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+# DATASET DISTILLATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Model distillation aims to distill the knowledge of a complex model into a simpler one. In this paper, we consider an alternative formulation called dataset distillation: we keep the model fixed and instead attempt to distill the knowledge from a large training dataset into a small one. The idea is to synthesize a small number of data points that do not need to come from the correct data distribution, but will, when given to the learning algorithm as training data, approximate the model trained on the original data. For example, we show that it is possible to compress 60, 000 MNIST training images into just 10 synthetic distilled images (one per class) and achieve close to original performance with only a few steps of gradient descent, given a particular fixed network initialization. We evaluate our method in a wide range of initialization settings and with different learning objectives. Experiments on multiple datasets show the advantage of our approach compared to alternative methods in most settings.
+
+# 1 INTRODUCTION
+
+Hinton et al. (2015) proposed network distillation as a way to transfer the knowledge from an ensemble of many separately-trained networks into a single, typically compact network, performing a type of model compression. In this paper, we are considering a related but orthogonal task: rather than distilling the model, we propose to distill the dataset. Unlike network distillation, we keep the model fixed but encapsulate the knowledge of the entire training dataset, which typically contains thousands to millions of images, into a small number of synthetic training images. In fact, we show that we can go as low as one synthetic image per category, training the same model to reach surprisingly good performance on these synthetic images. For example in Fig. 1a, we compress 60, 000 training images of MNIST digit dataset into only 10 synthetic images (one per class), given a fixed network initialization. Training the standard LENET (LeCun et al., 1998) architecture on these 10 images yields test-time MNIST recognition performance of $9 4 \%$ , compared to $9 9 \%$ for the original task. For networks with unknown random weights, 100 synthetic images train to $8 0 \%$ with a few gradient descent steps. We name our method Dataset Distillation and these images distilled images.
+
+But why is dataset distillation useful? There is the purely scientific question of how much data is really encoded in a given training set and how compressible it is? Moreover, given a few distilled images, we can now “load up" a given network with an entire dataset-worth of knowledge much more efficiently, compared to traditional training that often uses tens of thousands of gradient descent steps.
+
+A key question is whether it is even possible to compress a dataset into a small set of synthetic data samples. For example, is it possible to train an image classification model on synthetic images that are not on the natural image manifold? Conventional wisdom would suggest that the answer is no, as the synthetic training data may not follow the same distribution as the real test data. Yet, in this work, we show that this is indeed possible. We present a new optimization algorithm for synthesizing a small number of synthetic data samples not only capturing much of the original training data but also tailored explicitly for fast model training in only a few gradient steps. To achieve our goal, we first derive the network weights as a differentiable function of our synthetic training data. Given this connection, instead of optimizing the network weights for a particular training objective, we can optimize the pixel values of our distilled images. However, this formulation requires access to the initial network weights of the network. To relax this assumption, we develop a method for generating distilled images for networks with random initializations from a certain distribution. To further boost performance, we propose an iterative version, where we obtain a sequence of distilled images to train a model and each distilled image can be trained with multiple passes. Finally, we study the case of a simple linear model, deriving a lower bound on the size of distilled data required to achieve the same performance as training on the full dataset.
+
+![](images/8d775721ee9ef7c370a5410e69526b54af650a31e01a070c8e83eec5364a17b0.jpg)  
+Figure 1: Dataset Distillation: we distill the knowledge of tens of thousands of images into a few synthetic training images called distilled images. (a): On MNIST, 10 distilled images can train a standard LENET with a particular fixed initialization to $9 4 \%$ test accuracy (compared to $9 9 \%$ when fully trained). On CIFAR10, 100 distilled images can train a deep network with fixed initialization to $5 4 \%$ test accuracy (compared to $8 0 \%$ when fully trained). (b): Using pre-trained networks for SVHN, we can distill the domain difference between two SVHN and MNIST into 100 distilled images. These images can be used to quickly fine-tune networks trained for SVHN to achieve high accuracy on MNIST. (c): Training for a malicious objective, our formulation can be used to create adversarial attack images. If well-optimized networks retrained with these images for one single gradient step, they will catastrophically misclassify a particular targeted class.
+
+We demonstrate that a handful of distilled images can be used to train a model with a fixed initialization to achieve surprisingly high performance. For a network with unknown random weights pre-trained on other tasks, our method can still find distilled images for fast model fine-tuning. We further test our method on a wide range of initialization settings: fixed initialization, random initialization, fixed pre-trained weights, and random pre-trained weights, as well as two training objectives: image classification and malicious dataset poisoning attack. Extensive experiments on four publicly available datasets, MNIST (LeCun, 1998), CIFAR10 (Krizhevsky & Hinton, 2009), PASCAL-VOC (Everingham et al., 2010) and CUB-200 (Wah et al., 2011), show that our method often performs better than alternative methods and existing baselines. Our code and models will be available upon publication.
+
+# 2 RELATED WORK
+
+Knowledge distillation. The main inspiration for this paper is network distillation (Hinton et al., 2015), a widely used technique in ensemble learning (Radosavovic et al., 2018) and model compression (Ba & Caruana, 2014; Romero et al., 2015; Howard et al., 2017). While network distillation aims to distill the knowledge of multiple networks into a single model, our goal is to compress the knowledge of an entire dataset into a few synthetic training images. Our method is also related to the theoretical concept of teaching dimension, which specifies the size of dataset necessary to teach a target model (oracle) to a learner (Goldman & Kearns, 1995; Shinohara & Miyano, 1991). While these methods do not enforce the training data to be real, they need the existence of oracle models, which our method does not require.
+
+Dataset pruning, core-set construction, and instance selection. Another way to distill knowledge is to summarize the entire dataset by a small subset, either by only using the “valuable” data for model training (Angelova et al., 2005; Lapedriza et al., 2013; Felzenszwalb et al., 2010) or by only labeling the “valuable” data via active learning (Cohn et al., 1996; Tong & Koller, 2001). Similarly, core-set construction (Bachem et al., 2017; Tsang et al., 2005; Har-Peled & Kushal, 2007; Sener & Savarese, 2018) and instance selection (Olvera-López et al., 2010) methods aim to select a subset of the entire training data, such that models trained on the subset will perform as closely well as possible to the model trained on full dataset for faster training time. For example, solutions to many classical linear learning algorithms, e.g., Perceptron (Rosenblatt, 1957) and support vector machine (SVMs) (Hearst et al., 1998), are weighted sums of a subset of training examples, which can be viewed as core-sets. However, algorithms constructing these subsets require many more training examples per category than we do, in part because their “valuable” images have to be real, whereas our distilled images are exempt from this constraint.
+
+Gradient-based hyperparameter optimization. Our work bears similarity with the gradient-based hyperparameter optimization techniques, which compute the gradient of hyperparameter w.r.t. the final validation loss by reversing the entire training procedure (Bengio, 2000; Domke, 2012; Pedregosa, 2016; Maclaurin et al., 2015). We also backpropagate errors through optimization steps. However, we use only training set data and focus much more heavily on learning synthetic training data rather than tuning hyperparameters. To our knowledge, this direction has only been slightly touched on previously (Maclaurin et al., 2015). We explore it in much greater depth and demonstrate the idea of dataset distillation through various settings. More crucially, our distilled images can work well across random initialization weights, which cannot be achieved by any prior work.
+
+Understanding datasets. Researchers have presented various approaches for understanding and visualizing learned models (Zeiler & Fergus, 2014; Zhou et al., 2015; Mahendran & Vedaldi, 2015; Bau et al., 2017; Koh & Liang, 2017). Unlike these approaches, we are interested in understanding the intrinsic properties of the training data rather than a specific trained model. Analyzing training datasets has, in the past, been mainly focused on the investigation of bias in datasets (Ponce et al., 2006; Torralba & Efros, 2011). For example, Torralba & Efros (2011) proposed to quantify the “value” of dataset samples using cross-dataset generalization. Our method offers a new perspective for understanding datasets by distilling full datasets into few synthetic samples.
+
+# 3 APPROACH
+
+Given a model and a dataset, we aim to obtain a new, much-reduced synthetic dataset which performs almost as well as the original dataset. We first present our main optimization algorithm for training a network with a fixed initialization with one gradient descent (GD) step (Sec. 3.1). In Sec. 3.2, we derive the resolution to a more challenging case, where the initial weight is random rather than fixed. We also discuss the initial weights distribution where our method can work well. Furthermore, we study a linear network case to help the readers understand both the solution and limits of our method in Sec. 3.3. In Sec. 3.4, we extend our approach to more than one gradient descent steps and more than one passes. Finally, Sec. 3.5 and Sec. 3.6 demonstrate how to obtain distilled images with different initialization distributions and learning objectives.
+
+Consider a training dataset $\mathbf { x } = \{ x _ { i } \} _ { i = 1 } ^ { N }$ . We parameterize our neural network as $\theta$ and denote $\ell ( x _ { i } , \theta )$ as the loss function that represents the loss of this network on a data point $x _ { i }$ . Our task is to find the minimizer of the empirical error over the entire training data:
+
+$$
+\theta ^ { * } = \underset { \theta } { \arg \operatorname* { m i n } } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \ell ( x _ { i } , \theta ) = \underset { \theta } { \arg \operatorname* { m i n } } \ell ( \mathbf { x } , \theta ) ,
+$$
+
+where for notation simplicity we overload the $\ell ( \cdot )$ notation so that $\ell ( \mathbf { x } , \theta )$ represents the average error of $\theta$ over the entire dataset $\mathbf { x } = \{ x _ { i } \} _ { i = 1 } ^ { N }$ . We make the mild assumption that $\ell$ is twice-differentiable, which holds for the majority of modern machine learning models (e.g., most neural networks) and tasks.
+
+# Algorithm 1 Dataset Distillation
+
+Input: $p ( \theta _ { 0 } )$ : distribution of initial weights; $M$ : the number of distilled data   
+Input: $\alpha$ : step size; $n$ : batch size; $T$ : the number of optimization iterations; $\tilde { \eta } _ { 0 }$ : initial value for $\tilde { \eta }$   
+1: Initialize $\hat { \tilde { \mathbf { x } } } = \{ \tilde { x } _ { i } \} _ { i = 1 } ^ { M }$ randomly, $\tilde { \eta }  \tilde { \eta } _ { 0 }$   
+2: for each training step $t = 1$ to $T$ do   
+3: Get a minibatch of real data $\mathbf { x } _ { t } = \{ x _ { t , j } \} _ { j = 1 } ^ { n }$   
+4: Sample a batch of initial weights $\theta _ { 0 } ^ { ( j ) } \sim p ( \theta _ { 0 } )$   
+5: for each sampled $\theta _ { 0 } ^ { ( j ) }$ do   
+6: Compute updated parameter with GD: $\theta _ { 1 } ^ { ( j ) } = \theta _ { 0 } ^ { ( j ) } - \tilde { \eta } \nabla _ { \theta _ { 0 } ^ { ( j ) } } \ell ( \tilde { \mathbf { x } } , \theta _ { 0 } ^ { ( j ) } )$   
+7: Evaluate the objective function on real data: $\mathscr { L } ^ { ( j ) } = \ell ( \mathbf { x } _ { t } , \boldsymbol { \theta } _ { 1 } ^ { ( j ) } )$   
+8: end for   
+9: Update $\begin{array} { r } { \tilde { \mathbf { x } }  \tilde { \mathbf { x } } - \alpha \nabla _ { \tilde { \mathbf { x } } } \sum _ { j } \mathcal { L } ^ { ( j ) } } \end{array}$ , and $\begin{array} { r } { \tilde { \eta }  \tilde { \eta } - \alpha \nabla _ { \tilde { \eta } } \sum _ { j } \mathcal { L } ^ { ( j ) } } \end{array}$
+
+10: end for Output: distilled data $\tilde { \bf x }$ and the optimized learning rate $\tilde { \eta }$
+
+# 3.1 OPTIMIZING DISTILLED DATA
+
+Standard training usually applies minibatch stochastic gradient descent (SGD) or its variants. At each step $t$ , we sample a minibatch of training data $\mathbf { x } _ { t } = \{ x _ { t , j } \} _ { j = 1 } ^ { n }$ and update the current parameters as
+
+$$
+\theta _ { t + 1 } = \theta _ { t } - \eta \nabla _ { \theta _ { t } } \ell ( \mathbf { x } _ { t } , \theta _ { t } ) ,
+$$
+
+where $\eta$ is the learning rate. Such a training process often takes tens of thousands or even millions of above update steps to converge. Instead, we aim to learn a tiny set of synthetic distilled training data $\tilde { \mathbf { x } } = \{ \tilde { x } _ { i } \} _ { i = 1 } ^ { M }$ with $M \ll N$ and a corresponding learning rate $\tilde { \eta }$ so that a single GD step like
+
+$$
+\theta _ { 1 } = \theta _ { 0 } - \tilde { \eta } \nabla _ { \theta _ { 0 } } \ell ( \tilde { \mathbf { x } } , \theta _ { 0 } )
+$$
+
+using these learned synthetic data $\tilde { \mathbf { x } }$ greatly boosts performance on the real training dataset.
+
+Given an initialization $\theta _ { 0 }$ , we obtain these synthetic data and $\tilde { \eta }$ that minimize the objective below $\mathcal { L }$ :
+
+$$
+\tilde { \mathbf { x } } ^ { * } , \tilde { \eta } ^ { * } = \mathop { \mathrm { a r g } \operatorname* { m i n } } _ { \tilde { \mathbf { x } } , \tilde { \eta } } \mathcal { L } \big ( \tilde { \mathbf { x } } , \tilde { \eta } ; \theta _ { 0 } \big ) = \mathop { \mathrm { a r g } \operatorname* { m i n } } _ { \tilde { \mathbf { x } } , \tilde { \eta } } \ell ( \mathbf { x } , \theta _ { 1 } ) = \mathop { \mathrm { a r g } \operatorname* { m i n } } _ { \tilde { \mathbf { x } } , \tilde { \eta } } \ell \big ( \mathbf { x } , \theta _ { 0 } - \tilde { \eta } \nabla _ { \theta _ { 0 } } \ell \big ( \tilde { \mathbf { x } } , \theta _ { 0 } \big ) \big ) ,
+$$
+
+where we derive the new weights $\theta _ { 1 }$ as a function of distilled images $\tilde { \mathbf { x } }$ and learning rate $\tilde { \eta }$ using Eqn. 2 and then evaluate the new weights over all the training images $\mathbf { x }$ . Note that the loss $\mathcal { L } ( \tilde { \mathbf { x } } , \tilde { \eta } ; \theta _ { 0 } )$ is differentiable w.r.t. $\tilde { \mathbf { x } }$ and $\tilde { \eta }$ , and can thus be optimized using standard gradient-based algorithms. In many classification tasks, the data $\mathbf { x }$ may contain discrete parts, e.g., the class labels in data-label pairs. For such cases, we fix the discrete parts rather than learn them.
+
+# 3.2 DISTILLATION FOR RANDOM INITIALIZATIONS
+
+Unfortunately, the above distilled data optimized for a given initialization do not generalize well to other initialization weights. The distilled data often look like random noise (e.g., in Fig. 2a) as it encodes the information of both training dataset $\mathbf { x }$ and a particular network initialization $\theta _ { 0 }$ . To address the above issue, we turn to calculate a small number of distilled data that can work for networks with random initializations from a specific distribution. We formulate the optimization problem as follows:
+
+$$
+\tilde { \mathbf { x } } ^ { * } , \tilde { \eta } ^ { * } = \arg \operatorname* { m i n } _ { \tilde { \mathbf { x } } , \tilde { \eta } } \mathbb { E } _ { \theta _ { 0 } \sim p ( \theta _ { 0 } ) } \mathcal { L } ( \tilde { \mathbf { x } } , \tilde { \eta } ; \theta _ { 0 } ) ,
+$$
+
+where $\theta _ { 0 }$ is a randomly sampled network initialization from the distribution $p ( \theta _ { 0 } )$ . Algorithm 1 illustrates our main method. During optimization, the distilled data are optimized to work well for multiple networks whose initial weights are sampled from $p ( \theta _ { 0 } )$ . In practice, we observe that the final distilled data generalize well to the unseen initializations. Besides, these distilled images usually look quite informative, encoding the discriminative features of each category (Fig. 3).
+
+For distilled data to be properly learned, it turns out to be crucial for $\ell ( { \mathbf { x } } , \cdot )$ to share similar local conditions (e.g., output values, gradient magnitudes) over $\theta _ { 0 }$ sampled from $p ( \theta _ { 0 } )$ . In the next section, we derive a lower bound on the number of distilled data needed for a simple model with arbitrary initial $\theta _ { 0 }$ , and discuss its implications on choosing $p ( \theta _ { 0 } )$ .
+
+# 3.3 ANALYSIS OF A SIMPLE LINEAR CASE WITH QUADRATIC LOSS
+
+This section studies our formulation in a simple linear regression case. We derive the lower bound of the number of distilled images needed to achieve the same performance as training on full dataset for arbitrary initialization with one GD step. Consider a dataset $\mathbf { x }$ containing $N$ data-target pairs $\{ ( d _ { i } , t _ { i } ) \} _ { i = 1 } ^ { N }$ , where $d _ { i } \in \mathbb { R } ^ { D }$ and $t _ { i } \in \mathbb { R }$ , which we represent as two matrices: an $N \times D$ data matrix $\mathbf { d }$ and an $N \times 1$ target matrix $\mathbf { t }$ . Given the mean squared error and a $D \times 1$ weight matrix $\theta$ , we have
+
+$$
+\ell ( { \mathbf x } , \theta ) = \ell ( ( { \mathbf d } , { \mathbf t } ) , \theta ) = \frac { 1 } { 2 N } \| { \mathbf d } \theta - { \mathbf t } \| ^ { 2 } .
+$$
+
+We aim to learn $M$ synthetic data-target pairs $\tilde { \mathbf { x } } = ( \tilde { \mathbf { d } } , \tilde { \mathbf { t } } )$ , where $\tilde { \mathbf { d } }$ is an $M \times D$ matrix, $\tilde { \mathbf { t } }$ an $M \times 1$ matrix $M \ll N$ ), and $\tilde { \eta }$ the learning rate, to minimize $\ell ( \mathbf { x } , \theta _ { 0 } - \tilde { \eta } \nabla _ { \theta _ { 0 } } \ell ( \tilde { \mathbf { x } } , \theta _ { 0 } ) )$ . The updated weight matrix after one GD step with these distilled data is
+
+$$
+\theta _ { 1 } = \theta _ { 0 } - \tilde { \eta } \nabla _ { \theta _ { 0 } } \ell ( \tilde { \mathbf { x } } , \theta _ { 0 } ) = \theta _ { 0 } - \frac { \tilde { \eta } } { M } \tilde { \mathbf { d } } ^ { T } ( \tilde { \mathbf { d } } \theta _ { 0 } - \tilde { \mathbf { t } } ) = ( \mathbf { I } - \frac { \tilde { \eta } } { M } \tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { d } } ) \theta _ { 0 } + \frac { \tilde { \eta } } { M } \tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { t } } .
+$$
+
+Note that for such quadratic loss, there always exists some learned distilled data $\tilde { \mathbf { x } }$ allowing us to achieve the same performance as training on full dataset $\mathbf { x }$ (i.e., attaining the global minimum) for any initialization $\theta _ { 0 }$ .∗ But how small can $M$ , the size of distilled data, be? For such models, the global minimum is attained at any $\theta ^ { * }$ satisfying $\mathbf { d } ^ { T } \mathbf { d } \theta ^ { * } = \mathbf { d } ^ { T } \mathbf { t }$ . Substituting Eqn. (6) in, we have
+
+$$
+\mathbf { d } ^ { T } \mathbf { d } ( \mathbf { I } - \frac { \tilde { \eta } } { M } \tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { d } } ) \theta _ { 0 } + \frac { \tilde { \eta } } { M } \mathbf { d } ^ { T } \mathbf { d } \tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { t } } = \mathbf { d } ^ { T } \mathbf { t } .
+$$
+
+Here we make the mild assumption that the feature columns of the data matrix $\mathbf { d }$ are independent (i.e., ${ \bf d } ^ { T } { \bf d }$ has full rank). For a $\bar { \bf x } = ( \tilde { \bf d } , \tilde { \bf t } )$ to satisfy the above equation for any $\theta _ { 0 }$ , we must have
+
+$$
+\mathbf { I } - \frac { \widetilde { \eta } } { M } \widetilde { \mathbf { d } } ^ { T } \widetilde { \mathbf { d } } = \mathbf { 0 } ,
+$$
+
+which implies that $\tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { d } }$ has full rank and $M \geq D$
+
+Discussion. The analysis considers only a simple case but suggests that any small number of distilled data fails to generalize to arbitrary starting $\theta _ { 0 }$ . This is intuitively expected as the optimization target $\ell ( { \bf x } , \theta _ { 1 } ) = \ell ( { \bf x } , \theta _ { 0 } - \tilde { \eta } \nabla _ { \boldsymbol { \theta } _ { 0 } } \ell ( \tilde { \bf x } , \tilde { \theta _ { 0 } } ) )$ depends on the local behavior of $\ell ( { \mathbf { x } } , \cdot )$ around $\theta _ { 0 }$ , which can be drastically different across various $\theta _ { 0 }$ values. We note that the lower bound $M \geq D$ is a quite restricting one, considering that real datasets often have thousands to even hundreds of thousands of dimensions (e.g., image classification). This analysis motivates us to focus on $p ( \theta _ { 0 } )$ distributions that yield similar local conditions over the support. Sec. 3.5 discusses several practical choices explored in this paper. Additionally, to address the limitation of using a single GD step, we extend our method to multiple GD steps in the next section. In Sec. 4.1, we empirically verify that using multiple steps is much more effective than using just one on deep convolutional networks, with the total amount of distilled data fixed.
+
+# 3.4 MULTIPLE GRADIENT DESCENT STEPS AND MULTIPLE EPOCHS
+
+We can extend Algorithm 1 to more than one gradient descent steps by changing Line 6 to multiple sequential GD steps each on a different batch of distilled data and learning rate, i.e., each step $i$ is
+
+$$
+\theta _ { i + 1 } = \theta _ { i } - \tilde { \eta } _ { i } \nabla _ { \theta _ { i } } \ell \big ( \tilde { \mathbf { x } } _ { i } , \theta _ { i } \big ) ,
+$$
+
+and changing Line 9 to backpropagate through all steps. However, naively computing gradients is both memory-intensive and computationally-expensive. Therefore, we exploit a recent technique called back-gradient optimization, which allows for significantly faster gradient calculation of such updates in reverse-mode differentiation (i.e., backpropagation). Specifically, back-gradient optimization formulates the necessary second order terms into efficient Hessian-vector products (Pearlmutter, 1994), which can be easily calculated with modern automatic differentiation systems such as PyTorch (Paszke et al., 2017). For further algorithm details in this aspect, we refer readers to prior work (Domke, 2012; Maclaurin et al., 2015).
+
+Multiple epochs. To further improve the performance, we can train the network with the same distilled images for multiple epochs (passes) of the GD step(s). In particular, we tie the image pixels for the same distilled images used in different epochs. In other words, for each epoch, our method cycles through all GD steps, where each step is associated with a different batch of distilled data. We do not tie the trained learning rates across epochs as later epochs often use smaller learning rates.
+
+# 3.5 DISTILLATION WITH DIFFERENT INITIALIZATIONS
+
+Inspired by the analysis of the simple linear case in Sec. 3.3, we aim to focus on initial weights distributions $p ( \theta )$ that yield similar local conditions over the support. In this work, we focus on the following four practical choices:
+
+• Random initialization: Distribution over model weights initialized using methods that attempts to ensure gradient flow of constant magnitude, e.g., He Initialization (He et al., 2015) and Xavier Initialization (Glorot & Bengio, 2010) for convolutional neural networks (CNNs). • Fixed initialization: A fixed initial weights sampled using the method above. • Random pre-trained weights: Distribution over models pre-trained on other tasks and datasets, e.g., pre-trained ALEXNET (Krizhevsky et al., 2012) networks for ImageNet classification (Deng et al., 2009). Each network is pre-trained on the same task, but with different initializations. • Fixed pre-trained weights: A fixed model weights pre-trained on other tasks and datasets.
+
+Distillation for pre-trained weights. Such learned distilled data essentially fine-tunes weights pre-trained on one task to perform well for a new task, thus bridging the gap between two domains. Domain mismatch and dataset bias represent a challenging problem in machine learning today (Torralba & Efros, 2011). Extensive prior work has been proposed to adapt models to new tasks and datasets (Daume III, 2007; Saenko et al., 2010). In this work, we characterize the domain mismatch via distilled data. In Sec. 4.2, we show that a very small number of distilled images are sufficient to quickly adapt CNN models to new classification tasks.
+
+# 3.6 DISTILLATION WITH DIFFERENT OBJECTIVES
+
+Previous sections show that we can train distilled data to minimize the loss of the distilled task $\ell ( \mathbf { x } , \theta _ { 1 } )$ defined on the final updated weights $\theta _ { 1 }$ (Line 7 in Algorithm 1). Distilled images trained with different final learning objectives can train models to exhibit different desired behaviours. We have already mentioned image classification as one of the applications, where distilled images help train accurate classifiers. Below, we introduce a quite different training objective to further demonstrate the flexibility of our method.
+
+Distillation for a malicious data-poisoning objective. For example, our approach can be used to construct a new form of data poisoning attack. To illustrate this idea, we consider the following scenario. When a single GD step is applied with our synthetic adversarial data, a well-behaved image classifier catastrophically forgets a category but still maintains high performance on other categories.
+
+Formally, given an attacked category $K$ and a target category $T$ , we want the classifier to misclassify images from category $K$ to category $T$ . To achieve this, we consider a new final objective function $\ell _ { K  T } ( { \bf x } , \theta _ { 1 } )$ , which is a classification loss encouraging $\theta _ { 1 }$ to classify category $K$ images mistakenly as category $T$ while correctly predicting other images, e.g., a cross entropy loss with target labels of $K$ modified to $T$ . Then, the attacking distilled images can be obtained via optimizing
+
+$$
+\tilde { \mathbf { x } } ^ { * } , \tilde { \eta } ^ { * } = \underset { \tilde { \mathbf { x } } , \tilde { \eta } } { \arg \operatorname* { m i n } } \mathbb { E } _ { \theta _ { 0 } \sim p ( \theta _ { 0 } ) } \mathcal { L } _ { K  T } ( \tilde { \mathbf { x } } , \tilde { \eta } ; \theta _ { 0 } ) = \underset { \tilde { \mathbf { x } } , \tilde { \eta } } { \arg \operatorname* { m i n } } \mathbb { E } _ { \theta _ { 0 } \sim p ( \theta _ { 0 } ) } \ell _ { K  T } ( \mathbf { x } , \theta _ { 1 } ) ,
+$$
+
+ere $p ( \theta _ { 0 } )$ is the distribution over random pre-trained weights of well-optimized classifiers.
+
+Compared to prior data poisoning attacks (Biggio et al., 2012; Li et al., 2016; Muñoz-González et al., 2017; Koh & Liang, 2017), our approach crucially does not require the poisoned training data to be stored and trained on repeatedly. Instead, our method attacks the model training just in one iteration and with only a few data. This advantage makes our method effective for many online training algorithms and useful for the case where malicious users hijack the data feeding pipeline for only one gradient step (e.g., one network transmission). In Sec. 4.2, we show that a single batch of distilled data applied in one step can successfully attack well-optimized neural network models. This setting can be viewed as distilling dataset knowledge of a specific category into data.
+
+![](images/d9cc87466ca723438e80ecf7575814ebb58542895a77db8bf2c894b21e9c8d0b.jpg)
+
+(a) MNIST. These images train networks with a partic- (b) CIFAR10. These images train networks with a particular initialization from $\mathrm { 1 2 . 9 \% }$ test accuracy to $9 3 . 7 6 \%$ . ular initialization from $8 . 8 2 \%$ test accuracy to $5 4 . 0 3 \%$ .
+
+Figure 2: Distilled images trained for fixed initialization. MNIST distilled images use 1 GD step and 3 epochs (10 images in total). CIFAR10 distilled images use $1 0 ~ \mathrm { G D }$ steps and 3 epochs (100 images in total). For CIFAR10, only selected steps are shown. At left, we report the corresponding learning rates for all 3 epochs.
+
+![](images/6853e2a4c8a59b193aa8a6a786f362e4dcce1afc7a0786b4ecebdf0b08ac5450.jpg)
+
+![](images/b26b5fdf32d9c06bdc2cc0d9131a184e5bae06630f386df4201bf857ed822cbd.jpg)  
+(b) CIFAR10. These images train networks with unknown initialization to $3 6 . \bar { 7 } 9 \% \pm 1 . 1 8 \%$ test accuracy.
+
+(a) MNIST. These images train networks with unknown initialization to $7 9 . 5 0 \% \pm 8 . 0 8 \%$ test accuracy.
+
+Figure 3: Distilled images trained for random initialization with $1 0 \mathrm { G D }$ steps and 3 epochs. We show images from selected GD steps and corresponding trained learning rates for all 3 epochs.
+
+# 4 EXPERIMENTS
+
+We report image classification results on MNIST (LeCun, 1998) and CIFAR10 (Krizhevsky & Hinton, 2009). For MNIST, distilled images are trained with LENET (LeCun et al., 1998), which achieves about $9 9 \%$ test accuracy if fully trained. For CIFAR10, we use a network architecture following Krizhevsky (2012) which achieves around $8 0 \%$ test accuracy if fully trained. For random initializations and random pre-trained weights, we report means and standard deviations on 200 held-out models, unless otherwise specified.
+
+Baselines. For each experiment, in addition to baselines specific to the setting, we generally compare our method against baselines trained with data derived or selected from real images:
+
+• Random real images: We randomly sample the same number of real training images per category.   
+• Optimized real images: We sample sets of real images as above, and choose on the top $2 0 \%$ sets that perform the best training images.   
+• $k$ -means: For each category, we use $k$ -means to extract the same number of cluster centroids as the number of distilled images in our method.   
+• Average real images: We compute the average image of all the images in each category, which is reused in different GD steps.
+
+For these baselines, we perform each evaluation on 200 hold-out models with all combinations of learning rate $\in$ {learned learning rate with our method, $0 . 0 0 1 , 0 . 0 0 3 , 0 . 0 1 , 0 . 0 3 , 0 . 1 , 0 . 3 \}$ and #epochs $\in \{ 1 , 3 , 5 \}$ . We report results from the best performing combination. We run all the experiments on NVIDIA Titan $\mathrm { X p }$ and V100 GPUs. We use one GPU for fixed initial weights and four GPUs for random initial weights. Each training typically takes 1 to 4 hours. Please see supplemental material Sec. S-6.1 for more training and baseline details.
+
+# 4.1 DATASET DISTILLATION
+
+Fixed initialization. With access to initial network weights, distilled images can directly train a particular network to reach high performance. For example, 10 learned distilled images can boost the test accuracy of a neural network with an initial accuracy $1 2 . 9 0 \%$ to the final accuracy $9 3 . 7 6 \%$ on MNIST (Fig. 2a). Similarly, 100 images can train a network with an initial accuracy $8 . 8 2 \%$ to $5 4 . 0 3 \%$ test accuracy on CIFAR10 (Fig. 2b). This result suggests that even only a few distilled images have enough capacity to distill part of the dataset.
+
+Random initialization. Trained with randomly sampled initializations using Xavier initialization (Glorot & Bengio, 2010), the learned distilled images do not need to encode information tailored for a particular starting point and thus can represent meaningful content independent of network initializations. In Fig. 3, we see that such distilled images reveal discriminative features of the corresponding categories: e.g., the ship image in Fig. 3b. These 100 images can train randomly
+
+![](images/dfb4142a5480849c2ef6e92464747a0dbf26b2a4b8d991ba56a7eb884d064837.jpg)  
+Figure 4: Hyperparameter sensitivity studies on random initialization: (a) average test accuracy w.r.t. the number of gradient descent steps. The number of epochs is fixed to be 2. (b) average test accuracy w.r.t. the number of epochs. The number of steps is fixed to be 10, with each containing 10 images (one per category).
+
+![](images/484484dd4e342abf8dc288bf3f64c564707ea50cd7f4c6e03a5eea5e7bc0e14a.jpg)
+
+![](images/62303a7aea0dbd3b53ede2ff574250f1bf1d9af29a87b55eea1d7227727ad208.jpg)  
+Figure 5: Comparison between applying the same number of images in one versus multiple GD steps on random initialization, with the number of epochs fixed to 1. $N$ denotes the total number of images per category. For multiple steps runs, each of the $N$ steps applies one image per category.
+
+<table><tr><td rowspan=3 colspan=1></td><td rowspan=1 colspan=2>Ours</td><td rowspan=1 colspan=6>Baselines</td></tr><tr><td rowspan=2 colspan=1>Fixed init.</td><td rowspan=2 colspan=1>Random init.</td><td rowspan=1 colspan=4>Used as training data in same number of GD steps</td><td rowspan=1 colspan=2>Used as data for K-NN classification</td></tr><tr><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>k-means</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>k-means</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>96.6%</td><td rowspan=1 colspan=1>79.5% ± 8.1%</td><td rowspan=1 colspan=1>68.6% ±9.8%</td><td rowspan=1 colspan=1>73.0% ± 7.6%</td><td rowspan=1 colspan=1>76.4% ± 9.5%</td><td rowspan=1 colspan=1>77.1% ± 2.7%</td><td rowspan=1 colspan=1>71.5% ± 2.1%</td><td rowspan=1 colspan=1>92.2%±0.1%</td></tr><tr><td rowspan=1 colspan=1>CIFAR10</td><td rowspan=1 colspan=1>54.0%</td><td rowspan=1 colspan=1>36.8%±1.2%</td><td rowspan=1 colspan=1>21.3% ± 1.5%</td><td rowspan=1 colspan=1>23.4% ± 1.3%</td><td rowspan=1 colspan=1>22.5%±3.1%</td><td rowspan=1 colspan=1>22.3%±0.7%</td><td rowspan=1 colspan=1>18.8% ± 1.3%</td><td rowspan=1 colspan=1>29.4% ±0.3%</td></tr></table>
+
+Table 1: Comparison between our method trained for $1 0 \mathrm { \ G D }$ steps and 3 epochs and various baselines. For baselines using K-Nearest Neighbor (K-NN), best result among all combinations of distance metric $\in \{ l _ { 1 } , l _ { 2 } \}$ and $\mathtt { K } \in \{ 1 , 3 \}$ is reported. In K-NN and $k$ -means, K and $k$ can have different values. All methods use 10 images per class, except for the average real images baseline, which reuses the same images in different GD steps.
+
+initialized networks to $3 6 . 7 9 \%$ average test accuracy on CIFAR10. Similarly, for MNIST, the 100 distilled images shown in Fig. 3a can train randomly initialized networks to $\dot { 7 } 9 . 5 0 \%$ test accuracy.
+
+Multiple gradient descent steps and multiple epochs. In Fig. 3, we learn distilled images for 10 GD steps applied in 3 epochs, leading to a total of 100 images (with each step containing one image per category). In each epoch, these 10 steps are sequentially applied once. The early steps tend to look noisier, likely regularizing random weights to point easier for further optimization. In later steps, the images gradually look like real data and share the discriminative features for these categories. Fig. 4a shows that using more steps significantly improves the results. Fig. 4b shows a similar but slower trend as the number of epochs increases. We observe that longer training (i.e., more epochs) can help the model learn all the knowledge from the distilled images, but the performance is eventually limited by the capacity of the images (i.e., the number of total images). Alternatively, we can train the model with one GD step but a big batch size. Sec. 3.3 has shown theoretical limitations of using only one step in a simple linear case. In Fig. 5, we empirically verify that with convolutional networks, using multiple steps drastically outperforms single step method, with the same number of distilled images.
+
+Table 1 compares our method against several baselines. Our method with both fixed and random initialization outperform all the baselines on CIFAR10 and most of the baselines on MNIST.
+
+# 4.2 DISTILLATION FOR DIFFERENT INITIALIZATIONS AND OBJECTIVES
+
+Next, we show two extended settings of our main algorithm discussed in Sec. 3.5 and Sec. 3.6. Both cases assume that the initial weights are random but pre-trained on a different dataset. We train the distilled images on 2000 random pre-trained models, and then apply them on unseen models.
+
+Fixed and random pre-trained weights on digits. As shown in Sec. 3.5, we can optimize distilled images to quickly fine-tune pre-trained models for a new dataset. Table 2 shows that our method is more effective compared to various baseline on adaptation among three digits datasets: MNIST, USPS (Hull, 1994), and SVHN (Netzer et al., 2011). We also compared against a state-of-the-art few-short supervised domain adaptation method (Motiian et al., 2017). Although our method uses the entire training set to compute the distilled images, both methods use the same number of images to distill the knowledge of target dataset. Prior work (Motiian et al., 2017) is outperformed by our method with
+
+![](images/d8a4fe9defd345ba383d265477c6b60afccbd7b85bbe8e608eeab44a1d5ca65a.jpg)  
+Figure 6: Performance for our method and baselines with random pre-trained initialization and a malicious objective. Distilled images are trained for one GD step. For baselines, we use the same numbers of images with incorrect labels and also apply one GD step, and report the result that achieves the highest accuracy w.r.t. the incorrect labels while having $\geq 1 0 \%$ misclassification ratio on the attacked category, to avoid results with learning rates too low to change model behavior at all. (a) Our method slightly outperforms the best baseline in accuracy w.r.t. incorrect labels. (b) Our method performs similarly with some baselines in changing the prediction of the attacked category on MNIST, but is much better than all baselines on CIFAR10.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>.Ourswith fixedpre-trained</td><td rowspan=1 colspan=1>Ourswith randompre-trained</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>k-means</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Domain adaptationMotiian et al.(2017)</td><td rowspan=1 colspan=1>No adaptation</td><td rowspan=1 colspan=1>Train on fulldestinationtraining set</td></tr><tr><td rowspan=1 colspan=1>M→u</td><td rowspan=1 colspan=1>97.9%</td><td rowspan=1 colspan=1>95.4%±1.8%</td><td rowspan=1 colspan=1>94.9%±0.8%</td><td rowspan=1 colspan=1>95.2%±0.7%</td><td rowspan=1 colspan=1>92.2%±1.6%</td><td rowspan=1 colspan=1>93.9%±0.8%</td><td rowspan=1 colspan=1>96.7%±0.5%</td><td rowspan=1 colspan=1>90.4%±3.0%</td><td rowspan=1 colspan=1>97.3%±0.3%</td></tr><tr><td rowspan=1 colspan=1>u→M</td><td rowspan=1 colspan=1>93.2%</td><td rowspan=1 colspan=1>92.7% ± 1.4%</td><td rowspan=1 colspan=1>87.1% ±2.9%</td><td rowspan=1 colspan=1>87.6% ± 2.1%</td><td rowspan=1 colspan=1>85.6% ±3.1%</td><td rowspan=1 colspan=1>78.4% ±5.0%</td><td rowspan=1 colspan=1>89.2% ±2.4%</td><td rowspan=1 colspan=1>67.5%±3.9%</td><td rowspan=1 colspan=1>98.6%±0.5%</td></tr><tr><td rowspan=1 colspan=1>S→M</td><td rowspan=1 colspan=1>96.2%</td><td rowspan=1 colspan=1>85.2%±4.7%</td><td rowspan=1 colspan=1>84.6%±2.1%</td><td rowspan=1 colspan=1>85.2%±1.2%</td><td rowspan=1 colspan=1>85.8%±1.2%</td><td rowspan=1 colspan=1>74.9%± 2.6%</td><td rowspan=1 colspan=1>74.0%± 1.5%</td><td rowspan=1 colspan=1>51.6%±2.8%</td><td rowspan=1 colspan=1>98.6%±0.5%</td></tr></table>
+
+Table 2: Performance of our method and baselines in adapting models among MNIST $( \mathcal { M } )$ , USPS $( \mathcal { U } )$ , and SVHN $( S )$ . 100 distilled images are trained for $1 0 \mathrm { G D }$ steps and 3 epochs. Few-shot domain adaptation method by Motiian et al. (2017) and baselines use the same numbers image per class.
+
+Table 3: Performance of our method and baselines in adapting an ALEXNET pre-trained on ImageNet to PASCAL-VOC and CUB-200. Only one distilled image per class are trained to be applied in 1 GD step repeated for 3 epochs. Our method significantly outperforms the baselines. Results are collected over 10 runs.   
+
+<table><tr><td rowspan=1 colspan=1>Destination dataset</td><td rowspan=1 colspan=1>Ours</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Fine-tune on fulldestination training set</td></tr><tr><td rowspan=1 colspan=1>PASCAL-VOC</td><td rowspan=1 colspan=1>70.75%</td><td rowspan=1 colspan=1>19.41%± 3.73%</td><td rowspan=1 colspan=1>23.82%±3.66%</td><td rowspan=1 colspan=1>9.94%</td><td rowspan=1 colspan=1>75.57%±0.18%</td></tr><tr><td rowspan=1 colspan=1>CUB-200</td><td rowspan=1 colspan=1>38.76%</td><td rowspan=1 colspan=1>7.11%±0.66%</td><td rowspan=1 colspan=1>7.23%±0.78%</td><td rowspan=1 colspan=1>2.88%</td><td rowspan=1 colspan=1>41.21%±0.51%</td></tr></table>
+
+fixed pre-trained weights on all the tasks, and by our method with random pre-trained weights on two of the three tasks. This result shows that our distilled images indeed convey compressed information of the full dataset.
+
+Fixed pre-trained weights on ImageNet. In Table 3, we adapt a widely-used ALEXNET model (Krizhevsky, 2014) pre-trained on ImageNet (Deng et al., 2009) to perform image classification on PASCAL-VOC (Everingham et al., 2010) and CUB-200 (Wah et al., 2011) datasets. Using only 1 distilled image per category, our method outperforms the baselines significantly. Our result is also comparable to the accuracy of fine-tuning on the full datasets which contain thousands of images.
+
+Random Pre-trained weights and a malicious data-poisoning objective. Sec. 3.6 shows that our method can construct a new type of data poisoning, where the attacker can apply just one GD step with a few malicious data to manipulate a well-trained model. We train distilled images to make well-optimized neural networks to misclassify a particular attacked category as another target category within only one GD step. Our method requires no access to the exact weights of the model. In Fig. 6b, we evaluate our method on 200 held-out models, against various baselines using data derived from real images and incorrect labels. While some baselines perform similarly well as our method on MNIST, our method significantly outperforms all the baselines on CIFAR10.
+
+# 5 DISCUSSION
+
+In this paper, we present dataset distillation for compressing the knowledge of entire training data into a few synthetic training images. We can train a network to reach high performance with a small number of distilled images and several gradient descent steps. Finally, we demonstrate two applications including fast domain adaptation and effective data poisoning attack. In the future, we plan to extend our method to compress large-scale visual datasets such as ImageNet (Deng et al., 2009) and other types of data (e.g., audio and text). Also, our current method is sensitive to the initial weights distribution. We would like to investigate more on various initialization strategies, with which distilled images can work well.
+
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+
+# S-6 SUPPLEMENTARY MATERIAL
+
+# S-6.1 EXPERIMENT DETAILS
+
+For the networks used in our experiments, we disable dropout layers due to the randomness and computational cost they introduce in distillation. Moreover, we initialize the distilled learning rates as 0.02 and use Adam optimizer (Kingma & Ba, 2014) with a learning rate of 0.001. For random initialization and random pre-trained weights, we sample 4 to 16 initial weights in each step.
+
+Details of the baselines are listed below.
+
+• Random real images: We randomly sample the same number of real training images per category. 10 such set of sampled images are evaluated. • Optimized real images: We sampled 50 sets of real images using above procedure, and evaluate 10 sets that achieve best performance on 20 held-out models and 1024 training images. • $k$ -means: For each category, we use $k$ -means to extract the same number of cluster centroids as the number of distilled images in our method. 10 such set of sampled images are evaluated. • Average real images: We compute the average image of all the images in each category, which is reused in different GD steps. We evaluate the model only once because average images are deterministic.

