & Berg, 2009; Torralba & Efros, 2011; Weyand & Leibe, 2011; Mezuman & Weiss, 2012) we find that the ImageNet dataset is not invariant to translations and rescalings. Figure 5 shows the distribution of the distances between the eyes of a \"Tibetan terrier\" and the positions of the center point between the dog’s eyes. Notice that both distributions are far from uniform. Similar results are obtained for the other four categories. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/24fecbc7f848b672f4bee1c0d89154ae077eb2c305f5959d8742a51bfd5d9aca.jpg",
+        "image_caption": [
+            "Figure 4: Average pooling makes VGG representations approximately shiftable. Plotted are summation of feature maps from the last layer of VGG (spatial dimension $7 { \\bf x } 7 $ ) as an input image is vertically translated in the image plane (top), or rescaled (bottom). Left: the original VGG16 with its $2 \\mathbf { x } 2$ max pooling layers. Right: VGG16 where every $2 \\mathbf { x } 2$ max pooling layer was replaced by a 6x6 average pooling layer. More randomly selected images are shown in the appendix. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/086cee2ee3cec063d5efaeebe2f70147eff5ecac249bb8e40ca7d033388356ec.jpg",
+        "image_caption": [
+            "Figure 5: Photographer’s biases in the ImageNet’s “Tibetan terrier” category. Left: Example of the hand-labeling procedure. Middle: Positions of the middle point between the dog’s eyes. Right: Histogram of distances between the dog’s eyes. Notice the bias in both the object’s position and scale. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            494,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To be more quantitative, we used the available bounding-box labels, and extracted the center point of the bounding-box and its height as proxies for the object position and size respectively. We then applied a statistical significance test to ask whether object location and object sizes were uniform for that category. For more than 900 out of the 1000 categories we found that location and size were highly non uniform $( P < 1 0 ^ { - 1 0 }$ ). Given these strong biases, we cannot expect a learning system to learn to be invariant. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Even if the training set is not invariant, we can make it invariant using data augmentation. Will this make the CNN learn an invariant prediction? First, we note that we used pretrained networks and according to the authors’ description of the training procedure, all three networks were trained using data augmentation. Obviously, not any data augmentation is sufficient for the networks to learn invariances. To understand the failure of data augmentation, it is again instructive to consider the subsampling factor. Since in modern networks the subsampling factor is approximately 60, then for a system to learn complete invariance to translation only, it would need to see $6 0 ^ { \\mathrm { { 2 } } } = 3 6 0 0$ augmented versions of each training example, or it would need to have an inductive bias that allows it to generalize over transformations. If we also add invariance to rotations and scalings, the number grows exponentially with the number of irrelevant transformations. Engstrom et al. (Engstrom et al., 2017) suggest a sophisticated data augmentation method and show that it increases the invariance to translation and rotation. However, for challenging datasets such as ImageNet the lack of invariance largely persists. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/8687e7fea4e72e1813da5fc3562a8991c87990ba4dbd8ba66b1a09134f0d9887.jpg",
+        "image_caption": [
+            "Figure 6: The performance of modern CNNs on test images from ImageNet that are embedded in a random location in a larger image is quite poor (less than $50 \\%$ accuracy). Human performance is not affected. Right: An example of a full sized image and the same image resized to $1 0 0 \\mathrm { x } 1 0 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 IMPLICATIONS FOR PRACTICAL SYSTEMS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            398,
+            547,
+            414
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Although our results show that modern CNNs fail to generalize for small image transformations, their performance on the ImageNet test set is still amazingly good and far better than previous techniques. This is related to the fact that the ImageNet test set contains the same photographer’s biases as the training set, so generalization to very different sizes and locations is not required. To highlight this point, we created a new test set in which ImageNet images were embedded in a larger image in a random location (and the missing pixels were filled in using a simple inpainting algorithm). Figure 6 shows that human performance is not affected by the rescaling and random translations, while the performance of modern CNNs deteriorates dramatically. In fact, when images are scaled to half their original size and randomly translated, the accuracy of modern CNNs is less than $50 \\%$ , typically considered poor performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "One way in which modern systems partially address the lack of invariance is using test time augmentation in which the system output on a given image is computed by a majority vote among many random crops of the image. Clearly this is wasteful in resources and still only provides partial invariance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "CNN architectures were designed based on an intuition that the convolutional structure and pooling operations will give invariance to translations and small image deformations \"for free\". In this paper we have shown that this intuition breaks down once subsampling, or \"stride\" is used and we have presented empirical evidence that modern CNNs do not display the desired invariances since the architecture ignores the classic sampling theorem. This still leaves open the possibility of a CNN learning invariance from the data but we have shown that the ImageNet training and testing examples include significant photographer’s bias so that it is unlikely that a system will learn invariance using these examples. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In addition to pointing out these failures, the sampling theorem also suggests a way to impose translation invariance by ensuring that all representations are sufficiently blurred to overcome the subsampling. However, such blurred representations may lead to a decrease in performance, especially in datasets and benchmarks that contain photographer’s bias. Alternatively, one could use specially designed features in which invariance is hard coded or neural network architectures that explicitly enforce invariance (Sifre & Mallat, 2013; Gens & Domingos, 2014; Cheng et al., 2016a;b; Dieleman et al., 2016; 2015; Xu et al., 2014; Worrall et al., 2017; Cohen & Welling, 2016). Again, as long as the datasets contain significant photographer’s bias, such invariant approaches may lead to a decrease in performance. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A PIPELINE FOR PRODUCING THE BOTTOM ROW OF FIGURE 1 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            136,
+            694,
+            151
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We download this video: https://www.youtube.com/watch?v=0mgnf6t9VEc using an online downloader. We load the video frames and resize them to 299 by 299 as used by the standard Keras applications framework (https://keras.io/applications/). We preprocess the frames using the standard Keras preprocessing function. Finally, we use the predictions of the InceptionV3 model to demonstrate the jagged behavior shown in figure 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B OTHER SUPPLEMENTARY MATERIAL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            258,
+            509,
+            273
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/98c2e66b5716d5c3223c08b075ec93d411cd418415a1b94243db98e43db65a9a.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: The networks used (taken from (https://keras.io/applications/)) "
+        ],
+        "table_body": "<table><tr><td>Network</td><td>Top-1</td><td>Top-5</td><td>Parameters</td><td>Depth</td></tr><tr><td>VGG16 (Simonyan &amp; Zisserman, 2014)</td><td>0.715</td><td>0.901</td><td>138,357,544</td><td>16</td></tr><tr><td>ResNet50 (He et al., 2016)</td><td>0.759</td><td>0.929</td><td>25,636,712</td><td>50</td></tr><tr><td>InceptionResNetV2 (Szegedy et al., 2017)</td><td>0.804</td><td>0.953</td><td>55,873,736</td><td>134</td></tr></table>",
+        "bbox": [
+            210,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/a71df43d23932beae44db3fe1f86af62e842a039bccc1744ae6f6a5a13b43090.jpg",
+        "image_caption": [
+            "Figure 7: We also measure jaggedness using the Mean Absolute Difference (MAD) in the probability of the correct response as the image is shifted by a single pixel. Results are similar to those using the jaggedness measure described in the text. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            338,
+            409,
+            656,
+            578
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/b224b7ac821e3722354ddb36cd7f8e8fdcd75330482d741d76e42df5f1cbaba0.jpg",
+        "image_caption": [
+            "Figure 8: Modern deep convolutional neural networks are sensitive to small image translations (Without image downscaling). A) Example of InceptionResNetV2 sensitivity to very small horizontal translations. B) Comparison of three networks of various depths (16, 50, and 134 for VGG16, ResNet50, and InceptionResNetV2 respectively) in the task of horizontal image translation. Rows are sorted by sum. B) Modern networks have more jagged predictions. Jaggedness is calculated by counting the number of times the network’s predictions had the correct class in its top-5 and after just one pixel shift it moved outside of the top-5 (and also the opposite transition from non-top-5 to top5). Similar results can be seen with the alternative measure of jaggedness: Mean Absolute Difference (MAD) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            109,
+            820,
+            305
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/461b9bb8f402b1fe1034878c41ff3c84da22c54d2181d9efe5c17603bf7490dd.jpg",
+        "image_caption": [
+            "Figure 9: Modern deep convolutional neural networks are sensitive to small image rescalings A) Comparison of three networks of various depths (16, 50, and 134 for VGG16, ResNet50, and InceptionResNetV2 respectively) in the task of image rescaling. Rows are sorted by their sum. B) Quantification of the jagged behaviour of the deep neural networks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            478,
+            820,
+            809
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/a18d4c5c49d12344dd249d611faf0645c92d07b90746efe496d666cacc637d8a.jpg",
+        "image_caption": [
+            "Figure 10: Nonshiftability as a function of depth in the three networks. Nonshiftability is defined as the number of times the global sum of a feature map changes by more than $20 \\%$ of the mean as the input is translated. We only consider feature maps where the maximum response was above a threshold. According to our analysis, this measure should be zero if the representation is shiftable. Each line shows the nonshiftability in different layers in response to a randomly selected image. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/736a357904deebb3c77412a16d918f4ebba396b7bf0dd53a75b4bcd7b1f06c5a.jpg",
+        "image_caption": [
+            "Figure 11: Another randomly selected image. Plotted are summation of feature maps from the last layer of VGG (spatial dimension $7 \\mathrm { x } 7$ ) as the input is vertically translated in the image plane (top), or rescaled (bottom). Left: the original VGG16 with its $2 \\mathbf { x } 2$ max pooling layers. Right: VGG16 where every $2 \\mathbf { x } 2$ max pooling layer was replaced by a 6x6 average pooling layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            164,
+            725,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/d5f9473f2cfb1d47f8e298c4f5541896cfbe98ed884eb1d06c9c00cf27628e28.jpg",
+        "image_caption": [
+            "Figure 12: Another randomly selected image. Plotted are summation of feature maps from the last layer of VGG (spatial dimension $7 \\mathrm { x } 7$ ) as the input is vertically translated in the image plane (top), or rescaled (bottom). Left: the original VGG16 with its $2 \\mathbf { x } 2$ max pooling layers. Right: VGG16 where every 2x2 max pooling layer was replaced by a 6x6 average pooling layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            269,
+            578,
+            723,
+            785
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/d9682b73986d14dd062e66ccce5d52708494e1951d88ed769031a05502fe93d0.jpg",
+        "image_caption": [
+            "Figure 13: Another randomly selected image. Plotted are summation of feature maps from the last layer of VGG (spatial dimension $7 \\mathrm { x } 7$ ) as the input is vertically translated in the image plane (top), or rescaled (bottom). Left: the original VGG16 with its $2 \\mathbf { x } 2$ max pooling layers. Right: VGG16 where every $2 \\mathbf { x } 2$ max pooling layer was replaced by a 6x6 average pooling layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            269,
+            371,
+            723,
+            577
+        ],
+        "page_idx": 15
+    }
+]

parse/train/rk6qdGgCZ/rk6qdGgCZ.md

@@ -0,0 +1,252 @@
+# FIXING WEIGHT DECAY REGULARIZATION IN ADAM
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We note that common implementations of adaptive gradient algorithms, such as Adam, limit the potential benefit of weight decay regularization, because the weights do not decay multiplicatively (as would be expected for standard weight decay) but by an additive constant factor. We propose a simple way to resolve this issue by decoupling weight decay and the optimization steps taken w.r.t. the loss function. We provide empirical evidence that our proposed modification (i) decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam, and (ii) substantially improves Adam’s generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperformed by the latter). We also demonstrate that longer optimization runs require smaller weight decay values for optimal results and introduce a normalized variant of weight decay to reduce this dependence. Finally, we propose a version of Adam with warm restarts (AdamWR) that has strong anytime performance while achieving state-ofthe-art results on CIFAR-10 and ImageNet32x32. Our source code will become available after the review process.
+
+# 1 INTRODUCTION
+
+Adaptive gradient methods, such as AdaGrad (Duchi et al., 2011), RMSProp (Tieleman & Hinton, 2012), and Adam (Kingma & Ba, 2014) have become a default method of choice for training feedforward and recurrent neural networks (Xu et al., 2015; Gregor et al., 2015; Radford et al., 2015). Nevertheless, state-of-the-art results for popular image classification datasets, such as CIFAR-10 and CIFAR-100 Krizhevsky (2009), are still obtained by applying SGD with momentum (Huang et al., 2016; 2017; Loshchilov & Hutter, 2016; Gastaldi, 2017). Furthermore, Wilson et al. (2017) suggested that adaptive gradient methods do not generalize as well as SGD with momentum when tested on a diverse set of deep learning tasks such as image classification, character-level language modeling and constituency parsing. Different hypotheses about the origins of this worse generalization have been investigated, such as the presence of sharp local minima (Keskar et al., 2016; Dinh et al., 2017) and inherent problems of adaptive gradient methods (Wilson et al., 2017). In this paper, we show that a major factor in the poor generalization of the most popular adaptive gradient method, Adam, lies in its dysfunctional implementation of weight decay; the issue we identify in Adam also pertains to other adaptive gradient methods.
+
+Specifically, our analysis of Adam given in this paper leads to the following observations:
+
+# The standard way to implement $\mathbf { L } _ { 2 }$ regularization/weight decay in Adam is dysfunctional.
+
+One possible explanation why Adam and other adaptive gradient methods might be outperformed by SGD with momentum is that $\mathrm { L _ { 2 } }$ regularization/weight decay are implemented suboptimally in common deep learning libraries. Therefore, on tasks/datasets where the use of $\mathrm { L _ { 2 } }$ regularization is beneficial (e.g., on many popular image classification datasets), Adam leads to worse results than SGD with momentum (for which $\mathrm { L _ { 2 } }$ regularization behaves as expected).
+
+$\mathbf { L } _ { 2 }$ regularization and weight decay are not the same thing. Contrary to common belief, the two techniques are not equivalent. For SGD, they can be made equivalent by a reparameterization of the weight decay factor based on the learning rate; this is not the case for Adam. In particular, when combined with adaptive gradients, $\mathrm { L _ { 2 } }$ regularization leads to weights with large gradients being regularized less than they would be when using weight decay.
+
+Optimal weight decay is a function (among other things) of the total number of batch passes/weight updates.
+
+Our empirical analysis of Adam suggests that the longer the runtime/number of batch passes to be performed, the smaller the optimal weight decay. This effect tends to be neglected because hyperparameters are often tuned for a fixed or a comparable number of training epochs. As a result, the values of the weight decay found to perform best for short runs do not generalize to much longer runs.
+
+Our contributions are aimed at fixing the issues described above:
+
+Decoupling weight decay from the gradient-based update (Section 2). We suggest to decouple the gradient-based update from weight decay for both SGD and Adam. The resulting SGD version SGDW decouples optimal settings of the learning rate and the weight decay factor, and the resulting Adam version AdamW generalizes substantially better than Adam.
+
+Normalizing the values of weight decay (Section 3). We propose to parameterize the weight decay factor as a function of the total number of batch passes. This leads to a greater invariance of the hyperparameter settings in the sense that the values found to perform best for short runs also perform well for many times longer runs.
+
+Adam with warm restarts and normalized weight decay (Section 4). After we fix the weight decay in Adam and design AdamW, we introduce AdamWR to obtain strong anytime performance by performing warm restarts.
+
+The main motivation of this paper is to fix the weight decay in Adam to make it competitive w.r.t. SGD with momentum even for those problems where it did not use to be competitive. We hope that as a result, practitioners do not need to switch between Adam and SGD anymore, which in turn should help to reduce the common issue of selecting dataset/task-specific training algorithms and their hyperparameters.
+
+# 2 DECOUPLING THE WEIGHT DECAY FROM THE GRADIENT-BASED UPDATE
+
+In the weight decay described by Hanson & Pratt (1988), the weights $\boldsymbol { x }$ decay exponentially as
+
+$$
+\begin{array} { r } { \pmb { x } _ { t + 1 } = ( 1 - w _ { t } ) \pmb { x } _ { t } - \alpha _ { t } \nabla f _ { t } ( \pmb { x } _ { t } ) , } \end{array}
+$$
+
+where $w _ { t }$ defines the rate of the weight decay at time-step $t$ and $\nabla f _ { t } ( { \pmb x } _ { t } )$ is the $t { \cdot }$ -th batch gradient multiplied by a learning rate $\alpha _ { t }$ . Following Hanson & Pratt (1988), one can also modify the original batch loss $f _ { t } ( \pmb { x } _ { t } )$ and consider a bias term (also referred to as the regularization term) accounting for “costs” on weights which are, e.g., quadratic in the weight values as for $\mathrm { L _ { 2 } }$ regularization:
+
+$$
+f _ { t , r e g } ( \pmb { x } _ { t } ) = f _ { t } ( \pmb { x } _ { t } ) + \frac { w _ { t } } { 2 } \left\| \pmb { x } _ { t } \right\| _ { 2 } ^ { 2 } ,
+$$
+
+where $w _ { t }$ defines the impact of the $\mathrm { L _ { 2 } }$ regularization. In order to consider the weight decay regularization, one can reformulate the objective function as in Eq. (2) or directly adjust $\bar { \nabla } f _ { t } ( \pmb { x } _ { t } )$ as
+
+$$
+\nabla f _ { t , r e g } ( { \pmb x } _ { t } ) = \nabla f _ { t } ( { \pmb x } _ { t } ) + w _ { t } { \pmb x } _ { t } .
+$$
+
+Historically, stochastic gradient descent methods inherited this way of implementing the weight decay regularization.
