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+# A CRITICAL ANALYSIS OF SELF-SUPERVISION, ORWHAT WE CAN LEARN FROM A SINGLE IMAGE
+
+Yuki M. Asano Christian Rupprecht
+
+Andrea Vedaldi
+
+Visual Geometry Group   
+University of Oxford   
+{yuki,chrisr,vedaldi}@robots.ox.ac.uk
+
+# ABSTRACT
+
+We look critically at popular self-supervision techniques for learning deep convolutional neural networks without manual labels. We show that three different and representative methods, BiGAN, RotNet and DeepCluster, can learn the first few layers of a convolutional network from a single image as well as using millions of images and manual labels, provided that strong data augmentation is used. However, for deeper layers the gap with manual supervision cannot be closed even if millions of unlabelled images are used for training. We conclude that: (1) the weights of the early layers of deep networks contain limited information about the statistics of natural images, that (2) such low-level statistics can be learned through self-supervision just as well as through strong supervision, and that (3) the low-level statistics can be captured via synthetic transformations instead of using a large image dataset.
+
+# 1 INTRODUCTION
+
+Despite tremendous progress in supervised learning, learning without external supervision remains difficult. Self-supervision has recently emerged as one of the most promising approaches to address this limitation. Self-supervision builds on the fact that convolutional neural networks (CNNs) transfer well between tasks (Shin et al., 2016; Oquab et al., 2014; Girshick, 2015; Huh et al., 2016). The idea then is to pre-train networks via pretext tasks that do not require expensive manual annotations and can be automatically generated from the data itself. Once pre-trained, networks can be applied to a target task by using only a modest amount of labelled data.
+
+Early successes in self-supervision have encouraged authors to develop a large variety of pretext tasks, from colorization to rotation estimation and image autoencoding. Recent papers have shown performance competitive with supervised learning by learning complex neural networks on very large image datasets. Nevertheless, for a given model complexity, pre-training by using an off-theshelf annotated image datasets such as ImageNet remains much more efficient.
+
+In this paper, we aim to investigate the effectiveness of current self-supervised approaches by characterizing how much information they can extract from a given dataset of images. Since deep networks learn a hierarchy of representations, we further break down this investigation on a per-layer basis. We are motivated by the fact that the first few layers of most networks extract low-level information (Yosinski et al., 2014), and thus learning them may not require the high-level semantic information captured by manual labels.
+
+Concretely, in this paper we answer the following simple question: “is self-supervision able to exploit the information contained in a large number of images in order to learn different parts of a neural network?”
+
+We contribute two key findings. First, we show that as little as a single image is sufficient, when combined with self-supervision and data augmentation, to learn the first few layers of standard deep networks as well as using millions of images and full supervision (Figure 1). Hence, while selfsupervised learning works well for these layers, this may be due more to the limited complexity of such features than the strength of the supervisory technique. This also confirms the intuition that early layers in a convolutional network amounts to low-level feature extractors, analogous to early learned and hand-crafted features for visual recognition (Olshausen & Field, 1997; Lowe, 2004; Dalal & Triggs, 2005). Finally, it demonstrates the importance of image transformations in learning such low-level features as opposed to image diversity.1
+
+![](images/170a22e5e719a53efa60993ee606484b35b4465c1a6a9fe2204b521c8d69c3b5.jpg)
+
+![](images/f6e9438a75e6c4530cb249d7647d3dfd569bce7aadafc4b938fc22c16106a227.jpg)  
+Figure 1: Single-image self-supervision. We show that several self-supervision methods can be used to train the first few layers of a deep neural networks using a single training image, such as this Image A, B or even C (above), provided that sufficient data augmentation is used.
+
+Our second finding is about the deeper layers of the network. For these, self-supervision remains inferior to strong supervision even if millions of images are used for training. Our finding is that this is unlikely to change with the addition of more data. In particular, we show that training these layers with self-supervision and a single image already achieves as much as two thirds of the performance that can be achieved by using a million different images.
+
+We show that these conclusions hold true for three different self-supervised methods, BiGAN (Donahue et al., 2017), RotNet (Gidaris et al., 2018) and DeepCluster (Caron et al., 2018), which are representative of the spectrum of techniques that are currently popular. We find that performance as a function of the amount of data is dependent on the method, but all three methods can indeed leverage a single image to learn the first few layers of a deep network almost “perfectly”.
+
+Overall, while our results do not improve self-supervision per-se, they help to characterize the limitations of current methods and to better focus on the important open challenges.
+
+# 2 RELATED WORK
+
+Our paper relates to three broad areas of research: (a) self-supervised/unsupervised learning, (b) learning from a single sample, and (c) designing/learning low-level feature extractors. We discuss closely related work for each.
+
+Self-supervised learning: A wide variety of proxy tasks, requiring no manual annotations, have been proposed for the self-training of deep convolutional neural networks. These methods use various cues and tasks namely, in-painting (Pathak et al., 2016), patch context and jigsaw puzzles (Doersch et al., 2015; Noroozi & Favaro, 2016; Noroozi et al., 2018; Mundhenk et al., 2017), clustering (Caron et al., 2018), noise-as-targets (Bojanowski & Joulin, 2017), colorization (Zhang et al., 2016; Larsson et al., 2017), generation (Jenni & Favaro, 2018; Ren & Lee, 2018; Donahue et al., 2017), geometry (Dosovitskiy et al., 2016; Gidaris et al., 2018) and counting (Noroozi et al., 2017). The idea is that the pretext task can be constructed automatically and easily on images alone. Thus, methods often modify information in the images and require the network to recover them. Inpainting or colorization techniques fall in this category. However these methods have the downside that the features are learned on modified images which potentially harms the generalization to unmodified ones. For example, colorization uses a gray scale image as input, thus the network cannot learn to extract color information, which can be important for other tasks.
+
+Slightly less related are methods that use additional information to learn features. Here, often temporal information is used in the form of videos. Typical pretext tasks are based on temporalcontext (Misra et al., 2016; Wei et al., 2018; Lee et al., 2017; Sermanet et al., 2018), spatio-temporal cues (Isola et al., 2015; Gao et al., 2016; Wang et al., 2017), foreground-background segmentation via video segmentation (Pathak et al., 2017), optical-flow (Gan et al., 2018; Mahendran et al., 2018), future-frame synthesis (Srivastava et al., 2015), audio prediction from video (de Sa, 1994; Owens et al., 2016), audio-video alignment (Arandjelovic & Zisserman ´ , 2017), ego-motion estimation (Jayaraman & Grauman, 2015), slow feature analysis with higher order temporal coherence (Jayaraman & Grauman, 2016), transformation between frames (Agrawal et al., 2015) and patch tracking in videos (Wang & Gupta, 2015). Since we are interested in learning features from as little data as one image, we cannot make use of methods that rely on video input.
+
+Our contribution inspects three unsupervised feature learning methods that use very different means of extracting information from the data: BiGAN (Donahue et al., 2017) utilizes a generative adversarial task, RotNet (Gidaris et al., 2018) exploits the photographic bias in the dataset and DeepCluster (Caron et al., 2018) learns stable feature representations under a number of image transformations by proxy labels obtained from clustering. These are described in more detail in the Methods section.
+
+Learning from a single sample: In some applications of computer vision, the bold idea of learning from a single sample comes out of necessity. For general object tracking, methods such as max margin correlation filters (Rodriguez et al., 2013) learn robust tracking templates from a single sample of the patch. A single image can also be used to learn and interpolate multi-scale textures with a GAN framework (Rott Shaham et al., 2019). Single sample learning was pursued by the semi-parametric exemplar SVM model (Malisiewicz et al., 2011). They learn one SVM per positive sample separating it from all negative patches mined from the background. While only one sample is used for the positive set, the negative set consists of thousands of images and is a necessary component of their method. The negative space was approximated by a multi-dimensional Gaussian by the Exemplar LDA (Hariharan et al., 2012). These SVMs, one per positive sample, are pooled together using a max aggregation. We differ from both of these approaches in that we do not use a large collection of negative images to train our model. Instead we restrict ourselves to a single or a few images with a systematic augmentation strategy.
+
+Classical learned and hand-crafted low-level feature extractors: Learning and hand-crafting features pre-dates modern deep learning approaches and self-supervision techniques. For example the classical work of (Olshausen & Field, 1997) shows that edge-like filters can be learned via sparse coding of just 10 natural scene images. SIFT (Lowe, 2004) and HOG (Dalal & Triggs, 2005) have been used extensively before the advent of convolutional neural networks and, in many ways, they resemble the first layers of these networks. The scatter transform of Bruna & Mallat (2013); Oyallon et al. (2017) is an handcrafted design that aims at replacing at least the first few layers of a deep network. While these results show that effective low-level features can be handcrafted, this is insufficient to clarify the power and limitation of self-supervision in deep networks. For instance, it is not obvious whether deep networks can learn better low level features than these, how many images may be required to learn them, and how effective self-supervision may be in doing so. For instance, as we also show in the experiments, replacing low-level layers in a convolutional networks with handcrafted features such as Oyallon et al. (2017) may still decrease the overall performance of the model. Furthermore, this says little about deeper layers, which we also investigate.
+
+In this work we show that current deep learning methods learn slightly better low-level representations than hand crafted features such as the scattering transform. Additionally, these representations can be learned from one single image with augmentations and without supervision. The results show how current self-supervised learning approaches that use one million images yield only relatively small gains when compared to what can be achieved from one image and augmentations, and motivates a renewed focus on augmentations and incorporating prior knowledge into feature extractors.
+
+# 3 METHODS
+
+We discuss first our data and data augmentation strategy (section 3.1) and then we summarize the three different methods for unsupervised feature learning used in the experiments (section 3.2).
+
+# 3.1 DATA
+
+Our goal is to understand the performance of representation learning methods as a function of the image data used to train them. To make comparisons as fair as possible, we develop a protocol where only the nature of the training data is changed, but all other parameters remain fixed.
+
+In order to do so, given a baseline method trained on $d$ source images, we replace those with another set of $d$ images. Of these, now only $N \ll d$ are source images (i.e. i.i.d. samples), while the remaining $d - N$ are augmentations of the source ones. Thus, the amount of information in the training data is controlled by $N$ and we can generate a continuum of datasets that vary from one extreme, utilizing a single source image $N = 1$ , to the other extreme, using all $N { = } d$ original training set images. For example, if the baseline method is trained on ImageNet, then $d = 1 , 2 8 1 , 1 6 7$ . When $N = 1$ , it means that we train the method using a single source image and generate the remaining 1,281,166 images via augmentation. Other baselines use CIFAR-10/100 images, so in those cases $d = 5 0 , 0 0 0$ instead.
+
+The data augmentation protocol, is an extreme version of augmentations already employed by most deep learning protocols. Each method we test, in fact, already performs some data augmentation internally. Thus, when the method is applied on our augmented data, this can be equivalently thought of as incrementing these “native” augmentations by concatenating them with our own.
+
+Choice of augmentations. Next, we describe how the $N$ source images are expanded to additional $d - N$ images so that the models can be trained on exactly $d$ images, independent from the choice of $N$ . The idea is to use an aggressive form of data augmentation involving cropping, scaling, rotation, contrast changes, and adding noise. These transformations are representative of invariances that one may wish to incorporate in the features. Augmentation can be seen as imposing a prior on how we expect the manifold of natural images to look like. When training with very few images, these priors become more important since the model cannot extract them directly from data.
+
+Given a source image of size size $H \times W$ , we first extract a certain number of random patches of size $( w , h )$ , where $w \leq W$ and $h \leq H$ satisfy the additional constraints $\beta \le \frac { w h } { W H }$ and $\begin{array} { r } { \gamma \leq \frac { h } { w } \leq \gamma ^ { - 1 } } \end{array}$ . Thus, the smallest size of the crops is limited to be at least $\beta W H$ and at most the whole image. Additionally, changes to the aspect ratio are limited by $\gamma$ . In practice we use $\beta = 1 0 ^ { - 3 }$ and $\textstyle \gamma = { \frac { 3 } { 4 } }$ .
+
+Second, good features should not change much by small image rotations, so images are rotated (before cropping to avoid border artifacts) by $\alpha \in \mathsf { \Gamma } ( - 3 5 , 3 5 )$ degrees. Due to symmetry in image statistics, images are also flipped left-to-right with $50 \%$ probability.
+
+Illumination changes are common in natural images, we thus expect image features to be robust to color and contrast changes. Thus, we employ a set of linear transformations in RGB space to model this variability in real data. Additionally, the color/intensity of single pixels should not affect the feature representation, as this does not change the contents of the image. To this end, color jitter with additive brightness, contrast and saturation are sampled from three uniform distributions in (0.6, 1.4) and hue noise from $( - 0 . 1 , 0 . 1 )$ is applied to the image patches. Finally, the cropped and transformed patches are scaled to the color range $( - 1 , 1 )$ and then rescaled to full $S \times S$ resolution to be supplied to each representation learning method, using bilinear interpolation. This formulation ensures that the patches are created in the target resolution $S$ , independent from the size and aspect ratio $W , H$ of the source image.
+
+Real samples. The images used for the $N { = } 1$ and $N { = } 1 0$ experiments are shown in Figure 1 and the appendix respectively (this is all the training data used in such experiments). For the special case of using a single training image, i.e. $N { = } 1$ , we have chosen one photographic $( 2 5 6 0 \times 1 9 2 0 )$ ) and one drawn image $( 6 0 0 \times 2 2 5 )$ , which we call Image A and Image $B$ , respectively. The two images were manually selected as they contain rich texture and are diverse, but their choice was not optimized for performance. We test only two images due to the cost of running a full set of experiments (each image is expanded up to $1 . 2 \mathbf { M }$ times for training some of the models, as explained above). However, this is sufficient to prove our main points. We also test another $( 1 1 6 5 \times 5 8 5 )$ ) Image $C$ to ablate the “crowdedness” of an image, as this latter contains large areas covering no objects. While resolution matters to some extent as a bigger image contains more pixels, the information within is still far more correlated, and thus more redundant than sampling several smaller images. In particular, the resolution difference in Image $\mathtt { A }$ and $_ \mathrm { B }$ appears to be negligible in our experiments. For CIFAR-10, where $S = 3 2$ we only use Image B due to the resolution difference. In direct comparison, Image B is the size of about 132 CIFAR images which is still much less than $d = 5 0 { , } 0 0 0$ . For $N > 1$ , we select the source images randomly from each method’s training set.
+
+# 3.2 REPRESENTATION LEARNING METHODS
+
+Generative models. Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) learn to generate images using an adversarial objective: a generator network maps noise samples to image samples, approximating a target image distribution and a discriminator network is tasked with distinguishing generated and real samples. Generator and discriminator are pitched one against the other and learned together; when an equilibrium is reached, the generator produces images indistinguishable (at least from the viewpoint of the discriminator) from real ones.
+
+Bidirectional Generative Adversarial Networks (BiGAN) (Donahue et al., 2017; Dumoulin et al., 2016) are an extension of GANs designed to learn a useful image representation as an approximate inverse of the generator through joint inference on an encoding and the image. This method’s native augmentation uses random crops and random horizontal flips to learn features from $S = 1 2 8$ sized images. As opposed to the other two methods discussed below it employs leaky ReLU nonlinearities as is typical in GAN discriminators.
+
+Rotation. Most image datasets contain pictures that are ‘upright’ as this is how humans prefer to take and look at them. This photographer bias can be understood as a form of implicit data labelling. RotNet (Gidaris et al., 2018) exploits this by tasking a network with predicting the upright direction of a picture after applying to it a random rotation multiple of 90 degrees (in practice this is formulated as a 4-way classification problem). The authors reason that the concept of ‘upright’ requires learning high level concepts in the image and hence this method is not vulnerable to exploiting low-level visual information, encouraging the network to learn more abstract features. In our experiments, we test this hypothesis by learning from impoverished datasets that may lack the photographer bias. The native augmentations that RotNet uses on the $S { = } 2 5 6$ inputs only comprise horizontal flips and non-scaled random crops to $2 2 4 \times 2 2 4$ .
+
+Clustering. DeepCluster (Caron et al., 2018) is a recent state-of-the-art unsupervised representation learning method. This approach alternates $k$ -means clustering to produce pseudo-labels for the data and feature learning to fit the representation to these labels. The authors attribute the success of the method to the prior knowledge ingrained in the structure of the convolutional neural network (Ulyanov et al., 2018).
+
+The method alternatives between a clustering step, in which $k$ -means is applied on the PCA-reduced features with ID for each i $k = 1 0 ^ { 4 }$ , and a learning step, in which the network is trainedder a set of augmentations (random resized crops with usterand $\beta { \dot { = } } 0 . 0 8 , \gamma = { \textstyle \frac { 3 } { 4 } }$ horizontal flips) that constitute its native augmentations used on top of the $S { = } 2 5 6$ input images.
+
+# 4 EXPERIMENTS
+
+We evaluate the representation learning methods on ImageNet and CIFAR-10/100 using linear probes (Section 4.1). After ablating various choices of transformations in our augmentation protocol (Section 4.2), we move to the core question of the paper: whether a large dataset is beneficial to unsupervised learning, especially for learning early convolutional features (Section 4.3).
+
+# 4.1 LINEAR PROBES AND BASELINE ARCHITECTURE
+
+In order to quantify if a neural network has learned useful feature representations, we follow the standard approach of using linear probes (Zhang et al., 2017). This amounts to solving a difficult task such as ImageNet classification by training a linear classifier on top of pre-trained feature representations, which are kept fixed. Linear classifiers heavily rely on the quality of the representation since their discriminative power is low.
+
+We apply linear probes to all intermediate convolutional layers of networks and train on the ImageNet LSVRC-12 (Deng et al., 2009) and CIFAR-10/100 (Krizhevsky, 2009) datasets, which are the standard benchmarks for evaluation in self-supervised learning. Our base encoder architecture is AlexNet (Krizhevsky et al., 2012) with BatchNorm, since this is a good representative model and is most often used in other unsupervised learning work for the purpose of benchmarking. This model has five convolutional blocks (each comprising a linear convolution later followed by ReLU and optionally max pooling). We insert the probes right after the ReLU layer in each block, and denote these entry points conv1 to conv5. Applying the linear probes at each convolutional layer allows studying the quality of the representation learned at different depths of the network.
+
+Table 1: Ablating data augmentation using MonoGAN (left). Training a linear classifier on the features extracted at different depths of the network for CIFAR-10.   
+
+<table><tr><td></td><td colspan="4">CIFAR-10</td></tr><tr><td></td><td>conv1</td><td>conv2</td><td>conv3</td><td>conv4</td></tr><tr><td>(a） Fully sup.</td><td>66.5</td><td>70.1</td><td>72.4</td><td>75.9</td></tr><tr><td>(b) Random feat.</td><td>57.8</td><td>55.5</td><td>54.2</td><td>47.3</td></tr><tr><td>（c） No aug.</td><td>57.9</td><td>56.2</td><td>54.2</td><td>47.8</td></tr><tr><td>d） Jitter</td><td>58.9</td><td>58.0</td><td>57.0</td><td>49.8</td></tr><tr><td>通 Rotation</td><td>61.4</td><td>58.8</td><td>56.1</td><td>47.5</td></tr><tr><td>(f) Scale</td><td>67.9</td><td>69.3</td><td>67.9</td><td>59.1</td></tr><tr><td>（g） Rot.&amp;jitter</td><td>64.9</td><td>63.6</td><td>61.0</td><td>53.4</td></tr><tr><td>? Rot.&amp; scale</td><td>67.6</td><td>69.9</td><td>68.0</td><td>60.7</td></tr><tr><td>i Jitter &amp; scale</td><td>68.1</td><td>71.3</td><td>69.5</td><td>62.4</td></tr><tr><td>i All</td><td>68.1</td><td>72.3</td><td>70.8</td><td>63.5</td></tr></table>
+
+Table 2: ImageNet LSVRC-12 linear probing evaluation (below). A linear classifier is trained on the (downsampled) activations of each layer in the pretrained model. We report classification accuracy averaged over 10 crops. The ‡ indicated that numbers are taken from (Zhang et al., 2017).   
+
+<table><tr><td rowspan="2">Method, Reference</td><td rowspan="2"></td><td colspan="6">ILSVRC-12</td></tr><tr><td>#images</td><td>conv1</td><td>conv2</td><td>conv3</td><td>conv4</td><td>conv5</td></tr><tr><td>(a)</td><td>Full-supervision‡</td><td>1,281,167</td><td>19.3</td><td>36.3</td><td>44.2</td><td>48.3</td><td>50.5</td></tr><tr><td>(b)</td><td>(Oyallon et al., 2017): Scattering</td><td>0</td><td>-</td><td>18.9</td><td>-</td><td>1</td><td>1</td></tr><tr><td>(c)</td><td>Random*</td><td>0</td><td>11.6</td><td>17.1</td><td>16.9</td><td>16.3</td><td>14.1</td></tr><tr><td>(d)</td><td>(Krahenbuhl et al., 2016):k-means‡</td><td>~160</td><td>17.5</td><td>23.0</td><td>24.5</td><td>23.2</td><td>20.6</td></tr><tr><td>(e)</td><td>(Donahue et al., 2017): BiGAN‡</td><td>1,281,167</td><td>17.7</td><td>24.5</td><td>31.0</td><td>29.9</td><td>28.0</td></tr><tr><td>(f)</td><td>mono,Image A</td><td>1</td><td>20.4</td><td>30.9</td><td>33.4</td><td>28.4</td><td>16.0</td></tr><tr><td>（g)</td><td>mono, Image B</td><td>1</td><td>20.5</td><td>30.4</td><td>31.6</td><td>27.0</td><td>16.8</td></tr><tr><td>(h)</td><td>deka</td><td>10</td><td>16.2</td><td>16.5</td><td>16.5</td><td>13.1</td><td>7.5</td></tr><tr><td>(i</td><td>kilo</td><td>1,000</td><td>16.1</td><td>17.7</td><td>18.3</td><td>17.6</td><td>13.5</td></tr><tr><td>i</td><td>(Gidaris et al., 2018): RotNet</td><td>1,281,167</td><td>18.8</td><td>31.7</td><td>38.7</td><td>38.2</td><td>36.5</td></tr><tr><td>()</td><td>mono,Image A</td><td>1</td><td>19.9</td><td>30.2</td><td>30.6</td><td>27.6</td><td>21.9</td></tr><tr><td>1</td><td>mono, Image B</td><td>1</td><td>17.8</td><td>27.6</td><td>27.9</td><td>25.4</td><td>20.2</td></tr><tr><td>(m)</td><td>deka</td><td>10</td><td>19.6</td><td>30.7</td><td>32.6</td><td>28.9</td><td>22.6</td></tr><tr><td>(n)</td><td>kilo</td><td>1,000</td><td>21.0</td><td>33.5</td><td>36.5</td><td>34.0</td><td>29.4</td></tr><tr><td>(0)</td><td>(Caron et al., 2018): DeepCluster</td><td>1,281,167</td><td>18.0</td><td>32.5</td><td>39.2</td><td>37.2</td><td>30.6</td></tr><tr><td>(p)</td><td>mono, Image A</td><td>1</td><td>20.7</td><td>31.5</td><td>32.5</td><td>28.5</td><td>21.0</td></tr><tr><td>(q)</td><td>mono,Image B</td><td>1</td><td>19.7</td><td>30.1</td><td>31.6</td><td>28.5</td><td>20.4</td></tr><tr><td>(r)</td><td>mono, Image C</td><td>1</td><td>18.9</td><td>29.2</td><td>31.5</td><td>28.9</td><td>23.5</td></tr><tr><td>(s）</td><td>deka</td><td>10</td><td>18.5</td><td>29.0</td><td>31.1</td><td>28.2</td><td>21.9</td></tr><tr><td>(t</td><td>kilo</td><td>1,000</td><td>19.5</td><td>29.8</td><td>33.0</td><td>31.7</td><td>26.8</td></tr></table>
+
+Table 3: CIFAR-10/100. Accuracy of linear classifiers on different network layers.   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="4"></td><td colspan="4">CIFAR-100</td></tr><tr><td>conv1</td><td>conv2</td><td>CIFAR-10 conv3</td><td>conv4</td><td>conv1</td><td>conv2</td><td>conv3</td><td>conv4</td></tr><tr><td>Fully supervised</td><td>66.5</td><td>70.1</td><td>72.4</td><td>75.9</td><td>38.7</td><td>43.6</td><td>44.4</td><td>46.5</td></tr><tr><td>Random</td><td>57.8</td><td>55.5</td><td>54.2</td><td>47.3</td><td>30.9</td><td>29.8</td><td>28.6</td><td>24.1</td></tr><tr><td>RotNet</td><td>64.4</td><td>65.6</td><td>65.6</td><td>59.1</td><td>36.0</td><td>35.9</td><td>34.2</td><td>25.8</td></tr><tr><td>GAN (CIFAR-10)</td><td>67.7</td><td>73.0</td><td>72.5</td><td>69.2</td><td>39.6</td><td>46.0</td><td>45.1</td><td>39.9</td></tr><tr><td>GAN (CIFAR-100)</td><td>1</td><td>1</td><td>1</td><td>1</td><td>38.1</td><td>42.2</td><td>44.0</td><td>46.6</td></tr><tr><td>MonoGAN</td><td>68.1</td><td>72.3</td><td>70.8</td><td>63.5</td><td>39.9</td><td>46.9</td><td>44.5</td><td>38.8</td></tr></table>
+
+Details. While linear probes are conceptually straightforward, there are several technical details that affect the final accuracy by a few percentage points. Unfortunately, prior work has used several slightly different setups, so that comparing results of different publications must be done with caution. To make matters more difficult, not all papers released evaluation source code. We prove this standardized testing code here2.
+
+In our implementation, we follow the original proposal (Zhang et al., 2017) in pooling each representation to a vector with 9600, 9216, 9600, 9600, 9216 dimensions for $\mathtt { c o n v l - 5 }$ using adaptive max-pooling, and absorb the batch normalization weights into the preceding convolutions. For evaluation on ImageNet we follow RotNet to train linear probes: images are resized such that the shorter edge has a length of 256 pixels, random crops of $2 2 4 \times 2 2 4$ are computed and flipped horizontally with $5 0 \%$ probability. Learning lasts for 36 epochs and the learning rate schedule starts from 0.01 and is divided by five at epochs 5, 15 and 25. The top-1 accuracy of the linear classifier is then measured on the ImageNet validation subset. This uses DeepCluster’s protocol, extracting 10 crops for each validation image (four at the corners and one at the center along with their horizontal flips) and averaging the prediction scores before the accuracy is computed. For CIFAR-10/100 data, we follow the same learning rate schedule and for both training and evaluation we do not reduce the dimensionality of the representations and keep the images’ original size of $3 2 \times 3 2$ .
+
+# 4.2 EFFECT OF AUGMENTATIONS
+
+In order to better understand which image transformations are important to learn a good feature representations, we analyze the impact of augmentation settings. For speed, these experiments are conducted using the CIFAR-10 images $Q = 5 0$ , 000 in the training set) and with the smaller source Image $_ \mathrm { B }$ and a GAN using the Wasserstein GAN formulation with gradient penalty (Gulrajani et al., 2017). The encoder is a smaller AlexNet-like CNN consisting of four convolutional layers (kernel sizes: $7 , 5 , 3 , 3$ ; strides: 3, 2, 2, 1) followed by a single fully connected layer as the discriminator. Given that the GAN is trained on a single image (w/ augmentations), we call this setting MonoGAN.
+
+Table 1 reports all $2 ^ { 3 }$ combinations of the three main augmentations (scale, rotation, and jitter) and a randomly initialized network baseline (see Table 1 (b)) using the linear probes protocol discussed above. Without data augmentation the model only achieves marginally better performance than the random network (which also achieves a non-negligible level of performance (Ulyanov et al., 2017; Caron et al., 2018)). This is understandable since the dataset literally consists of a single training image cloned $d$ times. Color jitter and rotation slightly improve the performance of all probes by 1- $2 \%$ points, but random rescaling adds at least ten points at every depth (see Table 1 (f,h,i)) and is the most important single augmentation. A similar conclusion can be drawn when two augmentations are combined, although there are diminishing returns as more augmentations are combined. Overall, we find all three types of augmentations are of importance when training in the ultra-low data setting.
+
+# 4.3 BENCHMARK EVALUATION
+
+We analyze how performance varies as a function $N$ , the number of actual samples that are used to generated the augmented datasets, and compare it to the gold-standard setup (in terms of choice of training data) defined in the papers that introduced each method. The evaluation is again based on linear probes (Section 4.1).
+
+Mono is enough. From Table 2 we make the following observations. Training with just a single source image (f,g,k,l,p,q) is much better than random initialization (c) for all layers. Notably, these models also outperform Gabor-like filters from Scattering networks (Bruna & Mallat, 2013), which are hand crafted image features, replacing the first two convolutional layers as in (Oyallon et al., 2017). Using the same protocol as in the paper, this only achieves an accuracy of $1 8 . 9 \%$ compared to (p)’s conv $2 > 3 0 \%$ .
+
+More importantly, when comparing within pretext task, even with one image we are able to improve the quality of conv1–conv3 features compared to full (unsupervised) ImageNet training for GAN based self-supervision (e-i). For the other methods $( \mathrm { j \cdot }$ -n, o-s) we reach and also surpass the performance for the first layer and are within $1 . 5 \%$ points for the second. Given that the best unsupervised performance for conv2 is 32.5, our method using a single source Image A (Table 2, p) is remarkably close with 31.5.
+
+Image contents. While we surpass the GAN based approach of (Donahue et al., 2017) for both single source images, we find more nuanced results for the other two methods: For RotNet, as expected, the photographic bias cannot be extracted from a single image. Thus its performance is low with little training data and increases together with the number of images (Table 2, j-n). When comparing Image A and B trained networks for RotNet, we find that the photograph yields better performance than the hand drawn animal image. This indicates that the method can extract rotation information from low level image features such as patches which is at first counter intuitive. Considering that the hand-drawn image does not work well, we can assume that lighting and shadows even in small patches can indeed give important cues on the up direction which can be learned even from a single (real) image. DeepCluster shows poor performance in conv1 which we can improve upon in the single image setting (Table 2, o-r). Naturally, the image content matters: a trivial image without any image gradient (e.g. picture of a white wall) would not provide enough signal for any method. To better understand this issue, we also train DeepCluster on the much less cluttered Image C to analyze how much the image influences our claims. We find that even though this image contains large parts of sky and sea, the performance is only slightly lower than that of Image A. This finding indicates that the augmentations can even compensate for large untextured areas and the exact choice of image is not critical.
+
+![](images/47da3d26c9e9068651078be6be738408c464663bb72932691e02010a521e8e9a.jpg)  
+Figure 2: conv1 filters trained using a single image. The 96 learned $( 3 \times 1 1 \times 1 1 )$ filters for the first layer of AlexNet are shown for each single training image and method along with their linear classifier performance. For visualization, each filter is normalized to be in the range of $( - 1 , 1 )$ .
+
+More than one image. While BiGAN fails to converge for $N \in \{ 1 0 , 1 0 0 0 \}$ , most likely due to issues in learning from a distribution which is neither whole images nor only patches, we find that both RotNet and DeepCluster improve their performance in deeper layers when increasing the number of training images. However, for conv1 and conv2, a single image is enough. In deeper layers, DeepCluster seems to require large amounts of source images to yield the reported results as the deka- and kilo- variants start improving over the single image case (Table 2, o-t). This need for data also explains the gap between the two input images which have different resolutions. Summarizing Table 2, we can conclude that learning conv1, conv2 and for the most part conv3 (33.4 vs. 39.4) on over 1M images does not yield a significant performance increase over using one single training image — a highly unexpected result.
+
+Generalization. In Table 3, we show the results of training linear classifiers for the CIFAR-10 dataset and compare against various baselines. We find that the GAN trained on the smaller Image B outperforms all other methods including the fully-supervised trained one for the first convolutional layer. We also outperform the same architecture trained on the full CIFAR-10 training set using RotNet, which might be due to the fact that either CIFAR images do not contain much information about the orientation of the picture or because they do not contain as many objects as in ImageNet. While the GAN trained on the whole dataset outperforms the MonoGAN on the deeper layers, the gap stays very small until the last layer. These findings are also reflected in the experiments on the CIFAR-100 dataset shown in Table 3. We find that our method obtains the best performance for the first two layers, even against the fully supervised version. The gap between our mono variant and the other methods increases again with deeper layers, hinting to the fact that we cannot learn very high level concepts in deeper layers from just one single image. These results corroborate the finding that our method allows learning very generalizable early features that are not domain dependent.
+
+# 4.4 QUALITATIVE ANALYSIS
+
+Visual comparison of weights. In Figure 2, we compare the learned filters of all first-layer convolutions of an AlexNet trained with the different methods and a single image. First, we find that the filters closely resemble those obtained via supervised training: Gabor-like edge detectors and various color blobs. Second, we find that the look is not easily predictive of its performance, e.g.
+
+while generatively learned filters (BiGAN) show many edge detectors, its linear probes performance is about the same as that of DeepCluster which seems to learn many somewhat redundant point features. However, we also find that some edge detectors are required, as we can confirm from RotNet and DeepCluster trained on Image B, which yield less crisp filters and worse performances.
+
+Table 4: Finetuning experiments The pretrained model’s first two convolutions are left frozen (or replaced by the Scattering transform) and the nework is retrained using ImageNet LSVRC-12 training set.   
+
+<table><tr><td></td><td>Top-1</td></tr><tr><td>Full sup.</td><td>59.4</td></tr><tr><td>Random</td><td>42.6</td></tr><tr><td>Scattering</td><td>49.2</td></tr><tr><td>BiGAN, A</td><td>51.4</td></tr><tr><td>RotNet, A</td><td>49.5</td></tr><tr><td>DeepCluster A</td><td>52.5</td></tr></table>
+
+![](images/28641601f8b033094919f1f4d9ec2805f166320615575ce7240f102b217c3607.jpg)  
+Figure 3: Style transfer with single-image pretraining. We show two style transfer results using the Image A trained BiGAN and the ImageNet pretrained AlexNet.
+
+Fine-tuning instead of freezing. In Tab. 4, we show the results of retraining a network with the first two convolutional filters, or the scattering transform from (Oyallon et al., 2017), left frozen. We observe that our single image trained DeepCluster and BiGAN models achieve performances closes to the supervised benchmark. Notably, the scattering transform as a replacement for conv1-2 performs slightly worse than the analyzed single image methods. We also show in the appendix the results of retraining a network initialized with the first two convolutional layers obtained from a single image and subsequently linearly probing the model. The results are shown in Appendix Tab. 5 and we find that we can recover the performance of fully-supervised networks, i.e. the first two convolutional filters trained from just a single image generalize well and do not get stuck in an image specific minimum.
+
+Neural style transfer. Lastly, we show how our features trained on only a single image can be used for other applications. In Figure 3 we show two basic style transfers using the method of (Gatys et al., 2016) from an official PyTorch tutorial3. Image content and style are separated and the style is transferred from the source to target image using all CNN features, not just the shallow layers. We visually compare the results of using our features and from full ImageNet supervision. We find almost no visual differences in the stylized images and can conclude that our early features are equally powerful as fully supervised ones for this task.
+
+# 5 CONCLUSIONS
+
+We have made the surprising observation that we can learn good and generalizable features through self-supervision from one single source image, provided that sufficient data augmentation is used. Our results complement recent works (Mahajan et al., 2018; Goyal et al., 2019) that have investigated self-supervision in the very large data regime. Our main conclusion is that these methods succeed perfectly in capturing the simplest image statistics, but that for deeper layers a gap exist with strong supervision which is compensated only in limited manner by using large datasets. This novel finding motivates a renewed focus on the role of augmentations in self-supervised learning and critical rethinking of how to better leverage the available data.
+
+# ACKNOWLEDGEMENTS.
+
+We thank Aravindh Mahendran for fruitful discussions. Yuki Asano gratefully acknowledges support from the EPSRC Centre for Doctoral Training in Autonomous Intelligent Machines & Systems (EP/L015897/1). The work is supported by ERC IDIU-638009.
+
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+
+# A APPENDIX
+
+# A.1 IMAGENET TRAINING IMAGES
+
+![](images/046e21f8bf046efcf2184e9d1086c428f27cb732dddd0a050c1fa62dee848678.jpg)  
+Figure 4: ImageNet images for the $N { = } 1 0$ experiments.
+
+The images used for the $N { = } 1 0$ experiments are shown in fig. 4.
+
+# A.2 VISUAL COMPARISON OF FILTERS
+
+![](images/2f5f075fffec40a9fe974918f1470779b27b8abdc86af77cd7e6cd3a644973b5.jpg)  
+Figure 5: Filter visualization. We show activation maximization (left) and retrieval of top 9 activated images from the training set of ImageNet (right) for four random non-cherrypicked target filters. From top to bottom: conv1-5 of the BiGAN trained on a single image A. The filter visualization is obtained by learning a (regularized) input image that maximizes the response to the target filter using the library Lucid (Olah et al., 2018).
+
+In order to understand what deeper neurons are responding to in our model, we visualize random neurons via activation maximization (Erhan et al., 2009; Zeiler & Fergus, 2014) in each layer. Additionally, we retrieve the top-9 images in the ImageNet training set that activate each neuron most in Figure 5. Since the mono networks are not trained on the ImageNet dataset, it can be used here for visualization. From the first convolutional layer we find typical neurons strongly reacting to oriented edges. In layers 2-4 we find patterns such as grids (conv2:3), and textures such as leopard skin (conv2:2) and round grid cover (conv4:4). Confirming our hypothesis that the neural network is only extracting patterns and not semantic information, we do not find any neurons particularly specialized to certain objects even in higher levels as for example dog faces or similar which can be fund in supervised networks. This finding aligns with the observations of other unsupervised methods (Caron et al., 2018; Zhang et al., 2017). As most neurons extract simple patterns and textures, the surprising effectiveness of training a network using a single image can be explained by the recent finding that even CNNs trained on ImageNet rely on texture (as opposed to shape) information to classify (Geirhos et al., 2019).
+
+Table 5: Finetuning experiments Models are initialized using conv1 and conv2 from various single image trained models and the whole network is fine-tuned using ImageNet LSVRC-12 training set. Accuracy is averaged over 10 crops.   
+
+<table><tr><td>c1</td><td>c2 c3</td><td>c4</td><td>c5</td></tr><tr><td>Full sup.</td><td>19.3 36.3 44.2</td><td>48.3 50.5</td><td></td></tr><tr><td>BiGAN, A</td><td>22.5 37.6 44.2</td><td>47.6 48.3</td><td></td></tr><tr><td>RotNet, A</td><td>22.0 38.2</td><td>44.8 49.2 51.8</td><td></td></tr><tr><td>DeepCluster, A 21.8 35.9</td><td>43.6</td><td>48.8</td><td>50.4</td></tr></table>
+
+# A.3 RETRAINING FROM SINGLE IMAGE INITIALIZATION
+
+In Table 5, we initialize AlexNet models using the first two convolutional filters learned from a single image and retrain them using ImageNet. We find that the networks recover their performance fully and the first filters do not make the network stuck in a bad local minimum despite having been trained on a single image from a different distribution. The difference from the BiGAN to the full supervision model is likely due to it using a smaller input resolution (112 instead of 224), as the BiGAN’s output resolution is limited.
+
+# A.4 LINEAR PROBES ON IMAGENET
+
+We show two plots of the ImageNet linear probes results (Table 2 of the paper) in fig. 6. On the left we plot performance per layer in absolute scale. Naturally the performance of the supervised model improves with depth, while all unsupervised models degrade after conv3. From the relative plot on the right, it becomes clear that with our training scheme, we can even slightly surpass supervised performance on conv1 presumably since our model is trained with sometimes very small patches, thus receiving an emphasis on learning good low level filters. The gap between all self-supervised methods and the supervised baseline increases with depth, due to the fact that the supervised model is trained for this specific task, whereas the self-supervised models learn from a surrogate task without labels.
+
+![](images/9bb9bbb4170ebecfa72682345ec6c7569f3b6ee0c4cf53fa2ee208649efb83a1.jpg)  
+Figure 6: Linear Classifiers on ImageNet. Classification accuracies of linear classifiers trained on the representations from Table 2 are shown in absolute scale.
+
+# A.5 EXAMPLE AUGMENTED TRAINING DATA
+
+In figs. 7 to 10 we show example patches generated by our augmentation strategy for the datasets with different N. Even though the images and patches are very different in color and shape distribu
+
+tion, our model learns weights that perform similarly in the linear probes benchmark (see Table 2 in the paper).
+
+![](images/d7b4139a73a054c21258fc27b0228d0a962fd351f9792c2826ceaccf0656d21e.jpg)  
+Figure 7: Example crops of Image A ( $N = 1$ ) dataset.
+
+![](images/edd8e1f5572325341cf862f150b4183d8a280538461d8991f5f2d6aa06e81d59.jpg)  
+Figure 8: Example crops of Image B $N = 1$ ) dataset. 50 samples were selected randomly.
+
+![](images/181050cf9ad7ded511060047fb8ebdbc14b65be193fdd00679f0687df8f35d45.jpg)  
+Figure 9: Example crops of deka $N = 1 0$ ) dataset. 50 samples were selected randomly.
+
+![](images/5f1fd0a8bdc394bef8d19a152e1205bbd6f49971803be773b209278e19a9fa63.jpg)  
+Figure 10: Example crops of kilo ( $\overline { { N = 1 0 0 0 } }$ ) dataset. 50 samples were selected randomly.