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+        "text": "DATASET DISTILLATION ",
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "Model distillation aims to distill the knowledge of a complex model into a simpler one. In this paper, we consider an alternative formulation called dataset distillation: we keep the model fixed and instead attempt to distill the knowledge from a large training dataset into a small one. The idea is to synthesize a small number of data points that do not need to come from the correct data distribution, but will, when given to the learning algorithm as training data, approximate the model trained on the original data. For example, we show that it is possible to compress 60, 000 MNIST training images into just 10 synthetic distilled images (one per class) and achieve close to original performance with only a few steps of gradient descent, given a particular fixed network initialization. We evaluate our method in a wide range of initialization settings and with different learning objectives. Experiments on multiple datasets show the advantage of our approach compared to alternative methods in most settings. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            448,
+            336,
+            463
+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "Hinton et al. (2015) proposed network distillation as a way to transfer the knowledge from an ensemble of many separately-trained networks into a single, typically compact network, performing a type of model compression. In this paper, we are considering a related but orthogonal task: rather than distilling the model, we propose to distill the dataset. Unlike network distillation, we keep the model fixed but encapsulate the knowledge of the entire training dataset, which typically contains thousands to millions of images, into a small number of synthetic training images. In fact, we show that we can go as low as one synthetic image per category, training the same model to reach surprisingly good performance on these synthetic images. For example in Fig. 1a, we compress 60, 000 training images of MNIST digit dataset into only 10 synthetic images (one per class), given a fixed network initialization. Training the standard LENET (LeCun et al., 1998) architecture on these 10 images yields test-time MNIST recognition performance of $9 4 \\%$ , compared to $9 9 \\%$ for the original task. For networks with unknown random weights, 100 synthetic images train to $8 0 \\%$ with a few gradient descent steps. We name our method Dataset Distillation and these images distilled images. ",
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+        "type": "text",
+        "text": "But why is dataset distillation useful? There is the purely scientific question of how much data is really encoded in a given training set and how compressible it is? Moreover, given a few distilled images, we can now “load up\" a given network with an entire dataset-worth of knowledge much more efficiently, compared to traditional training that often uses tens of thousands of gradient descent steps. ",
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+        "text": "A key question is whether it is even possible to compress a dataset into a small set of synthetic data samples. For example, is it possible to train an image classification model on synthetic images that are not on the natural image manifold? Conventional wisdom would suggest that the answer is no, as the synthetic training data may not follow the same distribution as the real test data. Yet, in this work, we show that this is indeed possible. We present a new optimization algorithm for synthesizing a small number of synthetic data samples not only capturing much of the original training data but also tailored explicitly for fast model training in only a few gradient steps. To achieve our goal, we first derive the network weights as a differentiable function of our synthetic training data. Given this connection, instead of optimizing the network weights for a particular training objective, we can optimize the pixel values of our distilled images. However, this formulation requires access to the initial network weights of the network. To relax this assumption, we develop a method for generating distilled images for networks with random initializations from a certain distribution. To further boost performance, we propose an iterative version, where we obtain a sequence of distilled images to train a model and each distilled image can be trained with multiple passes. Finally, we study the case of a simple linear model, deriving a lower bound on the size of distilled data required to achieve the same performance as training on the full dataset. ",
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+        "img_path": "images/8d775721ee9ef7c370a5410e69526b54af650a31e01a070c8e83eec5364a17b0.jpg",
+        "image_caption": [
+            "Figure 1: Dataset Distillation: we distill the knowledge of tens of thousands of images into a few synthetic training images called distilled images. (a): On MNIST, 10 distilled images can train a standard LENET with a particular fixed initialization to $9 4 \\%$ test accuracy (compared to $9 9 \\%$ when fully trained). On CIFAR10, 100 distilled images can train a deep network with fixed initialization to $5 4 \\%$ test accuracy (compared to $8 0 \\%$ when fully trained). (b): Using pre-trained networks for SVHN, we can distill the domain difference between two SVHN and MNIST into 100 distilled images. These images can be used to quickly fine-tune networks trained for SVHN to achieve high accuracy on MNIST. (c): Training for a malicious objective, our formulation can be used to create adversarial attack images. If well-optimized networks retrained with these images for one single gradient step, they will catastrophically misclassify a particular targeted class. "
+        ],
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+        "text": "",
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+        "text": "We demonstrate that a handful of distilled images can be used to train a model with a fixed initialization to achieve surprisingly high performance. For a network with unknown random weights pre-trained on other tasks, our method can still find distilled images for fast model fine-tuning. We further test our method on a wide range of initialization settings: fixed initialization, random initialization, fixed pre-trained weights, and random pre-trained weights, as well as two training objectives: image classification and malicious dataset poisoning attack. Extensive experiments on four publicly available datasets, MNIST (LeCun, 1998), CIFAR10 (Krizhevsky & Hinton, 2009), PASCAL-VOC (Everingham et al., 2010) and CUB-200 (Wah et al., 2011), show that our method often performs better than alternative methods and existing baselines. Our code and models will be available upon publication. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Knowledge distillation. The main inspiration for this paper is network distillation (Hinton et al., 2015), a widely used technique in ensemble learning (Radosavovic et al., 2018) and model compression (Ba & Caruana, 2014; Romero et al., 2015; Howard et al., 2017). While network distillation aims to distill the knowledge of multiple networks into a single model, our goal is to compress the knowledge of an entire dataset into a few synthetic training images. Our method is also related to the theoretical concept of teaching dimension, which specifies the size of dataset necessary to teach a target model (oracle) to a learner (Goldman & Kearns, 1995; Shinohara & Miyano, 1991). While these methods do not enforce the training data to be real, they need the existence of oracle models, which our method does not require. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Dataset pruning, core-set construction, and instance selection. Another way to distill knowledge is to summarize the entire dataset by a small subset, either by only using the “valuable” data for model training (Angelova et al., 2005; Lapedriza et al., 2013; Felzenszwalb et al., 2010) or by only labeling the “valuable” data via active learning (Cohn et al., 1996; Tong & Koller, 2001). Similarly, core-set construction (Bachem et al., 2017; Tsang et al., 2005; Har-Peled & Kushal, 2007; Sener & Savarese, 2018) and instance selection (Olvera-López et al., 2010) methods aim to select a subset of the entire training data, such that models trained on the subset will perform as closely well as possible to the model trained on full dataset for faster training time. For example, solutions to many classical linear learning algorithms, e.g., Perceptron (Rosenblatt, 1957) and support vector machine (SVMs) (Hearst et al., 1998), are weighted sums of a subset of training examples, which can be viewed as core-sets. However, algorithms constructing these subsets require many more training examples per category than we do, in part because their “valuable” images have to be real, whereas our distilled images are exempt from this constraint. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Gradient-based hyperparameter optimization. Our work bears similarity with the gradient-based hyperparameter optimization techniques, which compute the gradient of hyperparameter w.r.t. the final validation loss by reversing the entire training procedure (Bengio, 2000; Domke, 2012; Pedregosa, 2016; Maclaurin et al., 2015). We also backpropagate errors through optimization steps. However, we use only training set data and focus much more heavily on learning synthetic training data rather than tuning hyperparameters. To our knowledge, this direction has only been slightly touched on previously (Maclaurin et al., 2015). We explore it in much greater depth and demonstrate the idea of dataset distillation through various settings. More crucially, our distilled images can work well across random initialization weights, which cannot be achieved by any prior work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Understanding datasets. Researchers have presented various approaches for understanding and visualizing learned models (Zeiler & Fergus, 2014; Zhou et al., 2015; Mahendran & Vedaldi, 2015; Bau et al., 2017; Koh & Liang, 2017). Unlike these approaches, we are interested in understanding the intrinsic properties of the training data rather than a specific trained model. Analyzing training datasets has, in the past, been mainly focused on the investigation of bias in datasets (Ponce et al., 2006; Torralba & Efros, 2011). For example, Torralba & Efros (2011) proposed to quantify the “value” of dataset samples using cross-dataset generalization. Our method offers a new perspective for understanding datasets by distilling full datasets into few synthetic samples. ",
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+    {
+        "type": "text",
+        "text": "3 APPROACH",
+        "text_level": 1,
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+        "text": "Given a model and a dataset, we aim to obtain a new, much-reduced synthetic dataset which performs almost as well as the original dataset. We first present our main optimization algorithm for training a network with a fixed initialization with one gradient descent (GD) step (Sec. 3.1). In Sec. 3.2, we derive the resolution to a more challenging case, where the initial weight is random rather than fixed. We also discuss the initial weights distribution where our method can work well. Furthermore, we study a linear network case to help the readers understand both the solution and limits of our method in Sec. 3.3. In Sec. 3.4, we extend our approach to more than one gradient descent steps and more than one passes. Finally, Sec. 3.5 and Sec. 3.6 demonstrate how to obtain distilled images with different initialization distributions and learning objectives. ",
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+        "text": "Consider a training dataset $\\mathbf { x } = \\{ x _ { i } \\} _ { i = 1 } ^ { N }$ . We parameterize our neural network as $\\theta$ and denote $\\ell ( x _ { i } , \\theta )$ as the loss function that represents the loss of this network on a data point $x _ { i }$ . Our task is to find the minimizer of the empirical error over the entire training data: ",
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+        "type": "equation",
+        "img_path": "images/e010d3489766ead9858144e6388b4eafc7db62a926aef6f69bdbf34a6a751813.jpg",
+        "text": "$$\n\\theta ^ { * } = \\underset { \\theta } { \\arg \\operatorname* { m i n } } \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\ell ( x _ { i } , \\theta ) = \\underset { \\theta } { \\arg \\operatorname* { m i n } } \\ell ( \\mathbf { x } , \\theta ) ,\n$$",
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+        "type": "text",
+        "text": "where for notation simplicity we overload the $\\ell ( \\cdot )$ notation so that $\\ell ( \\mathbf { x } , \\theta )$ represents the average error of $\\theta$ over the entire dataset $\\mathbf { x } = \\{ x _ { i } \\} _ { i = 1 } ^ { N }$ . We make the mild assumption that $\\ell$ is twice-differentiable, which holds for the majority of modern machine learning models (e.g., most neural networks) and tasks. ",
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+        "type": "text",
+        "text": "Algorithm 1 Dataset Distillation ",
+        "text_level": 1,
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+        "text": "Input: $p ( \\theta _ { 0 } )$ : distribution of initial weights; $M$ : the number of distilled data   \nInput: $\\alpha$ : step size; $n$ : batch size; $T$ : the number of optimization iterations; $\\tilde { \\eta } _ { 0 }$ : initial value for $\\tilde { \\eta }$   \n1: Initialize $\\hat { \\tilde { \\mathbf { x } } } = \\{ \\tilde { x } _ { i } \\} _ { i = 1 } ^ { M }$ randomly, $\\tilde { \\eta }  \\tilde { \\eta } _ { 0 }$   \n2: for each training step $t = 1$ to $T$ do   \n3: Get a minibatch of real data $\\mathbf { x } _ { t } = \\{ x _ { t , j } \\} _ { j = 1 } ^ { n }$   \n4: Sample a batch of initial weights $\\theta _ { 0 } ^ { ( j ) } \\sim p ( \\theta _ { 0 } )$   \n5: for each sampled $\\theta _ { 0 } ^ { ( j ) }$ do   \n6: Compute updated parameter with GD: $\\theta _ { 1 } ^ { ( j ) } = \\theta _ { 0 } ^ { ( j ) } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } ^ { ( j ) } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } ^ { ( j ) } )$   \n7: Evaluate the objective function on real data: $\\mathscr { L } ^ { ( j ) } = \\ell ( \\mathbf { x } _ { t } , \\boldsymbol { \\theta } _ { 1 } ^ { ( j ) } )$   \n8: end for   \n9: Update $\\begin{array} { r } { \\tilde { \\mathbf { x } }  \\tilde { \\mathbf { x } } - \\alpha \\nabla _ { \\tilde { \\mathbf { x } } } \\sum _ { j } \\mathcal { L } ^ { ( j ) } } \\end{array}$ , and $\\begin{array} { r } { \\tilde { \\eta }  \\tilde { \\eta } - \\alpha \\nabla _ { \\tilde { \\eta } } \\sum _ { j } \\mathcal { L } ^ { ( j ) } } \\end{array}$ ",
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+        "text": "10: end for Output: distilled data $\\tilde { \\bf x }$ and the optimized learning rate $\\tilde { \\eta }$ ",
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+        "type": "text",
+        "text": "3.1 OPTIMIZING DISTILLED DATA ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Standard training usually applies minibatch stochastic gradient descent (SGD) or its variants. At each step $t$ , we sample a minibatch of training data $\\mathbf { x } _ { t } = \\{ x _ { t , j } \\} _ { j = 1 } ^ { n }$ and update the current parameters as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d7dd981bfeefa7ce7e090d26dbaa2ecf8a5498512ed49672770118df8b365c31.jpg",
+        "text": "$$\n\\theta _ { t + 1 } = \\theta _ { t } - \\eta \\nabla _ { \\theta _ { t } } \\ell ( \\mathbf { x } _ { t } , \\theta _ { t } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\eta$ is the learning rate. Such a training process often takes tens of thousands or even millions of above update steps to converge. Instead, we aim to learn a tiny set of synthetic distilled training data $\\tilde { \\mathbf { x } } = \\{ \\tilde { x } _ { i } \\} _ { i = 1 } ^ { M }$ with $M \\ll N$ and a corresponding learning rate $\\tilde { \\eta }$ so that a single GD step like ",
+        "bbox": [
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+    {
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+        "img_path": "images/72559ca6a15ef375fb45021ba8fc8cf857f4f26fcfe046e69742b21f6b12a25f.jpg",
+        "text": "$$\n\\theta _ { 1 } = \\theta _ { 0 } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "using these learned synthetic data $\\tilde { \\mathbf { x } }$ greatly boosts performance on the real training dataset. ",
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+        "type": "text",
+        "text": "Given an initialization $\\theta _ { 0 }$ , we obtain these synthetic data and $\\tilde { \\eta }$ that minimize the objective below $\\mathcal { L }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/65b6fb3234300cf50678c2fbdd96dabadb0eba49e12c01332ea5a21953a2e842.jpg",
+        "text": "$$\n\\tilde { \\mathbf { x } } ^ { * } , \\tilde { \\eta } ^ { * } = \\mathop { \\mathrm { a r g } \\operatorname* { m i n } } _ { \\tilde { \\mathbf { x } } , \\tilde { \\eta } } \\mathcal { L } \\big ( \\tilde { \\mathbf { x } } , \\tilde { \\eta } ; \\theta _ { 0 } \\big ) = \\mathop { \\mathrm { a r g } \\operatorname* { m i n } } _ { \\tilde { \\mathbf { x } } , \\tilde { \\eta } } \\ell ( \\mathbf { x } , \\theta _ { 1 } ) = \\mathop { \\mathrm { a r g } \\operatorname* { m i n } } _ { \\tilde { \\mathbf { x } } , \\tilde { \\eta } } \\ell \\big ( \\mathbf { x } , \\theta _ { 0 } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } } \\ell \\big ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } \\big ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where we derive the new weights $\\theta _ { 1 }$ as a function of distilled images $\\tilde { \\mathbf { x } }$ and learning rate $\\tilde { \\eta }$ using Eqn. 2 and then evaluate the new weights over all the training images $\\mathbf { x }$ . Note that the loss $\\mathcal { L } ( \\tilde { \\mathbf { x } } , \\tilde { \\eta } ; \\theta _ { 0 } )$ is differentiable w.r.t. $\\tilde { \\mathbf { x } }$ and $\\tilde { \\eta }$ , and can thus be optimized using standard gradient-based algorithms. In many classification tasks, the data $\\mathbf { x }$ may contain discrete parts, e.g., the class labels in data-label pairs. For such cases, we fix the discrete parts rather than learn them. ",
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+    {
+        "type": "text",
+        "text": "3.2 DISTILLATION FOR RANDOM INITIALIZATIONS ",
+        "text_level": 1,
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+        "text": "Unfortunately, the above distilled data optimized for a given initialization do not generalize well to other initialization weights. The distilled data often look like random noise (e.g., in Fig. 2a) as it encodes the information of both training dataset $\\mathbf { x }$ and a particular network initialization $\\theta _ { 0 }$ . To address the above issue, we turn to calculate a small number of distilled data that can work for networks with random initializations from a specific distribution. We formulate the optimization problem as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/ec7b678c0f5b347d9c8c56d73be55e18b4a79a2e00ca0e0cf079d984203c6571.jpg",
+        "text": "$$\n\\tilde { \\mathbf { x } } ^ { * } , \\tilde { \\eta } ^ { * } = \\arg \\operatorname* { m i n } _ { \\tilde { \\mathbf { x } } , \\tilde { \\eta } } \\mathbb { E } _ { \\theta _ { 0 } \\sim p ( \\theta _ { 0 } ) } \\mathcal { L } ( \\tilde { \\mathbf { x } } , \\tilde { \\eta } ; \\theta _ { 0 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\theta _ { 0 }$ is a randomly sampled network initialization from the distribution $p ( \\theta _ { 0 } )$ . Algorithm 1 illustrates our main method. During optimization, the distilled data are optimized to work well for multiple networks whose initial weights are sampled from $p ( \\theta _ { 0 } )$ . In practice, we observe that the final distilled data generalize well to the unseen initializations. Besides, these distilled images usually look quite informative, encoding the discriminative features of each category (Fig. 3). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For distilled data to be properly learned, it turns out to be crucial for $\\ell ( { \\mathbf { x } } , \\cdot )$ to share similar local conditions (e.g., output values, gradient magnitudes) over $\\theta _ { 0 }$ sampled from $p ( \\theta _ { 0 } )$ . In the next section, we derive a lower bound on the number of distilled data needed for a simple model with arbitrary initial $\\theta _ { 0 }$ , and discuss its implications on choosing $p ( \\theta _ { 0 } )$ . ",
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+    {
+        "type": "text",
+        "text": "3.3 ANALYSIS OF A SIMPLE LINEAR CASE WITH QUADRATIC LOSS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This section studies our formulation in a simple linear regression case. We derive the lower bound of the number of distilled images needed to achieve the same performance as training on full dataset for arbitrary initialization with one GD step. Consider a dataset $\\mathbf { x }$ containing $N$ data-target pairs $\\{ ( d _ { i } , t _ { i } ) \\} _ { i = 1 } ^ { N }$ , where $d _ { i } \\in \\mathbb { R } ^ { D }$ and $t _ { i } \\in \\mathbb { R }$ , which we represent as two matrices: an $N \\times D$ data matrix $\\mathbf { d }$ and an $N \\times 1$ target matrix $\\mathbf { t }$ . Given the mean squared error and a $D \\times 1$ weight matrix $\\theta$ , we have ",
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+        "img_path": "images/03d4e19049a1c4cb29bf93ff0420bf85802ebb56a63202f3be0008bf54df741c.jpg",
+        "text": "$$\n\\ell ( { \\mathbf x } , \\theta ) = \\ell ( ( { \\mathbf d } , { \\mathbf t } ) , \\theta ) = \\frac { 1 } { 2 N } \\| { \\mathbf d } \\theta - { \\mathbf t } \\| ^ { 2 } .\n$$",
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+        "type": "text",
+        "text": "We aim to learn $M$ synthetic data-target pairs $\\tilde { \\mathbf { x } } = ( \\tilde { \\mathbf { d } } , \\tilde { \\mathbf { t } } )$ , where $\\tilde { \\mathbf { d } }$ is an $M \\times D$ matrix, $\\tilde { \\mathbf { t } }$ an $M \\times 1$ matrix $M \\ll N$ ), and $\\tilde { \\eta }$ the learning rate, to minimize $\\ell ( \\mathbf { x } , \\theta _ { 0 } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } ) )$ . The updated weight matrix after one GD step with these distilled data is ",
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+        "text": "$$\n\\theta _ { 1 } = \\theta _ { 0 } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } ) = \\theta _ { 0 } - \\frac { \\tilde { \\eta } } { M } \\tilde { \\mathbf { d } } ^ { T } ( \\tilde { \\mathbf { d } } \\theta _ { 0 } - \\tilde { \\mathbf { t } } ) = ( \\mathbf { I } - \\frac { \\tilde { \\eta } } { M } \\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { d } } ) \\theta _ { 0 } + \\frac { \\tilde { \\eta } } { M } \\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { t } } .\n$$",
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+        "type": "text",
+        "text": "Note that for such quadratic loss, there always exists some learned distilled data $\\tilde { \\mathbf { x } }$ allowing us to achieve the same performance as training on full dataset $\\mathbf { x }$ (i.e., attaining the global minimum) for any initialization $\\theta _ { 0 }$ .∗ But how small can $M$ , the size of distilled data, be? For such models, the global minimum is attained at any $\\theta ^ { * }$ satisfying $\\mathbf { d } ^ { T } \\mathbf { d } \\theta ^ { * } = \\mathbf { d } ^ { T } \\mathbf { t }$ . Substituting Eqn. (6) in, we have ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d896aafde41677c4b553da0f04f86e5a6b8bf9e2988262bfc48bae66852e843a.jpg",
+        "text": "$$\n\\mathbf { d } ^ { T } \\mathbf { d } ( \\mathbf { I } - \\frac { \\tilde { \\eta } } { M } \\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { d } } ) \\theta _ { 0 } + \\frac { \\tilde { \\eta } } { M } \\mathbf { d } ^ { T } \\mathbf { d } \\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { t } } = \\mathbf { d } ^ { T } \\mathbf { t } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here we make the mild assumption that the feature columns of the data matrix $\\mathbf { d }$ are independent (i.e., ${ \\bf d } ^ { T } { \\bf d }$ has full rank). For a $\\bar { \\bf x } = ( \\tilde { \\bf d } , \\tilde { \\bf t } )$ to satisfy the above equation for any $\\theta _ { 0 }$ , we must have ",
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+    {
+        "type": "equation",
+        "img_path": "images/5cd7e17f6e10952cf70f2e9869c4a269e03a24f7198894b92def4fed8255ccdc.jpg",
+        "text": "$$\n\\mathbf { I } - \\frac { \\widetilde { \\eta } } { M } \\widetilde { \\mathbf { d } } ^ { T } \\widetilde { \\mathbf { d } } = \\mathbf { 0 } ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "which implies that $\\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { d } }$ has full rank and $M \\geq D$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Discussion. The analysis considers only a simple case but suggests that any small number of distilled data fails to generalize to arbitrary starting $\\theta _ { 0 }$ . This is intuitively expected as the optimization target $\\ell ( { \\bf x } , \\theta _ { 1 } ) = \\ell ( { \\bf x } , \\theta _ { 0 } - \\tilde { \\eta } \\nabla _ { \\boldsymbol { \\theta } _ { 0 } } \\ell ( \\tilde { \\bf x } , \\tilde { \\theta _ { 0 } } ) )$ depends on the local behavior of $\\ell ( { \\mathbf { x } } , \\cdot )$ around $\\theta _ { 0 }$ , which can be drastically different across various $\\theta _ { 0 }$ values. We note that the lower bound $M \\geq D$ is a quite restricting one, considering that real datasets often have thousands to even hundreds of thousands of dimensions (e.g., image classification). This analysis motivates us to focus on $p ( \\theta _ { 0 } )$ distributions that yield similar local conditions over the support. Sec. 3.5 discusses several practical choices explored in this paper. Additionally, to address the limitation of using a single GD step, we extend our method to multiple GD steps in the next section. In Sec. 4.1, we empirically verify that using multiple steps is much more effective than using just one on deep convolutional networks, with the total amount of distilled data fixed. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.4 MULTIPLE GRADIENT DESCENT STEPS AND MULTIPLE EPOCHS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We can extend Algorithm 1 to more than one gradient descent steps by changing Line 6 to multiple sequential GD steps each on a different batch of distilled data and learning rate, i.e., each step $i$ is ",
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+    {
+        "type": "equation",
+        "img_path": "images/c9907d9410f57a8c52f160c5f9f94426af0848dd9f36822e560f18bb3dc94a1e.jpg",
+        "text": "$$\n\\theta _ { i + 1 } = \\theta _ { i } - \\tilde { \\eta } _ { i } \\nabla _ { \\theta _ { i } } \\ell \\big ( \\tilde { \\mathbf { x } } _ { i } , \\theta _ { i } \\big ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "and changing Line 9 to backpropagate through all steps. However, naively computing gradients is both memory-intensive and computationally-expensive. Therefore, we exploit a recent technique called back-gradient optimization, which allows for significantly faster gradient calculation of such updates in reverse-mode differentiation (i.e., backpropagation). Specifically, back-gradient optimization formulates the necessary second order terms into efficient Hessian-vector products (Pearlmutter, 1994), which can be easily calculated with modern automatic differentiation systems such as PyTorch (Paszke et al., 2017). For further algorithm details in this aspect, we refer readers to prior work (Domke, 2012; Maclaurin et al., 2015). ",
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+    {
+        "type": "text",
+        "text": "Multiple epochs. To further improve the performance, we can train the network with the same distilled images for multiple epochs (passes) of the GD step(s). In particular, we tie the image pixels for the same distilled images used in different epochs. In other words, for each epoch, our method cycles through all GD steps, where each step is associated with a different batch of distilled data. We do not tie the trained learning rates across epochs as later epochs often use smaller learning rates. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.5 DISTILLATION WITH DIFFERENT INITIALIZATIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Inspired by the analysis of the simple linear case in Sec. 3.3, we aim to focus on initial weights distributions $p ( \\theta )$ that yield similar local conditions over the support. In this work, we focus on the following four practical choices: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• Random initialization: Distribution over model weights initialized using methods that attempts to ensure gradient flow of constant magnitude, e.g., He Initialization (He et al., 2015) and Xavier Initialization (Glorot & Bengio, 2010) for convolutional neural networks (CNNs). • Fixed initialization: A fixed initial weights sampled using the method above. • Random pre-trained weights: Distribution over models pre-trained on other tasks and datasets, e.g., pre-trained ALEXNET (Krizhevsky et al., 2012) networks for ImageNet classification (Deng et al., 2009). Each network is pre-trained on the same task, but with different initializations. • Fixed pre-trained weights: A fixed model weights pre-trained on other tasks and datasets. ",
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+    },
+    {
+        "type": "text",
+        "text": "Distillation for pre-trained weights. Such learned distilled data essentially fine-tunes weights pre-trained on one task to perform well for a new task, thus bridging the gap between two domains. Domain mismatch and dataset bias represent a challenging problem in machine learning today (Torralba & Efros, 2011). Extensive prior work has been proposed to adapt models to new tasks and datasets (Daume III, 2007; Saenko et al., 2010). In this work, we characterize the domain mismatch via distilled data. In Sec. 4.2, we show that a very small number of distilled images are sufficient to quickly adapt CNN models to new classification tasks. ",
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+    {
+        "type": "text",
+        "text": "3.6 DISTILLATION WITH DIFFERENT OBJECTIVES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Previous sections show that we can train distilled data to minimize the loss of the distilled task $\\ell ( \\mathbf { x } , \\theta _ { 1 } )$ defined on the final updated weights $\\theta _ { 1 }$ (Line 7 in Algorithm 1). Distilled images trained with different final learning objectives can train models to exhibit different desired behaviours. We have already mentioned image classification as one of the applications, where distilled images help train accurate classifiers. Below, we introduce a quite different training objective to further demonstrate the flexibility of our method. ",
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+    {
+        "type": "text",
+        "text": "Distillation for a malicious data-poisoning objective. For example, our approach can be used to construct a new form of data poisoning attack. To illustrate this idea, we consider the following scenario. When a single GD step is applied with our synthetic adversarial data, a well-behaved image classifier catastrophically forgets a category but still maintains high performance on other categories. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Formally, given an attacked category $K$ and a target category $T$ , we want the classifier to misclassify images from category $K$ to category $T$ . To achieve this, we consider a new final objective function $\\ell _ { K  T } ( { \\bf x } , \\theta _ { 1 } )$ , which is a classification loss encouraging $\\theta _ { 1 }$ to classify category $K$ images mistakenly as category $T$ while correctly predicting other images, e.g., a cross entropy loss with target labels of $K$ modified to $T$ . Then, the attacking distilled images can be obtained via optimizing ",
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+    {
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+        "img_path": "images/5c348c409ef14948ca91df819d4e27937168257c5d6d29ac2458c8448faeac04.jpg",
+        "text": "$$\n\\tilde { \\mathbf { x } } ^ { * } , \\tilde { \\eta } ^ { * } = \\underset { \\tilde { \\mathbf { x } } , \\tilde { \\eta } } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { \\theta _ { 0 } \\sim p ( \\theta _ { 0 } ) } \\mathcal { L } _ { K  T } ( \\tilde { \\mathbf { x } } , \\tilde { \\eta } ; \\theta _ { 0 } ) = \\underset { \\tilde { \\mathbf { x } } , \\tilde { \\eta } } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { \\theta _ { 0 } \\sim p ( \\theta _ { 0 } ) } \\ell _ { K  T } ( \\mathbf { x } , \\theta _ { 1 } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "ere $p ( \\theta _ { 0 } )$ is the distribution over random pre-trained weights of well-optimized classifiers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Compared to prior data poisoning attacks (Biggio et al., 2012; Li et al., 2016; Muñoz-González et al., 2017; Koh & Liang, 2017), our approach crucially does not require the poisoned training data to be stored and trained on repeatedly. Instead, our method attacks the model training just in one iteration and with only a few data. This advantage makes our method effective for many online training algorithms and useful for the case where malicious users hijack the data feeding pipeline for only one gradient step (e.g., one network transmission). In Sec. 4.2, we show that a single batch of distilled data applied in one step can successfully attack well-optimized neural network models. This setting can be viewed as distilling dataset knowledge of a specific category into data. ",
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+    {
+        "type": "image",
+        "img_path": "images/d9cc87466ca723438e80ecf7575814ebb58542895a77db8bf2c894b21e9c8d0b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "(a) MNIST. These images train networks with a partic- (b) CIFAR10. These images train networks with a particular initialization from $\\mathrm { 1 2 . 9 \\% }$ test accuracy to $9 3 . 7 6 \\%$ . ular initialization from $8 . 8 2 \\%$ test accuracy to $5 4 . 0 3 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "Figure 2: Distilled images trained for fixed initialization. MNIST distilled images use 1 GD step and 3 epochs (10 images in total). CIFAR10 distilled images use $1 0 ~ \\mathrm { G D }$ steps and 3 epochs (100 images in total). For CIFAR10, only selected steps are shown. At left, we report the corresponding learning rates for all 3 epochs. ",
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+        "image_caption": [
+            "(b) CIFAR10. These images train networks with unknown initialization to $3 6 . \\bar { 7 } 9 \\% \\pm 1 . 1 8 \\%$ test accuracy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(a) MNIST. These images train networks with unknown initialization to $7 9 . 5 0 \\% \\pm 8 . 0 8 \\%$ test accuracy. ",
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+    {
+        "type": "text",
+        "text": "Figure 3: Distilled images trained for random initialization with $1 0 \\mathrm { G D }$ steps and 3 epochs. We show images from selected GD steps and corresponding trained learning rates for all 3 epochs. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We report image classification results on MNIST (LeCun, 1998) and CIFAR10 (Krizhevsky & Hinton, 2009). For MNIST, distilled images are trained with LENET (LeCun et al., 1998), which achieves about $9 9 \\%$ test accuracy if fully trained. For CIFAR10, we use a network architecture following Krizhevsky (2012) which achieves around $8 0 \\%$ test accuracy if fully trained. For random initializations and random pre-trained weights, we report means and standard deviations on 200 held-out models, unless otherwise specified. ",
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+    {
+        "type": "text",
+        "text": "Baselines. For each experiment, in addition to baselines specific to the setting, we generally compare our method against baselines trained with data derived or selected from real images: ",
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+    {
+        "type": "text",
+        "text": "• Random real images: We randomly sample the same number of real training images per category.   \n• Optimized real images: We sample sets of real images as above, and choose on the top $2 0 \\%$ sets that perform the best training images.   \n• $k$ -means: For each category, we use $k$ -means to extract the same number of cluster centroids as the number of distilled images in our method.   \n• Average real images: We compute the average image of all the images in each category, which is reused in different GD steps. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For these baselines, we perform each evaluation on 200 hold-out models with all combinations of learning rate $\\in$ {learned learning rate with our method, $0 . 0 0 1 , 0 . 0 0 3 , 0 . 0 1 , 0 . 0 3 , 0 . 1 , 0 . 3 \\}$ and #epochs $\\in \\{ 1 , 3 , 5 \\}$ . We report results from the best performing combination. We run all the experiments on NVIDIA Titan $\\mathrm { X p }$ and V100 GPUs. We use one GPU for fixed initial weights and four GPUs for random initial weights. Each training typically takes 1 to 4 hours. Please see supplemental material Sec. S-6.1 for more training and baseline details. ",
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+    {
+        "type": "text",
+        "text": "4.1 DATASET DISTILLATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Fixed initialization. With access to initial network weights, distilled images can directly train a particular network to reach high performance. For example, 10 learned distilled images can boost the test accuracy of a neural network with an initial accuracy $1 2 . 9 0 \\%$ to the final accuracy $9 3 . 7 6 \\%$ on MNIST (Fig. 2a). Similarly, 100 images can train a network with an initial accuracy $8 . 8 2 \\%$ to $5 4 . 0 3 \\%$ test accuracy on CIFAR10 (Fig. 2b). This result suggests that even only a few distilled images have enough capacity to distill part of the dataset. ",
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+    {
+        "type": "text",
+        "text": "Random initialization. Trained with randomly sampled initializations using Xavier initialization (Glorot & Bengio, 2010), the learned distilled images do not need to encode information tailored for a particular starting point and thus can represent meaningful content independent of network initializations. In Fig. 3, we see that such distilled images reveal discriminative features of the corresponding categories: e.g., the ship image in Fig. 3b. These 100 images can train randomly ",
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+    {
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+        "img_path": "images/dfb4142a5480849c2ef6e92464747a0dbf26b2a4b8d991ba56a7eb884d064837.jpg",
+        "image_caption": [
+            "Figure 4: Hyperparameter sensitivity studies on random initialization: (a) average test accuracy w.r.t. the number of gradient descent steps. The number of epochs is fixed to be 2. (b) average test accuracy w.r.t. the number of epochs. The number of steps is fixed to be 10, with each containing 10 images (one per category). "
+        ],
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+    {
+        "type": "image",
+        "img_path": "images/484484dd4e342abf8dc288bf3f64c564707ea50cd7f4c6e03a5eea5e7bc0e14a.jpg",
+        "image_caption": [],
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/62303a7aea0dbd3b53ede2ff574250f1bf1d9af29a87b55eea1d7227727ad208.jpg",
+        "image_caption": [
+            "Figure 5: Comparison between applying the same number of images in one versus multiple GD steps on random initialization, with the number of epochs fixed to 1. $N$ denotes the total number of images per category. For multiple steps runs, each of the $N$ steps applies one image per category. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/b1db4d77982a9e7047894721707c3acb9bb1af483909ee5de2f595655cd19436.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=3 colspan=1></td><td rowspan=1 colspan=2>Ours</td><td rowspan=1 colspan=6>Baselines</td></tr><tr><td rowspan=2 colspan=1>Fixed init.</td><td rowspan=2 colspan=1>Random init.</td><td rowspan=1 colspan=4>Used as training data in same number of GD steps</td><td rowspan=1 colspan=2>Used as data for K-NN classification</td></tr><tr><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>k-means</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>k-means</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>96.6%</td><td rowspan=1 colspan=1>79.5% ± 8.1%</td><td rowspan=1 colspan=1>68.6% ±9.8%</td><td rowspan=1 colspan=1>73.0% ± 7.6%</td><td rowspan=1 colspan=1>76.4% ± 9.5%</td><td rowspan=1 colspan=1>77.1% ± 2.7%</td><td rowspan=1 colspan=1>71.5% ± 2.1%</td><td rowspan=1 colspan=1>92.2%±0.1%</td></tr><tr><td rowspan=1 colspan=1>CIFAR10</td><td rowspan=1 colspan=1>54.0%</td><td rowspan=1 colspan=1>36.8%±1.2%</td><td rowspan=1 colspan=1>21.3% ± 1.5%</td><td rowspan=1 colspan=1>23.4% ± 1.3%</td><td rowspan=1 colspan=1>22.5%±3.1%</td><td rowspan=1 colspan=1>22.3%±0.7%</td><td rowspan=1 colspan=1>18.8% ± 1.3%</td><td rowspan=1 colspan=1>29.4% ±0.3%</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Comparison between our method trained for $1 0 \\mathrm { \\ G D }$ steps and 3 epochs and various baselines. For baselines using K-Nearest Neighbor (K-NN), best result among all combinations of distance metric $\\in \\{ l _ { 1 } , l _ { 2 } \\}$ and $\\mathtt { K } \\in \\{ 1 , 3 \\}$ is reported. In K-NN and $k$ -means, K and $k$ can have different values. All methods use 10 images per class, except for the average real images baseline, which reuses the same images in different GD steps. ",
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+    {
+        "type": "text",
+        "text": "initialized networks to $3 6 . 7 9 \\%$ average test accuracy on CIFAR10. Similarly, for MNIST, the 100 distilled images shown in Fig. 3a can train randomly initialized networks to $\\dot { 7 } 9 . 5 0 \\%$ test accuracy. ",
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+        "text": "Multiple gradient descent steps and multiple epochs. In Fig. 3, we learn distilled images for 10 GD steps applied in 3 epochs, leading to a total of 100 images (with each step containing one image per category). In each epoch, these 10 steps are sequentially applied once. The early steps tend to look noisier, likely regularizing random weights to point easier for further optimization. In later steps, the images gradually look like real data and share the discriminative features for these categories. Fig. 4a shows that using more steps significantly improves the results. Fig. 4b shows a similar but slower trend as the number of epochs increases. We observe that longer training (i.e., more epochs) can help the model learn all the knowledge from the distilled images, but the performance is eventually limited by the capacity of the images (i.e., the number of total images). Alternatively, we can train the model with one GD step but a big batch size. Sec. 3.3 has shown theoretical limitations of using only one step in a simple linear case. In Fig. 5, we empirically verify that with convolutional networks, using multiple steps drastically outperforms single step method, with the same number of distilled images. ",
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+        "type": "text",
+        "text": "Table 1 compares our method against several baselines. Our method with both fixed and random initialization outperform all the baselines on CIFAR10 and most of the baselines on MNIST. ",
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+        "type": "text",
+        "text": "4.2 DISTILLATION FOR DIFFERENT INITIALIZATIONS AND OBJECTIVES ",
+        "text_level": 1,
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+        "text": "Next, we show two extended settings of our main algorithm discussed in Sec. 3.5 and Sec. 3.6. Both cases assume that the initial weights are random but pre-trained on a different dataset. We train the distilled images on 2000 random pre-trained models, and then apply them on unseen models. ",
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+        "type": "text",
+        "text": "Fixed and random pre-trained weights on digits. As shown in Sec. 3.5, we can optimize distilled images to quickly fine-tune pre-trained models for a new dataset. Table 2 shows that our method is more effective compared to various baseline on adaptation among three digits datasets: MNIST, USPS (Hull, 1994), and SVHN (Netzer et al., 2011). We also compared against a state-of-the-art few-short supervised domain adaptation method (Motiian et al., 2017). Although our method uses the entire training set to compute the distilled images, both methods use the same number of images to distill the knowledge of target dataset. Prior work (Motiian et al., 2017) is outperformed by our method with ",
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+    {
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+        "img_path": "images/d8a4fe9defd345ba383d265477c6b60afccbd7b85bbe8e608eeab44a1d5ca65a.jpg",
+        "image_caption": [
+            "Figure 6: Performance for our method and baselines with random pre-trained initialization and a malicious objective. Distilled images are trained for one GD step. For baselines, we use the same numbers of images with incorrect labels and also apply one GD step, and report the result that achieves the highest accuracy w.r.t. the incorrect labels while having $\\geq 1 0 \\%$ misclassification ratio on the attacked category, to avoid results with learning rates too low to change model behavior at all. (a) Our method slightly outperforms the best baseline in accuracy w.r.t. incorrect labels. (b) Our method performs similarly with some baselines in changing the prediction of the attacked category on MNIST, but is much better than all baselines on CIFAR10. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "table",
+        "img_path": "images/056fe2ea35ee419a26732a50443aff3d4a4566714d753f60629e67007eb8a436.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>.Ourswith fixedpre-trained</td><td rowspan=1 colspan=1>Ourswith randompre-trained</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>k-means</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Domain adaptationMotiian et al.(2017)</td><td rowspan=1 colspan=1>No adaptation</td><td rowspan=1 colspan=1>Train on fulldestinationtraining set</td></tr><tr><td rowspan=1 colspan=1>M→u</td><td rowspan=1 colspan=1>97.9%</td><td rowspan=1 colspan=1>95.4%±1.8%</td><td rowspan=1 colspan=1>94.9%±0.8%</td><td rowspan=1 colspan=1>95.2%±0.7%</td><td rowspan=1 colspan=1>92.2%±1.6%</td><td rowspan=1 colspan=1>93.9%±0.8%</td><td rowspan=1 colspan=1>96.7%±0.5%</td><td rowspan=1 colspan=1>90.4%±3.0%</td><td rowspan=1 colspan=1>97.3%±0.3%</td></tr><tr><td rowspan=1 colspan=1>u→M</td><td rowspan=1 colspan=1>93.2%</td><td rowspan=1 colspan=1>92.7% ± 1.4%</td><td rowspan=1 colspan=1>87.1% ±2.9%</td><td rowspan=1 colspan=1>87.6% ± 2.1%</td><td rowspan=1 colspan=1>85.6% ±3.1%</td><td rowspan=1 colspan=1>78.4% ±5.0%</td><td rowspan=1 colspan=1>89.2% ±2.4%</td><td rowspan=1 colspan=1>67.5%±3.9%</td><td rowspan=1 colspan=1>98.6%±0.5%</td></tr><tr><td rowspan=1 colspan=1>S→M</td><td rowspan=1 colspan=1>96.2%</td><td rowspan=1 colspan=1>85.2%±4.7%</td><td rowspan=1 colspan=1>84.6%±2.1%</td><td rowspan=1 colspan=1>85.2%±1.2%</td><td rowspan=1 colspan=1>85.8%±1.2%</td><td rowspan=1 colspan=1>74.9%± 2.6%</td><td rowspan=1 colspan=1>74.0%± 1.5%</td><td rowspan=1 colspan=1>51.6%±2.8%</td><td rowspan=1 colspan=1>98.6%±0.5%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Performance of our method and baselines in adapting models among MNIST $( \\mathcal { M } )$ , USPS $( \\mathcal { U } )$ , and SVHN $( S )$ . 100 distilled images are trained for $1 0 \\mathrm { G D }$ steps and 3 epochs. Few-shot domain adaptation method by Motiian et al. (2017) and baselines use the same numbers image per class. ",
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+    {
+        "type": "table",
+        "img_path": "images/7d23accc4a633e265e002cbbb3f7e3f27e852db9ebfb4fd72f99661284118aea.jpg",
+        "table_caption": [
+            "Table 3: Performance of our method and baselines in adapting an ALEXNET pre-trained on ImageNet to PASCAL-VOC and CUB-200. Only one distilled image per class are trained to be applied in 1 GD step repeated for 3 epochs. Our method significantly outperforms the baselines. Results are collected over 10 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Destination dataset</td><td rowspan=1 colspan=1>Ours</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Fine-tune on fulldestination training set</td></tr><tr><td rowspan=1 colspan=1>PASCAL-VOC</td><td rowspan=1 colspan=1>70.75%</td><td rowspan=1 colspan=1>19.41%± 3.73%</td><td rowspan=1 colspan=1>23.82%±3.66%</td><td rowspan=1 colspan=1>9.94%</td><td rowspan=1 colspan=1>75.57%±0.18%</td></tr><tr><td rowspan=1 colspan=1>CUB-200</td><td rowspan=1 colspan=1>38.76%</td><td rowspan=1 colspan=1>7.11%±0.66%</td><td rowspan=1 colspan=1>7.23%±0.78%</td><td rowspan=1 colspan=1>2.88%</td><td rowspan=1 colspan=1>41.21%±0.51%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "fixed pre-trained weights on all the tasks, and by our method with random pre-trained weights on two of the three tasks. This result shows that our distilled images indeed convey compressed information of the full dataset. ",
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+    {
+        "type": "text",
+        "text": "Fixed pre-trained weights on ImageNet. In Table 3, we adapt a widely-used ALEXNET model (Krizhevsky, 2014) pre-trained on ImageNet (Deng et al., 2009) to perform image classification on PASCAL-VOC (Everingham et al., 2010) and CUB-200 (Wah et al., 2011) datasets. Using only 1 distilled image per category, our method outperforms the baselines significantly. Our result is also comparable to the accuracy of fine-tuning on the full datasets which contain thousands of images. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Random Pre-trained weights and a malicious data-poisoning objective. Sec. 3.6 shows that our method can construct a new type of data poisoning, where the attacker can apply just one GD step with a few malicious data to manipulate a well-trained model. We train distilled images to make well-optimized neural networks to misclassify a particular attacked category as another target category within only one GD step. Our method requires no access to the exact weights of the model. In Fig. 6b, we evaluate our method on 200 held-out models, against various baselines using data derived from real images and incorrect labels. While some baselines perform similarly well as our method on MNIST, our method significantly outperforms all the baselines on CIFAR10. ",
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we present dataset distillation for compressing the knowledge of entire training data into a few synthetic training images. We can train a network to reach high performance with a small number of distilled images and several gradient descent steps. Finally, we demonstrate two applications including fast domain adaptation and effective data poisoning attack. In the future, we plan to extend our method to compress large-scale visual datasets such as ImageNet (Deng et al., 2009) and other types of data (e.g., audio and text). Also, our current method is sensitive to the initial weights distribution. We would like to investigate more on various initialization strategies, with which distilled images can work well. ",
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+    {
+        "type": "text",
+        "text": "S-6 SUPPLEMENTARY MATERIAL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "S-6.1 EXPERIMENT DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            135,
+            390,
+            150
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the networks used in our experiments, we disable dropout layers due to the randomness and computational cost they introduce in distillation. Moreover, we initialize the distilled learning rates as 0.02 and use Adam optimizer (Kingma & Ba, 2014) with a learning rate of 0.001. For random initialization and random pre-trained weights, we sample 4 to 16 initial weights in each step. ",
+        "bbox": [
+            174,
+            161,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Details of the baselines are listed below. ",
+        "bbox": [
+            176,
+            218,
+            411,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Random real images: We randomly sample the same number of real training images per category. 10 such set of sampled images are evaluated. • Optimized real images: We sampled 50 sets of real images using above procedure, and evaluate 10 sets that achieve best performance on 20 held-out models and 1024 training images. • $k$ -means: For each category, we use $k$ -means to extract the same number of cluster centroids as the number of distilled images in our method. 10 such set of sampled images are evaluated. • Average real images: We compute the average image of all the images in each category, which is reused in different GD steps. We evaluate the model only once because average images are deterministic. ",
+        "bbox": [
+            196,
+            241,
+            826,
+            351
+        ],
+        "page_idx": 12
+    }
+]

parse/train/SyGjjsC5tQ/SyGjjsC5tQ.md

@@ -0,0 +1,782 @@
+# STABLE OPPONENT SHAPINGIN DIFFERENTIABLE GAMES
+
+Alistair Letcher1 Jakob Foerster1 David Balduzzi2 Tim Rocktaschel ¨ 3 Shimon Whiteson1 1University of Oxford 2DeepMind 3University College London
+
+# ABSTRACT
+
+A growing number of learning methods are actually differentiable games whose players optimise multiple, interdependent objectives in parallel – from GANs and intrinsic curiosity to multi-agent RL. Opponent shaping is a powerful approach to improve learning dynamics in these games, accounting for player influence on others’ updates. Learning with Opponent-Learning Awareness (LOLA) is a recent algorithm that exploits this response and leads to cooperation in settings like the Iterated Prisoner’s Dilemma. Although experimentally successful, we show that LOLA agents can exhibit ‘arrogant’ behaviour directly at odds with convergence. In fact, remarkably few algorithms have theoretical guarantees applying across all $\mathbf { \xi } _ { n }$ -player, non-convex) games. In this paper we present Stable Opponent Shaping (SOS), a new method that interpolates between LOLA and a stable variant named LookAhead. We prove that LookAhead converges locally to equilibria and avoids strict saddles in all differentiable games. SOS inherits these essential guarantees, while also shaping the learning of opponents and consistently either matching or outperforming LOLA experimentally.
+
+# 1 INTRODUCTION
+
+Problem Setting. While machine learning has traditionally focused on optimising single objectives, generative adversarial nets (GANs) (Goodfellow et al., 2014) have showcased the potential of architectures dealing with multiple interacting goals. They have since then proliferated substantially, including intrinsic curiosity (Pathak et al., 2017), imaginative agents (Racaniere et al., 2017), syn- \` thetic gradients (Jaderberg et al., 2017), hierarchical reinforcement learning (RL) (Wayne & Abbott, 2014; Vezhnevets et al., 2017) and multi-agent RL in general (Busoniu et al., 2008).
+
+These can effectively be viewed as differentiable games played by cooperating and competing agents – which may simply be different internal components of a single system, like the generator and discriminator in GANs. The difficulty is that each loss depends on all parameters, including those of other agents. While gradient descent on single functions has been widely successful, converging to local minima under rather mild conditions (Lee et al., 2017), its simultaneous generalisation can fail even in simple two-player, two-parameter zero-sum games. No algorithm has yet been shown to converge, even locally, in all differentiable games.
+
+Related Work. Convergence has widely been studied in convex $n$ -player games, see especially Rosen (1965); Facchinei & Kanzow (2007). However, the recent success of non-convex games exemplified by GANs calls for a better understanding of this general class where comparatively little is known. Mertikopoulos & Zhou (2018) recently prove local convergence of no-regreat learning to variationally stable equilibria, though under a number of regularity assumptions.
+
+Conversely, a number of algorithms have been successful in the non-convex setting for restricted classes of games. These include policy prediction in two-player two-action bimatrix games (Zhang & Lesser, 2010); WoLF in two-player two-action games (Bowling & Veloso, 2001); AWESOME in repeated games (Conitzer & Sandholm, 2007); Optimistic Mirror Descent in two-player bilinear zero-sum games (Daskalakis et al., 2018) and Consensus Optimisation (CO) in two-player zerosum games (Mescheder et al., 2017). An important body of work including Heusel et al. (2017); Nagarajan & Kolter (2017) has also appeared for the specific case of GANs.
+
+Working towards bridging this gap, some of the authors recently proposed Symplectic Gradient Adjustment (SGA), see Balduzzi et al. (2018). This algorithm is provably ‘attracted’ to stable fixed points while ‘repelled’ from unstable ones in all differentiable games $\stackrel { \cdot } { n }$ -player, non-convex). Nonetheless, these results are weaker than strict convergence guarantees. Moreover, SGA agents may act against their own self-interest by prioritising stability over individual loss. SGA was also discovered independently by Gemp & Mahadevan (2018), drawing on variational inequalities.