+
+The currently most common way (e.g., in popular libraries such as TensorFlow, Keras, PyTorch, Torch, and Lasagne) to introduce the weight decay regularization is to use the $\mathrm { L _ { 2 } }$ regularization term as in Eq. (2) or, often equivalently, to directly modify the gradient as in Eq. (3). Let’s first consider the simple case of SGD with momentum; Algorithm 1 demonstrates modifying the gradients directly in this method (see line 6). The weight decay term $w _ { t } \mathbf { x } _ { t - 1 }$ will first modify ${ \pmb g } _ { t }$ (see line 6) and then affect the momentum term $\pmb { m } _ { t }$ (see line 8). While the smoothing of the weight decay factor by $\beta _ { 1 }$
+
+1: given learning rate $\alpha _ { t } \in \mathbb { R }$ , momentum factor $\beta _ { 1 } \in \mathbb { R }$ , weight decay factor $w \in \mathbb { R }$   
+2: initialize time step $t \gets 0$ , parameter vector $\pmb { x } _ { t = 0 } ~ \in ~ \mathbb { R } ^ { n }$ , first moment vector $\pmb { m } _ { t = 0 } \gets \pmb { \theta }$ ,   
+schedule multiplier $\eta _ { t = 0 } \in \mathbb { R }$   
+3: repeat   
+4: $t \gets t + 1$   
+5: $\nabla f _ { t } ( { \pmb x } _ { t - 1 } ) \gets S e l e c t B a t c h ( { \pmb x } _ { t - 1 } )$ . select batch and return the corresponding gradient   
+6: $\pmb { \mathscr { g } } _ { t } \gets \nabla f _ { t } ( \pmb { x } _ { t - 1 } ) \ + w _ { t } \pmb { x } _ { t - 1 }$   
+7: ηt ← SetScheduleMultiplier(t) . can be fixed, decay, be used for warm restarts   
+8: ${ \pmb { m } } _ { t } \gets \beta _ { 1 } { \pmb { m } } _ { t - 1 } + \eta _ { t } \alpha _ { t } \pmb { g } _ { t }$   
+9: $\pmb { x } _ { t } \gets \pmb { x } _ { t - 1 } - \pmb { m } _ { t } \gets \eta _ { t } w _ { t } \pmb { x } _ { t - 1 }$
+
+<table><tr><td>Algorithm1 SGD with momentum</td><td>and SGDW with momentum</td><td></td></tr></table>
+
+10: until stopping criterion is met
+
+11: return optimized parameters $\mathbf { \boldsymbol { x } } _ { t }$
+
+# Algorithm 2 Adam and AdamW
+
+1: given $\alpha _ { t } = 0 . 0 0 1 , \beta _ { 1 } = 0 . 9 , \beta _ { 2 } = 0 . 9 9 9 , \epsilon = 1 0 ^ { - 8 } , w \in \mathbb { R }$   
+2: initialize time step $t \gets 0$ , parameter vector $\pmb { x } _ { t = 0 } \in \mathbb { R } ^ { n }$ , first moment vector $\pmb { m } _ { t = 0 }  \pmb { \theta }$ , second   
+moment vector $\pmb { \nu } _ { t = 0 }  \pmb { \theta }$ , schedule multiplier $\eta _ { t = 0 } \in \mathbb { R }$   
+3 : repeat   
+4: $t \gets t + 1$   
+5: $\nabla f _ { t } ( { \pmb x } _ { t - 1 } ) \gets S e l e c t B a t c h ( { \pmb x } _ { t - 1 } )$ . select batch and return the corresponding gradient   
+6: $\pmb { \mathrm { g } } _ { t }  \nabla f _ { t } ( \pmb { x } _ { t - 1 } ) \ + w _ { t } \pmb { x } _ { t - 1 }$   
+7: $\pmb { m } _ { t }  \beta _ { 1 } \pmb { m } _ { t - 1 } + \overline { { ( 1 - \beta _ { 1 } ) \pmb { g } _ { t } } }$ . here and below all operations are element-wise   
+8: $\pmb { \nu } _ { t } \gets \beta _ { 2 } \pmb { \nu } _ { t - 1 } + ( 1 - \beta _ { 2 } ) \pmb { g } _ { t } ^ { 2 }$   
+9: $\hat { \pmb { m } } _ { t } \gets \pmb { m } _ { t } / ( 1 - \beta _ { 1 } ^ { t } )$ $\triangleright$ here, $\beta _ { 1 }$ is taken to the power of $t$   
+10: $\hat { \pmb { { \nu } } } _ { t } \gets { \pmb { { \nu } } } _ { t } / ( 1 - \beta _ { 2 } ^ { t } )$ $\triangleright$ here, $\beta _ { 2 }$ is taken to the power of $t$   
+11: ηt ← SetScheduleMultiplier(t) . can be fixed, decay, be used for warm restarts   
+12: $\pmb { x } _ { t }  \pmb { x } _ { t - 1 } - \eta _ { t } ( \alpha _ { t } \hat { m } _ { t } / ( \sqrt { \hat { \nu } _ { t } } + \epsilon ) + w _ { t } \pmb { x } _ { t - 1 } )$   
+13: until stopping criterion is met   
+14: return optimized parameters $\mathbf { \boldsymbol { x } } _ { t }$
+
+(see line 8) might be a feature, we note (for simplicity, we omit $\eta _ { t }$ ) that $\mathbf { \boldsymbol { x } } _ { t }$ will decay by $\alpha _ { t } w _ { t } \pmb { x } _ { t - 1 }$ (see line 9) and not $w _ { t } \mathbf { x } _ { t - 1 }$ as one could expect according to the definition of the weight decay given by Eq. (1). Practically, if one wants to keep the actual weight decay $\alpha _ { t } w _ { t }$ fixed while changing $\alpha _ { t }$ to $\alpha _ { t } ^ { \prime }$ , then $w _ { t }$ should be modified to $\begin{array} { r } { w _ { t } ^ { \prime } = \frac { \alpha _ { t } w _ { t } } { \alpha _ { t } ^ { \prime } } } \end{array}$ αtwtα0 . This renders the problem of hyperparameter selection of $\alpha _ { t }$ and $w _ { t }$ non-separable.
+
+We propose to fix this problem by following the original definition of weight decay given by Eq. (1) and decay the weights simultaneously with the update of $\mathbf { \boldsymbol { x } } _ { t }$ based on gradient information in Line 9 of Algorithm 1. This yields our proposed SGD variant SGDW with momentum. Although the proposed simple modification explicitly decouples $w _ { t }$ and $\alpha _ { t }$ , some problem-dependent implicit coupling is likely to remain. In order to account for a possible scheduling of both $\alpha _ { t }$ and $w _ { t }$ , we introduce a scaling factor $\eta _ { t }$ delivered by a user-defined procedure SetScheduleMultiplier $( t )$ . It should be noted that when $\mathrm { L _ { 2 } }$ regularization is used, weight decay contributes to the batch gradient and thus effectively is scheduled in the same way as the learning rate. Now, since we decouple the two we should also remember to schedule both of them with $\eta _ { t }$ .
+
+Having shown that using $\mathrm { L _ { 2 } }$ regularization instead of weight decay already couples regularization and learning rate in the simple case of SGD with momentum, we now consider adaptive gradient optimizers, such as the Adam algorithm proposed by Kingma & Ba (2014), in which the coupling leads to even more unintended behavior. As an adaptive gradient method, Adam maintains a vector $\nu _ { t }$ responsible for storing smoothed amplitudes of parameter-wise gradients ${ \pmb g } _ { t } ^ { 2 }$ (see line 8 in Algorithm 2). These factors are used to control parameter-wise learning rates by normalizing parameter-wise√ gradients by $\sqrt { \hat { \nu _ { t } } } + \epsilon$ in line 12 of Algorithm 2. The common way to introduce the weight decay $w _ { t } \mathbf { x } _ { t - 1 }$ to Adam results in an update which only distantly resembles the original weight decay given by Eq. (1) because the $\nu _ { t }$ vectors are not only responsible for the parameter-wise amplitudes of $\pmb { g } _ { t }$ but also for the parameter-wise amplitudes of weights $\mathbf { \boldsymbol { x } } _ { t }$ . The amplitudes are then used to renormalize $\hat { \pmb { m } } _ { t }$ as given in line 12 of Algorithm 2. To gain a bit of intuition, let us consider the case when $t$ is large, causing $\beta _ { 1 } ^ { t }$ and $\beta _ { 2 } ^ { t }$ to go to zero and
+
+$$
+\boldsymbol { x } _ { t } \gets \boldsymbol { x } _ { t - 1 } - \eta _ { t } \alpha _ { t } \frac { \beta _ { 1 } m _ { t - 1 } + ( 1 - \beta _ { 1 } ) g _ { t } } { \sqrt { \beta _ { 2 } \nu _ { t - 1 } + ( 1 - \beta _ { 2 } ) g _ { t } ^ { 2 } } + \epsilon } , \ \mathrm { w i t h } \ g _ { t } = \nabla f _ { t } ( \boldsymbol { x } _ { t - 1 } ) + w _ { t } \boldsymbol { x } _ { t - 1 } ,
+$$
+
+where operations are performed parameter-wise. Not only the batch gradient $\nabla f _ { t } ( { \pmb x } _ { t - 1 } )$ is normalized but also the weight decay $w _ { t } \mathbf { x } _ { t - 1 }$ itself. Since this formula normalizes updates by their typical amplitudes, the decay of weights does not account for amplitudes anymore, leading to the relative decay being weaker for weights with large gradients. This is a correct implementation of $\mathbf { L } _ { 2 }$ regularization, but not of weight decay. Therefore, it might be misleading to use the two terms interchangeably, as is commonly done in the literature. We note that this difference between the two mechanisms for Adam has not been investigated and/or described before. As in the case of SGDW, we propose to follow the original definition of weight decay and perform it simultaneously with the gradient-based update as shown in line 12 of Algorithm 2 for AdamW. As we will demonstrate experimentally (in Section 5.2), AdamW generalizes much better than Adam.
+
+# 3 NORMALIZED WEIGHT DECAY
+
+Since our preliminary experiments showed that different weight decay factors are optimal for different computational budgets (defined in terms of the number of batch passes), we introduce a normalized weight decay to reduce this dependence. At iteration $t$ , $w _ { t }$ is set as follows:
+
+$$
+w _ { t } = w _ { n o r m } \sqrt { \frac { b _ { t } } { B T _ { i } } } ,
+$$
+
+where $b _ { t }$ is the batch size, $B$ is the total number of training points to be used in one epoch and $T _ { i }$ is the total number of epochs within the $i$ -th run/restart of the algorithm. Thus, $w _ { n o r m }$ can be interpreted as the weight decay to be used if only one batch pass is allowed. We note a recent relevant observation of Li et al. (2017) who demonstrated that a smaller batch size (for the same total number of epochs) leads to the shrinking effect of weight decay being more pronounced. Here, we propose to address that effect with normalized weight decay.
+
+# 4 ADAM WITH WARM RESTARTS AND NORMALIZED WEIGHT DECAY
+
+We now apply warm restarts to Adam, following the recent work of Loshchilov & Hutter (2016). There, the authors proposed Stochastic Gradient Descent with Warm Restarts (SGDR) to improve anytime performance of SGD by quickly cooling down the learning rate and periodically increasing it. SGDR has been successfully adopted to lead to new state-of-the-art results for popular image classification benchmarks (Huang et al., 2017; Gastaldi, 2017), and we therefore tried extending it to Adam. However, while our initial version of Adam with warm restarts had better anytime performance than Adam, it was not competitive with SGD with warm restarts, precisely because of Adam’s dysfunctional weight decay. Now, having fixed weight decay regularization (Section 2) and also having introduced normalized weight decay (Section 3), the work of Loshchilov & Hutter (2016) on warm restarts directly carries over, and we use it to construct AdamWR to fully benefit from warm restarts.
+
+In the interest of keeping the presentation self-contained, we briefly describe how SGDR schedules the change of the effective learning rate in order to accelerate the training of DNNs. Here, we decouple the initial learning rate and its multiplier $\eta _ { t }$ used to obtain the actual learning rate at iteration $t$ (see, e.g., line 8 in Algorithm 1). In SGDR, we simulate a new warm-started run/restart of SGD once $T _ { i }$ epochs are performed, where $i$ is the index of the run. Importantly, the restarts are not performed from scratch but emulated by increasing $\eta _ { t }$ while the old value of $\mathbf { } _ { \pmb { x } _ { t } }$ is used as an initial solution. The amount by which $\eta _ { t }$ is increases controls to which extent the previously acquired information (e.g., momentum) is used. Within the $i \cdot$ -th run, the value of $\eta _ { t }$ decays according to the cosine annealing (Loshchilov & Hutter, 2016) for each batch as follows:
+
+$$
+\eta _ { t } = \eta _ { m i n } ^ { ( i ) } + 0 . 5 ( \eta _ { m a x } ^ { ( i ) } - \eta _ { m i n } ^ { ( i ) } ) ( 1 + \cos ( \pi T _ { c u r } / T _ { i } ) ) ,
+$$
+
+where η min and $\eta _ { m a x } ^ { ( i ) }$ are ranges for the multiplier and $T _ { c u r }$ accounts for how many epochs have been performed since the last restart. $T _ { c u r }$ is updated at each batch iteration $t$ and is thus not constrained to integer values. Adjusting (e.g., decreasing) η(i)min and $\eta _ { m a x } ^ { ( i ) }$ at every $i$ -th restart (see also Smith (2016)) could potentially improve performance, but we do not consider that option in our experiments because it would involve additional hyperparameters. For $\eta _ { m a x } ^ { ( i ) } = 1$ and $\eta _ { m i n } ^ { ( i ) } = 0$ , one can simplify Eq. (6) to
+
+$$
+\eta _ { t } = 0 . 5 + 0 . 5 \cos ( \pi T _ { c u r } / T _ { i } ) .
+$$
+
+In order to maintain a good anytime performance, one can start with an initially small $T _ { i }$ (e.g., from $1 \%$ to $10 \%$ of the expected total budget) and multiply it by a factor of $T _ { m u l t }$ (e.g., $T _ { m u l t } = 2$ ) at every restart. The $( i + 1 )$ -th restart is triggered when $T _ { c u r } = T _ { i }$ by setting $T _ { c u r }$ to 0. An example setting of the schedule multiplier is given in Section 1.1 of the supplementary material. Note that the effective learning rate is controlled by $\eta _ { t } \alpha _ { t }$ where $\alpha _ { t }$ is set to the initial learning rate and stays constant in our experimental setup. The reason why we employ $\alpha _ { t }$ and not simply $\alpha$ is to account for possible practical extensions, e.g., to adapt $\alpha _ { t }$ as a function of batch size in (scheduled) large-batch settings.
+
+Our proposed AdamWR algorithm represents AdamW given in Algorithm 2 with $\eta _ { t }$ following Eq. (7) and $w _ { t }$ computed at each iteration using normalized weight decay according to Eq. (5). We note that normalized weight decay allowed us to use a constant parameter setting across short and long runs performed within AdamWR. Equivalently to AdamWR, we define SGDWR as SGDW with warm restarts.
+
+# 5 EXPERIMENTAL VALIDATION
+
+Our experimental setup follows that of Gastaldi (2017), who proposed, in addition to $\mathrm { L _ { 2 } }$ regularization, to apply the new Shake-Shake regularization to a 3-branch residual neural network. Gastaldi (2017) showed that this regularization allowed to achieve new state-of-the-art results of $2 . 8 6 \%$ on the CIFAR-10 dataset (Krizhevsky, 2009) and of $1 5 . 8 5 \%$ on CIFAR-100. The network was trained by SGDR with batch size 128 for 1800 epochs $\begin{array} { r } { T _ { 0 } = 1 8 0 0 \mathrm { \Omega } } \end{array}$ ) without restarts with the learning rate scheduled by Eq. (6). The regular data augmentation procedure used for the CIFAR datasets was applied. We used the same model/source code based on fb.resnet.torch 1. The base networks are a 26 2x64d ResNet (i.e. the network has a depth of 26, 2 residual branches and the first residual block has a width of 64) and $2 6 2 \mathrm { x } 9 6 \mathrm { d }$ ResNet with 11.6M and 25.6M parameters, respectively. For a detailed description of the network and the Shake-Shake method, we refer the interested reader to Gastaldi (2017).
+
+# 5.1 DECOUPLING THE WEIGHT DECAY AND INITIAL LEARNING RATE PARAMETERS
+
+In order to verify our hypothesis about the coupling of the initial learning rate $\alpha _ { t }$ and the weight decay factor $w _ { t }$ , we trained a 2x64d ResNet with cosine annealing for 100 epochs with different settings of $\alpha _ { t }$ and $w _ { t }$ . Throughout this paper, we scheduled the learning rate with cosine annealing because it leads to better results than a fixed learning rate (see SuppFigure 1 in the supplementary material). Figure 1 compares SGD vs. SGDW (top row) and Adam vs. AdamW (bottom row). For the case of SGD (Figure 1, top left), weight decay is not decoupled from the learning rate (the common way as described in Algorithm 1), and the figure clearly shows that the basin of best hyperparameter settings (depicted by color and top-10 hyperparameter settings by black circles) is not aligned with the $\mathbf { X }$ -axis or y-axis but lies on the diagonal. This suggests that the two hyperparameters are interdependent and need to be changed simultaneously, while only changing one of them might substantially worsen results. Consider, e.g., the setting at the top left black circle $( \alpha _ { t } = 1 / 2$ , $w _ { t } = 1 / 8 * 0 . 0 0 1 )$ ; only changing either $\alpha _ { t }$ or $w _ { t }$ by itself would worsen results, while changing both of them could still yield clear improvements. We note that this coupling of initial learning rate and weight decay factor might have contributed to SGD’s reputation of being very sensitive to its hyperparameter settings.
+
+![](images/7ad12dbf6e15c266b2455863a707c3f0bb3f131894189e9132a7f993e1ac46fb.jpg)  
+Figure 1: The Top-1 test error of a 26 2x64d ResNet on CIFAR-10 measured after 100 epochs. The proposed SGDW and AdamW (right column) have a more separable hyperparameter space.
+
+In contrast, the results for our new SGDW in Figure 1 (top right) show that SGDW decouples weight decay and initial learning rate. The proposed approach renders the two hyperparameters more separable: even if the learning rate is not well tuned yet (e.g., consider the value of 1/1024 in Figure 1, top right), leaving it fixed and only optimizing the weight decay factor would yield a good value (of $1 / 4 ^ { * } 0 . 0 0 1$ ). This is not the case for the original SGD shown in Figure 1 (top left).
+
+The results for different hyperparameter settings of the original Adam are given in Figure 1 (bottom left). Adam’s best hyperparameter settings performed clearly worse than SGD’s best ones (compare Figure 1, top left). While both methods use the original way to employ weight decay, the original Adam did not benefit from it at all: its best results obtained for non-zero weight decay values were comparable to the best ones obtained without the weight decay regularization, i.e., when $w _ { t } = 0$ . Similarly to the original SGD, the shape of the hyperparameter landscape suggests that the two hyperparameters are coupled.
+
+In contrast, the results for our new AdamW in Figure 1 (bottom right) show that AdamW largely decouples weight decay and learning rate. The results for the best hyperparameter settings were substantially better than the best ones of the original Adam and rivaled those of SGD and SGDW.
+
+![](images/52a47381f3c762e6730e56799945e627b42b9e17fe08beb91dbe05ad79f79083.jpg)  
+Figure 2: Learning curves (top row) and generalization results (bottom row) obtained by a $2 6 2 \mathrm { x } 9 6 \mathrm { d }$ ResNet trained with Adam and AdamW on CIFAR-10. See text for details.
+
+In summary, the experimental results in Figure 1 support our hypothesis that the weight decay and learning rate hyperparameters can be decoupled, and that this in turn simplifies the problem of hyperparameter tuning in SGD and improves Adam’s performance to be competitive w.r.t. SGD with momentum.
+
+# 5.2 BETTER GENERALIZATION OF ADAMW
+
+While the previous experiment suggested that the basin of optimal hyperparameters of AdamW is broader and deeper than the one of Adam, we next investigated the results for much longer runs of 1800 epochs to compare the generalization capabilities of AdamW and Adam.
+
+We fixed the initial learning rate to 0.001 which represents both the default learning rate for Adam and the one which showed reasonably good results in our experiments. Figure 2 shows the results for 12 settings of the weight decay of Adam and 7 settings of the normalized weight decay of AdamW. Interestingly, while the dynamics of the learning curves of Adam and AdamW often coincided for the first half of the training run, AdamW often led to lower training loss and test errors (see Figure 2 top left and top right, respectively). Importantly, the use of weight decay in Adam did not yield as good results as in AdamW (see also Figure 2, bottom left). Next, we investigated whether AdamW’s better results were only due to better convergence or due to better generalization. The results in Figure 2 (bottom right) for the best settings of Adam and AdamW suggest that AdamW did not only yield better training loss but also yielded better generalization performance for similar training loss values. The results on ImageNet32x32 (see SuppFigure 4 in the supplementary material) lead to the same conclusion of substantially improved generalization performance.