parse/train/SJxbu6VKDr/SJxbu6VKDr.md

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+# GATED CHANNEL TRANSFORMATION FOR VISUAL RECOGNITION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In this work, we propose a generally applicable transformation unit for visual recognition with deep convolutional neural networks. This transformation explicitly models channel relationships with explainable control variables. These variables determine the neuron behaviors of competition or cooperation, and they are jointly optimized with convolutional weights towards more accurate recognition. In Squeeze-and-Excitation (SE) Networks, the channel relationships are implicitly learned by fully connected layers, and the SE block is integrated at the block-level. We instead introduce a channel normalization layer to reduce the number of parameters and computational complexity. This lightweight layer incorporates a simple $l _ { 2 }$ normalization, enabling our transformation unit applicable to operator-level without much increase of additional parameters. Extensive experiments demonstrate the effectiveness of our unit with clear margins on many vision tasks, i.e., image classification on ImageNet, object detection and instance segmentation on COCO, video classification on Kinetics.
+
+# 1 INTRODUCTION
+
+Convolutional Neural Networks (CNNs) have proven to be critical and robust in visual recognition tasks, such as image classification (Huang et al., 2018), detection (Singh et al., 2018), and segmentation (Singh et al., 2018). Notably, a single convolutional layer operates only on a neighboring local context of each spatial position of a feature map, which could possibly lead to local ambiguities (Torralba, 2003). To relief this problem, VGGNets (Simonyan & Zisserman, 2015) were proposed to construct deep CNNs, using a series of convolutional layers with non-linear activation functions and downsampling operators to cover a large extent of context. Moreover, He et al. (2016a) introduced a residual connection to help CNNs benefit from deeper architectures further.
+
+Apart from improving the depth of CNNs, another branch of methods focuses on augmenting convolutional layer with modules that directly operate on context across large neighborhoods. Squeezeand-Excitation Networks (SE-Nets) (Hu et al., 2018b) leveraged globally embedding information to model channel relationship and modulate feature maps on the channel-wise level. Moreover, its following method, GE-Nets (Hu et al., 2018a), used largely neighboring embedding instead. These modules can be conveniently assembled into modern networks, such as ResNets (He et al., 2016a) and Inception (Szegedy et al., 2015) networks, to improve the representational ability of networks.
+
+However, the SE module uses two fully connected $( F C )$ layers to process channel-wise embeddings, which leads to two problems. First, the number of SE modules to be applied in CNNs is limited. In Hu et al. (2018b), SE module was applied at the block-level, i.e., a single SE module is utilized per Res-block (He et al., 2016a) or Inception-block (Szegedy et al., 2016). The dimension of the $F C$ layer is decreased to save the computational cost further. However, the designed $F C$ layers still hinder the wide deployment of SE modules across all layers. Second, due to the complexity of the parameters in $F C$ (or convolutional layer in GE), it is difficult to analyze the interactions among the channels at different layers. The channel relationships learned by convolution and FC operations are inherently implicit (Hu et al., 2018b), resulting in agnostic behaviors of the neuron outputs.
+
+In this paper, we propose a Gated Channel Transformation (GCT) for efficient and accurate contextual information modeling. First, we use a normalization component to replace the $F C$ layers. Normalization methods, e.g., Local Response Normalization (LRN) (Krizhevsky et al., 2012), create competitions among different neurons in neural networks. Batch normalization (Ioffe & Szegedy,
+
+2015) and its variants can smooth gradient and have been widely used in accelerating CNNs training process. We leverage a simple $l _ { 2 }$ normalization for modeling channel relationship, which is more stable and computationally efficient comparing to FC layers. Second, we introduce a few channelwise parameters to control the behavior of the gated adaptation of feature channels. Compared to the large number of parameters in $F C$ , our designed parameters are much more lightweight. Besides, the gating weight parameter is convenient for channel relationship analysis and is helpful to understand the effect of GCT modules across different layers. According to our visualization analysis, GCT prefers to encourage cooperation in shallower layers, but competition is enhanced in deeper layers.
+
+Our experiments show that GCT is a simple and effective architecture for modeling relationship among channels. It significantly improves the generalization capability of deep convolutional networks across visual recognition tasks and datasets.
+
+# 2 RELATED WORK
+
+Gating and attention mechanisms. Gating mechanisms have been successfully deployed in some recurrent neural network architectures. Long Short-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) introduced an input gate, output gate and forget gate, which are used to regulate the flow of information into and out of the module. Based on gating mechanisms, some attention methods focus on forcing computational resources towards the most informative components of features (Larochelle & Hinton, 2010; Mnih et al., 2014; Vaswani et al., 2017). Recent works introduce the attention mechanism into convolutional networks (e.g., (Gehring et al., 2017; Dauphin et al., 2017)). Following these studies, SE-Nets (Hu et al., 2018b) and its following work GE-Nets (Hu et al., 2018a) introduced a lightweight gating mechanism which focuses on enhancing the representational power of the convolutional network by modeling channel-wise relationship. Compared to the SE module, our GCT also pays attention to the cross-channel relationship but can achieve better performance gains with less computation and parameters.
+
+Normalization layers. In recent years, normalization layers have been widely used in deep networks to create competition between neurons (Krizhevsky et al., 2012) and produce smoother optimization surfaces (Ioffe & Szegedy, 2015). Local Response Normalization (LRN) (Krizhevsky et al., 2012) computes the statistics in a small neighborhood among channels for each pixel. Batch Normalization (BN) (Ioffe & Szegedy, 2015) utilizes global spatial information along the batch dimension and suggests to be deployed for all layers. Layer Normalization (LN) (Ba et al., 2016) computes along the channel dimension instead of the batch dimension. Group Normalization (GN) (Wu & He, 2018) differently divides the channels into groups and computes within each group the mean and variance for normalization. Similar to LRN, GN and LN, our GCT also utilizes channel-related information with normalization structure.
+
+Deep architectures. VGGNets (Simonyan & Zisserman, 2015) and Inception networks (Szegedy et al., 2015) demonstrated that it was significant to improve the quality of representation by increasing the depth of a network. ResNets (He et al., 2016a) utilized shortcut connections to identity-based skip connections, and proved that it was highly effective to build considerably deeper and stronger networks with them. Some other researchers focused on improving the representation ability of the computational elements contained within a network (Szegedy et al., 2016). The more diverse composition of operators within a computational element can be constructed with multi-branch convolutions or pooling layers. Other than this, grouped convolutions have proven to be a practical method to increase the cardinality of learned transformations (Xie et al., 2017). We build our GCT on these deep architectures. All these networks with GCT achieve promising performance improvements, but the growth of computational complexity is negligible.
+
+# 3 GATED CHANNEL TRANSFORMATION
+
+SE-Nets proposed a lightweight SE module to augment convolutional networks by operating on a global context. The SE module contains two operators, i.e., a “squeeze” operator to embed channel context and an “excitation” operator to modulate the feature maps. The architecture of our Gated Channel Transformation benefits from this framework. Differently, GCT leverages a normalization operator instead of the $F C$ layers in the SE module for channel relationship modeling. Notably, the normalization operator is parameter-free. To make GCT learnable, we redesign the structure of the “squeeze” and “excitation” operators. Our new operators contain three sets of channel-wise trainable parameters. Thus, GCT is more convenient to be deployed occupying a small number of parameters. The gating parameters can be visualized for easier analysis of GCT’s behavior, while the discriminative ability is maintained.
+
+![](images/35db85c1efdd0ee01b8a5c414d84000799edc706d13baf04e0ab8d2a4f6120c9.jpg)  
+Figure 1: An overview of the structure of Gated Channel Transformation (GCT).
+
+Let $\mathbf { x } \in \mathbb { R } ^ { C \times H \times W }$ be an activation feature in a convolutional network, where $H$ and $W$ are the spatial height and width, and $C$ is the number of channels. In general, GCT performs the following transformation:
+
+$$
+\begin{array} { r } { \hat { \mathbf { x } } = F ( \mathbf { x } | \alpha , \gamma , \beta ) , \alpha , \gamma , \beta \in \mathbb { R } ^ { C } . } \end{array}
+$$
+
+Here $\alpha , \gamma$ and $\beta$ are trainable parameters. Embedding weights $_ { \pmb { \alpha } }$ are responsible for adapting the embedding outputs. The gating weights $\gamma$ and biases $\beta$ control the activation of the gate. They determine the behavior of GCT in each channel. The parameter complexity of GCT is $\bar { O ( C ) }$ , which is smaller than the SE module $( O ( C ^ { 2 } ) )$ ( $\mathrm { H u }$ et al., 2018b). In SE-Net, two $F C$ layers are leveraged, which have the parameter complexity of $O ( C ^ { 2 } )$ .
+
+An illustration of the structure of GCT is shown in Fig. 1. Let $\textbf { x } = ~ [ x _ { 1 } , x _ { 2 } , . . . , x _ { C } ] , x _ { c } ~ =$ $[ x _ { c } ^ { i , j } ] _ { H \times W } ~ \in ~ \mathbb { R } ^ { H \times W } , c ~ \in ~ \{ 1 , 2 , . . . , C \}$ , where $x _ { c }$ is corresponding to each channel of $\mathbf { x }$ . The detailed transformation consists of following parts.
+
+Global Context Embedding. Global context embedding (GCE) aggregates global context in each channel. GCE can exploit global contextual information outside the small receptive fields of convolutional layers. Given the embedding weights ${ \pmb { \alpha } } = [ \alpha _ { 1 } , . . . , \alpha _ { C } ]$ , GCE is defined as:
+
+$$
+s _ { c } = \alpha _ { c } | | x _ { c } | | _ { 2 } = \alpha _ { c } \{ [ \sum _ { i = 1 } ^ { H } \sum _ { j = 1 } ^ { W } ( x _ { c } ^ { i , j } ) ^ { 2 } ] + \epsilon \} ^ { \frac { 1 } { 2 } } ,
+$$
+
+where $\epsilon$ is a small constant to avoid the problem of derivation at the zero point. Different from SE, GCT does not use global average pooling (GAP) to aggregate channel context. GAP might fail in some extreme cases. For example, if SE is deployed after the Instance Normalization (Ulyanov et al., 2016) layer that is popular in style transfer task, the output of GAP will be constant for any inputs since IN fixes the mean of each channel of features. To avoid this problem, we choose $\ell _ { p }$ -norm instead. It is worth noting that GCT is robust with different $\ell _ { p }$ -norms. In Sec. 4.5, we compare the performance of some popular $\ell _ { p }$ -norms and choose the best one, $\ell _ { 2 }$ -norm, to be our default setting. Notably, the performance of $\ell _ { 1 }$ -norm is very close to $\ell _ { 2 }$ -norm and $\ell _ { 1 }$ -norm can be equivalently replaced by GAP when the input of GCT is always non-negative (for example, after ReLU activation). In this case, $\ell _ { 1 }$ -norm is more computationally efficient.
+
+Besides, we use trainable parameters $\alpha _ { c }$ to adjust each channel because different channels should have different significance.
+
+Channel Normalization. Normalization methods can model relationship in visual or photographic features (Lyu & Simoncelli, 2008) with lightweight computing resource (e.g., Ioffe & Szegedy (2015)). Similar to LRN, we use a $\ell _ { 2 }$ normalization to operate across channels, namely channel normalization (CN). Let $\mathbf { s } = [ s _ { 1 } , . . . , s _ { C } ]$ , the formula of CN is:
+
+![](images/d6365d27fdc7c5de3b2e985053b879fc13c21f86cccfb69f27cebcf4ba7ab755.jpg)  
+Figure 2: Training curve comparisons for ResNets with different depth on ImageNet.
+
+$$
+\hat { s } _ { c } = \frac { \sqrt { C } s _ { c } } { | | \mathbf { s } | | _ { 2 } } = \frac { \sqrt { C } s _ { c } } { [ ( \displaystyle \sum _ { c = 1 } ^ { C } s _ { c } ^ { 2 } ) + \epsilon ] ^ { \frac { 1 } { 2 } } } ,
+$$
+
+where $\epsilon$ is a small constant. The scalar $\sqrt { C }$ is used to normalize the scale of $\hat { s } _ { c }$ , avoiding a too small scale of $\hat { s } _ { c }$ when $C$ is large. Compared to the $F C$ layers used by SE, our CN operator has less computational complexity $( O ( C ) )$ compared to the $F C$ layers $( O ( C ^ { 2 } ) )$ .
+
+Gating Adaptation. We employ a gating mechanism, namely gating adaptation, to adapt the original feature. By introducing the gating mechanism, our GCT can facilitate both competition and cooperation during the training process. Let the gating weights $\gamma = [ \gamma _ { 1 } , . . . , \gamma _ { C } ]$ and the gating biases $\beta = [ \beta _ { 1 } , . . . , \beta _ { C } ]$ , we design the following gating function:
+
+$$
+\hat { x } _ { c } = x _ { c } [ 1 + \operatorname { t a n h } ( \gamma _ { c } \hat { s } _ { c } + \beta _ { c } ) ] .
+$$
+
+The scale of each original channel $x _ { c }$ will be adapted by its corresponding gate, i.e., $1 + \operatorname { t a n h } ( \gamma _ { c } \hat { s } _ { c } +$ $\beta _ { c , \ - }$ ). The trainable $\gamma _ { c }$ and $\beta _ { c }$ are employed to control the activation of gate. LRN benefits from only the competitions among the neurons (Krizhevsky et al., 2012). However, the gating mechanism in GCT is able to create both competition and cooperation among different channels. This capability is more consistent with the training process in biological neural networks (Demin & Nekhaev, 2018). When the gating weight of one channel $( \gamma _ { c } )$ is activated positively, GCT promotes this channel to compete with the others as in LRN. When the gating weight is activated negatively, GCT encourages this channel to cooperate with the others. We analyze these adaptive channel relationships in Sec.4.4.
+
+Besides, this gate function allows original features to pass to the next layer when the gating weights and biases are zeros, which is
+
+$$
+\hat { \mathbf { x } } = F ( \mathbf { x } | \alpha , \mathbf { 0 } , \mathbf { 0 } ) = \mathbf { 1 } \mathbf { x } = \mathbf { x } .
+$$
+
+The ability of modeling identity mapping can effectively improve the robustness to the degradation problem in deep networks. ResNets also benefits from this idea. Therefore, we propose to initialize $\gamma$ and $\beta$ to 0 in the initialization of GCT layers. By doing this, the initial steps of the training process will be more stable, and the final performance of GCT will be better.
+
+# 4 EXPERIMENTS
+
+We apply GCT for all the convolutional layers in deep networks rather than block-level deployment in SE-Net. In all GCT counterparts, we employ one GCT layer before each convolutional layer. In the Kinetics experiments, we apply GCT at the last two convolutional layers in each Res-Block. More training details are shown in Appendix A.
+
+# 4.1 EXPERIMENTS ON IMAGENET
+
+We experiment on the ImageNet 2012 dataset (Russakovsky et al., 2015) with $1 , 0 0 0$ classes. We train all the models on the 1.28M training images and evaluate on the $5 0 , 0 0 0$ validation images. We also conduct classification experiments on CIFAR (Krizhevsky & Hinton, 2009) in Appendix B.
+
+Implementation details. In the training process of all the models, the input image is $2 2 4 \times 2 2 4$ randomly cropped from a resized image using the same augmentation in Szegedy et al. (2015). We use SGD with a mini-batch size of 256. For ResNet-152 and ResNeXt-50, we use half mini-batch size and double the training steps). The weight decay is 0.0001, and the momentum is 0.9. The base learning rate is 0.1, and we divide it by 10 every 30 epochs. All models are trained for 100 epochs from scratch, using the weight initialization strategy described in He et al. (2015). Besides, we start the training process with a learning rate of 0.01 for 1 epoch. After the warmup, we go back to the original learning rate schedule. In all comparisons, we evaluate the error on the single $2 2 4 \times 2 2 4$ center crop from an image whose shorter side is 256. For ResNet-200 (He et al., 2016b), we evaluate on $3 2 0 \times 3 2 0$ following He et al. (2016b).
+
+Table 1: Improvement in error performance $( \% )$ on ImageNet. The numbers in brackets denote the improvement in performance over the baselines. ResNet- ${ } ^ { 2 0 0 ^ { * } }$ means we follow the strategy in (He et al., 2016b) to train this model on $2 2 4 \times 2 2 4$ but evaluate on $3 2 0 \times 3 2 0$ .   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>original</td><td rowspan=1 colspan=2>GCT</td></tr><tr><td rowspan=1 colspan=1>Network</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>VGG-16 (Simonyan &amp; Zisserman,2015)</td><td rowspan=1 colspan=1>26.2</td><td rowspan=1 colspan=1>8.3</td><td rowspan=1 colspan=1>25.1(1.1)</td><td rowspan=1 colspan=1>7.5(0.8)</td></tr><tr><td rowspan=1 colspan=1>Inception-v3 (Szegedy et al., 2016)</td><td rowspan=1 colspan=1>24.3</td><td rowspan=1 colspan=1>7.3</td><td rowspan=1 colspan=1>23.7(0.6)</td><td rowspan=1 colspan=1>7.1(0.2)</td></tr><tr><td rowspan=1 colspan=1>ResNeXt-50 (Xie et al., 2017)</td><td rowspan=1 colspan=1>22.4</td><td rowspan=1 colspan=1>6.3</td><td rowspan=1 colspan=1>21.7(0.7)</td><td rowspan=1 colspan=1>6.0(0.3)</td></tr><tr><td rowspan=1 colspan=1>ResNet-50 (He et al., 2016a)</td><td rowspan=1 colspan=1>23.8</td><td rowspan=1 colspan=1>7.0</td><td rowspan=1 colspan=1>22.7(1.1)</td><td rowspan=1 colspan=1>6.3（0.7)</td></tr><tr><td rowspan=1 colspan=1>ResNet-101 (He et al.,2016a)</td><td rowspan=1 colspan=1>22.2</td><td rowspan=1 colspan=1>6.2</td><td rowspan=1 colspan=1>21.4(0.8)</td><td rowspan=1 colspan=1>5.9（0.3)</td></tr><tr><td rowspan=1 colspan=1>ResNet-152 (He et al.,2016a)</td><td rowspan=1 colspan=1>21.6</td><td rowspan=1 colspan=1>5.9</td><td rowspan=1 colspan=1>20.8(0.8)</td><td rowspan=1 colspan=1>5.5(0.4)</td></tr><tr><td rowspan=1 colspan=1>ResNet-200*(He et al.,2016b)</td><td rowspan=1 colspan=1>20.7</td><td rowspan=1 colspan=1>5.2</td><td rowspan=1 colspan=1>19.7(1.0)</td><td rowspan=1 colspan=1>4.8(0.4)</td></tr></table>
+
+Table 2: Compared to SE in different networks on ImageNet. We evaluate the models of error performance $( \% )$ , GFLOPs (G) and parameters (M). G/P means GFLOPs/parameters. In VGG-16 experiments, SE is employed for all the convolutional layers, which is the same as GCT. In other experiments, SE is only employed in block level (Res-Block or Inception-Block) as proposed (Hu et al., 2018b). This difference makes that SE uses comparable GFLOPs with GCT.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>original</td><td rowspan=1 colspan=2>SE</td><td rowspan=1 colspan=2>GCT (ours)</td></tr><tr><td rowspan=1 colspan=1>Network</td><td rowspan=1 colspan=1>top-1/5</td><td rowspan=1 colspan=1>G/P</td><td rowspan=1 colspan=1>top-1/5</td><td rowspan=1 colspan=1>G/P</td><td rowspan=1 colspan=1>top-1/5</td><td rowspan=1 colspan=1>G/P</td></tr><tr><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>23.8/7.0</td><td rowspan=1 colspan=1>3.879/25.61</td><td rowspan=1 colspan=1>22.9/6.6</td><td rowspan=1 colspan=1>3.893/28.14</td><td rowspan=1 colspan=1>22.7/6.3</td><td rowspan=1 colspan=1>3.900/25.68</td></tr><tr><td rowspan=1 colspan=1>ResNeXt-50</td><td rowspan=1 colspan=1>22.4/6.3</td><td rowspan=1 colspan=1>3.795/25.10</td><td rowspan=1 colspan=1>22.0/6.1</td><td rowspan=1 colspan=1>3.809/27.63</td><td rowspan=1 colspan=1>21.7/6.0</td><td rowspan=1 colspan=1>3.821/25.19</td></tr><tr><td rowspan=1 colspan=1>VGG-16</td><td rowspan=1 colspan=1>26.2/8.3</td><td rowspan=1 colspan=1>15.497/138.37</td><td rowspan=1 colspan=1>25.2/7.7</td><td rowspan=1 colspan=1>15.525/138.60</td><td rowspan=1 colspan=1>25.1/7.5</td><td rowspan=1 colspan=1>15.516/138.38</td></tr><tr><td rowspan=1 colspan=1>Inception-v3</td><td rowspan=1 colspan=1>24.3/7.3</td><td rowspan=1 colspan=1>2.847/23.87</td><td rowspan=1 colspan=1>24.0/7.2</td><td rowspan=1 colspan=1>2.851/25.53</td><td rowspan=1 colspan=1>23.7/7.1</td><td rowspan=1 colspan=1>2.862/23.99</td></tr></table>
+
+Integration with deep modern architectures. We study the effects of integrating GCT layers with some state-of-the-art backbone architectures, e.g., ResNets and ResNeXts (Xie et al., 2017), in which we apply GCT before all the convolutional layers. We report all these results in Table 1. Compared to original architectures, we observe significant performance improvements by introducing GCT into networks. Particularly, the top-1 error of GCT-ResNet-101 is ${ \bf 2 1 . 4 \% }$ , which is even better than the ResNet-152 baseline $( 2 1 . 6 \% )$ with a deeper network and much more parameters. In addition, GCT is able to bring stable improvement in ResNets with different depth ( ${ \bf \cdot 1 . 1 \% }$ top-1 improvement in ResNet-50, $\mathbf { 0 . 8 \bar { \% } }$ in ResNet-152 and $\mathbf { 1 . 0 \% }$ in ResNet-200). Besides, we observe a smooth improvement throughout the training schedule, which is shown in Fig.2.
+
+We also explore the improvement with GCT in non-residual networks (e.g., VGG-16 (Simonyan & Zisserman, 2015) and Inception-v3 (Szegedy et al., 2016)). To stabilize the training process, we employ BN (Ioffe & Szegedy, 2015) layers after every convolutional layer. Similar to the effectiveness in residual architectures, GCT layers bring promising improvements in non-residual structures.
+
+Compared to SE. We conduct experiments on ImageNet to compare SE with GCT in both residual and non-residual networks and the results are reported in Table 2. We follow the methods in (Hu et al., 2018b) to integrate SE into VGG-16 (Simonyan & Zisserman, 2015), Inception (Szegedy et al., 2016), ResNet-50 (He et al., 2016a) and ResNeXt-50 (Xie et al., 2017) and train these models in same training schedule. Compared to SE, GCT always achieves better improvement.
+
+In order to compare computational complexity, we calculate the GFLOPs and the number of parameters. In VGG-16 experiments, SE is employed for all the convolutional layers, which is the same as GCT. Under this fair condition, GCT achieves better performance with less increase in both GFLOPs (0.019G vs.0.028G) and parameters (0.01M vs.0.23M). In other experiments, SE is only employed in block-level (Res-Block or Inception-Block) as proposed (Hu et al., 2018b), which means the number of SE is smaller than GCT. However, the increase in parameters of GCT is still much less than SE, and the value of GFLOPs is comparable. Compared to SE, the increase in parameters of GCT is negligible, but the performance is better.
+
+Table 3: Improvement on COCO with Mask R-CNN framework. $\mathbf { B N } ^ { * }$ means BN is frozen. $^ +$ means increasing the training iterations from 90K to 270K. When using GN, we follow the strategy in the original paper (Wu & He, 2018).   
+
+<table><tr><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>box head</td><td rowspan=1 colspan=1>box AP</td><td rowspan=1 colspan=1>mask AP</td></tr><tr><td rowspan=1 colspan=1>ResNet-50BN*</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>37.8</td><td rowspan=1 colspan=1>34.2</td></tr><tr><td rowspan=1 colspan=1>ResNet-50BN*+GCT</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>39.8(2.0)</td><td rowspan=1 colspan=1>36.0(1.8)</td></tr><tr><td rowspan=1 colspan=1>ResNet-101BN*</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>40.1</td><td rowspan=1 colspan=1>36.1</td></tr><tr><td rowspan=2 colspan=1>ResNet-101BN*+GCT+ResNet-50 BN*</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>42.0(1.9)</td><td rowspan=1 colspan=1>37.7(1.6)</td></tr><tr><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>38.6</td><td rowspan=1 colspan=1>34.5</td></tr><tr><td rowspan=1 colspan=1>+ResNet-50 GN</td><td rowspan=1 colspan=1>GN</td><td rowspan=1 colspan=1>40.8(2.2)</td><td rowspan=1 colspan=1>36.1(1.6)</td></tr><tr><td rowspan=1 colspan=1>+ResNet-50 BN*+GCT</td><td rowspan=1 colspan=1>GN</td><td rowspan=1 colspan=1>41.6(3.0)</td><td rowspan=1 colspan=1>37.1(2.6)</td></tr><tr><td rowspan=1 colspan=1>+ResNet-50BN*+GCT</td><td rowspan=1 colspan=1>GN+GCT</td><td rowspan=1 colspan=1>41.8(3.2)</td><td rowspan=1 colspan=1>37.3(2.8)</td></tr><tr><td rowspan=1 colspan=1>+ResNet-101 BN*</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>40.9</td><td rowspan=1 colspan=1>36.4</td></tr><tr><td rowspan=1 colspan=1>+ResNet-101 GN</td><td rowspan=1 colspan=1>GN</td><td rowspan=1 colspan=1>42.3(1.4)</td><td rowspan=1 colspan=1>37.2(0.8)</td></tr><tr><td rowspan=1 colspan=1>ResNet-101BN*+GCT</td><td rowspan=1 colspan=1>GN</td><td rowspan=1 colspan=1>43.1(2.2)</td><td rowspan=1 colspan=1>38.3(1.9)</td></tr></table>
+
+# 4.2 EXPERIMENTS ON COCO
+
+Next we evaluate the generalizability on the COCO dataset (Lin et al., 2014). We train the models on the COCO train2017 set and evaluate on the COCO eval2017 set (a.k.a minival).
+
+Implementation details. We experiment on the Mask R-CNN baselines (He et al., 2017) and its GN counterparts (Wu & He, 2018). All the backbone models are pre-trained on ImageNet using the scale and aspect ratio augmentation in Szegedy et al. (2015) and fine-tune on COCO with a batch size of 16 (2 images/GPU). Besides, all these experiments use the Feature Pyramid Network (FPN) Lin et al. (2017). We also use the same hyperparameters and two training schedules used in (Wu & He, 2018). The short schedule includes 90K iterations, in which the learning rate is divided by 10 at 60K and 80K iterations. The long schedule increases the iterations to 270K, in which the learning rate is divided by 10 at 210K and 250K. The base learning rate is 0.02 in both schedules.
+
+Improvements on Mask R-CNN. Table 3 shows the comparison of $\mathbf { B N } ^ { * }$ (frozen BN), GN and $\mathbf { B } \mathbf { N } ^ { * } { + } \mathbf { G } \mathbf { C } \mathbf { T }$ (using GCT before all the convolutional layers of the backbones). First, we use a short training schedule to compare baselines and GCT counterparts. GCT shows stable and significant improvement in both ResNet-50 and ResNet-101. In ResNet-50, GCT improves detection AP by 2.0 and segmentation AP by 1.8. Moreover, in ResNet-101, GCT also improves detection AP by 1.9 and segmentation AP by 1.6. Then, we use the long schedule to compare GN and $\mathbf { B N ^ { * } { + } G C T }$ . GN is more effective than BN when batch size is small as in this case of detection and segmentation using Mask R-CNN. However, we deploy GCT together with BN into the backbone, and these $\mathbf { B } \mathbf { N } ^ { * } { + } \mathbf { G } \mathbf { C } \mathbf { T }$ counterparts achieve much better performance than GN backbones. Compared to GN in ResNet-101, $\mathrm { \mathbf { B } N ^ { * } { + } G C T }$ improves detection AP by 0.8 and segmentation AP by 1.1. In particular, ResNet-101 with $\mathrm { \mathbf { B } N ^ { * } { + } G C T }$ trained in the short schedule achieves better segmentation AP (37.7) than the GN counterpart (37.2) trained with the long schedule. This GN counterpart also uses GN in the backbone, the box heads, and the FPN. We also explore to combine GCT with GN by introducing GCT into GN box head. The results show $\mathrm { \bf G N { + } } \mathrm { \bf G C T }$ achieves a better performance. It demonstrates the benefits of integrating GCT with GN. We now have shown the effectiveness of GCT in working with both BN and GN.
+
+# 4.3 EXPERIMENTS ON KINETICS
+
+Our previous experiments demonstrate the effectiveness of GCT on image-related tasks. We now evaluate the generalizability in video understanding task of action recognition on the large scale Kinetics-400 (Kay et al., 2017) dataset. We employ the ResNet-50 (3D) and ResNet-101 (3D) as the backbone and apply GCT in the last two convolutional layers in each Res-Block. The backbone networks are pre-trained on ImageNet (Russakovsky et al., 2015).
+
+![](images/4f3eed4af43b00704fbd313bc531759518072555c6f428f60926c05148a110c1.jpg)  
+Figure 3: Analysis. The visulization of parameters of $\gamma$ (Fig.(a), (b)), and the ratio of variance of GCT output and input feature (Fig.(c)) in all the GCT layers in ResNet-50 on ImageNet.
+
+We compare with the state-of-the-art Non-Local Networks (NL-Net) (Wang et al., 2018). The results show that GCT counterparts consistently improves the recognition accuracy over both the ResNet50 and ResNet-101 baselines, as shown in Table 4. Because of our limited memory resource, we can NOT apply GCT in all the convolutional layers, which we believe can further improve the performance.
+
+In summary, extensive experiments demonstrate that GCT is effective across a wide range of architectures, tasks, and datasets.
+
+# 4.4 ANALYSIS
+
+To analyze the behavior of GCT in different layers, we visualize the distribution of the gating weights $( \gamma )$ of each GCT layer in ResNet-50 on ImageNet. Further, we sort these distributions according to their layer index in 3D space (Fig. 3a). The bigger layer index means it is closer to the network output. To make the visualization clearer, we re-scale the vertical $z$ axis with $\boldsymbol { l o g ( 1 + z ) }$ , which corresponds to the percentage density of $\gamma$ . We also calculate the mean and standard deviation (std) of $\gamma$ in each layer and show them in a bar chart (Fig. 3b). As shown in Fig. 3a and 3b, the mean of $\gamma$ tends to be less than 0 in the GCT layers far from the network output. Oppositely, in the layers close to the output, the mean tends to be greater than 0.
+
+According to Eq. 3 & 4, the adaptation of channel $x _ { c }$ is related to $\hat { s } _ { c }$ , which corresponds to the ratio of the weighted $\ell _ { 2 }$ -norm of $x _ { c }$ (i.e., $s _ { c . }$ ) and the average of all the $s _ { c }$ . When the gating weight $\gamma _ { c }$ is greater than 0, the adaptation is positively correlated to $\hat { s } _ { c }$ and increases the variance between $x _ { c }$ and others; When $\gamma _ { c }$ is lower than 0, the adaptation is negatively correlated and reduces the variance.
+
+Based on the analysis and the results we observe, we suppose that GCT tends to reduce the difference among channels in layers far away from the output. This behavior is helpful to encourage cooperation among channels and relieve overfitting. Apart from this, GCT tends to increase the difference among channels when close to the output. Here, GCT acts like the attention mechanism that is useful for creating competition.
+
+To further validate our hypothesis, we calculate the ratio of the variance of output and input feature of each GCT layer, which we show in Fig. 3c. More visualizations are shown in Appendix C. Generally, the shallow layers learn low-level features to capture general characteristics like textures, edges, and corners. The feature variances become larger in deeper layers, where the high-level features are more discriminative and task-related. As expected, in the layers close to network output, GCT tends to magnify the variance of input feature (the ratio is always greater than 1), but in the layers far away from the output, GCT tends to reduce the variance (the ratio is always less than 1). This phenomenon is consistent with our previous hypothesis and shows that GCT is effective in creating both competition and cooperation among channels. Our observation validates that GCT can adaptively learn the channel relationships at different layers.
+
+# 4.5 ABLATION STUDIES.
+
+In this section, we conduct a serial of ablation experiments to explain the relative importance of each operator in the GCT. At last, we show how the performance changes with regards to the GCT position in a network.
+
+Table 4: Improvement in top-1 accuracy $( \% )$ over the state-ofthe-art method on Kinetics.   
+
+<table><tr><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>NL-Net</td><td rowspan=1 colspan=1>GCT</td></tr><tr><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>74.6</td><td rowspan=1 colspan=1>75.1(0.5)</td></tr><tr><td rowspan=1 colspan=1>ResNet-101</td><td rowspan=1 colspan=1>75.7</td><td rowspan=1 colspan=1>76.2(0.5)</td></tr></table>
+
+Table 5: Ablation experiments. We evaluate error performance in GCT-ResNet-50 on ImageNet $( \% )$ . The ResNet-50 baseline achieves a top-1 of 23.8 and a top-5 of 7.0.
+
+(a) Embedding operator.
+
+(b) Normalization operator.   
+
+<table><tr><td rowspan=1 colspan=1>Normalization</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>mean+variance</td><td rowspan=1 colspan=1>23.7</td><td rowspan=1 colspan=1>7.1</td></tr><tr><td rowspan=1 colspan=1>l1</td><td rowspan=1 colspan=1>22.9</td><td rowspan=1 colspan=1>6.4</td></tr><tr><td rowspan=1 colspan=1>l2</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.3</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>Norm</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>lo</td><td rowspan=1 colspan=1>23.1</td><td rowspan=1 colspan=1>6.7</td></tr><tr><td rowspan=1 colspan=1>l1</td><td rowspan=1 colspan=1>22.8</td><td rowspan=1 colspan=1>6.3</td></tr><tr><td rowspan=1 colspan=1>l2</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.3</td></tr></table>
+
+(c) Adaptation operator.   
+
+<table><tr><td rowspan=1 colspan=1>Adaptation</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>Sigmoid</td><td rowspan=1 colspan=1>22.9</td><td rowspan=1 colspan=1>6.5</td></tr><tr><td rowspan=1 colspan=1>1+ELU</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.4</td></tr><tr><td rowspan=1 colspan=1>1+tanh</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.3</td></tr></table>
+
+(d) Application position.   
+
+<table><tr><td rowspan=1 colspan=1>Position</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>after BN</td><td rowspan=1 colspan=1>23.1</td><td rowspan=1 colspan=1>6.6</td></tr><tr><td rowspan=1 colspan=1>before BN</td><td rowspan=1 colspan=1>23.1</td><td rowspan=1 colspan=1>6.5</td></tr><tr><td rowspan=1 colspan=1>before Conv</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.3</td></tr></table>
+
+Table 6: Clock time comparison. We calculate average inference times (ms) per batch by using 1 GTX 1080Ti with 16 batch size for 1,000 iterations on ImageNet. For the sake of fairness, the modules are applied for all the Convs in VGG-16.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>baseline</td><td rowspan=1 colspan=1>+SE</td><td rowspan=1 colspan=1>+GCT(l1-norm)</td><td rowspan=1 colspan=1>+GCT(l2-norm)</td></tr><tr><td rowspan=1 colspan=1>time(ms)/batch</td><td rowspan=1 colspan=1>51.11</td><td rowspan=1 colspan=1>87.3136.20↑</td><td rowspan=1 colspan=1>59.468.35↑</td><td rowspan=1 colspan=1>59.738.62↑</td></tr></table>
+
+Embedding component. To explain the importance of $\ell _ { p }$ -norm in GCE, we compare embedding operators with different $\ell _ { p }$ norm. We report the results in Table 5a, which shows all the $\ell _ { p }$ -norms are effective in GCE, but the $\ell _ { 2 }$ -norm is slightly better than $\ell _ { 1 }$ -norm. The results demonstrate the GCE of GCT is robust to different $\ell _ { p }$ -norms. In addition, we make a clock time comparison between SE and GCTs with different embedding component. As shown in Table 6, $\ell _ { 2 }$ -norm is computationally similar to $\ell _ { 1 }$ -norm and GCT is much more efficient than SE $( { \bf 8 . 6 2 } m s \mathrm { ~ \uparrow ~ }$ vs. $3 6 . 2 0 m s \uparrow$ ).
+
+Normalization component. We also explore the significance of $\ell _ { p }$ -norm in CN by comparing $\ell _ { p }$ normalization with mean and variance normalization. The mean and variance normalization will normalize mean to 0 and variance to 1, which is widely used in normalization layers (e.g., Ioffe & Szegedy (2015)). We show all these results in Table 5b. Particularly, mean and variance normalization achieves a top-1 error of $2 3 . 7 \%$ , which is only slightly better than the ResNet-50 baseline $( 2 3 . 8 \% )$ . Both $\ell _ { 1 }$ and $\ell _ { 2 }$ normalization make more promising improvements, and $\ell _ { 2 }$ performs slightly better. $\ell _ { p }$ normalization is better at representation learning in channel normalization.
+
+Adaptation component. We replace the activation function of the gating adaptation with a few different non-linear activation functions and show the results in Table 5c. Compare to the baseline (top-1 of $2 3 . 8 \%$ ), all the non-linear adaptation operator achieves promising performance, and $1 +$ tanh achieves a slightly better improvement. Both $1 + t a n h$ and $1 +$ ELU (Clevert et al., 2016) can model identity mapping, and achieve better results than Sigmoid. These strong results show that GCT is also robust to the choice of activation functions and the identity mapping is important in training.
+
+Application position. To find the best way to deploy GCT layers, we conduct experiments in ResNet-50 architecture on ImageNet by separately applying GCT after all the BN layers, before all the BN layers, and before all the convolutional layers. The results are reported in Table 5d. All the placement methods are effective in using GCT to improve the representational power of networks. However, it is better to employ GCT before all the convolutional layers, which is similar to the strategy in (Krizhevsky et al., 2012) (normalization after ReLU).
+
+# 5 CONCLUSION AND FUTURE WORK
+
+In this paper, we propose GCT, a novel layer that effectively improves the discriminability of deep CNNs by leveraging the relationship among channels. Benefit from the design of combining normalization and gating mechanisms, GCT can facilitate two types of neuron relations, i.e., competition and cooperation, with negligible complexity of parameters. We conduct expensive experiments to show the effectiveness and robustness of GCT across a wide range of modern CNNs and datasets. In future work, we will study the feasibility to apply GCT into recurrent networks.
+
+# REFERENCES
+
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+Yanping Huang, Yonglong Cheng, Dehao Chen, HyoukJoong Lee, Jiquan Ngiam, Quoc V Le, and Zhifeng Chen. Gpipe: Efficient training of giant neural networks using pipeline parallelism. arXiv preprint arXiv:1811.06965, 2018.
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+Will Kay, Joao Carreira, Karen Simonyan, Brian Zhang, Chloe Hillier, Sudheendra Vijayanarasimhan, Fabio Viola, Tim Green, Trevor Back, Paul Natsev, et al. The kinetics human action video dataset. arXiv preprint arXiv:1705.06950, 2017.
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+
+# A TRAINING DETAILS
+
+Same to modern normalization layers (e.g., BN (Ioffe & Szegedy, 2015)), we propose to apply GCT for all convolutional layers in deep networks. However, there are many different points nearby one convolutional layer to employ GCT. In deep networks, each convolutional layer always works together with a normalization layer (e.g., BN (Ioffe & Szegedy, 2015)) and an activation layer (e.g., ReLU). For this reason, there are three possible points to deploy GCT layer, which are before the convolutional layer, before the normalization layer, and after the normalization layer. All these methods are effective, but we find to be better to employ GNC before the convolutional layer. In Sec. 4.5, we compare the performance of these three application methods.
+
+In the training process, we propose to use 1 to initialize $_ { \pmb { \alpha } }$ and use 0 to initialize all $\gamma$ and $\beta$ . By doing this, GCT will be initialized as an identity mapping module, which will make the training process more stable. Besides, to avoid the bad influence of unstable gradient on the GCT gate in initial training steps, we propose to use warmup method (to start training with a small learning rate). In all the experiments on ImageNet (Russakovsky et al., 2015) and CIFAR (Krizhevsky & Hinton, 2009), we start training with a learning rate of 0.01 for 1 epoch. After the warmup, we go back to the original learning rate schedule. Finally, we propose NOT to apply weight decay on $\beta$ parameters, which is possible to reduce the performance of GCT.
+
+# B EXPERIMENTS ON CIFAR
+
+We conduct more experiments on the CIFAR-10 and CIFAR-100 datasets (Krizhevsky & Hinton, 2009). We follow the same training and testing strategies in (He et al., 2016a) to conduct our experiments but with a different learning rate schedule. We use a base learning rate of 0.1 and take cosine decay method (Loshchilov & Hutter) to adjust the learning rate for 300 epochs, which achieves better baseline performance. Following the above training protocol, we start the training process with a learning rate of 0.01 for 1 epoch.
+
+We report the performance of ResNet-110 (He et al., 2016a) and its GCT counterpart in Table 7. To reduce the variances from different runs, we repeat all experiments for 5 times and report the averaged the results. As with the previous experiments, we observe promising improvements in performance, which shows that GCT can generalize to other image classification datasets.
+
+Table 7: Improvement in top-1 error $( \% )$ on the CIFAR-10 and CIFAR-100 datasets.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>ResNet-110</td><td rowspan=1 colspan=1>GCT</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>5.81</td><td rowspan=1 colspan=1>5.26(0.55)</td></tr><tr><td rowspan=1 colspan=1>CIFAR-100</td><td rowspan=1 colspan=1>26.66</td><td rowspan=1 colspan=1>25.77 (0.89)</td></tr></table>
+
+# C MORE VISUALIZATION RESULTS
+
+In ResNet-50 (He et al., 2016a) backbone, We visualize the channel activation before and after the GCT layer in both low-level stage (far away from the network output) and high-level stage (close to the output) in Fig.4 and 5, respectively. The input image is from the validation dataset of ImageNet. As we can see, for the stages far away from the network output, the proposed GCT layer tends to reduce the variance of input feature, which encourages cooperation among channels and avoids excessive activation values or loss of useful features. On the contrary, for those stages close to the output, GCT tends to magnify the variance. Here, GCT acts like the attention mechanism that is useful for creating competition.
+
+![](images/bba516bc4d3a63602666dab0b6c5ddc5af43ac6f9c2b8eb7fc1c547621d60211.jpg)  
+Figure 4: Visualization of the channel activation for a GCT layer in Stage 2 (low-level) of ResNet-50 on the validation dataset of ImageNet.
+
+![](images/cfb0388bec48aea32229c87f5fc1dc827cf9259d3d4ac76c3616a7ba8b94c909.jpg)  
+Figure 5: Visualization of the channel activation for a GCT layer in Stage 5 (high-level) of ResNet50 on the validation dataset of ImageNet.