+
+In a different direction, Learning with Opponent-Learning Awareness (LOLA) (Foerster et al., 2018) modifies the learning objective by predicting and differentiating through opponent learning steps. This is intuitively appealing and experimentally successful, encouraging cooperation in settings like the Iterated Prisoner’s Dilemma (IPD) where more stable algorithms like SGA defect. However, LOLA has no guarantees of converging or even preserving fixed points of the game.
+
+Contribution. We begin by constructing the first explicit tandem game where LOLA agents adopt ‘arrogant’ behaviour and converge to non-fixed points. We pinpoint the cause of failure and show that a natural variant named LookAhead (LA), discovered before LOLA by Zhang & Lesser (2010), successfully preserves fixed points. We then prove that LookAhead locally converges and avoids strict saddles in all differentiable games, filling a theoretical gap in multi-agent learning. This is enabled through a unified approach based on fixed-point iterations and dynamical systems. These techniques apply equally well to algorithms like CO and SGA, though this is not our present focus.
+
+While LookAhead is theoretically robust, the shaping component endowing LOLA with a capacity to exploit opponent dynamics is lost. We solve this dilemma with an algorithm named Stable Opponent Shaping (SOS), trading between stability and exploitation by interpolating between LookAhead and LOLA. Using an intuitive and theoretically grounded criterion for this interpolation parameter, SOS inherits both strong convergence guarantees from LA and opponent shaping from LOLA.
+
+On the experimental side, we show that SOS plays tit-for-tat in the IPD on par with LOLA, while all other methods mostly defect. We display the practical consequences of our theoretical guarantees in the tandem game, where SOS always outperforms LOLA. Finally we implement a more involved GAN setup, testing for mode collapse and mode hopping when learning Gaussian mixture distributions. SOS successfully spreads mass across all Gaussians, at least matching dedicated algorithms like CO, while LA is significantly slower and simultaneous gradient descent fails entirely.
+
+# 2 BACKGROUND
+
+# 2.1 DIFFERENTIABLE GAMES
+
+We frame the problem of multi-agent learning as a game. Adapted from Balduzzi et al. (2018), the following definition insists only on differentiability for gradient-based methods to apply. This concept is strictly more general than stochastic games, whose parameters are usually restricted to action-state transition probabilities or functional approximations thereof.
+
+Definition 1. A differentiable game is a set of $n$ players with parameters $\theta = ( \theta ^ { 1 } , \ldots , \theta ^ { n } ) \in \mathbb { R } ^ { d }$ and twice continuously differentiable losses $L ^ { i } : \mathbb { R } ^ { d } \overset { \cdot } {  } \mathbb { R }$ , where $\theta ^ { i } \in \mathbb { R } ^ { d _ { i } }$ for each $i$ and $\textstyle \sum _ { i } d _ { i } = d$ .
+
+Crucially, note that each loss is a function of all parameters. From the viewpoint of player $i$ , parameters can be written as $\theta = ( \theta ^ { i } , \theta ^ { - i } )$ where $\bar { \theta ^ { - i } }$ contains all other players’ parameters. We do not make the common assumption that each $L ^ { i }$ is convex as a function of $\theta ^ { i }$ alone, for any fixed opponent parameters $\theta ^ { - i }$ , nor do we restrict $\theta$ to the probability simplex – though this restriction can be recovered via projection or sigmoid functions $\bar { \sigma : \mathbb { R } \to [ 0 , 1 ] }$ . If $n = 1$ , the ‘game’ is simply to minimise a given loss function. In this case one can reach local minima by (possibly stochastic) gradient descent (GD). For arbitrary $n$ , the standard solution concept is that of Nash equilibria.
+
+Definition 2. A point $\bar { \theta } \in \mathbb { R } ^ { d }$ is a (local) Nash equilibrium if for each $i$ , there are neighbourhoods $U _ { i }$ of ${ \bar { \theta } } ^ { i }$ such that $L ^ { i } ( \theta ^ { i } , \bar { \theta } ^ { - i } ) \ge L ^ { i } ( \bar { \theta } )$ for all $\theta ^ { i } \in U _ { i }$ . In other words, each player’s strategy is a local best response to current opponent strategies.
+
+We write $\nabla _ { i } L ^ { k } = \nabla _ { \theta ^ { i } } L ^ { k }$ and $\nabla _ { i j } L ^ { k } = \nabla _ { \theta ^ { j } } \nabla _ { \theta ^ { i } } L ^ { k }$ for any $i , j , k$ . Define the simultaneous gradient of the game as the concatenation of each player’s gradient,
+
+$$
+\boldsymbol { \xi } = \left( \nabla _ { 1 } L ^ { 1 } , \ldots , \nabla _ { n } L ^ { n } \right) ^ { \boldsymbol { \mathsf { T } } } \in \mathbb { R } ^ { d } .
+$$
+
+The $i$ th component of $\xi$ is the direction of greatest increase in $L ^ { i }$ with respect to $\theta ^ { i }$ . If each agent minimises their loss independently from others, they perform GD on their component $\nabla _ { i } L ^ { i }$ with learning rate $\alpha _ { i }$ . Hence, the parameter update for all agents is given by $\theta  \theta - \alpha \odot \xi$ , where $\alpha = ( \alpha _ { 1 } , \ldots , \alpha _ { n } ) ^ { \intercal }$ and $\odot$ is element-wise multiplication. This is also called naive learning (NL), reducing to $\theta  \theta - \alpha \xi$ if agents have the same learning rate. This is assumed for notational simplicity, though irrelevant to our results. The following example shows that $\mathrm { N L }$ can fail to converge.
+
+Example 1. Consider $L ^ { 1 / 2 } = \pm x y$ , where players control the $x$ and $y$ parameters respectively. The origin is a (global and unique) Nash equilibrium. The simultaneous gradient is ${ \boldsymbol { \xi } } = ( y , - x )$ and cycles around the origin. Explicitly, a gradient step from $( x , y )$ yields
+
+$$
+( x , y )  ( x , y ) - \alpha ( y , - x ) = ( x - \alpha y , y + \alpha x )
+$$
+
+which has distance from the origin $( 1 + \alpha ^ { 2 } ) ( x ^ { 2 } + y ^ { 2 } ) > ( x ^ { 2 } + y ^ { 2 } )$ for any $\alpha > 0$ and $( x , y ) \neq 0$ . It follows that agents diverge away from the origin for any $\alpha > 0$ . The cause of failure is that $\xi$ is not the gradient of a single function, implying that each agent’s loss is inherently dependent on others. This results in a contradiction between the non-stationarity of each agent, and the optimisation of each loss independently from others. Failure of convergence in this simple two-player zero-sum game shows that gradient descent does not generalise well to differentiable games. We consider an alternative solution concept to Nash equilibria before introducing LOLA.
+
+# 2.2 STABLE FIXED POINTS
+
+Consider the game given by $L ^ { 1 } = L ^ { 2 } = x y$ where players control the $x$ and $y$ parameters respectively. The optimal solution is $( x , y ) \to \pm ( \infty , - \infty )$ , since then $L ^ { 1 } = L ^ { 2 }  - \infty$ . However the origin is a global Nash equilibrium, while also a saddle point of $x y$ . It is highly undesirable to converge to the origin in this game, since infinitely better losses can be reached in the anti-diagonal direction. In this light, Nash equilibria cannot be the right solution concept to aim for in multi-agent learning. To define stable fixed points, first introduce the ‘Hessian’ of the game as the block matrix
+
+$$
+\begin{array} { r } { H = \nabla \xi = \left( \begin{array} { c c c } { \nabla _ { 1 1 } L ^ { 1 } } & { \cdots } & { \nabla _ { 1 n } L ^ { 1 } } \\ { \vdots } & { \ddots } & { \vdots } \\ { \nabla _ { n 1 } L ^ { n } } & { \cdots } & { \nabla _ { n n } L ^ { n } } \end{array} \right) \in \mathbb { R } ^ { d \times d } . } \end{array}
+$$
+
+This can equivalently be viewed as the Jacobian of the vector field $\xi$ . Importantly, note that $H$ is not symmetric in general unless $n = 1$ , in which case we recover the usual Hessian ${ \bf \dot { \boldsymbol { H } } } = \nabla ^ { 2 } { \boldsymbol { L } }$ .
+
+Definition 3. A point $\bar { \theta }$ is a fixed point if $\xi ( \bar { \theta } ) = 0$ . It is stable if $H ( \bar { \theta } ) \succeq 0$ , unstable if $H ( \bar { \theta } ) \prec 0$ and a strict saddle if $H ( \bar { \theta } )$ has an eigenvalue with negative real part.
+
+The name ‘fixed point’ is coherent with GD, since $\xi ( \bar { \theta } ) = 0$ implies a fixed update ${ \bar { \theta } }  { \bar { \theta } } - \alpha \xi ( { \bar { \theta } } ) =$ $\bar { \theta }$ . Though Nash equilibria were shown to be inadequate above, it is not obvious that stable fixed points (SFPs) are a better solution concept. In Appendix A we provide intuition for why SFPs are both closer to local minima in the context of multi-loss optimisation, and more tractable for convergence proofs. Moreover, this definition is an improved variant on that in Balduzzi et al. (2018), assuming positive semi-definiteness only at $\bar { \theta }$ instead of holding in a neighbourhood. This makes the class of SFPs as large as possible, while sufficient for all our theoretical results.
+
+Assuming invertibility of $H ( \bar { \theta } )$ at SFPs is crucial to all convergence results in this paper. The same assumption is present in related work including Mescheder et al. (2017), and cannot be avoided. Even for single losses, a fixed point with singular Hessian can be a local minimum, maximum, or saddle point. Invertibility is thus necessary to ensure that SFPs really are ‘local minima’. This is omitted from now on. Finally note that unstable fixed points are a subset of strict saddles, making Theorem 6 both stronger and more general than results for SGA by Balduzzi et al. (2018).
+
+# 2.3 LEARNING WITH OPPONENT-LEARNING AWARENESS (LOLA)
+
+Accounting for nonstationarity, Learning with Opponent-Learning Awareness (LOLA) modifies the learning objective by predicting and differentiating through opponent learning steps (Foerster et al., 2018). For simplicity, if $n = 2$ then agent 1 optimises $L ^ { 1 } ( \theta ^ { 1 } , \theta ^ { 2 } + \Delta \theta ^ { 2 } )$ with respect to $\theta ^ { 1 }$ , where $\Delta \theta ^ { 2 }$ is the predicted learning step for agent 2. Foerster et al. (2018) assume that opponents are naive learners, namely $\Delta \theta ^ { 2 } = - \dot { \alpha } _ { 2 } \dot { \nabla _ { 2 } } L ^ { 2 }$ . After first-order Taylor expansion, the loss is approximately given by $L ^ { 1 } + \dot { \nabla } _ { 2 } L ^ { 1 } \cdot \Delta \theta ^ { 2 }$ . By minimising this quantity, agent 1 learns parameters that align the opponent learning step $\Delta \theta ^ { 2 }$ with the direction of greatest decrease in $L ^ { 1 }$ , exploiting opponent dynamics to further reduce one’s losses. Differentiating with respect to $\theta ^ { 1 }$ , the adjustment is
+
+$$
+\begin{array} { r } { \nabla _ { 1 } L ^ { 1 } + \left( \nabla _ { 2 1 } L ^ { 1 } \right) ^ { \top } \Delta \theta ^ { 2 } + \left( \nabla _ { 1 } \Delta \theta ^ { 2 } \right) ^ { \top } \nabla _ { 2 } L ^ { 1 } . } \end{array}
+$$
+
+By explicitly differentiating through $\Delta \theta ^ { 2 }$ in the rightmost term, LOLA agents actively shape opponent learning. This has proven effective in reaching cooperative equilibria in multi-agent learning, finding success in a number of games including tit-for-tat in the IPD. The middle term above was originally dropped by the authors because “LOLA focuses on this shaping of the learning direction of the opponent”. We choose not to eliminate this term, as also inherent in LOLA-DiCE (Foerster et al., 2018). Preserving both terms will in fact be key to developing stable opponent shaping.
+
+First we formulate $n$ -player LOLA in vectorial form. Let $H _ { d }$ and $H _ { o }$ be the matrices of diagonal and anti-diagonal blocks of $H$ , so that $H = H _ { d } + H _ { o }$ . Also define $L = ( L ^ { 1 } , \dots , L ^ { n } )$ and the operator diag : $\overline { { \mathbb { R } ^ { d \times n } } } \to \mathbb { R } ^ { d }$ constructing a vector from the block matrix diagonal, namely $\mathrm { d i a g } ( M ) _ { i } = M _ { i i }$ .
+
+Proposition 1 (Appendix B). Writing $\begin{array} { r } { \chi = \mathrm { d i a g } ( H _ { o } ^ { \intercal } \nabla L ) } \end{array}$ , the LOLA gradient adjustment is
+
+$$
+\begin{array} { r } { \mathrm { L O L A } = ( I - \alpha H _ { o } ) \xi - \alpha \chi . } \end{array}
+$$
+
+While experimentally successful, LOLA fails to preserve fixed points $\bar { \theta }$ of the game since
+
+$$
+( I - \alpha H _ { o } ) \xi ( \bar { \theta } ) - \alpha \chi ( \bar { \theta } ) = - \alpha \chi ( \bar { \theta } ) \neq 0
+$$
+
+in general. Even if $\bar { \theta }$ is a Nash equilibrium, the update ${ \bar { \theta } }  { \bar { \theta } } - \alpha \mathrm { { L O L A } } \neq { \bar { \theta } }$ can push them away despite parameters being optimal. This may worsen the losses for all agents, as in the game below.
+
+Example 2 (Tandem). Imagine a tandem controlled by agents facing opposite directions, who feed $x$ and $y$ force into their pedals respectively. Negative numbers correspond to pedalling backwards.
+
+Moving coherently requires $x \approx - y$ , embodied by a quadratic loss $( x + y ) ^ { 2 }$ . However it is easier for agents to pedal forwards, translated by linear losses $- 2 x$ and $- 2 y$ . The game is thus given by $L ^ { 1 } ( x , y ) = ( x + \mathbf { \bar { \psi } } y ) ^ { 2 } - 2 x$ and $L ^ { 2 } ( x , y ) = \bar { ( } x + y ) \bar { ^ { 2 } } - 2 y$ . These sub-goals are incompatible, so agents cannot simply accelerate forwards. The SFPs are given by $\{ x + y = 1 \}$ . Computing ${ \dot { \chi } } ( x , 1 - x ) \ = \ ( 4 , 4 ) \ \neq \ 0$ , none of these are preserved by LOLA. Instead, we show in Appendix C that LOLA can only converge to sub-optimal scenarios with worse losses for both agents, for any $\alpha$ .
+
+![](images/2950f3836b3a7ba8fe7acf716f21bf0146a2c51474c7a136f8a29ee19e2f5692.jpg)  
+Figure 1: Illustration of the tandem game.
+
+Intuitively, the root of failure is that LOLA agents try to shape opponent learning and enforce compliance by accelerating forwards, assuming a dynamic response from their opponent. The other agen does the same, so they become ‘arrogant’ and suffer by pushing strongly in opposite directions.
+
+# 3 METHOD
+
+# 3.1 LOOKAHEAD
+
+The shaping term $\chi$ prevents LOLA from preserving fixed points. Consider removing this component entirely, giving $( I - \alpha H _ { o } ) \xi$ . This variant preserves fixed points, but what does it mean from the perspective of each agent? Note that LOLA optimises $L ^ { 1 } ( { \theta } ^ { 1 } , { \theta } ^ { 2 } + \Delta { \theta } ^ { 2 } )$ with respect to $\theta ^ { 1 }$ , while $\bar { \Delta } \theta ^ { 2 }$ is a function of $\bar { \theta ^ { 1 } }$ . In other words, we assume that our opponent’s learning step depends on our current optimisation with respect to $\theta ^ { 1 }$ . This is inaccurate, since opponents cannot see our updated parameters until the next step. Instead, assume we optimise $L ^ { 1 } ( \theta ^ { 1 } , \hat { \theta } ^ { 2 } + \Delta \theta ^ { 2 } ( \hat { \theta } ^ { 1 } , \hat { \theta } ^ { 2 } ) )$ where $\hat { \theta } ^ { 1 } , \hat { \theta } ^ { 2 }$ are the current parameters. After Taylor expansion, the gradient with respect to $\theta ^ { 1 }$ is given by
+
+$$
+\nabla _ { 1 } L ^ { 1 } + \left( \nabla _ { 2 1 } L ^ { 1 } \right) ^ { \boldsymbol { \mathsf { T } } } \Delta \theta ^ { 2 }
+$$
+
+since $\Delta \theta ^ { 2 } ( \hat { \theta } ^ { 1 } , \hat { \theta } ^ { 2 } )$ does not depend on $\theta ^ { 1 }$ . In vectorial form, we recover the variant $( I - \alpha H _ { o } ) \xi$ since the shaping term corresponds precisely to differentiating through $\Delta \theta ^ { 2 }$ . We name this LookAhead, which was discovered before LOLA by Zhang & Lesser (2010) though not explicitly named. Using the stop-gradient operator $\perp ^ { 1 }$ , this can be reformulated as optimising $L ^ { 1 } ( \theta ^ { 1 } , \theta ^ { 2 } + \bot \Delta \theta ^ { 2 } )$ where $\perp$ prevents gradient flowing from $\Delta \theta ^ { 2 }$ upon differentiation.
+
+The main result of Zhang & Lesser (2010) is that LookAhead converges to Nash equilibria in the small class of two-player, two-action bimatrix games. We will prove local convergence to SFP and non-convergence to strict saddles in all differentiable games. On the other hand, by discarding the problematic shaping term, we also eliminated LOLA’s capacity to exploit opponent dynamics and encourage cooperation. This will be witnessed in the IPD, where LookAhead agents mostly defect.
+
+# 3.2 STABLE OPPONENT SHAPING (SOS)
+
+We propose Stable Opponent Shaping (SOS), an algorithm preserving both advantages at once. Define the partial stop-gradient operator $\perp ^ { p } : = p \perp + ( 1 - p ) I$ , where $I$ is the identity and $p$ stands for partial. A $p$ -LOLA agent optimises the modified objective
+
+$$
+L ^ { 1 } ( \theta ^ { 1 } , \theta ^ { 2 } + \perp ^ { 1 - p } \Delta \theta ^ { 2 } , \ldots , \theta ^ { n } + \perp ^ { 1 - p } \Delta \theta ^ { n } ) ,
+$$
+
+collapsing to LookAhead at $p = 0$ and LOLA at $p = 1$ . The resulting gradient is given by
+
+$$
+\xi _ { p } : = p \mathrm { - } \mathrm { L O L A } = ( I - \alpha H _ { o } ) \xi - p \alpha \chi
+$$
+
+with $\xi _ { 0 } = \mathrm { L A }$ . We obtain an algorithm trading between shaping and stability as a function of $p$ . Note however that preservation of fixed points only holds if $p$ is infinitesimal, in which case $p$ -LOLA is almost identical to LookAhead – losing the very purpose of interpolation. Instead we propose a two-part criterion for $p$ at each learning step, through which all guarantees descend.
+
+First choose $p$ such that $\xi _ { p }$ points in the same direction as LookAhead. This will not be enough to prove convergence itself, but prevents arrogant behaviour by ensuring convergence only to fixed points. Formally, the first criterion is given by $\langle \xi _ { p } , \xi _ { 0 } \rangle \geq 0$ . If $\langle - \alpha \chi , \bar { \xi } _ { 0 } \rangle \geq 0$ then $\langle \xi _ { p } , \xi _ { 0 } \rangle \geq 0$ automatically, so we choose $p = 1$ for maximal shaping. Otherwise choose
+
+$$
+p = \operatorname* { m i n } \left\{ 1 , \frac { - a \| \xi _ { 0 } \| ^ { 2 } } { \langle - \alpha \chi , \xi _ { 0 } \rangle } \right\}
+$$
+
+with any hyperparameter $0 < a < 1$ . This guarantees a positive inner product
+
+$$
+\begin{array} { r } { \langle \xi _ { p } , \xi _ { 0 } \rangle = p \langle - \alpha \chi , \xi _ { 0 } \rangle + \Vert \xi _ { 0 } \Vert ^ { 2 } \geq - a \Vert \xi _ { 0 } \Vert ^ { 2 } + \Vert \xi _ { 0 } \Vert ^ { 2 } = \Vert \xi _ { 0 } \Vert ^ { 2 } ( 1 - a ) > 0 . } \end{array}
+$$
+
+We complement this with a second criterion ensuring local convergence. The idea is to scale $p$ by a function of $\| \xi \|$ if $\| \xi \|$ is small enough, which certainly holds in neighbourhoods of fixed points. Let $0 < b < 1$ be a hyperparameter and take $p = \| \boldsymbol { \xi } \| ^ { \check { 2 } }$ if $\| \xi \| < b$ , otherwise $p = 1$ . Choosing $p _ { 1 }$ and $p _ { 2 }$ according to these criteria, the two-part criterion is $\ddot { p } = \operatorname* { m i n } \{ p _ { 1 } , p _ { 2 } \}$ . SOS is obtained by combining $p$ -LOLA with this criterion, as summarised in Algorithm 1. Crucially, all theoretical results in the next section are independent from the choice of hyperparameters $a$ and $b$ .
+
+# Algorithm 1: Stable Opponent Shaping
+
+<table><tr><td colspan="2">1 Initialise θ randomly and fix hyperparameters a,b ∈(0,1). 2 while not done do</td></tr><tr><td>3</td><td>Compute ε = (I-αHo)ξ and X= diag(HVL) at 0.</td></tr><tr><td>4</td><td>allsoll2</td></tr><tr><td>5</td><td>if|/ξl&lt;b then P2= |/ε|² else P2=1</td></tr><tr><td>6</td><td>Let p = min{p1,p2},compute $p = £o - pαX and assgn θ ← 0-αξp.</td></tr><tr><td colspan="2">7end</td></tr></table>
+
+# 4 THEORETICAL RESULTS
+
+Our central theoretical contribution is that LookAhead and SOS converge locally to SFP and avoid strict saddles in all differentiable games. Since the learning gradients involve second-order Hessian terms, our results assume thrice continuously differentiable losses (omitted hereafter). Losses which are $C ^ { 2 }$ but not $C ^ { 3 }$ are very degenerate, so this is a mild assumption. Statements made about SOS crucially hold for any hyperparameters $a , b \in ( 0 , 1 )$ . See Appendices $\mathrm { D }$ and E for detailed proofs.
+
+Convergence is proved using Ostrowski’s Theorem. This reduces convergence of a gradient adjustment $g$ to positive stability (eigenvalues with positive real part) of $\nabla g$ at stable fixed points.
+
+Theorem 2. Let $H \succeq 0$ be invertible with symmetric diagonal blocks. Then there exists $\epsilon > 0$ such that $( I - \alpha H _ { o } ) H$ is positive stable for all $0 < \alpha < \epsilon$ .
+
+This type of result would usually be proved either by analytical means showing positive definiteness and hence positive stability, or direct eigenvalue analysis. We show in Appendix D that $( I - \alpha H _ { o } ) H$ is not necessarily positive definite, while there is no necessary relationship between eigenpairs of $H$ and $H _ { o }$ . This makes our theorem all the more interesting and non-trivial. We use a similarity transformation trick to circumvent the dual obstacle, allowing for analysis of positive definiteness with respect to a new inner product. We obtain positive stability by invariance under change of basis.
+
+Corollary 3. LookAhead converges locally to stable fixed points for $\alpha > 0$ sufficiently small.
+
+Using the second criterion for $p$ , we prove local convergence of SOS in all differentiable games despite the presence of a shaping term (unlike LOLA).
+
+Theorem 4. SOS converges locally to stable fixed points for $\alpha > 0$ sufficiently small.
+
+# 4.2 AVOIDING STRICT SADDLES
+
+Using the first criterion for $p$ , we prove that SOS only converges to fixed points (unlike LOLA).   
+Proposition 5. If SOS converges to $\bar { \theta }$ and $\alpha > 0$ is small then $\bar { \theta }$ is a fixed point of the game.
+
+Now assume that $\theta$ is initialised randomly (or with arbitrarily small noise), as is standard in ML. Let $F ( \theta ) = \theta - \alpha \xi _ { p } ( \theta )$ be the SOS iteration. Using both the second criterion and the Stable Manifold Theorem from dynamical systems, we can prove that every strict saddle $\bar { \theta }$ has a neighbourhood $U$ such that $\{ \theta \in \bar { U } \mid F ^ { n } ( \theta ) \stackrel { . } {  } \bar { \theta }$ as $n  \infty \}$ has measure zero for $\alpha > 0$ sufficiently small. Since $\theta$ is initialised randomly, we obtain the following result.
+
+Theorem 6. SOS locally avoids strict saddles almost surely, for $\alpha > 0$ sufficiently small.
+
+This also holds for LookAhead, and could be strenghtened to global initialisations provided a strong boundedness assumption on $\| H \| _ { 2 }$ . This is trickier for SOS since $p ( \theta )$ is not globally continuous. Altogether, our results for LookAhead and the correct criterion for $p$ -LOLA lead to some of the strongest theoretical guarantees in multi-agent learning. Furthermore, SOS retains all of LOLA’s opponent shaping capacity while LookAhead does not, as shown experimentally in the next section.
+
+# 5 EXPERIMENTS AND DISCUSSION
+
+We evaluate the performance of SOS in three differentiable games. We first showcase opponent shaping and superiority over LA/CO/SGA/NL in the Iterated Prisoner’s Dilemma (IPD). This leaves SOS and LOLA, which have differed only in theory up to now. We bridge this gap by showing that SOS always outperforms LOLA in the tandem game, avoiding arrogant behaviour by decaying $p$ while LOLA overshoots. Finally we test SOS on a more involved GAN learning task, with results similar to dedicated methods like Consensus Optimisation.
+
+# 5.1 EXPERIMENTAL SETUP
+
+IPD: This game is an infinite sequence of the well-known Prisoner’s Dilemma, where the payoff is discounted by a factor $\gamma \in [ 0 , 1 )$ at each iteration. Agents are endowed with a memory of actions at the previous state. Hence there are 5 parameters for each agent $i$ : the probability $P ^ { i } \bar { ( \textit { C } | \textit { s t a t e } ) }$ of cooperating at start state $s _ { 0 } = \emptyset$ or state $s _ { t } = ( a _ { t - 1 } ^ { 1 } , a _ { t - 1 } ^ { 2 } )$ for $t > 0$ . One Nash equilibrium is to always defect (DD), with a normalised loss of 2. A better equilibrium with loss 1 is named tit-for-tat (TFT), where each player begins by cooperating and then mimicks the opponent’s previous action.
+
+We run 300 training episodes for SOS, LA, CO, SGA and NL. The parameters are initialised following a normal distribution around $1 / 2$ probability of cooperation, with unit variance. We fix $\alpha = 1$ and $\gamma = 0 . 9 6$ , following Foerster et al. (2018). We choose $a = 0 . 5$ and $b = 0 . 1$ for SOS. The first is a robust and arbitrary middle ground, while the latter is intentionally small to avoid poor SFP.
+
+![](images/cb0e38afb2d07627a526ffddf93c30890884749530316f308b2a07efd23e113a.jpg)  
+Figure 2: Results in the IPD. (A) Probability that agents cooperate, given memory state, at the end of 50 training runs. SOS and LOLA mostly play tit-for-tat, while others mostly defect. (B) Average loss at each step, across 300 runs, with shaded deviations. SOS and LOLA outperform all others.
+
+Tandem: Though local convergence is guaranteed for SOS, it is possible that SOS diverges from poor initialisations. This turns out to be impossible in the tandem game since the Hessian is globally positive semi-definite. We show this explicitly by running 300 training episodes for SOS and LOLA. Parameters are initialised following a normal distribution around the origin. We found performance to be robust to hyperparameters $a , b$ . Here we fix $a = b = 0 . 5$ and $\alpha = 0 . 1$ .
+
+Gaussian mixtures: We reproduce a setup from Balduzzi et al. (2018). The game is to learn a Gaussian mixture distribution using GANs. Data is sampled from a highly multimodal distribution designed to probe the tendency to collapse onto a subset of modes during training – see ground truth in Appendix F. The generator and discriminator networks each have 6 ReLU layers of 384 neurons, with 2 and 1 output neurons respectively. Learning rates are chosen by grid search at iteration 8k, with $a = 0 . 5$ and $b = 0 . 1$ for SOS, following the same reasoning as the IPD.
+
+# 5.2 RESULTS AND DISCUSSION
+
+IPD: Results are given in Figure 2. Parameters in part (A) are the end-run probabilities of cooperating for each memory state, encoded in different colours. Only 50 runs are shown for visibility. Losses at each step are displayed in part (B), averaged across 300 episodes with shaded deviations.
+
+SOS and LOLA mostly succeed in playing tit-for-tat, displayed by the accumulation of points in the correct corners of (A) plots. For instance, CC and CD points are mostly in the top right and left corners so agent 2 responds to cooperation with cooperation. Agents also cooperate at the start state, represented by $\mathcal { D }$ points all hidden in the top right corner. Tit-for-tat strategy is further indicated by the losses close to 1 in part (B). On the other hand, most points for LA/CO/SGA/NL are accumulated at the bottom left, so agents mostly defect. This results in poor losses, demonstrating the limited effectiveness of recent proposals like SGA and CO. Finally note that trained parameters and losses for SOS are almost identical to those for LOLA, displaying equal capacity in opponent shaping while also inheriting convergence guarantees and outperforming LOLA in the next experiment.
+
+Tandem: Results are given in Figure 3. SOS always succeeds in decreasing $p$ to reach the correct equilibria, with losses averaging at 0. LOLA fails to preserve fixed points, overshooting with losses averaging at $4 / 9$ . The criterion for SOS is shown in action in part (B), decaying $p$ to avoid overshooting. This illustrates that purely theoretical guarantees descend into practical outperformance. Note that SOS even gets away from the LOLA fixed points if initialised there (not shown), converging to improved losses using the alignment criterion with LookAhead.
+
+![](images/b01b67a7e21fa234a54d7ebc0e7880c31af82547e152fefa1b5abf23d7fc1875.jpg)  
+Figure 3: Results in the tandem game. (A) Average loss and $\mathbf { ( B ) }$ average $p$ at each learning step, across 300 runs, with shaded deviations. SOS decays $p$ to avoid arrogance and outperforms LOLA.
+
+![](images/fc2c6bc621ffa09e24ef5e43ffc41951b6c98c2d77bc5a4e99b8e1286b2f432f.jpg)  
+Figure 4: Generator distribution at sampled iterations. NL suffers from mode collapse and hopping, while CO and SOS learn the correct mixture of Gaussians. Below each plot: KL divergence $\mathcal { \bar { D } } _ { K L } ( P \left| \right| Q )$ from generator $P$ to ground truth $Q$ , estimated from 25600 samples. To the RHS of each row: learning rate $\alpha$ . Best result at each iteration shown in bold.
+
+Gaussian mixtures: The generator distribution and KL divergence are given at $\{ 2 \mathrm { k } , 4 \mathrm { k } , 6 \mathrm { k } , 8 \mathrm { k } \}$ iterations for NL, CO and SOS in Figure 4. Results for SGA, LOLA and LA are in Appendix F. SOS achieves convincing results by spreading mass across all Gaussians, as do CO/SGA/LOLA. LookAhead is significantly slower, while NL fails through mode collapse and hopping. Only visual inspection was used for comparison by Balduzzi et al. (2018), while KL divergence gives stronger numerical evidence here. SOS and CO are slightly superior to others with reference to this metric. However CO is aimed specifically toward two-player zero-sum GAN optimisation, while SOS is widely applicable with strong theoretical guarantees in all differentiable games.
+
+Theoretical results in machine learning have significantly helped understand the causes of success and failure in applications, from optimisation to architecture. While gradient descent on single losses has been studied extensively, algorithms dealing with interacting goals are proliferating, with little grasp of the underlying dynamics. The analysis behind CO and SGA has been helpful in this respect, though lacking either in generality or convergence guarantees. The first contribution of this paper is to provide a unified framework and fill this theoretical gap with robust convergence results for LookAhead in all differentiable games. Capturing stable fixed points as the correct solution concept was essential for these techniques to apply.
+
+Furthermore, we showed that opponent shaping is both a powerful approach leading to experimental success and cooperative behaviour – while at the same time preventing LOLA from preserving fixed points in general. This conundrum is solved through a robust interpolation between LookAhead and LOLA, giving birth to SOS through a robust criterion. This was partially enabled by choosing to preserve the ‘middle’ term in LOLA, and using it to inherit stability from LookAhead. This results in convergence guarantees stronger than all previous algorithms, but also in practical superiority over LOLA in the tandem game. Moreover, SOS fully preserves opponent shaping and outperforms SGA, CO, LA and NL in the IPD by encouraging tit-for-tat policy instead of defecting. Finally, SOS convincingly learns Gaussian mixtures on par with the dedicated CO algorithm.
+
+# 7 ACKNOWLEDGEMENTS
+
+This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 637713). It was also supported by the Oxford-Google DeepMind Graduate Scholarship.
+
+# REFERENCES
+
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+
+# APPENDIX
+
+# A STABLE FIXED POINTS
+
+In the main text we showed that Nash equilibria are inadequate in multi-agent learning, exemplified by the simple game given by $L ^ { 1 } = L ^ { 2 } = x y$ , where the origin is a global Nash equilibrium but a saddle point of the losses. It is not however obvious that SFP are a better solution concept. We begin by pointing out that for single losses, invertibility and symmetry of the Hessian imply positive definiteness at SFP. These are exactly local minima of $L$ detected by the second partial derivative test, namely those points provably attainable by gradient descent.
+
+To emphasise this, note that gradient descent does not converge locally to all local minima. This can be seen by considering the example $L ( x , y ) = y ^ { 2 }$ and the local (global) minimum $( 0 , 0 )$ . There is no neighbourhood for which gradient descent converges to $( 0 , 0 )$ , since initialising at $( x _ { 0 } , y _ { 0 } )$ will always converge to $( x _ { 0 } , 0 )$ for appropriate learning rates, with $x _ { 0 } \neq 0$ almost surely. This occurs precisely because the Hessian is singular at $( 0 , 0 )$ . Though a degenerate example, this suggests an important difference to make between the ideal solution concept (local minima) and that for which local convergence claims are possible to attain (local minima with invertible $H \succeq 0$ ).
+
+Accordingly, the definition of SFP is the immediate generalisation of ‘fixed points with positive semi-definite Hessian’, or in other words, ‘second-order-tractable local minima’. It is important to impose only positive semi-definiteness to keep the class as large as possible, despite strict positive definiteness holding for single losses due to symmetry. Imposing strict positivity would for instance exclude the origin in the cyclic game $L ^ { 1 } = x \dot { y } = - \dot { L ^ { 2 } }$ , a point certainly worthy of convergence.
+
+Note also that imposing a weaker condition than $H \succeq 0$ would be incorrect. Invertibility aside, local convergence of gradient descent on single functions cannot be guaranteed if $H \not \cong 0$ , since such points are strict saddles. These are almost always avoided by gradient descent, as proven by Lee et al. (2016) and Panageas & Piliouras (2017). It is thus necessary to impose $H \succeq 0$ as a minimal requirement in optimisation methods attempting to generalise gradient descent.
+
+Remark A.1. A matrix $H$ is positive semi-definite iff the same holds for its symmetric part $S =$ $( H + H ^ { \mathsf { T } } ) / 2$ , so SFP could equivalently be defined as $S ( \bar { \theta } ) \succeq 0$ . This is the original formulation given by part of the authors (Balduzzi et al., 2018), who also imposed the extra requirement $S ( \theta ) \succeq 0$ in a neighbourhood of $\bar { \theta }$ . After discussion we decided to drop this assumption, pointing out that it is 1) more restrictive, 2) superficial to all theoretical results and 3) weakens the analogy with tractable local minima. The only thing gained by imposing semi-positivity in a neighbourhood is that SFP become a subset of Nash equilibria.
+
+Regarding unstable fixed points and strict saddles, note that $H ( \bar { \theta } ) \succ 0$ implies $H ( \theta ) \succ 0$ in a neighbourhood, hence being equivalent to the definition in Balduzzi et al. (2018). It follows also that unstable points are a subset of strict saddles: if $H ( \bar { \theta } ) \prec 0$ then all eigenvalues are negative since any eigenpair $( v , \lambda )$ satisfies
+
+$$
+0 > \operatorname { R e } ( v ^ { \mathsf { T } } H v ) = \operatorname { R e } ( \lambda v ^ { \mathsf { T } } v ) = \operatorname { R e } ( \lambda ) .
+$$
+
+We introduced strict saddles in this paper as a generalisation of unstable FP, which are more difficult to handle but nonetheless tractable using dynamical systems. The name is chosen by analogy to the definition in Lee et al. (2016) for single losses.
+
+# B LOLA VECTORIAL FORM
+
+Proposition B.1. The LOLA gradient adjustment is
+
+$$
+\begin{array} { r } { \mathrm { L O L A } = ( I - \alpha H _ { o } ) \xi - \alpha \mathrm { d i a g } ( H _ { o } ^ { \intercal } \boldsymbol { \nabla } L ) . } \end{array}
+$$
+
+in the usual assumption of equal learning rates.
+
+Proof. Recall the modified objective
+
+$$
+L ^ { 1 } ( \theta ^ { 1 } , \theta ^ { 2 } - \alpha \nabla _ { 2 } L ^ { 2 } , \ldots , \theta ^ { n } - \alpha \nabla _ { n } L ^ { n } )
+$$
+
+for agent 1, and so on for each agent. First-order Taylor expansion yields
+
+$$
+L ^ { 1 } - \alpha \sum _ { j \neq 1 } ( \nabla _ { j } L ^ { 1 } ) ^ { \intercal } \nabla _ { j } L ^ { j }
+$$
+
+and similarly for each agent. Differentiating with respect to $\theta ^ { i }$ , the adjustment for player $i$ is
+
+$$
+\begin{array} { r l } & { \mathrm { L O A } _ { k } = \nabla _ { i } \left[ L ^ { i } - \alpha \displaystyle \sum _ { j \neq i } ( \nabla _ { j } L ^ { i } ) ^ { \top } \nabla _ { j } L ^ { j } \right] } \\ & { \quad = \nabla _ { i } L ^ { i } - \alpha \displaystyle \sum _ { j \neq i } ( \nabla _ { j i } L ^ { i } ) ^ { \top } \nabla _ { j } L ^ { j } + ( \nabla _ { j i } L ^ { i } ) ^ { \top } \nabla _ { j } L ^ { i } } \\ & { \quad = \nabla _ { i } L ^ { i } - \alpha \displaystyle \sum _ { j \neq i } \nabla _ { i } L ^ { i } \nabla _ { j } L ^ { i } - \alpha \displaystyle \sum _ { j \neq i } ( \nabla _ { j i } L ^ { i } ) ^ { \top } \nabla _ { j } L ^ { j } } \\ & { \quad = \xi _ { i } - \alpha \displaystyle \sum _ { j } ( H _ { o } ) _ { i j } \xi _ { j } - \alpha \displaystyle \sum _ { j } ( H _ { o } ^ { \top } ) _ { i j } ( \nabla L ) _ { j i } } \\ & { \quad = \xi _ { i } - \alpha ( H _ { o } ) _ { i j } - \alpha ( H _ { o } ^ { \top } \nabla L ) _ { i i } } \\ & { \quad = \left[ \xi - \alpha H _ { o } \xi _ { i } - \alpha \mathrm { i d s } ( H _ { o } ^ { \top } \nabla L ) \right] _ { i } , } \end{array}
+$$
+
+and thus
+
+$$
+\mathrm { L O L A } = ( I - \alpha H _ { o } ) \xi - \alpha \mathrm { d i a g } ( H _ { o } ^ { \mathsf { T } } \nabla L )
+$$
+
+as required.
+
+# C TANDEM GAME
+
+We provide a more detailed exposition of the tandem game in this section, including computation of fixed points for NL/LOLA and corresponding losses. Recall that the game is given by
+
+$$
+L ^ { 1 } ( x , y ) = ( x + y ) ^ { 2 } - 2 x \qquad { \mathrm { a n d } } \qquad L ^ { 2 } ( x , y ) = ( x + y ) ^ { 2 } - 2 y .
+$$
+
+Intuitively, agents wants to have $x \approx - y$ since $( x + y ) ^ { 2 }$ is the leading loss, but would also prefer to have positive $x$ and $y$ . These are incompatible, so the agents must not be ‘arrogant’ and instead make concessions. The fixed points are given by
+
+$$
+\xi = 2 ( x + y - 1 ) \binom { 1 } { 1 } = 0 ,
+$$
+
+namely any pair $( x , 1 - x )$ . The corresponding losses are $L ^ { 1 } = 1 - 2 x = - L ^ { 2 }$ , summing to 0 for any $x$ . We have
+
+$$
+H = 2 \left( \begin{array} { c c } { { 1 } } & { { 1 } } \\ { { 1 } } & { { 1 } } \end{array} \right) \succeq 0
+$$
+
+everywhere, so all fixed points are SFP. LOLA fails to preserve these, since
+
+$$
+\chi = \mathrm { d i a g } ( H _ { o } ^ { \mathsf { T } } \nabla L ) = 4 \mathrm { d i a g } \left( { 0 } \begin{array} { c c } { { 0 } } & { { 1 } } \\ { { 1 } } & { { 0 } } \end{array} \right) \left( x + y - 1 \begin{array} { c c } { { } } & { { x + y } } \\ { { x + y } } & { { x + y - 1 } } \end{array} \right) = 4 ( x + y ) \left( { 1 } \atop { 1 } \right)
+$$
+
+which is non-zero for any SFP $( x , 1 - x )$ . Instead, LOLA can only converge to points such that
+
+$$
+\mathrm { L O L A } = \xi - \alpha H _ { o } \xi - \alpha \chi = 0 .
+$$
+
+We solve this explicitly as follows:
+
+$$
+\begin{array} { l } { { \mathrm { L O L A } = 2 ( x + y - 1 ) \binom { 1 } { 1 } - 4 \alpha ( x + y - 1 ) \binom { 0 } { 1 } \left( { 1 } \atop { 1 } \right)}  \binom { 1 } { 1 } - 4 \alpha ( x + y ) \binom { 1 } { 1 } }  \\ { { \mathrm { ~ } } } \\ { { \mathrm { ~ } } } \\ { { \mathrm { ~ } } } \end{array}
+$$
+
+The fixed points for LOLA are thus pairs $( x , y )$ such that
+
+$$
+x + y = { \frac { 1 - 2 \alpha } { 1 - 4 \alpha } } ,
+$$
+
+noting that $( 1 - 2 \alpha ) / ( 1 - 4 \alpha ) > 1$ for all $\alpha > 0$ . This leads to worse losses
+
+$$
+L ^ { 1 } = \left( \frac { 1 - 2 \alpha } { 1 - 4 \alpha } \right) ^ { 2 } - 2 x > 1 - 2 x = L ^ { 1 } ( x , 1 - x )
+$$
+
+for agent 1 and similarly for agent 2. In particular, losses always sum to something greater than 0. This becomes negligible as the learning rate becomes smaller, but is always positive nonetheless Taking $\alpha$ arbitrarily small is not a viable solution since convergence will in turn be arbitrarily slow. LOLA is thus not a strong algorithm candidate for all differentiable games.
+
+# D CONVERGENCE PROOFS
+
+We use Ostrowski’s theorem as a unified framework for proving local convergence of gradient-based methods. This is a standard result on fixed-point iterations, adapted from (Ortega & Rheinboldt, 2000, 10.1.3). We also invoke and prove a topological result of our own, Lemma D.9, at the end of this section. This is useful in deducing local convergence, though not central to intuition.
+
+Theorem D.1 (Ostrowski). Let $F : \Omega  \mathbb { R } ^ { d }$ be continuously differentiable on an open subset $\Omega \subseteq \mathbb { R } ^ { d }$ , and assume $\bar { x } \in \Omega$ is a fixed point. If all eigenvalues of $\nabla F ( { \bar { x } } )$ are strictly in the unit circle of $\mathbb { C }$ , then there is an open neighbourhood $U$ of $\bar { x }$ such that for all $x _ { 0 } \in U$ , the sequence $F ^ { ( k ) } ( x _ { 0 } )$ converges to $\bar { x }$ . Moreover, the rate of convergence is at least linear in $k$ .
+
+Definition D.2. A matrix $M$ is called positive stable if all its eigenvalues have positive real part.
+
+Recall the simultaneous gradient $\xi$ and the Hessian $H$ defined for differentiable games. Let $X$ be any matrix with continuously differentiable entries.
+
+Corollary D.3. Assume $\bar { x }$ is a fixed point of a differentiable game such that $X H ( { \bar { x } } )$ is positive stable. Then the iterative procedure
+
+$$
+F ( x ) = x - \alpha X \xi ( x )
+$$
+
+converges locally to $\bar { x }$ for $\alpha > 0$ sufficiently small.
+
+Proof. By definition of fixed points, $\xi ( \bar { x } ) = 0$ and so
+
+$$
+\nabla [ X \xi ] ( \bar { x } ) = \nabla X ( \bar { x } ) \xi ( \bar { x } ) + X ( \bar { x } ) \nabla \xi ( \bar { x } ) = X H ( \bar { x } )
+$$
+
+is positive stable by assumption, namely has eigenvalues $a _ { k } + i b _ { k }$ with $a _ { k } > 0$ . It follows that
+
+$$
+\nabla F ( { \bar { x } } ) = I - \alpha \nabla [ X \xi ] ( { \bar { x } } )
+$$
+
+has eigenvalues $1 - \alpha a _ { k } - i \alpha b _ { k }$ , which are in the unit circle for small $\alpha$ . More precisely,
+
+$$
+\begin{array} { r l } & { | 1 - \alpha a _ { k } - i \alpha b _ { k } | ^ { 2 } < 1 } \\ { \iff } & { 1 - 2 \alpha a _ { k } + \alpha ^ { 2 } a _ { k } ^ { 2 } + \alpha ^ { 2 } b _ { k } ^ { 2 } < 1 } \\ { \iff } & { 0 < \alpha < \cfrac { 2 a _ { k } } { a _ { k } ^ { 2 } + b _ { k } ^ { 2 } } } \end{array}
+$$
+
+which is always possible for $a _ { k } > 0$ . Hence $\nabla F ( { \bar { x } } )$ has eigenvalues in the unit circle for $0 < \alpha <$ $\mathrm { m i n } _ { k } 2 a _ { k } / ( a _ { k } ^ { 2 } + \dot { b } _ { k } ^ { 2 } )$ , and we are done by Ostrowski’s Theorem since $\bar { x }$ is a fixed point of $F$ .
+
+We apply this corollary to LookAhead, which is given by
+
+$$
+F ( \theta ) = \theta - \alpha X \xi ( \theta )
+$$
+
+where $X = \left( I - \alpha H _ { o } \right)$ . It is thus sufficient to prove the following result.
+
+Theorem D.4. Let $H \succeq 0$ invertible with symmetric diagonal blocks. Then there exists $\epsilon > 0$ such that $( I - \alpha H _ { o } ) H$ is positive stable for all $0 < \alpha < \epsilon$ .
+
+Remark D.5. Note that $( I - \alpha H _ { o } ) H$ may fail to be positive definite, though true in the case of $2 \times 2$ matrices. This no longer holds in higher dimensions, exemplified by the Hessian
+
+$$
+H = \left( \begin{array} { l l l l } { 9 } & { - 4 } & { - 3 } & { - 3 } \\ { - 2 } & { 1 } & { 2 } & { 1 } \\ { - 3 } & { 0 } & { 1 } & { 0 } \\ { - 3 } & { 1 } & { 2 } & { 1 } \end{array} \right) .