+
+![](images/0ff069bf8c97d4f4a604ac3f1be2a4b774410f1171b5b75df88a8d5e5c87160c.jpg)  
+Figure 3: Top-1 test error on CIFAR-10 (left) and Top-5 test error on ImageNet32x32 (right).
+
+5.3 EASIER HYPERPARAMETER SELECTION DUE TO NORMALIZED WEIGHT DECAY
+
+Our experimental results with Adam and SGD suggested that the total runtime in terms of the number of epochs affect the basin of optimal hyperparameters (see SuppFigure 3 in the supplementary material). More specifically, the greater the total number of epochs the smaller the values of the weight decay should be. SuppFigure 3 shows that our remedy for this problem, the normalized weight decay defined in Eq. (7), simplifies hyperparameter selection because the optimal values observed for short runs are similar to the ones for much longer runs. While our initial experiments on CIFAR-10 suggested the square root fit we proposed in Eq. (7), to double-check that this is not a coincidence, we also performed experiments on the ImageNet32x32 dataset (Chrabaszcz et al., 2017), a downsampled version of the original ImageNet dataset with 1.2 million $3 2 \times 3 2$ pixels images, where an epoch is 24 times longer than on CIFAR-10. This experiment also supported the square root scaling: the best values of the normalized weight decay observed on CIFAR-10 represented nearly optimal values for ImageNet $3 2 \mathrm { x } 3 2 $ (see SuppFigure 3). In contrast, had we used the same raw weight decay values $w _ { t }$ for ImageNet32x32 as for CIFAR-10 and for the same number of epochs, without the proposed normalization, $w _ { t }$ would have been roughly 5 greater than optimal for ImageNet32x32, leading to much worse performance. The optimal normalized weight decay values were also very similar (e.g., $w _ { n o r m } = 0 . 0 2 5$ and $w _ { n o r m } = 0 . 0 5$ ) across SGDW and AdamW.
+
+We investigated whether the use of much longer runs (1800 epochs) of the original Adam with $\mathrm { L _ { 2 } }$ regularization makes the use of cosine annealing unnecessary. The results of Adam without cosine annealing (i.e., with fixed learning rate) for a 4 by 4 logarithmic grid of hyperparameter settings are given in SuppFigure 5 in the supplementary material. Even after taking into account the low resolution of the grid, the results appear to be at best comparable to the ones obtained with AdamW with 18 times less epochs and a smaller network (see SuppFigure 2). These results are not very surprising given Figure 1 (which demonstrates the effectiveness of AdamW) and SuppFigure 2 (which demonstrates the necessity to use some learning rate schedule such as cosine annealing).
+
+# 5.4 ADAMWR WITH WARM RESTARTS FOR BETTER ANYTIME PERFORMANCE
+
+Finally, we investigated the strong anytime performance AdamWR obtains from warm restarts (using normalized weight decay to avoid the need for a different weight decay factor for restarts with longer annealing schedules). As Figure 3 shows, AdamWR greatly sped up AdamW on CIFAR10 and ImageNet32x32, up to a factor of 10 (see the results at the first restart). For the default learning rate of 0.001, AdamW achieved $15 \%$ relative improvement in test errors compared to Adam both on CIFAR-10 (also see Figure 2) and ImageNet32x32 (also see SuppFigure 4). AdamWR achieved the same improved results but with a much better anytime performance. These improvements closed most of the gap between Adam and SGDWR on CIFAR-10 and yielded comparable performance on ImageNet32x32.
+
+# 6 DISCUSSION AND CONCLUSION
+
+Following suggestions that adaptive gradient methods such as Adam might lead to worse generalization than SGD with momentum (Wilson et al., 2017), we identified at least one possible explanation to this phenomenon: the dysfunctional use of $\mathrm { L _ { 2 } }$ regularization and weight decay. We proposed a simple fix to deal with this issue, yielding substantially better generalization performance in our AdamW variant. We also proposed normalized weight decay and warm restarts for Adam, showing that a more robust hyperparameteer selection and a better anytime performance can be achieved in our new AdamWR variant.
+
+Our preliminary results obtained with AdamW and AdamWR on image classification datasets must be verified on a wider range of tasks, especially the ones where the use of regularization is expected to be important. It would be interesting to integrate our findings on weight decay into other methods which attempt to improve Adam, e.g, normalized direction-preserving Adam (Zhang et al., 2017). While we focussed our experimental analysis on Adam, we believe that similar results also hold for other adaptive gradient methods, such as AdaGrad (Duchi et al., 2011) and RMSProp (Tieleman & Hinton, 2012).
+
+The results shown in Figure 2 suggest that Adam and AdamW follow very similar curves most of the time until the third phase of the run where AdamW starts to branch out to outperform Adam. As pointed out by an anonymous reviewer, it would be interesting to investigate what causes this branching and whether the desired effects are observed at the bottom of the landscape. One could investigate this using the approach of Im et al. (2016) to switch from Adam to AdamW at a given epoch index. Since it is quite possible that the effect of regularization is not that pronounced in the early stages of training, one could think of designing a version of Adam which exploits this by being fast in the early stages and well-regularized in the late stages of training. The latter might be achieved with a custom schedule of the weight decay factor.
+
+In this paper, we argue that the popular interpretation that weight decay $\mathbf { \tau } = \mathbf { L } _ { 2 }$ regularization is not precise. Instead, the difference between the two leads to the following important consequences. Two algorithms as different as SGD and Adam will exhibit different effective rates of weight decay even if the same regularization coefficient is used to include $\mathrm { L _ { 2 } }$ regularization in the objective function. Moreover, when decoupled weight decay is applied, two algorithms as different as SGDW and AdamW will optimize two effectively different objective functions even if the same weight decay factor is used. Our findings suggest that the original Adam algorithm with $\mathrm { L _ { 2 } }$ regularization affects effective rates of weight decay in a way that precludes effective regularization, and that effective regularization is achievable by decoupling the weight decay.
+
+Advani & Saxe (2017) analytically showed that in the limited data regime of deep networks the presence of eigenvalues that are zero forms a frozen subspace in which no learning occurs and thus smaller (e.g., zero) initial weight norms should be used to achieve best generalization results. Our future work shall consider adapting initial weight norms or weight norm constraints (Salimans & Kingma, 2016) at each warm restart. Kawaguchi et al. (2017) proposed a family of regularization techniques which are specific to the current batch and its size. Similarly to $\mathrm { L _ { 2 } }$ regularization and weight decay, the latter techniques might be attempted to be transformed to act directly on weights.
+
+# REFERENCES
+
+Madhu S. Advani and Andrew M. Saxe. High-dimensional dynamics of generalization error in neural networks. arXiv:1710.03667, 2017.   
+Patryk Chrabaszcz, Ilya Loshchilov, and Frank Hutter. A downsampled variant of ImageNet as an alternative to the CIFAR datasets. arXiv:1707.08819, 2017.   
+Laurent Dinh, Razvan Pascanu, Samy Bengio, and Yoshua Bengio. Sharp minima can generalize for deep nets. arXiv:1703.04933, 2017.   
+John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and stochastic optimization. The Journal of Machine Learning Research, 12:2121–2159, 2011.   
+Xavier Gastaldi. Shake-Shake regularization. arXiv preprint arXiv:1705.07485, 2017.
+
+Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, and Daan Wierstra. Draw: A recurrent neural network for image generation. arXiv:1502.04623, 2015.
+
+Stephen Jose Hanson and Lorien Y Pratt. Comparing biases for minimal network construction with ´ back-propagation. In Proceedings of the 1st International Conference on Neural Information Processing Systems, pp. 177–185, 1988.
+
+Gao Huang, Zhuang Liu, and Kilian Q Weinberger. Densely connected convolutional networks. arXiv:1608.06993, 2016.
+
+Gao Huang, Yixuan Li, Geoff Pleiss, Zhuang Liu, John E Hopcroft, and Kilian Q Weinberger. Snapshot ensembles: Train 1, get m for free. arXiv:1704.00109, 2017.
+
+Daniel Jiwoong Im, Michael Tao, and Kristin Branson. An empirical analysis of deep network loss surfaces. arXiv preprint arXiv:1612.04010, 2016.
+
+Kenji Kawaguchi, Leslie Pack Kaelbling, and Yoshua Bengio. Generalization in deep learning. arXiv:1710.05468, 2017.
+
+Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. arXiv:1609.04836, 2016.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv:1412.6980, 2014.
+
+Alex Krizhevsky. Learning multiple layers of features from tiny images. 2009.
+
+Hao Li, Zheng Xu, Gavin Taylor, and Tom Goldstein. Visualizing the loss landscape of neural nets. arXiv preprint arXiv:1712.09913, 2017.
+
+Ilya Loshchilov and Frank Hutter. SGDR: stochastic gradient descent with warm restarts. arXiv:1608.03983, 2016.
+
+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv:1511.06434, 2015.
+
+Tim Salimans and Diederik P Kingma. Weight normalization: A simple reparameterization to accelerate training of deep neural networks. In Advances in Neural Information Processing Systems, pp. 901–909, 2016.
+
+Leslie N Smith. Cyclical learning rates for training neural networks. arXiv:1506.01186v3, 2016.
+
+Tijmen Tieleman and Geoffrey Hinton. Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural networks for machine learning, 4(2):26– 31, 2012.
+
+Ashia C Wilson, Rebecca Roelofs, Mitchell Stern, Nathan Srebro, and Benjamin Recht. The marginal value of adaptive gradient methods in machine learning. arXiv:1705.08292, 2017.
+
+Kelvin Xu, Jimmy Ba, Ryan Kiros, Kyunghyun Cho, Aaron Courville, Ruslan Salakhudinov, Rich Zemel, and Yoshua Bengio. Show, attend and tell: Neural image caption generation with visual attention. In International Conference on Machine Learning, pp. 2048–2057, 2015.
+
+Zijun Zhang, Lin Ma, Zongpeng Li, and Chuan Wu. Normalized direction-preserving adam. arXiv:1709.04546, 2017.
+
+# 1 SUPPLEMENTARY MATERIAL
+
+# 1.1 AN EXAMPLE SETTING OF THE SCHEDULE MULTIPLIER
+
+An example schedule of the schedule multiplier $\eta _ { t }$ is given in SuppFigure 1 for $T _ { i = 0 } = 1 0 0$ and $T _ { m u l t } = 2$ . After the initial 100 epochs the learning rate will reach 0 because $\eta _ { t = 1 0 0 } = 0$ . Then, since $T _ { c u r } = T _ { i = 0 }$ , we restart by resetting $T _ { c u r } = 0$ , causing the multiplier $\eta _ { t }$ to be reset to 1 due to Eq. (7). This multiplier will then decrease again from 1 to 0, but now over the course of 200 epochs because $T _ { i = 1 } = T _ { i = 0 } T _ { m u l t } = 2 0 0$ . Solutions obtained right before the restarts, when $\eta _ { t } = 0$ (e.g., at epoch indexes 100, 300, 700 and 1500 as shown in SuppFigure 1) are recommended by the optimizer as the solutions, with more recent solutions prioritized.
+
+![](images/e1136b7a9c1678087dff60655ce9bbb521fd87b5f4b6da3cc91d9b66fe540e22.jpg)  
+SuppFigure 1: An example schedule of the learning rate multiplier as a function of epoch index. The first run is scheduled to converge at epoch $T _ { i = 0 } = 1 0 0$ , then the budget for the next run is doubled as $T _ { i = 1 } = T _ { i = 0 } T _ { m u l t } = 2 0 0$ , etc.
+
+![](images/17833324a93093bca68da92e2e53961c517952fd5102b4cf95bfdcb2b6964358.jpg)  
+SuppFigure 2: Adam with fixed learning rate (left) and with cosine annealing (right). We show the final test error of a $2 6 ~ 2 \mathrm { x } 6 4 \mathrm { d }$ ResNet on CIFAR-10 after 100 epochs of SGD with momentum. The results where the learning rate is fixed (left) are inferior to the ones where the learning rate is scheduled according to cosine annealing (right). Therefore, we schedule the learning rate with cosine annealing for all methods given in the paper.
+
+![](images/5f160ef2fc5e0bb5f0bd015a3408463f8f327bfc9a812fbf805295bea1d5d435.jpg)  
+SuppFigure 3: Effect of normalized weight decay. We show the final test Top-1 error on CIFAR10 (first two rows for AdamW without and with normalized weight decay) and Top-5 error on ImageNet32x32 (last two rows for AdamW and SGDW, both with normalized weight decay) of a $2 6 2 \mathrm { x } 6 4 \mathrm { d }$ ResNet after different numbers of epochs (see columns). While the optimal settings of the raw weight decay change significantly for different runtime budgets (see the first row), the values of the normalized weight decay remain very similar for different budgets (see the second row) and different datasets (here, CIFAR-10 and ImageNet32x32), and even across AdamW and SGDW.
+
+![](images/34a75388019587badb9d17928eefc833ecc3e6db8aa0e1a91ecb741aea7e654e.jpg)  
+SuppFigure 4: Learning curves (top row) and generalization results (Top-5 errors in bottom row) obtained by a $2 6 2 \mathrm { x } 9 6 \mathrm { d }$ ResNet trained with Adam and AdamW on ImageNet32x32.
+
+![](images/b8803b776bd6799e677439534292c7be94ce223d9f24b87e94e074f7d33cd8a5.jpg)  
+SuppFigure 5: Adam without cosine annealing, i.e., with fixed learning rate. We show the final test error of a $2 6 ~ 2 \mathrm { x } 9 6 \mathrm { d }$ ResNet on CIFAR-10 after 1800 epochs of the original Adam for different settings of learning rate and weight decay used for $\mathrm { L _ { 2 } }$ regularization. These results can be compared to the ones of AdamW shown in SuppFigure 3 (top row). The results of AdamW with only 100 epochs and a smaller network seem to be at least as good as the ones of Adam with 18 times as many epochs and a bigger network.