parse/train/SJxbu6VKDr/SJxbu6VKDr_content_list.json

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+    {
+        "type": "text",
+        "text": "GATED CHANNEL TRANSFORMATION FOR VISUAL RECOGNITION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
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+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a generally applicable transformation unit for visual recognition with deep convolutional neural networks. This transformation explicitly models channel relationships with explainable control variables. These variables determine the neuron behaviors of competition or cooperation, and they are jointly optimized with convolutional weights towards more accurate recognition. In Squeeze-and-Excitation (SE) Networks, the channel relationships are implicitly learned by fully connected layers, and the SE block is integrated at the block-level. We instead introduce a channel normalization layer to reduce the number of parameters and computational complexity. This lightweight layer incorporates a simple $l _ { 2 }$ normalization, enabling our transformation unit applicable to operator-level without much increase of additional parameters. Extensive experiments demonstrate the effectiveness of our unit with clear margins on many vision tasks, i.e., image classification on ImageNet, object detection and instance segmentation on COCO, video classification on Kinetics. ",
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            482,
+            336,
+            498
+        ],
+        "page_idx": 0
+    },
+    {
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+        "text": "Convolutional Neural Networks (CNNs) have proven to be critical and robust in visual recognition tasks, such as image classification (Huang et al., 2018), detection (Singh et al., 2018), and segmentation (Singh et al., 2018). Notably, a single convolutional layer operates only on a neighboring local context of each spatial position of a feature map, which could possibly lead to local ambiguities (Torralba, 2003). To relief this problem, VGGNets (Simonyan & Zisserman, 2015) were proposed to construct deep CNNs, using a series of convolutional layers with non-linear activation functions and downsampling operators to cover a large extent of context. Moreover, He et al. (2016a) introduced a residual connection to help CNNs benefit from deeper architectures further. ",
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+    {
+        "type": "text",
+        "text": "Apart from improving the depth of CNNs, another branch of methods focuses on augmenting convolutional layer with modules that directly operate on context across large neighborhoods. Squeezeand-Excitation Networks (SE-Nets) (Hu et al., 2018b) leveraged globally embedding information to model channel relationship and modulate feature maps on the channel-wise level. Moreover, its following method, GE-Nets (Hu et al., 2018a), used largely neighboring embedding instead. These modules can be conveniently assembled into modern networks, such as ResNets (He et al., 2016a) and Inception (Szegedy et al., 2015) networks, to improve the representational ability of networks. ",
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+    {
+        "type": "text",
+        "text": "However, the SE module uses two fully connected $( F C )$ layers to process channel-wise embeddings, which leads to two problems. First, the number of SE modules to be applied in CNNs is limited. In Hu et al. (2018b), SE module was applied at the block-level, i.e., a single SE module is utilized per Res-block (He et al., 2016a) or Inception-block (Szegedy et al., 2016). The dimension of the $F C$ layer is decreased to save the computational cost further. However, the designed $F C$ layers still hinder the wide deployment of SE modules across all layers. Second, due to the complexity of the parameters in $F C$ (or convolutional layer in GE), it is difficult to analyze the interactions among the channels at different layers. The channel relationships learned by convolution and FC operations are inherently implicit (Hu et al., 2018b), resulting in agnostic behaviors of the neuron outputs. ",
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+        "text": "In this paper, we propose a Gated Channel Transformation (GCT) for efficient and accurate contextual information modeling. First, we use a normalization component to replace the $F C$ layers. Normalization methods, e.g., Local Response Normalization (LRN) (Krizhevsky et al., 2012), create competitions among different neurons in neural networks. Batch normalization (Ioffe & Szegedy, ",
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+    {
+        "type": "text",
+        "text": "2015) and its variants can smooth gradient and have been widely used in accelerating CNNs training process. We leverage a simple $l _ { 2 }$ normalization for modeling channel relationship, which is more stable and computationally efficient comparing to FC layers. Second, we introduce a few channelwise parameters to control the behavior of the gated adaptation of feature channels. Compared to the large number of parameters in $F C$ , our designed parameters are much more lightweight. Besides, the gating weight parameter is convenient for channel relationship analysis and is helpful to understand the effect of GCT modules across different layers. According to our visualization analysis, GCT prefers to encourage cooperation in shallower layers, but competition is enhanced in deeper layers. ",
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+        "text": "Our experiments show that GCT is a simple and effective architecture for modeling relationship among channels. It significantly improves the generalization capability of deep convolutional networks across visual recognition tasks and datasets. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Gating and attention mechanisms. Gating mechanisms have been successfully deployed in some recurrent neural network architectures. Long Short-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) introduced an input gate, output gate and forget gate, which are used to regulate the flow of information into and out of the module. Based on gating mechanisms, some attention methods focus on forcing computational resources towards the most informative components of features (Larochelle & Hinton, 2010; Mnih et al., 2014; Vaswani et al., 2017). Recent works introduce the attention mechanism into convolutional networks (e.g., (Gehring et al., 2017; Dauphin et al., 2017)). Following these studies, SE-Nets (Hu et al., 2018b) and its following work GE-Nets (Hu et al., 2018a) introduced a lightweight gating mechanism which focuses on enhancing the representational power of the convolutional network by modeling channel-wise relationship. Compared to the SE module, our GCT also pays attention to the cross-channel relationship but can achieve better performance gains with less computation and parameters. ",
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+        "type": "text",
+        "text": "Normalization layers. In recent years, normalization layers have been widely used in deep networks to create competition between neurons (Krizhevsky et al., 2012) and produce smoother optimization surfaces (Ioffe & Szegedy, 2015). Local Response Normalization (LRN) (Krizhevsky et al., 2012) computes the statistics in a small neighborhood among channels for each pixel. Batch Normalization (BN) (Ioffe & Szegedy, 2015) utilizes global spatial information along the batch dimension and suggests to be deployed for all layers. Layer Normalization (LN) (Ba et al., 2016) computes along the channel dimension instead of the batch dimension. Group Normalization (GN) (Wu & He, 2018) differently divides the channels into groups and computes within each group the mean and variance for normalization. Similar to LRN, GN and LN, our GCT also utilizes channel-related information with normalization structure. ",
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+    {
+        "type": "text",
+        "text": "Deep architectures. VGGNets (Simonyan & Zisserman, 2015) and Inception networks (Szegedy et al., 2015) demonstrated that it was significant to improve the quality of representation by increasing the depth of a network. ResNets (He et al., 2016a) utilized shortcut connections to identity-based skip connections, and proved that it was highly effective to build considerably deeper and stronger networks with them. Some other researchers focused on improving the representation ability of the computational elements contained within a network (Szegedy et al., 2016). The more diverse composition of operators within a computational element can be constructed with multi-branch convolutions or pooling layers. Other than this, grouped convolutions have proven to be a practical method to increase the cardinality of learned transformations (Xie et al., 2017). We build our GCT on these deep architectures. All these networks with GCT achieve promising performance improvements, but the growth of computational complexity is negligible. ",
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+        "type": "text",
+        "text": "3 GATED CHANNEL TRANSFORMATION ",
+        "text_level": 1,
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+        "text": "SE-Nets proposed a lightweight SE module to augment convolutional networks by operating on a global context. The SE module contains two operators, i.e., a “squeeze” operator to embed channel context and an “excitation” operator to modulate the feature maps. The architecture of our Gated Channel Transformation benefits from this framework. Differently, GCT leverages a normalization operator instead of the $F C$ layers in the SE module for channel relationship modeling. Notably, the normalization operator is parameter-free. To make GCT learnable, we redesign the structure of the “squeeze” and “excitation” operators. Our new operators contain three sets of channel-wise trainable parameters. Thus, GCT is more convenient to be deployed occupying a small number of parameters. The gating parameters can be visualized for easier analysis of GCT’s behavior, while the discriminative ability is maintained. ",
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+        "image_caption": [
+            "Figure 1: An overview of the structure of Gated Channel Transformation (GCT). "
+        ],
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+        "type": "text",
+        "text": "Let $\\mathbf { x } \\in \\mathbb { R } ^ { C \\times H \\times W }$ be an activation feature in a convolutional network, where $H$ and $W$ are the spatial height and width, and $C$ is the number of channels. In general, GCT performs the following transformation: ",
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+    {
+        "type": "equation",
+        "img_path": "images/198293f243bf10818637a551926c7e860bb7a60d3844498ac7aa08cdcd3fc09a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\mathbf { x } } = F ( \\mathbf { x } | \\alpha , \\gamma , \\beta ) , \\alpha , \\gamma , \\beta \\in \\mathbb { R } ^ { C } . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here $\\alpha , \\gamma$ and $\\beta$ are trainable parameters. Embedding weights $_ { \\pmb { \\alpha } }$ are responsible for adapting the embedding outputs. The gating weights $\\gamma$ and biases $\\beta$ control the activation of the gate. They determine the behavior of GCT in each channel. The parameter complexity of GCT is $\\bar { O ( C ) }$ , which is smaller than the SE module $( O ( C ^ { 2 } ) )$ ( $\\mathrm { H u }$ et al., 2018b). In SE-Net, two $F C$ layers are leveraged, which have the parameter complexity of $O ( C ^ { 2 } )$ . ",
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+    {
+        "type": "text",
+        "text": "An illustration of the structure of GCT is shown in Fig. 1. Let $\\textbf { x } = ~ [ x _ { 1 } , x _ { 2 } , . . . , x _ { C } ] , x _ { c } ~ =$ $[ x _ { c } ^ { i , j } ] _ { H \\times W } ~ \\in ~ \\mathbb { R } ^ { H \\times W } , c ~ \\in ~ \\{ 1 , 2 , . . . , C \\}$ , where $x _ { c }$ is corresponding to each channel of $\\mathbf { x }$ . The detailed transformation consists of following parts. ",
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+    {
+        "type": "text",
+        "text": "Global Context Embedding. Global context embedding (GCE) aggregates global context in each channel. GCE can exploit global contextual information outside the small receptive fields of convolutional layers. Given the embedding weights ${ \\pmb { \\alpha } } = [ \\alpha _ { 1 } , . . . , \\alpha _ { C } ]$ , GCE is defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/6c844c1c41eb9290f9d8f1eafd09f9de4c365a5d441b589dea93abe070e1fa4c.jpg",
+        "text": "$$\ns _ { c } = \\alpha _ { c } | | x _ { c } | | _ { 2 } = \\alpha _ { c } \\{ [ \\sum _ { i = 1 } ^ { H } \\sum _ { j = 1 } ^ { W } ( x _ { c } ^ { i , j } ) ^ { 2 } ] + \\epsilon \\} ^ { \\frac { 1 } { 2 } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\epsilon$ is a small constant to avoid the problem of derivation at the zero point. Different from SE, GCT does not use global average pooling (GAP) to aggregate channel context. GAP might fail in some extreme cases. For example, if SE is deployed after the Instance Normalization (Ulyanov et al., 2016) layer that is popular in style transfer task, the output of GAP will be constant for any inputs since IN fixes the mean of each channel of features. To avoid this problem, we choose $\\ell _ { p }$ -norm instead. It is worth noting that GCT is robust with different $\\ell _ { p }$ -norms. In Sec. 4.5, we compare the performance of some popular $\\ell _ { p }$ -norms and choose the best one, $\\ell _ { 2 }$ -norm, to be our default setting. Notably, the performance of $\\ell _ { 1 }$ -norm is very close to $\\ell _ { 2 }$ -norm and $\\ell _ { 1 }$ -norm can be equivalently replaced by GAP when the input of GCT is always non-negative (for example, after ReLU activation). In this case, $\\ell _ { 1 }$ -norm is more computationally efficient. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Besides, we use trainable parameters $\\alpha _ { c }$ to adjust each channel because different channels should have different significance. ",
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+    {
+        "type": "text",
+        "text": "Channel Normalization. Normalization methods can model relationship in visual or photographic features (Lyu & Simoncelli, 2008) with lightweight computing resource (e.g., Ioffe & Szegedy (2015)). Similar to LRN, we use a $\\ell _ { 2 }$ normalization to operate across channels, namely channel normalization (CN). Let $\\mathbf { s } = [ s _ { 1 } , . . . , s _ { C } ]$ , the formula of CN is: ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/d6365d27fdc7c5de3b2e985053b879fc13c21f86cccfb69f27cebcf4ba7ab755.jpg",
+        "image_caption": [
+            "Figure 2: Training curve comparisons for ResNets with different depth on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/42137232532d56777079d1a7e564ddb03db8bd71f327d83e34e457835630645b.jpg",
+        "text": "$$\n\\hat { s } _ { c } = \\frac { \\sqrt { C } s _ { c } } { | | \\mathbf { s } | | _ { 2 } } = \\frac { \\sqrt { C } s _ { c } } { [ ( \\displaystyle \\sum _ { c = 1 } ^ { C } s _ { c } ^ { 2 } ) + \\epsilon ] ^ { \\frac { 1 } { 2 } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon$ is a small constant. The scalar $\\sqrt { C }$ is used to normalize the scale of $\\hat { s } _ { c }$ , avoiding a too small scale of $\\hat { s } _ { c }$ when $C$ is large. Compared to the $F C$ layers used by SE, our CN operator has less computational complexity $( O ( C ) )$ compared to the $F C$ layers $( O ( C ^ { 2 } ) )$ . ",
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+    {
+        "type": "text",
+        "text": "Gating Adaptation. We employ a gating mechanism, namely gating adaptation, to adapt the original feature. By introducing the gating mechanism, our GCT can facilitate both competition and cooperation during the training process. Let the gating weights $\\gamma = [ \\gamma _ { 1 } , . . . , \\gamma _ { C } ]$ and the gating biases $\\beta = [ \\beta _ { 1 } , . . . , \\beta _ { C } ]$ , we design the following gating function: ",
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+    {
+        "type": "equation",
+        "img_path": "images/798572ad83622725444aa018262a9d37c4fe9cb41f44318e9565c3c7f2981c8f.jpg",
+        "text": "$$\n\\hat { x } _ { c } = x _ { c } [ 1 + \\operatorname { t a n h } ( \\gamma _ { c } \\hat { s } _ { c } + \\beta _ { c } ) ] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The scale of each original channel $x _ { c }$ will be adapted by its corresponding gate, i.e., $1 + \\operatorname { t a n h } ( \\gamma _ { c } \\hat { s } _ { c } +$ $\\beta _ { c , \\ - }$ ). The trainable $\\gamma _ { c }$ and $\\beta _ { c }$ are employed to control the activation of gate. LRN benefits from only the competitions among the neurons (Krizhevsky et al., 2012). However, the gating mechanism in GCT is able to create both competition and cooperation among different channels. This capability is more consistent with the training process in biological neural networks (Demin & Nekhaev, 2018). When the gating weight of one channel $( \\gamma _ { c } )$ is activated positively, GCT promotes this channel to compete with the others as in LRN. When the gating weight is activated negatively, GCT encourages this channel to cooperate with the others. We analyze these adaptive channel relationships in Sec.4.4. ",
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+    {
+        "type": "text",
+        "text": "Besides, this gate function allows original features to pass to the next layer when the gating weights and biases are zeros, which is ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/01330840195eb129cb7f33959b136e08588985956fcde3ceb5e2bd4b1bed87e6.jpg",
+        "text": "$$\n\\hat { \\mathbf { x } } = F ( \\mathbf { x } | \\alpha , \\mathbf { 0 } , \\mathbf { 0 } ) = \\mathbf { 1 } \\mathbf { x } = \\mathbf { x } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The ability of modeling identity mapping can effectively improve the robustness to the degradation problem in deep networks. ResNets also benefits from this idea. Therefore, we propose to initialize $\\gamma$ and $\\beta$ to 0 in the initialization of GCT layers. By doing this, the initial steps of the training process will be more stable, and the final performance of GCT will be better. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We apply GCT for all the convolutional layers in deep networks rather than block-level deployment in SE-Net. In all GCT counterparts, we employ one GCT layer before each convolutional layer. In the Kinetics experiments, we apply GCT at the last two convolutional layers in each Res-Block. More training details are shown in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTS ON IMAGENET ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We experiment on the ImageNet 2012 dataset (Russakovsky et al., 2015) with $1 , 0 0 0$ classes. We train all the models on the 1.28M training images and evaluate on the $5 0 , 0 0 0$ validation images. We also conduct classification experiments on CIFAR (Krizhevsky & Hinton, 2009) in Appendix B. ",
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+        "type": "text",
+        "text": "Implementation details. In the training process of all the models, the input image is $2 2 4 \\times 2 2 4$ randomly cropped from a resized image using the same augmentation in Szegedy et al. (2015). We use SGD with a mini-batch size of 256. For ResNet-152 and ResNeXt-50, we use half mini-batch size and double the training steps). The weight decay is 0.0001, and the momentum is 0.9. The base learning rate is 0.1, and we divide it by 10 every 30 epochs. All models are trained for 100 epochs from scratch, using the weight initialization strategy described in He et al. (2015). Besides, we start the training process with a learning rate of 0.01 for 1 epoch. After the warmup, we go back to the original learning rate schedule. In all comparisons, we evaluate the error on the single $2 2 4 \\times 2 2 4$ center crop from an image whose shorter side is 256. For ResNet-200 (He et al., 2016b), we evaluate on $3 2 0 \\times 3 2 0$ following He et al. (2016b). ",
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+    {
+        "type": "table",
+        "img_path": "images/39d46841552b7d6b128eec627e94d6ff96aaf5d68893eb484944a38944f9c00d.jpg",
+        "table_caption": [
+            "Table 1: Improvement in error performance $( \\% )$ on ImageNet. The numbers in brackets denote the improvement in performance over the baselines. ResNet- ${ } ^ { 2 0 0 ^ { * } }$ means we follow the strategy in (He et al., 2016b) to train this model on $2 2 4 \\times 2 2 4$ but evaluate on $3 2 0 \\times 3 2 0$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>original</td><td rowspan=1 colspan=2>GCT</td></tr><tr><td rowspan=1 colspan=1>Network</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>VGG-16 (Simonyan &amp; Zisserman,2015)</td><td rowspan=1 colspan=1>26.2</td><td rowspan=1 colspan=1>8.3</td><td rowspan=1 colspan=1>25.1(1.1)</td><td rowspan=1 colspan=1>7.5(0.8)</td></tr><tr><td rowspan=1 colspan=1>Inception-v3 (Szegedy et al., 2016)</td><td rowspan=1 colspan=1>24.3</td><td rowspan=1 colspan=1>7.3</td><td rowspan=1 colspan=1>23.7(0.6)</td><td rowspan=1 colspan=1>7.1(0.2)</td></tr><tr><td rowspan=1 colspan=1>ResNeXt-50 (Xie et al., 2017)</td><td rowspan=1 colspan=1>22.4</td><td rowspan=1 colspan=1>6.3</td><td rowspan=1 colspan=1>21.7(0.7)</td><td rowspan=1 colspan=1>6.0(0.3)</td></tr><tr><td rowspan=1 colspan=1>ResNet-50 (He et al., 2016a)</td><td rowspan=1 colspan=1>23.8</td><td rowspan=1 colspan=1>7.0</td><td rowspan=1 colspan=1>22.7(1.1)</td><td rowspan=1 colspan=1>6.3（0.7)</td></tr><tr><td rowspan=1 colspan=1>ResNet-101 (He et al.,2016a)</td><td rowspan=1 colspan=1>22.2</td><td rowspan=1 colspan=1>6.2</td><td rowspan=1 colspan=1>21.4(0.8)</td><td rowspan=1 colspan=1>5.9（0.3)</td></tr><tr><td rowspan=1 colspan=1>ResNet-152 (He et al.,2016a)</td><td rowspan=1 colspan=1>21.6</td><td rowspan=1 colspan=1>5.9</td><td rowspan=1 colspan=1>20.8(0.8)</td><td rowspan=1 colspan=1>5.5(0.4)</td></tr><tr><td rowspan=1 colspan=1>ResNet-200*(He et al.,2016b)</td><td rowspan=1 colspan=1>20.7</td><td rowspan=1 colspan=1>5.2</td><td rowspan=1 colspan=1>19.7(1.0)</td><td rowspan=1 colspan=1>4.8(0.4)</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 2: Compared to SE in different networks on ImageNet. We evaluate the models of error performance $( \\% )$ , GFLOPs (G) and parameters (M). G/P means GFLOPs/parameters. In VGG-16 experiments, SE is employed for all the convolutional layers, which is the same as GCT. In other experiments, SE is only employed in block level (Res-Block or Inception-Block) as proposed (Hu et al., 2018b). This difference makes that SE uses comparable GFLOPs with GCT. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>original</td><td rowspan=1 colspan=2>SE</td><td rowspan=1 colspan=2>GCT (ours)</td></tr><tr><td rowspan=1 colspan=1>Network</td><td rowspan=1 colspan=1>top-1/5</td><td rowspan=1 colspan=1>G/P</td><td rowspan=1 colspan=1>top-1/5</td><td rowspan=1 colspan=1>G/P</td><td rowspan=1 colspan=1>top-1/5</td><td rowspan=1 colspan=1>G/P</td></tr><tr><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>23.8/7.0</td><td rowspan=1 colspan=1>3.879/25.61</td><td rowspan=1 colspan=1>22.9/6.6</td><td rowspan=1 colspan=1>3.893/28.14</td><td rowspan=1 colspan=1>22.7/6.3</td><td rowspan=1 colspan=1>3.900/25.68</td></tr><tr><td rowspan=1 colspan=1>ResNeXt-50</td><td rowspan=1 colspan=1>22.4/6.3</td><td rowspan=1 colspan=1>3.795/25.10</td><td rowspan=1 colspan=1>22.0/6.1</td><td rowspan=1 colspan=1>3.809/27.63</td><td rowspan=1 colspan=1>21.7/6.0</td><td rowspan=1 colspan=1>3.821/25.19</td></tr><tr><td rowspan=1 colspan=1>VGG-16</td><td rowspan=1 colspan=1>26.2/8.3</td><td rowspan=1 colspan=1>15.497/138.37</td><td rowspan=1 colspan=1>25.2/7.7</td><td rowspan=1 colspan=1>15.525/138.60</td><td rowspan=1 colspan=1>25.1/7.5</td><td rowspan=1 colspan=1>15.516/138.38</td></tr><tr><td rowspan=1 colspan=1>Inception-v3</td><td rowspan=1 colspan=1>24.3/7.3</td><td rowspan=1 colspan=1>2.847/23.87</td><td rowspan=1 colspan=1>24.0/7.2</td><td rowspan=1 colspan=1>2.851/25.53</td><td rowspan=1 colspan=1>23.7/7.1</td><td rowspan=1 colspan=1>2.862/23.99</td></tr></table>",
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+        "type": "text",
+        "text": "Integration with deep modern architectures. We study the effects of integrating GCT layers with some state-of-the-art backbone architectures, e.g., ResNets and ResNeXts (Xie et al., 2017), in which we apply GCT before all the convolutional layers. We report all these results in Table 1. Compared to original architectures, we observe significant performance improvements by introducing GCT into networks. Particularly, the top-1 error of GCT-ResNet-101 is ${ \\bf 2 1 . 4 \\% }$ , which is even better than the ResNet-152 baseline $( 2 1 . 6 \\% )$ with a deeper network and much more parameters. In addition, GCT is able to bring stable improvement in ResNets with different depth ( ${ \\bf \\cdot 1 . 1 \\% }$ top-1 improvement in ResNet-50, $\\mathbf { 0 . 8 \\bar { \\% } }$ in ResNet-152 and $\\mathbf { 1 . 0 \\% }$ in ResNet-200). Besides, we observe a smooth improvement throughout the training schedule, which is shown in Fig.2. ",
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+    {
+        "type": "text",
+        "text": "We also explore the improvement with GCT in non-residual networks (e.g., VGG-16 (Simonyan & Zisserman, 2015) and Inception-v3 (Szegedy et al., 2016)). To stabilize the training process, we employ BN (Ioffe & Szegedy, 2015) layers after every convolutional layer. Similar to the effectiveness in residual architectures, GCT layers bring promising improvements in non-residual structures. ",
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+    {
+        "type": "text",
+        "text": "Compared to SE. We conduct experiments on ImageNet to compare SE with GCT in both residual and non-residual networks and the results are reported in Table 2. We follow the methods in (Hu et al., 2018b) to integrate SE into VGG-16 (Simonyan & Zisserman, 2015), Inception (Szegedy et al., 2016), ResNet-50 (He et al., 2016a) and ResNeXt-50 (Xie et al., 2017) and train these models in same training schedule. Compared to SE, GCT always achieves better improvement. ",
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+    {
+        "type": "text",
+        "text": "In order to compare computational complexity, we calculate the GFLOPs and the number of parameters. In VGG-16 experiments, SE is employed for all the convolutional layers, which is the same as GCT. Under this fair condition, GCT achieves better performance with less increase in both GFLOPs (0.019G vs.0.028G) and parameters (0.01M vs.0.23M). In other experiments, SE is only employed in block-level (Res-Block or Inception-Block) as proposed (Hu et al., 2018b), which means the number of SE is smaller than GCT. However, the increase in parameters of GCT is still much less than SE, and the value of GFLOPs is comparable. Compared to SE, the increase in parameters of GCT is negligible, but the performance is better. ",
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+        "type": "table",
+        "img_path": "images/0f18880b14917db02da9170d7bf98031bc31db82cf562df90f3f0e67a163639a.jpg",
+        "table_caption": [
+            "Table 3: Improvement on COCO with Mask R-CNN framework. $\\mathbf { B N } ^ { * }$ means BN is frozen. $^ +$ means increasing the training iterations from 90K to 270K. When using GN, we follow the strategy in the original paper (Wu & He, 2018). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>box head</td><td rowspan=1 colspan=1>box AP</td><td rowspan=1 colspan=1>mask AP</td></tr><tr><td rowspan=1 colspan=1>ResNet-50BN*</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>37.8</td><td rowspan=1 colspan=1>34.2</td></tr><tr><td rowspan=1 colspan=1>ResNet-50BN*+GCT</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>39.8(2.0)</td><td rowspan=1 colspan=1>36.0(1.8)</td></tr><tr><td rowspan=1 colspan=1>ResNet-101BN*</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>40.1</td><td rowspan=1 colspan=1>36.1</td></tr><tr><td rowspan=2 colspan=1>ResNet-101BN*+GCT+ResNet-50 BN*</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>42.0(1.9)</td><td rowspan=1 colspan=1>37.7(1.6)</td></tr><tr><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>38.6</td><td rowspan=1 colspan=1>34.5</td></tr><tr><td rowspan=1 colspan=1>+ResNet-50 GN</td><td rowspan=1 colspan=1>GN</td><td rowspan=1 colspan=1>40.8(2.2)</td><td rowspan=1 colspan=1>36.1(1.6)</td></tr><tr><td rowspan=1 colspan=1>+ResNet-50 BN*+GCT</td><td rowspan=1 colspan=1>GN</td><td rowspan=1 colspan=1>41.6(3.0)</td><td rowspan=1 colspan=1>37.1(2.6)</td></tr><tr><td rowspan=1 colspan=1>+ResNet-50BN*+GCT</td><td rowspan=1 colspan=1>GN+GCT</td><td rowspan=1 colspan=1>41.8(3.2)</td><td rowspan=1 colspan=1>37.3(2.8)</td></tr><tr><td rowspan=1 colspan=1>+ResNet-101 BN*</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>40.9</td><td rowspan=1 colspan=1>36.4</td></tr><tr><td rowspan=1 colspan=1>+ResNet-101 GN</td><td rowspan=1 colspan=1>GN</td><td rowspan=1 colspan=1>42.3(1.4)</td><td rowspan=1 colspan=1>37.2(0.8)</td></tr><tr><td rowspan=1 colspan=1>ResNet-101BN*+GCT</td><td rowspan=1 colspan=1>GN</td><td rowspan=1 colspan=1>43.1(2.2)</td><td rowspan=1 colspan=1>38.3(1.9)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTS ON COCO ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next we evaluate the generalizability on the COCO dataset (Lin et al., 2014). We train the models on the COCO train2017 set and evaluate on the COCO eval2017 set (a.k.a minival). ",
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+    {
+        "type": "text",
+        "text": "Implementation details. We experiment on the Mask R-CNN baselines (He et al., 2017) and its GN counterparts (Wu & He, 2018). All the backbone models are pre-trained on ImageNet using the scale and aspect ratio augmentation in Szegedy et al. (2015) and fine-tune on COCO with a batch size of 16 (2 images/GPU). Besides, all these experiments use the Feature Pyramid Network (FPN) Lin et al. (2017). We also use the same hyperparameters and two training schedules used in (Wu & He, 2018). The short schedule includes 90K iterations, in which the learning rate is divided by 10 at 60K and 80K iterations. The long schedule increases the iterations to 270K, in which the learning rate is divided by 10 at 210K and 250K. The base learning rate is 0.02 in both schedules. ",
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+    {
+        "type": "text",
+        "text": "Improvements on Mask R-CNN. Table 3 shows the comparison of $\\mathbf { B N } ^ { * }$ (frozen BN), GN and $\\mathbf { B } \\mathbf { N } ^ { * } { + } \\mathbf { G } \\mathbf { C } \\mathbf { T }$ (using GCT before all the convolutional layers of the backbones). First, we use a short training schedule to compare baselines and GCT counterparts. GCT shows stable and significant improvement in both ResNet-50 and ResNet-101. In ResNet-50, GCT improves detection AP by 2.0 and segmentation AP by 1.8. Moreover, in ResNet-101, GCT also improves detection AP by 1.9 and segmentation AP by 1.6. Then, we use the long schedule to compare GN and $\\mathbf { B N ^ { * } { + } G C T }$ . GN is more effective than BN when batch size is small as in this case of detection and segmentation using Mask R-CNN. However, we deploy GCT together with BN into the backbone, and these $\\mathbf { B } \\mathbf { N } ^ { * } { + } \\mathbf { G } \\mathbf { C } \\mathbf { T }$ counterparts achieve much better performance than GN backbones. Compared to GN in ResNet-101, $\\mathrm { \\mathbf { B } N ^ { * } { + } G C T }$ improves detection AP by 0.8 and segmentation AP by 1.1. In particular, ResNet-101 with $\\mathrm { \\mathbf { B } N ^ { * } { + } G C T }$ trained in the short schedule achieves better segmentation AP (37.7) than the GN counterpart (37.2) trained with the long schedule. This GN counterpart also uses GN in the backbone, the box heads, and the FPN. We also explore to combine GCT with GN by introducing GCT into GN box head. The results show $\\mathrm { \\bf G N { + } } \\mathrm { \\bf G C T }$ achieves a better performance. It demonstrates the benefits of integrating GCT with GN. We now have shown the effectiveness of GCT in working with both BN and GN. ",
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+    {
+        "type": "text",
+        "text": "4.3 EXPERIMENTS ON KINETICS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our previous experiments demonstrate the effectiveness of GCT on image-related tasks. We now evaluate the generalizability in video understanding task of action recognition on the large scale Kinetics-400 (Kay et al., 2017) dataset. We employ the ResNet-50 (3D) and ResNet-101 (3D) as the backbone and apply GCT in the last two convolutional layers in each Res-Block. The backbone networks are pre-trained on ImageNet (Russakovsky et al., 2015). ",
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+    {
+        "type": "image",
+        "img_path": "images/4f3eed4af43b00704fbd313bc531759518072555c6f428f60926c05148a110c1.jpg",
+        "image_caption": [
+            "Figure 3: Analysis. The visulization of parameters of $\\gamma$ (Fig.(a), (b)), and the ratio of variance of GCT output and input feature (Fig.(c)) in all the GCT layers in ResNet-50 on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "We compare with the state-of-the-art Non-Local Networks (NL-Net) (Wang et al., 2018). The results show that GCT counterparts consistently improves the recognition accuracy over both the ResNet50 and ResNet-101 baselines, as shown in Table 4. Because of our limited memory resource, we can NOT apply GCT in all the convolutional layers, which we believe can further improve the performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In summary, extensive experiments demonstrate that GCT is effective across a wide range of architectures, tasks, and datasets. ",
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+    {
+        "type": "text",
+        "text": "4.4 ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To analyze the behavior of GCT in different layers, we visualize the distribution of the gating weights $( \\gamma )$ of each GCT layer in ResNet-50 on ImageNet. Further, we sort these distributions according to their layer index in 3D space (Fig. 3a). The bigger layer index means it is closer to the network output. To make the visualization clearer, we re-scale the vertical $z$ axis with $\\boldsymbol { l o g ( 1 + z ) }$ , which corresponds to the percentage density of $\\gamma$ . We also calculate the mean and standard deviation (std) of $\\gamma$ in each layer and show them in a bar chart (Fig. 3b). As shown in Fig. 3a and 3b, the mean of $\\gamma$ tends to be less than 0 in the GCT layers far from the network output. Oppositely, in the layers close to the output, the mean tends to be greater than 0. ",
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+    {
+        "type": "text",
+        "text": "According to Eq. 3 & 4, the adaptation of channel $x _ { c }$ is related to $\\hat { s } _ { c }$ , which corresponds to the ratio of the weighted $\\ell _ { 2 }$ -norm of $x _ { c }$ (i.e., $s _ { c . }$ ) and the average of all the $s _ { c }$ . When the gating weight $\\gamma _ { c }$ is greater than 0, the adaptation is positively correlated to $\\hat { s } _ { c }$ and increases the variance between $x _ { c }$ and others; When $\\gamma _ { c }$ is lower than 0, the adaptation is negatively correlated and reduces the variance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Based on the analysis and the results we observe, we suppose that GCT tends to reduce the difference among channels in layers far away from the output. This behavior is helpful to encourage cooperation among channels and relieve overfitting. Apart from this, GCT tends to increase the difference among channels when close to the output. Here, GCT acts like the attention mechanism that is useful for creating competition. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To further validate our hypothesis, we calculate the ratio of the variance of output and input feature of each GCT layer, which we show in Fig. 3c. More visualizations are shown in Appendix C. Generally, the shallow layers learn low-level features to capture general characteristics like textures, edges, and corners. The feature variances become larger in deeper layers, where the high-level features are more discriminative and task-related. As expected, in the layers close to network output, GCT tends to magnify the variance of input feature (the ratio is always greater than 1), but in the layers far away from the output, GCT tends to reduce the variance (the ratio is always less than 1). This phenomenon is consistent with our previous hypothesis and shows that GCT is effective in creating both competition and cooperation among channels. Our observation validates that GCT can adaptively learn the channel relationships at different layers. ",
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+    {
+        "type": "text",
+        "text": "4.5 ABLATION STUDIES. ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we conduct a serial of ablation experiments to explain the relative importance of each operator in the GCT. At last, we show how the performance changes with regards to the GCT position in a network. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 4: Improvement in top-1 accuracy $( \\% )$ over the state-ofthe-art method on Kinetics. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>NL-Net</td><td rowspan=1 colspan=1>GCT</td></tr><tr><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>74.6</td><td rowspan=1 colspan=1>75.1(0.5)</td></tr><tr><td rowspan=1 colspan=1>ResNet-101</td><td rowspan=1 colspan=1>75.7</td><td rowspan=1 colspan=1>76.2(0.5)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 5: Ablation experiments. We evaluate error performance in GCT-ResNet-50 on ImageNet $( \\% )$ . The ResNet-50 baseline achieves a top-1 of 23.8 and a top-5 of 7.0. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(a) Embedding operator. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ef443284127e660d17fe46e545061756b8bb101ee5e18cba630f920fd7a5ef51.jpg",
+        "table_caption": [
+            "(b) Normalization operator. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Normalization</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>mean+variance</td><td rowspan=1 colspan=1>23.7</td><td rowspan=1 colspan=1>7.1</td></tr><tr><td rowspan=1 colspan=1>l1</td><td rowspan=1 colspan=1>22.9</td><td rowspan=1 colspan=1>6.4</td></tr><tr><td rowspan=1 colspan=1>l2</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.3</td></tr></table>",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Norm</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>lo</td><td rowspan=1 colspan=1>23.1</td><td rowspan=1 colspan=1>6.7</td></tr><tr><td rowspan=1 colspan=1>l1</td><td rowspan=1 colspan=1>22.8</td><td rowspan=1 colspan=1>6.3</td></tr><tr><td rowspan=1 colspan=1>l2</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.3</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/b8fc59eadd4a47ae248082027cd02f9ca8d904b3515c1e5c77c72b9987a62e37.jpg",
+        "table_caption": [
+            "(c) Adaptation operator. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Adaptation</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>Sigmoid</td><td rowspan=1 colspan=1>22.9</td><td rowspan=1 colspan=1>6.5</td></tr><tr><td rowspan=1 colspan=1>1+ELU</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.4</td></tr><tr><td rowspan=1 colspan=1>1+tanh</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/684807e3851ac83f756d936d240b7acabf68e895949eabc70930168648a20695.jpg",
+        "table_caption": [
+            "(d) Application position. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Position</td><td rowspan=1 colspan=1>top-1</td><td rowspan=1 colspan=1>top-5</td></tr><tr><td rowspan=1 colspan=1>after BN</td><td rowspan=1 colspan=1>23.1</td><td rowspan=1 colspan=1>6.6</td></tr><tr><td rowspan=1 colspan=1>before BN</td><td rowspan=1 colspan=1>23.1</td><td rowspan=1 colspan=1>6.5</td></tr><tr><td rowspan=1 colspan=1>before Conv</td><td rowspan=1 colspan=1>22.7</td><td rowspan=1 colspan=1>6.3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Clock time comparison. We calculate average inference times (ms) per batch by using 1 GTX 1080Ti with 16 batch size for 1,000 iterations on ImageNet. For the sake of fairness, the modules are applied for all the Convs in VGG-16. ",
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+        "img_path": "images/cd57195b0d9ce936bbabd5134f4522503d2847bee9ac2da55c503f28bd57ef4b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>baseline</td><td rowspan=1 colspan=1>+SE</td><td rowspan=1 colspan=1>+GCT(l1-norm)</td><td rowspan=1 colspan=1>+GCT(l2-norm)</td></tr><tr><td rowspan=1 colspan=1>time(ms)/batch</td><td rowspan=1 colspan=1>51.11</td><td rowspan=1 colspan=1>87.3136.20↑</td><td rowspan=1 colspan=1>59.468.35↑</td><td rowspan=1 colspan=1>59.738.62↑</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Embedding component. To explain the importance of $\\ell _ { p }$ -norm in GCE, we compare embedding operators with different $\\ell _ { p }$ norm. We report the results in Table 5a, which shows all the $\\ell _ { p }$ -norms are effective in GCE, but the $\\ell _ { 2 }$ -norm is slightly better than $\\ell _ { 1 }$ -norm. The results demonstrate the GCE of GCT is robust to different $\\ell _ { p }$ -norms. In addition, we make a clock time comparison between SE and GCTs with different embedding component. As shown in Table 6, $\\ell _ { 2 }$ -norm is computationally similar to $\\ell _ { 1 }$ -norm and GCT is much more efficient than SE $( { \\bf 8 . 6 2 } m s \\mathrm { ~ \\uparrow ~ }$ vs. $3 6 . 2 0 m s \\uparrow$ ). ",
+        "bbox": [
+            173,
+            401,
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+            486
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Normalization component. We also explore the significance of $\\ell _ { p }$ -norm in CN by comparing $\\ell _ { p }$ normalization with mean and variance normalization. The mean and variance normalization will normalize mean to 0 and variance to 1, which is widely used in normalization layers (e.g., Ioffe & Szegedy (2015)). We show all these results in Table 5b. Particularly, mean and variance normalization achieves a top-1 error of $2 3 . 7 \\%$ , which is only slightly better than the ResNet-50 baseline $( 2 3 . 8 \\% )$ . Both $\\ell _ { 1 }$ and $\\ell _ { 2 }$ normalization make more promising improvements, and $\\ell _ { 2 }$ performs slightly better. $\\ell _ { p }$ normalization is better at representation learning in channel normalization. ",
+        "bbox": [
+            173,
+            492,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Adaptation component. We replace the activation function of the gating adaptation with a few different non-linear activation functions and show the results in Table 5c. Compare to the baseline (top-1 of $2 3 . 8 \\%$ ), all the non-linear adaptation operator achieves promising performance, and $1 +$ tanh achieves a slightly better improvement. Both $1 + t a n h$ and $1 +$ ELU (Clevert et al., 2016) can model identity mapping, and achieve better results than Sigmoid. These strong results show that GCT is also robust to the choice of activation functions and the identity mapping is important in training. ",
+        "bbox": [
+            173,
+            597,
+            825,
+            695
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Application position. To find the best way to deploy GCT layers, we conduct experiments in ResNet-50 architecture on ImageNet by separately applying GCT after all the BN layers, before all the BN layers, and before all the convolutional layers. The results are reported in Table 5d. All the placement methods are effective in using GCT to improve the representational power of networks. However, it is better to employ GCT before all the convolutional layers, which is similar to the strategy in (Krizhevsky et al., 2012) (normalization after ReLU). ",
+        "bbox": [
+            174,
+            700,
+            825,
+            785
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "In this paper, we propose GCT, a novel layer that effectively improves the discriminability of deep CNNs by leveraging the relationship among channels. Benefit from the design of combining normalization and gating mechanisms, GCT can facilitate two types of neuron relations, i.e., competition and cooperation, with negligible complexity of parameters. We conduct expensive experiments to show the effectiveness and robustness of GCT across a wide range of modern CNNs and datasets. In future work, we will study the feasibility to apply GCT into recurrent networks. ",
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+    {
+        "type": "text",
+        "text": "A TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            662,
+            370,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Same to modern normalization layers (e.g., BN (Ioffe & Szegedy, 2015)), we propose to apply GCT for all convolutional layers in deep networks. However, there are many different points nearby one convolutional layer to employ GCT. In deep networks, each convolutional layer always works together with a normalization layer (e.g., BN (Ioffe & Szegedy, 2015)) and an activation layer (e.g., ReLU). For this reason, there are three possible points to deploy GCT layer, which are before the convolutional layer, before the normalization layer, and after the normalization layer. All these methods are effective, but we find to be better to employ GNC before the convolutional layer. In Sec. 4.5, we compare the performance of these three application methods. ",
+        "bbox": [
+            173,
+            694,
+            825,
+            805
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In the training process, we propose to use 1 to initialize $_ { \\pmb { \\alpha } }$ and use 0 to initialize all $\\gamma$ and $\\beta$ . By doing this, GCT will be initialized as an identity mapping module, which will make the training process more stable. Besides, to avoid the bad influence of unstable gradient on the GCT gate in initial training steps, we propose to use warmup method (to start training with a small learning rate). In all the experiments on ImageNet (Russakovsky et al., 2015) and CIFAR (Krizhevsky & Hinton, 2009), we start training with a learning rate of 0.01 for 1 epoch. After the warmup, we go back to the original learning rate schedule. Finally, we propose NOT to apply weight decay on $\\beta$ parameters, which is possible to reduce the performance of GCT. ",
+        "bbox": [
+            174,
+            813,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENTS ON CIFAR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            424,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We conduct more experiments on the CIFAR-10 and CIFAR-100 datasets (Krizhevsky & Hinton, 2009). We follow the same training and testing strategies in (He et al., 2016a) to conduct our experiments but with a different learning rate schedule. We use a base learning rate of 0.1 and take cosine decay method (Loshchilov & Hutter) to adjust the learning rate for 300 epochs, which achieves better baseline performance. Following the above training protocol, we start the training process with a learning rate of 0.01 for 1 epoch. ",
+        "bbox": [
+            173,
+            133,
+            825,
+            217
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We report the performance of ResNet-110 (He et al., 2016a) and its GCT counterpart in Table 7. To reduce the variances from different runs, we repeat all experiments for 5 times and report the averaged the results. As with the previous experiments, we observe promising improvements in performance, which shows that GCT can generalize to other image classification datasets. ",
+        "bbox": [
+            173,
+            223,
+            825,
+            280
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/828b461a0cbcc50120f3507e2eb8f134dca128230134117c851112cd9a49bca5.jpg",
+        "table_caption": [
+            "Table 7: Improvement in top-1 error $( \\% )$ on the CIFAR-10 and CIFAR-100 datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>ResNet-110</td><td rowspan=1 colspan=1>GCT</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>5.81</td><td rowspan=1 colspan=1>5.26(0.55)</td></tr><tr><td rowspan=1 colspan=1>CIFAR-100</td><td rowspan=1 colspan=1>26.66</td><td rowspan=1 colspan=1>25.77 (0.89)</td></tr></table>",
+        "bbox": [
+            366,
+            319,
+            632,
+            364
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C MORE VISUALIZATION RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            387,
+            482,
+            405
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In ResNet-50 (He et al., 2016a) backbone, We visualize the channel activation before and after the GCT layer in both low-level stage (far away from the network output) and high-level stage (close to the output) in Fig.4 and 5, respectively. The input image is from the validation dataset of ImageNet. As we can see, for the stages far away from the network output, the proposed GCT layer tends to reduce the variance of input feature, which encourages cooperation among channels and avoids excessive activation values or loss of useful features. On the contrary, for those stages close to the output, GCT tends to magnify the variance. Here, GCT acts like the attention mechanism that is useful for creating competition. ",
+        "bbox": [
+            173,
+            417,
+            825,
+            531
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/bba516bc4d3a63602666dab0b6c5ddc5af43ac6f9c2b8eb7fc1c547621d60211.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of the channel activation for a GCT layer in Stage 2 (low-level) of ResNet-50 on the validation dataset of ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            220,
+            558,
+            779,
+            797
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/cfb0388bec48aea32229c87f5fc1dc827cf9259d3d4ac76c3616a7ba8b94c909.jpg",
+        "image_caption": [
+            "Figure 5: Visualization of the channel activation for a GCT layer in Stage 5 (high-level) of ResNet50 on the validation dataset of ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            338,
+            353,
+            658,
+            621
+        ],
+        "page_idx": 11
+    }
+]

parse/train/Syez3j0cKX/Syez3j0cKX.md

@@ -0,0 +1,320 @@
+# DISSECTING AN ADVERSARIAL FRAMEWORK FOR INFORMATION RETRIEVAL
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recent advances in Generative Adversarial Networks facilitated by improvements to the framework and successful application to various problems has resulted in extensions to multiple domains. IRGAN attempts to leverage the framework for Information-Retrieval (IR), a task that can be described as modeling the correct conditional probability distribution $p ( d | q )$ over the documents $( d )$ , given the query $( q )$ . The work that proposes IRGAN claims that optimizing their minimax loss function will result in a generator which can learn the distribution, but their setup and baseline term steer the model away from an exact adversarial formulation, and this work attempts to point out certain inaccuracies in their formulation. Analyzing their loss curves gives insight into possible mistakes in the loss functions and better performance can be obtained by using the co-training like setup we propose, where two models are trained in a co-operative rather than an adversarial fashion.
+
+# 1 INTRODUCTION
+
+Information-Retrieval (IR) involves providing a list of ranked documents $\{ d _ { 1 } , d _ { 2 } , \dots , d _ { k } \}$ in answer to a query $q$ . This general formulation can be extended to various tasks like web-search, where the documents are web pages and information needs are queries, content-recommendation, where the documents are items/content to suggest and queries are users, and Question-Answering, where the documents are answers and queries are questions. The retrieved list can also be viewed as a probability distribution over candidates, one example being $\begin{array} { r } { R a n k _ { q } ( d _ { i } ) \equiv p ( d _ { i } | q ) \propto ( \frac { 1 } { R a n k _ { q } ( d _ { i } ) } ) ^ { l } } \end{array}$ where $l$ is a hyperparameter. Even if the probability distribution is not explicit, it is desirable to retrieve a higher ranked document more often than a lower ranked document.
+
+GANs were proposed as alternatives to generative models and have been shown to be capable of modeling the true data well. High dimensional settings like images and word sequences have seen some success. Given that the generator in GANs tries to model the training data’s distribution, adversarial setups seem like a natural fit for IR. The learned distribution can then be used to retrieve relevant documents for incoming queries. IRGAN is a framework proposed by Wang et al. (2017), with the hope of giving Information-Retrieval, access to the large literature of GANs.
+
+IRGAN consists of a discriminator and a generator. Like in a typical setup, the discriminator learns to distinguish between documents produces by the real probability distribution or the real ranking and the generator’s probability distribution. It increases the likelihood of the former and decreases it for the latter. The generator tries to bring its probability distribution closer to the real one so that it increases the likelihood of confusing the discriminator into believing that it is the true distribution. Ideally, equilibrium is achieved when the generator manages to rank the documents according to the true distribution.
+
+However, the formulation and implementation of the loss function in the work seems to have a few issues. Specifically, the use of the baseline term recommended in the work results in pitting the loss functions of the discriminator and the generator directly against each other and this leads to issues that are conspicuous in the loss curves. The training starts off with a pre-trained discriminator and generator, and the performance of the generator decreases as the training proceeds, while you would actually expect the opposite. When pre-training is not used, the generator does not learn at all. This forces IRGAN to choose the generator or discriminator based on whichever has better performance, while it expected that the generator is chosen at equilibrium.
+
+Given the traction this paper has received since its inception (53 citations as of $2 7 ^ { t h }$ September 2018), it is important to critically analyze the work and attribute the claimed performance improvements correctly. To this end, we propose two models which outperform IRGAN on two of the three tasks and give a comparable performance on the third. They also serve as an ablation study by experimentally showing that the generator might not be playing a vital role during train or test time.
+
+The following contributions are made in this work • We propose a model motivated by Co-training which outperforms IRGANs • We point out inaccuracies in the minimax loss function used in IRGANs • We substantiate the same by drawing conclusions from the loss curves
+
+# 2 RELATED WORK
+
+# 2.1 GENERATIVE ADVERSARIAL NETWORKS
+
+Generative Adversarial Networks (GANs) (Goodfellow et al. (2014)) were proposed as an alternative to generative models (Salakhutdinov & Larochelle (2010)) which used Markov Chains or other approximations to compute intractable probability distributions. In essence, the generator tries to model the real data distribution and the discriminator learns to differentiate between real data points and generated data points. GANs are notoriously unstable to train and works like DCGANs (Radford et al. (2015)) and Wasserstein GAN (Arjovsky et al. (2017)) have successfully attempted to alleviate a few issues. Nonetheless, GANs have been widely applied to various problems like image generation, text generation, cross-modal retrieval and more niche ones like Interactive Image Generation (Zhu et al. (2016)), Text to Image (Zhang et al. (2017)), Image to Image style transfer (Isola et al. (2017)) and robotics (Bousmalis et al. (2017)).
+
+While GANs allow generation based on a random variable $z$ , Conditional GANs (Mirza & Osindero (2014)) partition the sample variable into two parts ( $z$ and $y$ ). $y$ is used to denote which part of the probability distribution the generator has to generate from, and $z$ plays the same role played in Vanilla GANs (Goodfellow et al. (2014)). Conditional GANs dovetail with IR because $y$ can be used to represent the query or its embedding, and in theory, the model should be able to generate the required document.
+
+$$
+y \sim q u e r y \qquad G ( z | y ) \sim p _ { \theta } ( d | z , q )
+$$
+
+We feel that an eventual adversarial formulation for IR will be similar to this in flavor.
+
+# 2.2 RETRIEVAL OF IMAGE RESPONSES
+
+Creswell & Bharath (2016) employed Sketch-GANs for the interesting task of retrieving similar merchant seals (images) based on an input image. DCGANs (Radford et al. (2015)) are used to generate an image, and post training, the last layer of the discriminator is popped off and the rest of it is used as an encoder. This model, however, is specifically for retrieving image responses.
+
+# 3 BACKGROUND
+
+In the subsequent sections, $D$ denotes the discriminator, $G$ the generator, $p _ { t r u e }$ the real probability distribution over documents, $\phi$ the parameters of the discriminator, $\theta$ the parameters of the generator, $d$ the document, $q$ the query and $r$ the rank of a document with respect to a query.
+
+The equations used to train the discriminator and generator in Goodfellow et al. (2014) are the following respectively.
+
+$$
+\begin{array} { r l r } {  { \nabla _ { \theta _ { d } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } [ \log D ( \pmb { x } ^ { ( i ) } ) + \log ( 1 - D ( G ( \pmb { z } ^ { ( i ) } ) ) ) ] } } \\ & { } & { \nabla _ { \theta _ { g } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \log ( 1 - D ( G ( \pmb { z } ^ { ( i ) } ) ) ) } \end{array}
+$$
+
+The discriminator minimizes the likelihood of a “generated” data point and maximizes it for a “real” data point, while the generator tries to generate data points which the discriminator thinks is “real”. The two models are trained alternatively and the procedure culminates in a generator which is able to produce data which looks like the real data.
+
+# 4 IRGAN FORMULATION
+
+This section elucidates the IRGAN formulation (Wang et al. (2017)). Comments by the authors are in italics (in this section alone), while normal typeface is a paraphrased version of IRGAN. IRGAN is motivated by the combination of two schools of thoughts, the generative retrieval model and the discriminative retrieval model.
+
+# 4.1 DISCRIMINATOR AND GENERATOR
+
+The generative retrieval model $p _ { \theta } ( d | q , r )$ tries to sample relevant documents from a candidate pool with the aim of cloning the true probability distribution $p _ { t r u e }$ . The discriminative retrieval model $f _ { \phi } ( q , d )$ , which is a binary classifier, tries to discriminate between real and generated pairs $( q , d )$ .
+
+Two different loss functions IRGAN-Pointwise and IRGAN-Pairwise are proposed.
+
+# 4.2 IRGAN-POINTWISE
+
+This is called so because each data point is independently used to train, unlike in IRGAN-Pairwise where pairs of points are used. The dataset is expected to have some cue with respect to how often a document is correctly retrieved for a query, if at all.
+
+$$
+J ^ { G ^ { * } , D ^ { * } } = \underset { \theta } { \mathrm { m i n } } \underset { \phi } { \mathrm { m a x } } \sum _ { n = 1 } ^ { N } ( E _ { d \sim p _ { t r u e } ( d | q _ { n } , r ) } [ ( \log D ( d | q _ { n } ) ) ] + E _ { d \sim p _ { \theta } ( d | q _ { n } , r ) } [ ( \log 1 - D ( d | q _ { n } ) ) ] )
+$$
+
+Note that the generator $G$ can alternately be written as $p _ { \theta } ( d | q _ { n } , r )$ , which denotes the modeled probability distribution, and $D ( d | q ) = \sigma ( \ ' f _ { \phi } ( d , q ) )$ represents the discriminator’s score.
+
+# 4.3 IRGAN-PAIRWISE
+
+In some IR problems the training data may not be a set of relevant documents for each query, but rather a set of ordered document pairs $R _ { n } = [ < d _ { i } , d _ { j } > | d _ { i } \succ d _ { j } ]$ , where $d _ { i } \succ d _ { j }$ means that the first document is more relevant for query $q _ { n }$ than the second document. $o$ represents a real pair $< d _ { u } , d _ { v } >$ and $o ^ { \prime }$ represents a generated pair $< d _ { u } ^ { \prime } , d _ { v } ^ { \prime } >$ . The discriminator’s goal in this setting is to discriminate between $o$ and $o ^ { \prime }$ , with $D ( o | q ) = \overset { \vartriangle } { \boldsymbol { \sigma } } ( \bar { f } _ { \phi } ( d _ { u } , q ) - f _ { \phi } ( d _ { v } , q ) )$
+
+$$
+J ^ { G ^ { * } , D ^ { * } } = \underset { \theta } { \mathrm { m i n } } \underset { \phi } { \mathrm { m a x } } \sum _ { n = 1 } ^ { N } ( E _ { o \sim p _ { t r u c } ( o | q _ { n } ) } [ ( \log D ( o | q _ { n } ) ) ] + E _ { o ^ { \prime } \sim p _ { \theta } ( o ^ { \prime } | q _ { n } ) } [ ( \log 1 - D ( o ^ { \prime } | q _ { n } ) ) ] )
+$$
+
+Note the similarity between this and the previous formula. The problem with this formula is that $D ( o | q )$ is actually supposed to denote the probability that the pair o is from the real data distribution and not the probability that the pair is correctly ranked, as mentioned in the paper.
+
+# 4.4 OPTIMIZING THE GENERATOR
+
+The generator samples documents from the candidate pool based on its belief (relevance score). This sampling has the downside that the gradients cannot be backpropagated, and policy gradients (Sutton et al. (2000)) have to be used. As an intuition, the documents can be considered as the arms of a contextual multi-arm bandit (Auer et al. (2002), Lu et al. (2010)), and picking an arm can be viewed as analogous to choosing the document as relevant. The policy discovered gives us the relevance of each document and $- \log ( 1 - D ( d | q ) )$ is the reward for picking that action/document $( d )$ . Let $J ^ { \tilde { G } }$ represent the objective function of the generator that it has to maximize. The policy gradient (REINFORCE) can be written as the following.
+
+$$
+\nabla _ { \theta } J ^ { G } ( q _ { n } ) = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \nabla _ { \theta } \log p _ { \theta } ( d _ { k } | q _ { n } , r ) \log ( 1 + e x p ( f _ { \phi } ( d _ { k } , q _ { n } ) ) )
+$$
+
+# 4.5 BASELINE TERM
+
+To reduce the variance in REINFORCE, a standard trick is to use the advantage function instead of just the reward. This does not change the optimal parameters.
+
+$$
+\log ( 1 + e x p ( f _ { \phi } ( d _ { k } , q _ { n } ) ) ) - \mathbb { E } _ { p _ { \theta } ( d _ { k } | q _ { n } ) } [ \log ( 1 + e x p ( f _ { \phi } ( d _ { k } , q _ { n } ) ) ) ]
+$$
+
+Another baseline term that is suggested for each query is $f _ { \phi } ( d _ { + } , q )$ , where $d _ { + }$ represents the positive document. This is legal because the term does not depend on the document (action). This is motivated by the belief of a larger generator score if $f _ { \phi } ( \bar { d _ { + } } , q )$ is large and lower if $f _ { \phi } ( d _ { + } , q )$ is low. This baseline term is used in two of their three tasks and causes the violation of adversarial formulation, as we show in the following section.
+
+# 5 INSIGHTS INTO IRGAN MINIMAX LOSS FUNCTION
+
+Having shown that the generator can be optimized using REINFORCE, we focus on the loss function and show how the baseline term exacerbates training. We consider Stochastic Gradient Descent updates for ease of illustration. Consider a triple $( q , d _ { r } , d _ { g } )$ , where $d _ { r }$ denotes the correct document according to the true distribution and $d _ { g }$ denotes the generated document. The discriminator’s updates are in the direction of $\nabla J ^ { D }$ , with the following definition.
+
+$$
+J ^ { D } = \log D ( d _ { r } | q ) + \log ( 1 - D ( d _ { g } | q ) )
+$$
+
+With the baseline term included, the generator’s updates are in the direction of $\nabla J ^ { G }$ , with the following definition.
+
+$$
+J ^ { G } = \log ( 1 - D ( d _ { r } | q ) ) + \log D ( d _ { g } | q )
+$$
+
+Since maximizing $\log ( 1 - z )$ with respect to $z$ is the same as maximizing $- \log z$ , we can write the following equivalent loss functions
+
+$$
+J ^ { D } = \log D ( d _ { r } | q ) - \log D ( d _ { g } | q ) \qquad J ^ { G } = - \log D ( d _ { r } | q ) + \log D ( d _ { g } | q )
+$$
+
+Note that this substitution is similar in principle to the substitution in Goodfellow et al. (2014), where the motivation is to allow easier flow of gradients. It is apparent that the discriminator and the generator are optimizing directly opposite loss functions and this detrimental to the performance of the models. We provide experimental proof later that the performance improvements shown in IRGAN are mainly because of the discriminator maximizing the likelihood of the real data and not because of the generator.
+
+# 6 PROPOSED MODELS
+
+We propose two models to compare and critically analyze performance gains facilitated by IRGAN and illustrate them in Figure 1.
+
+The first model increases the likelihood of the training data and decreases the likelihood of documents which are not relevant to the query but have a high score according to its own parameters. It maximizes the following, where the sampling for the second term is from a candidate pool with only negative answers (denoted by $p ^ { - } .$ ). Not following this will lead to undesirable updates because sampling positive documents for the second term will result in decreasing the likelihood of real data. $\psi$ denotes the parameters of the only model.
+
+$$
+J ^ { M o d e l 1 } = \operatorname* { m a x } _ { \psi } \sum _ { n = 1 } ^ { N } ( E _ { d \sim p _ { t r u e } ( d | q _ { n } , r ) } [ ( \log D ( d | q _ { n } ) ) ] + E _ { d \sim p _ { \psi } ^ { - } ( d | q _ { n } , r ) } [ ( \log 1 - D ( d | q _ { n } ) ) ] )
+$$
+
+![](images/90d72faff9848714db0f8a16698b2246a2947c87938645c125df29a2315c4b90.jpg)  
+Figure 1: Models
+
+To alleviate the pernicious loss function of IRGAN, we propose a model which uses two discriminators in a co-operative setup influenced by Co-training (Blum & Mitchell (1998)). Instead of using two different views $( x _ { 1 } , x _ { 2 } )$ as mentioned in the work, we use the same views for both the discriminators but let them influence each other in a feedback loop. Training is similar to Model 1 with the only difference being that each discriminator decreases the likelihood of documents relevant to the other discriminator rather than itself, as shown in the equation below. This model achieves better performance than IRGAN.
+
+$$
+\begin{array} { r l } & { J ^ { M o d e l 1 } = \displaystyle \operatorname* { m a x } _ { \psi _ { 1 } } \sum _ { n = 1 } ^ { N } ( E _ { d \sim p _ { t r u e } ( d | q _ { n } , r ) } [ ( \log D ( d | q _ { n } ) ) ] + E _ { d \sim p _ { \psi _ { 2 } } ^ { - } ( d | q _ { n } , r ) } [ ( \log 1 - D ( d | q _ { n } ) ) ] ) } \\ & { \displaystyle } \\ & { J ^ { M o d e l 2 } = \displaystyle \operatorname* { m a x } _ { \psi _ { 2 } } \sum _ { n = 1 } ^ { N } ( E _ { d \sim p _ { t r u e } ( d | q _ { n } , r ) } [ ( \log D ( d | q _ { n } ) ) ] + E _ { d \sim p _ { \psi _ { 1 } } ^ { - } ( d | q _ { n } , r ) } [ ( \log 1 - D ( d | q _ { n } ) ) ] ) } \end{array}
+$$
+
+# 7 EXPERIMENTAL SETUP
+
+This section describes the datasets, the task and hyperparameters.
+
+# 7.1 DATASETS
+
+We conduct experiments on three tasks, Web Search, Item Recommendation and Question Answering, using the same datasets mentioned in IRGAN.
+
+Table 1: Datasets   
+
+<table><tr><td>Task</td><td>Dataset</td></tr><tr><td>WebSearch</td><td>LETOR by Liu et al. (2007)</td></tr><tr><td>Item-Recommendation</td><td>Movielens</td></tr><tr><td>Question Answering</td><td>InsuranceQA by Feng et al. (2015)</td></tr></table>
+
+# 7.2 TASK
+
+In Web Search, the task is to retrieve the document which is most relevant to the query. Each query on average has around 5 positive documents. In Content Recommendation, users give ratings for movies and given a user, the task is to retrieve a movie that they would probably rate high. In IRGAN, any movie retrieved for which the user rating is greater than or equal to 4 (out of a scale of 5) is considered correct. Based on the dataset statistics, around $5 5 \%$ of the user-movie ratings are $\geq 5$ . This makes the problem easy to solve. In Question Answering, every query has just one relevant document in most cases. This is thus the hardest task.
+
+# 7.3 HYPERPARAMETERS
+
+The hyperparameters for the proposed model are the same except for absence of G Epochs, for obvious reasons. Information about hyperparameter tuning is mentioned in the Appendix.
+
+Table 2: Hyperparameters for IRGAN   
+
+<table><tr><td>Hyperparameter</td><td>Description</td></tr><tr><td>Learning Rate</td><td>Forboth generator and discriminator</td></tr><tr><td>Batch Size</td><td>Batch size for training</td></tr><tr><td>Embed Dim</td><td>Embedding dimension of query or document</td></tr><tr><td>Epochs</td><td>Number of epochs of training</td></tr><tr><td>DEpochs</td><td>Number of epochs the discriminator is trained per epoch</td></tr><tr><td>G_Epochs Temperature</td><td>Number of epochs the generator is trained per epoch Temperature parameter for softmax sampling of documents</td></tr></table>
+
+# 8 EXPERIMENTS AND DISCUSSION
+
+We report only the $\mathrm { P @ 5 }$ and ${ \mathrm { N D C G } } @ 5$ values because all other metrics follow the same trend.
+
+# 8.1 WEB SEARCH
+
+Table 3 reports the performance of various models. As can be seen, both the Single Discriminator and the Co-training models outperform IRGAN models. The fact that each query is associated approximately with 5 positive documents provides evidence that the proposed models can perform well in sparse reward settings.
+
+Table 3: Results on LETOR dataset used in IRGAN   
+
+<table><tr><td>Model</td><td>P@5</td><td>NDCG@5</td></tr><tr><td>RankNet (Burges et al. (2005))</td><td>0.1219</td><td>0.1709</td></tr><tr><td>LambdaRank (Burges et al. (2007))</td><td>0.1352</td><td>0.1920</td></tr><tr><td>IRGAN-pointwise</td><td>0.1657</td><td>0.2225</td></tr><tr><td>IRGAN-pairwise</td><td>0.1676</td><td>0.2154</td></tr><tr><td>Single Discriminator</td><td>0.1676</td><td>0.2190</td></tr><tr><td>Co-training</td><td>0.1733</td><td>0.2252</td></tr></table>
+
+# 8.2 ITEM-RECOMMENDATION
+
+This task, in contrast to the other two, has multiple relevant documents that can be retrieved for each query, making it slightly easier. Each user (query) rates a movie (document), and $5 5 \%$ of the entries in the train set and $5 6 \%$ in the test set are relevant pairs. It can be seen in Table 4 that the single discriminator model achieves only a slightly lower score, and given the small size of the dataset (943 users), it makes just 7 more mistakes when compared to IRGAN. This is not a statistically significant number, especially because the IRGAN generator is pre-initialized to a model which scores 0.34 but our model learns from scratch.
+
+# 8.3 QUESTION ANSWERING
+
+After close correspondence with the authors of IRGAN, we obtained all the hyperparameters required for the models. Multiple random seeds were used in vain, the results in the paper for Question-Answering tasks could not be replicated. We instead mention the best results out of all random seeds. We believe that if there is some random seed which gives better performance for IRGAN, it should do so for our model as well. The co-training model outperforms IRGAN-Pairwise.
+
+Table 4: Results on Movielens Dataset   
+
+<table><tr><td>Model</td><td>P@5</td><td>NDCG@5</td></tr><tr><td>BPR (Rendle et al. (2009))</td><td>0.3044</td><td>0.3245</td></tr><tr><td>LambdaFM (Yuan et al. (2016))</td><td>0.3474</td><td>0.3749</td></tr><tr><td>IRGAN-pointwise</td><td>0.3750</td><td>0.4099</td></tr><tr><td>Single Discriminator</td><td>0.3675</td><td>0.3925</td></tr><tr><td>Co-training</td><td>0.345</td><td>0.373</td></tr></table>
+
+Table 5: $\mathrm { P @ 1 }$ on InsuranceQA   
+
+<table><tr><td>Model</td><td>P@1</td></tr><tr><td>IRGAN-Pairwise Single Discriminator Co-training</td><td>0.616 0.614 0.623</td></tr></table>
+
+# 8.4 LOSS CURVES
+
+![](images/9d9f5cbe6e06a42653d99ecdd9ebfc40d91d7554d358d409e99c9bf6be2943b1.jpg)  
+Figure 2: Performance curves
+
+The loss curves in Figure 2 picked from IRGAN’s work show deteriorating performance of the generator, which is in contrast to what is observed in actual adversarial training. In the minimax setting, since the generator is expected to capture the real data distribution, its performance is supposed to improve and this can indirectly be seen in GANs and DCGANs where the samples generated look more and more like real-world data points. Further, a deteriorating generator implies that the discriminator’s improvement in performance is only because of the first term of $J ^ { D }$ , which hints that our proposed models might be able to do better than IRGAN. The reason offered in the paper is that “A worse generator could be the result of the sparsity of document distribution, i.e., each question usually has only one correct answer”. But this reason does not seem plausible, given that DCGANs have been able to model very high dimensional data, where the probability distribution is only a tiny part of the real space.
+
+Further, the increase in performance of the discriminator in all cases is coupled with a deteriorating generator. This substantiates our claim that the discriminator and the generator are optimizing directly opposite loss functions.
+
+Item-recommendation task is a little different from the other two tasks at hand because of a large number of positive answers. When the loss curves are plotted, though the generator’s performance improves, the discriminator’s loss remains high and almost constant throughout the procedure, as shown in Figure 3. This is another indication that the performance of IRGAN is not actually because of the adversarial setup, but because of the maximization of the likelihood of the real data.
+
+# 9 CONNECTIONS TO PREVIOUS WORK
+
+We have already shown in Section 2 that Conditional GANs are connected directly to Information Retrieval. The problem can also be viewed as a contextual multi-armed bandit problem (Li et al.
+
+![](images/6d5d7efb0cd7b45fd56ee19f6142f5ea5e2a0dd6ff2125a68691f433a1d369d9.jpg)  
+Figure 3: Discriminator Loss for Content-Recommendation
+
+(2010)), where each documents is an arm and the context $x _ { q , d }$ can be used to determine the actionvalue function $f _ { \theta } ( x _ { q , d } )$ . In previous works (Li et al. (2010)) $f$ has been considered to be linear, but recent studies Collier & Llorens (2018) have modeled them as deep neural networks.
+
+In Pfau & Vinyals (2016), a parallel is drawn between Actor-Critic algorithms (Konda & Tsitsiklis (2000)) and GANs. This is directly related to our work because REINFORCE (Sutton et al. (2000)) with a baseline can be connected to Actor-Critic algorithms when bootstrapping is used (Sutton & Barto (2018)). The work shows a restricted scenario which involves a stateless MDP, each action setting all the pixels of the image and cross-entropy loss instead of mean-squared Bellmann residual in which GANs are equivalent to Actor-Critic algorithms. But this equivalence holds only when the baseline term is not used so the formulation in IRGAN is not exactly equivalent to a GAN framework. Another study (Finn et al. (2016)) draws a parallel between Inverse Reinforcement Learning $\mathrm { N g }$ et al. (2000)) and GANs because both the methods try to “learn” the cost function to optimize for.
+
+# 10 CONCLUSION AND FUTURE WORK
+
+The experiments performed show that IRGAN is by no means state-of-the-art on those datasets. Further, the performance does not justify the large training time of 4 hours per generator epoch and 1 hour of discriminator epoch as opposed to 2 hours per epoch of the co-training model (11 GB GPU and Question Answering task). The shaky mathematical formulation renders the generator useless after training, and any gains in performance can be attributed directly to the first term of $J ^ { D }$ , where the likelihood of the real data is increased. We showed that the discriminator and generator are optimizing directly opposite loss functions and this is the cause of deleterious training.
+
+The poor performance of IRGAN on Web-Search and Question Answering and only a satisfactory performance on Content-Recommendation (which has dense rewards) lead us to speculate that it does not work well in sparse reward scenarios. This is similar to a well-known problem called the Sparse Reward Reinforcement Learning. We think that a correct formulation along with established techniques from the former, like reward shaping $\mathrm { N g }$ et al. (1999)) may lead to better performance. Newer methods like Hindsight Experience Replay (Andrychowicz et al. (2017)) which allow models to learn both from mistakes and rewards may further ameliorate learning.
+
+We would also like to explore in the direction of learning correct adversarial frameworks for more complex tasks like Image Retrieval and Question Answering which will involve learning end-toend trainable models. With advances in modeling sequences, this could also involve generation of documents rather than sampling them.
+
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+
+# APPENDIX A HYPERPARAMETERS
+
+The hyperparameters are mentioned in tables 6, 7, 8, 9 and 10. The following were the ranges of hyperparameter tuning, along with the best value. Gradient Descent Optimizer was used so that the comparison with IRGAN is fair.
+
+Table 6: Single Discriminator for Web-Search   
+
+<table><tr><td>Hyperparameter/Seed</td><td>Range/List</td><td>Best</td></tr><tr><td>Learning Rate</td><td>0.002-0.2</td><td>0.004</td></tr><tr><td>Batch Size</td><td>[8,16,32]</td><td>8</td></tr><tr><td>Feature Size</td><td>[46,92]</td><td>46</td></tr><tr><td>Random Seed</td><td>[20,40,60]</td><td>40</td></tr></table>
+
+For the co-training model, for every epoch, we optimize the two discriminators several times. We call these the outer and inner epochs in Table 7.
+
+Table 7: Co-training for Web-Search   
+
+<table><tr><td>Hyperparameter/Seed</td><td>Range/List</td><td>Best</td></tr><tr><td>Learning Rate Outer Epochs Inner Epochs Batch Size Feature Size Random Seed</td><td>0.002-0.2 [30,50] [30,50] [8,16,32] [46,92]</td><td>0.006 50 30 8 46</td></tr></table>
+
+DNS K in Table 8 represents the number of candidates that are chosen before performing the softmax. This is done to make the procedure computationally tractable. We use the value suggested in IRGAN.
+
+Table 8: Single Discriminator for Content Recommendation   
+
+<table><tr><td>Hyperparameter/Seed</td><td>Range/List</td><td>Best</td></tr><tr><td>Learning Rate</td><td>0.01-0.05</td><td>0.02</td></tr><tr><td rowspan="3">Batch Size Embedding Dimension Random Seed</td><td>10</td><td>10</td></tr><tr><td>[20,40,60]</td><td>20</td></tr><tr><td>70</td><td>70</td></tr><tr><td>DNS_K</td><td>5</td><td>5</td></tr></table>
+
+# APPENDIX B DESCRIPTION OF MODELS
+
+The discriminator and the generator have the same architecture in all the tasks. For the Web-retrieval task, the model has a single hidden layer 46 units.
+
+For the content-recommendation task, the model converts users and movies to a 5 dimensional embedding. This can be though to be a single hidden layer which compresses a one-hot user embedding to a 5 dimensional embedding.
+
+For the Question-Answering task, each word is initialized to a 100 dimensional random vector. A Convolutional Neural Network is then used and the window size of the convolutional kernel is (1,2,3,5). A max-pooling-over-time strategy is then used and the output is a 100 dimensional vector because each feature map is pooled to a scalar. Note that this architecture is the same as the one used in the IRGAN paper. We refer the user to that for further description.
+
+Table 9: Single Discriminator for Question Answering   
+
+<table><tr><td>Hyperparameter</td><td>Best</td></tr><tr><td>Learning Rate</td><td>0.05</td></tr><tr><td>Epochs</td><td>20</td></tr><tr><td>Batch Size</td><td>100</td></tr><tr><td>Embedding Dimension</td><td>100</td></tr></table>
+
+Table 10: Co-training for Question Answering   
+
+<table><tr><td>Hyperparameter</td><td>Best</td></tr><tr><td>Learning Rate</td><td>0.05</td></tr><tr><td>Outer Epochs</td><td>20</td></tr><tr><td>Inner Epochs</td><td>1</td></tr><tr><td>Batch Size</td><td>100</td></tr><tr><td>Embedding Dimension</td><td>100</td></tr></table>