+$$
+
+By direct computation (symbolic in $\alpha$ ), one can show that $G = ( I - \alpha H _ { o } ) H$ always has positive eigenvalues for small $\alpha > 0$ , whereas its symmetric part $S$ always has a negative eigenvalue with magnitude in the order of $\alpha$ . This implies that $S$ and in turn $G$ is not positive definite. As such, an analytical proof of the theorem involving bounds on the corresponding bilinear form will fail.
+
+This makes the result all the more interesting, but more involved. Central to the proof is a similarity transformation proving positive definiteness with respect to a different inner product, a novel technique we have not found in the multi-agent learning literature.
+
+Proof. We cannot study the eigenvalues of $G$ directly, since there is no necessary relationship between eigenpairs of $H$ and $H _ { o }$ . In the aim of using analytical tools, the trick is to find a positive definite matrix which is similar to $G$ , thus sharing the same positive eigenvalues. First define
+
+$$
+G _ { 1 } = ( I + \alpha H _ { d } ) H \qquad \mathrm { a n d } \qquad G _ { 2 } = - \alpha H ^ { 2 } ,
+$$
+
+where $H _ { d }$ is the sub-matrix of diagonal blocks,and rewrite
+
+$$
+G = ( I - \alpha H _ { o } ) H = ( I - \alpha ( H - H _ { d } ) ) H = ( I + \alpha H _ { d } ) H - \alpha H ^ { 2 } = G _ { 1 } + G _ { 2 } .
+$$
+
+Note that $H _ { d }$ is block diagonal with symmetric blocks $\nabla _ { i i } L ^ { i } \succeq 0$ , so $\left( I + \alpha H _ { d } \right)$ is symmetric and positive definite for all $\alpha \geq 0$ . In particular its principal square root
+
+$$
+M = ( I + \alpha H _ { d } ) ^ { 1 / 2 }
+$$
+
+is unique and invertible. Now note that
+
+$$
+M ^ { - 1 } G _ { 1 } M = M ^ { - 1 } M ^ { 2 } H M = M ^ { \mathsf { T } } H M ,
+$$
+
+which is positive semi-definite since
+
+$$
+u ^ { \mathsf { T } } M ^ { \mathsf { T } } H M u = ( M u ) ^ { \mathsf { T } } H ( M u ) \geq 0
+$$
+
+for all non-zero $u$ . In particular $M$ provides a similarity transformation which eliminates $H _ { d }$ from $G _ { 1 }$ while simultaneously delivering positive semi-definiteness. We can now prove that
+
+$$
+M ^ { - 1 } G M = M ^ { - 1 } G _ { 1 } M + M ^ { - 1 } G _ { 2 } M
+$$
+
+is positive definite, establishing positive stability of $G$ by similarity. Let $m = d - 1$ where $d$ is the vector space dimension, namely $\mathbf { \bar { \boldsymbol { H } } } \in \mathbb { R } ^ { d \times d }$ . Recall that the $m$ -sphere $S ^ { m } \subset \mathbb { R } ^ { d }$ is the space of unit vectors in $\mathbb { R } ^ { d }$ . Take any $u \in S ^ { m }$ and consider the quantity
+
+$$
+u ^ { \mathsf { T } } M ^ { - 1 } G M u .
+$$
+
+First note that a Taylor expansion of $M$ in $\alpha$ yields
+
+$$
+M = ( I + \alpha H _ { d } ) ^ { 1 / 2 } = I + O ( \alpha )
+$$
+
+and
+
+$$
+M ^ { - 1 } = ( I + \alpha H _ { d } ) ^ { - 1 / 2 } = I + O ( \alpha ) .
+$$
+
+This implies in turn that
+
+$$
+u ^ { \mathsf { T } } M ^ { - 1 } G M u = u ^ { \mathsf { T } } G u + O ( \alpha ) .
+$$
+
+There are two cases to distinguish. If $u ^ { \intercal } H u > 0$ then
+
+$$
+u ^ { \mathsf { T } } M ^ { - 1 } G M u = u ^ { \mathsf { T } } G u + O ( \alpha ) = u ^ { \mathsf { T } } G _ { 1 } u + O ( \alpha ) = u ^ { \mathsf { T } } H u + O ( \alpha ) > 0
+$$
+
+for $\alpha$ sufficiently small. Otherwise, $u ^ { \mathsf { T } } H u = 0$ and consider decomposing $H$ into symmetric and antisymmetric parts $S = ( H + H ^ { \intercal } ) / 2$ and $A = ( H - H ^ { \intercal } ) / 2$ , so that $H = S + A$ . By antisymmetry of $A$ we have $u ^ { \mathsf { T } } A u = 0$ and hence $u ^ { \mathsf { T } } H u = 0 = u ^ { \mathsf { T } } S u$ . Now $H \succeq 0$ implies $S \succeq 0$ , so by Cholesky decomposition of $S$ there exists a matrix $T$ such that $S = T ^ { \mathsf { T } } T$ . In particular $0 = u ^ { \mathsf { T } } S \dot { u } = \| T u \| ^ { 2 }$ implies $T u = 0$ , and in turn $S u = 0$ . Since $H$ is invertible and $u \ne 0$ , we have $0 \neq H u = A u$ and so $\lvert \lvert A u \rvert \rvert ^ { 2 } > 0$ . It follows in particular that
+
+$$
+- \alpha u ^ { \mathsf { T } } H ^ { 2 } u = - \alpha u ^ { \mathsf { T } } ( S ^ { \mathsf { T } } - A ^ { \mathsf { T } } ) ( S + A ) u = \alpha u ^ { \mathsf { T } } A ^ { \mathsf { T } } A u = \alpha \| A u \| ^ { 2 } > 0 .
+$$
+
+Using positive semi-definiteness of $M ^ { - 1 } G _ { 1 } M$ ,
+
+$$
+\begin{array} { r l } & { u ^ { \mathsf { T } } M ^ { - 1 } G M u = u ^ { \mathsf { T } } M ^ { - 1 } G _ { 1 } M u + u ^ { \mathsf { T } } M ^ { - 1 } G _ { 2 } M u } \\ & { \qquad \geq - \alpha u ^ { \mathsf { T } } M ^ { - 1 } H ^ { 2 } M u } \\ & { \qquad = - \alpha u ^ { \mathsf { T } } H ^ { 2 } u + O ( \alpha ^ { 2 } ) } \\ & { \qquad = \alpha \| A u \| ^ { 2 } + O ( \alpha ^ { 2 } ) > 0 } \end{array}
+$$
+
+for $\alpha > 0$ small enough. We conclude that for any $u \in S ^ { m }$ there is $\epsilon _ { u } > 0$ such that
+
+$$
+u ^ { \mathsf { T } } M ^ { - 1 } G M u > 0
+$$
+
+for all $0 < \alpha < \epsilon _ { u }$ , where $g ( \alpha , u ) = u ^ { \mathsf { T } } M ^ { - 1 } G M u$ is a function $g : \mathbb { R } ^ { + } \times S ^ { m }  \mathbb { R }$ with $S ^ { m }$ compact. By Lemma D.9, this can be extended uniformly with some $\epsilon > 0$ such that
+
+$$
+u ^ { \mathsf { T } } M ^ { - 1 } G M u > 0
+$$
+
+for all $u \in S ^ { m }$ and $0 < \alpha < \epsilon$ . It follows that $M ^ { - 1 } G M$ is positive definite for all $0 < \alpha < \epsilon$ and thus $G$ is positive stable for $\alpha$ in the same range, by similarity. □
+
+Corollary D.6. LookAhead converges locally to stable fixed points for $\alpha > 0$ sufficiently small.
+
+Proof. For any SFP $\bar { \theta }$ we have $\xi ( \bar { \theta } ) = 0$ and $H ( \bar { \theta } ) \succeq 0$ invertible by definition, with diagonal blocks $\nabla _ { i i } L ^ { i }$ symmetric by twice continuous differentiability. We are done by the result above and Corollary D.3. □
+
+We now prove that local convergence results descend to SOS. The following lemma establishes the crucial claim that our criterion for $p$ is $C ^ { 1 }$ in neighbourhoods of fixed points. This is necessary to invoke analytical arguments including Ostrowski’s Theorem, and would be untrue globally.
+
+Lemma D.7. If $\bar { \theta }$ is a fixed point and $\alpha$ is sufficiently small then $p = \| \boldsymbol { \xi } \| ^ { 2 }$ in a neighbourhood of $\bar { \theta }$
+
+Proof. First note that $\xi ( \bar { \theta } ) = 0$ , so there is a (bounded) neighbourhood $V$ of $\bar { \theta }$ such that $\| \xi ( \theta ) \| < b$ for all $\theta \ \in \ V$ , for any choice of hyperparameter $b \in \mathsf { \Gamma } ( 0 , 1 )$ . In particular $p _ { 2 } ( \underline { { \theta } } ) \ : = \ : \| \xi ( \theta ) \| ^ { 2 }$ by definition of the second criterion. We want to show that $\dot { p ( \theta ) } \stackrel { - } { = } p _ { 2 } ( \theta )$ near $\theta$ , or equivalently $p _ { 1 } ( \theta ) \geq p _ { 2 } ( \theta )$ . Since $p _ { 2 } ( \theta ) = \| \xi ( \theta ) \| ^ { 2 } < b ^ { 2 } < 1$ in $V$ , it remains only to show that
+
+$$
+\frac { - a \| \xi _ { 0 } \| ^ { 2 } } { \langle - \alpha \chi , \xi _ { 0 } \rangle } \geq \| \xi ( \theta ) \| ^ { 2 }
+$$
+
+in some neighbourhood $U \subseteq V$ of $\bar { \theta }$ , for any choice of hyperparameter $a \in ( 0 , 1 )$ . Now by boundedness of $V$ and continuity of $\chi$ , there exists $c > 0$ such that $\lVert - \alpha \boldsymbol { \chi } ( \theta ) \rVert = \alpha ^ { 2 } \lVert \boldsymbol { \chi } ( \theta ) \rVert < c$ for all $\theta \in V$ and bounded $\alpha$ . It follows by Cauchy-Schwartz that
+
+$$
+\frac { - a \| \xi _ { 0 } \| ^ { 2 } } { \langle - \alpha \chi , \xi _ { 0 } \rangle } \geq \frac { a \| \xi _ { 0 } \| } { \| - \alpha \chi \| } > a \| \xi _ { 0 } \| / c
+$$
+
+in $V$ . Now note that
+
+$$
+\| \xi _ { 0 } \| = \| ( I - \alpha H _ { o } ) \xi \| \ge d \| \xi \|
+$$
+
+in $V$ , for some $d > 0$ and $\alpha$ sufficiently small, by boundedness of $V$ and continuity of $H _ { o }$ . Finally there is a sub-neighbourhood $U \subset V$ such that $\| \xi ( \theta ) \| < a d / c$ for all $\theta \in U$ , so that $a d \| \xi \| / c >$ $\| \xi ( \theta ) \| ^ { 2 }$ and hence
+
+$$
+\frac { - a \| \xi _ { 0 } \| ^ { 2 } } { \langle - \alpha \chi , \xi _ { 0 } \rangle } > \| \xi \| ^ { 2 } = p _ { 2 }
+$$
+
+in $U$ . Hence $p ( \theta ) = \operatorname* { m i n } \{ p _ { 1 } ( \theta ) , p _ { 2 } ( \theta ) \} = p _ { 2 } ( \theta ) = \| \xi ( \theta ) \| ^ { 2 }$ for all $\theta \in U$ , as required.
+
+Theorem D.8. SOS converges locally to stable fixed points for $\alpha > 0$ sufficiently small.
+
+Proof. Though the criterion for $p$ is dual, we will only use the second part. More precisely,
+
+$$
+p = \operatorname* { m i n } \{ p _ { 1 } , p _ { 2 } \} \leq p _ { 2 } = \| \xi \|
+$$
+
+if $\| \xi \| < b$ . The aim is to show that if $\bar { \theta }$ is an SFP then $\nabla \xi _ { p } ( \bar { \theta } )$ is positive stable for small $\alpha$ , using Ostrowski to conclude as usual. The first problem we face is that $\nabla { \xi } _ { p }$ does not exist everywhere, since $p ( \theta )$ is not a continuous function. However we know by Lemma D.7 that $p = \| \xi \| ^ { 2 }$ in a neighbourhood $U$ of $\bar { \theta }$ , so $\xi _ { p }$ is continuously differentiable in $U$ . Moreover, $p ( { \bar { \theta } } ) = \| \xi ( { \bar { \theta } } ) \| ^ { 2 } = 0$ with gradient
+
+$$
+\nabla p (  { \bar { \theta } } ) = 2 H ^ { \mathsf { T } } \xi (  { \bar { \theta } } ) = 0
+$$
+
+by definition of fixed points. It follows that
+
+$$
+\nabla \xi _ { p } ( { \bar { \theta } } ) = ( I - \alpha H _ { o } ) H ( { \bar { \theta } } ) - \alpha \nabla p ( { \bar { \theta } } ) \chi ( { \bar { \theta } } ) - \alpha p ( { \bar { \theta } } ) \nabla \chi ( { \bar { \theta } } ) = ( I - \alpha H _ { o } ) H ( { \bar { \theta } } )
+$$
+
+which is identical to LookAhead. This is positive stable for all $0 < \alpha < \epsilon$ , and $\bar { \theta }$ is a fixed point of the iteration since
+
+$$
+\xi _ { p } ( \bar { \theta } ) = ( I - \alpha H _ { o } ) \xi ( \bar { \theta } ) - \alpha p ( \bar { \theta } ) \chi ( \bar { \theta } ) = 0 .
+$$
+
+We conclude by Corollary D.3 that SOS converges locally to SFP for any $a , b \in ( 0 , 1 )$ and $\alpha$ sufficiently small.
+
+Lemma D.9. Let $g : \mathbb { R } ^ { + } \times Y  Z$ continuous with $Y$ compact and $Z \subseteq \mathbb { R }$ . Assume that for any $u \in Y$ there is $\epsilon _ { u } > 0$ such that $g ( \alpha , u ) > 0$ for all $0 < \alpha < \epsilon _ { u }$ . Then there exists $\epsilon > 0$ such that $g ( \alpha , u ) > 0$ for all $0 < \alpha < \epsilon$ and $u \in Y$ .
+
+Proof. For any $u \in Y$ there is $\epsilon _ { u } > 0$ such that
+
+$$
+( 0 , \epsilon _ { u } ) \times \{ u \} \subseteq g ^ { - 1 } ( 0 , \infty ) .
+$$
+
+We would like to extend this uniformly in $u$ , namely prove that
+
+$$
+\left( 0 , \epsilon \right) \times Y \subseteq g ^ { - 1 } ( 0 , \infty ) .
+$$
+
+for some $\epsilon > 0$ . Now $g ^ { - 1 } ( 0 , \infty )$ is open by continuity of $g$ , so each $( 0 , \epsilon _ { u } ) \times \{ u \}$ has a neighbourhood $X _ { u }$ contained in $g ^ { - 1 } ( 0 , \infty )$ . Open sets in a product topology are unions of open products, so
+
+$$
+X _ { u } = \bigcup _ { x } U _ { x } \times V _ { x } .
+$$
+
+In particular $( 0 , \epsilon _ { u } ) \subseteq \bigcup _ { x } U _ { x }$ and at least one $V _ { x }$ contains $u$ , so we can take the open neighbourhood to be
+
+$$
+X _ { u } = ( 0 , \epsilon _ { u } ) \times V _ { u } \subseteq g ^ { - 1 } ( 0 , \infty )
+$$
+
+for some neighbourhood $V _ { u }$ of $u$ . In particular $Y \subseteq \bigcup _ { u \in Y } V _ { u }$ , and by compactness there is a finite cover $\textstyle Y \subseteq \bigcup _ { i = 1 } ^ { k } V _ { u _ { i } }$ . Letting $\epsilon = \mathrm { m i n } \{ \epsilon _ { i } \} _ { i = 1 } ^ { k } > 0$ , we obtain the required inclusion
+
+$$
+( 0 , \epsilon ) \times Y \subseteq ( 0 , \epsilon ) \times \bigcup _ { i = 1 } ^ { k } V _ { u _ { i } } = \bigcup _ { i = 1 } ^ { k } ( 0 , \epsilon ) \times V _ { u _ { i } } \subseteq \bigcup _ { i = 1 } ^ { k } ( 0 , \epsilon _ { i } ) \times V _ { u _ { i } } \subseteq g ^ { - 1 } ( 0 , \infty ) .
+$$
+
+# E NON-CONVERGENCE PROOFS
+
+Lemma E.1. Let $a _ { k }$ and $b _ { k }$ be sequences of real numbers, and define $c _ { k } = \operatorname* { m i n } \{ a _ { k } , b _ { k } \}$ . If
+
+$$
+L = \operatorname* { l i m } _ { k \to \infty } c _ { k } \qquad { \mathrm { a n d ~ } } \qquad L ^ { \prime } = \operatorname* { l i m } _ { k \to \infty } a _ { k }
+$$
+
+both exist then $L \leq L ^ { \prime }$ .
+
+Proof. Assume for contradiction that $L > L ^ { \prime }$ , then there exists $\delta > 0$ such that $L > L ^ { \prime } + \delta$ . By definition of limits, there exist $M , N \in \mathbb { N }$ such that
+
+$$
+| c _ { k } - L | < \delta / 2
+$$
+
+and
+
+$$
+| a _ { k ^ { \prime } } - L ^ { \prime } | < \delta / 2
+$$
+
+for all $k \geq M , k ^ { \prime } \geq N$ . Expanding the absolute value, this implies
+
+$$
+L - \delta / 2 < c _ { k } < L + \delta / 2 \qquad \mathrm { { a n d } } \qquad L ^ { \prime } - \delta / 2 < a _ { k } < L ^ { \prime } + \delta / 2
+$$
+
+for all $k \geq \operatorname* { m a x } \{ M , N \}$ . Now $c _ { k } \leq a _ { k }$ for all $k$ , hence
+
+$$
+L - \delta / 2 < c _ { k } \leq a _ { k } < L ^ { \prime } + \delta / 2
+$$
+
+which implies the contradiction
+
+$$
+L < L ^ { \prime } + \delta .
+$$
+
+Proposition E.2. If SOS converges to $\bar { \theta }$ and $\alpha > 0$ is small then $\bar { \theta }$ is a fixed point of the game.
+
+Proof. The iterative procedure is given by
+
+$$
+\theta _ { k + 1 } = F ( \theta _ { k } ) = \theta _ { k } - \alpha \xi _ { p } ( \theta _ { k } ) .
+$$
+
+If $\theta _ { k } \to \bar { \theta }$ as $k \to \infty$ then taking limits on both sides of the iteration yields
+
+$$
+\bar { \theta } = \bar { \theta } - \alpha \operatorname * { l i m } _ { k \to \infty } \xi _ { p } ( \theta _ { k } )
+$$
+
+and so $\operatorname* { l i m } _ { k } \xi _ { p } ( \theta _ { k } ) = 0$ , omitting $k \to \infty$ for convenience. It follows by continuity that
+
+$$
+\xi _ { 0 } ( \bar { \theta } ) + \operatorname * { l i m } _ { k } p ( \theta _ { k } ) - \alpha \chi ( \bar { \theta } ) = 0 ,
+$$
+
+noting that $p ( \theta )$ is not a globally continuous function. Assume for contradiction that $\xi _ { 0 } ( \bar { \theta } ) \not = 0$ .   
+There are two cases to distinguish for clarity.
+
+(i) First assume $\langle - \alpha \chi , \xi _ { 0 } \rangle ( \bar { \theta } ) \geq 0 .$ . Note that $\begin{array} { r } { \operatorname* { l i m } _ { k } p ( \theta _ { k } ) \ge 0 } \end{array}$ since $p ( \theta ) \geq 0$ for all $\theta$ , and so
+
+$$
+\langle \operatorname* { l i m } _ { k } \xi _ { p } ( \theta _ { k } ) , \xi _ { 0 } ( \bar { \theta } ) \rangle = \operatorname* { l i m } _ { k } p ( \theta _ { k } ) \langle - \alpha \chi , \xi _ { 0 } \rangle ( \bar { \theta } ) + \| \xi _ { 0 } ( \bar { \theta } ) \| ^ { 2 } > 0 .
+$$
+
+This is a contradiction since $\begin{array} { r } { \operatorname* { l i m } _ { k } \xi _ { p } ( \theta _ { k } ) = 0 } \end{array}$ .
+
+(ii) Otherwise, $\langle - \alpha \chi , \xi _ { 0 } \rangle ( \bar { \theta } ) < 0$ and hence $\langle - \alpha \chi , \xi _ { 0 } \rangle ( \theta ) < 0$ in a neighbourhood. In particular there exists $N \in \mathbb N$ such that
+
+$$
+\langle - \alpha \chi , \xi _ { 0 } \rangle ( \theta _ { k } ) < 0
+$$
+
+for all $k \geq N$ . In particular
+
+$$
+p _ { 1 } ( \theta _ { k } ) = \operatorname* { m i n } \left\{ 1 , \frac { - a \| \xi _ { 0 } ( \theta _ { k } ) \| ^ { 2 } } { \langle - \alpha \chi , \xi _ { 0 } \rangle ( \theta _ { k } ) } \right\}
+$$
+
+for all $k \geq N$ . Now notice that
+
+$$
+\operatorname* { l i m } _ { k } p ( \theta _ { k } ) = \operatorname* { l i m } _ { k } \operatorname* { m i n } \left\{ 1 , \frac { - a \lVert \xi _ { 0 } ( \theta _ { k } ) \rVert ^ { 2 } } { \langle - \alpha \mathcal { X } , \xi _ { 0 } \rangle ( \theta _ { k } ) } , p _ { 2 } ( \theta _ { k } ) \right\} ,
+$$
+
+which implies
+
+$$
+\operatorname* { l i m } _ { k } p ( \theta _ { k } ) \leq \operatorname* { l i m } _ { k } \frac { - a \| \xi _ { 0 } ( \theta _ { k } ) \| ^ { 2 } } { \langle - \alpha \chi , \xi _ { 0 } \rangle ( \theta _ { k } ) } = \frac { - a \| \xi _ { 0 } ( \bar { \theta } ) \| ^ { 2 } } { \langle - \alpha \chi , \xi _ { 0 } \rangle ( \bar { \theta } ) }
+$$
+
+by continuity and Lemma E.1. Finally we conclude
+
+$$
+\langle \operatorname* { l i m } _ { k } \xi _ { p } , \xi _ { 0 } \rangle ( \theta _ { k } ) = \operatorname* { l i m } _ { k } p ( \theta _ { k } ) \langle - \alpha \chi , \xi _ { 0 } \rangle ( { \bar { \theta } } ) + \| \xi _ { 0 } ( { \bar { \theta } } ) \| ^ { 2 } \geq - a \| \xi _ { 0 } ( { \bar { \theta } } ) \| ^ { 2 } + \| \xi _ { 0 } ( { \bar { \theta } } ) \| ^ { 2 } > 0
+$$
+
+for any $a \in ( 0 , 1 )$ , a contradiction.
+
+In both cases a contradiction is obtained, hence $\xi _ { 0 } ( \bar { \theta } ) = 0 = ( I - \alpha H _ { o } ) \xi ( \bar { \theta } )$ . Now note that $( I - \alpha H _ { o } ) ( \bar { \theta } )$ is singular iff $H _ { o } ( \bar { \theta } )$ has an eigenvalue $1 / \alpha$ , which is impossible for $\alpha$ sufficiently small. Hence $( I - \alpha { \bf \bar { { H } } } _ { o } ) \xi ( \bar { \theta } ) = 0$ implies $\xi ( \bar { \theta } ) = 0$ , as required. □
+
+Now assume that $\theta$ is initialised randomly (or with arbitrarily small noise around a point), as is standard in ML. We prove that SOS locally avoids strict saddles using the Stable Manifold Theorem, inspired from Lee et al. (2017).
+
+Theorem E.3 (Stable Manifold Theorem). Let $\bar { x }$ be a fixed point for the $C ^ { 1 }$ local diffeomorphism $F : U \to \mathbb { R } ^ { d }$ , where $U$ is a neighbourhood of $\bar { x }$ in $\mathbb { R } ^ { d }$ . Let $E ^ { s } \oplus E ^ { u }$ be the generalised eigenspaces of $\nabla F ( { \bar { x } } )$ corresponding to eigenvalues with $| \lambda | \le 1$ and $| \lambda | > 1$ respectively. Then there exists a local stable center manifold $W$ with tangent space $E ^ { s }$ at $\bar { x }$ and a neighbourhood $B$ of $\bar { x }$ such that $F ( W ) \cap B \subset W$ and $\cap _ { n = 0 } ^ { \infty } F ^ { - n } ( B ) \subset { \bar { W } }$ .
+
+In particular, if $\nabla F ( { \bar { x } } )$ has at least one eigenvalue $| \lambda | > 1$ then $E ^ { u }$ has dimension at least 1. Since $W$ has tangent space $E ^ { s }$ at $\bar { x }$ , with codimension at least one, it follows that $W$ has measure zero in $\mathbb { R } ^ { d }$ . This is central in proving that the set of initial points in a neighbourhood which converge through SOS to a given strict saddle $\bar { \theta }$ has measure zero.
+
+Theorem E.4. SOS locally avoids strict saddles almost surely, for $\alpha > 0$ sufficiently small.
+
+Proof. Let $\bar { \theta }$ a strict saddle and recall that SOS is given by
+
+$$
+{ \cal F } ( \theta ) = \theta - \alpha ( I - \alpha H _ { o } ) \xi ( \theta ) + \alpha ^ { 2 } p ( \theta ) \chi ( \theta ) .
+$$
+
+Recall by Lemma D.7 that $p ( \theta ) = \lVert \xi ( \theta ) \rVert ^ { 2 }$ for all $\theta$ in a neighbourhood $U$ of $\bar { \theta }$ . Restricting $F$ to $U$ all terms involved are continuously differentiable and
+
+$$
+\nabla F ( { \bar { \theta } } ) = I - \alpha ( I - \alpha H _ { o } ) H ( { \bar { \theta } } )
+$$
+
+by assumption that $\xi ( \bar { \theta } ) = 0$ . Since all terms except $I$ are of order at least $\alpha$ , $\nabla F ( { \bar { \theta } } )$ is invertible for all $\alpha$ sufficiently small. By the inverse function theorem, there exists a neighbourhood $V$ of $\bar { \theta }$ such that $F$ is has a continuously differentiable inverse on $V$ . Hence $F$ restricted to $U \cap V$ is a $C ^ { 1 }$ diffeomorphism with fixed point $\bar { \theta }$ .
+
+By definition of strict saddles, $H ( \bar { \theta } )$ has a negative eigenvalue. It follows by continuity that $( I -$ $\overset { \cdot } { \alpha { H _ { o } } } ) H ( \bar { \theta } )$ also has a negative eigenvalue $a + i b$ with $a < 0$ for $\alpha$ sufficiently small. Finally,
+
+$$
+\nabla F ( { \bar { \theta } } ) = I - \alpha ( I - \alpha H _ { o } ) H ( { \bar { \theta } } )
+$$
+
+has an eigenvalue $\lambda = 1 - \alpha a - i \alpha b$ with
+
+$$
+| \lambda | = 1 - 2 \alpha a + \alpha ^ { 2 } ( a ^ { 2 } + b ^ { 2 } ) \geq 1 - 2 \alpha a > 1 .
+$$
+
+It follows that $E ^ { s }$ has codimension at least one, implying in turn that the local stable set $W$ has measure zero. We can now prove that
+
+$$
+Z = \{ \theta \in U \cap V \mid \operatorname* { l i m } _ { n \to \infty } F ^ { n } ( \theta ) = { \bar { \theta } } \}
+$$
+
+has measure zero, or in other words, that local convergence to $\bar { \theta }$ occurs with zero probability. Let $B$ the neighbourhood guaranteed by the Stable Manifold Theorem, and take any $\theta \in Z$ . By definition of convergence there exists $N \in \mathbb N$ such that $F ^ { N + n } ( \theta ) \in B$ for all $n \geq 0$ , so that
+
+$$
+F ^ { N } ( \theta ) \in \cap _ { n = 0 } ^ { \infty } F ^ { - n } ( B ) \subset W
+$$
+
+by the Stable Manifold Theorem. This implies that $\theta \in F ^ { - N } ( W )$ , and finally $\theta \in \cup _ { n \in \mathbb { N } } F ^ { - n } ( W )$ . Since $\theta$ was arbitrary, we obtain the inclusion
+
+$$
+Z \subseteq \cup _ { n \in \mathbb { N } } F ^ { - n } ( W ) .
+$$
+
+Now $F ^ { - 1 }$ is $C ^ { 1 }$ , hence locally Lipschitz and thus preserves sets of measure zero, so that $F ^ { - n } ( W )$ has measure zero for each $n$ . Countable unions of measure zero sets are still measure zero, so we conclude that $Z$ also has measure zero. In other words, SOS converges to $\bar { \theta }$ with zero probability upon random initialisation of $\theta$ in $U$ . □
+
+![](images/543eef4b5a13920fd5d209b26d3ac2566960ad851eb14b0a84392a5710ed0b55.jpg)  
+Figure 5: Ground truth in the Gaussian mixture experiment.
+
+# F FURTHER GAUSSIAN MIXTURE EXPERIMENTS
+
+In the Gaussian mixture experiment, data is sampled from a highly multimodal distribution designed to probe the tendency to collapse onto a subset of modes during training, given in Figure 5.
+
+The generator distribution and KL divergence are given at $\{ 2 \mathrm { k } , 4 \mathrm { k } , 6 \mathrm { k } , 8 \mathrm { k } \}$ iterations for LA, LOLA and SGA in Figure 6. LOLA and SGA successfully spread mass across all Gaussians. LookAhead displays mode collapse and hopping in early stages, but begins to incorporate further mixtures near $8 k$ iterations. We ran further iterations and discovered that LookAhead eventually spreads mass across all mixtures, though very slowly. Comparing with results for NL/CO/SOS in the main text, we see that CO/SOS/LOLA/SGA are equally successful in qualitative terms.
+
+Note that SOS/CO are slightly superior with respect to KL divergence after $6 { - } 8 \mathbf { k }$ iterations, though LOLA is initially faster. This may be due only to random sampling. We also noticed experimentally that LOLA often moves away from the correct distribution after 8-10k iterations (not shown), while SOS stays stable in the long run. This may occur thanks to the two-part criterion encouraging convergence, while LOLA continually attempts to exploit opponent learning.
+
+Finally we plot $\| \xi \|$ at all iterations up to $1 2 k$ for SOS, LA and NL in Figure 7 (other algorithms are omitted for visibility). This gives further evidence of SOS converging quite rapidly to the correct distribution, while NL perpetually suffers from mode hopping and LA lags behind significantly.
+
+![](images/f175a0315d3a99e2dfdb6bf795e5ac1005836f6cc4952b35916b28a428a1bf31.jpg)  
+Figure 6: Generator distribution at sampled iterations for LA/LOLA/SGA. LA suffers in the early stages from mode collapse and hopping, but incorporates more mixtures later on. LOLA and SGA learn the correct mixture of Gaussians. Below each plot: KL divergence $D _ { K L } ( P \parallel Q )$ from generator $P$ to ground truth $Q$ , estimated from 25600 samples. Best result at each iteration in bold.
+
+![](images/92efd196354eae50fac088bf70cf3fb1c5cfa5c7a0c401934a7807cc33521d4a.jpg)  
+Figure 7: Semilog plot of $\| \xi \|$ at each iteration for SOS, LA and NL.

parse/train/r1gR2sC9FX/r1gR2sC9FX.md

@@ -0,0 +1,606 @@
+# ON THE SPECTRAL BIAS OF NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Neural networks are known to be a class of highly expressive functions able to fit even random input-output mappings with $1 0 0 \%$ accuracy. In this work we present properties of neural networks that complement this aspect of expressivity. By using tools from Fourier analysis, we show that deep ReLU networks are biased towards low frequency functions, meaning that they cannot have local fluctuations without affecting their global behavior. Intuitively, this property is in line with the observation that over-parameterized networks find simple patterns that generalize across data samples. We also investigate how the shape of the data manifold affects expressivity by showing evidence that learning high frequencies gets easier with increasing manifold complexity, and present a theoretical understanding of this behavior. Finally, we study the robustness of the frequency components with respect to parameter perturbation, to develop the intuition that the parameters must be finely tuned to express high frequency functions.
+
+# 1 INTRODUCTION
+
+While universal approximation properties of neural networks have been known since the early 90s (Hornik et al., 1989; Cybenko, 1989; Leshno et al., 1993; Barron, 1993), recent research has shed light on the mechanisms underlying such expressivity (Montufar et al., 2014; Raghu et al., 2016; Poole et al., 2016). At the same time, deep neural networks, despite being massively overparameterized, have been remarkably successful at generalizing to natural data. This fact is at odds with the traditional notions of model complexity and their empirically demonstrated ability to fit arbitrary random data to perfect accuracy (Zhang et al., 2017a; Arpit et al., 2017). It has prompted the recent investigations of possible implicit regularization mechanisms inherent in the learning process, inducing biases towards low complexity solutions (Soudry et al., 2017; Poggio et al., 2018; Neyshabur et al., 2017).
+
+In this work, our main goal is to expose one such bias by taking a closer look at neural networks through the lens of Fourier analysis1. We focus the discussion on ReLU networks, whose piecewise linear structure enables an analytic treatment. While they can approximate arbitrary functions, we find that these networks favour low frequency ones; in other words, they exhibit a bias towards smooth functions, a phenomenon we call the spectral bias2. We find that this bias manifests itself not just in the process of learning, but also in the parameterization of the model itself: in fact we show that the lower frequencies of trained networks are more robust with respect to random parameter perturbations. Finally, we also exhibit and analyze a rather intricate interplay between the spectral bias and the geometry of the data manifold: we show that high frequencies get easier to learn when the data lies on a lower dimensional manifold of complex shape embedded in the input space.
+
+# CONTRIBUTIONS
+
+1. We exploit the piecewise-linear structure of ReLU networks to evaluate and bound its Fourier spectrum.   
+2. We demonstrate the peculiar behaviour of neural networks with illustrative and minimal experiments and find evidence of a spectral bias: i.e. lower frequencies are learned first.   
+3. We illustrate how the manifold hypothesis adds a layer of subtlety by showing how the geometry of the data manifold attenuates the spectral bias in a non-trivial way. We present a theoretical analysis of this phenomenon and derive conditions on the manifolds that facilitate learning higher frequencies.   
+4. Given a trained network, we investigate the relative robustness of the lower frequencies with respect to random perturbations of the network parameters.
+
+The paper is organized as follows. In Section 2, we derive the Fourier spectrum of deep ReLU networks. Section 3 presents minimal experiments that demonstrate the spectral bias of ReLU networks. In Section 4, we study and discuss the role of the geometry of the data manifold. In Section 5, we empirically illustrate and theoretically explain our robustness result.
+
+# 2 FOURIER ANALYSIS OF RELU NETWORKS
+
+# 2.1 PRELIMINARIES
+
+Consider the class of scalar functions $f : \mathbb { R } ^ { d } \mapsto \mathbb { R }$ defined by a ReLU network with $L$ hidden layers of widths $d _ { 1 } , \cdots d _ { L }$ and a single output neuron:
+
+$$
+f ( \mathbf { x } ) = \left( T ^ { ( L + 1 ) } \circ \sigma \circ T ^ { ( L ) } \circ \cdots \circ \sigma \circ T ^ { ( 1 ) } \right) ( \mathbf { x } )
+$$
+
+where each $\boldsymbol { T } ^ { ( k ) } : \mathbb { R } ^ { d _ { k - 1 } }  \mathbb { R } ^ { d _ { k } }$ is an affine function $\ Q _ { 0 } \ = \ d$ and $d _ { L + 1 } = 1$ ) and $\sigma ( { \bf u } ) _ { i } { \mathbf \alpha } = { \mathbf \alpha }$ $\operatorname* { m a x } ( 0 , u _ { i } )$ denotes the ReLU activation function acting elementwise on a vector $\mathbf { u } = ( u _ { 1 } , \cdot \cdot \cdot u _ { n } )$ . In the standard basis, $T ^ { ( k ) } ( { \bf x } ) = W ^ { ( k ) } { \bf x } + { \bf b } ^ { ( k ) }$ for some weight matrix $W ^ { ( k ) }$ and bias vector $\mathbf { b } ^ { ( k ) }$ .
+
+ReLU networks are known to be continuous piece-wise linear (CPWL) functions, where the linear regions are convex polytopes (Raghu et al., 2016; Montufar et al., 2014; Zhang et al., 2018; Arora et al., 2018). Remarkably, the converse is also true: every CPWL function can be represented by a ReLU network (Arora et al., 2018, Theorem 2.1), which in turn endows ReLU networks with universal approximation properties. Given the ReLU network $f$ from Eqn. 1, we can make the piecewise linearity explicit by writing,
+
+$$
+f ( { \bf x } ) = \sum _ { \epsilon } 1 _ { P _ { \epsilon } } ( { \bf x } ) \left( W _ { \epsilon } { \bf x } + { \bf b } _ { \epsilon } \right)
+$$
+
+where $\epsilon$ is an index for the linear regions $P _ { \epsilon }$ and $1 _ { P _ { \epsilon } }$ is the indicator function on $P _ { \epsilon }$ . As shown in Appendix C in more detail, each region corresponds to an activation pattern3 of all hidden neurons of the network, which is a binary vector with components conditioned on the sign of the input of the respective neuron. The $1 \times d$ matrix $W _ { \epsilon }$ is given by
+
+$$
+W _ { \epsilon } = W ^ { ( L + 1 ) } W _ { \epsilon } ^ { ( L ) } \cdot \cdot \cdot W _ { \epsilon } ^ { ( 1 ) }
+$$
+
+where $W _ { \epsilon } ^ { ( k ) }$ is obtained from the original weight $W ^ { ( k ) }$ by setting its $j ^ { t h }$ column to zero whenever the neuron $j$ of the $k ^ { t h }$ layer is inactive.
+
+We will henceforth assume that the input data lies in a bounded domain of $\mathbb { R } ^ { d }$ , say $X = [ - A , A ] ^ { d }$ for some $A > 0$ and thus restrict ourselves to ReLU networks with bounded support4.
+
+# 2.2 FOURIER SPECTRUM
+
+In the following, we study the structure of ReLU networks in the Fourier domain, which is defined as:
+
+$$
+f ( { \bf x } ) = ( 2 \pi ) ^ { d / 2 } \int \tilde { f } ( { \bf k } ) e ^ { i { \bf k } \cdot { \bf x } } { \bf d } { \bf k } , \qquad \tilde { f } ( { \bf k } ) : = \int f ( { \bf x } ) e ^ { - i { \bf k } \cdot { \bf x } } { \bf d } { \bf x }
+$$
+
+where $\mathbf { d x } , \mathbf { d k }$ are the uniform Lebesgue measure on $\mathbb { R } ^ { d }$ and $\tilde { f }$ denotes the Fourier transform of $f$ (see Appendix B for a short recap of the Fourier transform). Lemmas 1 and 2 (proved in appendix D) yield the explicit form of the Fourier components.
+
+Lemma 1. The Fourier transform of ReLU networks decomposes as,
+
+$$
+\tilde { f } ( { \bf k } ) = i \sum _ { \epsilon } \frac { W _ { \epsilon } { \bf k } } { k ^ { 2 } } \tilde { 1 } _ { P _ { \epsilon } } ( { \bf k } )
+$$
+
+where $k = \| \mathbf { k } \|$ and $\begin{array} { r } { \tilde { 1 } _ { P } ( \mathbf { k } ) = \int _ { P } e ^ { - i \mathbf { k } \cdot \mathbf { x } } \mathbf { d } \mathbf { x } } \end{array}$ is the Fourier transform of the indicator function of $P$ .
+
+The Fourier transform of a polytope appearing in Eqn. 5 is a fairly intricate mathematical object; Diaz et al. (2016) develop an elegant procedure for evaluating it in arbitrary dimensions via a recursive application of Stokes theorem. We describe this procedure in detail in Appendix D.2, and present here its main corollary.
+
+Lemma 2. Let $P$ be a full dimensional polytope in $\mathbb { R } ^ { d }$ . The Fourier spectrum of its indicator function $\tilde { 1 } _ { P }$ satisfies the following:
+
+$$
+| \tilde { 1 } _ { P } ( \mathbf { k } ) | = \mathcal { O } \left( \frac { 1 } { k ^ { \Delta _ { \mathbf { k } } ^ { ( P ) } } } \right)
+$$
+
+where $1 \leq \Delta _ { \mathbf { k } } ^ { ( P ) } \leq d ,$ and $\Delta _ { \mathbf { k } } ^ { ( P ) } = j$ when $\mathbf { k }$ lies orthogonal to some $( d - j )$ -dimensional face of $P$
+
+Note that since a polytope has a finite number of facets (of any dimension), the $\mathbf { k }$ ’s for which $\Delta _ { \mathbf { k } } ^ { ( P ) } = j$ for some $j < d$ lie on a finite union of $j$ -dimensional subspaces of $\mathbb { R } ^ { d }$ . The Lebesgue measure of all such lower dimensional subspaces for all such $j$ equals 0, leading us to the conclusion that the spectrum decays as $\mathcal { O } ( k ^ { - d } )$ for almost all directions $\hat { \mathbf { k } }$ of the frequency vector $\mathbf { k }$ in $\mathbb { R } ^ { d }$ .
+
+Lemmas 1, 2 together yield the main result of this section. Given a ReLU network $f$ , its linear regions form a cell decomposition of $\mathbb { R } ^ { d }$ as union of polytopes; we denote by $\mathcal { F }$ the set of faces (of any dimension) of all such polytopes. For $\mathbf { k } \in \mathbb { R } ^ { d }$ , let $\Delta _ { \mathbf { k } }$ be the minimum integer $1 \leq j \leq d$ such that $\mathbf { k }$ lies orthogonal to some $( d - j )$ -dimensional face in $\mathcal { F }$ .
+
+Theorem 1. The Fourier components of the ReLU network $f$ satisfy the following:
+
+$$
+| \tilde { f } ( \mathbf { k } ) | = \mathcal { O } \left( \frac { N _ { f } L _ { f } } { k ^ { \Delta _ { \mathbf { k } } + 1 } } \right)
+$$
+
+where $N _ { f }$ is the number of linear regions and $L _ { f } = \operatorname* { m a x } _ { \epsilon } \| W _ { \epsilon } \| _ { 2 }$ is the Lipschitz constant of $f$ .
+
+Several remarks are in order:
+
+(a) The spectral decay of ReLU networks is highly anisotropic in large dimensions. In almost all directions $\hat { \mathbf { k } }$ of $\mathbb { R } ^ { d }$ , we have $\Delta _ { \mathbf { k } } = d$ , i.e. a $\mathcal { O } ( k ^ { - d - 1 } )$ decay. However, the decay can be as slow as $\mathcal { O } ( k ^ { - 2 } )$ in specific directions orthogonal to the facets bounding linear regions5.
+
+As we prove in Appendix D.3, the Lipschitz constant $L _ { f }$ can be bounded as,
+
+$$
+L _ { f } \le \prod _ { k = 1 } ^ { L + 1 } \| W ^ { ( k ) } \| \le \| W \| _ { \infty } ^ { L + 1 } \sqrt { d } \prod _ { k = 1 } ^ { L } d _ { k }
+$$
+
+where $\| \cdot \|$ is the spectral norm, $W$ is the ravelled parameter vector of the network and $\| \cdot \| _ { \infty }$ is the max-norm. This makes the bound on $L _ { f }$ scale exponentially in depth and polynomial in width. As for the number $N _ { f }$ of linear regions, Montufar et al. (2014) and Raghu et al. (2016) obtain tight bounds that exhibit the same scaling behaviour (Raghu et al., 2016, Theorem 1). This makes the overall bound in Eqn. 7 – and with it, the ability to express larger frequencies – scale exponentially in depth and polynomially in width6. This result complements the well-known universal approximation property of neural networks by explicitly incorporating a control on the capacity7 of the network, namely the width, depth and the norm of parameters. Architecture dependent controls on approximation have been formalized in the literature through approximation bounds and depth separation results, see e.g Barron (1993); Telgarsky (2016); Eldan & Shamir (2016).
+
+![](images/60b7da899e023d2e5531b01491b2e2ec8f955330412b71ebebd0299bd6c1101f.jpg)  
+Figure 1: Left (a, b): Evolution of the spectrum ( $\mathbf { \dot { x } }$ -axis for frequency) during training (y-axis). The colors show the measured amplitude of the network spectrum at the corresponding frequency, normalized by the target amplitude at the same frequency (i.e. $\rvert \tilde { f } _ { k _ { i } } \rvert / A _ { i } )$ and the colorbar is clipped between 0 and 1. Right (a, b): Evolution of the spectral norm $\mathbf { \widetilde { y } }$ -axis) of each layer during training $\mathbf { \bar { x } }$ -axis). Figure-set (a) shows the setting where all frequency components in the target function have the same amplitude, and (b) where higher frequencies have larger amplitudes. Gist: We find that even when higher frequencies have larger amplitudes, the model prioritizes learning lower frequencies first. We also find that the spectral norm of weights increases as the model fits higher frequency, which is what we expect from Theorem 1.
+
+(b) For a given architecture (i.e. fixed width and depth), the high frequency contributions of the network can be increased by increasing the norm of the parameters. Assuming the weight norm increases with training iterations, this suggests that the training of ReLU networks might be biased towards lower frequencies. We investigate this fact empirically in the next section.
+
+# 3 LOWER FREQUENCIES ARE LEARNED FIRST
+
+In this section and the ones that follow, we present experiments that illustrate the peculiar behaviour of deep ReLU networks in the Fourier domain. We begin with an experiment to demonstrate that networks tend to fit lower frequencies first during training. We refer to this phenomenon as the spectral bias, and discuss it in light of the results of Section 2.
+
+Experiment 1. The setup is as follows8 : Given frequencies $\kappa = ( k _ { 1 } , k _ { 2 } , . . . )$ with corresponding amplitudes ${ \boldsymbol { \alpha } } = ( A _ { 1 } , A _ { 2 } , \ldots )$ , and phases $\phi = ( \varphi _ { 1 } , \varphi _ { 2 } , \ldots )$ , we consider the mapping $\lambda : [ 0 , 1 ] \to \mathbb { R }$ given by
+
+$$
+\lambda ( z ) = \sum _ { i } A _ { i } \sin ( 2 \pi k _ { i } z + \varphi _ { i } ) .
+$$
+
+A 6-layer deep 256-unit wide ReLU network $f _ { \theta }$ is trained to regress $\lambda$ with $\kappa = ( 5 , 1 0 , . . . , 4 5 , 5 0 )$ and $N \ = \ 2 0 0$ input samples spaced equally over $[ 0 , 1 ]$ ; its spectrum $\tilde { f } _ { \theta } ( k )$ in expectation over $\varphi _ { i } \sim U ( 0 , 2 \pi )$ is monitored as training progresses. In the first setting, we set equal amplitude $A _ { i } = 1$ for all frequencies and in the second setting, the amplitude increases from $A _ { 1 } = 0 . 1$ to $A _ { 1 0 } = 1$ . Fig 1 shows the normalized magnitudes $| \tilde { f } _ { \theta } ( k _ { i } ) | / A _ { i }$ at various frequencies, as training progresses. The result is that lower frequencies (i.e. smaller $k _ { i }$ ’s) are regressed first, regardless of their amplitudes.