parse/train/rk6qdGgCZ/rk6qdGgCZ_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "FIXING WEIGHT DECAY REGULARIZATION IN ADAM",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+            145,
+            398,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
+            544,
+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We note that common implementations of adaptive gradient algorithms, such as Adam, limit the potential benefit of weight decay regularization, because the weights do not decay multiplicatively (as would be expected for standard weight decay) but by an additive constant factor. We propose a simple way to resolve this issue by decoupling weight decay and the optimization steps taken w.r.t. the loss function. We provide empirical evidence that our proposed modification (i) decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam, and (ii) substantially improves Adam’s generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperformed by the latter). We also demonstrate that longer optimization runs require smaller weight decay values for optimal results and introduce a normalized variant of weight decay to reduce this dependence. Finally, we propose a version of Adam with warm restarts (AdamWR) that has strong anytime performance while achieving state-ofthe-art results on CIFAR-10 and ImageNet32x32. Our source code will become available after the review process. ",
+        "bbox": [
+            233,
+            242,
+            764,
+            463
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            491,
+            336,
+            507
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adaptive gradient methods, such as AdaGrad (Duchi et al., 2011), RMSProp (Tieleman & Hinton, 2012), and Adam (Kingma & Ba, 2014) have become a default method of choice for training feedforward and recurrent neural networks (Xu et al., 2015; Gregor et al., 2015; Radford et al., 2015). Nevertheless, state-of-the-art results for popular image classification datasets, such as CIFAR-10 and CIFAR-100 Krizhevsky (2009), are still obtained by applying SGD with momentum (Huang et al., 2016; 2017; Loshchilov & Hutter, 2016; Gastaldi, 2017). Furthermore, Wilson et al. (2017) suggested that adaptive gradient methods do not generalize as well as SGD with momentum when tested on a diverse set of deep learning tasks such as image classification, character-level language modeling and constituency parsing. Different hypotheses about the origins of this worse generalization have been investigated, such as the presence of sharp local minima (Keskar et al., 2016; Dinh et al., 2017) and inherent problems of adaptive gradient methods (Wilson et al., 2017). In this paper, we show that a major factor in the poor generalization of the most popular adaptive gradient method, Adam, lies in its dysfunctional implementation of weight decay; the issue we identify in Adam also pertains to other adaptive gradient methods. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Specifically, our analysis of Adam given in this paper leads to the following observations: ",
+        "bbox": [
+            171,
+            724,
+            759,
+            738
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The standard way to implement $\\mathbf { L } _ { 2 }$ regularization/weight decay in Adam is dysfunctional. ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            752,
+            797,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One possible explanation why Adam and other adaptive gradient methods might be outperformed by SGD with momentum is that $\\mathrm { L _ { 2 } }$ regularization/weight decay are implemented suboptimally in common deep learning libraries. Therefore, on tasks/datasets where the use of $\\mathrm { L _ { 2 } }$ regularization is beneficial (e.g., on many popular image classification datasets), Adam leads to worse results than SGD with momentum (for which $\\mathrm { L _ { 2 } }$ regularization behaves as expected). ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { L } _ { 2 }$ regularization and weight decay are not the same thing. Contrary to common belief, the two techniques are not equivalent. For SGD, they can be made equivalent by a reparameterization of the weight decay factor based on the learning rate; this is not the case for Adam. In particular, when combined with adaptive gradients, $\\mathrm { L _ { 2 } }$ regularization leads to weights with large gradients being regularized less than they would be when using weight decay. ",
+        "bbox": [
+            174,
+            854,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Optimal weight decay is a function (among other things) of the total number of batch passes/weight updates. ",
+        "bbox": [
+            176,
+            103,
+            928,
+            118
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our empirical analysis of Adam suggests that the longer the runtime/number of batch passes to be performed, the smaller the optimal weight decay. This effect tends to be neglected because hyperparameters are often tuned for a fixed or a comparable number of training epochs. As a result, the values of the weight decay found to perform best for short runs do not generalize to much longer runs. ",
+        "bbox": [
+            232,
+            119,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are aimed at fixing the issues described above: ",
+        "bbox": [
+            174,
+            199,
+            598,
+            214
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Decoupling weight decay from the gradient-based update (Section 2). We suggest to decouple the gradient-based update from weight decay for both SGD and Adam. The resulting SGD version SGDW decouples optimal settings of the learning rate and the weight decay factor, and the resulting Adam version AdamW generalizes substantially better than Adam. ",
+        "bbox": [
+            174,
+            228,
+            825,
+            284
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Normalizing the values of weight decay (Section 3). We propose to parameterize the weight decay factor as a function of the total number of batch passes. This leads to a greater invariance of the hyperparameter settings in the sense that the values found to perform best for short runs also perform well for many times longer runs. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Adam with warm restarts and normalized weight decay (Section 4). After we fix the weight decay in Adam and design AdamW, we introduce AdamWR to obtain strong anytime performance by performing warm restarts. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The main motivation of this paper is to fix the weight decay in Adam to make it competitive w.r.t. SGD with momentum even for those problems where it did not use to be competitive. We hope that as a result, practitioners do not need to switch between Adam and SGD anymore, which in turn should help to reduce the common issue of selecting dataset/task-specific training algorithms and their hyperparameters. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 DECOUPLING THE WEIGHT DECAY FROM THE GRADIENT-BASED UPDATE",
+        "text_level": 1,
+        "bbox": [
+            173,
+            492,
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+    },
+    {
+        "type": "text",
+        "text": "In the weight decay described by Hanson & Pratt (1988), the weights $\\boldsymbol { x }$ decay exponentially as ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b60935a0cf5f14cba9594baa6ced130cb47befe3018f337df25795bf146934c3.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { x } _ { t + 1 } = ( 1 - w _ { t } ) \\pmb { x } _ { t } - \\alpha _ { t } \\nabla f _ { t } ( \\pmb { x } _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            566,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { t }$ defines the rate of the weight decay at time-step $t$ and $\\nabla f _ { t } ( { \\pmb x } _ { t } )$ is the $t { \\cdot }$ -th batch gradient multiplied by a learning rate $\\alpha _ { t }$ . Following Hanson & Pratt (1988), one can also modify the original batch loss $f _ { t } ( \\pmb { x } _ { t } )$ and consider a bias term (also referred to as the regularization term) accounting for “costs” on weights which are, e.g., quadratic in the weight values as for $\\mathrm { L _ { 2 } }$ regularization: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c24cbf1569d6cdff1c56458e7286c3c7715a7c5643de72e8f37d97737919f0b2.jpg",
+        "text": "$$\nf _ { t , r e g } ( \\pmb { x } _ { t } ) = f _ { t } ( \\pmb { x } _ { t } ) + \\frac { w _ { t } } { 2 } \\left\\| \\pmb { x } _ { t } \\right\\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            678,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { t }$ defines the impact of the $\\mathrm { L _ { 2 } }$ regularization. In order to consider the weight decay regularization, one can reformulate the objective function as in Eq. (2) or directly adjust $\\bar { \\nabla } f _ { t } ( \\pmb { x } _ { t } )$ as ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/89f3d8580e9a9d63ed9d2f9493cf5c91a7f741299c19896e025e7ed2138466b3.jpg",
+        "text": "$$\n\\nabla f _ { t , r e g } ( { \\pmb x } _ { t } ) = \\nabla f _ { t } ( { \\pmb x } _ { t } ) + w _ { t } { \\pmb x } _ { t } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Historically, stochastic gradient descent methods inherited this way of implementing the weight decay regularization. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The currently most common way (e.g., in popular libraries such as TensorFlow, Keras, PyTorch, Torch, and Lasagne) to introduce the weight decay regularization is to use the $\\mathrm { L _ { 2 } }$ regularization term as in Eq. (2) or, often equivalently, to directly modify the gradient as in Eq. (3). Let’s first consider the simple case of SGD with momentum; Algorithm 1 demonstrates modifying the gradients directly in this method (see line 6). The weight decay term $w _ { t } \\mathbf { x } _ { t - 1 }$ will first modify ${ \\pmb g } _ { t }$ (see line 6) and then affect the momentum term $\\pmb { m } _ { t }$ (see line 8). While the smoothing of the weight decay factor by $\\beta _ { 1 }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "1: given learning rate $\\alpha _ { t } \\in \\mathbb { R }$ , momentum factor $\\beta _ { 1 } \\in \\mathbb { R }$ , weight decay factor $w \\in \\mathbb { R }$   \n2: initialize time step $t \\gets 0$ , parameter vector $\\pmb { x } _ { t = 0 } ~ \\in ~ \\mathbb { R } ^ { n }$ , first moment vector $\\pmb { m } _ { t = 0 } \\gets \\pmb { \\theta }$ ,   \nschedule multiplier $\\eta _ { t = 0 } \\in \\mathbb { R }$   \n3: repeat   \n4: $t \\gets t + 1$   \n5: $\\nabla f _ { t } ( { \\pmb x } _ { t - 1 } ) \\gets S e l e c t B a t c h ( { \\pmb x } _ { t - 1 } )$ . select batch and return the corresponding gradient   \n6: $\\pmb { \\mathscr { g } } _ { t } \\gets \\nabla f _ { t } ( \\pmb { x } _ { t - 1 } ) \\ + w _ { t } \\pmb { x } _ { t - 1 }$   \n7: ηt ← SetScheduleMultiplier(t) . can be fixed, decay, be used for warm restarts   \n8: ${ \\pmb { m } } _ { t } \\gets \\beta _ { 1 } { \\pmb { m } } _ { t - 1 } + \\eta _ { t } \\alpha _ { t } \\pmb { g } _ { t }$   \n9: $\\pmb { x } _ { t } \\gets \\pmb { x } _ { t - 1 } - \\pmb { m } _ { t } \\gets \\eta _ { t } w _ { t } \\pmb { x } _ { t - 1 }$ ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/e96f99bc12a691e6993cd856ecc44e561f3d7c275cdc5441a2ac17b778dbd3d4.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "10: until stopping criterion is met "
+        ],
+        "table_body": "<table><tr><td>Algorithm1 SGD with momentum</td><td>and SGDW with momentum</td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "11: return optimized parameters $\\mathbf { \\boldsymbol { x } } _ { t }$ ",
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+            289,
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+            303
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Adam and AdamW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            324,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1: given $\\alpha _ { t } = 0 . 0 0 1 , \\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . 9 9 9 , \\epsilon = 1 0 ^ { - 8 } , w \\in \\mathbb { R }$   \n2: initialize time step $t \\gets 0$ , parameter vector $\\pmb { x } _ { t = 0 } \\in \\mathbb { R } ^ { n }$ , first moment vector $\\pmb { m } _ { t = 0 }  \\pmb { \\theta }$ , second   \nmoment vector $\\pmb { \\nu } _ { t = 0 }  \\pmb { \\theta }$ , schedule multiplier $\\eta _ { t = 0 } \\in \\mathbb { R }$   \n3 : repeat   \n4: $t \\gets t + 1$   \n5: $\\nabla f _ { t } ( { \\pmb x } _ { t - 1 } ) \\gets S e l e c t B a t c h ( { \\pmb x } _ { t - 1 } )$ . select batch and return the corresponding gradient   \n6: $\\pmb { \\mathrm { g } } _ { t }  \\nabla f _ { t } ( \\pmb { x } _ { t - 1 } ) \\ + w _ { t } \\pmb { x } _ { t - 1 }$   \n7: $\\pmb { m } _ { t }  \\beta _ { 1 } \\pmb { m } _ { t - 1 } + \\overline { { ( 1 - \\beta _ { 1 } ) \\pmb { g } _ { t } } }$ . here and below all operations are element-wise   \n8: $\\pmb { \\nu } _ { t } \\gets \\beta _ { 2 } \\pmb { \\nu } _ { t - 1 } + ( 1 - \\beta _ { 2 } ) \\pmb { g } _ { t } ^ { 2 }$   \n9: $\\hat { \\pmb { m } } _ { t } \\gets \\pmb { m } _ { t } / ( 1 - \\beta _ { 1 } ^ { t } )$ $\\triangleright$ here, $\\beta _ { 1 }$ is taken to the power of $t$   \n10: $\\hat { \\pmb { { \\nu } } } _ { t } \\gets { \\pmb { { \\nu } } } _ { t } / ( 1 - \\beta _ { 2 } ^ { t } )$ $\\triangleright$ here, $\\beta _ { 2 }$ is taken to the power of $t$   \n11: ηt ← SetScheduleMultiplier(t) . can be fixed, decay, be used for warm restarts   \n12: $\\pmb { x } _ { t }  \\pmb { x } _ { t - 1 } - \\eta _ { t } ( \\alpha _ { t } \\hat { m } _ { t } / ( \\sqrt { \\hat { \\nu } _ { t } } + \\epsilon ) + w _ { t } \\pmb { x } _ { t - 1 } )$   \n13: until stopping criterion is met   \n14: return optimized parameters $\\mathbf { \\boldsymbol { x } } _ { t }$ ",
+        "bbox": [
+            176,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "(see line 8) might be a feature, we note (for simplicity, we omit $\\eta _ { t }$ ) that $\\mathbf { \\boldsymbol { x } } _ { t }$ will decay by $\\alpha _ { t } w _ { t } \\pmb { x } _ { t - 1 }$ (see line 9) and not $w _ { t } \\mathbf { x } _ { t - 1 }$ as one could expect according to the definition of the weight decay given by Eq. (1). Practically, if one wants to keep the actual weight decay $\\alpha _ { t } w _ { t }$ fixed while changing $\\alpha _ { t }$ to $\\alpha _ { t } ^ { \\prime }$ , then $w _ { t }$ should be modified to $\\begin{array} { r } { w _ { t } ^ { \\prime } = \\frac { \\alpha _ { t } w _ { t } } { \\alpha _ { t } ^ { \\prime } } } \\end{array}$ αtwtα0 . This renders the problem of hyperparameter selection of $\\alpha _ { t }$ and $w _ { t }$ non-separable. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose to fix this problem by following the original definition of weight decay given by Eq. (1) and decay the weights simultaneously with the update of $\\mathbf { \\boldsymbol { x } } _ { t }$ based on gradient information in Line 9 of Algorithm 1. This yields our proposed SGD variant SGDW with momentum. Although the proposed simple modification explicitly decouples $w _ { t }$ and $\\alpha _ { t }$ , some problem-dependent implicit coupling is likely to remain. In order to account for a possible scheduling of both $\\alpha _ { t }$ and $w _ { t }$ , we introduce a scaling factor $\\eta _ { t }$ delivered by a user-defined procedure SetScheduleMultiplier $( t )$ . It should be noted that when $\\mathrm { L _ { 2 } }$ regularization is used, weight decay contributes to the batch gradient and thus effectively is scheduled in the same way as the learning rate. Now, since we decouple the two we should also remember to schedule both of them with $\\eta _ { t }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Having shown that using $\\mathrm { L _ { 2 } }$ regularization instead of weight decay already couples regularization and learning rate in the simple case of SGD with momentum, we now consider adaptive gradient optimizers, such as the Adam algorithm proposed by Kingma & Ba (2014), in which the coupling leads to even more unintended behavior. As an adaptive gradient method, Adam maintains a vector $\\nu _ { t }$ responsible for storing smoothed amplitudes of parameter-wise gradients ${ \\pmb g } _ { t } ^ { 2 }$ (see line 8 in Algorithm 2). These factors are used to control parameter-wise learning rates by normalizing parameter-wise√ gradients by $\\sqrt { \\hat { \\nu _ { t } } } + \\epsilon$ in line 12 of Algorithm 2. The common way to introduce the weight decay $w _ { t } \\mathbf { x } _ { t - 1 }$ to Adam results in an update which only distantly resembles the original weight decay given by Eq. (1) because the $\\nu _ { t }$ vectors are not only responsible for the parameter-wise amplitudes of $\\pmb { g } _ { t }$ but also for the parameter-wise amplitudes of weights $\\mathbf { \\boldsymbol { x } } _ { t }$ . The amplitudes are then used to renormalize $\\hat { \\pmb { m } } _ { t }$ as given in line 12 of Algorithm 2. To gain a bit of intuition, let us consider the case when $t$ is large, causing $\\beta _ { 1 } ^ { t }$ and $\\beta _ { 2 } ^ { t }$ to go to zero and ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
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+            160
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea9aa5c691010be5c6df39d4e10e31ac83b3aa14287be4778e047c963edd0ba5.jpg",
+        "text": "$$\n\\boldsymbol { x } _ { t } \\gets \\boldsymbol { x } _ { t - 1 } - \\eta _ { t } \\alpha _ { t } \\frac { \\beta _ { 1 } m _ { t - 1 } + ( 1 - \\beta _ { 1 } ) g _ { t } } { \\sqrt { \\beta _ { 2 } \\nu _ { t - 1 } + ( 1 - \\beta _ { 2 } ) g _ { t } ^ { 2 } } + \\epsilon } , \\ \\mathrm { w i t h } \\ g _ { t } = \\nabla f _ { t } ( \\boldsymbol { x } _ { t - 1 } ) + w _ { t } \\boldsymbol { x } _ { t - 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            185,
+            746,
+            222
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where operations are performed parameter-wise. Not only the batch gradient $\\nabla f _ { t } ( { \\pmb x } _ { t - 1 } )$ is normalized but also the weight decay $w _ { t } \\mathbf { x } _ { t - 1 }$ itself. Since this formula normalizes updates by their typical amplitudes, the decay of weights does not account for amplitudes anymore, leading to the relative decay being weaker for weights with large gradients. This is a correct implementation of $\\mathbf { L } _ { 2 }$ regularization, but not of weight decay. Therefore, it might be misleading to use the two terms interchangeably, as is commonly done in the literature. We note that this difference between the two mechanisms for Adam has not been investigated and/or described before. As in the case of SGDW, we propose to follow the original definition of weight decay and perform it simultaneously with the gradient-based update as shown in line 12 of Algorithm 2 for AdamW. As we will demonstrate experimentally (in Section 5.2), AdamW generalizes much better than Adam. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 NORMALIZED WEIGHT DECAY",
+        "text_level": 1,
+        "bbox": [
+            176,
+            396,
+            462,
+            411
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since our preliminary experiments showed that different weight decay factors are optimal for different computational budgets (defined in terms of the number of batch passes), we introduce a normalized weight decay to reduce this dependence. At iteration $t$ , $w _ { t }$ is set as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ab89fbc20262c106c2ec8af64c93f32520e0ae5c35fa9dae0b2c0fbd81a6029.jpg",
+        "text": "$$\nw _ { t } = w _ { n o r m } \\sqrt { \\frac { b _ { t } } { B T _ { i } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            496,
+            552,
+            531
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $b _ { t }$ is the batch size, $B$ is the total number of training points to be used in one epoch and $T _ { i }$ is the total number of epochs within the $i$ -th run/restart of the algorithm. Thus, $w _ { n o r m }$ can be interpreted as the weight decay to be used if only one batch pass is allowed. We note a recent relevant observation of Li et al. (2017) who demonstrated that a smaller batch size (for the same total number of epochs) leads to the shrinking effect of weight decay being more pronounced. Here, we propose to address that effect with normalized weight decay. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 ADAM WITH WARM RESTARTS AND NORMALIZED WEIGHT DECAY",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We now apply warm restarts to Adam, following the recent work of Loshchilov & Hutter (2016). There, the authors proposed Stochastic Gradient Descent with Warm Restarts (SGDR) to improve anytime performance of SGD by quickly cooling down the learning rate and periodically increasing it. SGDR has been successfully adopted to lead to new state-of-the-art results for popular image classification benchmarks (Huang et al., 2017; Gastaldi, 2017), and we therefore tried extending it to Adam. However, while our initial version of Adam with warm restarts had better anytime performance than Adam, it was not competitive with SGD with warm restarts, precisely because of Adam’s dysfunctional weight decay. Now, having fixed weight decay regularization (Section 2) and also having introduced normalized weight decay (Section 3), the work of Loshchilov & Hutter (2016) on warm restarts directly carries over, and we use it to construct AdamWR to fully benefit from warm restarts. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the interest of keeping the presentation self-contained, we briefly describe how SGDR schedules the change of the effective learning rate in order to accelerate the training of DNNs. Here, we decouple the initial learning rate and its multiplier $\\eta _ { t }$ used to obtain the actual learning rate at iteration $t$ (see, e.g., line 8 in Algorithm 1). In SGDR, we simulate a new warm-started run/restart of SGD once $T _ { i }$ epochs are performed, where $i$ is the index of the run. Importantly, the restarts are not performed from scratch but emulated by increasing $\\eta _ { t }$ while the old value of $\\mathbf { } _ { \\pmb { x } _ { t } }$ is used as an initial solution. The amount by which $\\eta _ { t }$ is increases controls to which extent the previously acquired information (e.g., momentum) is used. Within the $i \\cdot$ -th run, the value of $\\eta _ { t }$ decays according to the cosine annealing (Loshchilov & Hutter, 2016) for each batch as follows: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b46e90f4747ab62eafb0a7aae0bb8fb8e9fc6e5e2fb31ef928ccec9c73c9d06c.jpg",
+        "text": "$$\n\\eta _ { t } = \\eta _ { m i n } ^ { ( i ) } + 0 . 5 ( \\eta _ { m a x } ^ { ( i ) } - \\eta _ { m i n } ^ { ( i ) } ) ( 1 + \\cos ( \\pi T _ { c u r } / T _ { i } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            169,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where η min and $\\eta _ { m a x } ^ { ( i ) }$ are ranges for the multiplier and $T _ { c u r }$ accounts for how many epochs have been performed since the last restart. $T _ { c u r }$ is updated at each batch iteration $t$ and is thus not constrained to integer values. Adjusting (e.g., decreasing) η(i)min and $\\eta _ { m a x } ^ { ( i ) }$ at every $i$ -th restart (see also Smith (2016)) could potentially improve performance, but we do not consider that option in our experiments because it would involve additional hyperparameters. For $\\eta _ { m a x } ^ { ( i ) } = 1$ and $\\eta _ { m i n } ^ { ( i ) } = 0$ , one can simplify Eq. (6) to ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46cfd9b63909f3bebd69999cc1b60055442c3c0957399000f250dbf9cd286af5.jpg",