parse/train/Syez3j0cKX/Syez3j0cKX_content_list.json

@@ -0,0 +1,1644 @@
+[
+    {
+        "type": "text",
+        "text": "DISSECTING AN ADVERSARIAL FRAMEWORK FOR INFORMATION RETRIEVAL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent advances in Generative Adversarial Networks facilitated by improvements to the framework and successful application to various problems has resulted in extensions to multiple domains. IRGAN attempts to leverage the framework for Information-Retrieval (IR), a task that can be described as modeling the correct conditional probability distribution $p ( d | q )$ over the documents $( d )$ , given the query $( q )$ . The work that proposes IRGAN claims that optimizing their minimax loss function will result in a generator which can learn the distribution, but their setup and baseline term steer the model away from an exact adversarial formulation, and this work attempts to point out certain inaccuracies in their formulation. Analyzing their loss curves gives insight into possible mistakes in the loss functions and better performance can be obtained by using the co-training like setup we propose, where two models are trained in a co-operative rather than an adversarial fashion. ",
+        "bbox": [
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+            267,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            463,
+            336,
+            479
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Information-Retrieval (IR) involves providing a list of ranked documents $\\{ d _ { 1 } , d _ { 2 } , \\dots , d _ { k } \\}$ in answer to a query $q$ . This general formulation can be extended to various tasks like web-search, where the documents are web pages and information needs are queries, content-recommendation, where the documents are items/content to suggest and queries are users, and Question-Answering, where the documents are answers and queries are questions. The retrieved list can also be viewed as a probability distribution over candidates, one example being $\\begin{array} { r } { R a n k _ { q } ( d _ { i } ) \\equiv p ( d _ { i } | q ) \\propto ( \\frac { 1 } { R a n k _ { q } ( d _ { i } ) } ) ^ { l } } \\end{array}$ where $l$ is a hyperparameter. Even if the probability distribution is not explicit, it is desirable to retrieve a higher ranked document more often than a lower ranked document. ",
+        "bbox": [
+            174,
+            496,
+            825,
+            609
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "GANs were proposed as alternatives to generative models and have been shown to be capable of modeling the true data well. High dimensional settings like images and word sequences have seen some success. Given that the generator in GANs tries to model the training data’s distribution, adversarial setups seem like a natural fit for IR. The learned distribution can then be used to retrieve relevant documents for incoming queries. IRGAN is a framework proposed by Wang et al. (2017), with the hope of giving Information-Retrieval, access to the large literature of GANs. ",
+        "bbox": [
+            174,
+            617,
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+            700
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "IRGAN consists of a discriminator and a generator. Like in a typical setup, the discriminator learns to distinguish between documents produces by the real probability distribution or the real ranking and the generator’s probability distribution. It increases the likelihood of the former and decreases it for the latter. The generator tries to bring its probability distribution closer to the real one so that it increases the likelihood of confusing the discriminator into believing that it is the true distribution. Ideally, equilibrium is achieved when the generator manages to rank the documents according to the true distribution. ",
+        "bbox": [
+            174,
+            708,
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+            805
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, the formulation and implementation of the loss function in the work seems to have a few issues. Specifically, the use of the baseline term recommended in the work results in pitting the loss functions of the discriminator and the generator directly against each other and this leads to issues that are conspicuous in the loss curves. The training starts off with a pre-trained discriminator and generator, and the performance of the generator decreases as the training proceeds, while you would actually expect the opposite. When pre-training is not used, the generator does not learn at all. This forces IRGAN to choose the generator or discriminator based on whichever has better performance, while it expected that the generator is chosen at equilibrium. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Given the traction this paper has received since its inception (53 citations as of $2 7 ^ { t h }$ September 2018), it is important to critically analyze the work and attribute the claimed performance improvements correctly. To this end, we propose two models which outperform IRGAN on two of the three tasks and give a comparable performance on the third. They also serve as an ablation study by experimentally showing that the generator might not be playing a vital role during train or test time. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The following contributions are made in this work • We propose a model motivated by Co-training which outperforms IRGANs • We point out inaccuracies in the minimax loss function used in IRGANs • We substantiate the same by drawing conclusions from the loss curves ",
+        "bbox": [
+            174,
+            180,
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+            195
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            215,
+            205,
+            725,
+            257
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            276,
+            344,
+            292
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 GENERATIVE ADVERSARIAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            308,
+            500,
+            321
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Generative Adversarial Networks (GANs) (Goodfellow et al. (2014)) were proposed as an alternative to generative models (Salakhutdinov & Larochelle (2010)) which used Markov Chains or other approximations to compute intractable probability distributions. In essence, the generator tries to model the real data distribution and the discriminator learns to differentiate between real data points and generated data points. GANs are notoriously unstable to train and works like DCGANs (Radford et al. (2015)) and Wasserstein GAN (Arjovsky et al. (2017)) have successfully attempted to alleviate a few issues. Nonetheless, GANs have been widely applied to various problems like image generation, text generation, cross-modal retrieval and more niche ones like Interactive Image Generation (Zhu et al. (2016)), Text to Image (Zhang et al. (2017)), Image to Image style transfer (Isola et al. (2017)) and robotics (Bousmalis et al. (2017)). ",
+        "bbox": [
+            173,
+            333,
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+            473
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While GANs allow generation based on a random variable $z$ , Conditional GANs (Mirza & Osindero (2014)) partition the sample variable into two parts ( $z$ and $y$ ). $y$ is used to denote which part of the probability distribution the generator has to generate from, and $z$ plays the same role played in Vanilla GANs (Goodfellow et al. (2014)). Conditional GANs dovetail with IR because $y$ can be used to represent the query or its embedding, and in theory, the model should be able to generate the required document. ",
+        "bbox": [
+            173,
+            479,
+            825,
+            563
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a97be1c0f0e344a3ba3c85c727c92b24c26a9af1e7c6d45e829df950a7abe05.jpg",
+        "text": "$$\ny \\sim q u e r y \\qquad G ( z | y ) \\sim p _ { \\theta } ( d | z , q )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            560,
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+            579
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We feel that an eventual adversarial formulation for IR will be similar to this in flavor. ",
+        "bbox": [
+            173,
+            587,
+            735,
+            602
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 RETRIEVAL OF IMAGE RESPONSES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            618,
+            455,
+            633
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Creswell & Bharath (2016) employed Sketch-GANs for the interesting task of retrieving similar merchant seals (images) based on an input image. DCGANs (Radford et al. (2015)) are used to generate an image, and post training, the last layer of the discriminator is popped off and the rest of it is used as an encoder. This model, however, is specifically for retrieving image responses. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            720,
+            326,
+            736
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the subsequent sections, $D$ denotes the discriminator, $G$ the generator, $p _ { t r u e }$ the real probability distribution over documents, $\\phi$ the parameters of the discriminator, $\\theta$ the parameters of the generator, $d$ the document, $q$ the query and $r$ the rank of a document with respect to a query. ",
+        "bbox": [
+            174,
+            751,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The equations used to train the discriminator and generator in Goodfellow et al. (2014) are the following respectively. ",
+        "bbox": [
+            173,
+            800,
+            823,
+            829
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/44213bdfce8483287d68b89de4e3428b32e10fd92d67bf0cfc90c8558e02865a.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\nabla _ { \\theta _ { d } } \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } [ \\log D ( \\pmb { x } ^ { ( i ) } ) + \\log ( 1 - D ( G ( \\pmb { z } ^ { ( i ) } ) ) ) ] } } \\\\ & { } & { \\nabla _ { \\theta _ { g } } \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\log ( 1 - D ( G ( \\pmb { z } ^ { ( i ) } ) ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
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+            930
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The discriminator minimizes the likelihood of a “generated” data point and maximizes it for a “real” data point, while the generator tries to generate data points which the discriminator thinks is “real”. The two models are trained alternatively and the procedure culminates in a generator which is able to produce data which looks like the real data. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 IRGAN FORMULATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            179,
+            403,
+            196
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This section elucidates the IRGAN formulation (Wang et al. (2017)). Comments by the authors are in italics (in this section alone), while normal typeface is a paraphrased version of IRGAN. IRGAN is motivated by the combination of two schools of thoughts, the generative retrieval model and the discriminative retrieval model. ",
+        "bbox": [
+            174,
+            210,
+            825,
+            267
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1 DISCRIMINATOR AND GENERATOR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            284,
+            454,
+            299
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The generative retrieval model $p _ { \\theta } ( d | q , r )$ tries to sample relevant documents from a candidate pool with the aim of cloning the true probability distribution $p _ { t r u e }$ . The discriminative retrieval model $f _ { \\phi } ( q , d )$ , which is a binary classifier, tries to discriminate between real and generated pairs $( q , d )$ . ",
+        "bbox": [
+            174,
+            310,
+            825,
+            353
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Two different loss functions IRGAN-Pointwise and IRGAN-Pairwise are proposed. ",
+        "bbox": [
+            174,
+            358,
+            718,
+            375
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.2 IRGAN-POINTWISE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            390,
+            356,
+            405
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This is called so because each data point is independently used to train, unlike in IRGAN-Pairwise where pairs of points are used. The dataset is expected to have some cue with respect to how often a document is correctly retrieved for a query, if at all. ",
+        "bbox": [
+            174,
+            416,
+            825,
+            459
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/166a8ee4d133a85674ec16a7d7ee3a149ab24f15e65a962dec9f0b44dc4f9467.jpg",
+        "text": "$$\nJ ^ { G ^ { * } , D ^ { * } } = \\underset { \\theta } { \\mathrm { m i n } } \\underset { \\phi } { \\mathrm { m a x } } \\sum _ { n = 1 } ^ { N } ( E _ { d \\sim p _ { t r u e } ( d | q _ { n } , r ) } [ ( \\log D ( d | q _ { n } ) ) ] + E _ { d \\sim p _ { \\theta } ( d | q _ { n } , r ) } [ ( \\log 1 - D ( d | q _ { n } ) ) ] )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            465,
+            808,
+            508
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that the generator $G$ can alternately be written as $p _ { \\theta } ( d | q _ { n } , r )$ , which denotes the modeled probability distribution, and $D ( d | q ) = \\sigma ( \\ ' f _ { \\phi } ( d , q ) )$ represents the discriminator’s score. ",
+        "bbox": [
+            173,
+            522,
+            823,
+            551
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.3 IRGAN-PAIRWISE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            568,
+            346,
+            582
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In some IR problems the training data may not be a set of relevant documents for each query, but rather a set of ordered document pairs $R _ { n } = [ < d _ { i } , d _ { j } > | d _ { i } \\succ d _ { j } ]$ , where $d _ { i } \\succ d _ { j }$ means that the first document is more relevant for query $q _ { n }$ than the second document. $o$ represents a real pair $< d _ { u } , d _ { v } >$ and $o ^ { \\prime }$ represents a generated pair $< d _ { u } ^ { \\prime } , d _ { v } ^ { \\prime } >$ . The discriminator’s goal in this setting is to discriminate between $o$ and $o ^ { \\prime }$ , with $D ( o | q ) = \\overset { \\vartriangle } { \\boldsymbol { \\sigma } } ( \\bar { f } _ { \\phi } ( d _ { u } , q ) - f _ { \\phi } ( d _ { v } , q ) )$ ",
+        "bbox": [
+            174,
+            593,
+            825,
+            665
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d268131de5f91372211ed52026ae4c07b78369f0e040494bb5c6e26819c6a5e5.jpg",
+        "text": "$$\nJ ^ { G ^ { * } , D ^ { * } } = \\underset { \\theta } { \\mathrm { m i n } } \\underset { \\phi } { \\mathrm { m a x } } \\sum _ { n = 1 } ^ { N } ( E _ { o \\sim p _ { t r u c } ( o | q _ { n } ) } [ ( \\log D ( o | q _ { n } ) ) ] + E _ { o ^ { \\prime } \\sim p _ { \\theta } ( o ^ { \\prime } | q _ { n } ) } [ ( \\log 1 - D ( o ^ { \\prime } | q _ { n } ) ) ] )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note the similarity between this and the previous formula. The problem with this formula is that $D ( o | q )$ is actually supposed to denote the probability that the pair o is from the real data distribution and not the probability that the pair is correctly ranked, as mentioned in the paper. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.4 OPTIMIZING THE GENERATOR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            786,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The generator samples documents from the candidate pool based on its belief (relevance score). This sampling has the downside that the gradients cannot be backpropagated, and policy gradients (Sutton et al. (2000)) have to be used. As an intuition, the documents can be considered as the arms of a contextual multi-arm bandit (Auer et al. (2002), Lu et al. (2010)), and picking an arm can be viewed as analogous to choosing the document as relevant. The policy discovered gives us the relevance of each document and $- \\log ( 1 - D ( d | q ) )$ is the reward for picking that action/document $( d )$ . Let $J ^ { \\tilde { G } }$ represent the objective function of the generator that it has to maximize. The policy gradient (REINFORCE) can be written as the following. ",
+        "bbox": [
+            173,
+            811,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/53b34b332ce913e4b475d42e533df6dc5b532cf9c644611472c7b04d271caf89.jpg",
+        "text": "$$\n\\nabla _ { \\theta } J ^ { G } ( q _ { n } ) = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\nabla _ { \\theta } \\log p _ { \\theta } ( d _ { k } | q _ { n } , r ) \\log ( 1 + e x p ( f _ { \\phi } ( d _ { k } , q _ { n } ) ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            113,
+            722,
+            157
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.5 BASELINE TERM",
+        "text_level": 1,
+        "bbox": [
+            174,
+            171,
+            331,
+            185
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To reduce the variance in REINFORCE, a standard trick is to use the advantage function instead of just the reward. This does not change the optimal parameters. ",
+        "bbox": [
+            173,
+            196,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dc1513fcfaaaa9b5cffb2657f8a4d3bfee09e8dafac861f12cbb8eb0161d1e86.jpg",
+        "text": "$$\n\\log ( 1 + e x p ( f _ { \\phi } ( d _ { k } , q _ { n } ) ) ) - \\mathbb { E } _ { p _ { \\theta } ( d _ { k } | q _ { n } ) } [ \\log ( 1 + e x p ( f _ { \\phi } ( d _ { k } , q _ { n } ) ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            228,
+            717,
+            247
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Another baseline term that is suggested for each query is $f _ { \\phi } ( d _ { + } , q )$ , where $d _ { + }$ represents the positive document. This is legal because the term does not depend on the document (action). This is motivated by the belief of a larger generator score if $f _ { \\phi } ( \\bar { d _ { + } } , q )$ is large and lower if $f _ { \\phi } ( d _ { + } , q )$ is low. This baseline term is used in two of their three tasks and causes the violation of adversarial formulation, as we show in the following section. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "5 INSIGHTS INTO IRGAN MINIMAX LOSS FUNCTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Having shown that the generator can be optimized using REINFORCE, we focus on the loss function and show how the baseline term exacerbates training. We consider Stochastic Gradient Descent updates for ease of illustration. Consider a triple $( q , d _ { r } , d _ { g } )$ , where $d _ { r }$ denotes the correct document according to the true distribution and $d _ { g }$ denotes the generated document. The discriminator’s updates are in the direction of $\\nabla J ^ { D }$ , with the following definition. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9f1b36d78e0e646da138a5fcb3a103d6bfeaede671622dba461fb84690a1f349.jpg",
+        "text": "$$\nJ ^ { D } = \\log D ( d _ { r } | q ) + \\log ( 1 - D ( d _ { g } | q ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With the baseline term included, the generator’s updates are in the direction of $\\nabla J ^ { G }$ , with the following definition. ",
+        "bbox": [
+            169,
+            483,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9cf7ee5b67bad49b3996cbe3906ccc8847b5bc8d1af38faec03bf8e8f7ed3952.jpg",
+        "text": "$$\nJ ^ { G } = \\log ( 1 - D ( d _ { r } | q ) ) + \\log D ( d _ { g } | q )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            510,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since maximizing $\\log ( 1 - z )$ with respect to $z$ is the same as maximizing $- \\log z$ , we can write the following equivalent loss functions ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fbaf000615f3c276109857e9b7b692270f5d9de0a86b1ce47981cbda61e48207.jpg",
+        "text": "$$\nJ ^ { D } = \\log D ( d _ { r } | q ) - \\log D ( d _ { g } | q ) \\qquad J ^ { G } = - \\log D ( d _ { r } | q ) + \\log D ( d _ { g } | q )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that this substitution is similar in principle to the substitution in Goodfellow et al. (2014), where the motivation is to allow easier flow of gradients. It is apparent that the discriminator and the generator are optimizing directly opposite loss functions and this detrimental to the performance of the models. We provide experimental proof later that the performance improvements shown in IRGAN are mainly because of the discriminator maximizing the likelihood of the real data and not because of the generator. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "6 PROPOSED MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            713,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We propose two models to compare and critically analyze performance gains facilitated by IRGAN and illustrate them in Figure 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The first model increases the likelihood of the training data and decreases the likelihood of documents which are not relevant to the query but have a high score according to its own parameters. It maximizes the following, where the sampling for the second term is from a candidate pool with only negative answers (denoted by $p ^ { - } .$ ). Not following this will lead to undesirable updates because sampling positive documents for the second term will result in decreasing the likelihood of real data. $\\psi$ denotes the parameters of the only model. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd7ac91844814373750882eeca99abbb015582ff0357c6d7cbb93b32497a572b.jpg",
+        "text": "$$\nJ ^ { M o d e l 1 } = \\operatorname* { m a x } _ { \\psi } \\sum _ { n = 1 } ^ { N } ( E _ { d \\sim p _ { t r u e } ( d | q _ { n } , r ) } [ ( \\log D ( d | q _ { n } ) ) ] + E _ { d \\sim p _ { \\psi } ^ { - } ( d | q _ { n } , r ) } [ ( \\log 1 - D ( d | q _ { n } ) ) ] )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            199,
+            885,
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+            928
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/90d72faff9848714db0f8a16698b2246a2947c87938645c125df29a2315c4b90.jpg",
+        "image_caption": [
+            "Figure 1: Models "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            108,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To alleviate the pernicious loss function of IRGAN, we propose a model which uses two discriminators in a co-operative setup influenced by Co-training (Blum & Mitchell (1998)). Instead of using two different views $( x _ { 1 } , x _ { 2 } )$ as mentioned in the work, we use the same views for both the discriminators but let them influence each other in a feedback loop. Training is similar to Model 1 with the only difference being that each discriminator decreases the likelihood of documents relevant to the other discriminator rather than itself, as shown in the equation below. This model achieves better performance than IRGAN. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/775afd9ff2cdf176c2c0582a39e74b168f93e93ea46da558f387ceaeebdbd398.jpg",
+        "text": "$$\n\\begin{array} { r l } & { J ^ { M o d e l 1 } = \\displaystyle \\operatorname* { m a x } _ { \\psi _ { 1 } } \\sum _ { n = 1 } ^ { N } ( E _ { d \\sim p _ { t r u e } ( d | q _ { n } , r ) } [ ( \\log D ( d | q _ { n } ) ) ] + E _ { d \\sim p _ { \\psi _ { 2 } } ^ { - } ( d | q _ { n } , r ) } [ ( \\log 1 - D ( d | q _ { n } ) ) ] ) } \\\\ & { \\displaystyle } \\\\ & { J ^ { M o d e l 2 } = \\displaystyle \\operatorname* { m a x } _ { \\psi _ { 2 } } \\sum _ { n = 1 } ^ { N } ( E _ { d \\sim p _ { t r u e } ( d | q _ { n } , r ) } [ ( \\log D ( d | q _ { n } ) ) ] + E _ { d \\sim p _ { \\psi _ { 1 } } ^ { - } ( d | q _ { n } , r ) } [ ( \\log 1 - D ( d | q _ { n } ) ) ] ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            435,
+            800,
+            527
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "7 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            541,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This section describes the datasets, the task and hyperparameters. ",
+        "bbox": [
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+            588
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "7.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            604,
+            287,
+            619
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments on three tasks, Web Search, Item Recommendation and Question Answering, using the same datasets mentioned in IRGAN. ",
+        "bbox": [
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+            630,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/9c38b95a49166b43386fb53cc11bd66102ccdc7b7ee2cdbf8ea100ee7b0e2ee1.jpg",
+        "table_caption": [
+            "Table 1: Datasets "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td>Dataset</td></tr><tr><td>WebSearch</td><td>LETOR by Liu et al. (2007)</td></tr><tr><td>Item-Recommendation</td><td>Movielens</td></tr><tr><td>Question Answering</td><td>InsuranceQA by Feng et al. (2015)</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "7.2 TASK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Web Search, the task is to retrieve the document which is most relevant to the query. Each query on average has around 5 positive documents. In Content Recommendation, users give ratings for movies and given a user, the task is to retrieve a movie that they would probably rate high. In IRGAN, any movie retrieved for which the user rating is greater than or equal to 4 (out of a scale of 5) is considered correct. Based on the dataset statistics, around $5 5 \\%$ of the user-movie ratings are $\\geq 5$ . This makes the problem easy to solve. In Question Answering, every query has just one relevant document in most cases. This is thus the hardest task. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "7.3 HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            357,
+            117
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The hyperparameters for the proposed model are the same except for absence of G Epochs, for obvious reasons. Information about hyperparameter tuning is mentioned in the Appendix. ",
+        "bbox": [
+            173,
+            130,
+            823,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c2a43b621bff4c7cbd5abce861eef2f2468826dbe7a3ab2161d68a0b5ee24bda.jpg",
+        "table_caption": [
+            "Table 2: Hyperparameters for IRGAN "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Description</td></tr><tr><td>Learning Rate</td><td>Forboth generator and discriminator</td></tr><tr><td>Batch Size</td><td>Batch size for training</td></tr><tr><td>Embed Dim</td><td>Embedding dimension of query or document</td></tr><tr><td>Epochs</td><td>Number of epochs of training</td></tr><tr><td>DEpochs</td><td>Number of epochs the discriminator is trained per epoch</td></tr><tr><td>G_Epochs Temperature</td><td>Number of epochs the generator is trained per epoch Temperature parameter for softmax sampling of documents</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "8 EXPERIMENTS AND DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+            356,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We report only the $\\mathrm { P @ 5 }$ and ${ \\mathrm { N D C G } } @ 5$ values because all other metrics follow the same trend. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "8.1 WEB SEARCH ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            417,
+            312,
+            433
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 3 reports the performance of various models. As can be seen, both the Single Discriminator and the Co-training models outperform IRGAN models. The fact that each query is associated approximately with 5 positive documents provides evidence that the proposed models can perform well in sparse reward settings. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/bd612b52729167660f151032f2cfd67ac59d62d97eaa6d43b25d9319d725f222.jpg",
+        "table_caption": [
+            "Table 3: Results on LETOR dataset used in IRGAN "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>P@5</td><td>NDCG@5</td></tr><tr><td>RankNet (Burges et al. (2005))</td><td>0.1219</td><td>0.1709</td></tr><tr><td>LambdaRank (Burges et al. (2007))</td><td>0.1352</td><td>0.1920</td></tr><tr><td>IRGAN-pointwise</td><td>0.1657</td><td>0.2225</td></tr><tr><td>IRGAN-pairwise</td><td>0.1676</td><td>0.2154</td></tr><tr><td>Single Discriminator</td><td>0.1676</td><td>0.2190</td></tr><tr><td>Co-training</td><td>0.1733</td><td>0.2252</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "8.2 ITEM-RECOMMENDATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            674,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This task, in contrast to the other two, has multiple relevant documents that can be retrieved for each query, making it slightly easier. Each user (query) rates a movie (document), and $5 5 \\%$ of the entries in the train set and $5 6 \\%$ in the test set are relevant pairs. It can be seen in Table 4 that the single discriminator model achieves only a slightly lower score, and given the small size of the dataset (943 users), it makes just 7 more mistakes when compared to IRGAN. This is not a statistically significant number, especially because the IRGAN generator is pre-initialized to a model which scores 0.34 but our model learns from scratch. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "8.3 QUESTION ANSWERING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            814,
+            382,
+            829
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "After close correspondence with the authors of IRGAN, we obtained all the hyperparameters required for the models. Multiple random seeds were used in vain, the results in the paper for Question-Answering tasks could not be replicated. We instead mention the best results out of all random seeds. We believe that if there is some random seed which gives better performance for IRGAN, it should do so for our model as well. The co-training model outperforms IRGAN-Pairwise. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/cd240a3aca605b9f9c46c53c6e89f2e175abbf79add3883ab48214ae7696f6f4.jpg",
+        "table_caption": [
+            "Table 4: Results on Movielens Dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>P@5</td><td>NDCG@5</td></tr><tr><td>BPR (Rendle et al. (2009))</td><td>0.3044</td><td>0.3245</td></tr><tr><td>LambdaFM (Yuan et al. (2016))</td><td>0.3474</td><td>0.3749</td></tr><tr><td>IRGAN-pointwise</td><td>0.3750</td><td>0.4099</td></tr><tr><td>Single Discriminator</td><td>0.3675</td><td>0.3925</td></tr><tr><td>Co-training</td><td>0.345</td><td>0.373</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b9c3dfe73a8dd8b31416fdcddf7953b304919f9132bd313c2cf0b7133de8bdee.jpg",
+        "table_caption": [
+            "Table 5: $\\mathrm { P @ 1 }$ on InsuranceQA "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>P@1</td></tr><tr><td>IRGAN-Pairwise Single Discriminator Co-training</td><td>0.616 0.614 0.623</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "8.4 LOSS CURVES ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/9d9f5cbe6e06a42653d99ecdd9ebfc40d91d7554d358d409e99c9bf6be2943b1.jpg",
+        "image_caption": [
+            "Figure 2: Performance curves "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The loss curves in Figure 2 picked from IRGAN’s work show deteriorating performance of the generator, which is in contrast to what is observed in actual adversarial training. In the minimax setting, since the generator is expected to capture the real data distribution, its performance is supposed to improve and this can indirectly be seen in GANs and DCGANs where the samples generated look more and more like real-world data points. Further, a deteriorating generator implies that the discriminator’s improvement in performance is only because of the first term of $J ^ { D }$ , which hints that our proposed models might be able to do better than IRGAN. The reason offered in the paper is that “A worse generator could be the result of the sparsity of document distribution, i.e., each question usually has only one correct answer”. But this reason does not seem plausible, given that DCGANs have been able to model very high dimensional data, where the probability distribution is only a tiny part of the real space. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Further, the increase in performance of the discriminator in all cases is coupled with a deteriorating generator. This substantiates our claim that the discriminator and the generator are optimizing directly opposite loss functions. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Item-recommendation task is a little different from the other two tasks at hand because of a large number of positive answers. When the loss curves are plotted, though the generator’s performance improves, the discriminator’s loss remains high and almost constant throughout the procedure, as shown in Figure 3. This is another indication that the performance of IRGAN is not actually because of the adversarial setup, but because of the maximization of the likelihood of the real data. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "9 CONNECTIONS TO PREVIOUS WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We have already shown in Section 2 that Conditional GANs are connected directly to Information Retrieval. The problem can also be viewed as a contextual multi-armed bandit problem (Li et al. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/6d5d7efb0cd7b45fd56ee19f6142f5ea5e2a0dd6ff2125a68691f433a1d369d9.jpg",
+        "image_caption": [
+            "Figure 3: Discriminator Loss for Content-Recommendation "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            114,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "(2010)), where each documents is an arm and the context $x _ { q , d }$ can be used to determine the actionvalue function $f _ { \\theta } ( x _ { q , d } )$ . In previous works (Li et al. (2010)) $f$ has been considered to be linear, but recent studies Collier & Llorens (2018) have modeled them as deep neural networks. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Pfau & Vinyals (2016), a parallel is drawn between Actor-Critic algorithms (Konda & Tsitsiklis (2000)) and GANs. This is directly related to our work because REINFORCE (Sutton et al. (2000)) with a baseline can be connected to Actor-Critic algorithms when bootstrapping is used (Sutton & Barto (2018)). The work shows a restricted scenario which involves a stateless MDP, each action setting all the pixels of the image and cross-entropy loss instead of mean-squared Bellmann residual in which GANs are equivalent to Actor-Critic algorithms. But this equivalence holds only when the baseline term is not used so the formulation in IRGAN is not exactly equivalent to a GAN framework. Another study (Finn et al. (2016)) draws a parallel between Inverse Reinforcement Learning $\\mathrm { N g }$ et al. (2000)) and GANs because both the methods try to “learn” the cost function to optimize for. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "10 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+            614,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The experiments performed show that IRGAN is by no means state-of-the-art on those datasets. Further, the performance does not justify the large training time of 4 hours per generator epoch and 1 hour of discriminator epoch as opposed to 2 hours per epoch of the co-training model (11 GB GPU and Question Answering task). The shaky mathematical formulation renders the generator useless after training, and any gains in performance can be attributed directly to the first term of $J ^ { D }$ , where the likelihood of the real data is increased. We showed that the discriminator and generator are optimizing directly opposite loss functions and this is the cause of deleterious training. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The poor performance of IRGAN on Web-Search and Question Answering and only a satisfactory performance on Content-Recommendation (which has dense rewards) lead us to speculate that it does not work well in sparse reward scenarios. This is similar to a well-known problem called the Sparse Reward Reinforcement Learning. We think that a correct formulation along with established techniques from the former, like reward shaping $\\mathrm { N g }$ et al. (1999)) may lead to better performance. Newer methods like Hindsight Experience Replay (Andrychowicz et al. (2017)) which allow models to learn both from mistakes and rewards may further ameliorate learning. ",
+        "bbox": [
+            174,
+            762,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We would also like to explore in the direction of learning correct adversarial frameworks for more complex tasks like Image Retrieval and Question Answering which will involve learning end-toend trainable models. With advances in modeling sequences, this could also involve generation of documents rather than sampling them. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "APPENDIX A HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            473,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The hyperparameters are mentioned in tables 6, 7, 8, 9 and 10. The following were the ranges of hyperparameter tuning, along with the best value. Gradient Descent Optimizer was used so that the comparison with IRGAN is fair. ",
+        "bbox": [
+            174,
+            132,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/7a758d47850c1b299b7f5654f7b6960cfbeb52e8e84628a812e22a30c13e090d.jpg",
+        "table_caption": [
+            "Table 6: Single Discriminator for Web-Search "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter/Seed</td><td>Range/List</td><td>Best</td></tr><tr><td>Learning Rate</td><td>0.002-0.2</td><td>0.004</td></tr><tr><td>Batch Size</td><td>[8,16,32]</td><td>8</td></tr><tr><td>Feature Size</td><td>[46,92]</td><td>46</td></tr><tr><td>Random Seed</td><td>[20,40,60]</td><td>40</td></tr></table>",
+        "bbox": [
+            343,
+            219,
+            653,
+            292
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the co-training model, for every epoch, we optimize the two discriminators several times. We call these the outer and inner epochs in Table 7. ",
+        "bbox": [
+            171,
+            313,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/94370bc50feacafed87e9372094a050765746598986d531c7ca7008a70e16ca1.jpg",
+        "table_caption": [
+            "Table 7: Co-training for Web-Search "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter/Seed</td><td>Range/List</td><td>Best</td></tr><tr><td>Learning Rate Outer Epochs Inner Epochs Batch Size Feature Size Random Seed</td><td>0.002-0.2 [30,50] [30,50] [8,16,32] [46,92]</td><td>0.006 50 30 8 46</td></tr></table>",
+        "bbox": [
+            343,
+            385,
+            655,
+            486
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "DNS K in Table 8 represents the number of candidates that are chosen before performing the softmax. This is done to make the procedure computationally tractable. We use the value suggested in IRGAN. ",
+        "bbox": [
+            173,
+            505,
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+            547
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/e37b7fe125790f73459c7b27823dea348fad8453026269198cf58ec74d7c63b6.jpg",
+        "table_caption": [
+            "Table 8: Single Discriminator for Content Recommendation "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter/Seed</td><td>Range/List</td><td>Best</td></tr><tr><td>Learning Rate</td><td>0.01-0.05</td><td>0.02</td></tr><tr><td rowspan=\"3\">Batch Size Embedding Dimension Random Seed</td><td>10</td><td>10</td></tr><tr><td>[20,40,60]</td><td>20</td></tr><tr><td>70</td><td>70</td></tr><tr><td>DNS_K</td><td>5</td><td>5</td></tr></table>",
+        "bbox": [
+            339,
+            589,
+            658,
+            676
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX B DESCRIPTION OF MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            710,
+            519,
+            727
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The discriminator and the generator have the same architecture in all the tasks. For the Web-retrieval task, the model has a single hidden layer 46 units. ",
+        "bbox": [
+            171,
+            741,
+            823,
+            770
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the content-recommendation task, the model converts users and movies to a 5 dimensional embedding. This can be though to be a single hidden layer which compresses a one-hot user embedding to a 5 dimensional embedding. ",
+        "bbox": [
+            174,
+            790,
+            825,
+            833
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the Question-Answering task, each word is initialized to a 100 dimensional random vector. A Convolutional Neural Network is then used and the window size of the convolutional kernel is (1,2,3,5). A max-pooling-over-time strategy is then used and the output is a 100 dimensional vector because each feature map is pooled to a scalar. Note that this architecture is the same as the one used in the IRGAN paper. We refer the user to that for further description. ",
+        "bbox": [
+            174,
+            853,
+            823,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/fa9e70a372b5d2cf4b24e05f1aa20ab66bdf1ea0fadb1d1298c00b0e74cd3868.jpg",
+        "table_caption": [
+            "Table 9: Single Discriminator for Question Answering "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Best</td></tr><tr><td>Learning Rate</td><td>0.05</td></tr><tr><td>Epochs</td><td>20</td></tr><tr><td>Batch Size</td><td>100</td></tr><tr><td>Embedding Dimension</td><td>100</td></tr></table>",
+        "bbox": [
+            387,
+            277,
+            611,
+            351
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/784f1ec7f7afad31fe6f8beca79f253689be38d1d1aed1bed1024e2d8b773933.jpg",
+        "table_caption": [
+            "Table 10: Co-training for Question Answering "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Best</td></tr><tr><td>Learning Rate</td><td>0.05</td></tr><tr><td>Outer Epochs</td><td>20</td></tr><tr><td>Inner Epochs</td><td>1</td></tr><tr><td>Batch Size</td><td>100</td></tr><tr><td>Embedding Dimension</td><td>100</td></tr></table>",
+        "bbox": [
+            387,
+            684,
+            611,
+            772
+        ],
+        "page_idx": 11
+    }
+]