+
+Discussion. Multiple theoretical aspects may underlie these observations. First, for a fixed architecture, the bound in Theorem 1 allows for larger Fourier coefficients at higher frequencies if the parameter norm is large. However, the parameter norm can increase only gradually during training by gradient descent, which leads to the higher frequencies being learned late in the optimization process. To confirm that the bound indeed increases as the model fits higher frequencies, we plot in Fig 1 the spectral norm of weights of each layer during training for both cases of constant and increasing amplitudes.
+
+Second, consider the Mean Squared Error $\mathrm { M S E } [ f _ { \theta } , \lambda ]$ in terms of the Fourier components: letting $z _ { i } = i / N$ be the training sample points, we have:
+
+$$
+\mathbf { M S E } [ f _ { \theta } , \lambda ] = \frac { 1 } { N } \sum _ { i = 0 } ^ { N - 1 } | f _ { \theta } ( z _ { i } ) - \lambda ( z _ { i } ) | ^ { 2 } = \frac { 1 } { N } \sum _ { k = 0 } ^ { N - 1 } | \tilde { f } _ { \theta } ( k ) - \tilde { \lambda } ( k ) | ^ { 2 } = \mathbf { M S E } [ \tilde { f } _ { \theta } , \tilde { \lambda } ]
+$$
+
+where the second equality follows from Plancherel theorem. We make two observations – first, the square error in input space translates into square error in Fourier domain, with a priori no structural bias towards any particular frequency component9, i.e. all frequencies are weighted the same. Since the same cannot be said about e.g. cross-entropy loss, we use the MSE loss in most of our experiments to avoid a potential confounding factor. Second, the parameterization of the network can be exploited by considering the gradient of the MSE loss w.r.t. parameters,
+
+$$
+\frac { \partial } { \partial \theta } \mathrm { M S E } [ \tilde { f } _ { \theta } , \tilde { \lambda } ] = \frac { 2 } { N } \sum _ { k = 0 } ^ { N - 1 } \mathrm { R e } [ \tilde { f } _ { \theta } ( k ) - \tilde { \lambda } ( k ) ] \frac { \partial \tilde { f } _ { \theta } ( k ) } { \partial \theta } \leq \frac { 2 } { N } \sum _ { k = 0 } ^ { N - 1 } | \tilde { f } _ { \theta } ( k ) - \tilde { \lambda } ( k ) | \underbrace { \left| \frac { \partial \tilde { f } _ { \theta } ( k ) } { \partial \theta } \right| } _ { = \mathcal { O } ( k ^ { - \Delta - 1 } ) }
+$$
+
+where $\operatorname { R e } ( z )$ denotes the real part of $z$ . We find that a bias naturally emerges as a consequence of the spectral decay rate found in Theorem 1, in the sense that the magnitude of the residual $| \tilde { f } _ { \theta } ( k ) - \tilde { \lambda } ( k ) |$ contributes less to the net gradient for large $k$ . This generalizes the argument made in Xu (2018) for two layer sigmoid networks by observing that the gradient w.r.t parameters of the network function inherits the spectral decay rate of the function itself10. In Section 5, we use that the integral of $\tilde { f } _ { \theta }$ w.r.t. the standard measure in parameter space $d \theta$ also inherits the spectral decay rate of $f$ to make a statement about the robustness of $\tilde { f } _ { \theta } ( k )$ against random parameter perturbations.
+
+# 4 NOT ALL MANIFOLDS ARE LEARNED EQUAL
+
+In this section, we investigate the subtleties that arise when the data lies on a lower dimensional manifold embedded in the higher dimensional input space of the model (Goodfellow et al., 2016). We find that the shape of the data-manifold impacts the learnability of high frequencies in a nontrivial way. As we shall see, this is because low frequencies functions in the input space may have high frequency components when restricted to lower dimensional manifolds of complex shapes. To systematically investigate the impact of manifold shape on the spectral bias, we demonstrate results in an illustrative minimal setting11 free from unwanted confounding factors. We also present a mathematical exposition of the relationship between the Fourier spectrum of the network, the spectrum of the target function defined on the manifold, and the geometry of the manifold itself.
+
+![](images/9fd9898b2b04de42311764b720afd1b54750a79037eb3216f486dc23872fe061.jpg)  
+Figure 2: Functions learned by two identical networks (up to initialization) to classify the binarized value of a sine wave of frequency $k = 2 0 0$ defined on a $\gamma _ { L = 2 0 }$ manifold. Both yield close to perfect accuracy for the samples defined on the manifold (scatter plot), yet they differ significantly elsewhere. The shaded regions show the predicted class (Red or Blue) whereas contours show the confidence (absolute value of logits).
+
+Manifold hypothesis. We consider the case where the data lies on a lower dimensional data manifold $\mathcal { M } \subset \mathbf { \bar { \mathbb { R } } } ^ { d }$ embedded in input space, which we assume to be the image $\gamma ( [ 0 , 1 ] ^ { m } )$ of some injective mapping $\gamma : [ 0 , 1 ] ^ { m } \overset { \cdot } {  } \mathbb { R } ^ { d }$ defined on a lower dimensional latent space $[ 0 , 1 ] ^ { m }$ . Under this hypothesis and in the context of the standard regression problem, a target function $\dot { \tau : \mathcal { M } }  \mathbb { R }$ defined on data manifold can identified with a function $\lambda = \tau \circ \gamma$ defined on the latent space. Regressing $\tau$ is therefore equivalent to finding $f :  { \mathbb { R } ^ { d } } \to  { \mathbb { R } }$ such that $f \circ \gamma$ matches $\lambda$ . Further, assuming that the data probability distribution $\mu$ supported on $\mathcal { M }$ is induced by $\gamma$ from the uniform distribution $U$ in the latent space $[ 0 , 1 ] ^ { m }$ , the mean square error can be expressed as,
+
+$$
+\begin{array} { r } { \mathbf { M S E } _ { \mu } ^ { ( \mathbf { x } ) } [ f , \tau ] = \mathbb { E } _ { \mathbf { x } \sim \mu } | f ( \mathbf { x } ) - \tau ( \mathbf { x } ) | ^ { 2 } = \mathbb { E } _ { \mathbf { z } \sim U } | ( f ( \gamma ( \mathbf { z } ) ) - \lambda ( \mathbf { z } ) | ^ { 2 } = \mathbf { M S E } _ { U } ^ { ( \mathbf { z } ) } [ f \circ \gamma , \lambda ] } \end{array}
+$$
+
+Observe that there is a vast space of degenerate solutions $f$ that minimize the mean squared error – namely all functions on $\mathbb { R } ^ { d }$ that yield the same function when restricted to the data manifold $\mathcal { M }$ .
+
+Our findings from the previous section suggest that neural networks are biased towards expressing a particular subset of such solutions, namely those that are low frequency. It is also worth noting that there exist methods that restrict the space of solutions: notably adversarial training (Goodfellow et al., 2014) and Mixup (Zhang et al., 2017b).
+
+Experimental set up. The experimental setting is designed to afford control over both the shape of the data manifold and the target function defined on it. We will consider the family of curves in $\mathbb { R } ^ { 2 }$ generated by mappings $\gamma _ { L } : [ 0 , 1 ] \to \mathbb { R } ^ { 2 }$ given by
+
+$$
+\gamma _ { L } ( z ) = R _ { L } ( z ) ( \cos ( 2 \pi z ) , \sin ( 2 \pi z ) ) { \mathrm { ~ w h e r e ~ } } R _ { L } ( z ) = 1 + { \frac { 1 } { 2 } } \sin ( 2 \pi L z )
+$$
+
+Here, $\gamma _ { L } ( [ 0 , 1 ] )$ defines the data-manifold and corresponds to a flower-shaped curve with $L$ petals, or a unit circle when $L = 0$ (see e.g. Fig 2). Given a signal $\lambda : [ 0 , 1 ] \to \overline { { \mathbb { R } } }$ defined on the latent space [0, 1], the task entails learning a network $f : \mathbb { R } ^ { 2 }  \bar { \mathbb { R } }$ such that $f \circ \gamma _ { L }$ matches the signal $\lambda$ .
+
+Experiment 2. The set-up is similar to that of Experiment 1, and $\lambda$ is as defined in Eqn. 9 with frequencies $\kappa = ( 2 0 , 4 0 , . . . , 1 8 0 , 2 0 0 )$ , and amplitudes $A _ { i } = 1 \forall i$ . The model $f$ is trained on the dataset $\{ \gamma _ { L } ( z _ { i } ) , \lambda ( z _ { i } ) \} _ { i = 1 } ^ { N }$ with $N = 1 0 0 0$ uniformly spaced samples $z _ { i }$ between 0 and 1. The spectrum of $f \circ \gamma _ { L }$ in expectation over $\varphi _ { i } \sim U ( 0 , 2 \pi )$ is monitored as training progresses, and the result shown in Fig 3 for $L = 0 , 4 , 1 0 , 1 6$ . Fig 3e shows the corresponding mean squared error curves. More experimental details in appendix A.2.
+
+The results demonstrate a clear attenuation of the spectral bias as $L$ grows. Moreover, Fig 3e suggests that the larger the $L$ , the easier the learning task.
+
+Experiment 3. Here, we adapt the setting of Experiment 2 to binary classification by simply thresholding the function $\lambda$ at 0.5 to obtain a binary target signal. To simplify visualization, we only use signals with a single frequency mode $k$ , such that $\lambda ( z ) = \sin ( 2 \pi k z + \varphi )$ . We train the same network on the resulting classification task with cross-entropy $\mathrm { l o s s } ^ { 1 2 }$ for $k \in \{ 5 0 , 1 0 0 , . . . , 3 5 0 , 4 0 0 \}$ and $L ~ \in ~ \{ 0 , 2 , . . . , \bar { 1 } 8 , 2 0 \}$ . The heatmap in Fig 4 shows the classification accuracy for each $( k , L )$ pair. Fig 2 shows visualizations of the functions learned by the same network, trained on $( k , L ) = ( 2 0 0 , 2 0 )$ under identical conditions up to random initialization.
+
+![](images/3ee11f72f3bc8cd780a7301da488b98f42741daceb5f3003be64ea9c22b1ddd0.jpg)  
+Figure 3: (a,b,c,d): Evolution of the network spectrum $\mathbf { \widetilde { x } }$ -axis for frequency, colorbar for magnitude) during training $\mathbf { \widetilde { y } }$ -axis) for the same target functions defined on manifolds $\gamma _ { L }$ for various $L$ . Since the target function has amplitudes $A _ { i } = 1$ for all frequencies $k _ { i }$ plotted, the colorbar is clipped between 0 and 1. (e): Corresponding learning curves. Gist: Some manifolds (here with larger $L$ ) make it easier for the network to learn higher frequencies than others.
+
+![](images/d76406a83dbc862308e53af93f22589423e05fe04bf3a7ed8b91c429f13e9984.jpg)  
+Figure 4: Heatmap of training accuracies of a network trained to predict the binarized value of a sine wave of given frequency $\mathbf { \widetilde { x } }$ -axis) defined on $\gamma _ { L }$ for various $L$ ( $\mathbf { \widetilde { y } }$ -axis).
+
+Observe that increasing $L$ (i.e. going up a column in Fig 4) results in better (classification) performance for the same target signal. This is the same behaviour as we observed in Experiment 2 (Fig 3a-d), but now with binary cross-entropy loss instead of the MSE.
+
+Discussion. These experiments hint towards a rich interaction between the shape of the manifold and the effective difficulty of the learning task. The key technical reason underlying this phenomenon (as we formalize below) is that the relationship between frequency spectrum of the network $f$ and that of the fit $f \circ \gamma _ { L }$ is mediated by the embedding map $\gamma _ { L }$ . In particular, we will argue that a given signal defined on the manifold is easier to fit when the coordinate functions of the manifold embedding itself has high frequency components. Thus, in our experimental setting, the same signal embedded in a flower with more petals can be captured with lower frequencies of the network.
+
+To understand this mathematically, we address the following questions: given a target function $\lambda$ , how small can the frequencies of a solution $f$ be such that $f \circ \gamma = \lambda \colon$ And further, how does this relate to the geometry of the data-manifold $\mathcal { M }$ induced by $\gamma ?$ To find out, we write the Fourier transform of the composite function,
+
+$$
+\widetilde { ( f \circ \gamma ) } ( 1 ) = \int \mathbf { d k } \widetilde { f } ( \mathbf { k } ) P _ { \gamma } ( 1 , \mathbf { k } ) \quad \mathrm { w h e r e } \quad P _ { \gamma } ( 1 , \mathbf { k } ) = \int _ { [ 0 , 1 ] ^ { m } } \mathbf { d z } e ^ { i ( \mathbf { k } \cdot \gamma ( \mathbf { z } ) - 1 \cdot \mathbf { z } ) }
+$$
+
+The kernel $P _ { \gamma }$ depends on only $\gamma$ and elegantly encodes the correspondence between frequencies $\mathbf { k } \in \mathbb { R } ^ { d }$ in input space and frequencies $1 \in \mathbb { R } ^ { m }$ in the latent space $[ 0 , 1 ] ^ { m }$ . Following a procedure from Bergner et al., we can further investigate the behaviour of the kernel in the regime where the stationary phase approximation is applicable, i.e. when $l ^ { 2 } + k ^ { 2 } \to \infty$ (cf. section 3.2. of Bergner et al.). In this regime, the integral $P _ { \gamma }$ is dominated by critical points $\bar { \bf z }$ of its phase, which satisfy
+
+$$
+\mathbf { l } = J _ { \gamma } ( \bar { \mathbf { z } } ) \mathbf { k }
+$$
+
+where $J _ { \gamma } ( \mathbf { z } ) _ { i j } = \nabla _ { i } \gamma _ { j } ( \mathbf { z } )$ is the $m \times d$ Jacobian matrix of $\gamma$ . Non-zero values of the kernel correspond to pairs $( \mathbf { l } , \mathbf { k } )$ such that Eqn 15 has a solution. Further, given that the components of $\gamma$ (i.e. its coordinate functions) are defined on an interval $[ 0 , 1 ] ^ { m }$ , one can use their Fourier series representation together with Eqn 15 to obtain a condition on their frequencies (shown in appendix D.4). More precisely, we find that the $i$ -th component of the RHS in Eqn 15 is proportional to $\mathbf { p } \tilde { \gamma } _ { i } [ \mathbf { p } ] k _ { i }$ where $\mathbf { p } \in \mathbb { Z } ^ { m }$ is the frequency of the coordinate function $\gamma _ { i }$ . This yields that we can get arbitrarily large frequencies $l _ { i }$ if $\tilde { \gamma } _ { i } [ \mathbf { p } ]$ is large13 enough for large $\mathbf { p }$ , even when $k _ { i }$ is fixed.
+
+![](images/0bf4a723f12e5f29a32a01e41477da5aa49ac0445d5529a8130c70ef80d7c114.jpg)  
+Figure 5: Normalized spectrum of the model $\mathbf { \widetilde { x } }$ -axis for frequency, colorbar for magnitude) with perturbed parameters as a function of parameter perturbation (y-axis). The colormap is clipped between 0 and 1. Observe that the lower frequencies are more robust to parameter perturbations than the higher frequencies.
+
+This is precisely what Experiments 2 and 3 demonstrate in a minimal setting. From Eqn 13, observe that the coordinate functions have a frequency mode at $L$ . For increasing $L$ , it is apparent that the frequency magnitudes $l$ (in the latent space) that can be expressed with the same frequency $k$ (in the input space) increases with increasing $L$ . This allows the remarkable interpretation that the neural network function can express large frequencies on a manifold $( l )$ with smaller frequencies w.r.t its input domain $( k )$ , provided that the coordinate functions of the data manifold embedding itself has high-frequency components14.
+
+# 5 LOWER FREQUENCIES ARE MORE ROBUST
+
+The goal of this section is to show that lower frequency components of trained networks are more robust than their higher frequency counterparts with respect to random perturbations in parameter space. More precisely, we observe that in the neighbourhood of a solution in parameter space, the high frequency components decay faster than the low frequency ones. This property does not directly depend on the training process, but rather on the parametrization of the trained model. We present empirical evidence and a theoretical explanation of this phenomenon.
+
+Experiment 4. The set up is the same as in Experiment 1, where $\lambda$ is given by Eqn. 9. Training is performed for the frequencies $\kappa = ( 1 0 , 1 5 , 2 0 , . . . , 4 5 , 5 0 )$ and amplitudes $A _ { i } ~ = ~ 1 \forall i$ . After convergence to $\theta ^ { * }$ , we consider random (isotropic) perturbations $\theta = \theta ^ { * } + \delta \hat { \theta }$ of given magnitude $\delta$ , where $\hat { \theta } \sim U ( S ^ { \dim ( \theta ^ { * } ) } )$ is a unit vector. We evaluate the network function $f _ { \theta }$ at the perturbed parameters, and compute the magnitude of its discrete Fourier transform at frequencies $k _ { i }$ , $| \tilde { f } _ { \theta } ( k _ { i } ) |$ . We also average over 100 samples of $\hat { \theta }$ to obtain $| \tilde { f } _ { \mathbb { E } \theta } ( k _ { i } ) |$ , which we normalize by $| \tilde { f } _ { \theta * } ( k _ { i } ) |$ . The result, shown in Figure 5, demonstrate that higher frequencies are significantly less robust than the lower ones.
+
+Discussion. The interpretation is as follows: parameters that contribute towards expressing highfrequency components occupy a small volume in the parameter space. To formalize this intuition, given a bounded domain $\Theta$ of parameter space, let us define,
+
+$$
+\Xi _ { \epsilon } ( k ) = \{ \theta \in \Theta | \exists \mathbf { k } ^ { \prime } , k ^ { \prime } > k , | \tilde { f } _ { \theta } ( \mathbf { k } ^ { \prime } ) | > \epsilon \}
+$$
+
+to be the set of parameters such that $f _ { \theta }$ has Fourier components larger than $\epsilon$ for some $\mathbf { k } ^ { \prime }$ with larger norm than $k$ . Then the following Proposition holds (proved in appendix E).
+
+Proposition 1. The volume ratio,
+
+$$
+R ( k ) = \frac { V o l ( \Xi _ { \epsilon } ( k ) ) } { V o l ( \Theta ) }
+$$
+
+inherits the spectral decay rate of $| \tilde { f } _ { \theta } ( \mathbf { k } ) |$ , given by Theorem 1.
+
+Intuitively, expressing larger frequencies requires the parameters to be finely-tuned to work together.
+
+# 6 RELATED WORK
+
+While we focus on showing the spectral bias of deep ReLU networks towards learning functions with dominant lower frequency components, most of existing work has focused on showing that in theory, these networks are capable of learning arbitrarily complex functions. Hornik et al. (1989); Cybenko (1989); Leshno et al. (1993) have shown that neural networks can be universal approximators when given sufficient width; more recently, Lu et al. (2017) proved that this property holds also for width-bounded networks. Montufar et al. (2014) showed that the number of linear regions of deep ReLU networks grows polynomially with width and exponentially with depth; Raghu et al. (2016) generalized this result and provided asymptotically tight bounds. There have been various results of the benefits of depth for efficient approximation (Poole et al., 2016; Telgarsky, 2016; Eldan & Shamir, 2016). These analysis on the expressive power of deep neural networks can in part explain why over-parameterized networks can perfectly learn random input-output mappings (Zhang et al., 2017a). Our Fourier analysis of deep ReLU networks also reflects the width and depth dependence of their expressivity, but more interestingly reveals their spectral bias towards learning simple functions. Thus our work may be seen as a formalization of the findings of Arpit et al. (2017), where it is empirically shown that deep networks prioritize learning simple functions during training.
+
+A few other works studied neural networks through the lens of harmonic analysis. For example, Candes (1999) used the ridgelet transform to build constructive procedures for approximating a \` given function by neural networks, in the case of oscillatory activation functions. This approach has been recently generalized to unbounded activation functions by Sonoda & Murata (2017). Eldan & Shamir (2016) use insights on the support of the Fourier spectrum of two-layer networks to derive a worse-case depth-separation result. Barron (1993) makes use of Fourier space properties of the target function to derive an architecture-dependent approximation bound. In a work done independently from ours, and made available online almost at the same time, Xu et al. (2018) make the same observation that lower frequencies are learned first. The subsequent work by Xu (2018) proposes a theoretical analysis of the phenomenon in the case of 2-layer networks with sigmoid activation, based on the spectrum of the sigmoid function.
+
+In light of our findings, it is worth comparing the case of neural networks and other popular algorithms such that kernel machines (KM) and $K$ -nearest neighbor classifiers. We refer to the Appendix F for a detailed discussion and references. In summary, our discussion there suggests that 1. DNNs strike a good balance between function smoothness and expressivity/parameter-efficiency compared with KM; 2. DNNs learn a smoother function compared with $K \mathrm { N N s }$ since the spectrum of the DNN decays faster compared with $K \mathrm { N N s }$ in the experiments shown there.
+
+# 7 CONCLUSION
+
+We studied deep ReLU networks through the lens of Fourier analysis. Several conclusions can be drawn from our analysis. While neural networks can approximate arbitrary functions, we find that they favour low frequency ones – hence they exhibit a bias towards smooth functions – a phenomenon that we called spectral bias. We also illustrated how the geometry of the data manifold impacts expressivity in a non-trivial way, as high frequency functions defined on complex manifolds can be expressed by lower frequency network functions defined in input space. Finally, we found that the parameters contributing towards expressing lower frequencies are more robust to random perturbations than their higher frequency counterparts.
+
+We view future work that explore the properties of neural networks in Fourier domain as promising. For example, the Fourier transform affords a natural way of measuring how fast a function can change within a small neighborhood in its input domain ; as such, it is a strong candidate for quantifying and analyzing the sensitivity of a model – which in turn provides a natural measure of complexity (Novak et al., 2018). We hope to encourage more research in this direction.
+
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+
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+
+![](images/293f9ca766205feb18be9d7f61dff57e89ba99e9789569627e71f1d667fddba7.jpg)  
+Figure 6: The learnt function (green) overlayed on the target function (blue) as the training progresses. The target function is a superposition of sinusoids of frequencies $\kappa = ( 5 , 1 0 , . . . , 4 5 , 5 0 )$ , equal amplitudes and randomly sampled phases.
+
+![](images/d8be4a2f464dea3a472cd2a806e8dc72c605370a65943a0205761ce9096794f8.jpg)  
+Figure 7: Loss curves averaged over multiple runs. (cf. Experiment 1)
+
+# A EXPERIMENTAL DETAILS
+
+# A.1 EXPERIMENT 1
+
+We fit a 6 layer ReLU network with 256 units per layer $f _ { \theta }$ to the target function $\lambda$ , which is a superposition of sine waves with increasing frequencies:
+
+$$
+\lambda : [ 0 , 1 ] \to \mathbb { R } , \lambda ( z ) = \sum _ { i } A _ { i } \sin ( 2 \pi k _ { i } z + \varphi _ { i } )
+$$
+
+where $k _ { i } = ( 5 , 1 0 , 1 5 , . . . , 5 0 )$ , and $\varphi _ { i }$ is sampled from the uniform distribution $U ( 0 , 2 \pi )$ . In the first setting, we set equal amplitude for all frequencies, i.e. $A _ { i } = 1 \forall i$ , while in the second setting we assign larger amplitudes to the higher frequencies, i.e. $A _ { i } = ( 0 . 1 , 0 . 2 , . . . , 1 )$ . We sample $\lambda$ on 200 uniformly spaced points in [0, 1] and train the network for 80000 steps of full-batch gradient descent with Adam (Kingma & Ba, 2014). Note that we do not use stochastic gradient descent to avoid the stochasticity in parameter updates as a confounding factor. We evaluate the network on the same 200 point grid every 100 training steps and compute the magnitude of its (single-sided) discrete fourier transform at frequencies $k _ { i }$ which we denote with $\vert \tilde { f } _ { k _ { i } } \vert$ . Finally, we plot in figure 1 the normalized magnitudes $\frac { | \tilde { f } _ { k _ { i } } | } { A _ { i } }$ averaged over 10 runs (with different sets of sampled phases $\varphi _ { i }$ ). We also record the spectral norms of the weights at each layer as the training progresses, which we plot in figure 1 for both settings (the spectral norm is evaluated with 10 power iterations). In figure 6, we show an example target function and the predictions of the network trained on it (over the iterations), and in figure 7 we plot the loss curves.
+
+# A.2 EXPERIMENT 2
+
+We use the same 6-layer deep 256-unit wide network and define the target function
+
+$$
+\lambda : \mathcal { D } \to \mathbb { R } , z \mapsto \lambda ( z ) = \sum _ { i } A _ { i } \sin ( 2 \pi k _ { i } z + \varphi _ { i } )
+$$
+
+![](images/b3cb4740dede3f3ce0094d4ed55194ca9af0e32df33a75de7ce85512c7c9a2fa.jpg)  
+Figure 8: The target function used in Experiment 5.
+
+![](images/ae5571a3ba5f45db0905ecd4c10bdac6f02050b61f82a46a01948ff2b095b541.jpg)  
+Figure 9: Evolution with training iterations (y-axis) of the Fourier spectrum $\mathbf { \widetilde { x } }$ -axis for frequency, and colormap for magnitude) for a network with varying depth, width $= 1 6$ and weight clip $= 1 0$ . The spectrum of the target function is a constant 0.005 for all frequencies.
+
+where $k _ { i } = ( 2 0 , 4 0 , . . . , 1 8 0 , 2 0 0 )$ , $A _ { i } = 1 \forall i$ and $\varphi \sim U ( 0 , 2 \pi )$ . We sample $\phi$ on a grid with 1000 uniformly spaced points between 0 and 1 and map it to the input domain via $\gamma _ { L }$ to obtain a dataset $\{ ( \gamma _ { L } ( z _ { j } ) , \lambda \bar { ( } z _ { j } ) ) \} _ { j = 0 } ^ { \bar { 9 } 9 9 }$ , on which we train the network with 50000 full-batch gradient descent steps of Adam. On the same 1000-point grid, we evaluate the magnitude of the (single-sided) discrete Fourier transform of $f _ { \theta } \circ \gamma _ { L }$ every 100 training steps at frequencies $k _ { i }$ and average over 10 runs (each with a different set of sampled $z _ { i }$ ’s). Fig 3 shows the evolution of the spectrum as training progresses for $L = 0 , 4 , 1 0 , 1 6$ , and Fig 3e shows the corresponding loss curves.
+
+# A.3 QUALITATIVE ABLATION OVER ARCHITECTURES
+
+Theorem 1 exposes the relationship between the fourier spectrum of a network and its depth, width and max-norm of parameters. The following experiment is a qualitative ablation study over these variables.
+
+Experiment 5. In this experiment, we fit various networks to the $\delta$ -function at $x = 0 . 5$ (see Fig 8a). Its spectrum is constant for all frequencies $\left( \mathrm { F i g ~ 8 b } \right)$ , which makes it particularly useful for testing how well a given network can fit large frequencies. Fig 11 shows the ablation over weight clip (i.e. max parameter max-norm), Fig 9 over depth and Fig 10 over width. Fig 12 exemplarily shows how the network prediction evolves with training iterations. All networks are trained for 60K iterations of full-batch gradient descent under identical conditions (Adam optimizer with $l r = 0 . 0 0 0 3$ , no weight decay).
+
+We make the following observations.
+
+(a) Fig 9 shows that increasing the depth (for fixed width) significantly improves the network’s ability to fit higher frequencies (note that the depth increases linearly).   
+(b) Fig 10 shows that increasing the width (for fixed depth) also helps, but the effect is considerably weaker (note that the width increases exponentially).   
+(c) Fig 11 shows that increasing the weight clip (or the max parameter max-norm) also helps the network fit higher frequencies.
+
+![](images/c99360b10ee8c54ad481464c772e6e7c7469f8bad9329b150c886d1f3a6a6af6.jpg)  
+Figure 10: Evolution with training iterations (y-axis) of the Fourier spectrum ( $\mathbf { \widetilde { x } }$ -axis for frequency, and colormap for magnitude) for a network with varying width, depth $= 3$ and weight clip $= 1 0$ . The spectrum of the target function is a constant 0.005 for all frequencies.
+
+![](images/16de464320fa88f8cb5e99e7f8f548697630d7077fb5ac8e8073f27179ee98d2.jpg)  
+Figure 11: Evolution with training iterations (y-axis) of the Fourier spectrum ( $\mathbf { \widetilde { x } }$ -axis for frequency, and colormap for magnitude) for a network with varying weight clip, depth $= 6$ and width $= 6 4$ . The spectrum of the target function is a constant 0.005 for all frequencies.
+
+![](images/e0e8c0fbd8a46b0aa96223e94e28abc705e08e73c403031927b9fecd79e0cbb0.jpg)  
+Figure 12: Evolution with training iterations (y-axis) of the network prediction ( $\mathbf { \widetilde { x } }$ -axis for input, and colormap for predicted value) for a network with varying weight clip, depth $= 6$ and width $= 6 4$ . The target function is a $\delta$ peak at $x = 0 . 5$ .
+
+The above observations are all consistent with Theorem 1, and further show that lower frequencies are learned first (i.e. the spectral bias, cf. Experiment 1).
+
+# A.4 MNIST: A PROOF OF CONCEPT
+
+In the following experiment, we show that given two manifolds of the same dimension – one flat and the other not – the task of learning random labels is harder to solve if the input samples lie on the same manifold. We demonstrate on MNIST under the assumption that the manifold hypothesis is true, and use the fact that the spectrum of the target function we use (white noise) is constant in expectation, and therefore independent of the underlying coordinate system when defined on the manifold.
+
+Experiment 6. In this experiment, we investigate if it is easier to learn a signal on a more realistic data-manifold like that of MNIST (assuming the manifold hypothesis is true), and compare with a flat manifold of the same dimension. To that end, we use the 64-dimensional feature-space $\mathcal { E }$ of a denoising15 autoencoder as a proxy for the real data-manifold of unknown number of dimensions. The decoder functions as an embedding of $\mathcal { E }$ in the input space $X =  { \mathbb { R } } ^ { 7 8 4 }$ , which effectively amounts to training a network on the reconstructions of the autoencoder. For comparision, we use an injective embedding16 of a 64-dimensional hyperplane in $X$ . The latter is equivalent to sampling 784-dimensional vectors from $U ( [ 0 , 1 ] )$ and setting all but the first 64 components to zero. The target function is white-noise, sampled as scalars from the uniform distribution $U ( [ 0 , 1 ] )$ . Two identical networks are trained under identical conditions, and Fig 13 shows the resulting loss curves, each averaged over 10 runs.
+
+This result complements the findings of Arpit et al. (2017) and Zhang et al. (2017a), which show that it’s easier to fit random labels to random inputs if the latter is defined on the full dimensional input space (i.e. the dimension of the flat manifold is the same as that of the input space, and not that of the underlying data-manifold being used for comparison).
+
+![](images/a640435ee1ef724ce77dcbbfc76d39cdca3cbcb7fac88aaa26f6c0ff7a0e2cd9.jpg)  
+Figure 13: Loss curves of two identical networks trained to regress white-noise under identical conditions, one on MNIST reconstructions from a DAE with 64 encoder features (blue), and the other on 64-dimensional random vectors (green).
+
+# A.5 CIFAR-10: IT’S ALL CONNECTED
+
+We have seen that deep neural networks are biased towards learning low frequency functions. This should have as a consequence that isolated bubbles of constant prediction are rare. This in turn implies that given any two points in the input space and a network function that predicts the same class for the said points, there should be a path connecting them such that the network prediction does not change along the path. In the following, we present an experiment where we use a path finding method to find such a path between all Cifar-10 input samples indeed exist.
+
+Experiment 7. Using AutoNEB Kolsbjerg et al. (2016), we construct paths between (adversarial) Cifar-10 images that are classified by a ResNet20 to be all of the same target class. AutoNEB bends a linear path between points in some space $\mathbb { R } ^ { m }$ so that some maximum energy along the path is minimal. Here, the space is the input space of the neural network, i.e. the space of $3 2 \times 3 2 \times 3$ images and the logit output of the ResNet20 for a given class is minimized. We construct paths between the following points in image space:
+
+![](images/7ae5a6f666c849ca6dbb28e8d465616cf221a50bdf1dcf04bcccb4f933d187da.jpg)  
+Figure 14: Path between CIFAR-10 adversarial examples (e.g. “frog” and “automobile”, such that all images are classified as “airplane”).
+
+• From one training image to another, • from a training image to an adversarial, • from one adversarial to another.
+
+We only consider pairs of images that belong to the same class $c$ (or, for adversarials, that originate from another class $\neq c$ , but that the model classifies to be of the specified class $c$ ). For each class, we randomly select 50 training images and select a total of 50 random images from all other classes and generate adversarial samples from the latter. Then, paths between all pairs from the whole set of images are computed.
+
+The AutoNEB parameters are chosen as follows: We run four NEB iterations with 10 steps of SGD with learning rate 0.001 and momentum 0.9. This computational budget is similar to that required to compute the adversarial samples. The gradient for each NEB step is computed to maximize the logit output of the ResNet-20 for the specified target class $c$ . We use the formulation of NEB without springs Draxler et al. (2018).
+
+The result is very clear: We can find paths between all pairs of images for all CIFAR10 labels that do not cross a single decision boundary. This means that all paths belong to the same connected component regarding the output of the DNN. This holds for all possible combinations of images in the above list. Figure 15 shows connecting training to adversarial images and Figure 14 paths between pairs of adversarial images. Paths between training images are not shown, they provide no further insight. Note that the paths are strikingly simple: Visually, they are hard to distinguish from the linear interpolation. Quantitatively, they are essentially (but not exactly) linear, with an average length $( 3 . 0 \pm 0 . 3 ) \%$ longer than the linear connection.
+
+# B BRIEF RECAPITULATION OF FOURIER ANALYSIS
+
+The Fourier transform is a powerful mathematical tool used to represent functions as a weighted sum of oscillating functions, given that the function satisfies certain conditions. In the realm of signal processing and beyond, it is used to represent a time (space) domain signal $f$ as a sum of sinusoids of various (spatial) frequencies $\mathbf { k }$ , where the weights are referred to as the Fourier coefficients $\tilde { f } ( \mathbf { k } )$ .
+
+![](images/a10318e0f924d422d3b3a245bee4ee4c0c8ab42ab5429fe55c5024adaf2dc67a.jpg)  
+Figure 15: Each row is a path through the image space from an adversarial sample (right) to a true training image (left). All images are classified by a ResNet-20 to be of the class of the training sample on the right with at least $9 5 \%$ softmax certainty. This experiment shows we can find a path from adversarial examples (right, Eg. ”(cat)”) that are classified as a particular class (”airplane”) are connected to actual training samples from that class (left, ”airplane”) such that all samples along that path are also predicted by the network to be of the same class.
+
+Let $\begin{array} { r l r } { f } & { { } : } & { \mathbb { R } ^ { n } \quad \to \quad \mathbb { R } } \end{array}$ be a squared-integrable function17, i.e. such that $\textstyle \int _ { \mathbf { x } \in \mathbb { R } ^ { n } } | f ( \mathbf { x } ) | ^ { 2 } d \mathbf { x }$ is finite, or $\begin{array} { r l r l } { f } & { { } \in } & { } & { { } L ^ { 2 } ( \mathbb { R } ^ { n } ) } \end{array}$ . With the Fourier inversion theorem, it holds:
+
+$$
+f ( \mathbf { x } ) = { \frac { 1 } { 2 \pi } } \int _ { \mathbf { k } \in \mathbb { R } ^ { n } } { \tilde { f } } ( \mathbf { k } ) e ^ { i \mathbf { k } \cdot \mathbf { x } } d \mathbf { k } \qquad ( 1 7 ) \qquad { \tilde { f } } ( \mathbf { k } ) = \int _ { \mathbf { x } \in \mathbb { R } ^ { n } } f ( \mathbf { x } ) e ^ { - i \mathbf { k } \cdot \mathbf { x } } d \mathbf { x }
+$$
+
+Informally, equation 17 expresses the function $f ( \mathbf { x } )$ as a weighted sum (the integral) of plane waves $e ^ { \pm i \mathbf { k } \cdot \mathbf { x } }$ of the angular wavenumber $\mathbf { k }$ , where the unit vector $\hat { \mathbf { k } }$ gives the direction of the corresponding wave in $n$ -D space and the magnitude $k$ is inversely proportional to the wavelength. Equation 18 gives the expression for $\tilde { f } ( \mathbf { k } )$ , which is called the Fourier transform or the Fourier spectrum or simply the spectrum of $f$ . The $\scriptstyle { \frac { 1 } { 2 \pi } }$ coefficient and sign in the exponential functions are matters of convention.
+
+The asymptotic behaviour of $\tilde { f }$ for $k  \infty$ is a measure of smoothness of $f$ . In Bachmann-Landau or asymptotic notation18, we say $\tilde { f } = \mathcal { O } ( k ^ { - 1 } )$ if for $k  \infty$ , the function $\tilde { f }$ decays at least as fast as $\frac { 1 } { k }$ . A function whose spectrum is $\mathcal { O } ( k ^ { - 2 } )$ is in a sense smoother than one whose spectrum is $\mathcal { O } ( k ^ { - 1 } )$ , while the spectrum of an infinitely differentiable (or smooth) function must decay faster than any rational function of $k$ , assuming the function is integrable, i.e. the integral of its absolute value over its domain is finite (or the function is $L ^ { 1 }$ ). Intuitively, the higher-frequency oscillations in a smoother function must vanish faster. Formally, this is a straightforward consequence of the Riemann-Lebesgue lemma, stating that the spectrum of any $L ^ { 1 }$ function must vanish at infinity (potentially arbitrarily slowly), taken together with the well known property of the Fourier transform that it diagonalizes the differential operator i.e. $[ \widetilde { \nabla _ { \mathbf { x } } f } ] ( \mathbf { k } ) = \mathbf { k } \widetilde { f } ( \mathbf { k } )$ .
+
+# C THE CONTINUOUS PIECEWISE LINEAR STRUCTURE OF DEEP RELU NETWORKS
+
+We consider the class of ReLU network functions $f : \mathbb { R } ^ { d } \mapsto \mathbb { R }$ defined by Eqn. 1. Following the terminology of Raghu et al. (2016); Montufar et al. (2014), each linear region of the network then corresponds to a unique activation pattern, wherein each hidden neuron is assigned an activation variable $\epsilon \in \{ - 1 , 1 \}$ , conditioned on whether its input is positive or negative. ReLU networks can be explictly expressed as a sum over all possible activation patterns, as in the following lemma.
+
+Lemma 3. Given $L$ binary vectors $\epsilon ^ { ( 1 ) } , \cdot \cdot \cdot \epsilon ^ { ( L ) }$ with $\epsilon ^ { ( k ) } \in \{ - 1 , 1 \} ^ { d _ { k } }$ , let $T _ { \epsilon ^ { ( k ) } } ^ { ( k ) } : \mathbb { R } ^ { d _ { k - 1 } }  \mathbb { R } ^ { d _ { k } }$ the affine function defined by $T _ { \epsilon ^ { ( k ) } } ^ { ( k ) } ( { \mathbf { u } } ) _ { i } = ( T ^ { ( k ) } ( { \mathbf { u } } ) ) _ { i } i f ( \epsilon _ { k } ) _ { i } = 1$ , and 0 otherwise. ReLU network functions, as defined in Eqn. 1, can be expressed as
+
+$$
+f ( \mathbf { x } ) = \sum _ { \epsilon ^ { ( 1 ) } , \cdots \epsilon ^ { ( L ) } } 1 _ { P _ { f , \epsilon } } ( \mathbf { x } ) \left( T ^ { ( L + 1 ) } \circ T _ { \epsilon ^ { ( L ) } } ^ { ( L ) } \circ \cdots \circ T _ { \epsilon ^ { ( 1 ) } } ^ { ( 1 ) } \right) ( \mathbf { x } )
+$$
+
+where $1 _ { P }$ denotes the indicator function of the subset $P \subset \mathbb { R } ^ { d }$ , and $P _ { f , \epsilon }$ is the polytope defined as the set of solutions of the following linear inequalities (for all $k = 1 , \cdots , L )$ :
+
+$$
+\begin{array} { r } { ( \epsilon _ { k } ) _ { i } ( T ^ { ( k ) } \circ T _ { \epsilon ^ { ( k - 1 ) } } ^ { ( k - 1 ) } \circ \cdots \circ T _ { \epsilon ^ { ( 1 ) } } ^ { ( 1 ) } ) ( \mathbf { x } ) _ { i } \geq 0 , \quad i = 1 , \cdots d _ { k } } \end{array}
+$$
+
+$f$ is therefore affine on each of the polytopes $P _ { f , \epsilon }$ , which finitely partition the input space $\mathbb { R } ^ { d }$ to convex polytopes. Remarkably, the correspondence between ReLU networks and CPWL functions goes both ways: Arora et al. (2018) show that every CPWL function is be represented by a ReLU network, which in turn endows ReLU networks with the universal approximation property.
+
+Finally, in the standard basis, each affine map $T ^ { ( k ) } : \mathbb { R } ^ { d _ { k - 1 } }  \mathbb { R } ^ { d _ { k } }$ is specified by a weight matrix $W ^ { ( k ) } \in \mathbb { R } ^ { d _ { k - 1 } } \times \mathbb { R } ^ { d _ { k } }$ and a bias vector $b ^ { ( k ) } \in \mathbb { R } ^ { d _ { k } }$ . In the linear region $P _ { f , \epsilon }$ , $f$ can be expressed as $f _ { \epsilon } ( x ) = W _ { \epsilon } x + b _ { \epsilon }$ , where in particular
+
+$$
+W _ { \epsilon } = W ^ { ( L + 1 ) } W _ { \epsilon _ { L } } ^ { ( L ) } \cdot \cdot \cdot W _ { \epsilon _ { 1 } } ^ { ( 1 ) } \in \mathbb { R } ^ { 1 \times d } ,
+$$
+
+where $W _ { \epsilon } ^ { ( k ) }$ is obtained from $W ^ { ( k ) }$ by setting its $j$ th column to zero whenever $( \epsilon _ { k } ) _ { j } = - 1$
+
+# D FOURIER ANALYSIS OF RELU NETWORKS
+
+# D.1 PROOF OF LEMMA 1
+
+Proof. The vector-valued function $\mathbf { k } f ( \mathbf { x } ) e ^ { i \mathbf { k } \cdot \mathbf { x } }$ is continuous everywhere and has well-defined and continuous gradients almost everywhere. So by Stokes’ theorem (see e.g Spivak (2018)), the integral of its divergence is a pure boundary term. Since we restricted to functions with compact support, the theorem yields
+
+$$
+\int \nabla _ { \mathbf { x } } \cdot \left[ \mathbf { k } f ( \mathbf { x } ) e ^ { - i \mathbf { k } \cdot \mathbf { x } } \right] \mathbf { d x } = 0
+$$
+
+The integrand is $( \mathbf { k } \cdot ( \nabla _ { \mathbf { x } } f ) ( \mathbf { x } ) - i k ^ { 2 } f ( \mathbf { x } ) ) e ^ { - i \mathbf { k } \cdot \mathbf { x } }$ , so we deduce,
+
+$$
+{ \hat { f } } ( \mathbf { k } ) = { \frac { 1 } { - i k ^ { 2 } } } \mathbf { k } \cdot \int ( \nabla _ { \mathbf { x } } f ) ( \mathbf { x } ) e ^ { - i \mathbf { k } \cdot \mathbf { x } }
+$$
+
+Now, within each polytope of the decomposition (19), $f$ is affine so its gradient is a constant vector, $\nabla _ { \mathbf { x } } { f _ { \epsilon } } = W _ { \epsilon } ^ { T }$ , which gives the desired result (1). □
+
+# D.2 FOURIER TRANSFORM OF POLYTOPES
+
+# D.2.1 THEOREM 1 OF DIAZ ET AL. (2016)
+
+Let $F$ be a $m$ dimensional polytope in $\mathbb { R } ^ { d }$ , such that $1 \leq m \leq d$ . Denote by $\textbf { k } \in \mathbb { R } ^ { d }$ a vector in the Fourier space, by $\phi _ { \mathbf { k } } ( x ) = - \mathbf { k } \cdot \mathbf { x }$ the linear phase function, by $\tilde { F }$ the Fourier transform of the indicator function on $F$ , by $\partial { \cal F }$ the boundary of $F$ and by $\mathrm { v o l } _ { m }$ the $m$ -dimensional (Hausdorff) measure. Let $\mathrm { P r o j } _ { F } ( { \bf k } )$ be the orthogonal projection of $\mathbf { k }$ on to $F$ (obtained by removing all components of $\mathbf { k }$ orthogonal to $F$ ). Given a $m - 1$ dimensional facet $G$ of $F$ , let ${ \bf N } _ { F } ( G )$ be the unit normal vector to $G$ that points out of $F$ . It then holds:
+
+1. If ${ \sf P r o j } _ { F } ( { \bf k } ) = 0$ , then $\phi _ { \mathbf { k } } ( x ) = \Phi _ { \mathbf { k } }$ is constant on $F$ , and we have:
+
+$$
+\tilde { F } = { \mathrm { v o l } } _ { F } ( F ) e ^ { i \Phi _ { \mathbf { k } } }
+$$
+
+2. But if $\mathrm { P r o j } _ { F } ( { \bf k } ) \neq 0$ , then:
+
+$$
+\tilde { F } = i \sum _ { G \in \partial F } \frac { \mathrm { P r o j } _ { F } ( \mathbf { k } ) \cdot \mathbf { N } _ { F } ( G ) } { \| \mathrm { P r o j } _ { F } ( \mathbf { k } ) \| ^ { 2 } } \tilde { G } ( \mathbf { k } )
+$$
+
+# D.2.2 DISCUSSION
+
+The above theorem provides a recursive relation for computing the Fourier transform of an arbitrary polytope. More precisely, the Fourier transform of a $m$ -dimensional polytope is expressed as a sum of fourier transforms over the $m - 1$ dimensional boundaries of the said polytope (which are themselves polytopes) times a $\mathcal { O } ( k ^ { - 1 } )$ weight term (with $k = \| \mathbf { k } \| ,$ ). The recursion terminates if $\mathrm { P r o j } _ { F } ( { \bf k } ) = \bar { 0 }$ , which then yields a constant.
+
+To structure this computation, Diaz et al. (2016) introduce a book-keeping device called the face poset of the polytope. It can be understood as a weighted tree diagram with polytopes of various dimensions as its nodes. We start at the root node which is the full dimensional polytope $P$ (i.e. we initially set $m = n$ ). For all of the codimension-one boundary faces $F$ of $P$ , we then draw an edge from the root $P$ to node $F$ and weight it with a term given by:
+
+$$
+W _ { F , G } = i \frac { \mathrm { P r o j } _ { F } ( \mathbf { k } ) \cdot \mathbf { N } _ { F } ( G ) } { | | \mathrm { P r o j } _ { F } ( \mathbf { k } ) | | ^ { 2 } } \tilde { G } ( \mathbf { k } )
+$$
+
+and repeat the process iteratively for each $F$ . Note that the weight term is $\mathcal { O } ( k ^ { - 1 } )$ where $\mathrm { P r o j } _ { F } ( { \bf k } ) \ne$ 0. This process yields tree paths $T : P \to F _ { 1 } \to \dots \to F _ { q }$ where each $F _ { i + 1 } ~ \in ~ \partial F _ { i }$ has one dimension less than $F _ { i }$ . For a given path and $\mathbf { k }$ , the terminal node for this path, $F _ { q }$ , is the first polytope for which $\mathrm { P r o j } _ { F _ { q } } ( { \bf k } ) = 0$ . The final Fourier transform is obtained by multiplying the weights along each path and summing over all tree paths:
+
+$$
+\tilde { 1 } _ { P } ( \mathbf { k } ) = \sum _ { T } i ^ { q } \prod _ { i = 0 } ^ { q - 1 } \frac { \mathrm { P r o j } _ { F _ { i } } ( \mathbf { k } ) \cdot \mathbf { N } _ { F _ { i } } ( F _ { i + 1 } ) } { \lVert \mathrm { P r o j } _ { F _ { i } } ( \mathbf { k } ) \rVert ^ { 2 } } \mathrm { v o l } _ { F _ { q } } ( F _ { q } ) e ^ { i \Phi _ { \mathbf { k } } }
+$$
+
+where we wrote $F _ { 0 } = P$ . Together with Lemma 1, this gives the closed form expression of the Fourier transform of ReLU networks.