+        "text": "$$\n\\eta _ { t } = 0 . 5 + 0 . 5 \\cos ( \\pi T _ { c u r } / T _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In order to maintain a good anytime performance, one can start with an initially small $T _ { i }$ (e.g., from $1 \\%$ to $10 \\%$ of the expected total budget) and multiply it by a factor of $T _ { m u l t }$ (e.g., $T _ { m u l t } = 2$ ) at every restart. The $( i + 1 )$ -th restart is triggered when $T _ { c u r } = T _ { i }$ by setting $T _ { c u r }$ to 0. An example setting of the schedule multiplier is given in Section 1.1 of the supplementary material. Note that the effective learning rate is controlled by $\\eta _ { t } \\alpha _ { t }$ where $\\alpha _ { t }$ is set to the initial learning rate and stays constant in our experimental setup. The reason why we employ $\\alpha _ { t }$ and not simply $\\alpha$ is to account for possible practical extensions, e.g., to adapt $\\alpha _ { t }$ as a function of batch size in (scheduled) large-batch settings. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our proposed AdamWR algorithm represents AdamW given in Algorithm 2 with $\\eta _ { t }$ following Eq. (7) and $w _ { t }$ computed at each iteration using normalized weight decay according to Eq. (5). We note that normalized weight decay allowed us to use a constant parameter setting across short and long runs performed within AdamWR. Equivalently to AdamWR, we define SGDWR as SGDW with warm restarts. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL VALIDATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our experimental setup follows that of Gastaldi (2017), who proposed, in addition to $\\mathrm { L _ { 2 } }$ regularization, to apply the new Shake-Shake regularization to a 3-branch residual neural network. Gastaldi (2017) showed that this regularization allowed to achieve new state-of-the-art results of $2 . 8 6 \\%$ on the CIFAR-10 dataset (Krizhevsky, 2009) and of $1 5 . 8 5 \\%$ on CIFAR-100. The network was trained by SGDR with batch size 128 for 1800 epochs $\\begin{array} { r } { T _ { 0 } = 1 8 0 0 \\mathrm { \\Omega } } \\end{array}$ ) without restarts with the learning rate scheduled by Eq. (6). The regular data augmentation procedure used for the CIFAR datasets was applied. We used the same model/source code based on fb.resnet.torch 1. The base networks are a 26 2x64d ResNet (i.e. the network has a depth of 26, 2 residual branches and the first residual block has a width of 64) and $2 6 2 \\mathrm { x } 9 6 \\mathrm { d }$ ResNet with 11.6M and 25.6M parameters, respectively. For a detailed description of the network and the Shake-Shake method, we refer the interested reader to Gastaldi (2017). ",
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+    {
+        "type": "text",
+        "text": "5.1 DECOUPLING THE WEIGHT DECAY AND INITIAL LEARNING RATE PARAMETERS",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In order to verify our hypothesis about the coupling of the initial learning rate $\\alpha _ { t }$ and the weight decay factor $w _ { t }$ , we trained a 2x64d ResNet with cosine annealing for 100 epochs with different settings of $\\alpha _ { t }$ and $w _ { t }$ . Throughout this paper, we scheduled the learning rate with cosine annealing because it leads to better results than a fixed learning rate (see SuppFigure 1 in the supplementary material). Figure 1 compares SGD vs. SGDW (top row) and Adam vs. AdamW (bottom row). For the case of SGD (Figure 1, top left), weight decay is not decoupled from the learning rate (the common way as described in Algorithm 1), and the figure clearly shows that the basin of best hyperparameter settings (depicted by color and top-10 hyperparameter settings by black circles) is not aligned with the $\\mathbf { X }$ -axis or y-axis but lies on the diagonal. This suggests that the two hyperparameters are interdependent and need to be changed simultaneously, while only changing one of them might substantially worsen results. Consider, e.g., the setting at the top left black circle $( \\alpha _ { t } = 1 / 2$ , $w _ { t } = 1 / 8 * 0 . 0 0 1 )$ ; only changing either $\\alpha _ { t }$ or $w _ { t }$ by itself would worsen results, while changing both of them could still yield clear improvements. We note that this coupling of initial learning rate and weight decay factor might have contributed to SGD’s reputation of being very sensitive to its hyperparameter settings. ",
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+    {
+        "type": "image",
+        "img_path": "images/7ad12dbf6e15c266b2455863a707c3f0bb3f131894189e9132a7f993e1ac46fb.jpg",
+        "image_caption": [
+            "Figure 1: The Top-1 test error of a 26 2x64d ResNet on CIFAR-10 measured after 100 epochs. The proposed SGDW and AdamW (right column) have a more separable hyperparameter space. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "In contrast, the results for our new SGDW in Figure 1 (top right) show that SGDW decouples weight decay and initial learning rate. The proposed approach renders the two hyperparameters more separable: even if the learning rate is not well tuned yet (e.g., consider the value of 1/1024 in Figure 1, top right), leaving it fixed and only optimizing the weight decay factor would yield a good value (of $1 / 4 ^ { * } 0 . 0 0 1$ ). This is not the case for the original SGD shown in Figure 1 (top left). ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "The results for different hyperparameter settings of the original Adam are given in Figure 1 (bottom left). Adam’s best hyperparameter settings performed clearly worse than SGD’s best ones (compare Figure 1, top left). While both methods use the original way to employ weight decay, the original Adam did not benefit from it at all: its best results obtained for non-zero weight decay values were comparable to the best ones obtained without the weight decay regularization, i.e., when $w _ { t } = 0$ . Similarly to the original SGD, the shape of the hyperparameter landscape suggests that the two hyperparameters are coupled. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "In contrast, the results for our new AdamW in Figure 1 (bottom right) show that AdamW largely decouples weight decay and learning rate. The results for the best hyperparameter settings were substantially better than the best ones of the original Adam and rivaled those of SGD and SGDW. ",
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+        "page_idx": 5
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+    {
+        "type": "image",
+        "img_path": "images/52a47381f3c762e6730e56799945e627b42b9e17fe08beb91dbe05ad79f79083.jpg",
+        "image_caption": [
+            "Figure 2: Learning curves (top row) and generalization results (bottom row) obtained by a $2 6 2 \\mathrm { x } 9 6 \\mathrm { d }$ ResNet trained with Adam and AdamW on CIFAR-10. See text for details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "In summary, the experimental results in Figure 1 support our hypothesis that the weight decay and learning rate hyperparameters can be decoupled, and that this in turn simplifies the problem of hyperparameter tuning in SGD and improves Adam’s performance to be competitive w.r.t. SGD with momentum. ",
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+            825,
+            651
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 BETTER GENERALIZATION OF ADAMW ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            678,
+            486,
+            693
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "While the previous experiment suggested that the basin of optimal hyperparameters of AdamW is broader and deeper than the one of Adam, we next investigated the results for much longer runs of 1800 epochs to compare the generalization capabilities of AdamW and Adam. ",
+        "bbox": [
+            176,
+            708,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We fixed the initial learning rate to 0.001 which represents both the default learning rate for Adam and the one which showed reasonably good results in our experiments. Figure 2 shows the results for 12 settings of the weight decay of Adam and 7 settings of the normalized weight decay of AdamW. Interestingly, while the dynamics of the learning curves of Adam and AdamW often coincided for the first half of the training run, AdamW often led to lower training loss and test errors (see Figure 2 top left and top right, respectively). Importantly, the use of weight decay in Adam did not yield as good results as in AdamW (see also Figure 2, bottom left). Next, we investigated whether AdamW’s better results were only due to better convergence or due to better generalization. The results in Figure 2 (bottom right) for the best settings of Adam and AdamW suggest that AdamW did not only yield better training loss but also yielded better generalization performance for similar training loss values. The results on ImageNet32x32 (see SuppFigure 4 in the supplementary material) lead to the same conclusion of substantially improved generalization performance. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/0ff069bf8c97d4f4a604ac3f1be2a4b774410f1171b5b75df88a8d5e5c87160c.jpg",
+        "image_caption": [
+            "Figure 3: Top-1 test error on CIFAR-10 (left) and Top-5 test error on ImageNet32x32 (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            99,
+            825,
+            306
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 EASIER HYPERPARAMETER SELECTION DUE TO NORMALIZED WEIGHT DECAY",
+        "bbox": [
+            174,
+            366,
+            759,
+            380
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our experimental results with Adam and SGD suggested that the total runtime in terms of the number of epochs affect the basin of optimal hyperparameters (see SuppFigure 3 in the supplementary material). More specifically, the greater the total number of epochs the smaller the values of the weight decay should be. SuppFigure 3 shows that our remedy for this problem, the normalized weight decay defined in Eq. (7), simplifies hyperparameter selection because the optimal values observed for short runs are similar to the ones for much longer runs. While our initial experiments on CIFAR-10 suggested the square root fit we proposed in Eq. (7), to double-check that this is not a coincidence, we also performed experiments on the ImageNet32x32 dataset (Chrabaszcz et al., 2017), a downsampled version of the original ImageNet dataset with 1.2 million $3 2 \\times 3 2$ pixels images, where an epoch is 24 times longer than on CIFAR-10. This experiment also supported the square root scaling: the best values of the normalized weight decay observed on CIFAR-10 represented nearly optimal values for ImageNet $3 2 \\mathrm { x } 3 2 $ (see SuppFigure 3). In contrast, had we used the same raw weight decay values $w _ { t }$ for ImageNet32x32 as for CIFAR-10 and for the same number of epochs, without the proposed normalization, $w _ { t }$ would have been roughly 5 greater than optimal for ImageNet32x32, leading to much worse performance. The optimal normalized weight decay values were also very similar (e.g., $w _ { n o r m } = 0 . 0 2 5$ and $w _ { n o r m } = 0 . 0 5$ ) across SGDW and AdamW. ",
+        "bbox": [
+            174,
+            393,
+            825,
+            628
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We investigated whether the use of much longer runs (1800 epochs) of the original Adam with $\\mathrm { L _ { 2 } }$ regularization makes the use of cosine annealing unnecessary. The results of Adam without cosine annealing (i.e., with fixed learning rate) for a 4 by 4 logarithmic grid of hyperparameter settings are given in SuppFigure 5 in the supplementary material. Even after taking into account the low resolution of the grid, the results appear to be at best comparable to the ones obtained with AdamW with 18 times less epochs and a smaller network (see SuppFigure 2). These results are not very surprising given Figure 1 (which demonstrates the effectiveness of AdamW) and SuppFigure 2 (which demonstrates the necessity to use some learning rate schedule such as cosine annealing). ",
+        "bbox": [
+            174,
+            637,
+            825,
+            748
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 ADAMWR WITH WARM RESTARTS FOR BETTER ANYTIME PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            771,
+            722,
+            785
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Finally, we investigated the strong anytime performance AdamWR obtains from warm restarts (using normalized weight decay to avoid the need for a different weight decay factor for restarts with longer annealing schedules). As Figure 3 shows, AdamWR greatly sped up AdamW on CIFAR10 and ImageNet32x32, up to a factor of 10 (see the results at the first restart). For the default learning rate of 0.001, AdamW achieved $15 \\%$ relative improvement in test errors compared to Adam both on CIFAR-10 (also see Figure 2) and ImageNet32x32 (also see SuppFigure 4). AdamWR achieved the same improved results but with a much better anytime performance. These improvements closed most of the gap between Adam and SGDWR on CIFAR-10 and yielded comparable performance on ImageNet32x32. ",
+        "bbox": [
+            174,
+            799,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            468,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Following suggestions that adaptive gradient methods such as Adam might lead to worse generalization than SGD with momentum (Wilson et al., 2017), we identified at least one possible explanation to this phenomenon: the dysfunctional use of $\\mathrm { L _ { 2 } }$ regularization and weight decay. We proposed a simple fix to deal with this issue, yielding substantially better generalization performance in our AdamW variant. We also proposed normalized weight decay and warm restarts for Adam, showing that a more robust hyperparameteer selection and a better anytime performance can be achieved in our new AdamWR variant. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            231
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our preliminary results obtained with AdamW and AdamWR on image classification datasets must be verified on a wider range of tasks, especially the ones where the use of regularization is expected to be important. It would be interesting to integrate our findings on weight decay into other methods which attempt to improve Adam, e.g, normalized direction-preserving Adam (Zhang et al., 2017). While we focussed our experimental analysis on Adam, we believe that similar results also hold for other adaptive gradient methods, such as AdaGrad (Duchi et al., 2011) and RMSProp (Tieleman & Hinton, 2012). ",
+        "bbox": [
+            174,
+            238,
+            825,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The results shown in Figure 2 suggest that Adam and AdamW follow very similar curves most of the time until the third phase of the run where AdamW starts to branch out to outperform Adam. As pointed out by an anonymous reviewer, it would be interesting to investigate what causes this branching and whether the desired effects are observed at the bottom of the landscape. One could investigate this using the approach of Im et al. (2016) to switch from Adam to AdamW at a given epoch index. Since it is quite possible that the effect of regularization is not that pronounced in the early stages of training, one could think of designing a version of Adam which exploits this by being fast in the early stages and well-regularized in the late stages of training. The latter might be achieved with a custom schedule of the weight decay factor. ",
+        "bbox": [
+            173,
+            343,
+            825,
+            468
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we argue that the popular interpretation that weight decay $\\mathbf { \\tau } = \\mathbf { L } _ { 2 }$ regularization is not precise. Instead, the difference between the two leads to the following important consequences. Two algorithms as different as SGD and Adam will exhibit different effective rates of weight decay even if the same regularization coefficient is used to include $\\mathrm { L _ { 2 } }$ regularization in the objective function. Moreover, when decoupled weight decay is applied, two algorithms as different as SGDW and AdamW will optimize two effectively different objective functions even if the same weight decay factor is used. Our findings suggest that the original Adam algorithm with $\\mathrm { L _ { 2 } }$ regularization affects effective rates of weight decay in a way that precludes effective regularization, and that effective regularization is achievable by decoupling the weight decay. ",
+        "bbox": [
+            174,
+            474,
+            825,
+            601
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Advani & Saxe (2017) analytically showed that in the limited data regime of deep networks the presence of eigenvalues that are zero forms a frozen subspace in which no learning occurs and thus smaller (e.g., zero) initial weight norms should be used to achieve best generalization results. Our future work shall consider adapting initial weight norms or weight norm constraints (Salimans & Kingma, 2016) at each warm restart. Kawaguchi et al. (2017) proposed a family of regularization techniques which are specific to the current batch and its size. Similarly to $\\mathrm { L _ { 2 } }$ regularization and weight decay, the latter techniques might be attempted to be transformed to act directly on weights. ",
+        "bbox": [
+            174,
+            607,
+            825,
+            704
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "text": "Leslie N Smith. Cyclical learning rates for training neural networks. arXiv:1506.01186v3, 2016. ",
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+        "text": "Tijmen Tieleman and Geoffrey Hinton. Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural networks for machine learning, 4(2):26– 31, 2012. ",
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+    {
+        "type": "text",
+        "text": "Kelvin Xu, Jimmy Ba, Ryan Kiros, Kyunghyun Cho, Aaron Courville, Ruslan Salakhudinov, Rich Zemel, and Yoshua Bengio. Show, attend and tell: Neural image caption generation with visual attention. In International Conference on Machine Learning, pp. 2048–2057, 2015. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "1 SUPPLEMENTARY MATERIAL ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            102,
+            446,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "1.1 AN EXAMPLE SETTING OF THE SCHEDULE MULTIPLIER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            593,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "An example schedule of the schedule multiplier $\\eta _ { t }$ is given in SuppFigure 1 for $T _ { i = 0 } = 1 0 0$ and $T _ { m u l t } = 2$ . After the initial 100 epochs the learning rate will reach 0 because $\\eta _ { t = 1 0 0 } = 0$ . Then, since $T _ { c u r } = T _ { i = 0 }$ , we restart by resetting $T _ { c u r } = 0$ , causing the multiplier $\\eta _ { t }$ to be reset to 1 due to Eq. (7). This multiplier will then decrease again from 1 to 0, but now over the course of 200 epochs because $T _ { i = 1 } = T _ { i = 0 } T _ { m u l t } = 2 0 0$ . Solutions obtained right before the restarts, when $\\eta _ { t } = 0$ (e.g., at epoch indexes 100, 300, 700 and 1500 as shown in SuppFigure 1) are recommended by the optimizer as the solutions, with more recent solutions prioritized. ",
+        "bbox": [
+            173,
+            159,
+            826,
+            257
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/e1136b7a9c1678087dff60655ce9bbb521fd87b5f4b6da3cc91d9b66fe540e22.jpg",
+        "image_caption": [
+            "SuppFigure 1: An example schedule of the learning rate multiplier as a function of epoch index. The first run is scheduled to converge at epoch $T _ { i = 0 } = 1 0 0$ , then the budget for the next run is doubled as $T _ { i = 1 } = T _ { i = 0 } T _ { m u l t } = 2 0 0$ , etc. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            341,
+            282,
+            650,
+            467
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/17833324a93093bca68da92e2e53961c517952fd5102b4cf95bfdcb2b6964358.jpg",
+        "image_caption": [
+            "SuppFigure 2: Adam with fixed learning rate (left) and with cosine annealing (right). We show the final test error of a $2 6 ~ 2 \\mathrm { x } 6 4 \\mathrm { d }$ ResNet on CIFAR-10 after 100 epochs of SGD with momentum. The results where the learning rate is fixed (left) are inferior to the ones where the learning rate is scheduled according to cosine annealing (right). Therefore, we schedule the learning rate with cosine annealing for all methods given in the paper. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            560,
+            828,
+            825
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/5f160ef2fc5e0bb5f0bd015a3408463f8f327bfc9a812fbf805295bea1d5d435.jpg",
+        "image_caption": [
+            "SuppFigure 3: Effect of normalized weight decay. We show the final test Top-1 error on CIFAR10 (first two rows for AdamW without and with normalized weight decay) and Top-5 error on ImageNet32x32 (last two rows for AdamW and SGDW, both with normalized weight decay) of a $2 6 2 \\mathrm { x } 6 4 \\mathrm { d }$ ResNet after different numbers of epochs (see columns). While the optimal settings of the raw weight decay change significantly for different runtime budgets (see the first row), the values of the normalized weight decay remain very similar for different budgets (see the second row) and different datasets (here, CIFAR-10 and ImageNet32x32), and even across AdamW and SGDW. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            94,
+            828,
+            818
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/34a75388019587badb9d17928eefc833ecc3e6db8aa0e1a91ecb741aea7e654e.jpg",
+        "image_caption": [
+            "SuppFigure 4: Learning curves (top row) and generalization results (Top-5 errors in bottom row) obtained by a $2 6 2 \\mathrm { x } 9 6 \\mathrm { d }$ ResNet trained with Adam and AdamW on ImageNet32x32. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            282,
+            828,
+            696
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/b8803b776bd6799e677439534292c7be94ce223d9f24b87e94e074f7d33cd8a5.jpg",
+        "image_caption": [
+            "SuppFigure 5: Adam without cosine annealing, i.e., with fixed learning rate. We show the final test error of a $2 6 ~ 2 \\mathrm { x } 9 6 \\mathrm { d }$ ResNet on CIFAR-10 after 1800 epochs of the original Adam for different settings of learning rate and weight decay used for $\\mathrm { L _ { 2 } }$ regularization. These results can be compared to the ones of AdamW shown in SuppFigure 3 (top row). The results of AdamW with only 100 epochs and a smaller network seem to be at least as good as the ones of Adam with 18 times as many epochs and a bigger network. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            300,
+            304,
+            699,
+            617
+        ],
+        "page_idx": 13
+    }
+]

parse/train/x2TMPhseWAW/x2TMPhseWAW.md

@@ -0,0 +1,427 @@
+# Label Noise SGD Provably Prefers Flat Global Minimizers
+
+Alex Damian Princeton University ad27@princeton.edu
+
+Tengyu Ma Stanford University tengyuma@stanford.edu
+
+Jason Lee Princeton University jasonlee@princeton.edu
+
+# Abstract
+
+In overparametrized models, the noise in stochastic gradient descent (SGD) implicitly regularizes the optimization trajectory and determines which local minimum SGD converges to. Motivated by empirical studies that demonstrate that training with noisy labels improves generalization, we study the implicit regularization effect of SGD with label noise. We show that SGD with label noise converges to a stationary point of a regularized loss $L ( \theta ) + \lambda R ( \theta )$ , where $L ( \theta )$ is the training loss, $\lambda$ is an effective regularization parameter depending on the step size, strength of the label noise, and the batch size, and $R ( \theta )$ is an explicit regularizer that penalizes sharp minimizers. Our analysis uncovers an additional regularization effect of large learning rates beyond the linear scaling rule that penalizes large eigenvalues of the Hessian more than small ones. We also prove extensions to classification with general loss functions, significantly strengthening the prior work of Blanc et al. [3] to global convergence and large learning rates and of HaoChen et al. [12] to general models.
+
+# 1 Introduction
+
+One of the central questions in modern machine learning theory is the generalization capability of overparametrized models trained by stochastic gradient descent (SGD). Recent work identifies the implicit regularization effect due to the optimization algorithm as one key factor in explaining the generalization of overparameterized models [27, 11, 19, 10]. This implicit regularization is controlled by many properties of the optimization algorithm including search direction [11], learning rate [20], batch size [26], momentum [21] and dropout [22].