parse/train/_RnHyIeu5Y5/_RnHyIeu5Y5.md

@@ -0,0 +1,272 @@
+# ViTAE: Vision Transformer Advanced by Exploring Intrinsic Inductive Bias
+
+Yufei Xu1∗ Qiming Zhang1∗ Jing Zhang1 Dacheng Tao2,1
+
+1The University of Sydney, Australia, 2JD Explore Academy, China
+
+{yuxu7116,qzha2506}@uni.sydney.edu.au, jing.zhang1@sydney.edu.au, dacheng.tao@gmail.com
+
+# Abstract
+
+Transformers have shown great potential in various computer vision tasks owing to their strong capability in modeling long-range dependency using the self-attention mechanism. Nevertheless, vision transformers treat an image as 1D sequence of visual tokens, lacking an intrinsic inductive bias (IB) in modeling local visual structures and dealing with scale variance. Alternatively, they require large-scale training data and longer training schedules to learn the IB implicitly. In this paper, we propose a new Vision Transformer Advanced by Exploring intrinsic IB from convolutions, i.e., ViTAE. Technically, ViTAE has several spatial pyramid reduction modules to downsample and embed the input image into tokens with rich multi-scale context by using multiple convolutions with different dilation rates. In this way, it acquires an intrinsic scale invariance IB and is able to learn robust feature representation for objects at various scales. Moreover, in each transformer layer, ViTAE has a convolution block in parallel to the multi-head selfattention module, whose features are fused and fed into the feed-forward network. Consequently, it has the intrinsic locality IB and is able to learn local features and global dependencies collaboratively. Experiments on ImageNet as well as downstream tasks prove the superiority of ViTAE over the baseline transformer and concurrent works. Source code and pretrained models will be available at code.
+
+# 1 Introduction
+
+Transformers [79, 17, 40, 14, 46, 61] have shown a domination trend in NLP studies owing to their strong ability in modeling long-range dependencies by the self-attention mechanism [67, 81, 51]. Such success and good properties of transformers has inspired following many works that apply them in various computer vision tasks [19, 100, 97, 80, 7]. Among them, ViT [19] is the pioneering pure transformer model that embeds images into a sequence of visual tokens and models the global dependencies among them with stacked transformer blocks. Although it achieves promising performance on image classification, it requires large-scale training data and a longer training schedule. One important reason is that ViT
+
+![](images/4515266c49c6aff7e7d656459cb5d8b9e3662e222790882fee6a157e2ad56504.jpg)  
+Figure 1: Comparison of data and training efficiency of T2T-ViT-7 and ViTAE-T on ImageNet.
+
+lacks intrinsic inductive bias (IB) in modeling local visual structures (e.g., edges and corners) and dealing with objects at various scales like convolutions. Alternatively, ViT has to learn such IB implicitly from large-scale data.
+
+Unlike vision transformers, Convolution Neural Networks (CNNs) naturally equip with the intrinsic IBs of scale-invariance and locality and still serve as prevalent backbones in vision tasks [26, 70, 62, 8, 96]. The success of CNNs inspires us to explore intrinsic IBs in vision transformers. We start by analyzing the above two IBs of CNNs, i.e., locality and scale-invariance. Convolution that computes local correlation among neighbor pixels is good at extracting local features such as edges and corners. Consequently, CNNs can provide plentiful low-level features at the shallow layers [94], which are then aggregated into high-level features progressively by a bulk of sequential convolutions [32, 68, 71]. Moreover, CNNs have a hierarchy structure to extract multi-scale features at different layers [68, 38, 26]. Besides, intra-layer convolutions can also learn features at different scales by varying their kernel sizes and dilation rates [25, 70, 8, 45, 96]. Consequently, scale-invariant feature representation can be obtained via intra- or inter-layer feature fusion. Nevertheless, CNNs are not well suited to model long-range dependencies2, which is the key advantage of transformers. An interesting question comes up: Can we improve vision transformers by leveraging the good properties of CNNs? Recently, DeiT [76] explores the idea of distilling knowledge from CNNs to transformers to facilitate training and improve the performance. However, it requires an off-the-shelf CNN model as the teacher and consumes extra training cost.
+
+Different from DeiT, we explicitly introduce intrinsic IBs into vision transformers by re-designing the network structures in this paper. Current vision transformers always obtain tokens with singlescale context [19, 93, 80, 86, 47, 69, 77] and learn to adapt to objects at different scales from data. For example, T2T-ViT [93] improves ViT by delicately generating tokens in a soft split manner. Specifically, it uses a series of Tokens-to-Token transformation layers to aggregate single-scale neighboring contextual information and progressively structurizes the image to tokens. Motivated by the success of CNNs in dealing with scale variance, we explore a similar design in transformers, i.e., intra-layer convolutions with different receptive fields [70, 91], to embed multi-scale context into tokens. Such a design allows tokens to carry useful features of objects at various scales, thereby naturally having the intrinsic scale-invariance IB and explicitly facilitating transformers to learn scale-invariant features more efficiently from data. On the other hand, low-level local features are fundamental elements to generate high-level discriminative features. Although transformers can also learn such features at shallow layers from data, they are not skilled as convolutions by design. Recently, [89, 43, 21] stack convolutions and attention layers sequentially and demonstrate that locality is a reasonable compensation of global dependency. However, this serial structure ignores the global context during locality modeling (and vice versa). To avoid such a dilemma, we follow the “divide-and-conquer” idea and propose to model locality and long-range dependencies in parallel and then fuse the features to account for both. In this way, we empower transformers to learn local and long-range features within each block more effectively.
+
+Technically, we propose a new Vision Transformers Advanced by Exploring Intrinsic Inductive Bias $( V i T A E )$ , which is a combination of two types of basic cells, i.e., reduction cell (RC) and normal cell (NC). RCs are used to downsample and embed the input images into tokens with rich multi-scale context while NCs aim to jointly model locality and global dependencies in the token sequence. Moreover, these two types of cells share a simple basic structure, i.e., paralleled attention module and convolutional layers followed by a feed-forward network (FFN). It is noteworthy that RC has an extra pyramid reduction module with atrous convolutions of different dilation rates to embed multi-scale context into tokens. Following the setting in [93], we stack three reduction cells to reduce the spatial resolution by $1 / 1 6$ and a series of NCs to learn discriminative features from data. ViTAE outperforms representative vision transformers in terms of data efficiency and training efficiency (see Figure 1), as well as classification accuracy and generalization on downstream tasks.
+
+Our contributions are threefold. First, we explore two types of intrinsic IB in transformers, i.e., scale invariance and locality, and demonstrate the effectiveness of this idea in improving the feature learning ability of transformers. Second, we design a new transformer architecture named ViTAE based on two new reduction and normal cells to intrinsically incorporate the above two IBs. The proposed ViTAE embeds multi-scale context into tokens and learns both local and long-range features effectively. Third, ViTAE outperforms representative vision transformers regarding classification accuracy, data efficiency, training efficiency, and generalization on downstream tasks. ViTAE achieves $7 5 . 3 \%$ and $8 2 . 0 \%$ top-1 accuracy on ImageNet with 4.8M and 23.6M parameters, respectively.
+
+# 2 Related Work
+
+# 2.1 CNNs with intrinsic IB
+
+CNNs have led to a series of breakthroughs in image classification [38, 94, 26, 95, 87] and downstream computer vision tasks. The convolution operations in CNNs extract local features from the neighbor pixels within the receptive field determined by the kernel size [42]. Following the intuition that local pixels are more likely to be correlated in images [41], CNNs have the intrinsic IB in modeling locality. In addition to the locality, another critical topic in visual tasks is scale-invariance, where multi-scale features are needed to represent the objects at different scales effectively [49, 90]. For example, to effectively learn features of large objects, a large receptive field is needed by either using large convolution kernels [90, 91] or a series of convolution layers in deeper architectures [26, 32, 68, 71]. To construct multi-scale feature representation, the classical idea is using image pyramid [8, 1, 55, 4, 39, 16], where features are hand-crafted or learned from a pyramid of images at different resolutions respectively [44, 8, 52, 63, 35, 3]. Accordingly, features from the small scale image mainly encode the large objects while features from the large scale image respond more to small objects. In addition to the above inter-layer fusion way, another way is to aggregate multi-scale context by using multiple convolutions with different receptive fields within a single layer, i.e., intra-layer fusion [96, 71, 70, 70, 72]. Either inter-layer fusion or intra-layer fusion empower CNNs an intrinsic IB in modeling scale-invariance. This paper introduces such an IB to vision transformers by following the intra-layer fusion idea and utilizing multiple convolutions with different dilation rates in the reduction cells to encode multi-scale context into each visual token.
+
+# 2.2 Vision transformers with learned IB
+
+ViT [19] is the pioneering work that applies a pure transformer to vision tasks and achieves promising results. However, since ViT lacks intrinsic inductive bias in modeling local visual structures, it indeed learns the IB from amounts of data implicitly. Following works along this direction are to simplify the model structures with fewer intrinsic IBs and directly learn them from large scale data [50, 74, 75, 22, 18, 20, 27] which have achieved promising results and been studied actively. Another direction is to leverage the intrinsic IB from CNNs to facilitate the training of vision transformers, e.g., using less training data or shorter training schedules. For example, DeiT [76] proposes to distill knowledge from CNNs to transformers during training. However, it requires an off-the-shelf CNN model as a teacher, introducing extra computation cost during training. Recently, some works try to introduce the intrinsic IB of CNNs into vision transformers explicitly [23, 58, 21, 43, 15, 89, 83, 92, 6, 47, 11]. For example, [43, 21, 83] stack convolutions and attention layers sequentially, resulting in a serial structure and modeling the locality and global dependency accordingly. [80, 28] design sequential stage-wise structures while [47, 33] apply attention within local windows. However, these serial structure may ignore the global context during locality modeling (and vice versa). [88] establishes connection across different scales at the cost of heavy computation. Instead, we follow the “divide-and-conquer” idea and propose to model locality and global dependencies simultaneously via a parallel structure within each transformer layer. Conformer [58], the most relevant concurrent work to us, employs a unit to explore inter-block interactions between parallel convolution and transformer blocks. In contrast, in ViTAE, the convolution and attention modules are designed to be complementary to each other within the transformer block. In addition, Conformer is not designed to have inherent scale invariance IB.
+
+# 3 Methodology
+
+# 3.1 Revisit vision transformer
+
+We first give a brief review of vision transformer in this part. To adapt transformers to vision tasks, ViT [19] first splits an image $x \in R ^ { H \times W \times C }$ into tokens with a reduction ratio of $p$ (i.e., $x _ { t } \in R ^ { ( ( H \times W ) / p ^ { 2 } ) \times D } )$ , where $H , W$ and $C$ denote the height, width, and channel dimensions of the input image, $D = C p ^ { 2 }$ denotes the token dimension. Then, an extra class token is concatenated to the visual tokens before adding position embeddings in an element-wise manner. The resulting tokens are fed into the following transformer layers. Each transformer layer is composed of two parts, i.e., a multi-head self-attention module (MHSA) and a feed forward network (FFN).
+
+![](images/12ff638562a0b78bbcef09633d771c21fc798831f54a00e148f38d5e89e3596b.jpg)  
+Figure 2: The structure of the proposed ViTAE. It is constructed by stacking three RCs and several NCs. Both types of cells share a simple basic structure, i.e., an MHSA module and a parallel convolutional module followed by an FFN. In particular, RC has an extra pyramid reduction module using atrous convolutions with different dilation rates to embed multi-scale context into tokens.
+
+MHSA Multi-head self-attention extends single-head self-attention (SHSA) by using different projection matrices for each head. Specifically, the input tokens $x _ { t }$ are first projected to queries $( Q )$ , keys $( K )$ and values $( V )$ using projection matrices, i.e., $Q , K , V = x _ { t } W _ { Q } , x _ { t } Q _ { K } , x _ { t } Q _ { V }$ , where $W _ { Q / K / V } \in R ^ { D \times D }$ denotes the projection matrix for query, key, and value, respectively. Then, the self-attention operation is calculated as:
+
+$$
+A t t e n t i o n ( Q , K , V ) = s o f t m a x ( \frac { Q K ^ { T } } { \sqrt { D } } ) V .
+$$
+
+This SHSA module is repeated for $h$ times to formulate the MHSA module, where $h$ is the number of heads. The output features of the $h$ heads are concatenated along the channel dimension and formulate the output of the MHSA module.
+
+FFN FFN is placed on top of the MHSA module and applied to each token identically and separately. It consists of two linear transformations with an activation function in between. Besides, a layer normalization [2] and a shortcut are added before and aside from the MHSA and FFN, respectively.
+
+# 3.2 Overview architecture of ViTAE
+
+ViTAE aims to introduce the intrinsic IB in CNNs to vision transformers. As shown in Figure 2, ViTAE is composed of two types of cells, i.e., RCs and NCs. RCs are responsible for embedding multi-scale context and local information into tokens, and NCs are used to further model the locality and long-range dependencies in the tokens. Taken an image $x \in R ^ { H \times W \times C }$ as input, three RCs are used to gradually downsample $x$ by $4 \times , 2 \times$ , and $2 \times$ , respectively. Thereby, the output tokens of the RCs are of size $[ H / 1 6 , W / 1 6 , D ]$ where $D$ is the token dimension (64 in our experiments). The output tokens of RCs are then flattened as $R ^ { H W / 2 5 6 \times D }$ , concatenated with the class token, and added by the sinusoid position encoding. Next, the tokens are fed into the following NCs, which keep the length of the tokens. Finally, the prediction probability is obtained using a linear classification layer on the class token from the last NC.
+
+# 3.3 Reduction cell
+
+Instead of directly splitting and flatten images into visual tokens based on a linear image patch embedding layer, we devise the reduction cell to embed multi-scale context and local information into visual tokens, which introduces the intrinsic scale-invariance and locality IBs from convolutions. Technically, RC has two parallel branches responsible for modeling locality and long-range dependency, respectively, followed by an FFN for feature transformation. We denote the input feature of the $i _ { t h } \ : \mathrm { R C }$ as $f _ { i } \in \dot { R } ^ { H _ { i } \times W _ { i } \times D _ { i } }$ . The input of the first RC is the image $x$ . In the global dependencies branch, $f _ { i }$ is firstly fed into a Pyramid Reduction Module (PRM) to extract multi-scale context, i.e.,
+
+$$
+f _ { i } ^ { m s } \triangleq P R M _ { i } ( f _ { i } ) = C a t ( [ C o n v _ { i j } ( f _ { i } ; s _ { i j } , r _ { i } ) | s _ { i j } \in S _ { i } , r _ { i } \in  { \mathcal { R } } ] ) ,
+$$
+
+where $C o n v _ { i j } ( \cdot )$ indicates the $j$ th convolutional layer in the PRM $( P R M _ { i } ( \cdot ) )$ . It uses a dilation rate $s _ { i j }$ from the predefined dilation rate set $S _ { i }$ corresponding to the ith RC. Note that we use stride convolution to reduce the spatial dimension of features by a ratio $r _ { i }$ from the predefined reduction ratio set $\mathcal { R }$ . The conv features are concatenated along the channel dimension, i.e., $f _ { i } ^ { m s } \in$ $R ^ { ( W _ { i } / p ) \times ( H _ { i } / p ) \times ( | S _ { i } | D ) }$ , where $| { S _ { i } } |$ denotes the number of dilation rates in $S _ { i }$ . $f _ { i } ^ { m s }$ is then processed by an MHSA module to model long-range dependencies, i.e.,
+
+$$
+f _ { i } ^ { g } = M H S A _ { i } ( I m g 2 S e q ( f _ { i } ^ { m s } ) ) ,
+$$
+
+where $I m g 2 S e q ( \cdot )$ is a simple reshape operation to flatten the feature map to a 1D sequence. In this way, $f _ { i } ^ { g }$ embeds the multi-scale context in each token. In addition, we use a Parallel Convolutional Module (PCM) to embed local context within the tokens, which are fused with $f _ { i } ^ { g }$ as follows:
+
+$$
+f _ { i } ^ { l g } = f _ { i } ^ { g } + { \cal P } { \cal C } M _ { i } ( f _ { i } ) .
+$$
+
+Here, $P C M _ { i } ( \cdot )$ represents the PCM, which is composed of three stacked convolution layers and an $I m g 2 S e q ( \cdot )$ operation. It is noteworthy that the parallel convolution branch has the same spatial downsampling ratio as the PRM by using stride convolutions. In this way, the token features can carry both local and multi-scale context, implying that RC acquires the locality IB and scale-invariance IB by design. The fused tokens are then processed by the FFN, reshaped back to feature maps, and fed into the following RC or NC, i.e.,
+
+$$
+f _ { i + 1 } = S e q 2 I m g ( F F N _ { i } ( f _ { i } ^ { l g } ) + f _ { i } ^ { l g } ) ,
+$$
+
+where the $S e q 2 I m g ( \cdot )$ is a simple reshape operation to reshape a token sequence back to feature maps. $F F N _ { i } ( \cdot )$ represents the FFN in the ith RC. In our ViTAE, three RCs are stacked sequentially to gradually reduce the input image’s spatial dimension by $4 \times , 2 \times$ , and $2 \times$ , respectively. The feature maps generated by the last RC are of a size of $[ H / 1 6 , W / 1 6 , D ]$ , which are then flattened into visual tokens and fed into the following NCs.
+
+# 3.4 Normal cell
+
+As shown in the bottom right part of Figure 2, NCs share a similar structure with the reduction cell except for the absence of the PRM. Due to the relatively small $( \frac { 1 } { 1 6 } \times )$ spatial size of feature maps after RCs, it is unnecessary to use PRM in NCs. Given $f _ { 3 }$ from the third RC, we first concatenate it with the class token $t _ { c l s }$ , and then add it to the positional encodings to get the input tokens $t$ for the following NCs. Here we ignore the subscript for clarity since all NCs have an identical architecture but different learnable weights. $t _ { c l s }$ is randomly initialized at the start of training and fixed during the inference. Similar to the RC, the tokens are fed into the MHSA module, i.e., $t _ { g } = M H S A ( t )$ . Meanwhile, they are reshaped to 2D feature maps and fed into the PCM, i.e., $t _ { l } = \bar { I } m g 2 S e q ( P C M ( S e q 2 I m g ( t ) ) )$ . Note that the class token is discarded in PCM because it has no spatial connections with other visual tokens. To further reduce the parameters in NCs, we use group convolutions in PCM. The features from MHSA and PCM are then fused via element-wise sum, i.e., $t _ { l g } = t _ { g } + t _ { l }$ . Finally, $t _ { l g }$ are fed into the FFN to get the output features of NC, i.e., $t _ { n c } = F F N ( t _ { l g } ) \overline { { + } } t _ { l g }$ . Similar to ViT [19], we apply layer normalization to the class token generated by the last NC and feed it to the classification head to get the final classification result.
+
+# 3.5 Model details
+
+We use two variants of ViTAE in our experiments for a fair comparison of other models with similar model sizes. The details of them are summarized in Table 1. In the first RC, the default convolution kernel size is $7 \times 7$ with a
+
+Table 1: Model details of two variants of ViTAE.   
+
+<table><tr><td>Model</td><td>Reduction Cell Dilation</td><td>Cells</td><td>Normal Cell Heads Embed Cells</td><td></td><td>Params Macs (M)</td><td>(G)</td></tr><tr><td>ViTAE-T</td><td>[1,2,3,4] √</td><td>3</td><td>4 256</td><td>7</td><td>4.8</td><td>1.5</td></tr><tr><td>ViTAE-S</td><td>[1,2,3,4] √</td><td>3</td><td>6</td><td>384 14</td><td>23.6</td><td>5.6</td></tr></table>
+
+stride of 4 and dilation rates of $\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ . In the following two RCs, the convolution kernel size is $3 \times 3$ with a stride of 2 and dilation rates of $S _ { 2 } = [ 1 , 2 , 3 ]$ and $S _ { 3 } = [ 1 , 2 ]$ , respectively. Since the spatial dimension of tokens decreases, there is no need to use large kernels and dilation rates. PCM in both RCs and NCs comprises three convolutional layers with a kernel size of $3 \times 3$ .
+
+# 4 Experiments
+
+# 4.1 Implementation details
+
+We train and test the proposed ViTAE model on the standard ImageNet [38] dataset, which contains about 1.3 million images and covers 1k classes. Unless explicitly stated, the image size during training is set to $2 2 4 \times 2 2 4$ . We use the AdamW [48] optimizer with the cosine learning rate scheduler and uses the data augmentation strategy exactly the same as T2T [93] for a fair comparison, regarding the training strategies and the size of models. We use a batch size of 512 for training all our models and set the initial learning rate to be 5e-4. The results of our models can be found in Table 2, where all the models are trained for 300 epochs on 8 V100 GPUs. The models are built on PyTorch [57] and TIMM [82].
+
+# 4.2 Comparison with the state-of-the-art
+
+We compare our ViTAE with both CNN models and vision transformers with similar model sizes in Table 2. Both Top-1/5 accuracy and real Top-1 accuracy on the ImageNet validation set are reported. We categorize the methods into CNN models, vision transformers with learned IB, and vision transformers with introduced intrinsic IB. Compared with CNN models, our ViTAE-T achieves a $7 5 . 3 \%$ Top-1 accuracy, which is better than ResNet-18 with more parameters. The real Top-1 accuracy of the ViTAE model is $8 2 . 9 \%$ , which is comparable to ResNet-50 that has four more times of parameters than ours. Similarly, our ViTAE-S achieves $8 2 . 0 \%$ Top-1 accuracy with half of the parameters of ResNet-101 and ResNet-152, showing the superiority of learning both local and longrange features from specific structures with corresponding intrinsic IBs by design. Similar phenomena can also be observed when comparing ViTAE-T with MobileNetV1 [31] and MobileNetV2 [65], where ViTAE obtains better performance with fewer parameters. When compared with larger models which are searched according to NAS [73], our ViTAE-S achieves a similar performance when using $3 8 4 \times 3 8 4$ images as input, which further shows the potential of vision transformers with intrinsic IB.
+
+In addition, among the transformers with learned IB, ViT is the first pure transformer model for visual recognition. DeiT shares the same structure with ViT but uses different data augmentation and training strategies to facilitate the learning of transformers. DeiT⚗ denotes using an off-the-shelf CNN model as the teacher model to train DeiT, which introduces the intrinsic IB from CNN to transformer implicitly in a knowledge distillation manner, showing better performance than the vanilla ViT on the ImageNet dataset. It is exciting to see that our ViTAE-T with fewer parameters even outperforms the distilled model DeiT⚗, demonstrating the efficacy of introducing intrinsic IBs in transformers by design. Besides, compared with other transformers with explicit intrinsic IB, our ViTAE with fewer parameters also achieves comparable or better performance. For instance, ViTAE-T achieves comparable performance with LocalVit-T but has 1M fewer parameters, demonstrating the superiority of the proposed RCs and NCs in introducing intrinsic IBs.
+
+# 4.3 Ablation study
+
+We use T2T-ViT [93] as our baseline model in the following ablation study of our ViTAE. As shown in Table 3, we investigate the hyper-parameter settings in RCs and NCs by isolating them separately. All the models are trained for 100 epochs on ImageNet and follow the same training setting and data augmentation strategy as described in Section 4.1.
+
+Table 2: Comparison of ViTAE and SOTA methods on the ImageNet validation set.   
+
+<table><tr><td rowspan="2">Type Model</td><td rowspan="2">Params (M)</td><td rowspan="2">MACs (G)</td><td rowspan="2">Input Size</td><td colspan="3">ImageNet Real</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top-1</td></tr><tr><td rowspan="9">CNN</td><td>ResNet-18 [26]</td><td>11.7</td><td>3.6</td><td>224</td><td>70.3</td><td>86.7</td><td>77.3</td></tr><tr><td>ResNet-50 [26]</td><td>25.6</td><td>7.6</td><td>224</td><td>76.7</td><td>93.3</td><td>82.5</td></tr><tr><td>ResNet-101 [26]</td><td>44.5</td><td>15.2</td><td>224</td><td>78.3</td><td>94.1</td><td>83.7</td></tr><tr><td>ResNet-152 [26]</td><td>60.2</td><td>22.6</td><td>224</td><td>78.9</td><td>94.4</td><td>84.1</td></tr><tr><td>EfficientNet-B0 [73]</td><td>5.3</td><td>0.8</td><td>224</td><td>77.1</td><td>93.3</td><td>83.5</td></tr><tr><td>EfficientNet-B4 [73]</td><td>19.3</td><td>8.4</td><td>380</td><td>82.9</td><td>96.4</td><td>88.0</td></tr><tr><td>MobileNetV1 [31]</td><td>4.3</td><td>0.6</td><td>224</td><td>72.3</td><td>1</td><td>-</td></tr><tr><td>MobileNetV2(1.4) [65]</td><td>6.9</td><td>0.6</td><td>224</td><td>74.7</td><td>-</td><td>-</td></tr><tr><td>RegNetY-600M[62]</td><td>6.1</td><td>1.2</td><td>224</td><td>75.5</td><td>-</td><td>-</td></tr><tr><td>RegNetY-4GF[62] RegNetY-8GF[62]</td><td>20.6 39.2</td><td>8.0 16.0</td><td>224</td><td>80.0</td><td>1</td><td>86.4</td></tr><tr><td></td><td></td><td></td><td></td><td>224</td><td>81.7</td><td>1</td><td>87.4</td></tr><tr><td rowspan="14"></td><td>DeiT-T[76]</td><td>5.7</td><td>2.6</td><td>224</td><td>72.2</td><td>91.1</td><td>80.6</td></tr><tr><td>DeiT-T [76]</td><td>5.7</td><td>2.6</td><td>224</td><td>74.5</td><td>91.9</td><td>82.1</td></tr><tr><td>LocalViT-T[43]</td><td>5.9</td><td>2.6</td><td>224</td><td>74.8</td><td>92.6</td><td></td></tr><tr><td>LocalViT-T2T[43]</td><td>4.3</td><td>2.4</td><td>224</td><td>72.5</td><td>-</td><td>1 1</td></tr><tr><td>ConT-Ti [89]</td><td>5.8</td><td>1.6</td><td>224</td><td>74.9</td><td>-</td><td>-</td></tr><tr><td>PiT-Ti [29]</td><td>4.9</td><td>1.4</td><td>224</td><td>73.0</td><td>-</td><td>1</td></tr><tr><td>T2T-ViT-7 [93]</td><td>4.3</td><td>1.2</td><td>224</td><td>71.7</td><td>90.9</td><td>79.7</td></tr><tr><td>ViTAE-T</td><td>4.8</td><td>1.5</td><td>224</td><td>75.3</td><td>92.7</td><td>82.9</td></tr><tr><td>ViTAE-T ↑ 384</td><td>4.8</td><td>5.7</td><td>384</td><td>77.2</td><td>93.8</td><td>84.4</td></tr><tr><td>CeiT-T [92]</td><td>6.4</td><td>2.4</td><td>224</td><td>76.4</td><td>93.4</td><td>83.6</td></tr><tr><td>ConViT-Ti[15]</td><td>6.0</td><td>2.0</td><td>224</td><td>73.1</td><td>1</td><td>1</td></tr><tr><td>Cross ViT-Ti [6]</td><td>6.9</td><td>3.2</td><td>224</td><td>73.4</td><td>1</td><td>1</td></tr><tr><td>ViTAE-6M</td><td>6.5</td><td>2.0</td><td>224</td><td>77.9</td><td>94.1</td><td>84.9</td></tr><tr><td>PVT-T[80] LocalViT-PVT [43]</td><td>13.2</td><td>3.8</td><td>224</td><td>75.1</td><td>1</td><td></td></tr><tr><td rowspan="8">PiT-XS [29] ConT-M [89] ViTAE-13M DeiT-S [76]</td><td>13.5</td><td>9.6</td><td>224</td><td></td><td>94.2</td><td>-</td></tr><tr><td>ConViT-Ti+ [15] 10.0</td><td>4.0</td><td>224</td><td>78.2 76.7</td><td></td><td>1</td></tr><tr><td></td><td>2.8</td><td></td><td>78.1</td><td>1</td><td>-</td></tr><tr><td>10.6 19.2</td><td>6.2</td><td>224 224</td><td>80.2</td><td>-</td><td>1</td></tr><tr><td>13.2</td><td>3.4</td><td>224</td><td>81.0</td><td>- 95.4</td><td>- 86.8</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>22.1 DeiT-S [76] 22.1</td><td>9.8 9.8</td><td>224 224</td><td>79.9 81.2</td><td>95.0 95.4</td><td>85.7 86.8</td></tr><tr><td>PVT-S[80]</td><td>7.6</td><td>224</td><td>79.8</td><td>-</td><td></td></tr><tr><td></td><td>24.5 23.5</td><td>5.2</td><td></td><td>81.3</td><td></td><td>1</td></tr><tr><td>Conformer-Ti [58] Swin-T[47]</td><td></td><td></td><td>224</td><td></td><td>-</td><td></td></tr><tr><td>CeiT-S [92]</td><td>29.0</td><td>9.0</td><td>224</td><td>81.3</td><td>-</td><td>1</td></tr><tr><td>CvT-13 [83]</td><td>24.2 20.0</td><td>9.0</td><td>224</td><td>82.0 81.6</td><td>95.9</td><td>87.3 86.7</td></tr><tr><td>ConViT-S[15]</td><td>27.0</td><td>9.0 10.8</td><td>224 224</td><td>81.3</td><td>1 1</td><td>1</td></tr><tr><td>Cross ViT-S [6]</td><td>26.7</td><td>11.2</td><td>224</td><td>81.0</td><td>1</td><td>1</td></tr><tr><td>PiT-S [29]</td><td>23.5</td><td>4.8</td><td>224</td><td>80.9</td><td></td><td></td></tr><tr><td>TNT-S [23]</td><td></td><td></td><td></td><td></td><td>-</td><td>1</td></tr><tr><td>Twins-PCPVT-S[10]</td><td>23.8</td><td>10.4</td><td>224</td><td>81.3</td><td>95.6</td><td>-</td></tr><tr><td></td><td>24.1</td><td>7.4</td><td>224</td><td>81.2</td><td>-</td><td>-</td></tr><tr><td>Twins-SVT-S [10]</td><td>24.0</td><td>5.6</td><td>224</td><td>81.7</td><td>-</td><td>1</td></tr><tr><td>T2T-ViT-14 [93]</td><td>21.5</td><td>5.2</td><td>224</td><td>81.5</td><td>95.7</td><td>86.8</td></tr><tr><td>ViTAE-S</td><td>23.6</td><td>5.6</td><td>224</td><td>82.0</td><td>95.9</td><td>87.0</td></tr><tr><td>ViTAE-S ↑ 384</td><td>23.6</td><td>20.2</td><td>384</td><td>83.0</td><td>96.2</td><td>87.5</td></tr></table>
+
+We use $\checkmark$ and $\times$ to denote whether or not the corresponding module is enabled during the experiments. If all columns under the RC and NC are marked $\times$ as shown in the first row, the model becomes the standard T2T-ViT model. “Pre” indicates the output features of PCM and MHSA are fused before FFN while “Post” indicates a late fusion strategy correspondingly. “BN” indicates whether PCM uses BN after the convolutional layer or not. $\mathit { \Omega } ^ { 6 } \times 3 \mathit { \Omega } ^ { 5 }$ in the first column denotes that the dilation rate set is the same in the three RCs. “ $[ 1 , 2 , 3 , 4 ]$ $\downarrow ^ { \circ }$ denotes using lower dilation rates in deeper RCs, i.e., $\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ , $S _ { 2 } = [ 1 , 2 , 3 ]$ , $S _ { 3 } = [ 1 , 2 ]$ .
+
+As can be seen, using a pre-fusion strategy and BN achieves the best $6 9 . 9 \%$ Top-1 accuracy among other settings. It is noteworthy that all the variants of NC outperform the vanilla T2T-ViT, implying the effectiveness of PCM, which introduces the intrinsic locality IB in transformers. It can also be observed that BN plays an important role in improving the model’s performance as it can help to alleviate the scale deviation between convolution’s and attention’s features. For the RC, we first investigate the impact of using different dilation rates in the PRM, as shown in the first column. As can be seen, using larger dilation rates (e.g., 4 or 5) does not deliver better performance. We suspect that larger dilation rates may lead to plain features in the deeper RCs due to the smaller resolution of feature maps. To
+
+Table 3: Ablation Study of RCs and NCs in our ViTAE. “Pre” indicates the output features of PCM and MHSA are fused before FFN while “Post” indicates a late fusion strategy correspondingly. “BN” indicates whether PCM uses BN or not. “ $[ 1 , 2 , 3 , 4 ]$ $\downarrow ^ { \circ }$ denotes using smaller dilation rates in deeper RCs, i.e., $\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ , $\mathsf { \bar { S } } _ { 2 } = [ 1 , 2 , 3 ]$ , $ { S _ { 3 } } = [ 1 , 2 ]$ .   
+
+<table><tr><td>Reduction Cell</td><td></td><td>Normal Cell</td><td rowspan="2">Top-1</td></tr><tr><td>Dilation (S1 ~ S3) PCM</td><td>Pre</td><td>BN Post</td></tr><tr><td>× ×</td><td>×</td><td>× ×</td><td>68.7</td></tr><tr><td>× ×</td><td>√</td><td>× ×</td><td>69.1</td></tr><tr><td>× ×</td><td>×</td><td>√ ×</td><td>69.0</td></tr><tr><td>× ×</td><td>×</td><td>√ √</td><td>68.8</td></tr><tr><td>× ×</td><td>√</td><td>× √</td><td>69.9</td></tr><tr><td>[1,2]×3</td><td>× ×</td><td>×</td><td>× 69.5</td></tr><tr><td>[1,2,3]×3 ×</td><td>×</td><td>× ×</td><td>69.9</td></tr><tr><td>[1,2,3,4] × 3 ×</td><td>×</td><td>× ×</td><td>69.2</td></tr><tr><td>[1,2,3,4,5] × 3 ×</td><td>×</td><td>× ×</td><td>68.9</td></tr><tr><td>[1,2,3,4]↓ ×</td><td>×</td><td>× ×</td><td>69.8</td></tr><tr><td>[1,2,3,4]↓ √</td><td>×</td><td>× ×</td><td>71.7</td></tr><tr><td>[1,2,3,4]↓ √</td><td>√</td><td>× √</td><td>72.6</td></tr></table>
+
+validate the hypothesis, we use smaller dilation rates in deeper RCs as denoted by $[ 1 , 2 , 3 , 4 ] \downarrow$ . As can be seen, it achieves comparable performance as $[ 1 , 2 , 3 ] \times$ . However, compared with $[ 1 , 2 , 3 , 4 ] \downarrow$ , $[ 1 , 2 , 3 ] \times$ increases the amount of parameters from 4.35M to $4 . 6 \mathsf { M }$ . Therefore, we select $[ 1 , 2 , 3 , 4 ] \downarrow$ as the default setting. In addition, after using PCM in the RC, it introduces the intrinsic locality IB, and the performance increases to $7 1 . 7 \%$ Top-1 accuracy. Finally, the combination of RCs and NCs achieves the best accuracy at $7 2 . 6 \%$ , demonstrating the complementarity between our RCs and NCs.
+
+# 4.4 Data efficiency and training efficiency
+
+To validate the effectiveness of the introduced intrinsic IBs in improving data efficiency and training efficiency, we compare our ViTAE with T2T-ViT at different training settings: (a) training them using $20 \%$ , $60 \%$ , and $100 \%$ ImageNet training set for equivalent 100 epochs on the full ImageNet training set, e.g., we employ 5 times epochs when using $20 \%$ data for training compared with using $100 \%$ data; and (b) training them using the full ImageNet training set for 100, 200, and 300 epochs respectively. The results are shown in Figure 1. As can be seen, ViTAE consistently outperforms the T2T-ViT baseline by a large margin in terms of both data efficiency and training efficiency. For example, ViTAE using only $20 \%$ training data achieves comparable performance with T2T-ViT using all data. When $60 \%$ training data are used, ViTAE significantly outperforms T2T-ViT using all data by about an absolute $3 \%$ accuracy. It is also noteworthy that ViTAE trained for only 100 epochs has outperformed T2T-ViT trained for 300 epochs. After training ViTAE for 300 epochs, its performance is significantly boosted to $7 5 . 3 \%$ Top-1 accuracy. With the proposed RCs and NCs, the transformer layers in our ViTAE only need to focus on modeling long-range dependencies, leaving the locality and multi-scale context modeling to its convolution counterparts, i.e., PCM and PRM. Such a “divide-and-conquer” strategy facilitates the training of vision transformers, making it possible to learn more efficiently with less training data and fewer training epochs.
+
+To further validate the data efficiency of ViTAE model, we train the ViTAE model from scratch on the smaller datasets, i.e., Cifar10 and Cifar100. The results are summarized in Table 4. It can be viewed that with only $1 / 7$ number of epochs, the ViTAE-T model achieves better classification performance on Cifar10 dataset, with far fewer parameters (4.8M v.s. 86M), which further confirms ViTAE model’s data efficiency.
+
+Table 4: Results of training from scratch on Cifar10/100.   
+
+<table><tr><td>Model</td><td>Params (M)</td><td>Top-1 Acc</td><td>Epochs</td><td>Dataset</td></tr><tr><td>DeiT-B</td><td>86.0</td><td>97.5</td><td>7000</td><td>Cifar10</td></tr><tr><td>ViTAE-T</td><td>4.8</td><td>97.7</td><td>1000</td><td>Cifar10</td></tr><tr><td>ViTAE-T</td><td>4.8</td><td>85.0</td><td>1000</td><td>Cifar100</td></tr></table>
+
+# 4.5 Generalization on downstream tasks
+
+Table 5: Generalization of ViTAE and SOTA methods on different downstream tasks.   
+
+<table><tr><td>Model</td><td>Params (M)</td><td>Cifar10</td><td>Cifar100</td><td>iNat19</td><td>Cars</td><td>Flowers</td><td>Pets</td></tr><tr><td>Grafit ResNet-50 [78]</td><td>25.6</td><td>-</td><td>=</td><td>75.9</td><td>92.5</td><td>98.2</td><td>-</td></tr><tr><td>EfficientNet-B5 [73]</td><td>30</td><td>98.1</td><td>91.1</td><td>-</td><td>-</td><td>98.5</td><td>-</td></tr><tr><td>ViT-B/16 [19]</td><td>86.5</td><td>98.1</td><td>87.1</td><td>-</td><td>1</td><td>89.5</td><td>93.8</td></tr><tr><td>ViT-L/16 [19]</td><td>304.3</td><td>97.9</td><td>86.4</td><td>-</td><td>-</td><td>89.7</td><td>93.6</td></tr><tr><td>DeiT-B [76]</td><td>86.6</td><td>99.1</td><td>90.8</td><td>77.7</td><td>92.1</td><td>98.4</td><td>-</td></tr><tr><td>T2T-ViT-14 [93]</td><td>21.5</td><td>98.3</td><td>88.4</td><td>-</td><td>-</td><td>-</td><td>1</td></tr><tr><td>ViTAE-T</td><td>4.8</td><td>97.3</td><td>86.0</td><td>73.3</td><td>89.5</td><td>97.5</td><td>92.6</td></tr><tr><td>ViTAE-S</td><td>23.6</td><td>98.8</td><td>90.8</td><td>76.0</td><td>91.4</td><td>97.8</td><td>94.2</td></tr></table>
+
+We further investigate the generalization of the proposed ViTAE models on downstream tasks by finetuning them on the training sets of several fine-grained classification tasks3, including Flowers [53], Cars [36], Pets [56], and iNaturalist19. We also fine-tune the proposed ViTAE models on Cifar10 [37] and Cifar100 [37]. The results are shown in Table 5. It can be seen that ViTAE achieves SOTA performance on most of the datasets using comparable or fewer parameters. These results demonstrate that the good generalization ability of our ViTAE.
+
+# 4.6 Visual inspection of ViTAE
+
+To further analyze the property of our ViTAE, we first calculate the average attention distance of each layer in ViTAE-T and the baseline T2T-ViT-7 on the ImageNet test set, respectively. The results are shown in Figure 3. It can be observed that with the usage of PCM, which focuses on modeling locality, the transformer layers in the proposed NCs can better focus on modeling long-range dependencies, especially in shallow layers. In the deep layers, the average attention distances of ViTAE-T and T2T-ViT-7 are almost the same since modeling long-range dependencies is much more important.
+
+These results confirm the effectiveness of the adopted “divide-and-conquer” idea in the proposed ViTAE, i.e., introducing the intrinsic locality IB from convolutions into vision transformers makes it possible that transformer layers only need to be responsible to long-range dependencies, since locality can be well modeled by convolutions in PCM.
+
+![](images/0d27a138487be9062b7073116e5379b9d74b5e8122843b673cecf6fb02d9776d.jpg)  
+Figure 3: The average per-layer attention distance of T2T-ViT-7 and our ViTAE-T.
+
+Besides, we apply Grad-CAM [66] on the MHSA’s output in the last NC to qualitatively inspect ViTAE. The visualization results are provided in Figure 4. Compared with the baseline T2T-ViT, our ViTAE covers the single or multiple targets in the images more precisely and attends less to the background. Moreover, ViTAE can better handle the scale variance issue as shown in Figure 4(b). Namely, it can precisely cover the birds no matter they are in small, middle, or large size. Such observations demonstrate that introducing the intrinsic IBs of locality and scale-invariance from convolutions to transformers helps ViTAE learn more discriminate features than the pure transformers.
+
+![](images/c40cbf192eae3a11c74a33bbbb56c6df92c409dc0cc54401ef7f9004a71cc003.jpg)  
+Figure 4: Visual inspection of T2T-ViT and ViTAE using Grad-CAM [66]. (a) Images containing multiple or single objects and the heatmaps. (b) Images containing the same class of objects at different scales and the heatmaps (Best viewed in color).
+
+# 5 Limitation and discussion
+
+In this paper, we explore two types of IBs and incorporate them into transformers through the proposed reduction and normal cells. With the collaboration of these two cells, our ViTAE model achieves impressive performance on the ImageNet with fast convergence and high data efficiency. Nevertheless, due to computational resource constraints, we have not scaled the ViTAE model and train it on largesize dataset, e.g., ImageNet-21K [38] and JFT-300M [30]. Although it remains unclear by now, we are optimistic about its scale property from the following preliminary evidence. As illustrated in Figure 2, our ViTAE model can be viewed as an intra-cell ensemble of complementary transformer layers and convolution layers owing to the skip connection and parallel structure. According to the attention distance analysis shown in Figure 3, the ensemble nature enables the transformer layers and convolution layers to focus on what they are good at, i.e., modeling long-range dependencies and locality. Therefore, ViTAE is very likely to learn better feature representation from large-scale data. Besides, we only study two typical IBs in this paper. More kinds of IBs such as constituting viewpoint invariance [64] can be explored in the future study.
+
+# 6 Conclusion
+
+In this paper, we re-design the transformer block by proposing two basic cells (reduction cells and normal cells) to incorporate two types of intrinsic inductive bias (IB) into transformers, i.e., locality and scale-invariance, resulting in a simple yet effective vision transformer architecture named ViTAE. Extensive experiments show that ViTAE outperforms representative vision transformers in various respects including classification accuracy, data efficiency, training efficiency, and generalization ability on downstream tasks. We plan to scale ViTAE to the large or huge model size and train it on large-size datasets in the future study. In addition, other kinds of IBs will also be investigated. We hope that this study will provide valuable insights to the following studies of introducing intrinsic IB into vision transformers and understanding the impact of intrinsic and learned IBs.
+
+Acknowledgement Dr. Jing Zhang is supported by the ARC project FL-170100117.
+
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parse/train/rJe04p4YDB/rJe04p4YDB.md