+
+For a generic vector $\mathbf { k }$ , all paths terminate at the zero-dimensional vertices of the original polytope, i.e. $\mathrm { d i } \mathrm { \bar { m } } ( F _ { q } ) = 0$ , implying the length of the path $q$ equals the number of dimensions $d$ , yielding a $\mathcal { O } ( k ^ { - d } )$ spectrum. The exceptions occur if a path terminates prematurely, because $\mathbf { k }$ happens to lie orthogonal to some $d - r$ -dimensional face $F _ { r }$ in the path, in which case we are left with a $\mathcal { O } ( k ^ { - r } )$ term (with $r \ < \ d )$ which dominates asymptotically. Note that all vectors orthogonal to the $d - r$ dimensional face $F _ { r }$ lie on a $r$ -dimensional subspace of $\mathbb { R } ^ { d }$ . Since a polytope has a finite number of faces (of any dimension), the $\mathbf { k }$ ’s for which the Fourier transform is $\bar { \mathcal { O } } ( \bar { k } ^ { - \bar { r } } )$ (instead of $\mathcal { O } ( k ^ { - d } ) )$ lies on a finite union of closed subspaces of dimension $r$ (with $r < d$ ). The Lebesgue measure of all such lower dimensional subspaces for all such $r$ is 0, leading us to the conclusion that the spectrum decays as $\mathcal { O } ( k ^ { - d } )$ for almost all $\mathbf { k }$ ’s. We formalize this in the following corollary.
+
+Corollary 1. Let $P$ be a full dimensional polytope in $\mathbb { R } ^ { n }$ . The Fourier spectrum of its indicator function $1 _ { P }$ satisfies the following:
+
+$$
+\big | \tilde { 1 } _ { P } ( \mathbf { k } ) \big | = \mathcal { O } \left( \frac { 1 } { k ^ { \Delta _ { \mathbf { k } } } } \right)
+$$
+
+where $1 \leq \Delta _ { \mathbf { k } } \leq n$ , and $\Delta _ { \mathbf { k } } = j$ for k on a finite union of $j$ -dimensional subspaces of $\mathbb { R } ^ { n }$ .
+
+D.3 PROOF OF THE LIPSCHTIZ BOUND
+
+Proposition 2. The Lipschitz constant $L _ { f }$ of the ReLU network $f$ is bound as follows (for all $\epsilon$ ):
+
+$$
+\| W _ { \epsilon } \| \leq L _ { f } \leq \prod _ { k = 1 } ^ { L + 1 } \| W ^ { ( k ) } \| \leq \| \theta \| _ { \infty } ^ { L + 1 } \sqrt { d } \prod _ { k = 1 } ^ { L } d _ { k }
+$$
+
+Proof. The first equality is simply the fact that $L _ { f } = \operatorname* { m a x } _ { \epsilon } \| W _ { \epsilon } \|$ , and the second inequality follows trivially from the parameterization of a ReLU network as a chain of function compositions19, together with the fact that the Lipschitz constant of the ReLU function is 1 (cf. Miyato et al. (2018), equation 7). To see the third inequality, consider the definition of the spectral norm of a $I \times J$ matrix $W$ :
+
+$$
+\| W \| = \operatorname* { m a x } _ { \| \mathbf { h } \| = 1 } \| W \mathbf { h } \|
+$$
+
+Now, $\begin{array} { r } { \| W \mathbf { h } \| = \sqrt { \sum _ { i } | \mathbf { w } _ { i } \cdot \mathbf { h } | } } \end{array}$ , where $\mathbf { w } _ { i }$ is the $i$ -th row of the weight matrix $W$ and $i = 1 , . . . , I$ . Further, if $\| \mathbf { h } \| = 1$ , we have $| \mathbf { w } _ { i } \cdot \mathbf { h } | \leq \| \mathbf { w } _ { i } \| \| \mathbf { h } \| = \| \mathbf { w } _ { i } \|$ . Since $\left. \mathbf { w } _ { i } \right. = \sqrt { \sum _ { j } | w _ { i j } | }$ (with $j = 1 , . . . , J )$ and $| w _ { i j } | \leq \| \theta \| _ { \infty }$ , we find that $\lVert \mathbf { w } _ { i } \rVert \leq \sqrt { J } \lVert \theta \rVert _ { \infty }$ . Consequently, $\begin{array} { r } { \sqrt { \sum _ { i } | \mathbf { w } _ { i } \cdot \mathbf { h } | } \leq } \end{array}$ $\sqrt { I J } \| \theta \| _ { \infty }$ and we obtain:
+
+$$
+\| W \| \leq \sqrt { I J } \| \theta \| _ { \infty }
+$$
+
+Now for $W = W ^ { ( k ) }$ , we have $I = d _ { k - 1 }$ and $J = d _ { k }$ . In the product over $k$ , every $d _ { k }$ except the first and the last occur in pairs, which cancels the square root. For $k = 1$ , $d _ { k - 1 } = d$ (for the $d$ input neurons) and for $k = L + 1$ , $d _ { k } = 1$ (for a single output neuron). The final inequality now follows. □
+
+# D.4 THE FOURIER TRANSFORM OF A FUNCTION COMPOSITION
+
+Consider Equation 14. The general idea is to investigate the behaviour of $P _ { \gamma } ( 1 , { \bf k } )$ for large frequencies l on manifold but smaller frequencies $\mathbf { k }$ in the input domain. In particular, we are interested in the regime where the stationary phase approximation is applicable to $P _ { \gamma }$ , i.e. when $l ^ { 2 } + k ^ { 2 } \to \infty$ (cf. section 3.2. of Bergner et al.). In this regime, the integrand in $P _ { \gamma } ( \mathbf { k } , \mathbf { l } )$ oscillates fast enough such that the only constructive contribution originates from where the phase term $u ( \mathbf { z } ) = \mathbf { k } \cdot \boldsymbol { \gamma } ( \mathbf { z } ) - 1 \cdot \mathbf { z }$ does not change with changing $\mathbf { z }$ . This yields the condition that $\begin{array} { r } { \bar { \nabla _ { \mathbf { z } } } u ( \mathbf { z } ) = 0 } \end{array}$ , which translates to the condition (with Einstein summation convention implied and $\partial _ { \nu } = \partial / \partial x _ { \nu }$ ):
+
+$$
+l _ { \nu } = k _ { \mu } \partial _ { \nu } \gamma _ { \mu } ( { \bf z } )
+$$
+
+Now, we impose periodic boundary conditions20 on the components of $\gamma$ , and without loss of generality we let the period be $2 \pi$ . Further, we require that the manifold be contained in a $\bar { \mathsf { b } } \mathsf { o x } ^ { 2 1 }$ of some size in $\mathbb { R } ^ { d }$ . The $\mu$ -th component $\gamma _ { \mu }$ can now be expressed as a Fourier series:
+
+$$
+\gamma _ { \mu } ( { \mathbf { z } } ) = \sum _ { { \mathbf { p } } \in \mathbb { Z } ^ { m } } \widetilde { \gamma } _ { \mu } [ { \mathbf { p } } ] e ^ { - i p _ { \rho } z _ { \rho } } \qquad ( 3 3 ) \qquad \quad \partial _ { \nu } \gamma _ { \mu } ( { \mathbf { z } } ) = \sum _ { { \mathbf { p } } \in \mathbb { Z } ^ { m } } - i p _ { \nu } \widetilde { \gamma } _ { \mu } [ { \mathbf { p } } ] e ^ { - i p _ { \rho } z _ { \rho } }
+$$
+
+Equation 34 can be substituted in equation 32 to obtain:
+
+$$
+l \hat { l } _ { \nu } = - i k \sum _ { \mathbf { p } \in \mathbb { Z } ^ { m } } p _ { \nu } \hat { k } _ { \mu } \tilde { \gamma } _ { \mu } [ \mathbf { p } ] e ^ { - i p _ { \rho } z _ { \rho } }
+$$
+
+where we have split $k _ { \mu }$ and $l _ { \nu }$ in to their magnitudes $k$ and $l$ and directions $\hat { k } _ { \nu }$ and $\hat { l } _ { \mu }$ (respectively). We are now interested in the conditions on $\gamma$ under which the RHS can be large in magnitude, even when $k$ is fixed. Recall that $\gamma$ is constrained to a box – consequently, we can not arbitrarily scale up $\tilde { \gamma } _ { \mu }$ . However, if $\tilde { \gamma } _ { \mu } [ \mathbf { p } ]$ decays slowly enough with increasing $\mathbf { p }$ , the RHS can be made arbitrarily large (for certain conditions on $\mathbf { z }$ , $\hat { l } _ { \mu }$ and $\hat { k } _ { \nu }$ ).
+
+# E VOLUME IN PARAMETER SPACE AND PROOF OF PROPOSITION 1
+
+For a given neural network, we now show that the volume of the parameter space containing parameters that contribute $\epsilon$ -non-negligibly to frequency components of magnitude $k ^ { \prime }$ above a certain
+
+cut-off $k$ decays with increasing $k$ . For notational simplicity and without loss of generality, we absorb the direction $\hat { \mathbf { k } }$ of $\mathbf { k }$ in the respective mappings and only deal with the magnitude $k$ .
+
+Definition 1. Given a ReLU network $f _ { \theta }$ of fixed depth, width and weight clip $K$ with parameter vector $\theta$ , an $\epsilon > 0$ and $\Theta = B _ { K } ^ { \infty } ( 0 )$ a $L ^ { \infty }$ ball around 0, we define:
+
+$$
+\Xi _ { \epsilon } ( k ) = \{ \theta \in \Theta | \exists k ^ { \prime } > k , | \tilde { f } _ { \theta } ( k ^ { \prime } ) | > \epsilon \}
+$$
+
+as the set of all parameters vectors $\theta \in \Xi _ { \epsilon } ( k )$ that contribute more than an $\epsilon$ in expressing one or more frequencies $k ^ { \prime }$ above a cut-off frequency $k$ .
+
+Remark 1. If $k _ { 2 } \geq k _ { 1 }$ , we have $\Xi _ { \epsilon } ( k _ { 2 } ) \subseteq \Xi _ { \epsilon } ( k _ { 1 } )$ and consequently $\nu o l ( \Xi _ { \epsilon } ( k _ { 2 } ) ) \leq \nu o l ( \Xi _ { \epsilon } ( k _ { 1 } ) )$ , where vol is the Lebesgue measure.
+
+Lemma 4. Let $1 _ { k } ^ { \epsilon } ( \theta )$ be the indicator function on $\Xi _ { \epsilon } ( k )$ . Then:
+
+$$
+\exists \kappa > 0 : \forall k \geq \kappa , 1 _ { k } ^ { \epsilon } ( \theta ) = 0
+$$
+
+Proof. From theorem 1, we know that22 $| \tilde { f } _ { \theta } ( k ) | = \mathcal { O } ( k ^ { - \Delta - 1 } )$ for an integer $1 \leq \Delta \leq d$ . In the worse case where $\Delta = 1$ , we have that $\begin{array} { r } { \exists M < \infty : | \tilde { f } _ { \theta } ( k ) | < \frac { M } { k ^ { 2 } } } \end{array}$ . Now, simply select a $\kappa > \sqrt { \frac { M } { \epsilon } }$ such that $\textstyle { \frac { M } { \kappa ^ { 2 } } } \ < \ \epsilon$ . This yields that $\begin{array} { r } { | \tilde { f } _ { \theta } ( \kappa ) | < \frac { M } { \kappa ^ { 2 } } < \epsilon } \end{array}$ , and given that $\begin{array} { r } { \frac { M } { \kappa ^ { 2 } } \le \frac { M } { k ^ { 2 } } \forall k \ge \kappa } \end{array}$ , we find $| \tilde { f } _ { \theta } ( k ) | < \epsilon \forall k \ge \kappa$ . Now by definition 1, $\theta \not \in \Xi _ { \epsilon } ( \kappa )$ , and since $\Xi _ { \epsilon } ( k ) \subseteq \Xi _ { \epsilon } ( \kappa )$ (see remark 1), we have $\theta \not \in \Xi _ { \epsilon } ( k )$ , implying $1 _ { k } ^ { \dot { \epsilon } } ( \theta ) = 0 \forall k \geq \kappa$ . □
+
+Remark 2. We have $1 _ { k } ^ { \epsilon } ( \theta ) \leq | \tilde { f } _ { \theta } ( k ) |$ for large enough $k$ (i.e. for $k \geq \kappa _ { \mathrm { * } }$ ), since $| \tilde { f } _ { \theta } ( k ) | \geq 0$ .
+
+Proposition 1. The relative volume of $\Xi _ { \epsilon } ( k )$ w.r.t. $\Theta$ is $\mathcal { O } ( k ^ { - \Delta - 1 } )$ where $1 \leq \Delta \leq d .$ .
+
+Proof. The volume is given by the integral over the indicator function, i.e.
+
+$$
+\operatorname { v o l } ( \Xi _ { \epsilon } ( k ) ) = \int _ { \theta \in \Theta } 1 _ { k } ^ { \epsilon } ( \theta ) d \theta
+$$
+
+For a large enough $k$ , we have from remark 2, the monotonicity of the Lebesgue integral and theorem 1 that:
+
+$$
+\begin{array} { r l r } {  { \mathrm { v o l } ( \Xi _ { \epsilon } ( k ) ) = \int _ { \theta \in \Theta } 1 _ { k } ^ { \epsilon } ( \theta ) d \theta \leq \int _ { \theta \in \Theta } | \tilde { f } _ { \theta } ( k ) | d \theta = \mathcal { O } ( k ^ { - d + \Delta - 1 } ) \mathrm { v o l } ( \Theta ) } } \\ & { } & { \implies \frac { \mathrm { v o l } ( \Xi _ { \epsilon } ( k ) ) } { \mathrm { v o l } ( \Theta ) } = \mathcal { O } ( k ^ { - \Delta - 1 } ) } \end{array}
+$$
+
+# F KERNEL MACHINES AND KNNS
+
+In this section, in light of our findings, we want to compare DNNs with K-nearest neighbor (k-NN) classifier and kernel machines which are also popular learning algorithms, but are, in contrast to DNNs, better understood theoretically.
+
+# F.1 KERNEL MACHINES VS DNNS
+
+Given that we study why DNNs are biased towards learning smooth functions, we note that kernel machines (KM) are also highly Lipschitz smooth (Eg. for Gaussian kernels all derivatives are bounded). However there are crutial differences between the two. While kernel machines can approximate any target function in principal (Hammer & Gersmann, 2003), the number of Gaussian kernels needed scales linearly with the number of sign changes in the target function (Bengio et al., 2009). Ma & Belkin (2017) have further shown that for smooth kernels, a target function cannot be approximated within $\epsilon$ precision in any polynomial of $1 / \epsilon$ steps by gradient descent.
+
+Deep networks on the other hand are also capable of approximating any target function (as shown by the universal approximation theorems Hornik et al. (1989); Cybenko (1989)), but they are also parameter efficient in contrast to KM. For instance, we have seen that deep ReLU networks separate the input space into number of linear regions that grow polynomially in width of layers and exponentially in the depth of the network (Montufar et al., 2014; Raghu et al., 2016). A similar result on the exponentially growing expressive power of networks in terms of their depth is also shown in (Poole et al., 2016). In this paper we have further shown that DNNs are inherently biased towards lower frequency (smooth) functions over a finite parameter space. This suggests that DNNs strike a good balance between function smoothness and expressibility/parameter-efficiency compared with KM.
+
+# F.2 K-NN CLASSIFIER VS. DNN CLASSIFIER
+
+$K$ -nearest neighbor $( K \mathrm { N N } )$ also has a historical importance as a classification algorithm due to its simplicity. It has been shown to be a consistent approximator Devroye et al. (1996), i.e., asymptotically its empirical risk goes to zero as $K  \infty$ and $K / N  0$ , where $N$ is the number of training samples. However, because it is a memory based algorithm, it is prohibitively slow for large datasets. Since the smoothness of a $K \mathbf { N N }$ prediction function is not well studied, we compare the smoothness between $K \mathbf { N N }$ and DNN. For various values of $K$ , we train a $K \mathbf { N N }$ classifier on a $k = 1 5 0$ frequency signal (which is binarized) defined on the $L = 2 0$ manifold (see section 4), and extract probability predictions on a box interval in $\mathbb { R } ^ { 2 }$ . On this interval, we evaluate the 2D FFT and integrate out the angular components to obtain $\zeta ( k )$ :
+
+$$
+\zeta ( k ) = \frac { d } { d k } \int _ { 0 } ^ { k } d k ^ { \prime } k ^ { \prime } \int _ { 0 } ^ { 2 \pi } d \varphi | \widetilde { f } ( k ^ { \prime } , \varphi ) |
+$$
+
+Finally, we plot $\zeta ( k )$ for various $K$ in figure 16e. Furthermore, we train a DNN on the very same dataset and overlay the radial spectrum of the resulting probability map on the same plot. We find that while DNN’s are as expressive as a $K = 1$ KNN classifier at lower (radial) frequencies, the frequency spectrum of DNNs decay faster than KNN classifier for all values of $K$ considered, indicating that the DNN is smoother than the KNNs considered. We also repeat the experiment corresponding to Fig. 4 with KNNs (see Fig. 16) for various $K$ ’s, to find that unlike DNNs, KNNs do not necessarily perform better for larger $L$ ’s, suggesting that KNNs do not exploit the geometry of the manifold like DNNs do.
+
+![](images/a61e34d18fca4ecdf47998c63304fd63d6d01956d5b6498f16f37028a3da5935.jpg)  
+Figure 16: (a,b,c,d): Heatmaps of training accuracies $L$ -vs- $k$ ) of KNNs for various $K$ . When comparing with figure 4, note that the y-axis is flipped. (e): The frequency spectrum of $K \mathrm { N N s }$ with different values of $K$ , and a DNN. The DNN learns a smoother function compared with the KNNs considered since the spectrum of the DNN decays faster compared with $K \mathrm { N N s }$ .

parse/train/rkzfuiA9F7/rkzfuiA9F7.md

@@ -0,0 +1,248 @@
+# PROJECTIVE SUBSPACE NETWORKS FOR FEW-SHOT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Generalization from limited examples, usually studied under the umbrella of metalearning, equips learning techniques with the ability to adapt quickly in dynamical environments and proves to be an essential aspect of lifelong learning. In this paper, we introduce the Projective Subspace Networks (PSN), a deep learning paradigm that learns non-linear embeddings from limited supervision. In contrast to previous studies, the embedding in PSN deems samples of a given class to form an affine subspace. We will show that such modeling leads to robust solutions, yielding competitive results on supervised and semi-supervised few-shot classification. Moreover, our PSN approach has the ability of end-to-end learning. In contrast to previous works, our projective subspace can be thought of as a richer representation capturing higher-order information datapoints for modeling new concepts.
+
+# 1 INTRODUCTION
+
+Supervised learning with deep architectures, though achieving remarkable results in many areas, requires large amount of annotated data. Various studies show that many deep learning techniques in computer vision, speech recognition and natural language understanding, to name a few methods stated in Hinton et al. (2012); Krizhevsky et al. (2012), will fail to produce reliable models that generalize well if limited data annotations are available. Aside from the labor associated with annotating data, precise annotation can become ill-posed in some cases. One prime example of such a difficulty is object detection labeling which requires annotating bounding boxes of objects as explained in Alexe et al. (2012).
+
+In contrast to the current trend in deep learning, humans can learn new concepts from only a few examples. This in turn provides humans with lifelong learning abilities. Inspired by such learning abilities, several approaches are developed to study learning from limited examples Lake et al. (2015); Lazaridou et al. (2017); Vinyals et al. (2016); Triantafillou et al. (2017); Xu et al. (2017); Finn et al. (2017); Ravi & Larochelle (2017); Wang et al. (2018); Mishra et al. (2018); Qiao et al. (2018); Neill & Buitelaar (2018). In machine learning, the diverse ideas in this context include embedding features through metric learning (Koch et al. (2015), Vinyals et al. (2016)), optimization technique(Finn et al. (2017), Ravi & Larochelle (2017)), and generative models(Fei-Fei et al. (2006), Lake et al. (2015)).
+
+In this work, we propose a deep model that learns new concepts from limited data to address two challenging learning problems; namely: (i) few-shot classification and (ii) semi-supervised few-shot learning. The goal of few-shot classification is to learn a model that can discriminate a given query by comparing it to a few of samples (a.k.a. the support set). An example is to classify a motorcycle by viewing some different types of motorcycles (or other vehicles). The goal of semi-supervised few-shot learning is to additionally benefit from unlabeled data to boost the performance of the model.
+
+Our method, coined Projective Subspace Networks or PSN for short, learns non-linear embeddings using subspaces, in a sense that samples of a class are modeled as a low-dimensional affine subspace. The use of subspaces to model images and sets has a long history in computer vision and machine learning. For example, it has been proved that the set of all reflectance functions (the mapping from surface normals to intensities) produced by Lambertian objects lie close to a low-dimensional linear subspace (Basri & Jacobs (2003)).
+
+![](images/aced738d53e52c6897ce3211e737ab8e69a525b38abb35f46da9b6a0ccbbd0dc.jpg)  
+Figure 1: Feature embedding in (a) Matching Networks (Vinyals et al. (2016)), (b) Prototypical Networks(Snell et al. (2017)), and (c) our PSN method.
+
+In spite of its intriguing properties, to the best of our knowledge, the subspace modeling has never been used previously to address few-shot learning problems. This makes our paper distinct and novel as compared to former studies (e.g., Vinyals et al. (2016); Snell et al. (2017)). Fig. 1 provides a conceptual illustration of our approach.
+
+We empirically observed that embeddings tailored towards capturing the structure of each class through low-dimensional affine subspaces could lead to discriminative models. Interestingly, such models can be built with minimum overheads and without opting for advanced methods in subspace creation (e.g., such as the notion of sparsity). Our conjecture here is that subspaces are less sensitive to perturbations such as outliers and noise compared to other embedding techniques for few-shot learning as illustratively shown in Fig. 2.
+
+Our contributions in this work are:
+
+i. Few-shot learning is formulated as an embedding problem through subspaces. We rely on a well-established concept stating that samples of a class (and hence variations such as pose and illumination) can be effectively captured by a low-dimensional affine space.   
+ii. Adaptation from few-shot learning to semi-supervised learning is performed with a refinement through soft-assignment. The robustness of such a model is shown in our experiments.   
+iii. We also introduce an evaluation mechanism to assess the generalization ability over unseen classes during test time.
+
+# 2 RELATED WORK
+
+In this section, we briefly review the literature on few-shot learning and subspace clustering. Fewshot learning was originally introduced to imitate human learning capabilities in classification. Some of the early works made use of generative models and similarity learning to capture the variation within parts and geometric configurations of objects (Fei-Fei et al. (2006); Lake et al. (2015); Torralba et al. (2007)). As of late, few-shot problems are mainly addressed through meta-learning techniques. For example, in Koch et al. (2015); Vinyals et al. (2016); Snell et al. (2017); Garcia & Bruna (2018), the problem of few-shot learning is formulated as non-linear embedding or representation learning.
+
+The closest approach to our proposed work is the Prototypical Networks (PN hereafter) model ( Snell et al. (2017)). It uses the random choice of episode images which form the prototype center per class. In addition, recent metric learning approaches use complex architectures and pipelines in convolutional networks of Gidaris & Komodakis (2018) and Wang et al. (2018). In contrast, our projective subspace concept is extremely simple by design as it constitutes just a single layer of the network.
+
+Another common meta-learning approach to few-shot classification uses meta-learning and manipulates gradient updates to train the model parameters. Ravi & Larochelle (2017) utilized Long-Short Term Memory (LSTM)-based learner to optimize the model parameters and their gradients. Moreover, the Model Agnostic Meta-Learner(MAML) proposed by Finn et al. (2017) used gradients per task as well as a meta-gradient from combined tasks, and, as a result, it outperformed LSTM-based learner. Mishra et al. (2018) employed ResNets to model few-shot classification as a temporal solution with aggregation, thus, in meta-learning context, their model can refer to the past experience.
+
+![](images/8b18d1883eb76ee7a0fab0f20f0d7c7690caf88b5ae77000cd19ce3193a7baaa.jpg)  
+Figure 2: The effect of outliers on prototypes and subspaces. The odd rows show the decision boundaries obtained by prototypes (with and without outliers) for two- and three-class problems. The even rows depict how subspaces behave for the same problems. While affected, in general, subspaces show better resilience to perturbations and attain higher discriminatory power in comparison to prototypes. Best viewed in color.
+
+Our idea is based on the assumption that samples from any class in the support set form a lowdimensional subspace. As we will show, to train PSN, backpropagation through Singular Value Decomposition (SVD) is required. Backpropagation through matrix decomposition such as SVD is a well-studied problem Ionescu et al. (2015) with applications ranging from semantic segmentation to classification and visual recognition Ionescu et al. (2015); Li et al. (2017a); Gou et al. (2018).
+
+# 3 PROBLEM SET-UP
+
+We start by defining the terminology used in few-shot learning. The problem of $N$ -way $K$ -shot classification (e.g., 5-way 1-shot) is defined as classifying queries belonging to $N$ classes by seeing only $K$ samples from each class. For example, in 5-way 1-shot setting, the model needs to identify a query among five classes by seeing only one sample from each class. To obtain a reliable model, so-called episodes are used in training. An episode $\mathcal { T } _ { i }$ consists of two sets, the support set $S$ and the query set $Q$ . The system is then trained by minimizing a classification loss over episodes, simulating the scenarios it will encounter at test time. This episode setting is the same as proposed by Vinyals et al. (2016).
+
+A related problem is semi-supervised few-shot setting learning where unlabeled data is provided to the model. In the literature, various configurations are considered for semi-supervised few-shot learning (e.g., Garcia & Bruna (2018); Boney & Ilin (2017); Ren et al. (2018)). In this work, we follow the challenging protocol in Ren et al. (2018) where so-called distractors are introduced. Here, an episode includes the support set $S$ , query set $Q$ , and unlabeled set $\mathcal { R }$ . The support (labeled) $S$ and query $Q$ sets are configured as in few-shot learning. Additionally, an unlabeled set $\mathcal { R }$ is provided to assist the classification task within an episode. In the unlabeled set, there are examples from two different sources: the support classes and the distractor classes. As the name implies, examples from distractor classes are irrelevant to the classification task and represent classes outside the support set.
+
+# 4 PROJECTIVE SUBSPACE NETWORKS
+
+![](images/7e7053d564bc77d42de76a84bc25bc4d6910a79e3d512b4e043d57233a207903.jpg)  
+Figure 3: Projective Subspace Networks Architecture
+
+In what follows, we introduce our PSN approach. The whole framework is trained end-to-end (see Fig. 3) which enables PSN to be used with various deep architectures. As an example, in $\ S$ , we use PSN on top of the WideResNets (Zagoruyko & Komodakis (2016)) for few-shot classification.
+
+We start by introducing our notations for the $N$ -way, $K$ -shot) few-shot learning. Each episode or task $\mathcal { T } _ { i }$ is composed of the support set $S = \{ ( \pmb { x } _ { 1 , 1 } , c _ { 1 , 1 } ) , ( \pmb { x } _ { 1 , 2 } , c _ { 1 , 2 } ) , \cdots , ( \pmb { x } _ { N , K } , c _ { N , K } ) \}$ and the query set $Q = \{ \pmb { q } _ { 1 } , \cdots , \pmb { q } _ { N \times M } \}$ . Here, $\mathbf { \delta } _ { \mathbf { x } _ { i , j } }$ denotes the $j$ -th sample from class $i$ and $c _ { i , j } ~ \in$ $\{ 1 , \cdots , N \}$ . In the semi-supervised setting, there is an unlabeled set $\mathcal { R } \ = \ \{ \boldsymbol { r } _ { 1 } , . . , \boldsymbol { r } _ { U } \}$ within an episode. We propose to model points by subspaces $\{ Z _ { i } \} _ { i = 1 } ^ { N }$ . Each subspace $\boldsymbol { Z } _ { i }$ has a basis represented by $\mathbb { R } ^ { D \times n } \ni P _ { i } = [ \pmb { p } _ { 1 } , \cdot \cdot \cdot , \pmb { p } _ { n } ] ; n \leq D$ , with $P _ { i } ^ { \top } P _ { i } = \mathbf { I } _ { n }$ .
+
+# 4.1 PSN FOR FEW-SHOT CLASSIFICATION
+
+Let $f _ { \Theta } : \mathcal { X } \xrightarrow { } \mathbb { R } ^ { D }$ be a mapping from the input space $\mathcal { X }$ to some $D$ -dimensional representation realized by a neural network. Our goal is to learn $\Theta$ , i.e., the embedding function in a way that the resulting space is suitable for subspace representation. For simplicity, we assume that every class in an episode can be described by just one subspace. Extension to multiple subspaces per class is straightforward though. Define $\mu _ { k }$ as the mean of class $k$ in the embedded space. That is,
+
+$$
+\pmb { \mu } _ { k } = \frac { 1 } { K } \sum _ { i , \ c _ { i } = k } f _ { \Theta } ( \pmb { x } _ { i } ) .
+$$
+
+A basis for the subspace representing class $k$ can be obtained by Singular Value Decomposition (SVD). To be specific, we define $\mathbf { } X _ { k } = [ \pmb { x } _ { k , 1 } - \pmb { \mu } _ { k } , \cdot \cdot \cdot , \pmb { x } _ { k , K } - \pmb { \mu } _ { k } ]$ . Applying truncated SVD on $X _ { k }$ provides us with $P _ { k }$ . We emphasize that more involved techniques to obtain robust subspaces from $X _ { k }$ can potentially improve the PSN. Nevertheless, our goal is to assess whether the concept of subspace modeling for few-shot learning is justified or not and thus we opt for truncated SVD in our implementation. Now a query $\mathbf { \Delta } \mathbf { q } _ { j }$ can be projected onto $P _ { k }$ which yields:
+
+$$
+\pmb { y } _ { j , k } = \pmb { P } _ { k } ^ { \top } ( f _ { \Theta } ( \pmb { q } _ { j } ) - \pmb { \mu } _ { k } ) .
+$$
+
+The distance from the query to $P _ { k }$ is:
+
+$$
+d _ { j , k } = \| f _ { \Theta } ( \pmb q _ { j } ) - \pmb \mu _ { k } - P _ { k } \pmb y _ { j , k } \| .
+$$
+
+We define the probability of the query assigned to class $k$ using a softmax function as:
+
+$$
+p _ { \Theta } ( c = k | \pmb { q } _ { j } ) = \frac { \exp \big ( - d _ { j , k } ^ { 2 } \big ) } { \sum _ { k ^ { \prime } } \exp \big ( - d _ { j , k ^ { \prime } } ^ { 2 } \big ) } .
+$$
+
+Now, we can minimize the negative log of Eqn. 4 to obtain $\Theta$ . To train the whole framework, backpropagation through SVD is required which is available in modern deep learning packages such as PyTorch (Paszke et al. (2017)). Algorithm 1 explains the steps of training our PSN. The code will be released online on Github1.
+
+# Algorithm 1 Train Projective Subspace Networks
+
+<table><tr><td rowspan="16">Input: Each episode Ti with S = {(x1,1,C1,1), , (xN,K,cN,K)} and Q = {q1,,qN×M}</td><td>1:Oo← random initialization</td></tr><tr><td>2: for t in{Ti,..,TNT} do 3: Lt←0 4:</td></tr><tr><td>for k in {1,., N} do X←Sk</td></tr><tr><td>Get examples in the support set from class k Mean from the support set</td></tr><tr><td>μk ← k∑x∈x fe(x) Kμk +∑imife(ri) μk←</td></tr><tr><td>Refined mean(only semi-supervised learning) K+∑mi</td></tr><tr><td>8: X ←[xi - μk,..,xk- μk] 9:</td></tr><tr><td>[u,∑,vT] ← SVD(X) &gt;Matrix factorization using SVD</td></tr><tr><td>10: Pk ←U1....n Truncate the matrix 11: for qj in Qk do</td></tr><tr><td>12: Yj,k ←PT(fe(qj)-μk) Query projection</td></tr><tr><td>13: dj,k ← |lfe(qj)-μk -Pkyj,kll Distance calculation</td></tr><tr><td>exp(-d,k） 14: Pj,k← Softmax on distance scores</td></tr><tr><td>∑k exp(-d,)</td></tr><tr><td>15: end for</td></tr><tr><td>16: end for 17:</td></tr><tr><td>Lt ←N2m∑k∑;-log (pj,k)</td></tr><tr><td>18: Update Θ using VLt 19: end for</td></tr></table>
+
+# 4.2 PSN FOR SEMI-SUPERVISED FEW-SHOT LEARNING
+
+In what follows, we extend the model developed in $\ S 4 . 1$ to address semi-supervised few-shot learning. In doing so, we need to take advantage of the unlabeled data to fit better subspaces to our data. We achieve this by refining the center of each class according to
+
+$$
+\tilde { \mu } _ { k } = \frac { K \mu _ { k } + \sum _ { i } m _ { i } f _ { \Theta } ( r _ { i } ) } { K + \sum _ { i } m _ { i } } , \quad \mathrm { w h e r e } \quad m _ { i } = \frac { \exp ( - \| f _ { \Theta } ( r _ { i } ) - \mu _ { k } ) \| ^ { 2 } ) } { \sum _ { k ^ { \prime } } \exp ( - \| f _ { \Theta } ( r _ { i } ) - \mu _ { k ^ { \prime } } ) \| ^ { 2 } ) } \ .
+$$
+
+Here, $m _ { i }$ is the soft-assignment score for unlabeled samples. To work at the presence of distractors, we use a fake class with zero mean. We empirically observed that such a simple modification to the means can improve the results without the need of refining the SVD step.
+
+# 5 EXPERIMENTS
+
+Below we contrast and assess our method against state-of-the-art techniques on two challenging datasets, namely Mini-ImageNet (Ravi & Larochelle (2017)) and Tiered-ImageNet (Ren et al.
+
+Table 1: Few-shot classification results on Mini-ImageNet using 4-convolutional stages with $9 5 \%$ confidence intervals.   
+
+<table><tr><td>Models</td><td>5-way 5-shot</td><td>20-way 5-shot</td></tr><tr><td>Matching Nets (Vinyals et al. (2016))</td><td>55.31 ± 0.73%</td><td>22.69±0.20%</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>63.11 ± 0.92%</td><td>19.29 ± 0.29%</td></tr><tr><td>Meta-LearnerLSTM(Ravi &amp; Larochelle (2017))</td><td>60.60 ± 0.71%</td><td>26.06 ± 0.25%</td></tr><tr><td>Meta-SGD (Li et al. (2017b))</td><td>64.03 ± 0.94%</td><td>28.92 ± 0.35%</td></tr><tr><td>PN(Snell et al. (2017))</td><td>65.49 ± 0.25%</td><td>37.23 ± 0.21%</td></tr><tr><td>PSN</td><td>66.62± 0.69%</td><td>38.26 ± 0.23%</td></tr></table>
+
+(2018)). As the rule of thumb, the parameters of subspace dimension $( n )$ that we used in the experiments are $K$ -1 for training and two for testing stage.
+
+Mini-ImageNet. The Mini-ImageNet(Ravi & Larochelle (2017)) contains 60,000 images of the ImageNet(Russakovsky et al. (2015)) datasets. Images in the Mini-ImageNet are of size $8 4 \times 8 4$ and represent 100 classes with 64, 16, and 20 classes used for training, validation, and testing, respectively.
+
+Tiered-ImageNet. This dataset is also derived from ImageNet but contains a broader set of classes compared to the Mini-ImageNet. There are 351 classes from 20 different categories for training, 97 classes from 6 different categories for validation, and 160 classes from 8 different categories for testing. In contrast to the Mini-ImageNet, the training and test sets in Tiered-ImageNet represent distinct classes. Moreover, in designing the Tiered-ImageNet, the problem of few-shot learning with unlabeled data was taken into account and the labeled data is only within a small percentage when performing semi-supervised learning. In this large dataset, learning from the labeled data is still sufficient to produce reasonable representations even though the unlabeled data is set in huge portion e.g., $9 0 \%$ .
+
+# 5.1 FEW-SHOT LEARNING
+
+We follow the general practice and evaluate our method on the Mini-ImageNet dataset when it comes to few-shot learning and classification. Various procedures such as pre-training using 64 classes (e.g., Qiao et al. (2018)) and training with more classes/way have shown to increase the overall accuracy. Nevertheless, we avoid such procedures deliberately as we are mainly interested in contrasting the core idea, i.e., the role of subspaces in few-shot learning. So, we trained on 5- way 5-shot and 20-way 5-shot, then applied the same classification task setup during testing. The CNN architecture is the same as the one used in Snell et al. (2017) with 4-convolutional stages. We also use WideResNets (Zagoruyko & Komodakis (2016)) with 16 depth, 6 widening factor, and 0.3 dropout rate to compare with ResNets (He et al. (2016)) solution reported by Mishra et al. (2018). The feature dimensions from both architectures are 1600 and 384 respectively. We used ADAM ( Kingma & Ba (2015)) for optimizing our model and set the learning rate to 0.001 and cut it to half every 2.5K episodes for both architectures.
+
+Results. By design, the method cannot accommodate learning with exactly one example, hence, the comparison here is provided only for 5-shot in 5-way and 20-way. In every episode, the query set contains 15 samples from each class. Our method outperforms the previous methods with 4- convolutional stages shown in Table 1. We also implemented WideResNets (Zagoruyko & Komodakis (2016)) using our method and obtained the performance for 5-way 5-shot: ${ \bf 6 9 . 9 2 \pm 0 . 6 4 \% }$ and 20-way 5-shot: $\mathbf { 4 1 . 8 4 \pm 0 . 2 4 \% }$ that can outperform ResNets-based approach proposed by Mishra et al. (2018) with $6 8 . 8 8 \pm 0 . 9 2 \%$ in 5-way and 5-shot classification task.
+
+# 5.2 SEMI-SUPERVISED FEW-SHOT LEARNING
+
+In this experiment, the embedding architecture has 4-convolutional layers as PN (Snell et al. (2017)). We follow the experimental setup proposed by Ren et al. (2018). The episode composition for labeled or support set and query set is similar to the few-shot learning classification task, but there is an additional unlabeled set provided in each episode. Our model is trained on 100K episodes for
+
+Table 2: Semi-supervised few-shot classification results on the Mini-ImageNet and Tiered-ImageNet with $4 0 \%$ and $1 0 \%$ labeled data, respectively. We show the classification results with and without distractors. We compare our results to PN on semi-supervised learning (PN-SSL) with soft $K$ -means (non-masked) and masked $K$ -means (masked), as proposed by Ren et al. (2018).   
+
+<table><tr><td rowspan=1 colspan=1>Models</td><td rowspan=1 colspan=2>Mini-ImageNet5-way 5-shot</td><td rowspan=1 colspan=2>Tiered-ImageNet5-way 5-shot</td></tr><tr><td rowspan=1 colspan=1>PN,SupervisedPSN, Supervised</td><td rowspan=1 colspan=2>59.08±0.22%63.43 ± 0.61%</td><td rowspan=1 colspan=2>66.15±0.22%68.72 ±0.49%</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>w/0Distractors</td><td rowspan=1 colspan=1>w/Distractors</td><td rowspan=1 colspan=1>w/oDistractors</td><td rowspan=1 colspan=1>w/Distractors</td></tr><tr><td rowspan=1 colspan=1>PN-SSL,Non-MaskedPN-SSL,MaskedPSN, Semi-Supervised</td><td rowspan=1 colspan=1>64.59±0.28%64.39 ± 0.24%68.12 ± 0.67%</td><td rowspan=1 colspan=1>63.55±0.28%62.96 ± 0.14%66.10 ± 0.66%</td><td rowspan=1 colspan=1>70.25 ± 0.31%69.88 ± 0.20%71.15 ± 0.67%</td><td rowspan=1 colspan=1>68.32 ± 0.22%69.08 ± 0.25%69.15 ± 0.51%</td></tr></table>
+
+Mini-ImageNet   
+Table 3: Generalization performance of PSN and PN with varying number of way $( N )$ and shot $( K )$ at the test time.   
+
+<table><tr><td rowspan=1 colspan=1>Way(N)</td><td rowspan=1 colspan=1>Shot(K)</td><td rowspan=1 colspan=1>PSN</td><td rowspan=1 colspan=1>PN</td><td rowspan=1 colspan=1>Way(N)</td><td rowspan=1 colspan=1>Shot(K)</td><td rowspan=1 colspan=1>PSN</td><td rowspan=1 colspan=1>PN</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>35101520</td><td rowspan=1 colspan=1>61.26±0.79%66.62 ± 0.69%71.88 ±0.59%73.50 ±0.58%74.88± 0.63%</td><td rowspan=1 colspan=1>60.36 ±0.31%65.49 ±0.25%69.80 ±0.30%71.92 ±0.26%73.01± 0.31%</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>35101520</td><td rowspan=1 colspan=1>45.46±0.44%51.49 ± 0.46%57.14±0.41%60.20 ±0.39%61.38 ± 0.44%</td><td rowspan=1 colspan=1>44.09±0.28%49.06± 0.25%55.23± 0.27%58.06 ±0.22%59.27±0.22%</td></tr><tr><td rowspan=3 colspan=1>15</td><td rowspan=3 colspan=1>35101520</td><td rowspan=3 colspan=1>37.07 ±0.33%43.18 ±0.32%49.44±0.31%51.96 ± 0.32%54.23±0.34%</td><td rowspan=3 colspan=1>36.24±0.22%40.94±0.24%47.19 ±0.21%49.85 ±0.25%51.37 ± 0.20%</td><td rowspan=3 colspan=1>20</td><td rowspan=3 colspan=1>35101520</td><td rowspan=1 colspan=1>32.13±0.25%37.71± 0.23%</td><td rowspan=2 colspan=1>31.43±0.18%35.58 ±0.17%41.69 ± 0.16%</td></tr><tr><td rowspan=1 colspan=1>44.09±0.22%</td><td rowspan=1 colspan=1>41.6</td></tr><tr><td rowspan=1 colspan=1>47.27±0.22%49.00 ±0.23%</td><td rowspan=1 colspan=1>44.37± 0.17%45.90± 0.18%</td></tr></table>
+
+Mini-ImageNet and Tiered-ImageNet with $4 0 \%$ and $1 0 \%$ of labeled data, respectively. We used the ADAM solver (Kingma & Ba (2015)), the set the learning rate to 0.001 with the weight decay and cut the rate to half every 10K episodes. We trained in two settings: (i) supervised setting, where only labeled data is taken into account, and (ii) semi-supervised setting for which the unlabeled set is also used. The unlabeled set is composed of the examples from the classes in the support set and distractor classes. The number of supporting classes and distractor classes is set to five for training and testing. In the training stage, the number of examples in the unlabeled set is 50 consisting of five examples from each class. In the testing stage, the unlabeled set consists of 20 examples from each class. We also define the query set to have 20 examples per class for testing purpose.
+
+Results. In these experimental results, the performance is counted over 600 episodes. The results are averaged over 10 random splits of labeled and unlabeled sets. The supervised experiment shows that our method learns robust feature embedding from a small portion of labeled data. With the help of soft-assignment over unlabeled datapoints, the semi-supervised experiment detailed in Table 2 is demonstrated to outperform Prototypical Networks for Semi-Supervised Learning (PN-SSL) proposed by Ren et al. (2018).
+
+# 5.3 GENERALIZATION BEYOND $N$ -WAY $K$ -SHOT
+
+As a measure of generalization ability, we propose to evaluate algorithms beyond the somehow inflexible testing protocol of few-shot learning. In particular, we assess whether a model trained on low number of classes can generalize well to classification tasks involving large number of classes. To gain more insights, we further study how models trained with the $K$ -shot assumption will perform if extra examples are available at the test time.
+
+In the evaluation below, all models are trained with the 5-way 5-shot setting. At the test time, the models face the test protocol of $N$ -way $K$ -shot learning with $N ~ \in ~ \overline { { \{ 5 , 1 0 , 1 5 , 2 0 \} } }$ and $K \in \{ 3 , 5 , 1 0 , 1 5 , 2 0 \}$ . Table 3 contrasts the performance of PSN against PN. The table is selfexplanatory. In all experiments, PSN outperforms PN with the gap widened with more challenging settings (e.g., 20-way 20-shot).
+
+Robustness to Perturbations. Our motivation to develop PSN is to devise a model that shows better resilience to perturbations. Intuitively, to have a noticeable change in the orientation of a subspace, one needs to induce drastic changes to the set. To empirically verify our claim, we assess how robust is PSN in comparison to PN (Snell et al. (2017)) in the presence of perturbations. In particular, we considered the problems of 5-way 5-shot and 5-way 10-shot learning and introduced two types of perturbations at the test time to the trained models. Firstly, we randomly sampled examples from classes not presented in the support sets and included them in the support examples. Secondly, noisy examples are generated randomly using a multivariate Gaussian distribution with random mean and variance of $\sigma = \{ 0 . 1 5 , 0 . 3 , 0 . 4 \}$ . The results of this study are presented in Appendix A due to page limitations. To summarize, our experiments show that both PSN and PN are affected by outliers negatively. That said, the PSN exhibits a much better degree of resilience to outliers. When additive noise is considered, PSN behaves robustly for a wide-range of contamination. In contrast, the performance of PN drops rapidly and significantly in the presence of noise, reinforcing our idea that the use of subspaces indeed leads to a more robust model for the task in hand.
+
+# 6 DISCUSSION
+
+Conceptually, the PN Snell et al. (2017) is the closest work to ours as both solutions obtain a refined representation for each class in the support sets, with the former using the class mean as the representation while in PSN affine subspaces model classes. That said, PSN also uses the mean of each class towards identifying their representative subspaces. Aside from the theoretical properties of affine subspaces2, we empirically observed that for more challenging setups (e.g., 20-way 20-shot), utilizing the mean leads to a better performance. This is because prototypes lie on subspaces in the feature space. Additionally, prototypes in the PN can be easily utilized to design a hybrid with our approach.
+
+Remark 1. Interestingly, in special cases, PSN simplifies to PN. For example, if samples of a class span a infinitesimal region, at the limit, collapsing to a point, then an affine subspace reduces to a point, recovering PN from PSN.
+
+Remark 2. Since it is not possible to build an affine subspace from only one example, PSN cannot address $^ { l }$ -shot learning problems per se. However, simply applying augmentations to support images facilitates 1-shot learning. However, this issue is beyond the point we make in this work.
+
+Subspace Dimension. In comparison to other models such as matching networks or PN, PSN comes with an extra hyper-parameter, the dimensionality of the subspaces $( i . e . , n )$ . As a rule of thumb, we recommend to use $n = K - 1$ to train the model, while $n = 2$ at the test time gives reasonable and robust results. That said, we thoroughly studied the effect of this parameter and summarized our findings in Appendix B. In short, PSN exhibits a great degree of robustness to $n$ , which in turns, makes training of them painless.