+
+The parameter-dependent noise distribution in SGD is a crucial source of regularization [16, 18]. Blanc et al. [3] initiated the study of the regularization effect of label noise SGD with square loss1 by characterizing the local stability of global minimizers of the training loss. By identifying a data-dependent regularizer $R ( \theta )$ , Blanc et al. [3] proved that label noise SGD locally diverges from the global minimizer $\theta ^ { * }$ if and only if $\theta ^ { * }$ is not a first-order stationary point of minθ $R ( \theta )$ subject to $\bar { \cal L ( \theta ) } = 0$ . The analysis is only able to demonstrate that with sufficiently small step size $\eta$ , label noise SGD initialized at $\theta ^ { * }$ locally diverges by a distance of $\eta ^ { 0 . 4 }$ and correspondingly decreases the regularizer by $\eta ^ { 0 . 4 }$ . This is among the first results that establish that the noise distribution alters the local stability of stochastic gradient descent. However, the parameter movement of $\eta ^ { 0 . 4 }$ is required to be inversely polynomially small in dimension and condition number and is thus too small to affect the predictions of the model.
+
+HaoChen et al. [12], motivated by the local nature of Blanc et al. [3], analyzed label noise SGD in the quadratically-parametrized linear regression model [29, 32, 23]. Under a well-specified sparse linear regression model and with isotropic features, HaoChen et al. [12] proved that label noise SGD recovers the sparse ground-truth despite overparametrization, which demonstrated a global implicit bias towards sparsity in the quadratically-parametrized linear regression model.
+
+This work seeks to identify the global implicit regularization effect of label noise SGD. Our primary result, which supports Blanc et al. [3], proves that label noise SGD converges to a stationary point of $L ( \theta ) + \lambda R ( \theta )$ , where the regularizer $R ( \theta )$ penalizes sharp regions of the loss landscape.
+
+The focus of this paper is on label noise SGD due to its strong regularization effects in both real and synthetic experiments [25, 28, 31]. Furthermore, label noise is used in large-batch training as an additional regularizer [25] when the regularization from standard regularizers (e.g. mini-batch, batch-norm, and dropout) is not sufficient. Label noise SGD is also known to be less sensitive to initialization, as shown in HaoChen et al. [12]. In stark contrast, mini-batch SGD remains stuck when initialized at any poor global minimizer. Our analysis demonstrates a global regularization effect of label noise SGD by proving it converges to a stationary point of a regularized loss $L ( \theta ) + \lambda R ( \theta )$ , even when initialized at a zero error global minimum.
+
+The learning rate and minibatch size in SGD are known to be important sources of regularization [9]. Our main theorem highlights the importance of learning rate and batch size as the hyperparameters that control the balance between the loss and the regularizer – larger learning rates and smaller batch sizes lead to stronger regularization.
+
+Section 2 reviews the notation and assumptions used throughout the paper. Section 2.4 formally states the main result and Section 3 sketches the proof. Section 4 presents experimental results which support our theory. Finally, Section 6 discusses the implications of this work.
+
+# 2 Problem Setup and Main Result
+
+Section 2.1 describes our notation and the SGD with label noise algorithm. Section 2.2 introduces the explicit formula for the regularizer $R ( \theta )$ . Sections 2.3 and 2.4 formally state our main result.
+
+# 2.1 Notation
+
+We focus on the regression setting (see Appendix $\mathrm { E }$ for the extension to the classification setting). Let $\{ ( x _ { i } , y _ { i } ) \} _ { i \in [ n ] }$ be $n$ datapoints with $x _ { i } \in \mathcal { D }$ and $y _ { i } \in \mathbb { R }$ . Let $f : \mathcal { D } \times \mathbb { R } ^ { d } \to \mathbb { R }$ and let $f _ { i } ( \theta ) = f ( x _ { i } , \theta )$ denote the value of $f$ on the datapoint $x _ { i }$ . Define $\begin{array} { r } { \ell _ { i } ( \theta ) = \frac { 1 } { 2 } \left( f _ { i } ( \theta ) - y _ { i } \right) ^ { 2 } } \end{array}$ and $\begin{array} { r } { L ( \theta ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell _ { i } ( \theta ) } \end{array}$ Then we will follow Algorithm 1 which adds fresh additive noise to the labels $y _ { i }$ at every step before computing the gradient:
+
+# Algorithm 1: SGD with Label Noise
+
+<table><tr><td>Input: 0o,step size n, noise variance g²,batch size B,steps T fork=0toT-1do</td></tr><tr><td>Sample batch B(k) C[n}B uniformly and label noise e(𝑘）~{-σ,σ} for i ∈ B(𝑘).</td></tr><tr><td>Lete((0)=2(() -y - () and L(k)= B∑i∈B(k) ）.</td></tr><tr><td></td></tr><tr><td>0k+1←0k-n∀L(k)(0k)</td></tr><tr><td>end</td></tr></table>
+
+Note that $\sigma$ controls the strength of the label noise and will control the strength of the implicit regularization in Theorem 1. Throughout the paper we will use $\| \cdot \| = \| \cdot \| _ { 2 }$ . We make the following standard assumption on $f$ :
+
+Assumption 1 (Smoothness). We assume that each $f _ { i }$ is $\ell _ { f }$ -Lipschitz, $\nabla f _ { i }$ is $\rho _ { f }$ -Lipschitz, and $\nabla ^ { 2 } f _ { i }$ is $\kappa _ { f }$ -Lipschitz with respect to $\parallel \cdot \parallel _ { 2 } f o r i = 1 , \ldots , n$ .
+
+We will define $\ell = \ell _ { f } ^ { 2 }$ to be an upper bound on $\begin{array} { r } { \| \frac { 1 } { n } \sum _ { i } \nabla f _ { i } ( \theta ) \nabla f _ { i } ( \theta ) ^ { T } \| _ { 2 } } \end{array}$ , which is equal to $\| \nabla ^ { 2 } L ( \theta ) \| _ { 2 }$ at any global minimizer $\theta$ . Our results extend to any learning rate $\eta \in ( 0 , \frac { 2 } { \ell } )$ . However, they do not extend to the limit as $\begin{array} { r } { \eta \to \frac { 2 } { \ell } } \end{array}$ . Because we still want to track the dependence on $\frac { 1 } { \eta }$ , we do not assume $\eta$ is a fixed constant and instead assume some constant separation:
+
+Assumption 2 (Learning Rate Separation). There exists a constant $\nu \in ( 0 , 1 )$ such that $\begin{array} { r } { \eta \le \frac { 2 - \nu } { \ell } } \end{array}$
+
+In addition, we make the following local Kurdyka-Łojasiewicz assumption (KL assumption) which ensures that there are no regions where the loss is very flat. The KL assumption is very general and holds for some $\delta > 0$ for any analytic function defined on a compact domain (see Lemma 17).
+
+Assumption 3 (KL). Let $\theta ^ { * }$ be any global minimizer of $L$ . Then there exist $\epsilon _ { K L } > 0 , \mu > 0$ and $0 < \delta \le 1 / 2$ such that if $L ( \theta ) - L ( \theta ^ { * } ) \leq \epsilon _ { K L } ,$ , then $L ( \theta ) - L ( \theta ^ { * } ) \leq \mu \| \nabla L ( \theta ) \| ^ { 1 + \delta }$ .
+
+We assume $L ( \theta ^ { * } ) = 0$ for any global minimizer $\theta ^ { * }$ . Note that if $L$ satisfies Assumption 3 for some $\delta$ then it also satisfies Assumption 3 for any $\delta ^ { \prime } < \delta$ . Assumption 3 with $\delta = 1$ is equivalent to the much stronger Polyak-Łojasiewicz condition which is equivalent to local strong convexity.
+
+We will use $O , \Theta , \Omega$ to hide any polynomial dependence on $\mu , \ell _ { f } , \rho _ { f } , \kappa _ { f } , \nu , 1 / \sigma , n , d$ and $\tilde { O }$ to hide additional polynomial dependence on $\log { 1 / \eta } , \log { B }$ .
+
+# 2.2 The Implicit Regularizer $R ( \theta )$
+
+For $L , \sigma ^ { 2 } , B , \eta$ as defined above, we define the implicit regularizer $R ( \theta )$ , the effective regularization parameter $\lambda$ , and the regularized loss $\tilde { L } ( \theta )$ :
+
+$$
+R ( \theta ) = - \frac { 1 } { 2 \eta } \mathrm { t r } \log \left( 1 - \frac { \eta } { 2 } \nabla ^ { 2 } L ( \theta ) \right) , \qquad \lambda = \frac { \eta \sigma ^ { 2 } } { B } , \qquad \tilde { L } ( \theta ) = L ( \theta ) + \lambda R ( \theta ) .
+$$
+
+Here log refers to the matrix logarithm. To better understand the regularizer $R ( \theta )$ , let $\lambda _ { 1 } , \ldots , \lambda _ { d }$ be the eigenvalues of $\nabla ^ { 2 } L ( \theta )$ and let $\begin{array} { r } { R ( \lambda _ { i } ) = - \frac { 1 } { 2 \eta } \log ( 1 - \frac { \eta \lambda _ { i } } { 2 } ) } \end{array}$ . Then,
+
+$$
+R ( \theta ) = \sum _ { i = 1 } ^ { d } R ( \lambda _ { i } ) = \sum _ { i = 1 } ^ { d } \left( \frac { \lambda _ { i } } { 4 } + \frac { \eta \lambda _ { i } ^ { 2 } } { 1 6 } + \frac { \eta ^ { 2 } \lambda _ { i } ^ { 3 } } { 4 8 } + . . . \right) .
+$$
+
+In the limit as $\eta  0$ , $R ( \theta ) \to { \textstyle { \frac { 1 } { 4 } } } \mathrm { t r } \nabla ^ { 2 } L ( \theta )$ , which matches the regularizer in Blanc et al. [3] for infinitesimal learning rate near a global minimizer. However, in additional to the linear scaling rule, which is implicit in our definition of $\lambda$ , our analysis uncovers an additional regularization effect of large learning rates that penalizes larger eigenvalues more than smaller ones (see Figure 1 and Section 6.1).
+
+The goal of this paper is to show that Algorithm 1 converges to a stationary point of the regularized loss $\tilde { L } = L + \lambda R$ . In particular, we will show convergence to an $( \epsilon , \gamma )$ -stationary point, which is defined in the next section.
+
+![](images/533db9b5302ff86c0b05b716f9e5d4b521eb2b08072bddaf7d6c34dfb76ab5bb.jpg)  
+Figure 1: Regularization strength as a function of $\eta$
+
+# 2.3 $( \epsilon , \gamma )$ -Stationary Points
+
+We begin with the standard definition of an approximate stationary point:
+
+Definition 1 ( $\epsilon$ -stationary point). $\theta$ is an $\epsilon$ -stationary point of $f i f \| \nabla f ( \theta ) \| \leq \epsilon .$
+
+In stochastic gradient descent it is often necessary to allow λ = ησ2B to scale with $\epsilon$ to reach an $\epsilon$ -stationary point [8, 15] (e.g., $\lambda$ may need to be less than $\epsilon ^ { 2 }$ ). However, for $\lambda = { \cal O } ( \epsilon )$ , any local minimizer $\theta ^ { * }$ is an $\epsilon \cdot$ -stationary point of $\tilde { L } = L + \lambda R$ . Therefore, reaching a $\epsilon$ -stationary point of $\tilde { L }$ would be equivalent to finding a local minimizer and would not be evidence for implicit regularization. To address this scaling issue, we consider the rescaled regularized loss:
+
+$$
+\frac { 1 } { \lambda } \tilde { L } = \frac { 1 } { \lambda } L + R .
+$$
+
+![](images/8ed418606a90c8d4ef26d37a8d70c17e8997842fd1e1d9b212259490983abca0.jpg)  
+Figure 2: Local Coupling: We decompose $\theta$ as the sum of a regularized trajectory $\Phi _ { \tau _ { 1 } } ( \theta _ { 0 } ^ { * } )$ , a mean zero oscillating process $\xi _ { \tau _ { 1 } }$ , and an error term $\Delta _ { 1 }$ . Global Convergence: We repeat the coupling with a sequence of reference points $\{ \theta _ { m } ^ { * } \} _ { m }$ to prove convergence to a stationary point of $\textstyle { \frac { 1 } { \lambda } } { \tilde { L } }$ .
+
+Reaching an $\epsilon$ -stationary point of $\textstyle { \frac { 1 } { \lambda } } { \tilde { L } }$ requires non-trivially taking the regularizer $R$ into account. However, it is not possible for Algorithm 1 to reach an $\epsilon$ -stationary point of $\textstyle { \frac { 1 } { \lambda } } { \tilde { L } }$ even in the ideal setting when $\theta$ is initialized near a global minimizer $\theta ^ { * }$ of $\tilde { L }$ . The label noise will cause fluctuations of order $\sqrt { \lambda }$ around $\theta ^ { * }$ (see section 3) so $\Vert \nabla L \Vert$ will remain around $\sqrt { \lambda }$ . This causes $\scriptstyle { \frac { 1 } { \lambda } } \nabla L$ to become unbounded for $\lambda$ (and therefore $\epsilon$ ) sufficiently small, and thus Algorithm 1 cannot converge to an $\epsilon$ -stationary point. We therefore prove convergence to an $( \epsilon , \gamma )$ -stationary point:
+
+Definition 2 $( ( \epsilon , \gamma )$ -stationary point). $\theta$ is an $( \epsilon , \gamma )$ -stationary point of $f$ if there exists some $\theta ^ { * }$ such that $\| \nabla f ( \theta ^ { * } ) \| \le \epsilon$ and $\lVert \theta - \theta ^ { * } \rVert \leq \gamma$ .
+
+Intuitively, Algorithm 1 converges to an $( \epsilon , \gamma )$ -stationary point when it converges to a neighborhood of some $\epsilon$ -stationary point $\theta ^ { * }$ .
+
+# 2.4 Main Result
+
+Having defined an $( \epsilon , \gamma )$ -stationary point we can now state our main result:
+
+Theorem 1. Assume that $f$ satisfies Assumption $I , ~ \eta$ satisfies Assumption 2, and $L$ satisfies Assumption $3$ , i.e. $L ( \theta ) \ \overset { \cdot } { \leq } \ \mu \Vert \dot { \nabla } L ( \theta ) \Vert ^ { 1 + \delta }$ for $L ( \theta ) ~ \le ~ \epsilon _ { K L }$ . Let $\eta , B$ be chosen such that $\begin{array} { r } { \lambda : = \frac { \eta \sigma ^ { 2 } } { B } = \tilde { \Theta } ( \operatorname* { m i n } ( \epsilon ^ { 2 / \delta } , \gamma ^ { 2 } ) ) } \end{array}$ , and let $T = \tilde { \Theta } ( \eta ^ { - 1 } \lambda ^ { - 1 - \delta } ) = \mathrm { p o l y } ( \eta ^ { - 1 } , \gamma ^ { - 1 } )$ . Assume that $\theta$ is initialized within $O ( \sqrt { \lambda ^ { 1 + \delta } } )$ of some $\theta ^ { * }$ satisfying $L ( \theta ^ { * } ) = O ( \lambda ^ { 1 + \delta } )$ . Then for any $\zeta \in ( 0 , 1 )$ , with probability at least $1 - \zeta$ , if $\{ \theta _ { k } \}$ follows Algorithm $^ { l }$ with parameters $\eta , \sigma , T ,$ , there exists $k < T$ such that $\theta _ { k }$ is an $( \epsilon , \gamma )$ -stationary point of $\textstyle { \frac { 1 } { \lambda } } { \tilde { L } }$ .
+
+Theorem 1 guarantees that Algorithm 1 will hit an $( \epsilon , \gamma )$ -stationary point of $\textstyle { \frac { 1 } { \lambda } } { \tilde { L } }$ within a polynomial number of steps in $\epsilon ^ { - 1 } , \gamma ^ { - 1 }$ . In particular, when $\begin{array} { r } { \delta = \frac { 1 } { 2 } } \end{array}$ , Theorem 1 guarantees convergence within ${ \tilde { O } } ( \epsilon ^ { - 6 } + \gamma ^ { - 3 } )$ steps. The condition that $\theta _ { 0 }$ is close to an approximate global minimizer $\theta ^ { * }$ is not a strong assumption as recent methods have shown that overparameterized models can easily achieve zero training loss in the kernel regime (see Appendix C). However, in practice these minimizers of the training loss generalize poorly [1]. Theorem 1 shows that Algorithm 1 can then converge to a stationary point of the regularized loss which has better generalization guarantees (see Section 6.2). Theorem 1 also generalizes the local analysis in Blanc et al. [3] to a global result with weaker assumptions on the learning rate $\eta$ . For a full comparison with Blanc et al. [3], see section 3.1.
+
+# 3 Proof Sketch
+
+The proof of convergence to an $( \epsilon , \varphi )$ -stationary point of $\textstyle { \frac { 1 } { \lambda } } { \tilde { L } }$ has two components. In Section 3.1, we pick a reference point $\theta ^ { * }$ and analyze the behavior of Algorithm 1 in a neighborhood of $\theta ^ { * }$ . In Section 3.2, we repeat this local analysis with a sequence of reference points $\{ \bar { \theta } _ { m } ^ { * } \}$ .
+
+# 3.1 Local Coupling
+
+Let $\Phi _ { k } ( \cdot )$ denote $k$ steps of gradient descent on the regularized loss $\tilde { L }$ , i.e.
+
+$$
+\Phi _ { 0 } ( \theta ) = \theta \qquad \mathrm { a n d } \qquad \Phi _ { k + 1 } ( \theta ) = \Phi _ { k } ( \theta ) - \eta \nabla \tilde { L } ( \Phi _ { k } ( \theta ) ) ,
+$$
+
+where $\tilde { L } ( \theta ) = L ( \theta ) + \lambda R ( \theta )$ is the regularized loss defined in Equation (1). Lemma 1 states that if $\theta$ is initialized at an approximate global minimizer $\theta ^ { * }$ and follows Algorithm 1, there is a small mean zero random process $\xi$ such that $\theta _ { k } \approx \Phi _ { k } ( \theta ^ { * } ) + \xi _ { k }$ :
+
+# Lemma 1. Let
+
+$$
+\iota = c \log \frac { d } { \lambda \zeta } , \quad \mathcal { X } = \sqrt { \frac { 2 \lambda n d \iota } { \nu } } , \quad \mathcal { L } = c \lambda ^ { 1 + \delta } , \quad \mathcal { D } = c \sqrt { \mathcal { L } } \iota , \quad \mathcal { M } = \frac { \mathcal { D } } { \nu } , \quad \mathcal { T } = \frac { 1 } { c ^ { 2 } \eta \mathcal { X } \iota } ,
+$$
+
+where c is a sufficiently large constant. Assume $f$ satisfies Assumption $I$ and $\eta$ satisfies Assumption 2. Let θ follow Algorithm $^ { l }$ starting at $\theta ^ { * }$ and assume that $L ( \theta ^ { * } ) \leq \mathcal { L }$ for some $0 < \delta \le 1 / 2$ . Then there exists a random process $\{ \xi _ { k } \}$ such that for any $\tau \leq \mathcal { T }$ satisfying $\begin{array} { r } { \operatorname* { m a x } _ { k \leq \tau } \| \Phi _ { k } ( \theta ^ { * } ) - \theta ^ { * } \| \leq 8 \mathcal { M } , } \end{array}$ , with probability at least $1 - 1 0 d \tau e ^ { - \iota }$ we have simultaneously for all $k \leq \tau$ ,
+
+$$
+\begin{array} { r } { \| \theta _ { k } - \xi _ { k } - \Phi _ { k } ( \theta ^ { * } ) \| \le \mathcal { D } , \qquad \mathbb { E } [ \xi _ { k } ] = 0 , \qquad a n d \qquad \| \xi _ { k } \| \le \mathcal { X } . } \end{array}
+$$
+
+Note that because $\mathcal { M } \geq \mathcal { D }$ , the error term $\mathcal { D }$ is at least 8 times smaller than the movement in the direction of the regularized trajectory $\Phi _ { \tau } ( \theta ^ { * } )$ , which will allow us to prove convergence to an $( \epsilon , \gamma )$ -stationary point of $\textstyle { \frac { 1 } { \lambda } } { \tilde { L } }$ in Section 3.2.