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+# SEMI-SUPERVISED LEARNING BY COACHING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recent semi-supervised learning (SSL) methods often have a teacher to train a student in order to propagate labels from labeled data to unlabeled data. We argue that a weakness of these methods is that the teacher does not learn from the student’s mistakes during the course of student’s learning. To address this weakness, we introduce Coaching, a framework where a teacher generates pseudo labels for unlabeled data, from which a student will learn and the student’s performance on labeled data will be used as reward to train the teacher using policy gradient.
+
+Our experiments show that Coaching significantly improves over state-of-the-art SSL baselines. For instance, on CIFAR-10, with only 4,000 labeled examples, a WideResNet-28-2 trained by Coaching achieves $9 6 . 1 1 \%$ accuracy, which is better than $9 4 . 9 \%$ achieved by the same architecture trained with 45,000 labeled. On ImageNet with $10 \%$ labeled examples, Coaching trains a ResNet-50 to $7 2 . 9 4 \%$ top-1 accuracy, comfortably outperforming the existing state-of-the-art by more than $4 \%$ . Coaching also scales successfully to the high data regime with full ImageNet. Specifically, with additional 9 million unlabeled images from OpenImages, Coaching trains a ResNet-50 to $8 2 . 3 4 \%$ top-1 accuracy, setting a new state-of-the-art for the architecture on ImageNet without using extra labeled data.1
+
+# 1 INTRODUCTION
+
+Professional players in competitive sports such as chess, tennis, or swimming often have coaches to help improving their performance. Although coaches typically do not play as well as the players, they observe the players and provide instructions to improve the players’ performance. Modern semi-supervised learning (SSL) algorithms do not follow this strategy. They instead have a teacher model that generates pseudo labels for unlabeled data, from which a student model learns by imitation (e.g., Lee (2013); Tarvainen & Valpola (2017); Laine & Aila (2017)). A weakness of these methods is that the teacher does not adjust itself based on the student’s performance and cannot adapt to make the student better over time, unlike professional sport coaches develop their players.
+
+Here, we propose a new semi-supervised learning method, called Coaching as shown in Figure 1, where the teacher learns throughout the course of student’s training. In our method, a teacher generates pseudo labels for unlabeled data, from which the student will learn. The student’s performance on labeled data will be used as reward to train the teacher with policy gradient.
+
+![](images/3a0ffb52634b9d076504729ad6bafe34cf33d3a55de49387d0cf2e344d54bd51.jpg)  
+Figure 1: Each step of gradient descent in Coaching consists of two steps. Updating the Student (top): The teacher network $T$ samples the labels $\hat { y }$ of unlabeled data $x _ { \mathrm { u n l } }$ for the student $S$ to learn from. Updating the Teacher (bottom): The teacher updates itself using policy gradient to improve the student’s performance on labeled data $x _ { \mathrm { l a b } }$ .
+
+Experiments show that our method achieves significant improvements over state-of-the-art semisupervised learning baselines and can be up to $1 0 \times$ more data efficient than supervised learning. For instance, with CIFAR-10, only using 4,000 labeled examples, a WideResNet-28-2 can be coached to $9 6 . 1 1 \%$ accuracy, outperforming the same model trained with 45,000 labeled examples which achieves $9 4 . 9 \%$ . Meanwhile, on ImageNet, using ResNet-50 with only $1 0 \%$ labeled examples, our method achieves $7 2 . 9 4 \%$ top-1 accuracy, outperforming all existing semi-supervised learning methods with the same amount of labeled data, and approaching the top-1 accuracy of $7 6 . 3 \%$ of the same ResNet-50 trained with all labels. Coaching also scales to the high data regime. In particular, with all 1.28 million labeled examples from ImageNet, plus 9 million unlabeled and potentially out-of-distribution data from OpenImages (Kuznetsova et al., 2018), a ResNet-50 can be coached to the accuracy of $8 2 . 3 4 \%$ , which is a new state-of-the-art for the architecture without using extra labeled data.
+
+# 2 METHOD
+
+Notations. Let $T$ , $S$ respectively be the teacher network and the student network in Coaching, and $\theta _ { T }$ , $\theta _ { S }$ be their corresponding parameters. Since we work with both labeled data and unlabeled data, we use $( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } )$ to refer to a pair of an input and its corresponding label, and use $x _ { \mathrm { u n l } }$ to refer to an unlabeled example. In addition, we use $\ell ( x , y ; \theta )$ to denote the cross entropy loss computed on input $x$ by with parameter $\theta$ on label $y$ .
+
+As shown in Figure 1, each training step in Coaching consists of two phases:
+
+Phase 1: The student learns from data pseudo labeled by the teacher. In this phase, the teacher $T$ first performs a forward pass on $x _ { \mathrm { u n l } }$ to compute the class distribution $P ( \cdot | x _ { \mathrm { u n l } } ; \theta _ { T } )$ . From this distribution, the teacher samples a pseudo label $\hat { y } _ { \mathrm { u n l } } \sim P ( \cdot | x _ { \mathrm { u n l } } ; \theta _ { T } )$ . The pair $x _ { \mathrm { u n l } } , \hat { y } _ { \mathrm { u n l } }$ is then shown to the student $S$ to make an update on its parameters $\theta _ { S }$ . The update is based on the gradient computed by back-propagating from the cross entropy loss. For instance, if $\theta _ { S }$ is updated using SGD, then:
+
+$$
+\theta _ { S } ^ { ( t + 1 ) } : = \theta _ { S } ^ { t } - \eta \cdot \underbrace { \frac { \partial \ell ( x _ { \mathrm { u n l } } , \hat { y } _ { \mathrm { u n l } } ; \theta _ { S } ) } { \partial \theta _ { S } } } _ { \xrightarrow [ ] { \Delta } } \bigg | _ { \theta _ { S } = \theta _ { S } ^ { ( t ) } } = \theta _ { S } ^ { ( t ) } - \eta \cdot g _ { S } ^ { ( t ) } ,
+$$
+
+where $\eta$ is the learning rate.
+
+Phase 2: The teacher learns from the student’s loss. After the student updates its parameters   
+as in Equation 1, its parameters entropy loss. The goal of the testudent is updated as in Equation $\theta _ { S } ^ { ( t + 1 ) }$ ) led example xlab, ylab is evaluated on a laben Coaching is to give tn the cross entropy loss using the crosssuch that if theill be minimized. $\hat { y } _ { \mathrm { u n l } }$ $^ { l }$ $\ell ( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \theta _ { S } ^ { ( t + 1 ) } )$
+
+Clearly, $\ell ( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \theta _ { S } ^ { ( t + 1 ) } )$ depends on $\theta _ { S } ^ { ( t + 1 ) }$ , which in turn depends on the pseudo label $\hat { y } _ { \mathrm { u n l } }$ that the teacher samples. From the perspective of reinforcement learning, $\hat { y } _ { \mathrm { u n l } }$ can be treated as an onpolicy action of the teacher, which leads to the reward of $- \ell ( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \theta _ { S } ^ { ( t + 1 ) } )$ . In this perspective, we propose to train $\theta _ { T }$ to minimize the value of $\ell ( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \bar { \theta } _ { S } ^ { ( t + 1 ) } )$ , where $\bar { \theta } _ { S } ^ { ( t + 1 ) }$ is the expected destination that the teacher will guide the student to. This expectation is taken over all possible pseudo labels $\hat { y } _ { \mathrm { u n l } }$ . Formally,
+
+$$
+\theta _ { T } ^ { * } = \operatorname * { a r g m i n } _ { \theta _ { T } } R ( \theta _ { T } ) \mathrm { ~ w h e r e ~ } R ( \theta _ { T } ) = \ell \left( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \mathbb { E } _ { \hat { y } _ { \mathrm { t u n l } } \sim P ( \cdot | x _ { \mathrm { l a b } } ; \theta _ { T } ) } \left[ \theta _ { S } ^ { ( t + 1 ) } \right] \right)
+$$
+
+To find $\theta _ { T } ^ { * }$ , we differentiate $R ( \theta _ { T } )$ in Equation 2 with respect to $\theta _ { T }$ . Here, we present the resulting gradient $g _ { T } ^ { ( t ) }$ , which has the form
+
+$$
+g _ { T } ^ { ( t ) } \approx \eta \cdot \left[ \left( g _ { S } ^ { ( t ) } \right) ^ { \top } \cdot \left( \frac { \partial \ell ( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \theta _ { S } ) } { \partial \theta _ { S } } \bigg | _ { \theta _ { S } = \theta _ { S } ^ { ( t + 1 ) } } \right) ^ { \top } \right] \cdot \left( \frac { \partial \ell ( x _ { \mathrm { u n l } } , \hat { y } _ { \mathrm { u n l } } ; \theta _ { T } ) } { \partial \theta _ { T } } \bigg | _ { \theta _ { T } = \theta _ { T } ^ { ( t ) } } \right)
+$$
+
+The full derivation can be found in Appendix A, but intuitively, the differentiation depends on two tools. The first tool is the is the chain rule, which we leverage to differentiate $R ( \theta _ { T } )$ with respect to $\theta _ { T }$ . The second tool is the REINFORCE equation (Williams, 1992), which we leverage to establish the relationship between $\mathbb { E } _ { \hat { y } _ { \mathrm { u n l } } } \left[ \theta _ { S } ^ { ( t + 1 ) } \right]$ and $\theta _ { T }$ .
+
+Coaching combines the two steps above in an SGD step. We summarize the method in Algorithm 1.
+
+# Algorithm 1 The Coaching method.
+
+Input :Labeled data $x _ { \mathrm { l a b } }$ , $y _ { \mathrm { l a b } }$ and unlabeled data $x _ { \mathrm { u n l } }$ .   
+1 Initialize $\theta _ { T } ^ { ( 0 ) }$ and $\theta _ { S } ^ { ( 0 ) }$   
+2 for $t = 0$ to $N - 1$ do   
+3 Sample an unlabeled example $x _ { \mathrm { u n l } }$ and a labeled example $x _ { \mathrm { l a b } }$ , $y _ { \mathrm { l a b } }$   
+4 Sample $\hat { y } _ { \mathrm { u n l } } \sim P ( \cdot | x _ { \mathrm { u n l } } ; \theta _ { T } )$   
+5 $\theta _ { S } ^ { ( t + 1 ) } : = \theta _ { S } ^ { ( t ) } - \eta \cdot g _ { S } ^ { ( t ) }$ . Compute $g _ { S } ^ { ( t ) }$ with pseudo labels as in Equation 1 and update $\theta _ { S }$   
+6 Sθ(t+1)T : $\theta _ { T } ^ { ( t + 1 ) } : = \theta _ { T } ^ { ( t ) } - \eta \cdot h ^ { ( t ) } \cdot g _ { T } ^ { ( t ) }$ . Compute the gradient $g _ { T } ^ { ( t ) }$ as in Equation 3 and update $\theta _ { T }$   
+7 end   
+8 return $\theta _ { S } ^ { ( N ) }$ . Only the student model is used for predictions and evaluations
+
+Generalize to an arbitrary batch size. Above, we have only discussed Coaching for a single unlabeled data $x _ { \mathrm { u n l } }$ and a single labeled data $x _ { \mathrm { l a b } }$ , $y _ { \mathrm { l a b } }$ . Now, we describe how to scale Coaching to an arbitrary batch size. Scaling $x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } }$ to a minibatch of labeled example, $X _ { \mathrm { l a b } }$ , $Y _ { \mathrm { l a b } }$ is straightforward, as we can simply replace all computations of the cross entropy $\ell ( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \theta _ { S } ^ { ( t + 1 ) } )$ with the average cross entropy on the minibatch \`(Xlab, Ylab; θ(t+1)S ). T(1) (2) o scale a single unlabeled example xunl to a minibatch of unlabeled examples $X _ { \mathrm { u n l } } = \{ x _ { \mathrm { u n l } } ^ { ( 1 ) } , x _ { \mathrm { u n l } } ^ { ( 2 ) } , . . . , x _ { \mathrm { u n l } } ^ { ( B ) } \}$ , we treat each batch of pseudo labels $\hat { Y } _ { \mathrm { u n l } }$ as a compound action sampled from the joint distribution
+
+$$
+P \left( \hat { Y } _ { \mathrm { u n l } } \middle | X _ { \mathrm { u n l } } ; \theta _ { T } \right) = P \left( \hat { y } _ { \mathrm { u n l } } ^ { ( 1 ) } , \hat { y } _ { \mathrm { u n l } } ^ { ( 2 ) } , \ldots , \hat { y } _ { \mathrm { u n l } } ^ { ( B ) } \middle | X _ { \mathrm { u n l } } ; \theta _ { T } \right) = \prod _ { i = 1 } ^ { B } P \left( \hat { y } _ { \mathrm { u n l } } ^ { ( i ) } \middle | x _ { \mathrm { u n l } } ^ { ( i ) } ; \theta _ { T } \right)
+$$
+
+Since every pseudo label $\hat { y } _ { \mathrm { u n l } } ^ { ( i ) }$ is sampled independently, applying REINFORCE as in Equation 3 simply factors the per-instance cross entropy into the batch cross entropy $\begin{array} { r l } { ~ } & { { } \sum _ { i = 1 } ^ { B } \ell ( x _ { \mathrm { u n l } } ^ { ( i ) } , \hat { y } _ { \mathrm { u n l } } ^ { ( i ) } ; \theta _ { T } ) } \end{array}$ .
+
+# 3 EXPERIMENTS
+
+Compared to other methods that use both labeled data and unlabeled data, Coaching has three main advantages:
+
+1. The teacher does not only demonstrate its knowledge to the student but also adjusts its teaching strategy in an adaptive manner with the student, throughout the course of the student’s learning.   
+2. The teacher in Coaching can benefit from advanced SSL techniques such as consistency regularization.   
+3. The student in Coaching never learns directly from labeled data. This does not only prevent overfitting when limited labeled data is available, but also allows us to finetune the trained student in Coaching directly on labeled data to further boost the student’s performance.
+
+We perform experiments to verify the strength of Coaching. In Section 3.1, we consider the low data regime with typical benchmarks for SSL methods. After that, in Section 3.2, we consider the high data regime which contains potentially out-of-distribution data.
+
+Model Architectures. In our experiments, our teacher model and our student model always have the same architecture but with different weights. For CIFAR-10 and SVHN, we use the WideResNet28-2 (Zagoruyko & Komodakis, 2016), which has 1.45 million parameters. For ImageNet, we use a ResNet-50 (He et al., 2016), which has 25.5 million parameters. For experiments that train only one model, we apply exponential moving average with a decay rate of 0.99 on the weights of the model. For experiments that have a teacher model and a student model, we apply this exponential moving average on the weights of the student model only.
+
+Additional Implementation Details. To improve the stability and accuracy of the method, we apply a few minor enhancements to the teacher:
+
+1. Use cosine distance instead of dot product. As the dot product $h ^ { ( t ) }$ in Equation 3 has a large value range, in order to stabilize training, we compute $h ^ { ( t ) }$ using the gradients’ cosine distance.   
+2. Use a baseline for $h ^ { ( t ) }$ . To further reduce the variance of $h ^ { ( t ) }$ , we maintain a moving average $b$ of $h ^ { ( t ) }$ and subtract $b$ from $h ^ { ( t ) }$ every time we compute $g _ { T } ^ { ( t ) }$ as in Equation 3.   
+3. Additional supervised loss for the teacher. We find that adding the supervised loss $\ell ( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \theta _ { T } )$ to the teacher’s objective results in a faster learning and better student.   
+4. Consistently regularize the teacher. In the low data regime, consistency regularization improves the teacher and the student. More details are in Section 3.1.   
+5. Pre-training the teacher. When the number of classes is large, it is beneficial to initialize the teacher with a trained model so that the pseudo labels are better than random at the beginning of the student’s learning. If we pre-train the teacher, Point 3 has minimal effect.   
+6. Finetuning the student. Since the student in Coaching only learns from unlabeled data and pseudo labels generated by the teacher, finetuning a converged student on labeled data often improves the student’s performance.
+
+The details mentioned above are mutually orthogonal. Since (1) and (2) are crucial to stabilize the Coaching process, they are always used in our experiments. In addition, we apply (3) and (4) to the low data regime, and apply (5) for the high data regime for computational efficiency and strong performance. We will explain these decisions in the corresponding sections.
+
+# 3.1 RESULTS ON LOW DATA REGIME
+
+Datasets. We consider three datasets with reduced numbers of labeled instances: CIFAR10 (Krizhevsky, 2009) with 4,000 labeled examples, SVHN (Netzer et al., 2011) with 1,000 labeled examples, and ImageNet (Russakovsky et al., 2015) with 128,000 labeled examples, which is approximately $1 0 \%$ of the whole ImageNet. All images in these datasets are used as unlabeled examples, which means that even the labeled images can be used as unlabeled examples. We use the image size of $3 2 \times 3 2$ for CIFAR-10 and SVHN, and the image size of $2 2 4 \times 2 2 4$ for ImageNet. These datasets, label reductions, and image sizes are standard for low data image classification.
+
+Baselines. We compare Coaching against 3 baseline training algorithms Purely Supervised, PseudoLabel (Lee, 2013), and Unsupervised Data Augmentation (UDA; Xie et al. (2019)). We discuss these baselines more in Section 4. We choose these baselines for three reasons. First, the purely supervised baseline serves to verify our implementation and to demonstrate the overfitting of our models when labeled data is scarce. Second, comparing Coaching with Pseudo-Label confirms the benefits of continuing to train the teacher throughout the course of the student’s learning. Finally, we compare against UDA because is the state-of-the-art on the datasets that we consider.
+
+To ensure a fair comparison, we re-implement these baselines in our environment. We follow Oliver et al. (2018)’s train/eval/test splitting, and we use the same amount of resources to tune hyperparameters for our baselines as well as for Coaching. More details are in Appendix C.
+
+Additional baselines. In addition to the three main baselines discussed above, we also include four other baselines: Temporal Ensemble (Laine & Aila, 2017), Mean Teacher (Tarvainen & Valpola, 2017), VAT (Miyato et al., 2018), LGA (Jackson & Schulman, 2019), ICT (Verma et al., 2019), and MixMatch (Berthelot et al., 2019). We use results reported by Oliver et al. (2018). Since these methods do not share the same controlled environment, the comparison to them is not direct, and should be contextualized as suggested by Oliver et al. (2018).
+
+Data augmentations. In our implementation of UDA and Coaching, we use RandomAugment, which is a randomized augmentation strategy over all the operations in the search space of AutoAugment (Cubuk et al., 2019). We use RandomAugment because it is simple to implement, requires no expensive search, and achieves similar performance compared to UDA with AutoAugment. More details of RandomAugment can be found in Appendix C.2.   
+Table 1: Image Classification Accuracy on reduced CIFAR-10, SVHN, and ImageNet. Higher is better. For CIFAR-10 and SVHN, we report mean $\pm$ std over 10 runs, while for ImageNet, we report Top-1/Top-5 accuracy of a single run. Results in the second block are taken from past papers, while the rest shares the same environment and hyper-parameter settings. All methods share the same model architecture: WideResNet-28-2 for CIFAR-10 and SVHN, and ResNet-50 for ImageNet.   
+
+<table><tr><td>Methods</td><td>CIFAR-10 (4,000)</td><td>SVHN (1,000)</td><td>ImageNet (10%)</td></tr><tr><td>Purely Supervised on full dataset</td><td>94.92 ± 0.17</td><td>97.41 ± 0.16</td><td>76.89/93.27</td></tr><tr><td>Temporal Ensemble</td><td>83.63 ±0.63</td><td>92.81± 0.27</td><td></td></tr><tr><td>Mean Teacher</td><td>84.13± 0.28</td><td>94.35 ± 0.47</td><td></td></tr><tr><td>VAT+EntMin</td><td>86.87± 0.39</td><td>94.65 ± 0.19</td><td>-/83.39</td></tr><tr><td>LGA +VAT</td><td>87.94 ± 0.19</td><td>93.42 ± 0.36</td><td>1</td></tr><tr><td>ICT</td><td>92.71±0.02</td><td>96.11 ± 0.04</td><td></td></tr><tr><td>MixMatch</td><td>93.76±0.06</td><td>96.73 ± 0.31</td><td></td></tr><tr><td>Purely Supervised</td><td>82.14±0.25</td><td>88.17 ±0.47</td><td>57.75/80.23</td></tr><tr><td>Pseudo Labels</td><td>83.79 ± 0.11</td><td>89.81± 0.41</td><td>58.21/82.19</td></tr><tr><td>UDA (our implementation)</td><td>94.53 ±0.18</td><td>97.11 ± 0.17</td><td>68.07/88.19</td></tr><tr><td>Coaching</td><td>95.60 ±0.19</td><td>97.79 ± 0.11</td><td>72.39/90.52</td></tr><tr><td>Coaching + Finetune</td><td>96.11 ± 0.07</td><td>98.01 ± 0.07</td><td>72.94/90.80</td></tr></table>
+
+Main results. In Table 1, we present our main results before and after finetuning the student on labeled data. The results confirm that Coaching significantly outperforms UDA and other strong baselines in semi-supervised learning.
+
+On CIFAR-10 and SVHN, compared to the state-of-the-art UDA, Coaching’s error rate reduction are roughly $3 0 \%$ and $1 0 \%$ . As UDA’s accuracy is already relatively high, such error reductions are significant. On CIFAR-10, Coaching is also the first approach to exceed supervised learning on the all labels by using merely 4,000 labeled examples. Meanwhile, on ImageNet- $10 \%$ , Coaching outperforms UDA by almost $5 \%$ in top-1 accuracy, going from $6 8 . 0 7 \%$ to ${ \bar { 7 } } 2 . 9 4 \%$ . Even prior to finetuning on labeled data, Coaching still outperforms UDA and other baselines.
+
+Comparing to existing state-of-the-art methods. To the best of our knowledge, Coaching has achieved new state-of-the-art performances among the same model architectures on three datasets considered in this section.
+
+For CIFAR-10 and SVHN, all existing better results use a larger model and more advanced regularization techniques. For instance, Xie et al. (2019) reports $9 7 . 3 \%$ with UDA (Xie et al., 2019), but their backbone model is PyramidNet, which has $1 8 \times$ more parameters than WideResNet-28-2 and they train with Shake-Drop regularization (Yamada et al., 2018). Similarly, for ImageNet- $10 \%$ , the only better published result is $7 3 . 2 1 \%$ top-1 accuracy, achieved by MOAM- $S ^ { 4 } L$ (Zhai et al., 2019). This accuracy is only slightly better than Coaching’s $7 2 . 9 4 \%$ , but uses a $4 \times$ wider ResNet-50. We believe that the enhancements in architectures, regularization techniques, and model sizes, can be applied to Coaching to further improve our results.
+
+# 3.2 RESULTS ON HIGH DATA REGIME
+
+We have seen Coaching achieves strong performance for low data image classification tasks. Another aspect of these tasks is that the unlabeled data also come from the same domain as the labeled data, which is a restricted assumption. In this section, we show that Coaching also excels in the regime where we have a large labeled dataset and an order of magnitude more unlabeled data. In this regime, we also test the performance of our method when the unlabeled set may have out-of-domain images, i.e., the images belong to categories that do not exist in ImageNet.
+
+Datasets. We experiment with all labeled examples in ImageNet. Additionally, we take unlabeled images from the entire $4 ^ { \mathrm { t h } }$ version of OpenImages dataset (Kuznetsova et al., 2018), which has 9 million natural images. A few samples from OpenImages can be found in Figure 2. Unless otherwise specified, for both datasets, we use the image size of $2 2 4 \times 2 2 4$ .
+
+Baselines. Since this regime of high data has not been extensively studied, we are only aware of two relevant, strong baselines. Our first baseline is Billion-scale Semi-supervised Learning (Billion-scale SSL; Yalniz et al. (2019)). Billion-scale SSL uses unlabeled data from the YFCC100M dataset (Thomee et al., 2015), studies several self-training settings, with various model architectures for teachers and students. Here, we restrict our comparison to the settings that use ResNet-50 for both the teacher and the student. Our second baseline is UDA (Xie et al., 2019), for which the authors select unlabeled images algorithmically from the JFT dataset.2
+
+Other than these baselines, we compare Coaching to techniques that enhance supervised learning, such as DropBlock (Ghiasi et al., 2018), CutMix (Yun et al., 2019), and FixRes (Touvron et al., 2019).
+
+Implementation details. We implement Coaching the same as in Section 3.1, except for one part: Instead of directly training and consistently regularizing the teacher, we initialize the teacher using a pre-trained ResNet-50 (pre-trained on full ImageNet). Then, throughout the course of the student’s learning, we only train the teacher to minimize the student’s cross entropy loss. We do not use additional supervised loss for the teacher because because once the teacher is pre-trained, adding another loss to the teacher has minimal effect. We do not consistently regularize the teacher because Xie et al. (2019) has found that consistency regularization requires in-domain data, while we do not filter our unlabeled images from OpenImages.
+
+<table><tr><td rowspan="2">Methods</td><td rowspan="2">Unlabeled images</td><td colspan="2">Image size</td><td rowspan="2">Top-1</td><td rowspan="2">Top-5</td></tr><tr><td>Train</td><td>Test</td></tr><tr><td>Supervised</td><td>None</td><td>224</td><td>224</td><td>76.89</td><td>93.27</td></tr><tr><td>DropBlock</td><td>None</td><td>224</td><td>224</td><td>78.35</td><td>94.15</td></tr><tr><td>FixRes +CutMix</td><td>None</td><td>224</td><td>320</td><td>79.8</td><td>94.9</td></tr><tr><td>Coaching</td><td>OpenImages</td><td>224</td><td>320</td><td>79.80</td><td>94.87</td></tr><tr><td rowspan="2">FixRes Coaching</td><td>None</td><td>224</td><td>384</td><td>79.1</td><td>94.6</td></tr><tr><td>OpenImages</td><td>224</td><td>384</td><td>80.10</td><td>95.07</td></tr><tr><td>Billion-scale SSL</td><td>YFCC100M</td><td>224</td><td>224</td><td>77.6</td><td></td></tr><tr><td>Coaching</td><td>OpenImages</td><td>224</td><td>224</td><td>78.62</td><td>94.26</td></tr><tr><td>UDA</td><td>JFT</td><td>331</td><td>331</td><td>79.04</td><td>94.45</td></tr><tr><td>Coaching</td><td>OpenImages</td><td>224</td><td>331</td><td>79.86</td><td>94.92</td></tr><tr><td>Coaching+iterative</td><td>OpenImages</td><td>224</td><td>331</td><td>82.34</td><td>96.09</td></tr></table>
+
+Table 2: Image classification accuracy with full ImageNet plus unlabeled images. Results are organized by image size because image size has a strong impact on models’ performance.
+
+Results. We present our results in Table 2. As can be seen, Coaching outperforms all relevant SSL baselines. Specifically, for the image size of 224, Coaching outperforms Billion-scale SSL by about $1 \%$ top-1 accuracy, even though Billion-scale SSL uses 10 times more unlabeled data. Meanwhile, for the image size of 331, Coaching achieves the top-1 accuracy of $7 9 . 8 6 \%$ , comfortably outperforming the top-1 accuracy of $7 9 . 0 4 \%$ by UDA. This improvement is particularly significant, since Coaching simply uses all data from OpenImages, while UDA has to select and balance the class distribution of their unlabeled data using a pre-trained teacher. This difference suggests that the teacher in Coaching can give helpful pseudo labels to the student, even on potentially out-of-distribution data.
+
+It is worth mentioning that Coaching also outperforms the strong supervised baselines of DropBlock and FixRes, and is on par with FixRes+CutMix. However, DropBlock and CutMix are both regularization techniques orthogonal to Coaching. Similar to consistency regularization in Section 3.1, these techniques can be incorporated into the teacher in Coaching to improve performance.
+
+Comparing to state-of-the-art SSL results. Yalniz et al. (2019) reports the top-1 accuracy of $8 1 . 2 \%$ for a ResNet-50 student. However, they need to pre-train a much bigger network ResNext-101-32x48 teacher (829 million parameters, $3 2 \mathrm { x }$ larger than ResNet-50) on 1 billion Instagram images with weak labels (Mahajan et al., 2018). Then, they use the pseudo-labels from this teacher to train a ResNet-50 student for 2 billion steps. The fact that they use weakly labeled data from Instagram, much bigger architecture in ResNext-101-32x48 makes their results not directly comparable to ours.
+
+Meanwhile, without the need of a much bigger dataset and architecture as used in Yalniz et al. (2019), Coaching achieves almost as good top-1 accuracy. To achieve this, we iterate the process of Coaching by turning the student into the teacher after convergence. After 17 iterations, our final student achieves $8 2 . 3 4 \%$ top-1 accuracy on ImageNet, outperforming Yalniz et al. (2019)’s $8 1 . 2 \%$ , even though we do not have the weakly labeled data from Instagram.
+
+Insights about Coaching on OpenImages. Figure 2 shows five images taken from OpenImages, along with their OpenImages tags and the top 5 classes predicted by a teacher trained on ImageNet. From the figure, we can see that there are non-trivial overlapping contents between the OpenImages tags and the ImageNet top classes, such as sunglasses in the first image. We also see that for the images whose contents match stronger with an ImageNet class, such as the first and the third image, the entropy of the teacher’s prediction is smaller. As a result, when the teacher samples a pseudo label from these distribution, contents similar to an ImageNet class will receive more consistent labels, while content alien to ImageNet will have higher entropy on their labels. We suspect this is why a teacher trained on ImageNet can teach a student via pseudo labels on OpenImages.
+
+![](images/3f36c32f4813ff69e0c535abbb87f4e59c10b06addd499c4b452791024d90158.jpg)  
+Figure 2: An illustration of why OpenImages help ImageNet classification. Top: OpenImages tags. Middle: A sample image from OpenImages. Bottom: Top 5 labels for the image predicted by a teacher ResNet-50 trained on ImageNet. Some OpenImages tags overlap significantly with some ImageNet classes, such as wheel and car wheel in the second image. The class predictions also have a higher entropy when the ImageNet classes overlap less with the OpenImages contents (images 2, 4, 5), than when the ImageNet classes overlap more (images 1, 3).
+
+# 3.3 ANALYSIS
+
+Ablation Study of Implementation Details. To understand the contribution of each implementation detail of Coaching, we study their contributions on top of a purely supervised model. We conduct this study on ImageNet- $10 \%$ and visualize the results in Figure 3. From the figure we see that RandomAugment and UDA both improve the final accuracy significantly, respectively by $3 . 1 3 \%$ and $7 . 1 9 \%$ top-1 accuracy. On top of UDA, Coaching delivers a smaller improvement of $4 . 3 2 \%$ top-1 accuracy. However, since UDA’s accuracy is already high, we believe that the improvement of $4 . 3 2 \%$ top-1 accuracy is significant. Finally, finetuning only slightly improves over Coaching. However, this extra boost is a unique advantage of Coaching: it is possible for the student in Coaching to finetune on labeled data because the student never directly learns from these labeled data.
+
+Coaching overfits less than Supervised Learning. In our Coaching framework, the student never directly learns from labeled data. This behavior is helps the student to avoid overfitting, especially when labeled data is scarce. In Figure 4, we visualize the training accuracy of Coaching and Supervised Learning on CIFAR-10 with 4,000 labels and on ImageNet with $10 \%$ labels. As shown, the training accuracy of both the teacher and the student of Coaching stay relatively low. Meanwhile, the training accuracy of the supervised model eventually reaches $1 0 0 \%$ and causes overfitting.
+
+![](images/ed6b51d0471ea4fd1866849074a61ccddd659333af9621335de5b95481ca6918.jpg)  
+Figure 3: Breakdown of the gains of different components in Coaching. The gain of Coaching over UDA, albeit smaller than the gain of UDA over RandomAugment, is significant as UDA is already very strong.
+
+![](images/daed95c2a2f864200b064f0f14afc1c16ae289b2406a7dcf06a279f17506cb5e.jpg)  
+Figure 4: Training accuracy of Coaching and of supervised learning on CIFAR-10-4,000 and ImageNet- $10 \%$ . Both the teacher and the student in Coaching have lower training accuracy, effectively avoiding overfitting.
+
+# 4 RELATED WORK
+
+Pseudo-Label. Pseudo-Label (Lee, 2013) is one of the simplest semi-supervised learning algorithms: First, a teacher model is trained on labeled data. Then, the converged teacher model generates pseudo labels for unlabeled data. These unlabeled data and their pseudo labels are combined with the labeled data to train another model, which is called the student model. An inherent weakness of Pseudo-Label is that once the teacher generates an incorrect pseudo label for an unlabeled datum, the student can only naively learn from this wrong label. This phenomenon is called the confirmation bias. Arazo et al. (2019) addressed the confirmation bias by generating soft labels from the teacher and by adding noise to these labels. However, this is a manual fix from an outside model designer. The main difference between Pseudo-Label and Coaching is that in Coaching, the teacher is trained along with the student throughout the course of training. This allows wrong knowledge learned by the teacher to be fixed in an end-to-end manner, leading to stronger performances.
+
+Semi-supervised Learning (SSL). Pseudo-Label belongs to a more general group of algorithms known as Semi-supervised Learning. Unlike Pseudo-Label, typical SSL methods combine both labeled and unlabeled data to train a single model. Hence, the objective function of SSL is typically the sum of a supervised loss and an unsupervised loss. The supervised loss is often the cross-entropy computed on the labeled data. Meanwhile, the unsupervised loss can be a self-supervised loss (Rasmus et al., 2015; Noroozi & Favaro, 2018; Gidaris et al., 2018), or consistency regularization (Laine & Aila, 2017; Tarvainen & Valpola, 2017; Miyato et al., 2018; Berthelot et al., 2019; Xie et al., 2019). Self-supervised losses typically encourage the model to develop a common sense about the images. Meanwhile, consistency regularization enforces that the model is invariant against certain transformations of the data. The main difference between Coaching and SSL methods is that the student in Coaching never learns directly from labeled data. This helps the student in Coaching to avoid overfitting to labeled data, especially when labeled data is limited.
+
+Meta Learning. In Meta Learning, there is typically an outer loop that optimizes the performance of a model trained in an inner loop (Finn et al., 2017; Metz et al., 2019). Meta Learning has been applied to perform self-training and SSL in the low data regime (Agarwal et al., 2019; Ren et al., 2018; Boney & Ilin, 2018; Hsu et al., 2019). A crucial difference between Coaching and Meta Learning is that in Coaching, the pseudo labels are chosen to improve the student, and hence there is no need for an outer loop. We suspect this is an advantage of our method, since gradients to be very powerful for models to navigate in the parameter space.
+
+# 5 CONCLUSION
+
+In this paper, we proposed the Coaching method for semi-supervised learning. Key to Coaching is the idea that the teacher learns from the student’s loss and improves itself to generate pseudo labels in a way that helps student’s learning the most. The learning process in Coaching consists of two main updates: updating the student based on the pseudo labeled data produced by the teacher and updating the teacher based on the student’s performance. Experiments on standard CIFAR-10 and SVHN show that Coaching is much better than supervised learning and consistenly better than other semi-supervised learning methods. Coaching scales well to large problems, and successfully uses out-of-domain data to improve ImageNet classification.
+
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+
+Mehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In IEEE Conference on Computer Vision and Pattern Recognition, 2018. 8
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+Avital Oliver, Augustus Odena, Colin Raffel, Ekin D. Cubuk, and Ian J. Goodfellow. Realistic evaluation of deep semi-supervised learning algorithms. In Advances in Neural Information Processing Systems, 2018. 4, 14
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+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision, 2015. 4
+
+Antti Tarvainen and Harri Valpola. Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. In Advances in Neural Information Processing Systems, 2017. 1, 4, 8
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+
+# A DERIVATION OF THE TEACHER’S UPDATE RULE
+
+In this section, we present the detailed derivation of the Teacher’s update rule in Equation 3 from Section 2.
+
+Mathematical Notations and Conventions. Since we will work with the chain rule, we use the standard Jacobian notations.3 Specifically, for a differentiable function $f : \mathbb { R } ^ { m }  \mathbb { R } ^ { n }$ , and for a vector $x \in \mathbb { R } ^ { m }$ , we use the notation ∂f∂x ∈ Rn×m to denote the Jacobian matrix of f, whose dimension is $n \times m$ . Additionally, when we mention the Jacobian of a function $f$ at multiple points such as x1 and x2, we will use the notations of ∂f∂x $\left. { \frac { \partial f } { \partial x } } \right| _ { x = x _ { 1 } }$ and $\left. \frac { \partial f } { \partial x } \right| _ { x = x _ { 2 } }$
+
+Furthermore, by mathematical conventions, a vector $v \in \mathbb { R } ^ { n }$ is treated as a column matrix – that is, a matrix of size $n \times 1$ . For this reason, the gradient vector of a multi-variable real-valued function is actually the transpose of of its Jacobian matrix.
+
+Finally, all multiplications in this section are standard matrix multiplications. If an operand is a vector, then as discussed in the previous paragraph, the operand is treated as a column matrix.
+
+Dimension Annotations. Understanding that these notations and conventions might cause confusions, in the derivation below, we annotate the dimensions of the computed quantities to ensure that there is no confusion caused to our readers. To this end, we respectively use $| S |$ and $| T |$ to denote the dimensions of the parameters $\theta _ { S } , \theta _ { T }$ . That is, $\theta _ { S } \in \mathbb { R } ^ { | S | \times 1 }$ and $\boldsymbol { \theta _ { T } } \in \mathbb { R } ^ { | T | \times 1 }$ .
+
+We now present the derivation. We need to compute:
+
+$$
+\underbrace { \frac { \partial R } { \partial \theta _ { T } } } _ { 1 \times | T | } = \frac { \partial } { \partial \theta _ { T } } \ell \left( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \mathbb { E } _ { \hat { y } _ { \mathrm { u n l } } \sim P ( \cdot | x _ { \mathrm { u n l } } ; \theta _ { T } ) } \left[ \theta _ { S } ^ { ( t + 1 ) } \right] \right)
+$$
+
+To simplify our notation, let us define
+
+$$
+\underbrace { \bar { \theta } _ { S } ^ { ( t + 1 ) } } _ { | S | \times 1 } \triangleq \mathbb { E } _ { \hat { y } _ { \mathrm { u n l } } \sim P ( \cdot | x _ { \mathrm { u n l } } ; \theta _ { T } ) } \left[ \theta _ { S } ^ { ( t + 1 ) } \right]
+$$
+
+Then, by the chain rule, we have
+
+$$
+\begin{array} { l } { { \displaystyle \frac { \partial R } { \partial \theta _ { T } } = \frac { \partial } { \partial \theta _ { T } } \ell \left( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \mathbb { E } _ { \hat { y } _ { \mathrm { a n l } } \sim P ( \cdot \vert x _ { \mathrm { a n l } } ; \theta _ { T } ) } \left[ \theta _ { S } ^ { ( t + 1 ) } \right] \right) } } \\  { \displaystyle { \mathrm {  ~ \Lambda ~ } } } \\ { { \displaystyle = \frac { \partial } { \partial \theta _ { T } } \ell \left( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \bar { \theta } _ { S } ^ { ( t + 1 ) } \right) } } \\  { \displaystyle ~ = \underbrace { \frac { \partial \ell \left( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \theta _ { S } \right) } { \partial \theta _ { S } } \Big \vert _ { \theta _ { S } = \bar { \theta } _ { S } ^ { ( t + 1 ) } } \cdot \frac { \partial \bar { \theta } _ { S } ^ { ( t + 1 ) } } { \partial \theta _ { T } } } } \end{array}
+$$
+
+The first factor in Equation 7 can be simply computed via back-propagation. We now focus on the second term. We have
+
+$$
+\begin{array} { r l r } {  { \underbrace { \frac { \partial \bar { \theta } _ { S } ^ { ( t + 1 ) } } { \partial \theta _ { T } } } } = \frac { \partial } { \partial \theta _ { T } } \mathbb { E } _ { \hat { y } _ { \mathrm { u n l } } \sim P ( \cdot | x _ { \mathrm { u n l } } ; \theta _ { T } ) } [ \theta _ { S } ^ { ( t + 1 ) } ] }  \\ & { } & \\ & { } & { = \frac { \partial } { \partial \theta _ { T } } \mathbb { E } _ { \hat { y } _ { \mathrm { u n l } } \sim P ( \cdot | x _ { \mathrm { u n l } } ; \theta _ { T } ) } [ \theta _ { S } ^ { ( t ) } - \eta \cdot (  \frac { \partial \ell ( x _ { \mathrm { u n l } } , \hat { y } _ { \mathrm { u n l } } ; \theta _ { S } ) } { \partial \theta _ { S } } | _ { \theta _ { S } = \theta _ { S } ^ { ( t ) } } ) ^ { \top } ] } \end{array}
+$$
+
+3Standard: https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
+
+Note that in Equation 8 above, the Jacobian of $\ell ( x _ { \mathrm { u n l } } , \hat { y } _ { \mathrm { u n l } } ; \theta _ { S } )$ , which has dimension $1 \times | S |$ , needs to be transposed to match the dimension of $\theta _ { S } ^ { ( t ) }$ , which, as we discussed above, conventionally has
+
+Now, since $\theta _ { S } ^ { ( t ) }$ in Equation 8 does not depend on $\theta _ { T }$ , we can leave it out of subsequent derivations. Also, to simplify notations, let us define the gradient
+
+$$
+\underbrace { g _ { S } ^ { ( t ) } ( \hat { y } _ { \mathrm { u n l } } ) } _ { | S | \times | 1 | } \triangleq \left( \left. \frac { \partial \ell \left( x _ { \mathrm { u n l } } , \hat { y } _ { \mathrm { u n l } } ; \theta _ { S } \right) } { \partial \theta _ { S } } \right| _ { \theta _ { S } = \theta _ { S } ^ { ( t ) } } \right) ^ { \top }
+$$
+
+Then, Equation 8 becomes
+
+$$
+\underbrace { \frac { \partial \bar { \theta } _ { S } ^ { ( t + 1 ) } } { \partial \theta _ { T } } } _ { | S | \times | T | } = - \eta \cdot \frac { \partial } { \partial \theta _ { T } } \mathbb { E } _ { \hat { y } _ { \mathrm { u n l } } \sim P ( \cdot | x _ { \mathrm { u n l } } ; \theta _ { T } ) } \left[ \underbrace { g _ { S } ^ { ( t ) } ( \hat { y } _ { \mathrm { u n l } } ) } _ { | S | \times 1 } \right]
+$$
+
+Since $g _ { S } ^ { ( t ) } ( \hat { y } _ { \mathrm { u n l } } )$ has no dependency on on $\theta _ { T }$ , except for via $\hat { y } _ { \mathrm { u n l } }$ , we can apply the REINFORCE equation (Williams, 1992) to achieve
+
+$$
+\begin{array} { r l } & { \underbrace { \frac { \partial \bar { \theta } _ { S } ^ { ( t + 1 ) } } { \partial \theta _ { T } } } _ { | S | \times | T | } = - \eta \cdot \frac { \partial } { \partial \theta _ { T } } \mathbb { E } _ { \hat { y } _ { \mathrm { a n } } \times P ( \cdot | x _ { \mathrm { m i } } \times \theta _ { T } ) } \left[ g _ { S } ^ { ( t ) } ( \hat { y } _ { \mathrm { m i } } ) \right] } \\ & { = - \eta \cdot \mathbb { E } _ { \hat { y } _ { \mathrm { a n } } \times P ( \cdot | x _ { \mathrm { m i } } \times \theta _ { T } ) } \left[ \underbrace { g _ { S } ^ { ( t ) } ( \hat { y } _ { \mathrm { m i } } ) } _ { | S | \times 1 } \cdot \underbrace { \frac { \partial \log P \left( \hat { y } _ { \mathrm { m i } } | x _ { \mathrm { m i } } ; \theta _ { T } \right) } { \partial \theta _ { T } } } _ { 1 \times | T | } \right] } \\ & { = \eta \cdot \mathbb { E } _ { \hat { y } _ { \mathrm { a n } } \sim P ( \cdot | x _ { \mathrm { m i } } ; \theta _ { T } ) } \left[ \underbrace { g _ { S } ^ { ( t ) } ( \hat { y } _ { \mathrm { m i } } ) } _ { | S | \times 1 } \cdot \underbrace { \frac { \partial \ell \left( x _ { \mathrm { m i } } , \hat { y } _ { \mathrm { m i } } ; \theta _ { T } \right) } { \partial \theta _ { T } } } _ { 1 \times | T | } \right] } \end{array}
+$$
+
+Here, the last equality in Equation 11 is is due to the definition of the cross entropy loss, which is the negative of the log-prob term in the previous line.
+
+Now, we can substitute Equation 11 into Equation 7 to obtain
+
+$$
+\begin{array} { l } { \displaystyle \frac { \partial R } { \partial \theta _ { T } } = \underbrace { \frac { \partial \ell \left( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \theta _ { S } \right) } { \partial \theta _ { S } } \bigg | _ { \theta _ { S } = \bar { \theta } _ { S } ^ { ( t + 1 ) } } } _ { = \eta \cdot \underbrace { 1 \times | S | } _ { = \eta \cdot \underbrace { 1 \times | S | } _ { = \theta _ { S } } } \bigg | _ { \theta _ { S } = \bar { \theta } _ { S } ^ { ( t + 1 ) } } } \cdot \underbrace { \frac { \partial \bar { \theta } _ { S } ^ { ( t + 1 ) } } { \partial \theta _ { T } } } _ { | S | \times \left| T \right| } } \\ { = \eta \cdot \underbrace { \frac { \partial \ell \left( x _ { \mathrm { l a b } } , y _ { \mathrm { l a b } } ; \theta _ { S } \right) } { \partial \theta _ { S } } \bigg | _ { \theta _ { S } = \bar { \theta } _ { S } ^ { ( t + 1 ) } } } _ { \mathrm { 1 \times | S | } } \cdot \mathbb { E } _ { \hat { y } _ { \mathrm { l a n } } \sim P \cdot ( \cdot | x _ { \mathrm { m i } } ; \theta _ { T } ) } \left[ \underbrace { g _ { S } ^ { ( t ) } ( \hat { y } _ { \mathrm { u n } } ) } _ { | S | \times 1 } \cdot \underbrace { \frac { \partial \ell \left( x _ { \mathrm { u n l } } , \hat { y } _ { \mathrm { u n l } } ; \theta _ { T } \right) } { \partial \theta _ { T } } } _ { \mathrm { 1 \times | T | } } \right] } \end{array}
+$$
+
+Finally, if we use Monte Carlo approximation for every term in Equation 12 using the sampled $\hat { y } _ { \mathrm { u n l } }$ then we have Equation 3 from Section 2. Note that in Section 2, we use the gradient notation, which results in the transposes.
+
+# B TRAINING SPEED
+
+Coaching performs up to 5 forward passes and 3 backward passes. Compared to vanilla backpropagation training, this is more 3 than times more expensive in FLOPs. However, many computations in Coaching are parallelizable. For example, the forward pass of the student and the forward pass for the teacher on unlabeled data (the top half of Figure 1), can be run in parallel since they do not depend on each other. Therefore, on computing hardware with sufficient memory, we find Coaching to be between 2 and 2.5 times slower than standard back-propagation training.
+
+# C EXPERIMENTAL DETAILS
+
+# C.1 DATASET SPLITS
+
+We describe how we select the reduced datasets for the experiments on low data image classification in Section 3.1.
+
+For CIFAR-10, we download the five training data batch files from www.cs.toronto.edu/ \~kriz/cifar.html. Then, we load all the images into a list of 50,000 images, keeping the order as downloaded. The fisrt 5,000 images ares reserved for validation. The next 4,000 images are used as labeled data. For SVHN, we download the data from the mat files on ufldl.stanford.edu/ housenumbers/, and follow the same procedure as with CIFAR-10. We note that this selection process leads to a slight imbalance in the class distribution for both CIFAR-10 and SVHN, but the settings are the same for all of our experiments.
+
+For ImageNet, we follow the procedure in github.com/tensorflow/models/blob/ master/research/inception/inception/data/download_and_preprocess imagenet.sh. This results in 1,024 training TFRecord shards of approximately the same size. The order of the images in these shards are deterministic. For ImageNet- $10 \%$ , we use the first 102 shards; for ImageNet- $20 \%$ , we use the first 204 shards; and so on. The last 20 shards, corresponding to roughly 25,000 images, are reserved for hyper-parameters tuning.
+
+# C.2 RANDOMAUGMENT: A DATA AUGMENTATION POLICY
+
+We develop a data augmentation policy that achieves similarly high performance with AutoAugment (Cubuk et al., 2019) in a few cases that we consider, but which does not require learning a controller to generate policies. We names our policy RandomAugment. Our goal when developing RandomAugment is not to outperform AutoAugment, which is why we do not conduct extensive experiments with RandomAugment. Instead, we simply want to avoid indirectly using labeled data for our experiments, especially for the low data regime experiments in Section 3.1.
+
+Each policy of RandomAugment consists of two operations that applied sequentially on an image. Each operation applies a uniformly sampled transformation with probability 0.5, and with a level uniformly chosen between 1 and 10. For a more comprehensive discussion of the probability and the level of a transformation, we refer readers to the AutoAugment paper (Cubuk et al., 2019).
+
+Table 3: Transformations that RandomAugment uniformly samples for our datasets. We refer our readers to Cubuk et al. (2019) for the detailed descriptions of these transformations.   
+
+<table><tr><td>CIFAR-10 and ImageNet</td><td>SVHN</td></tr><tr><td>AutoContrast</td><td>AutoContrast</td></tr><tr><td>Brightness</td><td>Brightness</td></tr><tr><td>Color</td><td>Color</td></tr><tr><td>Contrast</td><td>Contrast</td></tr><tr><td>Equalize</td><td>Equalize</td></tr><tr><td>Invert</td><td>Invert</td></tr><tr><td>Sharpness</td><td>Sharpness</td></tr><tr><td>Posterize</td><td>Posterize</td></tr><tr><td>Sample Pairing</td><td>Solarize</td></tr><tr><td>Solarize</td><td>ShearX</td></tr><tr><td>Rotate</td><td>ShearY</td></tr><tr><td>ShearX</td><td>TranslateY</td></tr><tr><td>ShearY</td><td></td></tr><tr><td>TranslateX</td><td></td></tr><tr><td></td><td></td></tr><tr><td>TranslateY</td><td></td></tr></table>
+
+We manually design the set of transformations for each of our datasets: CIFAR-10, ImageNet, and SVHN. The set of transformations for CIFAR-10 and for ImageNet are the same, and are slightly different from the set of transformation for SVHN. This is because the numbers in the SVHN have a different requirement for invariant. For instance, numbers should not be invariant against rotations like 6 and 9, and should not be invariant against horizontal translation like 3 and 8. Table 3 presents the transformation for our dataset. In addition to these operations, we only allow RandomAugment to select the three transformations AutoContrast, Brightness, and Invert in the first augmenting transformation. This is to avoid a few degenerating cases. For instance, when Brightness is applied twice on an image, both times with small levels, the image will become almost black.
+
+In our experiments, RandomAugment’s performance is not far behind compared to AutoAugment. For example, on full ImageNet with ResNet-50, RandomAugment achives $\mathrm { \bar { 7 } 7 . 9 8 \% }$ top-1 accuracy, which is close to the top-1 accuracy of $7 7 . 6 \%$ reported by Cubuk et al. (2019).
+
+# C.3 HYPER-PARAMETERS
+
+To tune hyper-parameters, we follow Oliver et al. (2018) and allow each method to have 128 trials of hyper-parameters. When we tune, we let each model train for up to 50,000 steps. The optimal hyper-parameters are then used to run experiments that last for much more steps, as we report below. In our experiments with Coaching, training for more steps typically leads to stronger results. We stop at 1 million steps for CIFAR-10 and SVHN, and at 0.5 million steps for ImageNet simply because otherwise, these experiments will take too long. Meanwhile, in our experiments with purely supervised learning, Pseudo-Labels, and UDA, training for more steps overfits the models, and we have to employ early stopping.
+
+We report the hyper-parameters for our baselines and for Coaching in Section 3.1. For the highresource experiments in Section 3.2, we use the same hyper-parameters, because tuning them is too expensive. Our hyper-parameters can be found in Table 4, 5, 6.
+
+We note that our settings for UDA is different from originally reported by Xie et al. (2019). In their work, Xie et al. (2019) use a much larger batch size for their UDA objective. In our implementation of UDA, we keep these batch sizes the same. This leads to a much easier implementation of data parallelism in our framework, TensorFlow (Abadi et al., 2016) running on TPU big pods. To compensate for the difference, we train all UDA baselines for much longer than Xie et al. (2019). During the training process, we also mask out the supervised examples with high confidence. Effectively, our UDA model receives roughly the same amount of training with labeled examples and unlabeled examples as the models in Xie et al. (2019). We have also verified that on ImageNet- $10 \%$ with the augmentation policy from AutoAugment (Cubuk et al., 2019), our UDA implementation achives $6 8 . 7 7 \%$ top-1 accuracy, which is similar to $6 8 . 6 6 \%$ that Xie et al. (2019) reported.
+
+Table 4: Hyper-parameters for supervised learning and Pseudo-Labels.   
+
+<table><tr><td>Hyper-parameter</td><td>CIFAR-10</td><td>SVHN</td><td>ImageNet</td></tr><tr><td>Weight decay</td><td>0.0005</td><td>0.001</td><td>0.0002</td></tr><tr><td>Label smoothing</td><td>0</td><td>0</td><td>0.1</td></tr><tr><td>Batch normalization decay</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Learning rate</td><td>0.4</td><td>0.05</td><td>1.28</td></tr><tr><td>Number of training steps</td><td>50,000</td><td>50,000</td><td>40,000</td></tr><tr><td>Number of warm up steps</td><td>2500</td><td>0</td><td>2000</td></tr><tr><td>Batch size</td><td>1024</td><td>128</td><td>2048</td></tr><tr><td>Dropout rate</td><td>0.4</td><td>0.5</td><td>0.2</td></tr><tr><td>Pseudo label threshold</td><td>0.95</td><td>0.975</td><td>0.7</td></tr></table>
+
+Table 5: Hyper-parameters for UDA. Unlike originally done by Xie et al. (2019), we do not use a larger batch size for the UDA objective. Instead, we use the same batch size for both the labeled objective and the unlabeled objective. This is to avoid instances where some particularly small batch sizes for the labeled objective cannot be split on our computational hardware.   
+
+<table><tr><td>Hyper-parameter</td><td>CIFAR-10</td><td>SVHN</td><td>ImageNet</td></tr><tr><td>Weight decay</td><td>0.0005</td><td>0.0005</td><td>0.0002</td></tr><tr><td>Label smoothing</td><td>0</td><td>0</td><td>0.1</td></tr><tr><td>Batch normalization decay</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Learning rate</td><td>0.3</td><td>0.4</td><td>1.28</td></tr><tr><td>Number of training steps</td><td>1,000,000</td><td>1,000,000</td><td>500.000</td></tr><tr><td>Number of warm up steps</td><td>5,000</td><td>5.000</td><td>5,000</td></tr><tr><td>Batch size</td><td>128</td><td>128</td><td>2048</td></tr><tr><td>Dropout rate</td><td>0.5</td><td>0.6</td><td>0.25</td></tr><tr><td>UDA factor</td><td>2.5</td><td>1</td><td>20</td></tr><tr><td>UDA temperature</td><td>0.7</td><td>0.8</td><td>0.7</td></tr></table>
+
+Table 6: Hyper-parameters for Coaching.   
+
+<table><tr><td></td><td>Hyper-parameter</td><td>CIFAR-10</td><td>SVHN</td><td>ImageNet</td></tr><tr><td rowspan="5">Common</td><td>Weight decay</td><td>0.0005</td><td>0.0005</td><td>0.0002</td></tr><tr><td>Label smoothing</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Batch normalization decay</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Number of training steps</td><td>1,000,000</td><td>1,000,000</td><td>500,000</td></tr><tr><td>Number of warm up steps</td><td>2.000</td><td>2,000</td><td>1,000</td></tr><tr><td rowspan="3">Student</td><td>Learning rate</td><td>0.3</td><td>0.15</td><td>0.8</td></tr><tr><td>Batch size</td><td>128</td><td>128</td><td>2048</td></tr><tr><td>Dropout rate</td><td>0.35</td><td>0.45</td><td>0.1</td></tr><tr><td rowspan="5">Teacher</td><td>Learning rate</td><td>0.125</td><td>0.05</td><td>0.5</td></tr><tr><td>Batch size</td><td>128</td><td>128</td><td>2048</td></tr><tr><td>Dropout rate</td><td>0.5</td><td>0.65</td><td>0.1</td></tr><tr><td>UDA factor</td><td>1.0</td><td>2.5</td><td>16.0</td></tr><tr><td>UDA temperature</td><td>0.8</td><td>1.25</td><td>0.75</td></tr></table>