+
+Computational Complexity. The computational complexity of our PSN approach is $\mathcal { O } ( \operatorname* { m i n } ( N D ^ { 2 } K , N D K ^ { 2 } ) )$ , where $K$ , $N$ , and $D$ are the number of shot, way, and feature dimensionality respectively. Compared to the complexity of the PN algorithm, i.e., $\mathcal { O } ( N D K )$ , our algorithm is somehow slower due to the involvement of the SVD step. If the complexity is of a concern, fast approximate algorithms for SVD (e.g., Menon & Elkan (2011)) can be considered.
+
+# 7 CONCLUSIONS
+
+This paper presents the PSN, a novel approach for few-shot learning that employs nonlinear embeddings and modeling via affine subspaces. Empirically, we showed that the representations learned via PSN were expressive across a wide-range of supervised and semi-supervised few-shot problems.
+
+In PSN, each class is represented by the subspace formed by all its examples, meaning that each class is modeled by the span of its samples. One possibility to extend the PSN modeling is to benefit -cleverly- from the null space associated with each class, in the hope of designing a more discriminative model.
+
+# REFERENCES
+
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+
+# Appendices
+
+# A PERTURBATIONS EFFECT ON PERFORMANCE
+
+In this section, we study the effect of perturbation on the performance of PN and PSN. More specifically, we considered the problems of 5-way 5-shot and 5-way 10-shot learning and introduced two types of perturbations at the test time with the trained models. Note that, the model is obtained from 5-way 5-shot training without perturbations. Firstly, we randomly sampled examples from classes not presented in the support sets. Secondly, additive noise is generated randomly using a multivariate Gaussian distribution with random mean and variance of $\sigma = \{ 0 . 1 5 , 0 . 3 , 0 . 4 \}$ . Both examples in these two types are included in the support examples for prototypes and subspaces creation.
+
+The results of this study are depicted in Fig. 4. To summarize, our experiments show that both PSN and PN are affected by outliers negatively. That said, the PSN exhibits a much better degree of resilience to outliers. For example, modelling with PN leads to a drop of $19 \%$ and $12 \%$ percentage points when each support set has 20 outliers for the problem of 5-way 5-shot and 5-way 10-shot respectively. For the same experiment, the PSN modelling only suffers $11 \%$ and $9 \%$ percentage points of performance drop. When additive noise is considered, PSN behaves robustly for a widerange of contamination. In contrast, the performance of PN drops rapidly and significantly in the presence of noise, reinforcing our idea that subspaces form indeed a more robust model for the task in hand.
+
+![](images/3a9ce6abe6c975f9f6f8ff0a106e3d983aa7f834324e724ad6393fa82656ef31.jpg)  
+Figure 4: The charts show 5-way 5-shot and 5-way 10-shot results of the PSN and the PN algorithms in the presence of outliers and additive noise. In the first column, sub-plots show the effect of introducing outliers among support samples (the classes of outliers are disjoint with the support classes of samples). The second to fourth columns show the effect of introducing noisy examples generated randomly from Gaussian distributions with random means and variance of $\dot { \sigma } = \{ \bar { 0 . 1 5 } , \bar { 0 . 3 } , 0 . 4 \}$ , respectively. The performance is measured with increasing number of outliers and noisy examples (X-axes).
+
+# B SUBSPACE DIMENSION EXPERIMENT
+
+In this section, we report the results of experiments conducted to assess the effect of the subspace dimensionality $( i . e . , n )$ on the overall accuracy of the algorithm. In particular, we considered two few-shot problems, namely 5-way 5-shot and 5-way 20-shot. For the former, we varied the subspace dimensionality $n \in \{ 2 , 3 , 4 , 5 \}$ and considered all combinations during training and testing (for example, $n = 4$ during training and $n = 3$ at the test time). For the experiment on 5-way 20- shot problem, we varied the subspace dimensionality $n \in \{ 5 , 1 0 , 1 5 , 2 0 \}$ and again considered all combinations.
+
+Our experiments demonstrate that PSN behaves robustly over a wide-range of subspace dimensionality. For example, the lowest and highest performances for the 5-way 5-shot learning are $6 6 . 0 8 \pm 0 . 6 7 \%$ and $6 6 . 6 2 \pm 0 . 6 9 \%$ , respectively. For the problem of 5-way 20-shot learning, the lowest and highest performances are $7 5 . 0 3 \pm 0 . 5 7 \%$ and $7 5 . 5 8 \pm 0 . 5 7 \%$ , respectively.

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+        "type": "text",
+        "text": "PROJECTIVE SUBSPACE NETWORKS FOR FEW-SHOT LEARNING ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            251
+        ],
+        "page_idx": 0
+    },
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+        "text": "Generalization from limited examples, usually studied under the umbrella of metalearning, equips learning techniques with the ability to adapt quickly in dynamical environments and proves to be an essential aspect of lifelong learning. In this paper, we introduce the Projective Subspace Networks (PSN), a deep learning paradigm that learns non-linear embeddings from limited supervision. In contrast to previous studies, the embedding in PSN deems samples of a given class to form an affine subspace. We will show that such modeling leads to robust solutions, yielding competitive results on supervised and semi-supervised few-shot classification. Moreover, our PSN approach has the ability of end-to-end learning. In contrast to previous works, our projective subspace can be thought of as a richer representation capturing higher-order information datapoints for modeling new concepts. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            336,
+            470
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Supervised learning with deep architectures, though achieving remarkable results in many areas, requires large amount of annotated data. Various studies show that many deep learning techniques in computer vision, speech recognition and natural language understanding, to name a few methods stated in Hinton et al. (2012); Krizhevsky et al. (2012), will fail to produce reliable models that generalize well if limited data annotations are available. Aside from the labor associated with annotating data, precise annotation can become ill-posed in some cases. One prime example of such a difficulty is object detection labeling which requires annotating bounding boxes of objects as explained in Alexe et al. (2012). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrast to the current trend in deep learning, humans can learn new concepts from only a few examples. This in turn provides humans with lifelong learning abilities. Inspired by such learning abilities, several approaches are developed to study learning from limited examples Lake et al. (2015); Lazaridou et al. (2017); Vinyals et al. (2016); Triantafillou et al. (2017); Xu et al. (2017); Finn et al. (2017); Ravi & Larochelle (2017); Wang et al. (2018); Mishra et al. (2018); Qiao et al. (2018); Neill & Buitelaar (2018). In machine learning, the diverse ideas in this context include embedding features through metric learning (Koch et al. (2015), Vinyals et al. (2016)), optimization technique(Finn et al. (2017), Ravi & Larochelle (2017)), and generative models(Fei-Fei et al. (2006), Lake et al. (2015)). ",
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+        "text": "In this work, we propose a deep model that learns new concepts from limited data to address two challenging learning problems; namely: (i) few-shot classification and (ii) semi-supervised few-shot learning. The goal of few-shot classification is to learn a model that can discriminate a given query by comparing it to a few of samples (a.k.a. the support set). An example is to classify a motorcycle by viewing some different types of motorcycles (or other vehicles). The goal of semi-supervised few-shot learning is to additionally benefit from unlabeled data to boost the performance of the model. ",
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+        "text": "Our method, coined Projective Subspace Networks or PSN for short, learns non-linear embeddings using subspaces, in a sense that samples of a class are modeled as a low-dimensional affine subspace. The use of subspaces to model images and sets has a long history in computer vision and machine learning. For example, it has been proved that the set of all reflectance functions (the mapping from surface normals to intensities) produced by Lambertian objects lie close to a low-dimensional linear subspace (Basri & Jacobs (2003)). ",
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+    {
+        "type": "image",
+        "img_path": "images/aced738d53e52c6897ce3211e737ab8e69a525b38abb35f46da9b6a0ccbbd0dc.jpg",
+        "image_caption": [
+            "Figure 1: Feature embedding in (a) Matching Networks (Vinyals et al. (2016)), (b) Prototypical Networks(Snell et al. (2017)), and (c) our PSN method. "
+        ],
+        "image_footnote": [],
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+        "text": "In spite of its intriguing properties, to the best of our knowledge, the subspace modeling has never been used previously to address few-shot learning problems. This makes our paper distinct and novel as compared to former studies (e.g., Vinyals et al. (2016); Snell et al. (2017)). Fig. 1 provides a conceptual illustration of our approach. ",
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+    {
+        "type": "text",
+        "text": "We empirically observed that embeddings tailored towards capturing the structure of each class through low-dimensional affine subspaces could lead to discriminative models. Interestingly, such models can be built with minimum overheads and without opting for advanced methods in subspace creation (e.g., such as the notion of sparsity). Our conjecture here is that subspaces are less sensitive to perturbations such as outliers and noise compared to other embedding techniques for few-shot learning as illustratively shown in Fig. 2. ",
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+    {
+        "type": "text",
+        "text": "Our contributions in this work are: ",
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+        "text": "i. Few-shot learning is formulated as an embedding problem through subspaces. We rely on a well-established concept stating that samples of a class (and hence variations such as pose and illumination) can be effectively captured by a low-dimensional affine space.   \nii. Adaptation from few-shot learning to semi-supervised learning is performed with a refinement through soft-assignment. The robustness of such a model is shown in our experiments.   \niii. We also introduce an evaluation mechanism to assess the generalization ability over unseen classes during test time. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we briefly review the literature on few-shot learning and subspace clustering. Fewshot learning was originally introduced to imitate human learning capabilities in classification. Some of the early works made use of generative models and similarity learning to capture the variation within parts and geometric configurations of objects (Fei-Fei et al. (2006); Lake et al. (2015); Torralba et al. (2007)). As of late, few-shot problems are mainly addressed through meta-learning techniques. For example, in Koch et al. (2015); Vinyals et al. (2016); Snell et al. (2017); Garcia & Bruna (2018), the problem of few-shot learning is formulated as non-linear embedding or representation learning. ",
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+        "type": "text",
+        "text": "The closest approach to our proposed work is the Prototypical Networks (PN hereafter) model ( Snell et al. (2017)). It uses the random choice of episode images which form the prototype center per class. In addition, recent metric learning approaches use complex architectures and pipelines in convolutional networks of Gidaris & Komodakis (2018) and Wang et al. (2018). In contrast, our projective subspace concept is extremely simple by design as it constitutes just a single layer of the network. ",
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+    {
+        "type": "text",
+        "text": "Another common meta-learning approach to few-shot classification uses meta-learning and manipulates gradient updates to train the model parameters. Ravi & Larochelle (2017) utilized Long-Short Term Memory (LSTM)-based learner to optimize the model parameters and their gradients. Moreover, the Model Agnostic Meta-Learner(MAML) proposed by Finn et al. (2017) used gradients per task as well as a meta-gradient from combined tasks, and, as a result, it outperformed LSTM-based learner. Mishra et al. (2018) employed ResNets to model few-shot classification as a temporal solution with aggregation, thus, in meta-learning context, their model can refer to the past experience. ",
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+        "img_path": "images/8b18d1883eb76ee7a0fab0f20f0d7c7690caf88b5ae77000cd19ce3193a7baaa.jpg",
+        "image_caption": [
+            "Figure 2: The effect of outliers on prototypes and subspaces. The odd rows show the decision boundaries obtained by prototypes (with and without outliers) for two- and three-class problems. The even rows depict how subspaces behave for the same problems. While affected, in general, subspaces show better resilience to perturbations and attain higher discriminatory power in comparison to prototypes. Best viewed in color. "
+        ],
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Our idea is based on the assumption that samples from any class in the support set form a lowdimensional subspace. As we will show, to train PSN, backpropagation through Singular Value Decomposition (SVD) is required. Backpropagation through matrix decomposition such as SVD is a well-studied problem Ionescu et al. (2015) with applications ranging from semantic segmentation to classification and visual recognition Ionescu et al. (2015); Li et al. (2017a); Gou et al. (2018). ",
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+        "type": "text",
+        "text": "3 PROBLEM SET-UP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We start by defining the terminology used in few-shot learning. The problem of $N$ -way $K$ -shot classification (e.g., 5-way 1-shot) is defined as classifying queries belonging to $N$ classes by seeing only $K$ samples from each class. For example, in 5-way 1-shot setting, the model needs to identify a query among five classes by seeing only one sample from each class. To obtain a reliable model, so-called episodes are used in training. An episode $\\mathcal { T } _ { i }$ consists of two sets, the support set $S$ and the query set $Q$ . The system is then trained by minimizing a classification loss over episodes, simulating the scenarios it will encounter at test time. This episode setting is the same as proposed by Vinyals et al. (2016). ",
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+        "type": "text",
+        "text": "A related problem is semi-supervised few-shot setting learning where unlabeled data is provided to the model. In the literature, various configurations are considered for semi-supervised few-shot learning (e.g., Garcia & Bruna (2018); Boney & Ilin (2017); Ren et al. (2018)). In this work, we follow the challenging protocol in Ren et al. (2018) where so-called distractors are introduced. Here, an episode includes the support set $S$ , query set $Q$ , and unlabeled set $\\mathcal { R }$ . The support (labeled) $S$ and query $Q$ sets are configured as in few-shot learning. Additionally, an unlabeled set $\\mathcal { R }$ is provided to assist the classification task within an episode. In the unlabeled set, there are examples from two different sources: the support classes and the distractor classes. As the name implies, examples from distractor classes are irrelevant to the classification task and represent classes outside the support set. ",
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+    {
+        "type": "text",
+        "text": "4 PROJECTIVE SUBSPACE NETWORKS ",
+        "text_level": 1,
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+        "img_path": "images/7e7053d564bc77d42de76a84bc25bc4d6910a79e3d512b4e043d57233a207903.jpg",
+        "image_caption": [
+            "Figure 3: Projective Subspace Networks Architecture "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "In what follows, we introduce our PSN approach. The whole framework is trained end-to-end (see Fig. 3) which enables PSN to be used with various deep architectures. As an example, in $\\ S$ , we use PSN on top of the WideResNets (Zagoruyko & Komodakis (2016)) for few-shot classification. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We start by introducing our notations for the $N$ -way, $K$ -shot) few-shot learning. Each episode or task $\\mathcal { T } _ { i }$ is composed of the support set $S = \\{ ( \\pmb { x } _ { 1 , 1 } , c _ { 1 , 1 } ) , ( \\pmb { x } _ { 1 , 2 } , c _ { 1 , 2 } ) , \\cdots , ( \\pmb { x } _ { N , K } , c _ { N , K } ) \\}$ and the query set $Q = \\{ \\pmb { q } _ { 1 } , \\cdots , \\pmb { q } _ { N \\times M } \\}$ . Here, $\\mathbf { \\delta } _ { \\mathbf { x } _ { i , j } }$ denotes the $j$ -th sample from class $i$ and $c _ { i , j } ~ \\in$ $\\{ 1 , \\cdots , N \\}$ . In the semi-supervised setting, there is an unlabeled set $\\mathcal { R } \\ = \\ \\{ \\boldsymbol { r } _ { 1 } , . . , \\boldsymbol { r } _ { U } \\}$ within an episode. We propose to model points by subspaces $\\{ Z _ { i } \\} _ { i = 1 } ^ { N }$ . Each subspace $\\boldsymbol { Z } _ { i }$ has a basis represented by $\\mathbb { R } ^ { D \\times n } \\ni P _ { i } = [ \\pmb { p } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { p } _ { n } ] ; n \\leq D$ , with $P _ { i } ^ { \\top } P _ { i } = \\mathbf { I } _ { n }$ . ",
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+    {
+        "type": "text",
+        "text": "4.1 PSN FOR FEW-SHOT CLASSIFICATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Let $f _ { \\Theta } : \\mathcal { X } \\xrightarrow { } \\mathbb { R } ^ { D }$ be a mapping from the input space $\\mathcal { X }$ to some $D$ -dimensional representation realized by a neural network. Our goal is to learn $\\Theta$ , i.e., the embedding function in a way that the resulting space is suitable for subspace representation. For simplicity, we assume that every class in an episode can be described by just one subspace. Extension to multiple subspaces per class is straightforward though. Define $\\mu _ { k }$ as the mean of class $k$ in the embedded space. That is, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f263c393fde716e88b9af94737856f1e5a0f5be3ea4d0cd359eb53bc4eb771f0.jpg",
+        "text": "$$\n\\pmb { \\mu } _ { k } = \\frac { 1 } { K } \\sum _ { i , \\ c _ { i } = k } f _ { \\Theta } ( \\pmb { x } _ { i } ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "A basis for the subspace representing class $k$ can be obtained by Singular Value Decomposition (SVD). To be specific, we define $\\mathbf { } X _ { k } = [ \\pmb { x } _ { k , 1 } - \\pmb { \\mu } _ { k } , \\cdot \\cdot \\cdot , \\pmb { x } _ { k , K } - \\pmb { \\mu } _ { k } ]$ . Applying truncated SVD on $X _ { k }$ provides us with $P _ { k }$ . We emphasize that more involved techniques to obtain robust subspaces from $X _ { k }$ can potentially improve the PSN. Nevertheless, our goal is to assess whether the concept of subspace modeling for few-shot learning is justified or not and thus we opt for truncated SVD in our implementation. Now a query $\\mathbf { \\Delta } \\mathbf { q } _ { j }$ can be projected onto $P _ { k }$ which yields: ",
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+        "type": "equation",
+        "img_path": "images/7f46e875cd60447196cb9a512dd9c8680d6b7375385c82a1dce472a8d58026a0.jpg",
+        "text": "$$\n\\pmb { y } _ { j , k } = \\pmb { P } _ { k } ^ { \\top } ( f _ { \\Theta } ( \\pmb { q } _ { j } ) - \\pmb { \\mu } _ { k } ) .\n$$",
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+        "type": "text",
+        "text": "The distance from the query to $P _ { k }$ is: ",
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+    {
+        "type": "equation",
+        "img_path": "images/88d98919b89766208cb304089ee012102cd6674c906ec3df07bc8d5536a691be.jpg",
+        "text": "$$\nd _ { j , k } = \\| f _ { \\Theta } ( \\pmb q _ { j } ) - \\pmb \\mu _ { k } - P _ { k } \\pmb y _ { j , k } \\| .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We define the probability of the query assigned to class $k$ using a softmax function as: ",
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+        "type": "equation",
+        "img_path": "images/dbb795ae13bc8f5ce8bfb15b9ecd12aa18c74c48ae98612d4b71850b940f9ba6.jpg",
+        "text": "$$\np _ { \\Theta } ( c = k | \\pmb { q } _ { j } ) = \\frac { \\exp \\big ( - d _ { j , k } ^ { 2 } \\big ) } { \\sum _ { k ^ { \\prime } } \\exp \\big ( - d _ { j , k ^ { \\prime } } ^ { 2 } \\big ) } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Now, we can minimize the negative log of Eqn. 4 to obtain $\\Theta$ . To train the whole framework, backpropagation through SVD is required which is available in modern deep learning packages such as PyTorch (Paszke et al. (2017)). Algorithm 1 explains the steps of training our PSN. The code will be released online on Github1. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Train Projective Subspace Networks ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/5295b5a514131598bdf72c81259f8a5389fb003bf6220a11ed8c4c8c6d47e2ec.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"16\">Input: Each episode Ti with S = {(x1,1,C1,1), , (xN,K,cN,K)} and Q = {q1,,qN×M}</td><td>1:Oo← random initialization</td></tr><tr><td>2: for t in{Ti,..,TNT} do 3: Lt←0 4:</td></tr><tr><td>for k in {1,., N} do X←Sk</td></tr><tr><td>Get examples in the support set from class k Mean from the support set</td></tr><tr><td>μk ← k∑x∈x fe(x) Kμk +∑imife(ri) μk←</td></tr><tr><td>Refined mean(only semi-supervised learning) K+∑mi</td></tr><tr><td>8: X ←[xi - μk,..,xk- μk] 9:</td></tr><tr><td>[u,∑,vT] ← SVD(X) &gt;Matrix factorization using SVD</td></tr><tr><td>10: Pk ←U1....n Truncate the matrix 11: for qj in Qk do</td></tr><tr><td>12: Yj,k ←PT(fe(qj)-μk) Query projection</td></tr><tr><td>13: dj,k ← |lfe(qj)-μk -Pkyj,kll Distance calculation</td></tr><tr><td>exp(-d,k） 14: Pj,k← Softmax on distance scores</td></tr><tr><td>∑k exp(-d,)</td></tr><tr><td>15: end for</td></tr><tr><td>16: end for 17:</td></tr><tr><td>Lt ←N2m∑k∑;-log (pj,k)</td></tr><tr><td>18: Update Θ using VLt 19: end for</td></tr></table>",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4.2 PSN FOR SEMI-SUPERVISED FEW-SHOT LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In what follows, we extend the model developed in $\\ S 4 . 1$ to address semi-supervised few-shot learning. In doing so, we need to take advantage of the unlabeled data to fit better subspaces to our data. We achieve this by refining the center of each class according to ",
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+    {
+        "type": "equation",
+        "img_path": "images/9e016400eeff15303430588ab76be474540e36a243db4ab4488b0d37a68e0bc0.jpg",
+        "text": "$$\n\\tilde { \\mu } _ { k } = \\frac { K \\mu _ { k } + \\sum _ { i } m _ { i } f _ { \\Theta } ( r _ { i } ) } { K + \\sum _ { i } m _ { i } } , \\quad \\mathrm { w h e r e } \\quad m _ { i } = \\frac { \\exp ( - \\| f _ { \\Theta } ( r _ { i } ) - \\mu _ { k } ) \\| ^ { 2 } ) } { \\sum _ { k ^ { \\prime } } \\exp ( - \\| f _ { \\Theta } ( r _ { i } ) - \\mu _ { k ^ { \\prime } } ) \\| ^ { 2 } ) } \\ .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, $m _ { i }$ is the soft-assignment score for unlabeled samples. To work at the presence of distractors, we use a fake class with zero mean. We empirically observed that such a simple modification to the means can improve the results without the need of refining the SVD step. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Below we contrast and assess our method against state-of-the-art techniques on two challenging datasets, namely Mini-ImageNet (Ravi & Larochelle (2017)) and Tiered-ImageNet (Ren et al. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/a3855d4cd20555ee505ed9a74bb7c61a9a3a678d8d7f9b27d6480a49df2b31d2.jpg",
+        "table_caption": [
+            "Table 1: Few-shot classification results on Mini-ImageNet using 4-convolutional stages with $9 5 \\%$ confidence intervals. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>5-way 5-shot</td><td>20-way 5-shot</td></tr><tr><td>Matching Nets (Vinyals et al. (2016))</td><td>55.31 ± 0.73%</td><td>22.69±0.20%</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>63.11 ± 0.92%</td><td>19.29 ± 0.29%</td></tr><tr><td>Meta-LearnerLSTM(Ravi &amp; Larochelle (2017))</td><td>60.60 ± 0.71%</td><td>26.06 ± 0.25%</td></tr><tr><td>Meta-SGD (Li et al. (2017b))</td><td>64.03 ± 0.94%</td><td>28.92 ± 0.35%</td></tr><tr><td>PN(Snell et al. (2017))</td><td>65.49 ± 0.25%</td><td>37.23 ± 0.21%</td></tr><tr><td>PSN</td><td>66.62± 0.69%</td><td>38.26 ± 0.23%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(2018)). As the rule of thumb, the parameters of subspace dimension $( n )$ that we used in the experiments are $K$ -1 for training and two for testing stage. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Mini-ImageNet. The Mini-ImageNet(Ravi & Larochelle (2017)) contains 60,000 images of the ImageNet(Russakovsky et al. (2015)) datasets. Images in the Mini-ImageNet are of size $8 4 \\times 8 4$ and represent 100 classes with 64, 16, and 20 classes used for training, validation, and testing, respectively. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Tiered-ImageNet. This dataset is also derived from ImageNet but contains a broader set of classes compared to the Mini-ImageNet. There are 351 classes from 20 different categories for training, 97 classes from 6 different categories for validation, and 160 classes from 8 different categories for testing. In contrast to the Mini-ImageNet, the training and test sets in Tiered-ImageNet represent distinct classes. Moreover, in designing the Tiered-ImageNet, the problem of few-shot learning with unlabeled data was taken into account and the labeled data is only within a small percentage when performing semi-supervised learning. In this large dataset, learning from the labeled data is still sufficient to produce reasonable representations even though the unlabeled data is set in huge portion e.g., $9 0 \\%$ . ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 FEW-SHOT LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We follow the general practice and evaluate our method on the Mini-ImageNet dataset when it comes to few-shot learning and classification. Various procedures such as pre-training using 64 classes (e.g., Qiao et al. (2018)) and training with more classes/way have shown to increase the overall accuracy. Nevertheless, we avoid such procedures deliberately as we are mainly interested in contrasting the core idea, i.e., the role of subspaces in few-shot learning. So, we trained on 5- way 5-shot and 20-way 5-shot, then applied the same classification task setup during testing. The CNN architecture is the same as the one used in Snell et al. (2017) with 4-convolutional stages. We also use WideResNets (Zagoruyko & Komodakis (2016)) with 16 depth, 6 widening factor, and 0.3 dropout rate to compare with ResNets (He et al. (2016)) solution reported by Mishra et al. (2018). The feature dimensions from both architectures are 1600 and 384 respectively. We used ADAM ( Kingma & Ba (2015)) for optimizing our model and set the learning rate to 0.001 and cut it to half every 2.5K episodes for both architectures. ",
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+    {
+        "type": "text",
+        "text": "Results. By design, the method cannot accommodate learning with exactly one example, hence, the comparison here is provided only for 5-shot in 5-way and 20-way. In every episode, the query set contains 15 samples from each class. Our method outperforms the previous methods with 4- convolutional stages shown in Table 1. We also implemented WideResNets (Zagoruyko & Komodakis (2016)) using our method and obtained the performance for 5-way 5-shot: ${ \\bf 6 9 . 9 2 \\pm 0 . 6 4 \\% }$ and 20-way 5-shot: $\\mathbf { 4 1 . 8 4 \\pm 0 . 2 4 \\% }$ that can outperform ResNets-based approach proposed by Mishra et al. (2018) with $6 8 . 8 8 \\pm 0 . 9 2 \\%$ in 5-way and 5-shot classification task. ",
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+    {
+        "type": "text",
+        "text": "5.2 SEMI-SUPERVISED FEW-SHOT LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this experiment, the embedding architecture has 4-convolutional layers as PN (Snell et al. (2017)). We follow the experimental setup proposed by Ren et al. (2018). The episode composition for labeled or support set and query set is similar to the few-shot learning classification task, but there is an additional unlabeled set provided in each episode. Our model is trained on 100K episodes for ",
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+    {
+        "type": "table",
+        "img_path": "images/388d32c66834da50099ccaf8635d24afadd99c6ddeb3d5b6400fd3945d7d7e51.jpg",
+        "table_caption": [
+            "Table 2: Semi-supervised few-shot classification results on the Mini-ImageNet and Tiered-ImageNet with $4 0 \\%$ and $1 0 \\%$ labeled data, respectively. We show the classification results with and without distractors. We compare our results to PN on semi-supervised learning (PN-SSL) with soft $K$ -means (non-masked) and masked $K$ -means (masked), as proposed by Ren et al. (2018). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Models</td><td rowspan=1 colspan=2>Mini-ImageNet5-way 5-shot</td><td rowspan=1 colspan=2>Tiered-ImageNet5-way 5-shot</td></tr><tr><td rowspan=1 colspan=1>PN,SupervisedPSN, Supervised</td><td rowspan=1 colspan=2>59.08±0.22%63.43 ± 0.61%</td><td rowspan=1 colspan=2>66.15±0.22%68.72 ±0.49%</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>w/0Distractors</td><td rowspan=1 colspan=1>w/Distractors</td><td rowspan=1 colspan=1>w/oDistractors</td><td rowspan=1 colspan=1>w/Distractors</td></tr><tr><td rowspan=1 colspan=1>PN-SSL,Non-MaskedPN-SSL,MaskedPSN, Semi-Supervised</td><td rowspan=1 colspan=1>64.59±0.28%64.39 ± 0.24%68.12 ± 0.67%</td><td rowspan=1 colspan=1>63.55±0.28%62.96 ± 0.14%66.10 ± 0.66%</td><td rowspan=1 colspan=1>70.25 ± 0.31%69.88 ± 0.20%71.15 ± 0.67%</td><td rowspan=1 colspan=1>68.32 ± 0.22%69.08 ± 0.25%69.15 ± 0.51%</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [
+            "Mini-ImageNet ",
+            "Table 3: Generalization performance of PSN and PN with varying number of way $( N )$ and shot $( K )$ at the test time. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Way(N)</td><td rowspan=1 colspan=1>Shot(K)</td><td rowspan=1 colspan=1>PSN</td><td rowspan=1 colspan=1>PN</td><td rowspan=1 colspan=1>Way(N)</td><td rowspan=1 colspan=1>Shot(K)</td><td rowspan=1 colspan=1>PSN</td><td rowspan=1 colspan=1>PN</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>35101520</td><td rowspan=1 colspan=1>61.26±0.79%66.62 ± 0.69%71.88 ±0.59%73.50 ±0.58%74.88± 0.63%</td><td rowspan=1 colspan=1>60.36 ±0.31%65.49 ±0.25%69.80 ±0.30%71.92 ±0.26%73.01± 0.31%</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>35101520</td><td rowspan=1 colspan=1>45.46±0.44%51.49 ± 0.46%57.14±0.41%60.20 ±0.39%61.38 ± 0.44%</td><td rowspan=1 colspan=1>44.09±0.28%49.06± 0.25%55.23± 0.27%58.06 ±0.22%59.27±0.22%</td></tr><tr><td rowspan=3 colspan=1>15</td><td rowspan=3 colspan=1>35101520</td><td rowspan=3 colspan=1>37.07 ±0.33%43.18 ±0.32%49.44±0.31%51.96 ± 0.32%54.23±0.34%</td><td rowspan=3 colspan=1>36.24±0.22%40.94±0.24%47.19 ±0.21%49.85 ±0.25%51.37 ± 0.20%</td><td rowspan=3 colspan=1>20</td><td rowspan=3 colspan=1>35101520</td><td rowspan=1 colspan=1>32.13±0.25%37.71± 0.23%</td><td rowspan=2 colspan=1>31.43±0.18%35.58 ±0.17%41.69 ± 0.16%</td></tr><tr><td rowspan=1 colspan=1>44.09±0.22%</td><td rowspan=1 colspan=1>41.6</td></tr><tr><td rowspan=1 colspan=1>47.27±0.22%49.00 ±0.23%</td><td rowspan=1 colspan=1>44.37± 0.17%45.90± 0.18%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Mini-ImageNet and Tiered-ImageNet with $4 0 \\%$ and $1 0 \\%$ of labeled data, respectively. We used the ADAM solver (Kingma & Ba (2015)), the set the learning rate to 0.001 with the weight decay and cut the rate to half every 10K episodes. We trained in two settings: (i) supervised setting, where only labeled data is taken into account, and (ii) semi-supervised setting for which the unlabeled set is also used. The unlabeled set is composed of the examples from the classes in the support set and distractor classes. The number of supporting classes and distractor classes is set to five for training and testing. In the training stage, the number of examples in the unlabeled set is 50 consisting of five examples from each class. In the testing stage, the unlabeled set consists of 20 examples from each class. We also define the query set to have 20 examples per class for testing purpose. ",
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+    {
+        "type": "text",
+        "text": "Results. In these experimental results, the performance is counted over 600 episodes. The results are averaged over 10 random splits of labeled and unlabeled sets. The supervised experiment shows that our method learns robust feature embedding from a small portion of labeled data. With the help of soft-assignment over unlabeled datapoints, the semi-supervised experiment detailed in Table 2 is demonstrated to outperform Prototypical Networks for Semi-Supervised Learning (PN-SSL) proposed by Ren et al. (2018). ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 GENERALIZATION BEYOND $N$ -WAY $K$ -SHOT ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As a measure of generalization ability, we propose to evaluate algorithms beyond the somehow inflexible testing protocol of few-shot learning. In particular, we assess whether a model trained on low number of classes can generalize well to classification tasks involving large number of classes. To gain more insights, we further study how models trained with the $K$ -shot assumption will perform if extra examples are available at the test time. ",
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+    {
+        "type": "text",
+        "text": "In the evaluation below, all models are trained with the 5-way 5-shot setting. At the test time, the models face the test protocol of $N$ -way $K$ -shot learning with $N ~ \\in ~ \\overline { { \\{ 5 , 1 0 , 1 5 , 2 0 \\} } }$ and $K \\in \\{ 3 , 5 , 1 0 , 1 5 , 2 0 \\}$ . Table 3 contrasts the performance of PSN against PN. The table is selfexplanatory. In all experiments, PSN outperforms PN with the gap widened with more challenging settings (e.g., 20-way 20-shot). ",
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+    {
+        "type": "text",
+        "text": "Robustness to Perturbations. Our motivation to develop PSN is to devise a model that shows better resilience to perturbations. Intuitively, to have a noticeable change in the orientation of a subspace, one needs to induce drastic changes to the set. To empirically verify our claim, we assess how robust is PSN in comparison to PN (Snell et al. (2017)) in the presence of perturbations. In particular, we considered the problems of 5-way 5-shot and 5-way 10-shot learning and introduced two types of perturbations at the test time to the trained models. Firstly, we randomly sampled examples from classes not presented in the support sets and included them in the support examples. Secondly, noisy examples are generated randomly using a multivariate Gaussian distribution with random mean and variance of $\\sigma = \\{ 0 . 1 5 , 0 . 3 , 0 . 4 \\}$ . The results of this study are presented in Appendix A due to page limitations. To summarize, our experiments show that both PSN and PN are affected by outliers negatively. That said, the PSN exhibits a much better degree of resilience to outliers. When additive noise is considered, PSN behaves robustly for a wide-range of contamination. In contrast, the performance of PN drops rapidly and significantly in the presence of noise, reinforcing our idea that the use of subspaces indeed leads to a more robust model for the task in hand. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Conceptually, the PN Snell et al. (2017) is the closest work to ours as both solutions obtain a refined representation for each class in the support sets, with the former using the class mean as the representation while in PSN affine subspaces model classes. That said, PSN also uses the mean of each class towards identifying their representative subspaces. Aside from the theoretical properties of affine subspaces2, we empirically observed that for more challenging setups (e.g., 20-way 20-shot), utilizing the mean leads to a better performance. This is because prototypes lie on subspaces in the feature space. Additionally, prototypes in the PN can be easily utilized to design a hybrid with our approach. ",
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+    {
+        "type": "text",
+        "text": "Remark 1. Interestingly, in special cases, PSN simplifies to PN. For example, if samples of a class span a infinitesimal region, at the limit, collapsing to a point, then an affine subspace reduces to a point, recovering PN from PSN. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Remark 2. Since it is not possible to build an affine subspace from only one example, PSN cannot address $^ { l }$ -shot learning problems per se. However, simply applying augmentations to support images facilitates 1-shot learning. However, this issue is beyond the point we make in this work. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Subspace Dimension. In comparison to other models such as matching networks or PN, PSN comes with an extra hyper-parameter, the dimensionality of the subspaces $( i . e . , n )$ . As a rule of thumb, we recommend to use $n = K - 1$ to train the model, while $n = 2$ at the test time gives reasonable and robust results. That said, we thoroughly studied the effect of this parameter and summarized our findings in Appendix B. In short, PSN exhibits a great degree of robustness to $n$ , which in turns, makes training of them painless. ",
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+    {
+        "type": "text",
+        "text": "Computational Complexity. The computational complexity of our PSN approach is $\\mathcal { O } ( \\operatorname* { m i n } ( N D ^ { 2 } K , N D K ^ { 2 } ) )$ , where $K$ , $N$ , and $D$ are the number of shot, way, and feature dimensionality respectively. Compared to the complexity of the PN algorithm, i.e., $\\mathcal { O } ( N D K )$ , our algorithm is somehow slower due to the involvement of the SVD step. If the complexity is of a concern, fast approximate algorithms for SVD (e.g., Menon & Elkan (2011)) can be considered. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This paper presents the PSN, a novel approach for few-shot learning that employs nonlinear embeddings and modeling via affine subspaces. Empirically, we showed that the representations learned via PSN were expressive across a wide-range of supervised and semi-supervised few-shot problems. ",
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+    {
+        "type": "text",
+        "text": "In PSN, each class is represented by the subspace formed by all its examples, meaning that each class is modeled by the span of its samples. One possibility to extend the PSN modeling is to benefit -cleverly- from the null space associated with each class, in the hope of designing a more discriminative model. ",
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+        "type": "text",
+        "text": "Appendices ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            341,
+            127
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A PERTURBATIONS EFFECT ON PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            148,
+            581,
+            166
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this section, we study the effect of perturbation on the performance of PN and PSN. More specifically, we considered the problems of 5-way 5-shot and 5-way 10-shot learning and introduced two types of perturbations at the test time with the trained models. Note that, the model is obtained from 5-way 5-shot training without perturbations. Firstly, we randomly sampled examples from classes not presented in the support sets. Secondly, additive noise is generated randomly using a multivariate Gaussian distribution with random mean and variance of $\\sigma = \\{ 0 . 1 5 , 0 . 3 , 0 . 4 \\}$ . Both examples in these two types are included in the support examples for prototypes and subspaces creation. ",
+        "bbox": [
+            173,
+            180,
+            825,
+            279
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The results of this study are depicted in Fig. 4. To summarize, our experiments show that both PSN and PN are affected by outliers negatively. That said, the PSN exhibits a much better degree of resilience to outliers. For example, modelling with PN leads to a drop of $19 \\%$ and $12 \\%$ percentage points when each support set has 20 outliers for the problem of 5-way 5-shot and 5-way 10-shot respectively. For the same experiment, the PSN modelling only suffers $11 \\%$ and $9 \\%$ percentage points of performance drop. When additive noise is considered, PSN behaves robustly for a widerange of contamination. In contrast, the performance of PN drops rapidly and significantly in the presence of noise, reinforcing our idea that subspaces form indeed a more robust model for the task in hand. ",
+        "bbox": [
+            173,
+            285,
+            825,
+            410
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/3a9ce6abe6c975f9f6f8ff0a106e3d983aa7f834324e724ad6393fa82656ef31.jpg",
+        "image_caption": [
+            "Figure 4: The charts show 5-way 5-shot and 5-way 10-shot results of the PSN and the PN algorithms in the presence of outliers and additive noise. In the first column, sub-plots show the effect of introducing outliers among support samples (the classes of outliers are disjoint with the support classes of samples). The second to fourth columns show the effect of introducing noisy examples generated randomly from Gaussian distributions with random means and variance of $\\dot { \\sigma } = \\{ \\bar { 0 . 1 5 } , \\bar { 0 . 3 } , 0 . 4 \\}$ , respectively. The performance is measured with increasing number of outliers and noisy examples (X-axes). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            422,
+            820,
+            642
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B SUBSPACE DIMENSION EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            516,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we report the results of experiments conducted to assess the effect of the subspace dimensionality $( i . e . , n )$ on the overall accuracy of the algorithm. In particular, we considered two few-shot problems, namely 5-way 5-shot and 5-way 20-shot. For the former, we varied the subspace dimensionality $n \\in \\{ 2 , 3 , 4 , 5 \\}$ and considered all combinations during training and testing (for example, $n = 4$ during training and $n = 3$ at the test time). For the experiment on 5-way 20- shot problem, we varied the subspace dimensionality $n \\in \\{ 5 , 1 0 , 1 5 , 2 0 \\}$ and again considered all combinations. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            231
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Our experiments demonstrate that PSN behaves robustly over a wide-range of subspace dimensionality. For example, the lowest and highest performances for the 5-way 5-shot learning are $6 6 . 0 8 \\pm 0 . 6 7 \\%$ and $6 6 . 6 2 \\pm 0 . 6 9 \\%$ , respectively. For the problem of 5-way 20-shot learning, the lowest and highest performances are $7 5 . 0 3 \\pm 0 . 5 7 \\%$ and $7 5 . 5 8 \\pm 0 . 5 7 \\%$ , respectively. ",
+        "bbox": [
+            174,
+            238,
+            825,
+            294
+        ],
+        "page_idx": 11
+    }
+]

parse/train/vQTYEUtSUr/vQTYEUtSUr.md

@@ -0,0 +1,471 @@
+# Learning Local-Global Contextual Adaptation for Fully End-to-End Bottom-Up Human Pose Estimation
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+This paper presents a method of learning LOcal-GlObal Contextual Adaptation for fully end-to-end and fast bottom-up human Pose estimation, dubbed as LOGO$C A P$ . It is built on the conceptually simple center-offset formulation that lacks inaccuracy for pose estimation. When revisiting the bottom-up human pose estimation with the thought of “thinking, fast and slow” by D. Kahneman, we introduce a “slow keypointer” to remedy the lack of sufficient accuracy of the “fast keypointer”. In learning the “slow keypointer”, the proposed LOGO-CAP lifts the initial “fast” keypoints by offset predictions to keypoint expansion maps (KEMs) to counter their uncertainty in two modules. Firstly, the local KEMs (e.g. $1 1 \times 1 1$ ) are extracted from a low-dimensional feature map. A proposed convolutional message passing module learns to “re-focus” the local KEMs to the keypoint attraction maps (KAMs) by accounting for the structured output prediction nature of human pose estimation, which is directly supervised by the object keypoint similarity (OKS) loss in training. Secondly, the global KEMs are extracted, with a sufficiently large region-of-interest (e.g., $9 7 \times 9 7$ ), from the keypoint heatmaps that are computed by a direct map-to-map regression. Then, a local-global contextual adaptation module is proposed to convolve the global KEMs using the learned KAMs as the kernels. This convolution can be understood as the learnable offsets guided deformable and dynamic convolution in a pose-sensitive way. The proposed method is end-to-end trainable with near real-time inference speed, obtaining state-of-the-art performance on the COCO keypoint benchmark for bottom-up human pose estimation. With the COCO trained model, our LOGO-CAP also outperforms prior arts by a large margin on the challenging OCHuman dataset.
+
+# 24 1 Introduction
+
+# 1.1 Motivation and Objective
+
+Human pose is highly articulated with large structural and appearance variations. 2D human pose estimation in images is a classic structured output prediction problem, and remains a challenging one in computer vision and machine learning. Human pose estimation has numerous applications such as people-centered image understanding, autonomous driving and Augmented Reality (AR). With the recent resurgence of deep neural networks (DNNs), the performance of human pose estimation has witnessed remarkable improvement [12, 3, 15, 22, 11]. This paper focuses on the deep learning based problem formulation.
+
+33 There are two deep learning based paradigms for human pose estimation in the literature. The top  
+34 down paradigm consists of human detection and single human pose estimation in each detected   
+35 human bounding box [12]. The bottom-up paradigm also includes two components: human pose   
+36 keypoint detection and keypoint grouping [3]. The top-down paradigm often obtains better accuracy   
+37 performance, but suffers from its inferior efficiency since the computational cost of the single human   
+38 pose estimation component is linearly scaled with respect to the number of detected human bounding   
+39 boxes in an image. It is also largely affected by the performance of the human detection component   
+40 (e.g., not handling occlusion very well). Thanks to its efficiency, especially in real-time applications,   
+41 the bottom-up paradigm becomes more and more attractive. For both paradigms, state-of-the-art   
+42 methods often are not fully end-to-end trained and utilize different post-hoc processing modules   
+43 to improve the performance. This paper is interested in developing a fully end-to-end bottom-up   
+44 paradigm and aims at bridging its performance gap with the top-down paradigm.   
+45 For the bottom-up paradigm, the recently proposed center-offset approach [6, 28, 26, 11] is a con  
+46 ceptually simple formulation (see the left of Fig. 1 for an illustrative example and Fig. 3 for the   
+47 detailed workflow). It alleviates the need of sophisticated keypoint grouping. When introducing   
+48 human keypoints centers (i.e., anchors) by treating objects as points [35], it encodes a human pose   
+49 as a star structure using the offset vectors of human keypoints relative to the anchors. The main   
+50 challenge of the center-offset regression paradigm lies in the difficulty of accurately learning offset   
+51 vectors with large structural variations, especially the long-range ones, which also leads to inferior   
+52 performance. This paper builds on the center-offset approach and addresses its drawback.
+
+![](images/332f6e4b0367da1e63007c0e2069374f76b9f748ac02428f4ce01e345df82927.jpg)  
+Figure 1: Illustration of the proposed LOGO-CAP for bottom-up human pose estimation. It is built on the center-offset representation. See text for detail.
+
+# 53 1.2 Method Overview
+
+To address the drawback of the center-offset formulation, we build the intuitive idea of “Keypointing, fast and slow”, by analogy to the modes of thought suggested by Daniel Kahneman in “Thinking, fast and slow” [14]: (i) Fast Keypointer: We treat the vanilla center-offset based estimation [35] as the Fast Initializer of pose estimation. (ii) Slow Keypointer: The lack of localization accuracy in the Fast Initializer entails a Slow Solver that learns to refine the “fast” keypoints. By slow, it is only relatively speaking. The Slow Keypointer is actually fast with near real-time speed.
+
+60 To realize the Slow Keypointer, as illustrated in Fig. 1   
+61 and Fig. 3, this paper presents a method of learning   
+62 LOcal-GlObal Contextual Adaptation for fully end-to  
+63 end and fast bottom-up human Pose estimation, dubbed   
+64 as LOGO-CAP. To quantitatively motivate the proposed   
+65 method, we first present a surprisingly strong observa  
+66 tion for a vanilla center-offset regression method (Ta  
+67 ble 1) in the fully-annotated subset of the COCO val-2017   
+68 dataset.Specifically, the vanilla regression method utilizes   
+69 the HRNet-W32 [27] as the feature backbone to directly   
+70 predict keypoints center heatmap and the offset vectors.   
+71 This vanilla center-offset model obtains 60.1 average pre  
+72 cision (AP), which is not great, but reasonably good. It   
+73 clearly shows that the pose keypoints center and the offset vectors can be learned reasonably well.   
+74 Instead of directly utilizing the learned offset vectors for human pose estimation, we treat them as   
+75 human pose keypoint initialization and do a local window search to compute the empirical upper  
+76 bound of performance. More detailed, based on the predicted human poses, by introducing a local   
+77 window (e.g., $1 1 \times 1 1$ ) centered at each detected key point and by computing the single keypoint   
+78 similarity with the ground-truth keypoint, an empirical upper-bound of 88.9 AP is obtained, which   
+79 is significantly higher than the state of the art and shows the potential of improving the vanilla   
+80 center-offset regression paradigm.   
+81 Motivated by the above observation, a straightforward way is just to learn a local heatmap (e.g.,   
+82 $1 1 \times 1 1$ ) for each human pose keypoint based on the learned center and offset vectors, and then to   
+83 compute the refined keypoints by taking arg max within the local heatmap. Although appealing,   
+84 this does not work as observed during our development of the LOGO-CAP. The underlying reason   
+85 is easy to understand: if this can work, the original offset vector regression should work at the   
+86 first place since no additional information is introduced through learning the local heatmap. We   
+87 hypothesize that on the one hand, on top of the local heatmap, the structural relationship between   
+88 different keypoints of a human pose needs to be taken into account, and on the other hand, the   
+89 intrinsic uncertainty of the local information in a local heatmap needs to be resolved. The former   
+90 is the key challenge of structured output prediction problems. Many message passing algorithms   
+91 have been developed in the literature. The latter can not be addressed by simply increasing the local   
+92 window size. It entails learning stronger local-global information interaction and adaptation,.   