+
+Toward simplifying the update in Algorithm 1, we define $L ^ { ( k ) }$ to be the true loss without label noise on batch $B ^ { ( k ) }$ . The label-noise update $\hat { L } ^ { ( k ) } ( \theta _ { k } )$ is an unbiased perturbation of the mini-batch update: $\begin{array} { r } { \nabla \hat { L } ^ { ( k ) } ( \theta _ { k } ) = \nabla L ^ { ( k ) } ( \theta _ { k } ) - \frac { 1 } { B } \sum _ { i \in \mathcal { B } ^ { ( k ) } } \epsilon _ { i } ^ { ( k ) } \nabla f _ { i } ( \theta _ { k } ) } \end{array}$ (k)i ∇fi(θk). We decompose the update rule into three parts:
+
+$$
+\theta _ { k + 1 } = \theta _ { k } - \underbrace { \eta \nabla L ( \theta _ { k } ) } _ { \mathrm { g r a d i e n t \ d e s c e n t } } - \underbrace { \eta [ \nabla L ^ { ( k ) } ( \theta _ { k } ) - \nabla L ( \theta _ { k } ) ] } _ { \mathrm { m i n i b a t e h \ n o i s e } } + \underbrace { \eta \sum _ { i } \epsilon _ { i } ^ { ( k ) } \nabla f _ { i } ( \theta _ { k } ) } _ { \substack { \mathrm { i } \in \mathcal { B } ^ { ( k ) } } } .
+$$
+
+Let $m _ { k } = - \eta [ \nabla L ^ { ( k ) } ( \theta _ { k } ) - \nabla L ( \theta _ { k } ) ]$ denote the minibatch noise. Throughout the proof we will show that the minibatch noise is dominated by the label noise. We will also decompose the label noise into two terms. The first, $\epsilon _ { k } ^ { * }$ , will represent the label noise if the gradient were evaluated at $\theta ^ { * }$ whose distribution does not vary with $k$ . The other term, $z _ { k }$ represents the change in the noise due to evaluating the gradient at $\theta _ { k }$ rather than $\theta ^ { * }$ . More precisely, we have
+
+$$
+\epsilon _ { k } ^ { * } = \frac { \eta } { B } \sum _ { i \in \mathcal { B } ^ { ( k ) } } \epsilon _ { i } ^ { ( k ) } \nabla f _ { i } ( \theta ^ { * } ) \qquad \mathrm { a n d } \qquad z _ { k } = \frac { \eta } { B } \sum _ { i \in \mathcal { B } ^ { ( k ) } } \epsilon _ { i } ^ { ( k ) } [ \nabla f _ { i } ( \theta _ { k } ) - \nabla f _ { i } ( \theta ^ { * } ) ] .
+$$
+
+We define $\begin{array} { r } { G ( \theta ) = \frac { 1 } { n } \sum _ { i } \nabla f _ { i } ( \theta ) \nabla f _ { i } ( \theta ) ^ { T } } \end{array}$ to be the covariance of the model gradients. Note that $\epsilon _ { k } ^ { * }$ has covariance $\eta \lambda \dot { G } ( \theta ^ { \ast } )$ . To simplify notation in the Taylor expansions, we will use the following shorthand to refer to various quantities evaluated at $\theta ^ { * }$ :
+
+$$
+G = G ( \theta ^ { * } ) , \qquad \nabla ^ { 2 } L = \nabla ^ { 2 } L ( \theta ^ { * } ) , \qquad \nabla ^ { 3 } L = \nabla ^ { 3 } L ( \theta ^ { * } ) , \qquad \nabla R = \nabla R ( \theta ^ { * } ) .
+$$
+
+First we need the following standard decompositions of the Hessian:
+
+Proposition 1. For any $\theta \in \mathbb { R } ^ { d }$ we can decompose $\nabla ^ { 2 } L ( \theta ) = G ( \theta ) + E ( \theta )$ where $E ( \theta ) =$ $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f _ { i } ( \theta ) - y _ { i } ) \nabla ^ { 2 } f _ { i } ( \theta ) } \end{array}$ satisfies $\lVert E ( { \boldsymbol { \theta } } ) \rVert \leq \sqrt { 2 \rho _ { f } L ( { \boldsymbol { \theta } } ) }$ where $\rho _ { f }$ is defined in Assumption $^ { l }$ .
+
+The matrix $G$ in Proposition 1 is known as the Gauss-Newton term of the Hessian. We can now Taylor expand Algorithm 1 and Equation (2) to first order around $\theta ^ { * }$ :
+
+$$
+\begin{array} { c } { { \Phi _ { k + 1 } \bigl ( \theta ^ { * } \bigr ) \approx \Phi _ { k } \bigl ( \theta ^ { * } \bigr ) - \eta \bigl [ \nabla L + \nabla ^ { 2 } L \bigl ( \Phi _ { k } \bigl ( \theta ^ { * } \bigr ) - \theta ^ { * } \bigr ) \bigr ] , } } \\ { { \theta _ { k + 1 } \approx \theta _ { k } - \eta \bigl [ \nabla L + \nabla ^ { 2 } L \bigl ( \theta _ { k } - \theta ^ { * } \bigr ) \bigr ] + \epsilon _ { k } ^ { * } . } } \end{array}
+$$
+
+We define $v _ { k } = \theta _ { k } - \Phi _ { k } ( \theta ^ { * } )$ to be the deviation from the regularized trajectory. Then subtracting these two equations gives
+
+$$
+v _ { k + 1 } \approx ( I - \eta \nabla ^ { 2 } L ) v _ { k } + \epsilon _ { k } ^ { * } \approx ( I - \eta G ) v _ { k } + \epsilon _ { k } ^ { * } ,
+$$
+
+where we used Proposition 1 to replace $\nabla ^ { 2 } L$ with $G$ . Temporarily ignoring the higher order terms, we define the random process $\xi$ by
+
+$$
+\xi _ { k + 1 } = ( I - \eta G ) \xi _ { k } + \epsilon _ { k } ^ { * } \qquad \mathrm { a n d } \qquad \xi _ { 0 } = 0 .
+$$
+
+The process $\xi$ is referred to as an Ornstein Uhlenbeck process and it encodes the movement of $\theta$ to first order around $\theta ^ { * }$ . We defer the proofs of the following properties of $\xi$ to Appendix B:
+
+Proposition 2. For any $k \geq 0$ , with probability at least $1 - 2 d e ^ { - \iota }$ , $\| \xi _ { k } \| \le \mathcal { X }$ . In addition, as $k \to \infty$ , $\mathbb { E } [ \xi _ { k } \xi _ { k } ^ { T } ]  \lambda \dot { \Pi _ { G } } ( 2 - \eta G ) ^ { - 1 }$ where $\Pi _ { G }$ is the projection onto the span of $G$ .
+
+We can now analyze the effect of $\xi _ { k }$ on the second order Taylor expansion. Let $r _ { k } = \theta _ { k } - \Phi _ { k } ( \theta ^ { * } ) - \xi _ { k }$ be the deviation of $\theta$ from the regularized trajectory after removing the Ornstein Uhlenbeck process $\xi$ . Lemma 1 is equivalent to $\mathrm { P r } [ \| r _ { \tau } \| \geq \mathcal { D } ] \stackrel { . } { \leq } 1 0 \tau \dot { d } e ^ { - \iota }$ .
+
+We will prove by induction that $\| r _ { k } \| \le \mathcal { D }$ for all $k \leq t$ with probability at least $1 - 1 0 t d e ^ { - \iota }$ for all $t \leq \tau$ . The base case follows from $r _ { 0 } = 0$ so assume the result for some $t \geq 0$ . The remainder of this section will be conditioned on the event $\| r _ { k } \| \le \mathcal { D }$ for all $k \leq t . { \cal O } ( \cdot )$ notation will only be used to hide absolute constants that do not change with $t$ and will additionally not hide dependence on the absolute constant $c$ . The following proposition fills in the missing second order terms in the Taylor expansion around $\theta ^ { * }$ of $r _ { k }$ :
+
+Proposition 3. With probability at least $1 - 2 d e ^ { - \iota }$ ,
+
+$$
+r _ { k + 1 } = ( I - \eta G ) r _ { k } - \eta \left[ \frac { 1 } { 2 } \nabla ^ { 3 } L ( \xi _ { k } , \xi _ { k } ) - \lambda \nabla R \right] + m _ { k } + z _ { k } + \tilde { O } \left( c ^ { 5 / 2 } \eta \lambda ^ { 1 + \delta } \right)
+$$
+
+The intuition for the implicit regularizer $R ( \theta )$ is that by Propositions 1 and 2,
+
+$$
+\begin{array} { r } { \mathbb { E } [ \xi _ { k } \xi _ { k } ^ { T } ] \to \Pi _ { G } \lambda ( 2 - \eta G ) ^ { - 1 } \approx \lambda ( 2 - \eta \nabla ^ { 2 } L ) ^ { - 1 } . } \end{array}
+$$
+
+Therefore, when averaged over long timescales,
+
+$$
+\operatorname { \mathbb { E } } [ \nabla ^ { 3 } L ( \xi _ { k } , \xi _ { k } ) ] \approx \frac { \lambda } { 2 } \nabla ^ { 3 } L \left[ ( 2 - \eta \nabla ^ { 2 } L ) ^ { - 1 } \right] = \lambda \nabla \left[ - \frac { 1 } { 2 \eta } \operatorname { t r } \log \left( 1 - \frac { \eta } { 2 } \nabla ^ { 2 } L ( \theta ) \right) \right] \bigg | _ { \theta = \theta ^ { * } } = \lambda \nabla R .
+$$
+
+The second equality follows from the more general equality that for any matrix function $A$ and any scalar function $h$ that acts independently on each eigenvalue, $\nabla ( \mathrm { t r } h ( A ( \theta ) ) ) = ( \nabla A ( \theta ) ) ( h ^ { \prime } ( A ( \theta ) ) )$ which follows from the chain rule. The above equality is the special case when $A ( \theta ) = \nabla ^ { 2 } L ( \theta )$ and $\begin{array} { r } { h ( x ) = - \frac { 1 } { \eta } \log { \left( 1 - \frac { \eta } { 2 } x \right) } } \end{array}$ , which satisfies $\begin{array} { r } { h ^ { \prime } ( x ) = \frac { 1 } { 2 - \eta x } } \end{array}$ .
+
+The remaining details involve concentrating the mean zero error terms $m _ { k } , z _ { k }$ and showing that $\mathbb { E } [ \xi _ { k } \xi _ { k } ^ { T } ]$ does concentrate in the directions with large eigenvalues and that the directions with small eigenvalues, in which the covariance does not concentrate, do not contribute much to the error. This yields the following bound:
+
+Proposition 4. With probability at least $1 - 1 0 d e ^ { - \iota }$ , $\begin{array} { r } { \| r _ { t + 1 } \| = \tilde { O } \Big ( \frac { \lambda ^ { 1 / 2 + \delta / 2 } } { \sqrt { c } } \Big ) . } \end{array}$
+
+The proof of Proposition 4 can be found in Appendix B. Finally, because $\mathcal { D } = \tilde { O } ( c ^ { 5 / 2 } \lambda ^ { 1 / 2 + \delta / 2 } )$ , $\| r _ { t + 1 } \| \leq \mathcal { D }$ for sufficiently large $c$ . This completes the induction and the proof of Lemma 1.
+
+Comparison with Blanc et al. [3] Like Blanc et al. [3], Lemma 1 shows that $\theta$ locally follows the trajectory of gradient descent on an implicit regularizer $R ( \theta )$ . However, there are a few crucial differences:
+
+• Because we do not assume we start near a global minimizer where $L \ = \ 0$ , we couple to a regularized loss $\tilde { L } = L + \lambda R$ rather than just the regularizer $R ( \theta )$ . In this setting there is an additional correction term to the Hessian (Proposition 1) that requires carefully controlling the value of the loss across reference points to prove convergence to a stationary point. • The analysis in Blanc et al. [3] requires $\eta , \tau$ to be chosen in terms of the condition number of $\nabla ^ { 2 } L$ which can quickly grow during training as $\nabla ^ { 2 } L$ is changing. This makes it impossible to directly repeat the argument. We avoid this by precisely analyzing the error incurred by small eigenvalues, allowing us to prove convergence to an $( \epsilon , \gamma )$ stationary point of $\textstyle { \frac { 1 } { \lambda } } { \tilde { L } }$ for fixed $\eta , \lambda$ even if the smallest nonzero eigenvalue of $\nabla ^ { 2 } L$ converges to 0 during training. Unlike in Blanc et al. [3], we do not require the learning rate $\eta$ to be small. Instead, we only require that $\lambda$ scales with $\epsilon$ which can be accomplished either by decreasing the learning rate $\eta$ or increasing the batch size $B$ . This allows for stronger implicit regularization in the setting when $\eta$ is large (see Section 6.1). In particular, our regularizer $R ( \theta )$ changes with $\eta$ and is only equal to the regularizer in Blanc et al. [3] in the limit $\eta  0$ .
+
+# 3.2 Global Convergence
+
+In order to prove convergence to an $( \epsilon , \gamma )$ -stationary point of $\begin{array} { r l } {  { \frac { 1 } { \eta } \nabla \tilde { L } } } & { { } } \end{array}$ , we will define a sequence of reference points $\theta _ { m } ^ { * }$ and coupling times $\{ \tau _ { m } \}$ and repeatedly use a version of Lemma 1 to describe the long term behavior of $\theta$ . For notational simplicity, given a sequence of coupling times $\{ \tau _ { m } \}$ , define $\begin{array} { r } { \bar { T } _ { m } = \sum _ { k < m } \tau _ { k } } \end{array}$ to be the total number of steps until we have reached the reference point $\theta _ { m } ^ { * }$
+
+To be able to repeat the local analysis in Lemma 1 with multiple reference points, we need a more general coupling lemma that allows the random process $\xi$ defined in each coupling to continue where the random process in the previous coupling ended. To accomplish this, we define $\xi$ outside the scope of the local coupling lemma:
+
+Definition 3. Given a sequence of reference points $\{ \theta _ { m } ^ { * } \}$ and a sequence of coupling times $\{ \tau _ { m } \}$ , we define the random process $\xi$ by $\xi _ { 0 } = 0$ , and for $k \in [ T _ { m } , T _ { m + 1 } )$ ,
+
+$$
+\epsilon _ { k } ^ { * } = \frac { \eta } { B } \sum _ { i \in \mathcal { B } ^ { ( k ) } } \epsilon _ { i } ^ { ( k ) } \nabla f _ { i } ( \theta _ { m } ^ { * } ) \qquad a n d \qquad \xi _ { k + 1 } = ( I - \eta G ( \theta _ { m } ^ { * } ) ) \xi _ { k } + \epsilon _ { k } ^ { * } .
+$$
+
+Then we can prove the following more general coupling lemma:
+
+Lemma 2. Let $\mathcal { X } , \mathcal { L } , \mathcal { D } , \mathcal { M } , \mathcal { T }$ be defined as in Lemma $^ { l }$ . Assume $f$ satisfies Assumption $^ { l }$ and $\eta$ satisfies Assumption 2. Let $\Delta _ { m } = \theta _ { T _ { m } } - \xi _ { T _ { m } } - \theta _ { m } ^ { * }$ and assume that $\| \Delta _ { m } \| \leq \mathcal { D }$ and $L ( \theta _ { m } ^ { * } ) \leq \mathcal { L }$ for some $0 < \delta \le 1 / 2$ . Then for any $\tau _ { m } \leq \mathcal { T }$ satisfying $\begin{array} { r l } { \operatorname* { m a x } _ { k \in [ T _ { m } , T _ { m + 1 } ) } \left. \Phi _ { k - T _ { m } } ( \theta _ { m } ^ { * } + \Delta _ { m } ) - \theta _ { m } ^ { * } \right. \le } & { { } \quad } \end{array}$ $8 \mathcal { M }$ , with probability at least $1 - 1 0 d \tau _ { m } e ^ { - \iota }$ we have simultaneously for all $k \in ( T _ { m } , T _ { m + 1 } ]$ ,
+
+$$
+\begin{array} { r } { \| \theta _ { k } - \xi _ { k } - \Phi _ { k - T _ { m } } ( \theta _ { m } ^ { * } + \Delta _ { m } ) \| \le \mathcal { D } , \quad \quad \mathbb { E } [ \xi _ { k } ] = 0 , \quad \quad a n d \quad \quad \| \xi _ { k } \| \le \mathcal { X } . } \end{array}
+$$
+
+Unlike in Lemma 1, we couple to the regularized trajectory starting at $\theta _ { m } ^ { * } + \Delta _ { m }$ rather than at $\theta _ { m } ^ { * }$ to avoid accumulating errors (see Figure 2). The proof is otherwise identical to that of Lemma 1.