parse/train/rJe04p4YDB/rJe04p4YDB_content_list.json

@@ -0,0 +1,1805 @@
+[
+    {
+        "type": "text",
+        "text": "SEMI-SUPERVISED LEARNING BY COACHING",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            714,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            145,
+            400,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
+            544,
+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent semi-supervised learning (SSL) methods often have a teacher to train a student in order to propagate labels from labeled data to unlabeled data. We argue that a weakness of these methods is that the teacher does not learn from the student’s mistakes during the course of student’s learning. To address this weakness, we introduce Coaching, a framework where a teacher generates pseudo labels for unlabeled data, from which a student will learn and the student’s performance on labeled data will be used as reward to train the teacher using policy gradient. ",
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+        "type": "text",
+        "text": "Our experiments show that Coaching significantly improves over state-of-the-art SSL baselines. For instance, on CIFAR-10, with only 4,000 labeled examples, a WideResNet-28-2 trained by Coaching achieves $9 6 . 1 1 \\%$ accuracy, which is better than $9 4 . 9 \\%$ achieved by the same architecture trained with 45,000 labeled. On ImageNet with $10 \\%$ labeled examples, Coaching trains a ResNet-50 to $7 2 . 9 4 \\%$ top-1 accuracy, comfortably outperforming the existing state-of-the-art by more than $4 \\%$ . Coaching also scales successfully to the high data regime with full ImageNet. Specifically, with additional 9 million unlabeled images from OpenImages, Coaching trains a ResNet-50 to $8 2 . 3 4 \\%$ top-1 accuracy, setting a new state-of-the-art for the architecture on ImageNet without using extra labeled data.1 ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Professional players in competitive sports such as chess, tennis, or swimming often have coaches to help improving their performance. Although coaches typically do not play as well as the players, they observe the players and provide instructions to improve the players’ performance. Modern semi-supervised learning (SSL) algorithms do not follow this strategy. They instead have a teacher model that generates pseudo labels for unlabeled data, from which a student model learns by imitation (e.g., Lee (2013); Tarvainen & Valpola (2017); Laine & Aila (2017)). A weakness of these methods is that the teacher does not adjust itself based on the student’s performance and cannot adapt to make the student better over time, unlike professional sport coaches develop their players. ",
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+        "text": "Here, we propose a new semi-supervised learning method, called Coaching as shown in Figure 1, where the teacher learns throughout the course of student’s training. In our method, a teacher generates pseudo labels for unlabeled data, from which the student will learn. The student’s performance on labeled data will be used as reward to train the teacher with policy gradient. ",
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+    {
+        "type": "image",
+        "img_path": "images/3a0ffb52634b9d076504729ad6bafe34cf33d3a55de49387d0cf2e344d54bd51.jpg",
+        "image_caption": [
+            "Figure 1: Each step of gradient descent in Coaching consists of two steps. Updating the Student (top): The teacher network $T$ samples the labels $\\hat { y }$ of unlabeled data $x _ { \\mathrm { u n l } }$ for the student $S$ to learn from. Updating the Teacher (bottom): The teacher updates itself using policy gradient to improve the student’s performance on labeled data $x _ { \\mathrm { l a b } }$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Experiments show that our method achieves significant improvements over state-of-the-art semisupervised learning baselines and can be up to $1 0 \\times$ more data efficient than supervised learning. For instance, with CIFAR-10, only using 4,000 labeled examples, a WideResNet-28-2 can be coached to $9 6 . 1 1 \\%$ accuracy, outperforming the same model trained with 45,000 labeled examples which achieves $9 4 . 9 \\%$ . Meanwhile, on ImageNet, using ResNet-50 with only $1 0 \\%$ labeled examples, our method achieves $7 2 . 9 4 \\%$ top-1 accuracy, outperforming all existing semi-supervised learning methods with the same amount of labeled data, and approaching the top-1 accuracy of $7 6 . 3 \\%$ of the same ResNet-50 trained with all labels. Coaching also scales to the high data regime. In particular, with all 1.28 million labeled examples from ImageNet, plus 9 million unlabeled and potentially out-of-distribution data from OpenImages (Kuznetsova et al., 2018), a ResNet-50 can be coached to the accuracy of $8 2 . 3 4 \\%$ , which is a new state-of-the-art for the architecture without using extra labeled data. ",
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+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Notations. Let $T$ , $S$ respectively be the teacher network and the student network in Coaching, and $\\theta _ { T }$ , $\\theta _ { S }$ be their corresponding parameters. Since we work with both labeled data and unlabeled data, we use $( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } )$ to refer to a pair of an input and its corresponding label, and use $x _ { \\mathrm { u n l } }$ to refer to an unlabeled example. In addition, we use $\\ell ( x , y ; \\theta )$ to denote the cross entropy loss computed on input $x$ by with parameter $\\theta$ on label $y$ . ",
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+    {
+        "type": "text",
+        "text": "As shown in Figure 1, each training step in Coaching consists of two phases: ",
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+        "type": "text",
+        "text": "Phase 1: The student learns from data pseudo labeled by the teacher. In this phase, the teacher $T$ first performs a forward pass on $x _ { \\mathrm { u n l } }$ to compute the class distribution $P ( \\cdot | x _ { \\mathrm { u n l } } ; \\theta _ { T } )$ . From this distribution, the teacher samples a pseudo label $\\hat { y } _ { \\mathrm { u n l } } \\sim P ( \\cdot | x _ { \\mathrm { u n l } } ; \\theta _ { T } )$ . The pair $x _ { \\mathrm { u n l } } , \\hat { y } _ { \\mathrm { u n l } }$ is then shown to the student $S$ to make an update on its parameters $\\theta _ { S }$ . The update is based on the gradient computed by back-propagating from the cross entropy loss. For instance, if $\\theta _ { S }$ is updated using SGD, then: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b2504ad1262838f3f2f309f4e52a11943d8bf6fe25d78b618564f386ba04c99.jpg",
+        "text": "$$\n\\theta _ { S } ^ { ( t + 1 ) } : = \\theta _ { S } ^ { t } - \\eta \\cdot \\underbrace { \\frac { \\partial \\ell ( x _ { \\mathrm { u n l } } , \\hat { y } _ { \\mathrm { u n l } } ; \\theta _ { S } ) } { \\partial \\theta _ { S } } } _ { \\xrightarrow [ ] { \\Delta } } \\bigg | _ { \\theta _ { S } = \\theta _ { S } ^ { ( t ) } } = \\theta _ { S } ^ { ( t ) } - \\eta \\cdot g _ { S } ^ { ( t ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\eta$ is the learning rate. ",
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+    },
+    {
+        "type": "text",
+        "text": "Phase 2: The teacher learns from the student’s loss. After the student updates its parameters   \nas in Equation 1, its parameters entropy loss. The goal of the testudent is updated as in Equation $\\theta _ { S } ^ { ( t + 1 ) }$ ) led example xlab, ylab is evaluated on a laben Coaching is to give tn the cross entropy loss using the crosssuch that if theill be minimized. $\\hat { y } _ { \\mathrm { u n l } }$ $^ { l }$ $\\ell ( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\theta _ { S } ^ { ( t + 1 ) } )$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Clearly, $\\ell ( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\theta _ { S } ^ { ( t + 1 ) } )$ depends on $\\theta _ { S } ^ { ( t + 1 ) }$ , which in turn depends on the pseudo label $\\hat { y } _ { \\mathrm { u n l } }$ that the teacher samples. From the perspective of reinforcement learning, $\\hat { y } _ { \\mathrm { u n l } }$ can be treated as an onpolicy action of the teacher, which leads to the reward of $- \\ell ( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\theta _ { S } ^ { ( t + 1 ) } )$ . In this perspective, we propose to train $\\theta _ { T }$ to minimize the value of $\\ell ( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\bar { \\theta } _ { S } ^ { ( t + 1 ) } )$ , where $\\bar { \\theta } _ { S } ^ { ( t + 1 ) }$ is the expected destination that the teacher will guide the student to. This expectation is taken over all possible pseudo labels $\\hat { y } _ { \\mathrm { u n l } }$ . Formally, ",
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+    {
+        "type": "equation",
+        "img_path": "images/db239e95251018e168f171bd8479515d00f03e0f473cbab14735f5de7d715e49.jpg",
+        "text": "$$\n\\theta _ { T } ^ { * } = \\operatorname * { a r g m i n } _ { \\theta _ { T } } R ( \\theta _ { T } ) \\mathrm { ~ w h e r e ~ } R ( \\theta _ { T } ) = \\ell \\left( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\mathbb { E } _ { \\hat { y } _ { \\mathrm { t u n l } } \\sim P ( \\cdot | x _ { \\mathrm { l a b } } ; \\theta _ { T } ) } \\left[ \\theta _ { S } ^ { ( t + 1 ) } \\right] \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To find $\\theta _ { T } ^ { * }$ , we differentiate $R ( \\theta _ { T } )$ in Equation 2 with respect to $\\theta _ { T }$ . Here, we present the resulting gradient $g _ { T } ^ { ( t ) }$ , which has the form ",
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+    {
+        "type": "equation",
+        "img_path": "images/6ad3d8058f3553c02b7f5398b0370f29836b31d3391e7050ee8c5b3150a53682.jpg",
+        "text": "$$\ng _ { T } ^ { ( t ) } \\approx \\eta \\cdot \\left[ \\left( g _ { S } ^ { ( t ) } \\right) ^ { \\top } \\cdot \\left( \\frac { \\partial \\ell ( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\theta _ { S } ) } { \\partial \\theta _ { S } } \\bigg | _ { \\theta _ { S } = \\theta _ { S } ^ { ( t + 1 ) } } \\right) ^ { \\top } \\right] \\cdot \\left( \\frac { \\partial \\ell ( x _ { \\mathrm { u n l } } , \\hat { y } _ { \\mathrm { u n l } } ; \\theta _ { T } ) } { \\partial \\theta _ { T } } \\bigg | _ { \\theta _ { T } = \\theta _ { T } ^ { ( t ) } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The full derivation can be found in Appendix A, but intuitively, the differentiation depends on two tools. The first tool is the is the chain rule, which we leverage to differentiate $R ( \\theta _ { T } )$ with respect to $\\theta _ { T }$ . The second tool is the REINFORCE equation (Williams, 1992), which we leverage to establish the relationship between $\\mathbb { E } _ { \\hat { y } _ { \\mathrm { u n l } } } \\left[ \\theta _ { S } ^ { ( t + 1 ) } \\right]$ and $\\theta _ { T }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Coaching combines the two steps above in an SGD step. We summarize the method in Algorithm 1. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 The Coaching method. ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Input :Labeled data $x _ { \\mathrm { l a b } }$ , $y _ { \\mathrm { l a b } }$ and unlabeled data $x _ { \\mathrm { u n l } }$ .   \n1 Initialize $\\theta _ { T } ^ { ( 0 ) }$ and $\\theta _ { S } ^ { ( 0 ) }$   \n2 for $t = 0$ to $N - 1$ do   \n3 Sample an unlabeled example $x _ { \\mathrm { u n l } }$ and a labeled example $x _ { \\mathrm { l a b } }$ , $y _ { \\mathrm { l a b } }$   \n4 Sample $\\hat { y } _ { \\mathrm { u n l } } \\sim P ( \\cdot | x _ { \\mathrm { u n l } } ; \\theta _ { T } )$   \n5 $\\theta _ { S } ^ { ( t + 1 ) } : = \\theta _ { S } ^ { ( t ) } - \\eta \\cdot g _ { S } ^ { ( t ) }$ . Compute $g _ { S } ^ { ( t ) }$ with pseudo labels as in Equation 1 and update $\\theta _ { S }$   \n6 Sθ(t+1)T : $\\theta _ { T } ^ { ( t + 1 ) } : = \\theta _ { T } ^ { ( t ) } - \\eta \\cdot h ^ { ( t ) } \\cdot g _ { T } ^ { ( t ) }$ . Compute the gradient $g _ { T } ^ { ( t ) }$ as in Equation 3 and update $\\theta _ { T }$   \n7 end   \n8 return $\\theta _ { S } ^ { ( N ) }$ . Only the student model is used for predictions and evaluations ",
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+    {
+        "type": "text",
+        "text": "Generalize to an arbitrary batch size. Above, we have only discussed Coaching for a single unlabeled data $x _ { \\mathrm { u n l } }$ and a single labeled data $x _ { \\mathrm { l a b } }$ , $y _ { \\mathrm { l a b } }$ . Now, we describe how to scale Coaching to an arbitrary batch size. Scaling $x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } }$ to a minibatch of labeled example, $X _ { \\mathrm { l a b } }$ , $Y _ { \\mathrm { l a b } }$ is straightforward, as we can simply replace all computations of the cross entropy $\\ell ( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\theta _ { S } ^ { ( t + 1 ) } )$ with the average cross entropy on the minibatch \\`(Xlab, Ylab; θ(t+1)S ). T(1) (2) o scale a single unlabeled example xunl to a minibatch of unlabeled examples $X _ { \\mathrm { u n l } } = \\{ x _ { \\mathrm { u n l } } ^ { ( 1 ) } , x _ { \\mathrm { u n l } } ^ { ( 2 ) } , . . . , x _ { \\mathrm { u n l } } ^ { ( B ) } \\}$ , we treat each batch of pseudo labels $\\hat { Y } _ { \\mathrm { u n l } }$ as a compound action sampled from the joint distribution ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/648aa72b9563b3d171274f52cde37c53c00f6d6431253048c631283e45832a9c.jpg",
+        "text": "$$\nP \\left( \\hat { Y } _ { \\mathrm { u n l } } \\middle | X _ { \\mathrm { u n l } } ; \\theta _ { T } \\right) = P \\left( \\hat { y } _ { \\mathrm { u n l } } ^ { ( 1 ) } , \\hat { y } _ { \\mathrm { u n l } } ^ { ( 2 ) } , \\ldots , \\hat { y } _ { \\mathrm { u n l } } ^ { ( B ) } \\middle | X _ { \\mathrm { u n l } } ; \\theta _ { T } \\right) = \\prod _ { i = 1 } ^ { B } P \\left( \\hat { y } _ { \\mathrm { u n l } } ^ { ( i ) } \\middle | x _ { \\mathrm { u n l } } ^ { ( i ) } ; \\theta _ { T } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since every pseudo label $\\hat { y } _ { \\mathrm { u n l } } ^ { ( i ) }$ is sampled independently, applying REINFORCE as in Equation 3 simply factors the per-instance cross entropy into the batch cross entropy $\\begin{array} { r l } { ~ } & { { } \\sum _ { i = 1 } ^ { B } \\ell ( x _ { \\mathrm { u n l } } ^ { ( i ) } , \\hat { y } _ { \\mathrm { u n l } } ^ { ( i ) } ; \\theta _ { T } ) } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Compared to other methods that use both labeled data and unlabeled data, Coaching has three main advantages: ",
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+    {
+        "type": "text",
+        "text": "1. The teacher does not only demonstrate its knowledge to the student but also adjusts its teaching strategy in an adaptive manner with the student, throughout the course of the student’s learning.   \n2. The teacher in Coaching can benefit from advanced SSL techniques such as consistency regularization.   \n3. The student in Coaching never learns directly from labeled data. This does not only prevent overfitting when limited labeled data is available, but also allows us to finetune the trained student in Coaching directly on labeled data to further boost the student’s performance. ",
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+    {
+        "type": "text",
+        "text": "We perform experiments to verify the strength of Coaching. In Section 3.1, we consider the low data regime with typical benchmarks for SSL methods. After that, in Section 3.2, we consider the high data regime which contains potentially out-of-distribution data. ",
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+    {
+        "type": "text",
+        "text": "Model Architectures. In our experiments, our teacher model and our student model always have the same architecture but with different weights. For CIFAR-10 and SVHN, we use the WideResNet28-2 (Zagoruyko & Komodakis, 2016), which has 1.45 million parameters. For ImageNet, we use a ResNet-50 (He et al., 2016), which has 25.5 million parameters. For experiments that train only one model, we apply exponential moving average with a decay rate of 0.99 on the weights of the model. For experiments that have a teacher model and a student model, we apply this exponential moving average on the weights of the student model only. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Additional Implementation Details. To improve the stability and accuracy of the method, we apply a few minor enhancements to the teacher: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "1. Use cosine distance instead of dot product. As the dot product $h ^ { ( t ) }$ in Equation 3 has a large value range, in order to stabilize training, we compute $h ^ { ( t ) }$ using the gradients’ cosine distance.   \n2. Use a baseline for $h ^ { ( t ) }$ . To further reduce the variance of $h ^ { ( t ) }$ , we maintain a moving average $b$ of $h ^ { ( t ) }$ and subtract $b$ from $h ^ { ( t ) }$ every time we compute $g _ { T } ^ { ( t ) }$ as in Equation 3.   \n3. Additional supervised loss for the teacher. We find that adding the supervised loss $\\ell ( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\theta _ { T } )$ to the teacher’s objective results in a faster learning and better student.   \n4. Consistently regularize the teacher. In the low data regime, consistency regularization improves the teacher and the student. More details are in Section 3.1.   \n5. Pre-training the teacher. When the number of classes is large, it is beneficial to initialize the teacher with a trained model so that the pseudo labels are better than random at the beginning of the student’s learning. If we pre-train the teacher, Point 3 has minimal effect.   \n6. Finetuning the student. Since the student in Coaching only learns from unlabeled data and pseudo labels generated by the teacher, finetuning a converged student on labeled data often improves the student’s performance. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The details mentioned above are mutually orthogonal. Since (1) and (2) are crucial to stabilize the Coaching process, they are always used in our experiments. In addition, we apply (3) and (4) to the low data regime, and apply (5) for the high data regime for computational efficiency and strong performance. We will explain these decisions in the corresponding sections. ",
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+    {
+        "type": "text",
+        "text": "3.1 RESULTS ON LOW DATA REGIME ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Datasets. We consider three datasets with reduced numbers of labeled instances: CIFAR10 (Krizhevsky, 2009) with 4,000 labeled examples, SVHN (Netzer et al., 2011) with 1,000 labeled examples, and ImageNet (Russakovsky et al., 2015) with 128,000 labeled examples, which is approximately $1 0 \\%$ of the whole ImageNet. All images in these datasets are used as unlabeled examples, which means that even the labeled images can be used as unlabeled examples. We use the image size of $3 2 \\times 3 2$ for CIFAR-10 and SVHN, and the image size of $2 2 4 \\times 2 2 4$ for ImageNet. These datasets, label reductions, and image sizes are standard for low data image classification. ",
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+        "type": "text",
+        "text": "Baselines. We compare Coaching against 3 baseline training algorithms Purely Supervised, PseudoLabel (Lee, 2013), and Unsupervised Data Augmentation (UDA; Xie et al. (2019)). We discuss these baselines more in Section 4. We choose these baselines for three reasons. First, the purely supervised baseline serves to verify our implementation and to demonstrate the overfitting of our models when labeled data is scarce. Second, comparing Coaching with Pseudo-Label confirms the benefits of continuing to train the teacher throughout the course of the student’s learning. Finally, we compare against UDA because is the state-of-the-art on the datasets that we consider. ",
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+    {
+        "type": "text",
+        "text": "To ensure a fair comparison, we re-implement these baselines in our environment. We follow Oliver et al. (2018)’s train/eval/test splitting, and we use the same amount of resources to tune hyperparameters for our baselines as well as for Coaching. More details are in Appendix C. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Additional baselines. In addition to the three main baselines discussed above, we also include four other baselines: Temporal Ensemble (Laine & Aila, 2017), Mean Teacher (Tarvainen & Valpola, 2017), VAT (Miyato et al., 2018), LGA (Jackson & Schulman, 2019), ICT (Verma et al., 2019), and MixMatch (Berthelot et al., 2019). We use results reported by Oliver et al. (2018). Since these methods do not share the same controlled environment, the comparison to them is not direct, and should be contextualized as suggested by Oliver et al. (2018). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c64986282079eace632772d192c63c8f2498a70ec73340c3c8d9e3ed78358674.jpg",
+        "table_caption": [
+            "Data augmentations. In our implementation of UDA and Coaching, we use RandomAugment, which is a randomized augmentation strategy over all the operations in the search space of AutoAugment (Cubuk et al., 2019). We use RandomAugment because it is simple to implement, requires no expensive search, and achieves similar performance compared to UDA with AutoAugment. More details of RandomAugment can be found in Appendix C.2. ",
+            "Table 1: Image Classification Accuracy on reduced CIFAR-10, SVHN, and ImageNet. Higher is better. For CIFAR-10 and SVHN, we report mean $\\pm$ std over 10 runs, while for ImageNet, we report Top-1/Top-5 accuracy of a single run. Results in the second block are taken from past papers, while the rest shares the same environment and hyper-parameter settings. All methods share the same model architecture: WideResNet-28-2 for CIFAR-10 and SVHN, and ResNet-50 for ImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>CIFAR-10 (4,000)</td><td>SVHN (1,000)</td><td>ImageNet (10%)</td></tr><tr><td>Purely Supervised on full dataset</td><td>94.92 ± 0.17</td><td>97.41 ± 0.16</td><td>76.89/93.27</td></tr><tr><td>Temporal Ensemble</td><td>83.63 ±0.63</td><td>92.81± 0.27</td><td></td></tr><tr><td>Mean Teacher</td><td>84.13± 0.28</td><td>94.35 ± 0.47</td><td></td></tr><tr><td>VAT+EntMin</td><td>86.87± 0.39</td><td>94.65 ± 0.19</td><td>-/83.39</td></tr><tr><td>LGA +VAT</td><td>87.94 ± 0.19</td><td>93.42 ± 0.36</td><td>1</td></tr><tr><td>ICT</td><td>92.71±0.02</td><td>96.11 ± 0.04</td><td></td></tr><tr><td>MixMatch</td><td>93.76±0.06</td><td>96.73 ± 0.31</td><td></td></tr><tr><td>Purely Supervised</td><td>82.14±0.25</td><td>88.17 ±0.47</td><td>57.75/80.23</td></tr><tr><td>Pseudo Labels</td><td>83.79 ± 0.11</td><td>89.81± 0.41</td><td>58.21/82.19</td></tr><tr><td>UDA (our implementation)</td><td>94.53 ±0.18</td><td>97.11 ± 0.17</td><td>68.07/88.19</td></tr><tr><td>Coaching</td><td>95.60 ±0.19</td><td>97.79 ± 0.11</td><td>72.39/90.52</td></tr><tr><td>Coaching + Finetune</td><td>96.11 ± 0.07</td><td>98.01 ± 0.07</td><td>72.94/90.80</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Main results. In Table 1, we present our main results before and after finetuning the student on labeled data. The results confirm that Coaching significantly outperforms UDA and other strong baselines in semi-supervised learning. ",
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+    {
+        "type": "text",
+        "text": "On CIFAR-10 and SVHN, compared to the state-of-the-art UDA, Coaching’s error rate reduction are roughly $3 0 \\%$ and $1 0 \\%$ . As UDA’s accuracy is already relatively high, such error reductions are significant. On CIFAR-10, Coaching is also the first approach to exceed supervised learning on the all labels by using merely 4,000 labeled examples. Meanwhile, on ImageNet- $10 \\%$ , Coaching outperforms UDA by almost $5 \\%$ in top-1 accuracy, going from $6 8 . 0 7 \\%$ to ${ \\bar { 7 } } 2 . 9 4 \\%$ . Even prior to finetuning on labeled data, Coaching still outperforms UDA and other baselines. ",
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+    {
+        "type": "text",
+        "text": "Comparing to existing state-of-the-art methods. To the best of our knowledge, Coaching has achieved new state-of-the-art performances among the same model architectures on three datasets considered in this section. ",
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+    {
+        "type": "text",
+        "text": "For CIFAR-10 and SVHN, all existing better results use a larger model and more advanced regularization techniques. For instance, Xie et al. (2019) reports $9 7 . 3 \\%$ with UDA (Xie et al., 2019), but their backbone model is PyramidNet, which has $1 8 \\times$ more parameters than WideResNet-28-2 and they train with Shake-Drop regularization (Yamada et al., 2018). Similarly, for ImageNet- $10 \\%$ , the only better published result is $7 3 . 2 1 \\%$ top-1 accuracy, achieved by MOAM- $S ^ { 4 } L$ (Zhai et al., 2019). This accuracy is only slightly better than Coaching’s $7 2 . 9 4 \\%$ , but uses a $4 \\times$ wider ResNet-50. We believe that the enhancements in architectures, regularization techniques, and model sizes, can be applied to Coaching to further improve our results. ",
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+    {
+        "type": "text",
+        "text": "3.2 RESULTS ON HIGH DATA REGIME ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We have seen Coaching achieves strong performance for low data image classification tasks. Another aspect of these tasks is that the unlabeled data also come from the same domain as the labeled data, which is a restricted assumption. In this section, we show that Coaching also excels in the regime where we have a large labeled dataset and an order of magnitude more unlabeled data. In this regime, we also test the performance of our method when the unlabeled set may have out-of-domain images, i.e., the images belong to categories that do not exist in ImageNet. ",
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+    {
+        "type": "text",
+        "text": "Datasets. We experiment with all labeled examples in ImageNet. Additionally, we take unlabeled images from the entire $4 ^ { \\mathrm { t h } }$ version of OpenImages dataset (Kuznetsova et al., 2018), which has 9 million natural images. A few samples from OpenImages can be found in Figure 2. Unless otherwise specified, for both datasets, we use the image size of $2 2 4 \\times 2 2 4$ . ",
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+    {
+        "type": "text",
+        "text": "Baselines. Since this regime of high data has not been extensively studied, we are only aware of two relevant, strong baselines. Our first baseline is Billion-scale Semi-supervised Learning (Billion-scale SSL; Yalniz et al. (2019)). Billion-scale SSL uses unlabeled data from the YFCC100M dataset (Thomee et al., 2015), studies several self-training settings, with various model architectures for teachers and students. Here, we restrict our comparison to the settings that use ResNet-50 for both the teacher and the student. Our second baseline is UDA (Xie et al., 2019), for which the authors select unlabeled images algorithmically from the JFT dataset.2 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Other than these baselines, we compare Coaching to techniques that enhance supervised learning, such as DropBlock (Ghiasi et al., 2018), CutMix (Yun et al., 2019), and FixRes (Touvron et al., 2019). ",
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+    },
+    {
+        "type": "text",
+        "text": "Implementation details. We implement Coaching the same as in Section 3.1, except for one part: Instead of directly training and consistently regularizing the teacher, we initialize the teacher using a pre-trained ResNet-50 (pre-trained on full ImageNet). Then, throughout the course of the student’s learning, we only train the teacher to minimize the student’s cross entropy loss. We do not use additional supervised loss for the teacher because because once the teacher is pre-trained, adding another loss to the teacher has minimal effect. We do not consistently regularize the teacher because Xie et al. (2019) has found that consistency regularization requires in-domain data, while we do not filter our unlabeled images from OpenImages. ",
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+    {
+        "type": "table",
+        "img_path": "images/30bc755e86f1c9639dc7544e66731fa2a0d9ba43347961ece705589c19771adb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Methods</td><td rowspan=\"2\">Unlabeled images</td><td colspan=\"2\">Image size</td><td rowspan=\"2\">Top-1</td><td rowspan=\"2\">Top-5</td></tr><tr><td>Train</td><td>Test</td></tr><tr><td>Supervised</td><td>None</td><td>224</td><td>224</td><td>76.89</td><td>93.27</td></tr><tr><td>DropBlock</td><td>None</td><td>224</td><td>224</td><td>78.35</td><td>94.15</td></tr><tr><td>FixRes +CutMix</td><td>None</td><td>224</td><td>320</td><td>79.8</td><td>94.9</td></tr><tr><td>Coaching</td><td>OpenImages</td><td>224</td><td>320</td><td>79.80</td><td>94.87</td></tr><tr><td rowspan=\"2\">FixRes Coaching</td><td>None</td><td>224</td><td>384</td><td>79.1</td><td>94.6</td></tr><tr><td>OpenImages</td><td>224</td><td>384</td><td>80.10</td><td>95.07</td></tr><tr><td>Billion-scale SSL</td><td>YFCC100M</td><td>224</td><td>224</td><td>77.6</td><td></td></tr><tr><td>Coaching</td><td>OpenImages</td><td>224</td><td>224</td><td>78.62</td><td>94.26</td></tr><tr><td>UDA</td><td>JFT</td><td>331</td><td>331</td><td>79.04</td><td>94.45</td></tr><tr><td>Coaching</td><td>OpenImages</td><td>224</td><td>331</td><td>79.86</td><td>94.92</td></tr><tr><td>Coaching+iterative</td><td>OpenImages</td><td>224</td><td>331</td><td>82.34</td><td>96.09</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 2: Image classification accuracy with full ImageNet plus unlabeled images. Results are organized by image size because image size has a strong impact on models’ performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results. We present our results in Table 2. As can be seen, Coaching outperforms all relevant SSL baselines. Specifically, for the image size of 224, Coaching outperforms Billion-scale SSL by about $1 \\%$ top-1 accuracy, even though Billion-scale SSL uses 10 times more unlabeled data. Meanwhile, for the image size of 331, Coaching achieves the top-1 accuracy of $7 9 . 8 6 \\%$ , comfortably outperforming the top-1 accuracy of $7 9 . 0 4 \\%$ by UDA. This improvement is particularly significant, since Coaching simply uses all data from OpenImages, while UDA has to select and balance the class distribution of their unlabeled data using a pre-trained teacher. This difference suggests that the teacher in Coaching can give helpful pseudo labels to the student, even on potentially out-of-distribution data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "It is worth mentioning that Coaching also outperforms the strong supervised baselines of DropBlock and FixRes, and is on par with FixRes+CutMix. However, DropBlock and CutMix are both regularization techniques orthogonal to Coaching. Similar to consistency regularization in Section 3.1, these techniques can be incorporated into the teacher in Coaching to improve performance. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Comparing to state-of-the-art SSL results. Yalniz et al. (2019) reports the top-1 accuracy of $8 1 . 2 \\%$ for a ResNet-50 student. However, they need to pre-train a much bigger network ResNext-101-32x48 teacher (829 million parameters, $3 2 \\mathrm { x }$ larger than ResNet-50) on 1 billion Instagram images with weak labels (Mahajan et al., 2018). Then, they use the pseudo-labels from this teacher to train a ResNet-50 student for 2 billion steps. The fact that they use weakly labeled data from Instagram, much bigger architecture in ResNext-101-32x48 makes their results not directly comparable to ours. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Meanwhile, without the need of a much bigger dataset and architecture as used in Yalniz et al. (2019), Coaching achieves almost as good top-1 accuracy. To achieve this, we iterate the process of Coaching by turning the student into the teacher after convergence. After 17 iterations, our final student achieves $8 2 . 3 4 \\%$ top-1 accuracy on ImageNet, outperforming Yalniz et al. (2019)’s $8 1 . 2 \\%$ , even though we do not have the weakly labeled data from Instagram. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Insights about Coaching on OpenImages. Figure 2 shows five images taken from OpenImages, along with their OpenImages tags and the top 5 classes predicted by a teacher trained on ImageNet. From the figure, we can see that there are non-trivial overlapping contents between the OpenImages tags and the ImageNet top classes, such as sunglasses in the first image. We also see that for the images whose contents match stronger with an ImageNet class, such as the first and the third image, the entropy of the teacher’s prediction is smaller. As a result, when the teacher samples a pseudo label from these distribution, contents similar to an ImageNet class will receive more consistent labels, while content alien to ImageNet will have higher entropy on their labels. We suspect this is why a teacher trained on ImageNet can teach a student via pseudo labels on OpenImages. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/3f36c32f4813ff69e0c535abbb87f4e59c10b06addd499c4b452791024d90158.jpg",
+        "image_caption": [
+            "Figure 2: An illustration of why OpenImages help ImageNet classification. Top: OpenImages tags. Middle: A sample image from OpenImages. Bottom: Top 5 labels for the image predicted by a teacher ResNet-50 trained on ImageNet. Some OpenImages tags overlap significantly with some ImageNet classes, such as wheel and car wheel in the second image. The class predictions also have a higher entropy when the ImageNet classes overlap less with the OpenImages contents (images 2, 4, 5), than when the ImageNet classes overlap more (images 1, 3). "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "3.3 ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Ablation Study of Implementation Details. To understand the contribution of each implementation detail of Coaching, we study their contributions on top of a purely supervised model. We conduct this study on ImageNet- $10 \\%$ and visualize the results in Figure 3. From the figure we see that RandomAugment and UDA both improve the final accuracy significantly, respectively by $3 . 1 3 \\%$ and $7 . 1 9 \\%$ top-1 accuracy. On top of UDA, Coaching delivers a smaller improvement of $4 . 3 2 \\%$ top-1 accuracy. However, since UDA’s accuracy is already high, we believe that the improvement of $4 . 3 2 \\%$ top-1 accuracy is significant. Finally, finetuning only slightly improves over Coaching. However, this extra boost is a unique advantage of Coaching: it is possible for the student in Coaching to finetune on labeled data because the student never directly learns from these labeled data. ",
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+    {
+        "type": "text",
+        "text": "Coaching overfits less than Supervised Learning. In our Coaching framework, the student never directly learns from labeled data. This behavior is helps the student to avoid overfitting, especially when labeled data is scarce. In Figure 4, we visualize the training accuracy of Coaching and Supervised Learning on CIFAR-10 with 4,000 labels and on ImageNet with $10 \\%$ labels. As shown, the training accuracy of both the teacher and the student of Coaching stay relatively low. Meanwhile, the training accuracy of the supervised model eventually reaches $1 0 0 \\%$ and causes overfitting. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/ed6b51d0471ea4fd1866849074a61ccddd659333af9621335de5b95481ca6918.jpg",
+        "image_caption": [
+            "Figure 3: Breakdown of the gains of different components in Coaching. The gain of Coaching over UDA, albeit smaller than the gain of UDA over RandomAugment, is significant as UDA is already very strong. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/daed95c2a2f864200b064f0f14afc1c16ae289b2406a7dcf06a279f17506cb5e.jpg",
+        "image_caption": [
+            "Figure 4: Training accuracy of Coaching and of supervised learning on CIFAR-10-4,000 and ImageNet- $10 \\%$ . Both the teacher and the student in Coaching have lower training accuracy, effectively avoiding overfitting. "
+        ],
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Pseudo-Label. Pseudo-Label (Lee, 2013) is one of the simplest semi-supervised learning algorithms: First, a teacher model is trained on labeled data. Then, the converged teacher model generates pseudo labels for unlabeled data. These unlabeled data and their pseudo labels are combined with the labeled data to train another model, which is called the student model. An inherent weakness of Pseudo-Label is that once the teacher generates an incorrect pseudo label for an unlabeled datum, the student can only naively learn from this wrong label. This phenomenon is called the confirmation bias. Arazo et al. (2019) addressed the confirmation bias by generating soft labels from the teacher and by adding noise to these labels. However, this is a manual fix from an outside model designer. The main difference between Pseudo-Label and Coaching is that in Coaching, the teacher is trained along with the student throughout the course of training. This allows wrong knowledge learned by the teacher to be fixed in an end-to-end manner, leading to stronger performances. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Semi-supervised Learning (SSL). Pseudo-Label belongs to a more general group of algorithms known as Semi-supervised Learning. Unlike Pseudo-Label, typical SSL methods combine both labeled and unlabeled data to train a single model. Hence, the objective function of SSL is typically the sum of a supervised loss and an unsupervised loss. The supervised loss is often the cross-entropy computed on the labeled data. Meanwhile, the unsupervised loss can be a self-supervised loss (Rasmus et al., 2015; Noroozi & Favaro, 2018; Gidaris et al., 2018), or consistency regularization (Laine & Aila, 2017; Tarvainen & Valpola, 2017; Miyato et al., 2018; Berthelot et al., 2019; Xie et al., 2019). Self-supervised losses typically encourage the model to develop a common sense about the images. Meanwhile, consistency regularization enforces that the model is invariant against certain transformations of the data. The main difference between Coaching and SSL methods is that the student in Coaching never learns directly from labeled data. This helps the student in Coaching to avoid overfitting to labeled data, especially when labeled data is limited. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Meta Learning. In Meta Learning, there is typically an outer loop that optimizes the performance of a model trained in an inner loop (Finn et al., 2017; Metz et al., 2019). Meta Learning has been applied to perform self-training and SSL in the low data regime (Agarwal et al., 2019; Ren et al., 2018; Boney & Ilin, 2018; Hsu et al., 2019). A crucial difference between Coaching and Meta Learning is that in Coaching, the pseudo labels are chosen to improve the student, and hence there is no need for an outer loop. We suspect this is an advantage of our method, since gradients to be very powerful for models to navigate in the parameter space. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we proposed the Coaching method for semi-supervised learning. Key to Coaching is the idea that the teacher learns from the student’s loss and improves itself to generate pseudo labels in a way that helps student’s learning the most. The learning process in Coaching consists of two main updates: updating the student based on the pseudo labeled data produced by the teacher and updating the teacher based on the student’s performance. Experiments on standard CIFAR-10 and SVHN show that Coaching is much better than supervised learning and consistenly better than other semi-supervised learning methods. Coaching scales well to large problems, and successfully uses out-of-domain data to improve ImageNet classification. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            285,
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+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A DERIVATION OF THE TEACHER’S UPDATE RULE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            607,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present the detailed derivation of the Teacher’s update rule in Equation 3 from Section 2. ",
+        "bbox": [
+            174,
+            133,
+            823,
+            162
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Mathematical Notations and Conventions. Since we will work with the chain rule, we use the standard Jacobian notations.3 Specifically, for a differentiable function $f : \\mathbb { R } ^ { m }  \\mathbb { R } ^ { n }$ , and for a vector $x \\in \\mathbb { R } ^ { m }$ , we use the notation ∂f∂x ∈ Rn×m to denote the Jacobian matrix of f, whose dimension is $n \\times m$ . Additionally, when we mention the Jacobian of a function $f$ at multiple points such as x1 and x2, we will use the notations of ∂f∂x $\\left. { \\frac { \\partial f } { \\partial x } } \\right| _ { x = x _ { 1 } }$ and $\\left. \\frac { \\partial f } { \\partial x } \\right| _ { x = x _ { 2 } }$ ",
+        "bbox": [
+            173,
+            178,
+            825,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, by mathematical conventions, a vector $v \\in \\mathbb { R } ^ { n }$ is treated as a column matrix – that is, a matrix of size $n \\times 1$ . For this reason, the gradient vector of a multi-variable real-valued function is actually the transpose of of its Jacobian matrix. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Finally, all multiplications in this section are standard matrix multiplications. If an operand is a vector, then as discussed in the previous paragraph, the operand is treated as a column matrix. ",
+        "bbox": [
+            173,
+            318,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dimension Annotations. Understanding that these notations and conventions might cause confusions, in the derivation below, we annotate the dimensions of the computed quantities to ensure that there is no confusion caused to our readers. To this end, we respectively use $| S |$ and $| T |$ to denote the dimensions of the parameters $\\theta _ { S } , \\theta _ { T }$ . That is, $\\theta _ { S } \\in \\mathbb { R } ^ { | S | \\times 1 }$ and $\\boldsymbol { \\theta _ { T } } \\in \\mathbb { R } ^ { | T | \\times 1 }$ . ",
+        "bbox": [
+            173,
+            362,
+            826,
+            421
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We now present the derivation. We need to compute: ",
+        "bbox": [
+            173,
+            426,
+            519,
+            441
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a86d11a8b62e62f0c4b3504b4805ccc35c001d1f5ffc37d09f3a00fa7a1c5d37.jpg",
+        "text": "$$\n\\underbrace { \\frac { \\partial R } { \\partial \\theta _ { T } } } _ { 1 \\times | T | } = \\frac { \\partial } { \\partial \\theta _ { T } } \\ell \\left( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\mathbb { E } _ { \\hat { y } _ { \\mathrm { u n l } } \\sim P ( \\cdot | x _ { \\mathrm { u n l } } ; \\theta _ { T } ) } \\left[ \\theta _ { S } ^ { ( t + 1 ) } \\right] \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            449,
+            671,
+            502
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To simplify our notation, let us define ",
+        "bbox": [
+            173,
+            508,
+            421,
+            523
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f24b34568724d3bfa00375484a9a302d6e8805e735f4ab70eb4beb921953e49.jpg",
+        "text": "$$\n\\underbrace { \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } _ { | S | \\times 1 } \\triangleq \\mathbb { E } _ { \\hat { y } _ { \\mathrm { u n l } } \\sim P ( \\cdot | x _ { \\mathrm { u n l } } ; \\theta _ { T } ) } \\left[ \\theta _ { S } ^ { ( t + 1 ) } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            531,
+            612,
+            574
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Then, by the chain rule, we have ",
+        "bbox": [
+            173,
+            580,
+            388,
+            595
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24c997d06fe283a1cb04fb04f1f1040abac1cd205917985376ad06797370c3cb.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\frac { \\partial R } { \\partial \\theta _ { T } } = \\frac { \\partial } { \\partial \\theta _ { T } } \\ell \\left( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\mathbb { E } _ { \\hat { y } _ { \\mathrm { a n l } } \\sim P ( \\cdot \\vert x _ { \\mathrm { a n l } } ; \\theta _ { T } ) } \\left[ \\theta _ { S } ^ { ( t + 1 ) } \\right] \\right) } } \\\\  { \\displaystyle { \\mathrm {  ~ \\Lambda ~ } } } \\\\ { { \\displaystyle = \\frac { \\partial } { \\partial \\theta _ { T } } \\ell \\left( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\bar { \\theta } _ { S } ^ { ( t + 1 ) } \\right) } } \\\\  { \\displaystyle ~ = \\underbrace { \\frac { \\partial \\ell \\left( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\theta _ { S } \\right) } { \\partial \\theta _ { S } } \\Big \\vert _ { \\theta _ { S } = \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } \\cdot \\frac { \\partial \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } { \\partial \\theta _ { T } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            603,
+            669,
+            751
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The first factor in Equation 7 can be simply computed via back-propagation. We now focus on the second term. We have ",
+        "bbox": [
+            173,
+            757,
+            823,
+            786
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e038df5c3b9ebf193a1219b85aa02c12b1dd584cda8f50d486b738c5be2089b5.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\underbrace { \\frac { \\partial \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } { \\partial \\theta _ { T } } } } = \\frac { \\partial } { \\partial \\theta _ { T } } \\mathbb { E } _ { \\hat { y } _ { \\mathrm { u n l } } \\sim P ( \\cdot | x _ { \\mathrm { u n l } } ; \\theta _ { T } ) } [ \\theta _ { S } ^ { ( t + 1 ) } ] }  \\\\ & { } & \\\\ & { } & { = \\frac { \\partial } { \\partial \\theta _ { T } } \\mathbb { E } _ { \\hat { y } _ { \\mathrm { u n l } } \\sim P ( \\cdot | x _ { \\mathrm { u n l } } ; \\theta _ { T } ) } [ \\theta _ { S } ^ { ( t ) } - \\eta \\cdot (  \\frac { \\partial \\ell ( x _ { \\mathrm { u n l } } , \\hat { y } _ { \\mathrm { u n l } } ; \\theta _ { S } ) } { \\partial \\theta _ { S } } | _ { \\theta _ { S } = \\theta _ { S } ^ { ( t ) } } ) ^ { \\top } ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            790,
+            754,
+            900
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "3Standard: https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant ",
+        "bbox": [
+            186,
+            910,
+            799,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Note that in Equation 8 above, the Jacobian of $\\ell ( x _ { \\mathrm { u n l } } , \\hat { y } _ { \\mathrm { u n l } } ; \\theta _ { S } )$ , which has dimension $1 \\times | S |$ , needs to be transposed to match the dimension of $\\theta _ { S } ^ { ( t ) }$ , which, as we discussed above, conventionally has ",
+        "bbox": [
+            173,
+            102,
+            826,
+            150
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Now, since $\\theta _ { S } ^ { ( t ) }$ in Equation 8 does not depend on $\\theta _ { T }$ , we can leave it out of subsequent derivations. Also, to simplify notations, let us define the gradient ",
+        "bbox": [
+            173,
+            159,
+            825,
+            189
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/879d4a4b8473c2d7a3f7ea984a8ab8d2a983297dcda5abc07d81de38b1203bbd.jpg",
+        "text": "$$\n\\underbrace { g _ { S } ^ { ( t ) } ( \\hat { y } _ { \\mathrm { u n l } } ) } _ { | S | \\times | 1 | } \\triangleq \\left( \\left. \\frac { \\partial \\ell \\left( x _ { \\mathrm { u n l } } , \\hat { y } _ { \\mathrm { u n l } } ; \\theta _ { S } \\right) } { \\partial \\theta _ { S } } \\right| _ { \\theta _ { S } = \\theta _ { S } ^ { ( t ) } } \\right) ^ { \\top }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            195,
+            643,
+            250
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Then, Equation 8 becomes ",
+        "bbox": [
+            174,
+            257,
+            349,
+            272
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b655ca44908938b3b3d48c80ed0657f167949dac0c54a00cf1cc00e54f5b855b.jpg",
+        "text": "$$\n\\underbrace { \\frac { \\partial \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } { \\partial \\theta _ { T } } } _ { | S | \\times | T | } = - \\eta \\cdot \\frac { \\partial } { \\partial \\theta _ { T } } \\mathbb { E } _ { \\hat { y } _ { \\mathrm { u n l } } \\sim P ( \\cdot | x _ { \\mathrm { u n l } } ; \\theta _ { T } ) } \\left[ \\underbrace { g _ { S } ^ { ( t ) } ( \\hat { y } _ { \\mathrm { u n l } } ) } _ { | S | \\times 1 } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            280,
+            655,
+            337
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Since $g _ { S } ^ { ( t ) } ( \\hat { y } _ { \\mathrm { u n l } } )$ has no dependency on on $\\theta _ { T }$ , except for via $\\hat { y } _ { \\mathrm { u n l } }$ , we can apply the REINFORCE equation (Williams, 1992) to achieve ",
+        "bbox": [
+            176,
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+            377
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b562a2c720ababed4ed162b6406fec77a99fd222d2f0bbdcf3c6cda2cc9721e8.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underbrace { \\frac { \\partial \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } { \\partial \\theta _ { T } } } _ { | S | \\times | T | } = - \\eta \\cdot \\frac { \\partial } { \\partial \\theta _ { T } } \\mathbb { E } _ { \\hat { y } _ { \\mathrm { a n } } \\times P ( \\cdot | x _ { \\mathrm { m i } } \\times \\theta _ { T } ) } \\left[ g _ { S } ^ { ( t ) } ( \\hat { y } _ { \\mathrm { m i } } ) \\right] } \\\\ & { = - \\eta \\cdot \\mathbb { E } _ { \\hat { y } _ { \\mathrm { a n } } \\times P ( \\cdot | x _ { \\mathrm { m i } } \\times \\theta _ { T } ) } \\left[ \\underbrace { g _ { S } ^ { ( t ) } ( \\hat { y } _ { \\mathrm { m i } } ) } _ { | S | \\times 1 } \\cdot \\underbrace { \\frac { \\partial \\log P \\left( \\hat { y } _ { \\mathrm { m i } } | x _ { \\mathrm { m i } } ; \\theta _ { T } \\right) } { \\partial \\theta _ { T } } } _ { 1 \\times | T | } \\right] } \\\\ & { = \\eta \\cdot \\mathbb { E } _ { \\hat { y } _ { \\mathrm { a n } } \\sim P ( \\cdot | x _ { \\mathrm { m i } } ; \\theta _ { T } ) } \\left[ \\underbrace { g _ { S } ^ { ( t ) } ( \\hat { y } _ { \\mathrm { m i } } ) } _ { | S | \\times 1 } \\cdot \\underbrace { \\frac { \\partial \\ell \\left( x _ { \\mathrm { m i } } , \\hat { y } _ { \\mathrm { m i } } ; \\theta _ { T } \\right) } { \\partial \\theta _ { T } } } _ { 1 \\times | T | } \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            385,
+            725,
+            550
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here, the last equality in Equation 11 is is due to the definition of the cross entropy loss, which is the negative of the log-prob term in the previous line. ",
+        "bbox": [
+            174,
+            556,
+            821,
+            585
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Now, we can substitute Equation 11 into Equation 7 to obtain ",
+        "bbox": [
+            173,
+            592,
+            576,
+            607
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ebcefd3a05ce8831e8ca86102fe7d8501b98d62b12c119aaaec2c0e05b42a9e3.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { \\partial R } { \\partial \\theta _ { T } } = \\underbrace { \\frac { \\partial \\ell \\left( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\theta _ { S } \\right) } { \\partial \\theta _ { S } } \\bigg | _ { \\theta _ { S } = \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } } _ { = \\eta \\cdot \\underbrace { 1 \\times | S | } _ { = \\eta \\cdot \\underbrace { 1 \\times | S | } _ { = \\theta _ { S } } } \\bigg | _ { \\theta _ { S } = \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } } \\cdot \\underbrace { \\frac { \\partial \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } { \\partial \\theta _ { T } } } _ { | S | \\times \\left| T \\right| } } \\\\ { = \\eta \\cdot \\underbrace { \\frac { \\partial \\ell \\left( x _ { \\mathrm { l a b } } , y _ { \\mathrm { l a b } } ; \\theta _ { S } \\right) } { \\partial \\theta _ { S } } \\bigg | _ { \\theta _ { S } = \\bar { \\theta } _ { S } ^ { ( t + 1 ) } } } _ { \\mathrm { 1 \\times | S | } } \\cdot \\mathbb { E } _ { \\hat { y } _ { \\mathrm { l a n } } \\sim P \\cdot ( \\cdot | x _ { \\mathrm { m i } } ; \\theta _ { T } ) } \\left[ \\underbrace { g _ { S } ^ { ( t ) } ( \\hat { y } _ { \\mathrm { u n } } ) } _ { | S | \\times 1 } \\cdot \\underbrace { \\frac { \\partial \\ell \\left( x _ { \\mathrm { u n l } } , \\hat { y } _ { \\mathrm { u n l } } ; \\theta _ { T } \\right) } { \\partial \\theta _ { T } } } _ { \\mathrm { 1 \\times | T | } } \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            189,
+            614,
+            779,
+            736
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, if we use Monte Carlo approximation for every term in Equation 12 using the sampled $\\hat { y } _ { \\mathrm { u n l } }$ then we have Equation 3 from Section 2. Note that in Section 2, we use the gradient notation, which results in the transposes. ",
+        "bbox": [
+            174,
+            742,
+            825,
+            786
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B TRAINING SPEED ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            806,
+            354,
+            824
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Coaching performs up to 5 forward passes and 3 backward passes. Compared to vanilla backpropagation training, this is more 3 than times more expensive in FLOPs. However, many computations in Coaching are parallelizable. For example, the forward pass of the student and the forward pass for the teacher on unlabeled data (the top half of Figure 1), can be run in parallel since they do not depend on each other. Therefore, on computing hardware with sufficient memory, we find Coaching to be between 2 and 2.5 times slower than standard back-propagation training. ",
+        "bbox": [
+            173,
+            839,
+            826,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            419,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C.1 DATASET SPLITS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            334,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We describe how we select the reduced datasets for the experiments on low data image classification in Section 3.1. ",
+        "bbox": [
+            176,
+            160,
+            823,
+            188
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For CIFAR-10, we download the five training data batch files from www.cs.toronto.edu/ \\~kriz/cifar.html. Then, we load all the images into a list of 50,000 images, keeping the order as downloaded. The fisrt 5,000 images ares reserved for validation. The next 4,000 images are used as labeled data. For SVHN, we download the data from the mat files on ufldl.stanford.edu/ housenumbers/, and follow the same procedure as with CIFAR-10. We note that this selection process leads to a slight imbalance in the class distribution for both CIFAR-10 and SVHN, but the settings are the same for all of our experiments. ",
+        "bbox": [
+            173,
+            194,
+            826,
+            292
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For ImageNet, we follow the procedure in github.com/tensorflow/models/blob/ master/research/inception/inception/data/download_and_preprocess imagenet.sh. This results in 1,024 training TFRecord shards of approximately the same size. The order of the images in these shards are deterministic. For ImageNet- $10 \\%$ , we use the first 102 shards; for ImageNet- $20 \\%$ , we use the first 204 shards; and so on. The last 20 shards, corresponding to roughly 25,000 images, are reserved for hyper-parameters tuning. ",
+        "bbox": [
+            173,
+            299,
+            826,
+            383
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C.2 RANDOMAUGMENT: A DATA AUGMENTATION POLICY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            400,
+            586,
+            414
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We develop a data augmentation policy that achieves similarly high performance with AutoAugment (Cubuk et al., 2019) in a few cases that we consider, but which does not require learning a controller to generate policies. We names our policy RandomAugment. Our goal when developing RandomAugment is not to outperform AutoAugment, which is why we do not conduct extensive experiments with RandomAugment. Instead, we simply want to avoid indirectly using labeled data for our experiments, especially for the low data regime experiments in Section 3.1. ",
+        "bbox": [
+            173,
+            426,
+            825,
+            510
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Each policy of RandomAugment consists of two operations that applied sequentially on an image. Each operation applies a uniformly sampled transformation with probability 0.5, and with a level uniformly chosen between 1 and 10. For a more comprehensive discussion of the probability and the level of a transformation, we refer readers to the AutoAugment paper (Cubuk et al., 2019). ",
+        "bbox": [
+            174,
+            516,
+            825,
+            571
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/2218819223d375901793134d2bfabb04a1e7ed414bce30750d6f006805884d99.jpg",
+        "table_caption": [
+            "Table 3: Transformations that RandomAugment uniformly samples for our datasets. We refer our readers to Cubuk et al. (2019) for the detailed descriptions of these transformations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CIFAR-10 and ImageNet</td><td>SVHN</td></tr><tr><td>AutoContrast</td><td>AutoContrast</td></tr><tr><td>Brightness</td><td>Brightness</td></tr><tr><td>Color</td><td>Color</td></tr><tr><td>Contrast</td><td>Contrast</td></tr><tr><td>Equalize</td><td>Equalize</td></tr><tr><td>Invert</td><td>Invert</td></tr><tr><td>Sharpness</td><td>Sharpness</td></tr><tr><td>Posterize</td><td>Posterize</td></tr><tr><td>Sample Pairing</td><td>Solarize</td></tr><tr><td>Solarize</td><td>ShearX</td></tr><tr><td>Rotate</td><td>ShearY</td></tr><tr><td>ShearX</td><td>TranslateY</td></tr><tr><td>ShearY</td><td></td></tr><tr><td>TranslateX</td><td></td></tr><tr><td></td><td></td></tr><tr><td>TranslateY</td><td></td></tr></table>",
+        "bbox": [
+            385,
+            587,
+            611,
+            763
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We manually design the set of transformations for each of our datasets: CIFAR-10, ImageNet, and SVHN. The set of transformations for CIFAR-10 and for ImageNet are the same, and are slightly different from the set of transformation for SVHN. This is because the numbers in the SVHN have a different requirement for invariant. For instance, numbers should not be invariant against rotations like 6 and 9, and should not be invariant against horizontal translation like 3 and 8. Table 3 presents the transformation for our dataset. In addition to these operations, we only allow RandomAugment to select the three transformations AutoContrast, Brightness, and Invert in the first augmenting transformation. This is to avoid a few degenerating cases. For instance, when Brightness is applied twice on an image, both times with small levels, the image will become almost black. ",
+        "bbox": [
+            173,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, RandomAugment’s performance is not far behind compared to AutoAugment. For example, on full ImageNet with ResNet-50, RandomAugment achives $\\mathrm { \\bar { 7 } 7 . 9 8 \\% }$ top-1 accuracy, which is close to the top-1 accuracy of $7 7 . 6 \\%$ reported by Cubuk et al. (2019). ",
+        "bbox": [
+            176,
+            138,
+            823,
+            181
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.3 HYPER-PARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            200,
+            366,
+            214
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To tune hyper-parameters, we follow Oliver et al. (2018) and allow each method to have 128 trials of hyper-parameters. When we tune, we let each model train for up to 50,000 steps. The optimal hyper-parameters are then used to run experiments that last for much more steps, as we report below. In our experiments with Coaching, training for more steps typically leads to stronger results. We stop at 1 million steps for CIFAR-10 and SVHN, and at 0.5 million steps for ImageNet simply because otherwise, these experiments will take too long. Meanwhile, in our experiments with purely supervised learning, Pseudo-Labels, and UDA, training for more steps overfits the models, and we have to employ early stopping. ",
+        "bbox": [
+            174,
+            227,
+            825,
+            339
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We report the hyper-parameters for our baselines and for Coaching in Section 3.1. For the highresource experiments in Section 3.2, we use the same hyper-parameters, because tuning them is too expensive. Our hyper-parameters can be found in Table 4, 5, 6. ",
+        "bbox": [
+            174,
+            345,
+            825,
+            387
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We note that our settings for UDA is different from originally reported by Xie et al. (2019). In their work, Xie et al. (2019) use a much larger batch size for their UDA objective. In our implementation of UDA, we keep these batch sizes the same. This leads to a much easier implementation of data parallelism in our framework, TensorFlow (Abadi et al., 2016) running on TPU big pods. To compensate for the difference, we train all UDA baselines for much longer than Xie et al. (2019). During the training process, we also mask out the supervised examples with high confidence. Effectively, our UDA model receives roughly the same amount of training with labeled examples and unlabeled examples as the models in Xie et al. (2019). We have also verified that on ImageNet- $10 \\%$ with the augmentation policy from AutoAugment (Cubuk et al., 2019), our UDA implementation achives $6 8 . 7 7 \\%$ top-1 accuracy, which is similar to $6 8 . 6 6 \\%$ that Xie et al. (2019) reported. ",
+        "bbox": [
+            173,
+            393,
+            825,
+            534
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/a041a6887140c4e00ef7446066920f8cdf1b4a29b3c204939a95222d8502c2dc.jpg",
+        "table_caption": [
+            "Table 4: Hyper-parameters for supervised learning and Pseudo-Labels. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyper-parameter</td><td>CIFAR-10</td><td>SVHN</td><td>ImageNet</td></tr><tr><td>Weight decay</td><td>0.0005</td><td>0.001</td><td>0.0002</td></tr><tr><td>Label smoothing</td><td>0</td><td>0</td><td>0.1</td></tr><tr><td>Batch normalization decay</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Learning rate</td><td>0.4</td><td>0.05</td><td>1.28</td></tr><tr><td>Number of training steps</td><td>50,000</td><td>50,000</td><td>40,000</td></tr><tr><td>Number of warm up steps</td><td>2500</td><td>0</td><td>2000</td></tr><tr><td>Batch size</td><td>1024</td><td>128</td><td>2048</td></tr><tr><td>Dropout rate</td><td>0.4</td><td>0.5</td><td>0.2</td></tr><tr><td>Pseudo label threshold</td><td>0.95</td><td>0.975</td><td>0.7</td></tr></table>",
+        "bbox": [
+            331,
+            549,
+            665,
+            672
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/c1fc39d7c90ec78080fcd39352fe4d31803d4a7a1cb43dae57c96e8600d74030.jpg",
+        "table_caption": [
+            "Table 5: Hyper-parameters for UDA. Unlike originally done by Xie et al. (2019), we do not use a larger batch size for the UDA objective. Instead, we use the same batch size for both the labeled objective and the unlabeled objective. This is to avoid instances where some particularly small batch sizes for the labeled objective cannot be split on our computational hardware. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyper-parameter</td><td>CIFAR-10</td><td>SVHN</td><td>ImageNet</td></tr><tr><td>Weight decay</td><td>0.0005</td><td>0.0005</td><td>0.0002</td></tr><tr><td>Label smoothing</td><td>0</td><td>0</td><td>0.1</td></tr><tr><td>Batch normalization decay</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Learning rate</td><td>0.3</td><td>0.4</td><td>1.28</td></tr><tr><td>Number of training steps</td><td>1,000,000</td><td>1,000,000</td><td>500.000</td></tr><tr><td>Number of warm up steps</td><td>5,000</td><td>5.000</td><td>5,000</td></tr><tr><td>Batch size</td><td>128</td><td>128</td><td>2048</td></tr><tr><td>Dropout rate</td><td>0.5</td><td>0.6</td><td>0.25</td></tr><tr><td>UDA factor</td><td>2.5</td><td>1</td><td>20</td></tr><tr><td>UDA temperature</td><td>0.7</td><td>0.8</td><td>0.7</td></tr></table>",
+        "bbox": [
+            325,
+            731,
+            671,
+            858
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/6d7d0e772dbc4fe21a44fba0453c0f26490bb9eaa2429b506291b4c3f7a9137b.jpg",
+        "table_caption": [
+            "Table 6: Hyper-parameters for Coaching. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Hyper-parameter</td><td>CIFAR-10</td><td>SVHN</td><td>ImageNet</td></tr><tr><td rowspan=\"5\">Common</td><td>Weight decay</td><td>0.0005</td><td>0.0005</td><td>0.0002</td></tr><tr><td>Label smoothing</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Batch normalization decay</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Number of training steps</td><td>1,000,000</td><td>1,000,000</td><td>500,000</td></tr><tr><td>Number of warm up steps</td><td>2.000</td><td>2,000</td><td>1,000</td></tr><tr><td rowspan=\"3\">Student</td><td>Learning rate</td><td>0.3</td><td>0.15</td><td>0.8</td></tr><tr><td>Batch size</td><td>128</td><td>128</td><td>2048</td></tr><tr><td>Dropout rate</td><td>0.35</td><td>0.45</td><td>0.1</td></tr><tr><td rowspan=\"5\">Teacher</td><td>Learning rate</td><td>0.125</td><td>0.05</td><td>0.5</td></tr><tr><td>Batch size</td><td>128</td><td>128</td><td>2048</td></tr><tr><td>Dropout rate</td><td>0.5</td><td>0.65</td><td>0.1</td></tr><tr><td>UDA factor</td><td>1.0</td><td>2.5</td><td>16.0</td></tr><tr><td>UDA temperature</td><td>0.8</td><td>1.25</td><td>0.75</td></tr></table>",
+        "bbox": [
+            294,
+            101,
+            702,
+            270
+        ],
+        "page_idx": 14
+    }
+]

parse/train/rkhlb8lCZ/rkhlb8lCZ.md

@@ -0,0 +1,303 @@
+# WAVELET POOLING FOR CONVOLUTIONAL NEURAL NETWORKS
+
+Travis Williams Department of Electrical Engineering North Carolina A&T State University Greensboro, NC 27410, USA tlwilli3@aggies.ncat.edu
+
+Robert Li   
+Department of Electrical Engineering   
+North Carolina A&T State University   
+Greensboro, NC 27410, USA   
+eeli@ncat.edu
+
+# ABSTRACT
+
+Convolutional Neural Networks continuously advance the progress of 2D and 3D image and object classification. The steadfast usage of this algorithm requires constant evaluation and upgrading of foundational concepts to maintain progress. Network regularization techniques typically focus on convolutional layer operations, while leaving pooling layer operations without suitable options. We introduce Wavelet Pooling as another alternative to traditional neighborhood pooling. This method decomposes features into a second level decomposition, and discards the first-level subbands to reduce feature dimensions. This method addresses the overfitting problem encountered by max pooling, while reducing features in a more structurally compact manner than pooling via neighborhood regions. Experimental results on four benchmark classification datasets demonstrate our proposed method outperforms or performs comparatively with methods like max, mean, mixed, and stochastic pooling.
+
+# 1 INTRODUCTION
+
+Convolutional Neural Networks (CNNs) have become the standard-bearer in image and object classification (Nielsen, 2015). Due to the layer structures conforming to the shape of the inputs, CNNs consistently classify images, objects, videos, etc. at a higher accuracy rate than vector-based deep learning techniques (Nielsen, 2015). The strength of this algorithm motivates researchers to constantly evaluate and upgrade foundational concepts to continue growth and progress. The key components of CNN, the convolutional layer and pooling layer, consistently undergo modifications and innovations to elevate accuracy and efficiency of CNNs beyond previous benchmarks.
+
+Pooling has roots in predecessors to CNN such as Neocognitron, which manual subsampling by the user occurs (Fukushima, 1979), and Cresceptron, which introduces the first max pooling operation in deep learning (Weng et al., 1992). Pooling subsamples the results of the convolutional layers, gradually reducing spatial dimensions of the data throughout the network. The benefits of this operation are to reduce parameters, increase computational efficiency, and regulate overfitting (Boureau et al., 2010).
+
+Methods of pooling vary, with the most popular form being max pooling, and secondarily, average pooling (Nielsen, 2015; Lee et al., 2016). These forms of pooling are deterministic, efficient, and simple, but have weaknesses hindering the potential for optimal network learning (Lee et al., 2016; Yu et al., 2014). Other pooling operations, notably mixed pooling and stochastic pooling, use probabilistic approaches to correct some of the issues of the prior methods (Yu et al., 2014; Zeiler & Fergus, 2013).
+
+However, one commonality all these pooling operations employ a neighborhood approach to subsampling, reminiscent of nearest neighbor interpolation in image processing. Neighborhood interpolation techniques perform fast, with simplicity and efficiency, but introduce artifacts such as edge halos, blurring, and aliasing (Parker et al., 1983). Minimizing discontinuities in the data are critical to aiding in network regularization, and increasing classification accuracy.
+
+We propose a wavelet pooling algorithm that uses a second-level wavelet decomposition to subsample features. Our approach forgoes the nearest neighbor interpolation method in favor of an organic, subband method that more accurately represents the feature contents with less artifacts. We compare our proposed pooling method to max, mean, mixed, and stochastic pooling to verify its validity, and ability to produce near equal or superior results. We test these methods on benchmark image classification datasets such as Mixed National Institute of Standards and Technology (MNIST) (Lecun et al., 1998), Canadian Institute for Advanced Research (CIFAR-10) (Krizhevsky, 2009), Street House View Numbers (SHVN) (Netzer et al., 2011), and Karolinska Directed Emotional Faces (KDEF) (Lundqvist et al., 1998). We perform all simulations in MATLAB R2016b.
+
+The rest of this paper organizes as follows: Section 2 gives the background, Section 3 describes the proposed methods, Section 4 discusses the experimental results, and Section 5 gives the summary and conclusion.
+
+# 2 BACKGROUND
+
+Pooling is another term for subsampling. In this layer, the dimensions of the output of the convolutional layer are condensed. The dimensionality reduction happens by summarizing a region into one neuron value, and this occurs until all neurons have been affected. The two most popular forms of pooling are max pooling and average pooling (Nielsen, 2015; Lee et al., 2016). Max pooling involves taking the maximum value of a region $R _ { i j }$ and selecting it for the condensed feature map. Average pooling involves calculating the average value of a region and selecting it for the condensed feature map. The max pooling function is expressed as:
+
+$$
+a _ { k i j } = \operatorname* { m a x } _ { ( p , q ) \in R _ { i j } } ( a _ { k p q } )
+$$
+
+While average pooling is shown by the following equation:
+
+$$
+a _ { k i j } = \frac { 1 } { | R _ { i j } | } \sum _ { ( p , q ) \in R _ { i j } } a _ { k p q }
+$$
+
+Where $a _ { k i j }$ is the output activation of the $k ^ { t h }$ feature map at $( i , j ) , a _ { k p q }$ is the input activation at $( p , q )$ within $R _ { i j }$ , and $| R _ { i j } |$ is the size of the pooling region. An illustration of both of these pooling methods is expressed in Figure 1 (Williams & Li, 2016):
+
+<table><tr><td rowspan=1 colspan=8>Convolution Output</td></tr><tr><td rowspan=1 colspan=1>55</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>87</td><td rowspan=1 colspan=1>46</td><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>59</td><td rowspan=1 colspan=1>76</td><td rowspan=1 colspan=1>58</td></tr><tr><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1>61</td><td rowspan=1 colspan=1>29</td><td rowspan=1 colspan=1>34</td><td rowspan=1 colspan=1>79</td><td rowspan=1 colspan=1>94</td><td rowspan=1 colspan=1>15</td></tr><tr><td rowspan=1 colspan=1>29</td><td rowspan=1 colspan=1>35</td><td rowspan=1 colspan=1>71</td><td rowspan=1 colspan=1>63</td><td rowspan=1 colspan=1>94</td><td rowspan=1 colspan=1>73</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>17</td></tr><tr><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>41</td><td rowspan=1 colspan=1>98</td><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>45</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>76</td></tr><tr><td rowspan=1 colspan=1>92</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>。</td><td rowspan=1 colspan=1>46</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>7</td></tr><tr><td rowspan=1 colspan=1>38</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>86</td><td rowspan=1 colspan=1>97</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>13</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>48</td></tr><tr><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>19</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>42</td><td rowspan=1 colspan=1>45</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>56</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>。</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>66</td><td rowspan=1 colspan=1>44</td><td rowspan=1 colspan=1>72</td><td rowspan=1 colspan=1>8</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=4>Max Pooling</td></tr><tr><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>87</td><td rowspan=1 colspan=1>79</td><td rowspan=1 colspan=1>94</td></tr><tr><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>98</td><td rowspan=1 colspan=1>94</td><td rowspan=1 colspan=1>76</td></tr><tr><td rowspan=1 colspan=1>92</td><td rowspan=1 colspan=1>97</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>48</td></tr><tr><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>42</td><td rowspan=1 colspan=1>66</td><td rowspan=1 colspan=1>72</td></tr></table>
+
+Mean Pooling   
+Figure 1: Example of Max & Average Pooling with Stride of 2   
+
+<table><tr><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>56</td><td rowspan=1 colspan=1>51</td><td rowspan=1 colspan=1>61</td></tr><tr><td rowspan=1 colspan=1>36</td><td rowspan=1 colspan=1>68</td><td rowspan=1 colspan=1>61</td><td rowspan=1 colspan=1>30</td></tr><tr><td rowspan=1 colspan=1>35</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>52</td><td rowspan=1 colspan=1>19</td></tr><tr><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>39</td><td rowspan=1 colspan=1>34</td></tr></table>
+
+While max and average pooling both are effective, simple methods, they also have shortcomings. Max pooling, depending on the data, can erase details from an image (Yu et al., 2014; Zeiler & Fergus, 2013). This happens if the main details have less intensity than the insignificant details. In addition, max pooling commonly overfits training data (Yu et al., 2014; Zeiler & Fergus, 2013). Average pooling, depending on the data, can dilute pertinent details from an image. The averaging of data with values much lower than significant details causes this action (Yu et al., 2014; Zeiler & Fergus, 2013). Figure 2 illustrates these shortcomings using the toy image example:
+
+To combat these issues, researchers have created probabilistic pooling methods. Mixed pooling combines max and average pooling by randomly selecting one method over the other during training (Yu et al., 2014). There is no set way to perform mixed pooling. This method is applied arbitrarily in three different ways (1) for all features within a layer, (2) mixed between features within a layer, or (3) mixed between regions for different features within a layer (Lee et al., 2016; Yu et al., 2014). Mixed pooling is shown in the following equation:
+
+![](images/6836bbc3d6044d3b5f1a5e0d927f4633a7a7e6ef1b5ae5384f1dd2e7b57d9e59.jpg)  
+Figure 2: Shortcomings of Max & Average Pooling using Toy Image
+
+$$
+a _ { k i j } = \lambda \cdot \operatorname* { m a x } _ { ( p , q ) \in R _ { i j } } ( a _ { k p q } ) + ( 1 - \lambda ) \cdot \frac { 1 } { | R _ { i j } | } \sum _ { ( p , q ) \in R _ { i j } } a _ { k p q }
+$$
+
+where $\lambda$ is a random value 0 or 1, indicating max or average pooling for a particular region/feature/layer.
+
+Another probabilistic pooling method, called stochastic pooling, improves upon max pooling by randomly sampling from neighborhood regions based on the probability values of each activation (Zeiler & Fergus, 2013). These probabilities $p$ for each region are calculated by normalizing the activations within the region:
+
+$$
+p _ { p q } = \frac { a _ { p q } } { \sum _ { ( p , q ) \in R _ { i j } } a _ { p q } }
+$$
+
+The pooled activation is sampled from a multinomial distribution based on $p$ to pick a location $l$ within the region (Zeiler & Fergus, 2013). The process is captured in the following equation (Zeiler & Fergus, 2013):
+
+$$
+a _ { k i j } = a _ { l } \quad w h e r e \quad l \sim P ( p _ { 1 } , . . . , p _ { | R _ { i j } | } )
+$$
+
+![](images/2ca0fb375b792ba9a9b71febe50ad09430c488f886e3669eda98dc3d1a51f74a.jpg)  
+Figure 3 displays a visual example of stochastic pooling on a $3 { \bf x } 3$ region:   
+Figure 3: Stochastic Pooling Example
+
+In Figure 3, a region of activations are shown, and in the adjacent region, their corresponding probabilities based on Equation 4. In any given region, the activations with the highest probabilities have the higher chance of selection. However, any activation can be chosen. In this example, the stochastic pooling method selects the midrange activation with a probability of $13 \%$ . By being based off probability, and not deterministic, stochastic pooling avoids the shortcomings of max and average pooling, while enjoying some of the advantages of max pooling (Zeiler & Fergus, 2013).
+
+# 3 PROPOSED METHOD
+
+The previously highlighted pooling methods use neighborhoods to subsample, almost identical to nearest neighbor interpolation. Previous studies explore the possibilities of wavelets in image interpolation versus traditional methods (Dumic et al., 2007). Our proposed pooling method uses wavelets to reduce the dimensions of the feature maps. We propose using the wavelet transform to minimize artifacts resulting from neighborhood reduction (Parker et al., 1983). We postulate that our approach, which discards the first-order subbands, more organically captures the data compression. This organic reduction therefore lessens the creation of jagged edges and other artifacts that may impede correct image classification.
+
+# 3.1 FORWARD PROPAGATION
+
+The proposed wavelet pooling scheme pools features by performing a 2nd order decomposition in the wavelet domain according to the fast wavelet transform (FWT) (Mallat, 1989; Nason & Silverman, 1995; Strang & Nguyen, 1996; Burrus et al., 1998), which is a more efficient implementation of the two-dimensional discrete wavelet transform (DWT) as follows (Chui, 1992; Strang & Strela, 1995; Rieder et al., 1994):
+
+$$
+W _ { \varphi } [ j + 1 , k ] = h _ { \varphi } [ - n ] * W _ { \varphi } [ j , n ] | _ { n = 2 k , k \le 0 }
+$$
+
+$$
+W _ { \psi } [ j + 1 , k ] = h _ { \psi } [ - n ] * W _ { \psi } [ j , n ] | _ { n = 2 k , k \le 0 }
+$$
+
+where $\varphi$ is the approximation function, and $\psi$ is the detail function, $W _ { \varphi }$ , $W _ { \psi }$ are called approximation and detail coefficients. $h _ { \varphi } [ - n ]$ and $h _ { \psi } [ - n ]$ are the time reversed scaling and wavelet vectors, (n) represents the sample in the vector, while (j) denotes the resolution level. When using the FWT on images, we apply it twice (once on the rows, then again on the columns). By doing this in combination, we obtain our detail subbands (LH, HL, HH) at each decomposition level, and our approximation subband (LL) for the highest decomposition level.
+
+After performing the 2nd order decomposition, we reconstruct the image features, but only using the 2nd order wavelet subbands. This method pools the image features by a factor of 2 using the inverse FWT (IFWT) (Mallat, 1989; Nason & Silverman, 1995; Strang & Nguyen, 1996; Burrus et al., 1998), which is based off of the inverse DWT (IDWT) (Chui, 1992; Strang & Strela, 1995; Rieder et al., 1994):
+
+$$
+W _ { \varphi } [ j , k ] = h _ { \varphi } [ - n ] * W _ { \varphi } [ j + 1 , n ] + h _ { \psi } [ - n ] * W _ { \psi } [ j + 1 , n ] | _ { n = { \frac { k } { 2 } } , k \leq 0 }
+$$
+
+![](images/4f3d158ff784108d752855ef0f3ac75a0287d53a4f9a1c33a8ce754c00e1ff07.jpg)  
+Figure 4 gives an illustration of the algorithm for the forward propagation of wavelet pooling:   
+Figure 4: Wavelet Pooling Forward Propagation Algorithm
+
+# 3.2 BACKPROPAGATION
+
+The proposed wavelet pooling algorithm performs backpropagation by reversing the process of its forward propagation. First, the image feature being back propagated undergoes $1 ^ { s t }$ order wavelet decomposition. After decomposition, the detail coefficient subbands upsample by a factor of 2 to create a new $1 ^ { s t }$ level decomposition. The initial decomposition then becomes the $2 ^ { n d }$ level decomposition. Finally, this new $2 ^ { n d }$ order wavelet decomposition reconstructs the image feature for further backpropagation using the IDWT. Figure 5 details the backpropagation algorithm of wavelet pooling:
+
+![](images/67febc9a06058a1f18b93c877fafd23b66b72930b4b25034992e16feaa33623b.jpg)  
+Figure 5: Wavelet Pooling Backpropagation Algorithm
+
+# 4 RESULTS AND DISCUSSION
+
+All CNN experiments use MatConvNet (Vedaldi & Lenc, 2015). All training uses stochastic gradient descent (Bottou, 2010). For our proposed method, the wavelet basis is the Haar wavelet, mainly for its even, square subbands. All experiments are run on a 64-bit operating system, with an Intel Core i7-6800k CPU $\textcircled { a } ~ 3 . 4 0 ~ \mathrm { G H z }$ processor, with 64.0 GB of RAM. We utilize two GeForce Titan X Pascal GPUs with 12 GB of video memory for all training. All CNN structures except for MNIST use a network loosely based on Zeilers network (Zeiler & Fergus, 2013). We repeat the experiments with Dropout (Srivastava, 2013) and replace Local Response Normalization (Krizhevsky, 2009) with Batch Normalization (Ioffe & Szegedy, 2015) for CIFAR-10 and SHVN (Dropout only) to examine how these regularization techniques change the pooling results. To test the effectiveness of each pooling method on each dataset, we solely pool with that method for all pooling layers in that network. All pooling methods use a $2 \mathbf { x } 2$ window for an even comparison to the proposed method. Figure 6 gives a selection of each of the datasets.
+
+# 4.1 MNIST
+
+The network architecture is based on the example MNIST structure from MatConvNet, with batch normalization inserted. All other parameters are the same. Figure 7 shows our network structure for the MNIST experiments:
+
+![](images/ed5f7af51d677a75f6562aab9d4777531ec7fc85c14ca7ba987adc63d00a60a6.jpg)  
+Figure 6: Selection of Image Datasets
+
+![](images/95c98f5bb8cd108027f9a420c1a864090ebfbba981f7d03348eb869f3d9d69e0.jpg)  
+Figure 7: CNN MNIST Structure Block Diagram
+
+The input training data and test data come from the MNIST database of handwritten digits. The full training set of 60,000 images is used, as well as the full testing set of 10,000 images. Table 1 shows our proposed method outperforms all methods. Given the small number of epochs, max pooling is the only method to start to overfit the data during training. Mixed and stochastic pooling show a rocky trajectory, but do not overfit. Average and wavelet pooling show a smoother descent in learning and error reduction. Figure 8 shows the energy of each method per epoch.
+
+![](images/deb898ce72595fd7b606ef3ff92190e8d12d0e163a108fe312e27b2c0706e6b5.jpg)  
+Figure 8: MNIST Pooling Method Energy Performance of Training & Validation Sets
+
+Table 1 shows the accuracy of each method:
+
+Table 1: MNIST Performance of Pooling Methods   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Average</td><td rowspan=1 colspan=1>Max</td><td rowspan=1 colspan=1>Mixed</td><td rowspan=1 colspan=1>Stochastic</td><td rowspan=1 colspan=1>Wavelet</td></tr><tr><td rowspan=1 colspan=1>Accuracy (%)</td><td rowspan=1 colspan=1>98.72</td><td rowspan=1 colspan=1>98.80</td><td rowspan=1 colspan=1>98.86</td><td rowspan=1 colspan=1>98.90</td><td rowspan=1 colspan=1>99.01</td></tr></table>
+
+# 4.2 CIFAR-10
+
+We run two sets of experiments with the pooling methods. The first is a regular network structure with no dropout layers. We use this network to observe each pooling method without extra regularization. The second uses dropout and batch normalization, and performs over 30 more epochs to observe the effects of these changes. Figure 9 shows our network structure for the CIFAR-10 experiments:
+
+The input training and test data come from the CIFAR-10 dataset. The full training set of 50,000 images is used, as well as the full testing set of 10,000 images. For both cases, with no dropout, and with dropout, Table 2 and Table 3 show our proposed method has the second highest accuracy. Max pooling overfits fairly quickly, while wavelet pooling resists overfitting. The change in learning rate prevents our method from overfitting, and it continues to show a slower propensity for learning. Mixed and stochastic pooling maintain a consistent progression of learning, and their validation sets trend at a similar, but better rate than our proposed method. Average pooling shows the smoothest descent in learning and error reduction, especially in the validation set. Figure 10 shows the energy of each method per epoch.
+
+![](images/eab1af088fdeeed9fe5e1fdc17353c47fe3cad910a704148ae6af01cf8847a1f.jpg)  
+Figure 9: CNN CIFAR-10 Structure Block Diagram
+
+![](images/8343d05229f091b0ccaf3b71410317e193a25d016e7d738194f7228e11c02b31.jpg)  
+Figure 10: CIFAR-10 Pooling Method Energy Performance of Training & Validation Sets
+
+Tables 2 and 3 show the accuracy of each method:
+
+Table 2: CIFAR-10 Performance of Pooling Methods   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Average</td><td rowspan=1 colspan=1>Max</td><td rowspan=1 colspan=1>Mixed</td><td rowspan=1 colspan=1>Stochastic</td><td rowspan=1 colspan=1>Wavelet</td></tr><tr><td rowspan=1 colspan=1>Accuracy (%)</td><td rowspan=1 colspan=1>76.51</td><td rowspan=1 colspan=1>71.42</td><td rowspan=1 colspan=1>73.77</td><td rowspan=1 colspan=1>73.03</td><td rowspan=1 colspan=1>74.42</td></tr></table>
+
+Table 3: CIFAR-10 Performance of Pooling Methods $^ +$ Dropout   
+
+<table><tr><td></td><td>Average</td><td>Max</td><td>Mixed</td><td>Stochastic</td><td>Wavelet</td></tr><tr><td>Accuracy (%)</td><td>81.15</td><td>80.30</td><td>79.21</td><td>80.09</td><td>80.28</td></tr></table>
+
+# 4.3 SHVN
+
+We run two sets of experiments with the pooling methods. The first is a regular network structure with no dropout layers. We use this network to observe each pooling method without extra regularization. The second uses dropout to observe the effects of this change. Figure 11 shows our network structure for the SHVN experiments:
+
+The input training and test data come from the SHVN dataset. For the case with no dropout, we use 55,000 images from the training set. For the case with dropout, we use the full training set of 73,257 images, a validation set of 30,000 images we extract from the extra training set of 531,131 images, as well as the full testing set of 26,032 images. For both cases, with no dropout, and with dropout,
+
+![](images/b3a9dba3655fb309e5a32d14191f972a1e36fd28e5668c13d8d56ef140140af0.jpg)  
+Figure 11: CNN SHVN Structure Block Diagram
+
+Table 4 and Table 5 show our proposed method has the second lowest accuracy. Max and wavelet pooling both slightly overfit the data. Our method follows the path of max pooling, but performs slightly better in maintaining some stability. Mixed, stochastic, and average pooling maintain a slow progression of learning, and their validation sets trend at near identical rates. Figure 12 shows the energy of each method per epoch.
+
+![](images/5cc6b495708b04e274769b4f7b6e0755d6376f1b03d688956c266d1d52014244.jpg)  
+Figure 12: SHVN Pooling Method Energy Performance of Training & Validation Sets
+
+Tables 4 and 5 shows the accuracy of each method:
+
+Table 4: SHVN Performance of Pooling Methods   
+
+<table><tr><td></td><td>Average</td><td>Max</td><td>Mixed</td><td>Stochastic</td><td>Wavelet</td></tr><tr><td>Accuracy (%)</td><td>89.83</td><td>88.09</td><td>89.25</td><td>89.97</td><td>88.51</td></tr></table>
+
+Table 5: SHVN Performance of Pooling Methods $^ +$ Dropout   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Average</td><td rowspan=1 colspan=1>Max</td><td rowspan=1 colspan=1>Mixed</td><td rowspan=1 colspan=1>Stochastic</td><td rowspan=1 colspan=1>Wavelet</td></tr><tr><td rowspan=1 colspan=1>Accuracy (%)</td><td rowspan=1 colspan=1>92.80</td><td rowspan=1 colspan=1>92.18</td><td rowspan=1 colspan=1>92.13</td><td rowspan=1 colspan=1>91.04</td><td rowspan=1 colspan=1>91.10</td></tr></table>
+
+# 4.4 KDEF
+
+We run one set of experiments with the pooling methods that includes dropout. Figure 13 shows our network structure for the KDEF experiments:
+
+The input training and test data come from the KDEF dataset. This dataset contains 4,900 images of 35 people displaying seven basic emotions (afraid, angry, disgusted, happy, neutral, sad, and surprised) using facial expressions. They display emotions at five poses (full left and right profiles, half left and right profiles, and straight).
+
+![](images/5313b03d57e276b0ac92f5d863296f151505f94fa4cade18ea3fc474dc79355c.jpg)  
+Figure 13: CNN KDEF Structure Block Diagram
+
+This dataset contains a few errors that we fix (missing or corrupted images, uncropped images, etc.). All of the missing images are at angles of -90, -45, 45, or 90 degrees. We fix the missing and corrupt images by mirroring their counterparts in MATLAB and adding them back to the dataset. We manually crop the images that need to match the dimensions set by the creators $( 7 6 2 \mathrm { ~ x ~ } 5 6 2 )$ . KDEF does not designate a training or test data set. We shuffle the data and separate 3,900 images as training data, and 1,000 images as test data. We resize the images to $1 2 8 \mathrm { x } 1 2 8$ because of memory and time constraints.
+
+The dropout layers regulate the network and maintain stability in spite of some pooling methods known to overfit. Table 6 shows our proposed method has the second highest accuracy. Max pooling eventually overfits, while wavelet pooling resists overfitting. Average and mixed pooling resist overfitting, but are unstable for most of the learning. Stochastic pooling maintains a consistent progression of learning. Wavelet pooling also follows a smoother, consistent progression of learning. Figure 14 shows the energy of each method per epoch.
+
+![](images/537316cbfa6f3889562dfa12841eb7528f683ecceedd841395fefc3334bf1b8a.jpg)  
+Figure 14: KDEF Pooling Method Energy Performance of Training & Validation Sets
+
+Table 6 shows the accuracy of each method:
+
+Table 6: KDEF Performance of Pooling Methods $^ +$ Dropout   
+
+<table><tr><td></td><td>Average</td><td>Max</td><td>Mixed</td><td>Stochastic</td><td>Wavelet</td></tr><tr><td>Accuracy (%)</td><td>76.5</td><td>75.6</td><td>72.6</td><td>72.7</td><td>75.9</td></tr></table>
+
+# 4.5 COMPUTATIONAL COMPLEXITY
+
+Our construction and implementation of wavelet pooling is not efficient. We present this proposed methods as a proof-of-concept, to show its potential and validity, and also to be open to massive improvements. The main area of improvement is computational efficiency. As a proof-of-concept, the code written to implement this method is not at its peak form. Additionally, we did not have the time, space, or resources to optimize the code. We view the accuracy results and novelty as a starting point to spawn improvements, both from our own research as well as other researchers.
+
+We calculate efficiency in terms of mathematical operations (multiplications, additions, logical, etc.) that each method utilizes to complete its algorithm. For max pooling, we calculate operations based on the worst-case scenarios for each neighborhood in finding the maximum value. For average pooling, we calculate the number of additions and division for each neighborhood. Mixed pooling is the mean value of both average and max pooling. We calculate operations for stochastic pooling by counting the number of mathematical operations as well as the random selection of the values based on probability (Roulette Wheel Selection). For wavelet pooling, we calculate the number of operations for each subband at each level, in both decomposition and reconstruction.
+
+Table 7 shows the number of mathematical operations for one image in forward propagation. This table shows that for all methods, average pooling has the least number of computations, followed by mixed pooling, with max pooling not far behind. Stochastic pooling is the least computationally efficient pooling method out of the neighborhood-based methods. It uses about $3 \mathbf { x }$ more mathematical operations than average pooling, the most computationally efficient.
+
+However, wavelet pooling by far is the least computationally efficient method, using 54 to $2 1 3 \mathrm { x }$ more mathematical operations than average pooling. This is partially due to the implementation of the subband coding, which did not implement multidimensional decomposition and reconstruction.
+
+Table 7: Number of Mathematical Operations for Each Method According to Dataset   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>SHVN</td><td rowspan=1 colspan=1>KDEF</td></tr><tr><td rowspan=1 colspan=1>Max</td><td rowspan=1 colspan=1>6.2K</td><td rowspan=1 colspan=1>13K</td><td rowspan=1 colspan=1>26K</td><td rowspan=1 colspan=1>50K</td></tr><tr><td rowspan=1 colspan=1>Avg</td><td rowspan=1 colspan=1>3.5K</td><td rowspan=1 colspan=1>7.4K</td><td rowspan=1 colspan=1>15K</td><td rowspan=1 colspan=1>29K</td></tr><tr><td rowspan=1 colspan=1>Mix</td><td rowspan=1 colspan=1>4.8K</td><td rowspan=1 colspan=1>10K</td><td rowspan=1 colspan=1>20K</td><td rowspan=1 colspan=1>40K</td></tr><tr><td rowspan=1 colspan=1>Stoch</td><td rowspan=1 colspan=1>10.6K</td><td rowspan=1 colspan=1>22K</td><td rowspan=1 colspan=1>45K</td><td rowspan=1 colspan=1>86K</td></tr><tr><td rowspan=1 colspan=1>Wav</td><td rowspan=1 colspan=1>110K</td><td rowspan=1 colspan=1>405K</td><td rowspan=1 colspan=1>810K</td><td rowspan=1 colspan=1>6.2M</td></tr></table>
+
+Nonetheless, by implementing our method through good coding practices (vectorization, architecture, etc.), GPUs, and an improved FTW algorithm, this method can prove to be a viable option. There exists a few improvements to the FTW algorithm that utilize multidimensional wavelets (Karlsson & Vetterli, 1988; Weeks & Bayoumi, 1998), lifting (Valens, 1999), parallelization Holmstrom (1995), as well as other methods that boast of improving the efficiency in speed and memory ¨ (Oliver & Malumbres, 2008; Khoromskij & Miao, 2014; Kopenkov, 2008)
+
+# 5 CONCLUSION
+
+We prove wavelet pooling has potential to equal or eclipse some of the traditional methods currently utilized in CNNs. Our proposed method outperforms all others in the MNIST dataset, outperforms all but one in the CIFAR-10 and KDEF datasets, and performs within respectable ranges of the pooling methods that outdo it in the SHVN dataset. The addition of dropout and batch normalization show our proposed methods response to network regularization. Like the non-dropout cases, it outperforms all but one in both the CIFAR- $1 0 ~ \&$ KDEF datasets, and performs within respectable ranges of the pooling methods that outdo it in the SHVN dataset. Our results confirm previous studies proving that no one pooling method is superior, but some perform better than others depending on the dataset and network structure Boureau et al. (2010); Lee et al. (2016). Furthermore, many networks alternate between different pooling methods to maximize the effectiveness of each method.
+
+Future work and improvements in this area could be to vary the wavelet basis to explore which basis performs best for the pooling. Altering the upsampling and downsampling factors in the decomposition and reconstruction can lead to better image feature reductions outside of the $2 \mathbf { x } 2$ scale. Retention of the subbands we discard for the backpropagation could lead to higher accuracies and fewer errors. Improving the method of FTW we use could greatly increase computational efficiency. Finally, analyzing the structural similarity (SSIM) of wavelet pooling versus other methods could further prove the vitality of using our approach.
+
+# ACKNOWLEDGMENTS
+
+This research is supported by the Title III HBGI PhD Fellowship grant from the U.S. Department of Education.
+
+# REFERENCES
+
+Leon Bottou. Large–scale machine learning with stochastic gradient descent. In ´ Proceedings of COMPSTAT 2010, pp. 177–186. Springer, 2010.
+
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