+
+The proof of Theorem 1 easily follows from the following lemma which states that we decrease the regularized loss $\tilde { L }$ by at least $\mathcal { F }$ after every coupling:
+
+Lemma 3. Let $\begin{array} { r } { \mathcal { F } = \frac { { \mathcal { D } } ^ { 2 } } { \eta \nu \mathcal { T } } } \end{array}$ . Let $\Delta _ { m } = \theta _ { T _ { m } } - \xi _ { T _ { m } } - \theta _ { m } ^ { * }$ and assume $\| \Delta _ { m } \| \leq \mathcal { D }$ and $L ( \theta _ { m } ^ { * } ) \leq \mathcal { L }$ Then if $\theta _ { T _ { m } }$ is not an $( \epsilon , \gamma )$ -stationary point, there exists some $\tau _ { m } < \mathcal { T }$ such that if we define
+
+$$
+\begin{array} { r } { \theta _ { m + 1 } ^ { * } = \Phi _ { \tau _ { n } } \bigl ( \theta _ { m } ^ { * } + \Delta _ { m } \bigr ) \qquad a n d \qquad \Delta _ { m + 1 } = \theta _ { T _ { m + 1 } } - \xi _ { T _ { m + 1 } } - \theta _ { m + 1 } ^ { * } , } \end{array}
+$$
+
+then with probability $1 - 1 0 d \tau _ { m } e ^ { - \iota }$
+
+$$
+\begin{array} { r } { \tilde { L } ( \theta _ { m + 1 } ^ { * } ) \leq L ( \theta _ { m } ^ { * } ) - \mathcal { F } , \qquad \| \Delta _ { m + 1 } \| \leq \mathcal { D } \qquad a n d \qquad L ( \theta _ { m + 1 } ^ { * } ) \leq \mathcal { L } . } \end{array}
+$$
+
+We defer the proofs of Lemma 2 and Lemma 3 to Appendix B. Theorem 1 now follows directly from repeated applications of Lemma 3:
+
+Proof of Theorem $^ { l }$ . By assumption there exists some $\theta _ { 0 } ^ { * }$ such that $L ( \theta _ { 0 } ^ { * } ) \leq \mathcal { L }$ and $\lVert { \boldsymbol { \theta } } _ { 0 } - { \boldsymbol { \theta } } _ { 0 } ^ { * } \rVert \leq \mathcal { D }$ . Then so long as $\theta _ { T _ { m } }$ is not an $( \epsilon , \gamma )$ -stationary point, we can inductively apply Lemma 3 to get the existence of coupling times $\{ \tau _ { m } \}$ and reference points $\{ \theta _ { m } ^ { * } \}$ such that for any $m \geq 0$ , with probability $1 - 1 0 d T _ { m } e ^ { - \iota }$ we have $\tilde { L } ( \theta _ { m } ^ { * } ) \leq \tilde { L } ( \theta _ { 0 } ^ { * } ) - m \mathcal { \bar { F } }$ . As $\tilde { L } ( \theta _ { 0 } ^ { * } ) - \tilde { L } ( \theta _ { m } ^ { * } ) = O ( \lambda )$ , this can happen for at most $\begin{array} { r } { m = O \left( \frac { \lambda } { \mathcal { F } } \right) } \end{array}$ reference points, so at most $\begin{array} { r } { T = O \left( \frac { \lambda \mathcal { T } } { \mathcal { F } } \right) = \tilde { O } \left( \eta ^ { - 1 } \lambda ^ { - 1 - \delta } \right) } \end{array}$ iterations of Algorithm 1. By the choice of $\iota$ , this happens with probability $1 - 1 0 d T e ^ { - \iota } \geq 1 - \zeta$ .
+
+# 4 Experiments
+
+In order to test the ability of SGD with label noise to escape poor global minimizers and converge to better minimizers, we initialize Algorithm 1 at global minimizers of the training loss which achieve $1 0 0 \%$ training accuracy yet generalize poorly to the test set. Minibatch SGD would remain fixed at these initializations because both the gradient and the noise in minibatch SGD vanish at any global minimizer of the training loss. We show that SGD with label noise escapes these poor initializations and converges to flatter minimizers that generalize well, which supports Theorem 1. We run experiments with two initializations:
+
+![](images/ad91c4cdaa99ae899b2045a803ab46a7fa428a14bbab9226a753198f9983dc5d.jpg)  
+Figure 3: Label Noise SGD escapes poor global minimizers. The left column displays the training accuracy over time, the middle column displays the value of $\operatorname { t r } \nabla ^ { 2 } L ( \theta )$ over time which we use to approximate the implicit regularizer $R ( \theta )$ , and the right column displays their correlation. The horizontal dashed line represents the minibatch SGD baseline with random initialization. We report the median results over 3 random seeds and shaded error bars denote the $\operatorname* { m i n } / \operatorname* { m a x }$ over the three runs. The correlation plot uses a running average of 100 epochs for visual clarity.
+
+Full Batch Initialization: We run full batch gradient descent with random initialization until convergence to a global minimizer. We call this minimizer the full batch initialization. The final test accuracy of the full batch initialization was $76 \%$ .
+
+Adversarial Initialization: Following Liu et al. [21], we generate an adversarial initialization with final test accuracy $4 8 \%$ that achieves zero training loss by first teaching the network to memorize random labels and then training it on the true labels. See Appendix D for full details.
+
+Experiments were run with ResNet18 on CIFAR10 [17] without data augmentation or weight decay. The experiments were conducted with randomized label flipping with probability 0.2 (see Appendix E for the extension of Theorem 1 to classification with label flipping), cross entropy loss, and batch size 256. Because of the difficulty in computing the regularizer $R ( \theta )$ , we approximate it by its lower bound $\operatorname { t r } \nabla ^ { 2 } L ( \theta )$ . Figure 3 shows the test accuracy and $\mathrm { t r } \nabla ^ { 2 } L$ throughout training.
+
+SGD with label noise escapes both zero training loss initializations and converges to flatter minimizers that generalize much better, reaching the SGD baseline from the fullbatch initialization and getting within $1 \%$ of the baseline from the adversarial initialization. The test accuracy in both cases is strongly correlated with $\mathrm { t r } \nabla ^ { 2 } L$ . The strength of the regularization is also strongly correlated with $\eta$ which supports Theorem 1.
+
+# 5 Extensions
+
+# 5.1 SGD with momentum
+
+We replace the update in Algorithm 1 with heavy ball momentum with parameter $\beta$ :
+
+$$
+\theta _ { k + 1 } = \theta _ { k } - \eta \nabla \hat { L } ^ { ( k ) } ( \theta _ { k } ) + \beta ( \theta _ { k } - \theta _ { k - 1 } ) .
+$$
+
+We define:
+
+$$
+R ( \theta ) = \frac { 1 + \beta } { 2 \eta } \mathrm { t r } \log \left( 1 - \frac { \eta } { 2 ( 1 + \beta ) } \nabla ^ { 2 } L ( \theta ) \right) , \qquad \lambda = \frac { \eta \sigma ^ { 2 } } { B ( 1 - \beta ) } ,
+$$
+
+and as before $\tilde { L } ( \theta ) = L ( \theta ) + \lambda R ( \theta )$ . Let
+
+$$
+\Phi _ { 0 } ( \theta ) = \theta , \qquad \Phi _ { k + 1 } ( \theta ) = \Phi _ { k } ( \theta ) - \eta \nabla \tilde { L } ( \Phi _ { k } ( \theta ) ) + \beta ( \Phi _ { k } ( \theta ) - \Phi _ { k - 1 } ( \theta ) )
+$$
+
+represent gradient descent with momentum on $\tilde { L }$ . Then we have the following local coupling lemma:
+
+Lemma 4. Let
+
+$$
+\mathcal { X } = \sqrt { \frac { 2 \lambda n ^ { 2 } \iota } { \nu } } , \qquad \mathcal { L } = c \lambda ^ { 1 + \delta } , \qquad \mathcal { D } = c \sqrt { \mathcal { L } } \iota , \qquad \mathcal { T } = \frac { 1 } { c ^ { 2 } \eta \mathcal { X } \iota } ,
+$$
+
+where c is a sufficfollow Algorithm ntly large consta with momentum . Assume starting $f$ tisfieand $^ { l }$ and r so $\begin{array} { r } { \eta \le \frac { ( 2 - \nu ) ( 1 + \beta ) } { \ell } } \end{array}$ . Let . The $\theta$ $^ { l }$ $\beta$ $\theta ^ { * }$ $L ( \theta ^ { * } ) \leq \mathcal { L }$ $0 < \delta \le 1 / 2$ there exists a random process $\{ \xi _ { k } \}$ such that for any $\tau \leq \mathcal { T }$ satisfying $\begin{array} { r } { \operatorname* { m a x } _ { k \leq \tau } \| \Phi _ { k } ( \theta ^ { * } ) - \theta ^ { * } \| \leq 8 \mathcal { D } } \end{array}$ with probability at least $1 - 1 0 d \tau e ^ { - \iota }$ we have simultaneously for all $k \leq \tau$ ,
+
+$$
+\begin{array} { r } { \| \theta _ { k } - \xi _ { k } - \Phi _ { k } ( \theta ^ { * } ) \| \le \mathcal { D } , \qquad \mathbb { E } [ \xi _ { k } ] = 0 , \qquad a n d \qquad \| \xi _ { k } \| \le \mathcal { X } . } \end{array}
+$$
+
+As in Lemma 1, the error is 8 times smaller than the maximum movement of the regularized trajectory. Note that momentum increases the regularization parameter $\lambda$ by $\frac { 1 } { 1 - \beta }$ . For the commonly used momentum parameter $\beta = 0 . 9$ , this represents a $1 0 \times$ increase in regularization, which is likely the cause of the improved performance in Figure 4 $\beta = 0 . 9$ ) over Figure 3 $\beta = 0$ ).
+
+# 5.2 Arbitrary Noise Covariances
+
+The analysis in Section 3.1 is not specific to label noise SGD and can be carried out for arbitrary noise schemes. Let $\theta$ follow $\theta _ { k + 1 } \overset { \cdot } { = } \theta _ { k } - \eta \nabla L ( \theta _ { k } ) + \epsilon _ { k }$ starting at $\theta _ { 0 }$ where $\epsilon _ { k } \sim { \cal N } ( 0 , \eta \lambda \Sigma ( \theta _ { k } ) )$ and $\Sigma ^ { 1 / 2 }$ is Lipschitz. Given a matrix $S$ we define the regularizer $R _ { S } ( \theta ) = \left. S , \nabla ^ { 2 } L ( \theta ) \right.$ . The matrix $S$ controls the weight of each eigenvalue. As before we can define $\tilde { L } _ { S } ( \theta ) = L ( \theta ) + \lambda R _ { S } ( \theta )$ and $\Phi _ { k + 1 } ^ { S } ( \theta ) = \Phi _ { k } ^ { S } ( \theta ) - \eta \nabla \tilde { L } _ { S } ( \Phi _ { k } ( \theta ) )$ to be the regularized loss and the regularized trajectory respectively. Then we have the following version of Lemma 1:
+
+Proposition 5. Let $\theta$ be initialized at a minimizer $\theta ^ { * }$ of $L$ . Assume $\nabla ^ { 2 } L$ is Lipschitz, let $H = \nabla ^ { 2 } L ( \theta ^ { * } )$ and assume that $\Sigma ( \theta ^ { * } ) \preceq C H$ for some absolute constant $C$ . Let $\begin{array} { r } { \mathcal { X } = \sqrt { \frac { C d \lambda \iota } { \nu } } } \end{array}$ , $\mathcal { D } = c \lambda ^ { 3 / 4 } \iota ,$ , and $\begin{array} { r } { \mathcal { T } = \frac { 1 } { c ^ { 2 } \eta \mathcal { X } \iota } } \end{array}$ for a sufficiently large constant c. Then there exists a mean zero random process $\xi$ such that for any $\tau \leq \mathcal { T }$ satisfying $\begin{array} { r } { \operatorname* { m a x } _ { k < \tau } \| \Phi _ { k } ( \theta ^ { * } ) - \theta ^ { * } \| \leq 8 \mathcal { D } } \end{array}$ and with probability $1 - 1 0 d \tau e ^ { - \iota }$ , we have simultaneously for all $k \leq \tau$ :
+
+$$
+\begin{array} { r } { \| \theta _ { k } - \xi _ { k } - \Phi _ { k } ^ { S } ( \theta _ { 0 } ) \| \le \mathcal { D } \qquad a n d \qquad \| \xi _ { k } \| \le \mathcal { X } , } \end{array}
+$$
+
+where $S$ is the unique fixed point of $S \gets ( I - \eta H ) S ( I - \eta H ) + \eta \lambda \Sigma ( \theta ^ { * } )$ restricted to span $( H )$
+
+As in Lemma 1, the error is 8 times smaller than the maximum movement of the regularized trajectory. Although Proposition 5 couples to gradient descent on $R _ { S }$ , $S$ is defined in terms of the Hessian and the noise covariance at $\theta ^ { * }$ and therefore depends on the choice of reference point. Because $R _ { S }$ is changing, we cannot repeat Proposition 5 as in Section 3.2 to prove convergence to a stationary point because there is no fixed potential. Although it is sometimes possible to relate $R _ { S }$ to a fixed potential $R$ , we show in Appendix F.2 that this is not generally possible by providing an example where minibatch SGD perpetually cycles. Exploring the properties of these continuously changing potentials and their connections to generalization is an interesting avenue for future work.
+
+# 6 Discussion
+
+# 6.1 Sharpness and the Effect of Large Learning Rates
+
+Various factors can control the strength of the implicit regularization in Theorem 1. Most important is the implicit regularization parameter $\begin{array} { r } { \lambda = \frac { \eta \sigma ^ { 2 } } { | B | } } \end{array}$ . This supports the hypothesis that large learning rates and small batch sizes are necessary for implicit regularization [9, 26], and agrees with the standard linear scaling rule which proposes that for constant regularization strength, the learning rate $\eta$ needs to be inversely proportional to the batch size $| B |$ .
+
+However, our analysis also uncovers an additional regularization effect of large learning rates. Unlike the regularizer in Blanc et al. [3], the implicit regularizer $R ( \theta )$ defined in Equation (1) is dependent on $\eta$ . It is not possible to directly analyze the behavior of $R ( \theta )$ as $\eta  2 / \lambda _ { 1 }$ where $\lambda _ { 1 }$ is the largest eigenvalue of $\nabla ^ { 2 } L$ , as in this regime, $R ( \theta ) \to \infty$ (see Figure 1). If we let $\begin{array} { r } { \eta = \frac { 2 - \nu } { \lambda _ { 1 } } } \end{array}$ 2−ν , then we can better understand the behavior of $R ( \theta )$ by normalizing it by $\log 2 / \nu$ . This gives2
+
+$$
+\frac { R ( \theta ) } { \log 2 / \nu } = \sum _ { i } \frac { R ( \lambda _ { i } ) } { \log 2 / \nu } = \| \nabla ^ { 2 } L ( \theta ) \| _ { 2 } + O \left( \frac { 1 } { \log 2 / \nu } \right) \xrightarrow { \nu \to 0 } \| \nabla ^ { 2 } L ( \theta ) \| _ { 2 }
+$$
+
+so after normalization, $R ( \theta )$ becomes a better and better approximation of the spectral norm $\| \nabla ^ { 2 } L ( \theta ) \|$ as $\eta  2 / \lambda _ { 1 }$ . $R ( \theta )$ can therefore be seen as interpolating between $\mathrm { t r } \hat { \nabla } ^ { 2 } L ( \theta )$ , when $\eta \approx 0$ , and $\| \nabla ^ { 2 } L ( \theta ) \| _ { 2 }$ when $\eta \approx 2 / \lambda _ { 1 }$ . This also suggests that SGD with large learning rates may be more resilient to the edge of stability phenomenon observed in Cohen et al. [4] as the implicit regularization works harder to control eigenvalues approaching $2 / \eta$ .
+
+The sharpness-aware algorithm (SAM) of [7] is also closely related to $R ( \theta )$ . SAM proposes to minimize $\begin{array} { r } { \operatorname* { m a x } _ { \parallel \delta \parallel _ { 2 } } \le _ { \epsilon } L ( \theta + \delta ) } \end{array}$ . At a global minimizer of the training loss,
+
+$$
+\operatorname* { m a x } _ { \| \delta \| _ { 2 } \leq \epsilon } L ( \theta ^ { * } + \delta ) = \operatorname* { m a x } _ { \| \delta \| _ { 2 } \leq \epsilon } \frac { 1 } { 2 } \delta ^ { \top } \nabla ^ { 2 } L ( \theta ^ { * } ) \delta + O ( \epsilon ^ { 3 } ) \approx \frac { \epsilon ^ { 2 } } { 2 } \| \nabla ^ { 2 } L ( \theta ^ { * } ) \| _ { 2 } .
+$$
+
+The SAM algorithm is therefore explicitly regularizing the spectral norm of $\nabla ^ { 2 } L ( \theta )$ , which is closely connected to the large learning rate regularization effect of $R ( \theta )$ when $\eta \approx 2 / \lambda _ { 1 }$ .
+
+# 6.2 Generalization Bounds
+
+The implicit regularizer $R ( \theta )$ is intimately connected to data-dependent generalization bounds, which measure the Lipschitzness of the network via the network Jacobiapropose the all-layer margin, which bounds the generalization error $\begin{array} { r } { \lesssim \frac { \sum _ { l = 1 } ^ { L } \mathcal { C } _ { l } } { \sqrt { n } } \sqrt { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { 1 } { m _ { F } ( x _ { i } , y _ { i } ) ^ { 2 } } } } \end{array}$ is the all-layer margin. The norm of the parameters is generally controlled by weight decay regularization, so we focus our discussion on the all-layer margin. Ignoring higher-order secondary terms, Wei and Ma [30, Heuristic derivation of Lemma 3.1] showed for a feed-forward network $\begin{array} { r } { \dot { f } ( \theta ; x ) = \theta _ { L } \sigma ( \theta _ { L - 1 } \dots \sigma ( \theta _ { 1 } x ) ) } \end{array}$ , the all-layer margin satisfies3:
+
+$$
+\frac { 1 } { m _ { F } ( x , y ) } \lesssim \frac { \| \{ \frac { \partial f } { \partial \theta _ { l } } \} _ { l \in [ L ] } \| _ { 2 } } { \mathrm { o u t p u t ~ m a r g i n ~ o f ~ } ( x , y ) } \implies \mathrm { g e n e r a l i z a t i o n ~ e r r o r } \lesssim \frac { \sum _ { l = 1 } ^ { L } \mathcal { C } _ { l } } { \sqrt { n } } \sqrt { \frac { R ( \theta ) } { \mathrm { o u t p u t ~ m a r g i n } } }
+$$
+
+as $R ( \theta )$ is an upper bound on the squared norm of the Jacobian at any global minimizer $\theta$ . We emphasize this bound is informal as we discarded the higher-order terms in controlling the all-layer margin, but it accurately reflects that the regularizer $R ( \bar { \theta ) }$ lower bounds the all-layer margin $m _ { F }$ up to higher-order terms. Therefore SGD with label noise implicitly regularizes the all-layer margin.
+
+# Acknowledgments and Disclosure of Funding
+
+AD acknowledges support from a NSF Graduate Research Fellowship. TM acknowledges support of Google Faculty Award and NSF IIS 2045685. JDL acknowledges support of the ARO under MURI Award W911NF-11-1-0303, the Sloan Research Fellowship, NSF CCF 2002272, and an ONR Young Investigator Award.
+
+The experiments in this paper were performed on computational resources managed and supported by Princeton Research Computing, a consortium of groups including the Princeton Institute for Computational Science and Engineering (PICSciE) and the Office of Information Technology’s High Performance Computing Center and Visualization Laboratory at Princeton University.
+
+We would also like to thank Honglin Yuan and Jeff Z. HaoChen for useful discussions throughout various stages of the project.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 2 for a list of assumptions made in this paper and see Appendix A for a full discussion.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] This work is mainly theoretical and focuses on understanding an existing algorithm (Label Noise SGD).   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 2 for a list of assumptions made in this paper.   
+(b) Did you include complete proofs of all theoretical results? [Yes] A proof sketch of Theorem 1 is provided in Section 3 however full proofs of all claims in the paper can be found in Appendix B.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The instructions needed to reproduce the experiments in Section 4 can be found in Appendix D. Code will be submitted through the supplementary material and will be made available (through Github) upon acceptance.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4 and Appendix D.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Figure 3 and Figure 4.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix D.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] For CIFAR10 we cite Krizhevsky [17], as requested by the creators on https://www.cs.toronto.edu/ kriz/cifar.html. In Appendix D we additionally cite PyTorch [24], PyTorch Lightning [6], and Wandb [2].   
+(b) Did you mention the license of the assets? [Yes] We mention the MIT license for CIFAR10 in Appendix D.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]