+93 Along with the two hypotheses, the proposed LOGO  
+94 CAP lifts the initial keypoints via the center-offset pre  
+95 diction to keypoint expansion maps (KEMs) to counter   
+96 their lack of localization accuracy in two modules (Sec  
+97 tion 3.2). The KEMs extend the star-structured repre  
+98 sentation of the center-offset formulation to the pictorial   
+99 structure representation [10, 8]. The first module com  
+100 putes local KEMs and learns to account for the struc  
+101 tured output prediction nature of the human pose esti  
+102 mation problem, leading to the keypoint attraction maps   
+103 (KAMs). The second computes global KEMs and learns   
+104 to refine the global KEMs by leveraging the KAMs.   
+105 Our LOGO-CAP is a fully end-to-end bottom-up hu  
+106 man pose estimation method with near real-time infer  
+107 ence speed. It obtains $7 0 . 0 \mathrm { A P }$ in the fully-annotated sub  
+108 set of the COCO val-2017 dataset, which is an absolute   
+109 increase of $9 . 9 \mathrm { \ A P }$ compared to the vanilla center-offset   
+110 method, making a significant step forward. Fig. 1 shows a   
+111 pose estimation example. Fig. 2 shows the advantage of the proposed LOGO-CAP in terms of over  
+112 all speed-accuracy comparisons between our LOGO-CAP and prior arts. Meanwhile, we should   
+113 notice that there is also a significant gap compared to the empirical upper bound (Table 1), which   
+114 encourages more work to be investigated.
+
+Table 1: The performance of a vanilla center-offset regression approach, its empirical upper bound, and the performance of our proposed LOGO-CAP using HRNet-W32 [27] as the feature backbone. See text for detail.   
+
+<table><tr><td></td><td>Baseline</td><td>Emp.Bound</td><td>LOGO-CAP</td></tr><tr><td>AP</td><td>60.1</td><td>88.9</td><td>70.0</td></tr><tr><td>Ap50</td><td>85.2</td><td>93.1</td><td>88.2</td></tr><tr><td>AP75</td><td>66.7</td><td>90.6</td><td>76.4</td></tr><tr><td>APM</td><td>53.7</td><td>87.7</td><td>64.4</td></tr><tr><td>APL</td><td>71.5</td><td>90.2</td><td>78.4</td></tr></table>
+
+![](images/b00b95f5ab5d5768b4a9f2ddd1e45c959b1453ed3578645717e9e5960e8f33aa.jpg)  
+Figure 2: Speed-accuracy comparisons between our LOGO-CAP and prior arts on the COCO val-2017 dataset. $\mathrm { W } x .$ - $Y$ (e.g. W32-384) means that a model uses the backbone HRNet- $. \mathrm { W } x$ (HRNetW32) and is tested with the image resolution $Y$ in the short side.
+
+# 2 Related Works and Our Contributions
+
+116 There is a vast body of literature for human pose estimation. Many elegant representation schema   
+117 have been developed for modeling articulated human pose in the traditional approaches such as the   
+118 well-known pictorial structure model [10, 8] and its many variants [24, 1, 23, 33, 25]. Most of them   
+119 focused on single person pose estimation. They perform inference over a combination of local ob  
+120 servations on body parts (i.e., the data term) and the spatial dependencies between them (i.e., the   
+121 spring or clique term). The spatial dependencies are captured either using directed and acyclic struc  
+122 tures that facilitate the global optimization by dynamic programming [2, 9], or using structures with   
+123 loop introduced (for high-order part relationship modeling) which resort to approximate inference   
+124 by loopy belief propagation [19]. The bottleneck of the traditional methods lies in the data term   
+125 which is often based on hand-crafted features. With the resurgence of DNNs and the end-to-end   
+126 learning, the data term has been largely improved. We briefly review the recent deep learning based   
+127 approaches for bottom-up human pose estimation.   
+128 Limb-based Grouping Approaches have been extensively developed due to the naturalness of   
+129 modeling limbs based on keypoints. Given a predefined limb configuration (e.g., the COCO person   
+130 skeleton template consisting of 19 limbs based on 17 keypoints), the grouping can be addressed by   
+131 Part affinity field (PAF) [4, 3], Associative Embedding (AE) [20], mid-range offset fields in Person  
+132 Lab [22] and the fields of Part Intensity and Association [15]. Typically, sophisticated designs are   
+133 entailed to achieve good performance. For example, a bipartite graph matching is used in Open  
+134 Pose [3]. In addition to be computationally expensive, another drawback of these methods is not   
+135 fully end-to-end trainable. More recently, the differentiability issue was studied by the Hierarchical   
+136 Graph Clustering (HGG) method [13], which utilizes graph convolution networks to repeatedly de  
+137 lineate pose parameters of multiple persons from a keypoint graph. HGG improves the performance   
+138 compared to its baseline, the Associative Embedding method [20] at the expense of significantly   
+39 increased computational cost. In contrast to thoses approaches, our proposed LOGO-CAP is fully   
+40 end-to-end trainable and achieves near real-time inference speed.   
+141 Direct Regression based Approaches have attracted much attention due to their conceptually sim  
+142 ple formulation [6, 28, 26, 11, 30]. These center-offset based formulation are inspired by the re  
+143 cent remarkable success of direct bounding box regression in object detection such as the FCOS   
+144 method [29] and CenterNets [35, 6]. As aforementioned, one main challenge is the difficulty of ac  
+145 curately regress the offset vectors, especially for the long-range keypoints with respect to the center.   
+146 Sophisticated post-processing schema are often entailed to improve the performance. For example,   
+147 a method of matching the directly regressed poses to the nearest keypoints that are extracted from   
+148 the global keypiont heatmaps is used in [35]. Although being simple, the performance of this line of   
+149 work is usually inferior to the limb-based approaches. The mixture regression network [30] allevi  
+150 ated the issue of regression quality to some extent, but still remained an indispensable performance   
+151 gap comparing with the grouping-based approaches. Most recently, Geng et al. presented the first   
+152 competitive direct method, DEKR [11] with a novel pose-specific neural architecture for disentan  
+153 gled keypoint regression. To improve the performance, the DEKR method utilizes a lightweight   
+154 rescoring network to recalibrate the pose scores that are computed based on the keypoint heatmaps.   
+155 Despite good performance, the DEKR method entails the additional rescoring stage in both training   
+156 and testing, and thus is not fully end-to-end. The proposed LOGO-CAP retains the simplicity of the   
+157 vanilla center-offset formulation and enjoys fully end-to-end training and fast inference speed.   
+158 Our Contributions. The proposed LOGO-CAP makes three main contributions to the field of   
+159 bottom-up human pose estimation: (i) It addresses the drawback of the vanilla center-offset for  
+160 mulation while retaining its efficiency. It proposes the key idea of lifting a keypoint to a keypoint   
+161 expansion map to counter the lack of localization accuracy. To our knowledge, it is the first fully   
+162 end-to-end trainable method that achieves state-of-the-art performance. (ii) It presents a novel local  
+163 global contextual adaptation formulation that accounts for the nature of structured output predic  
+164 tion in human pose estimation and harnesses local-global structural information integration. (iii) It   
+165 obtains state-of-the-art performance in the COCO val-2017 and test-2017 datasets. It also shows   
+166 state-of-the-art transferability performance in the OCHuman dataset.
+
+# 3 Approach
+
+# 3.1 Problem Formulation
+
+169 We follow the COCO protocol of defining the human pose. It consists of 17 human pose keypoints:   
+170 8 pairs of symmetric keypoints (hips, ankles, knees, shoulders, elbows, wrists, ears and eyes) and   
+171 the nose keypoint. Let $P = \{ 1 , \cdots 1 7 \}$ be the set of keypoint indexes using a predefined order.   
+172 Let $\Lambda$ be an image lattice of the spatial size $H \times W$ (e.g., $5 1 2 \times 5 1 2$ ), and $I$ be an image defined   
+173 on $\Lambda$ . Let $P _ { I } ^ { n }$ be the set of keypoint indexes for a human pose instance $n$ in an image $I$ and we   
+174 have $P _ { I } ^ { n } \subseteq { \bar { P } }$ . For example, in COCO, we typically have $1 \leq n \leq 3 0$ , and different human pose   
+175 instances have different number of visible keypoints due to occlusion and/or truncation. Denote by   
+176 $L _ { I } ^ { n } = \{ ( x _ { i } , y _ { i } ) ; i \in P _ { I } ^ { n } \}$ the keypoint locations of a human pose instance $n$ in an image $I$ , where   
+177 $( x _ { i } , y _ { i } ) \in \Lambda$ . In the center-offset formulation, we introduce the keypoints center (i.e., the anchor),   
+178 $( x _ { c } , y _ { c } )$ based on a given $L _ { I } ^ { n }$ and we have,
+
+$$
+\bar { x _ { c } } = 1 / | L _ { I } ^ { n } | \cdot \sum _ { i \in P _ { I } ^ { n } } x _ { i } , \quad y _ { c } = 1 / | L _ { I } ^ { n } | \cdot \sum _ { i \in P _ { I } ^ { n } } y _ { i } .
+$$
+
+179 With the anchor, a keypoint $( x _ { i } , y _ { i } )$ is equivalently defined by its offset/displacement, denoted by   
+180 $( \Delta x _ { i } , \Delta y _ { i } )$ with $\Delta x _ { i } = x _ { i } - x _ { c }$ and $\Delta y _ { i } = y _ { i } - y _ { c }$ . So, $L _ { I } ^ { n }$ can also be equivalently expressed as   
+181 $L _ { I } ^ { n } = \{ ( x _ { c } , y _ { c } ) , ( \Delta x _ { i } , \Delta y _ { i } ) ; i \in P _ { I } ^ { n } \}$ .   
+182 The objective of human pose estimation is to recover $L _ { I } ^ { n } = \{ ( x _ { i } , y _ { i } ) ; i \in P _ { I } ^ { n } \}$ for all human pose   
+183 instances in an image. Denote by $\hat { L } _ { I } ^ { n } = \{ ( \hat { x } _ { i } , \hat { y } _ { i } ) ; i \in P _ { I } ^ { n } \}$ the estimated human pose. Following   
+184 the COCO protocol, the object keypoint similarity (OKS) is used to evaluate the accuracy,
+
+$$
+\ell _ { O K S } ( \hat { L } _ { I } ^ { n } , L _ { I } ^ { n } ) = 1 / | P _ { I } ^ { n } | \cdot \sum _ { i \in P _ { I } ^ { n } } \exp { ( - d _ { i } ^ { 2 } / 2 s ^ { 2 } \kappa _ { i } ^ { 2 } ) } ,
+$$
+
+185 where $d _ { i }$ is the Euclidian distance between the ground-truth keypoint $( x _ { i } , y _ { i } )$ and the predicted one   
+186 $( \hat { x } _ { i } , \hat { y } _ { i } )$ . $s$ is the square root of the human segment area, and $\kappa$ per-keypoint constant that controls   
+187 fall-off in evaluation. We have $\ell _ { O K S } ( \hat { L } _ { I } ^ { n } , L _ { I } ^ { n } ) \in [ 0 , 1 ]$ . The OKS metric is to evaluate the distance   
+188 between predicted keypoints and ground-truth keypoints normalized by the scale of the person with   
+189 the importance of keypoints equalized. In benchmarking different methods, the average precision
+
+![](images/2b074b3329aecd5a3e9d70b3bb08f97224a4ef88c225012fc3267bf474e158dd.jpg)  
+Figure 3: Illustration of the network and algorithmic flow of the proposed LOGO-CAP for bottomup human pose estimation. See text for detail.
+
+(AP) at190 $\mathrm { O K S } { = } 0 . 5 0 : 0 . 0 5 : 0 . 9 5$ is used as the primary metric, together with $A P ^ { 5 0 }$ at $\mathrm { O K S = 0 . 5 0 }$ , 191 $\dot { A } P ^ { 7 5 }$ at $\mathrm { O K S = 0 . 7 5 }$ , and AP across medium and large scales, $A { \tilde { P } } ^ { M }$ and $A P ^ { L }$ respectively.
+
+# 3.2 The Proposed LOGO-CAP
+
+We first present the network and the inference of LOGO-CAP, and then give details of the training. We keep different modules of the proposed LOGO-CAP simple, which in turn highlights the effectiveness of the proposed representation and algorithmic flow.
+
+# 3.2.1 The Network and the Inference
+
+As illustrated in Fig. 3, the proposed LOGO-CAP consists of four components as follows.
+
+i) A convolution neural network feature backbone. Given an input image $I$ , the output of the feature backbone is a $C$ -dim feature map, denoted by $F \in R ^ { C \times h \times w }$ , where $C$ is the feature dimension of the last convolutional layer in the feature backbone, and the spatial size $h \times w$ depends on the total stride in the feature backbone. We use off-the-shelf HRNets [27] in our experiments.
+
+202 ii) A parallel keypoint-offset regression module. Given the feature map $F$ , the output of keypoint   
+203 regression is an 18-dim feature map (i.e., heatmaps) for the 17 keypoints and the keypoints center   
+204 respectively. Denote by $\mathcal { H } \in \mathcal { R } ^ { 1 8 \times h \times w }$ the heatmaps, and by $\bar { \mathcal { H } } ^ { \dagger ^ { \star } } \in \mathit { R } ^ { 1 8 \times H \times W }$ the up-sampled   
+205 heatmaps (using bi-linear interpolation in our experiments). The output of offset regression is a   
+206 34-dim feature map (i.e., the offset vector fields) for the 17 keypoints. Denote by $O \in R ^ { 3 4 \times h \times w }$   
+207 the offset fields. We adopt a minimally-simple design in realizing the regression modules using a   
+208 channel-wise multi-layer perceptron (MLP). In implementation, we first apply dimension reduction   
+209 to the feature map $F$ using a $1 \times 1$ convolution followed by a Batch Normalization (BN) and a Rec  
+210 tified Linear Unit (ReLU). Then, the output is computed by a $1 \times 1$ convolution. More specifically,   
+211 we have the two parallel branches as follows,
+
+$$
+\begin{array} { r } { F _ { C \times h \times w } \xrightarrow [ C \times 1 \times 1 \times C _ { 1 } ] { C o n v + B N + R e L U } F _ { C _ { 1 } \times h \times w } ^ { \mathcal { H } } \xrightarrow [ C _ { 1 } \times 1 \times 1 \times 1 \mathrm { S } ] { C o n v } \mathcal { H } _ { 1 8 \times h \times w } \xrightarrow [ ] { U p S a m p l i n g } \mathcal { H } _ { 1 8 \times H \times W } ^ { \uparrow } , } \end{array}
+$$
+
+$$
+\begin{array} { r } { F _ { C \times h \times w } \xrightarrow [ { C \times 1 \times 1 \times C _ { 2 } } ] { C o n v + B N + R e L U } F _ { C _ { 2 } \times h \times w } ^ { \mathcal { O } } \xrightarrow [ { C _ { 2 } \times 1 \times 1 \times 3 \mathrm { { \ell } } ] { C o n v } } { \mathcal { O } } _ { 3 4 \times h \times w } , } \end{array}
+$$
+
+212 where $C _ { 1 }$ and $C _ { 2 }$ are predefined (e.g., $C _ { 1 } = 3 2$ and $C _ { 2 } = 2 5 6$ are typically used).
+
+213 Initial pose estimation via the center-offset approach. Based on the computed keypoints center   
+214 heatmap H↑(18) and offset fields $\mathcal { O }$ , a predefined maximum number of pose candidates is computed   
+215 as done in the vanilla center-offset approach. A non-maximum suppression (NMS) with a $3 \times 3$   
+216 window is applied in $\mathcal { H } _ { ( 1 8 ) } ^ { \uparrow }$ and then the top- $. N$ keypoints centers are selected (e.g., $N = 3 0$ in our   
+217 experiments). The $N$ pose instances are computed by retrieving their offset vectors in $\mathcal { O }$ based on   
+218 the selected $N$ keypoints centers. The $N$ pose instances are further pruned by thresholding their   
+219 confidence scores in $\mathcal { H } _ { ( 1 8 ) } ^ { \uparrow }$ with a predefined threshold (e.g., 0.01 used in our experiments). Without   
+220 confusion in the context, we still use $N$ to denote the number of poses instances by this initial pose   
+221 estimation step. We obtain the set of estimated keypoints centers, denoted by $\mathcal { C } _ { N \times 3 }$ each row of   
+222 which represents the position coordinates and the confidence score.   
+223 Lifting a keypoint to a keypoint expansion map (KEM) by imposing a mesh. For each of the $N$   
+224 pose instances, each of the 17 keypoints are placed in a local geometric mesh (e.g., $1 1 \times 1 1 )$ ) with the   
+225 estimated location as the mesh center, capturing the uncertainty of the center-offset pose estimation   
+226 as aforementioned in the introduction. This mesh can thus be interpreted as keypoint expansion   
+227 map (KEM), accounting for competency-aware representations. The entire mesh is denoted by   
+228 $\mathcal { M } _ { N \times 1 7 \times 1 1 \times 1 1 \times 2 }$ , which is used in computing the empirical upper bound in Table 1. We have,
+
+$$
+\{ \mathcal { H } _ { ( 1 8 ) } ^ { \uparrow } , \mathcal { O } _ { 3 4 \times h \times w } \} \xrightarrow { \mathrm { i n i t i a l p o s e ~ e s t i m a t i o n } } \{ \mathcal { C } _ { N \times 3 } , \mathcal { M } _ { N \times 1 7 \times 1 1 \times 1 1 \times 2 } \}
+$$
+
+229 iii) A convolution message passing module. We first encode the geometric mesh $\mathcal { M } _ { N \times 1 7 \times 1 1 \times 1 1 \times 2 }$   
+230 in a latent space with the dimensionality $C _ { 3 }$ (e.g., 64 in our experiments), computed based on the   
+231 feature backbone output. Then, a keypoint is represented by a $C _ { 3 } \times 1 1 \times 1 1$ local feature map. A   
+232 pose instance is represented by concatenating all the 17 keypoints. We have,
+
+$$
+\begin{array} { r } { F _ { C \times h \times w } \xrightarrow [ C \times 1 \times 1 \times C _ { 3 } ] { C o n v + B N + R e L U } F _ { C _ { 3 } \times h \times w } ^ { M } \xrightarrow [ ] { \overset { M _ { N \times 1 7 \times 1 1 \times 1 1 \times 2 } } { \underbrace { \mathrm { b i } \cdot \mathrm { l i n e a r } } } } \mathcal { K } _ { N \times ( 1 7 \times C _ { 3 } ) \times 1 1 \times 1 1 } , } \end{array}
+$$
+
+233 where the bi-linear interpolation is used due to the sub-pixel based locations in the mesh and for   
+234 better feature alignment.   
+235 To facilitate the structural information flow between different latent codes of the keypoints of a pose   
+236 instance, we propose a simple convolutional message passing (CMP) module with three layers of   
+237 Conv $\mathbf { + B N + }$ ReLU operations,
+
+$$
+\begin{array} { r } { K _ { N \times ( 1 7 \times C _ { 3 } ) \times 1 1 \times 1 1 } \Rightarrow [ \frac { C o n v + B N + R e L U } { C _ { i n } \times 3 \times 3 \times C _ { o u t } } ] _ { \times 3 } \Rightarrow \cdot \xrightarrow [ C _ { 6 } \times 1 \times 1 \times 1 7 ] { C o n v } K _ { N \times 1 7 \times 1 1 \times 1 1 } , } \end{array}
+$$
+
+238 where $C _ { i n } \in \{ ( 1 7 \times C _ { 3 } ) , C _ { 4 } , C _ { 5 } \}$ and $C _ { o u t } \in \{ C _ { 4 } , C _ { 5 } , C _ { 6 } \}$ (e.g., $C _ { 4 } = 5 1 2 , C _ { 5 } = 2 5 6 , C _ { 6 } = 1 2 8$   
+239 in our experiments). The resulting $K _ { N \times 1 7 \times 1 1 \times 1 1 }$ can be interpreted as keypoint attraction maps   
+240 (KAMs) which are “re-focused” based on the KEMs by the CMP. To account for the specificity of   
+241 different pose instances in the CMP, we adopt the Attention Normalization [17] to replace the BN in   
+242 the second Conv+BN+ReLU layer, which further improves the performance in our experiments.
+
+Through the CMP, we obtain the dynamic (a.k.a., data-driven) kernels for the 17 keypoints in a pose instance-sensitive way, which are used to refine the global heatmaps $\mathcal { H } ^ { \uparrow }$ for the 17 keypoints.
+
+iv) A local-global contextual adaptation module. We first compute another geometric mesh with enlarged mesh window $a \times a$ (e.g., $a = 9 7$ ) for each keypoint of the $N$ pose instances, and the entire mesh is denoted by $\mathcal { M } _ { N \times 1 7 \times a \times a \times 2 } ^ { L }$ , as done in Eqn. 5. The mesh can be interpreted as the global KEM. It is then instantiated with appearance features extracted from the global heatmaps H↑(1:17), similar to Eqn. 6, and we have,
+
+$$
+\begin{array} { r } { \mathcal { H } _ { ( 1 : 1 7 ) } ^ { \uparrow } \xrightarrow { \mathcal { M } _ { N \times 1 7 \times a \times a \times 2 } ^ { L } } \mathbb { H } _ { N \times 1 7 \times a \times a } \xrightarrow [ \mathrm { r e w e i g h i n g } ] { \mathcal { G } _ { a \times a } ( 0 , \sigma ) } \bar { \mathbb { H } } _ { N \times 1 7 \times a \times a } . } \end{array}
+$$
+
+where to encode the Gaussian prior of keypoint heatmaps, the resulting pose-guided heatmaps $\mathbb { H }$ is reweighed by a Gaussian kernel $\begin{array} { r } { \mathcal { G } _ { a \times a } ( 0 , \dot { \sigma } = \frac { a - 1 } { 2 \times 3 } ) } \end{array}$ (e.g., $\sigma = 1 6$ when $a = 9 7$ ) in an element-wise way. By doing so, it means that the enlarged mesh follows the $3 \sigma$ principle.
+
+Then, we apply the learned keypoint $1 1 \times 1 1$ kernels $K _ { n , i }$ ’s (Eqn. 7) to convolve the reweighed $a \times a$ heatmap $\bar { \mathbb H } _ { n , i }$ (Eqn. 8) in a pose instance-sensitive and keypoint-specific way, leading to LOcal-GlObal Contextual Adaptation,
+
+$$
+\mathbb { \tilde { H } } _ { N \times 1 7 \times a \times a } \xrightarrow [ \mathrm { L O G O - C A } ] { K _ { N \times 1 7 \times 1 1 \times 1 1 } } \mathbb { \tilde { H } } _ { N \times 1 7 \times a \times a } ,
+$$
+
+256 which represents the refined heatmaps for the 17 human pose keypoints.
+
+The Pose Estimation Output. With the local-global contextually adpated heatmaps $\tilde { \mathbb { H } } _ { N \times 1 7 \times a \times a }$ , we maintain the top-2 locations for each keypoint within the $a \times a$ heatmap, and then utilize a convex average of the top-2 locations as the final predicted offset vectors (i.e. $( \Delta x _ { i } ^ { \prime } , \Delta y _ { i } ^ { \prime } )$ ’s in Fig. 3), and of their confidence scores as the prediction score, with a predefined weight $\lambda$ for the top-1 location (0.75 in our experiments). Together with the predicted keypoints centers $\mathcal { C } _ { N \times 3 }$ (Eqn. 5), the final prediction score for each keypoint is the product between the convex average confidence score and the center confidence score. We keep the keypoints whose final scores are greater than 0. We have,
+
+$$
+\left\{ \mathcal C _ { N \times 3 } , \tilde { \mathbb H } _ { N \times 1 7 \times a \times a } \right\} \xrightarrow [ \mathrm { S c o r e ~ t h r e s h o l d i n g } ] { \mathrm { O u t p u t } } \left\{ \hat { L } _ { I } ^ { n } ; n = 1 , \cdots N ^ { \prime } \right\} ,
+$$
+
+where 264 $N ^ { \prime }$ is the number of the final predicted pose instances in an image $I$ .
+
+# 3.2.2 Loss Functions in Training
+
+In the fully end-to-end training, we need to define loss functions for the global heatmap $\mathcal { H }$ (Eqn. 3), the refined local heatmap $\tilde { \mathbb { H } }$ (Eqn. 9), the offset field $\mathcal { O }$ (Eqn. 4), and the keypoint kernels (Eqn. 7).
+
+$\mathcal { H } _ { 1 8 \times h \times w } ^ { G T }$ map Loss. The widely adopted mean squared error (MSE) loss is used. Denoted bythe ground truth heatmaps in which each keypoint (including the center) is modeled by a 2-D Gaussian with dataset-provided mean and variance. Let $\mathbf { p } = ( i , \mathbf { x } )$ be the index of the domain $D$ of dimensions $1 8 \times h \times w$ . For the predicted heatmaps $\mathcal { H } _ { 1 8 \times h \times w }$ , the MSE loss is defined by,
+
+$$
+\mathcal { L } _ { \mathcal { H } } = 1 / | D | \cdot \sum _ { \mathbf { p } \in D } \| w ( \mathbf { x } ) ( \mathcal { H } ( \mathbf { p } ) - \mathcal { \hat { H } } ( \mathbf { p } ) ) \| _ { 2 } ^ { 2 } ,
+$$
+
+272 where $w ( \mathbf { x } )$ represents the weight for the foreground and the background pixels. The foreground   
+273 mask is provided by the dataset annotation. In our experiment, we set $w ( \mathbf { x } ) = 1$ for a foreground   
+274 pixel and $w ( \mathbf { x } ) = 0 . 1$ for a background pixel.
+
+In defining the loss function $\mathcal { L } _ { \tilde { \mathbb { H } } }$ for the refined local heatmap $\tilde { \mathbb { H } }$ (Eqn. 9), the ground-truth heatmap $\tilde { \mathbb { H } } ^ { G T }$ is generated on-the-fly based on the mesh s using a Gaussian model with mean bein $\mathcal { M } _ { N \times 1 7 \times a \times a } ^ { L }$ (Eqn. 8) and the ground-truth key-ment between the current predicted keypoints and the ground-truth ones, and variance $\sigma$ (i.e., the standard deviation of the reweighing Gaussion prior model in Eqn. 8).
+
+The Offset Field Loss. The widely adopted SmoothL1 loss $[ ]$ is used. Let $\mathcal { O } _ { 3 4 \times h \times w } ^ { G T }$ be the groundtruth offset field, and be the non-empty set of ground-truth keypoints centers (Eqn. 1). For the predicted offset field $\mathcal { O } _ { 3 4 \times h \times w }$ (Eqn. 4), we have,
+
+$$
+\mathcal { L } _ { \mathcal { O } } = 1 / | \mathcal { C } ^ { G T } | \cdot \sum _ { \mathbf { p } \in \mathcal { C } ^ { G T } } \mathcal { A } ( \mathbf { p } ) \cdot \mathrm { S m o o t h L 1 } \left( \mathcal { O } ( \cdot , \mathbf { p } ) , \mathcal { O } ^ { G T } ( \cdot , \mathbf { p } ) ; \beta \right) ,
+$$
+
+where $\scriptstyle A ( \mathbf { p } )$ is the area of the person centered at the pixel $\mathbf { p }$ , and $\beta$ the cutting-off threshold (e.g., $\frac { 1 } { 9 }$ in our experiments), and SmoothL1 $( a , b ; \beta ) = 0 . 5 \times | a - b | ^ { 2 } / \beta$ if $| a - b | \le \beta$ , otherwise $| a - b | { - } 0 . 5 { \times } \beta$ .
+
+The OKS Loss for the Keyoint Kernels. Consider a single predicted pose instance, learning the keypoint kernels, $K _ { 1 7 \times 1 1 \times 1 1 }$ (Eqn. 7) is the key to facilitate the local-global contextual adaptation. To that end, the figure of merits of the KEF, $\mathcal { M } _ { 1 7 \times 1 1 \times 1 1 \times 2 }$ (Eqn. 5) needs to directly reflect the task loss, i.e., the OKS loss (Eqn. 2). With respect to the $N ^ { G T }$ ground-truth pose instances in an image, we can compute the similarity score per keypoint candidate in the KEF, and obtain the score tensor $S _ { 1 7 \times 1 1 \times 1 1 \times N ^ { G T } }$ . The score tensor is further clamped with a threshold 0.5, i.e., $S _ { 1 7 \times 1 1 \times 1 1 \times N ^ { G T } } =$ $\operatorname* { m a x } ( S _ { 1 7 \times 1 1 \times 1 1 \times N ^ { G T } } , 0 . 5 )$ . A mean reduction is applied to the first three dimensions of the clamped score tensor to compute the matching score for each of the $N ^ { G T }$ pose instance. Then, the best ground-truth pose instance indexed by $n ^ { * }$ is selected in terms of the matching score, and its matching score is denoted by $s _ { n ^ { * } }$ . Based on the selected ground-truth pose instance, we compute the perkeypoint similarity score for the predicted pose instance at hand, denoted by $s _ { k }$ ( $k \in [ 1 , 1 7 ] ,$ . Then, the loss function fo the keypoint kernels are defined by,
+
+$$
+\mathcal { L } _ { K } = s _ { n ^ { * } } \cdot \sum _ { k , i , j } s _ { k } \cdot | K _ { k , i , j } - S _ { k , i , j , n ^ { * } } | ^ { 2 } .
+$$
+
+97 The Total Loss is then defined by $\mathcal { L } = \mathcal { L } _ { \mathcal { H } } + \mathcal { L } _ { \tilde { \mathbb { H } } } + \lambda \cdot \left( \mathcal { L } _ { \mathcal { O } } + \mathcal { L } _ { K } \right)$ , where the trade-off parameter   
+98 $\lambda$ is used to balance the different loss items $\lambda = 0 . 0 1$ in our experiments).
+
+# 4 Experiments
+
+In this section, we present detailed experimental results and analyses of the proposed LOGO-CAP.   
+Our PyTorch source code will be released for reproducibility.
+
+Datasets. We use two datasets in our experiments: The COCO dataset [18] is the most popular testbed for human pose estimation. It consists of $6 5 \mathrm { k }$ , 5k and $2 0 \mathrm { k }$ images with human pose well-annotated in the training, validation and testing datasets respectively. In all experiments, the proposed LOGO-CAP is trained using the 65k training images. The OCHuman dataset [34] is one popular testing-only dataset for evaluating human pose estimation under the occlusion scenarios. It consists of a total number of 4713 images with 8110 detailed annotated human pose instances using the COCO keypoint configuration. All the annotated 8110 human pose instances have occlusions with the maxIOU $\geq 0 . 5$ . Furthermore, $3 2 \%$ instances are more challenging with the maxIOU $\geq 0 . 7 5$ .
+
+![](images/0e72cc55f2e31a42274abfd31b7efb1e906ecac81785af11a053ab3d68f99595.jpg)  
+Figure 4: Examples of human pose estimation in the COCO val-2017 dataset by the proposed LOGO-CAP with the HRNet-W32 backbone. Top: The COCO skeleton template based visualization. Bottom: The close-up visualization and OKS comparisons between the initial center-offset estimation and the refined keypoints.
+
+Table 2: Evaluation results on the COCO-val-2017 and COCO-testdev-2017 dataset. For HGG [13] and SimplePose [16], the multi-scale inference† is applied on the testdev-2017 dataset. For DEKR [11] that uses an rescoring network to get the final predictions, we report both the performance with and without rescoring (which is the fair baseline for our LOGO-CAP). The numbers of SPM [21] and HGG [13] are extracted from their papers.
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Method</td><td rowspan="2">Backbone</td><td colspan="5">COCO-val-2017</td><td colspan="5">COCO-testdev-2017</td></tr><tr><td>AP[%]</td><td>AP50[%]</td><td>|AP75[%]|APM[%]|</td><td></td><td>| AP𝐿 [%]</td><td>AP[%]</td><td>AP50[%]|AP75[%]</td><td></td><td>|APM[%]</td><td>APL[%]</td></tr><tr><td rowspan="5">Cerrnonn</td><td>OpenPose [35]</td><td>VGG-19</td><td>61.0</td><td>84.9</td><td>67.5</td><td>56.3</td><td>69.3</td><td>61.8</td><td>84.9</td><td>67.5</td><td>57.1</td><td>68.2</td></tr><tr><td>PifPaf[15]</td><td>ResNet-152</td><td>67.4</td><td>86.9</td><td>73.8</td><td>63.1</td><td>74.1</td><td>66.7</td><td>87.8</td><td>73.6</td><td>62.4</td><td>72.9</td></tr><tr><td>PersonLab [22]</td><td>ResNet-152</td><td>66.5</td><td>86.2</td><td>71.9</td><td>62.3</td><td>73.2</td><td>66.5</td><td>88.0</td><td>72.6</td><td>62.4</td><td>72.3</td></tr><tr><td>AE [20, 5]</td><td>HrHRNet-W32</td><td>67.1</td><td>86.2</td><td>73.0</td><td>61.5</td><td>76.1</td><td>66.4</td><td>87.5</td><td>72.8</td><td>61.2</td><td>74.2</td></tr><tr><td>HGG [13]</td><td>HrHRNet-W48</td><td>69.9</td><td>87.2</td><td>76.1</td><td>65.4</td><td>76.4</td><td>68.4</td><td>88.2</td><td>75.1</td><td>64.4</td><td>74.2</td></tr><tr><td rowspan="8"></td><td>SimplePose [16]</td><td>Hourglass IMHN</td><td>60.4 66.1</td><td>83.0 85.9</td><td>66.2 71.6</td><td>59.8</td><td>1 76.2</td><td>67.6† 68.5t</td><td>85.1 86.7†</td><td>73.7† 74.9†</td><td>62.7 66.4†</td><td>74.6t 71.9†</td></tr><tr><td>SPM[21]</td><td>Hourglass</td><td>一</td><td></td><td></td><td>一</td><td>一</td><td>66.9</td><td>88.5</td><td>72.9</td><td>62.6</td><td>0.731</td></tr><tr><td>CenterNet [35]</td><td>Hourglass</td><td>64.0</td><td>85.6</td><td>70.2</td><td>59.4</td><td>72.1</td><td>63.0</td><td>86.8</td><td>69.6</td><td>58.9</td><td>70.4</td></tr><tr><td>DEKR[11]</td><td>HRNet-W32</td><td>68.0</td><td>86.7</td><td>74.5</td><td>62.1</td><td>77.7</td><td>67.3</td><td>87.9</td><td>74.1</td><td>61.5</td><td>76.1</td></tr><tr><td>(w. Rescoring)</td><td>HRNet-W48</td><td>71.0</td><td>88.3</td><td>77.4</td><td>66.7</td><td>78.5</td><td>70.0</td><td>89.4</td><td>77.3</td><td>65.7</td><td>76.9</td></tr><tr><td>DEKR [11]</td><td>HRNet-W32</td><td>67.2</td><td>86.3</td><td>73.8</td><td>61.7</td><td>77.1</td><td>66.6</td><td>87.6</td><td>73.5</td><td>61.2</td><td>75.6</td></tr><tr><td>(w.o. Rescoring)</td><td>HRNet-W48</td><td>70.3</td><td>87.9</td><td>76.8</td><td>66.3</td><td>78.0</td><td>69.3</td><td>89.1</td><td>76.7</td><td>65.3</td><td>76.4</td></tr><tr><td>LOGO-CAP (Ours)</td><td>HRNet-W32 HRNet-W48</td><td>69.6 72.2</td><td>87.5 88.9</td><td>75.9 78.9</td><td>64.1 68.1</td><td>78.0 78.9</td><td>68.2 70.8</td><td>88.7 89.7</td><td>74.9 77.8</td><td>62.8 66.7</td><td>76.0 77.0</td></tr></table>
+
+# 10 4.1 Results on the COCO dataset
+
+Fig. 4 shows some qualitative examples of human pose estimation by the proposed LOGO-CAP.   
+More examples will be provided in the supplementary material.
+
+The proposed LOGO-CAP is compared with prior arts including OpenPose [3], PifPaf [15], PersonLab [22], AE [20] and DEKR [11]. As reported in Table 2, the proposed LOGO-CAP outperforms all of them on both both validation and test-dev datasets.
+
+In comparisons to the best-performing grouping approach, AE [20] with a larger backbone HrHRNet-W48 [5], our LOGO-CAP obtains competitive performance with a smaller HRNet-32 backbone, and improves the AP score with HRNet-W48 backbone on the validation and testdev datasets by 2.3 and 2.5 points, respectively. For the fully differentiable grouping approach HGG [13], our LOGO-CAP achieves better performance by a significantly large margin, more than 9.2 points on the validation set under the single-scale testing. Although the performance of HGG is improved by the multi-scale testing on the test-dev set, the performance of our LOGO-CAP is still significantly better without using the multi-scale testing scheme.
+
+In comparisons to the direct regression based approaches, our LOGO-CAP obtains the best results without incurring either the matching scheme used in CenterNet [35] or the additional rescoring network used in DEKR [11]. When we disable the rescoring network for DEKR [11] for fair comparisons, our LOGO-CAP significantly improves the AP on the validation and testdev datasets by 2.4 points and 1.6 points respectively when HRNet-W32 is used as backbone. The larger backbone is beneficial for both DEKR and our method, which further improves the AP score of our LOGO-CAP to 72.2 and 70.8 on the validation and test-dev dataset respectively, outperforming DEKR by 1.9 and 1.5 respectively.
+
+Table 3: Results on the OCHuman validation and testing datasets [34].   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>Val.AP[%]</td><td rowspan=1 colspan=1>TestAP [%]</td></tr><tr><td rowspan=1 colspan=1>umop-doL</td><td rowspan=1 colspan=1>RMPE [7]SBL [32]SBL [32]</td><td rowspan=1 colspan=1>HourglassResNet-50ResNet-152</td><td rowspan=1 colspan=1>38.837.841.0</td><td rowspan=1 colspan=1>30.730.433.3</td></tr><tr><td rowspan=2 colspan=1>dn-uonog</td><td rowspan=1 colspan=1>AE [20]HGG [20]DEKR [11]</td><td rowspan=1 colspan=1>HourglassHourglassHRNet-W32HRNet-W48</td><td rowspan=1 colspan=1>32.135.637.938.8</td><td rowspan=1 colspan=1>29.534.836.538.2</td></tr><tr><td rowspan=1 colspan=1>LOGO-CAP(Ours)</td><td rowspan=1 colspan=1>HRNet-W32HRNet-W48</td><td rowspan=1 colspan=1>39.041.2</td><td rowspan=1 colspan=1>38.140.4</td></tr></table>
+
+Table 4: The single image inference speed comparison for bottom-up human pose estimation approaches.   
+
+<table><tr><td>Method</td><td>AP [%]</td><td>Backbone</td><td>Time↓ [ms]</td><td>FPS个</td></tr><tr><td>PifPaf[15]</td><td>67.4</td><td>ResNet-152</td><td>213</td><td>4.68</td></tr><tr><td>AE[20,5]</td><td>67.1</td><td>HrHRNet-W32</td><td>560</td><td>1.78</td></tr><tr><td>CenterNet [35]</td><td>64.0</td><td>Hourglass</td><td>147</td><td>6.80</td></tr><tr><td>DEKR[11]</td><td>68.0</td><td>HRNet-W32</td><td>63</td><td>15.8</td></tr><tr><td>DEKR [11]</td><td>71.0</td><td>HRNet-W48</td><td>139</td><td>7.21</td></tr><tr><td>LOGO-CAP</td><td>69.6</td><td>HRNet-W32</td><td>48</td><td>20.7</td></tr><tr><td>LOGO-CAP</td><td>72.2</td><td>HRNet-W48</td><td>112</td><td>8.95</td></tr></table>
+
+# 4.2 Results on the OCHuman dataset
+
+Table 3 shows that our LOGO-CAP achieves the best AP performance on both the validation and testing datasets by significant margins of 2.4 and 2.2 points in comparing with the bottom-up approaches. For the top-down approaches, although they obtain strong AP scores on the validation split, there exists a large performance gap between the validation and testing sets. In comparisons to DEKR [11] (with the rescoring network), our LOGO-CAP improves the performance from 37.9 to 39.0 and from 36.5 to 38.1 on the validation and testing splits with the same backbone HRNetW32, respectively. The similar improvement is observed when the HRNet-W48 backbone is used, outperforming both bottom-up and top-down approaches.
+
+# 4.3 Inference Speed
+
+In comparing the inference speed, we test all the models on a single TITAN RTX GPU for its popularity in practice. The average inference speed, FPS (frames per second), over the 5000 images in COCO-val-2017 is used for the comparison. For DEKR [11], we re-implement their inference code with better speed obtained for fair comparisons at the algorithm level. For methods that have post-processing schema on CPU, only one thread is used. As shown in Table 4, our LOGO-CAP runs significantly faster than PifPaf [15] and AE [20]. The CenterNet [35] runs slower than DEKR and our LOCO-CAP as it requires a post-processing scheme to match the predicted offsets to the keypoints obtained from heatmaps. Comparing with DEKR, the speed improvement of our LOGOCAP is from the lightweight design of head modules since the same backbones are used. For the comparisons in Table 2, we run the models with different resolutions of testing images.
+
+# 4.4 Potentials and Limitations of the Proposed LOGO-CAP
+
+Consider the generic applicability of the center-offset formulation to many computer vision tasks as demonstrated in [35], we hypothesize that the proposed LOGO-CAP has a great potential to remedy the lack of sufficient accuracy using the vanilla center-offset method in those tasks. We also notice that the minimally-simple design in learning the “Slow Keypointer” can be relaxed for different accuracy-speed trade-offs in practice. For example, for the convolutional message passing module, an alternative method could be the Transformer model [31], which potentially will further improve the performance at the expense of inference speed. We leave these for future work.
+
+# 5 Conclusion
+
+This paper focuses on deep learning based formulation for bottom-up human pose estimation. It presents a method of learning LOcal-GlObal Contextual Adaptation for Pose estimation, dubbed as LOGO-CAP. The proposed LOGO-CAP is built on the conceptually simple center-offset paradigm and addresses its drawback of lacking the capability of accurately localizing human pose keypoints. The key idea of our LOG-CAP is to lift the center-offset predicted keypoints to keypoint expansion maps (KEMs),which counters the inaccuracy and uncertainty of the initial keypoints. Two types of KEMs are introduced in two parallel modules on top of the feature backbone. Local KEMs are used to learn keypoint attraction maps (KAMs) via a convolutional message passing module that accounts for the structured output prediction nature of human pose estimation. Global KEMs are used to learn local-global contextual adaptation which convolves global KEMs using the KAMs as kernels. The refined global KEMs are used in computing the final human pose estimation. The proposed LOGO-CAP obtains state-of-the-art performance in COCO val-2017 and test-dev 2017 datasets for bottom-up human pose estimation. It also achieves state-of-the-art transferability performance in the OCHuman dataset with the COCO trained models.
+
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+466 2, 3, 4, 8, 9
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 4.4.   
+(c) Did you discuss any potential negative societal impacts of your work? [No]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See Section 4.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 4 and the supplementary material.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We cited the COCO dataset [18] and the OCHuman dataset [34].   
+(b) Did you mention the license of the assets? [Yes] We mention the licenses in our source code.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] We briefly discussed it in Section 4.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "Learning Local-Global Contextual Adaptation for Fully End-to-End Bottom-Up Human Pose Estimation ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "This paper presents a method of learning LOcal-GlObal Contextual Adaptation for fully end-to-end and fast bottom-up human Pose estimation, dubbed as LOGO$C A P$ . It is built on the conceptually simple center-offset formulation that lacks inaccuracy for pose estimation. When revisiting the bottom-up human pose estimation with the thought of “thinking, fast and slow” by D. Kahneman, we introduce a “slow keypointer” to remedy the lack of sufficient accuracy of the “fast keypointer”. In learning the “slow keypointer”, the proposed LOGO-CAP lifts the initial “fast” keypoints by offset predictions to keypoint expansion maps (KEMs) to counter their uncertainty in two modules. Firstly, the local KEMs (e.g. $1 1 \\times 1 1$ ) are extracted from a low-dimensional feature map. A proposed convolutional message passing module learns to “re-focus” the local KEMs to the keypoint attraction maps (KAMs) by accounting for the structured output prediction nature of human pose estimation, which is directly supervised by the object keypoint similarity (OKS) loss in training. Secondly, the global KEMs are extracted, with a sufficiently large region-of-interest (e.g., $9 7 \\times 9 7$ ), from the keypoint heatmaps that are computed by a direct map-to-map regression. Then, a local-global contextual adaptation module is proposed to convolve the global KEMs using the learned KAMs as the kernels. This convolution can be understood as the learnable offsets guided deformable and dynamic convolution in a pose-sensitive way. The proposed method is end-to-end trainable with near real-time inference speed, obtaining state-of-the-art performance on the COCO keypoint benchmark for bottom-up human pose estimation. With the COCO trained model, our LOGO-CAP also outperforms prior arts by a large margin on the challenging OCHuman dataset. ",
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+    {
+        "type": "text",
+        "text": "24 1 Introduction ",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "1.1 Motivation and Objective ",
+        "text_level": 1,
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+        "text": "Human pose is highly articulated with large structural and appearance variations. 2D human pose estimation in images is a classic structured output prediction problem, and remains a challenging one in computer vision and machine learning. Human pose estimation has numerous applications such as people-centered image understanding, autonomous driving and Augmented Reality (AR). With the recent resurgence of deep neural networks (DNNs), the performance of human pose estimation has witnessed remarkable improvement [12, 3, 15, 22, 11]. This paper focuses on the deep learning based problem formulation. ",
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+        "text": "33 There are two deep learning based paradigms for human pose estimation in the literature. The top  \n34 down paradigm consists of human detection and single human pose estimation in each detected   \n35 human bounding box [12]. The bottom-up paradigm also includes two components: human pose   \n36 keypoint detection and keypoint grouping [3]. The top-down paradigm often obtains better accuracy   \n37 performance, but suffers from its inferior efficiency since the computational cost of the single human   \n38 pose estimation component is linearly scaled with respect to the number of detected human bounding   \n39 boxes in an image. It is also largely affected by the performance of the human detection component   \n40 (e.g., not handling occlusion very well). Thanks to its efficiency, especially in real-time applications,   \n41 the bottom-up paradigm becomes more and more attractive. For both paradigms, state-of-the-art   \n42 methods often are not fully end-to-end trained and utilize different post-hoc processing modules   \n43 to improve the performance. This paper is interested in developing a fully end-to-end bottom-up   \n44 paradigm and aims at bridging its performance gap with the top-down paradigm.   \n45 For the bottom-up paradigm, the recently proposed center-offset approach [6, 28, 26, 11] is a con  \n46 ceptually simple formulation (see the left of Fig. 1 for an illustrative example and Fig. 3 for the   \n47 detailed workflow). It alleviates the need of sophisticated keypoint grouping. When introducing   \n48 human keypoints centers (i.e., anchors) by treating objects as points [35], it encodes a human pose   \n49 as a star structure using the offset vectors of human keypoints relative to the anchors. The main   \n50 challenge of the center-offset regression paradigm lies in the difficulty of accurately learning offset   \n51 vectors with large structural variations, especially the long-range ones, which also leads to inferior   \n52 performance. This paper builds on the center-offset approach and addresses its drawback. ",
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+        "img_path": "images/332f6e4b0367da1e63007c0e2069374f76b9f748ac02428f4ce01e345df82927.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of the proposed LOGO-CAP for bottom-up human pose estimation. It is built on the center-offset representation. See text for detail. "
+        ],
+        "image_footnote": [],
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+        "text": "53 1.2 Method Overview ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "To address the drawback of the center-offset formulation, we build the intuitive idea of “Keypointing, fast and slow”, by analogy to the modes of thought suggested by Daniel Kahneman in “Thinking, fast and slow” [14]: (i) Fast Keypointer: We treat the vanilla center-offset based estimation [35] as the Fast Initializer of pose estimation. (ii) Slow Keypointer: The lack of localization accuracy in the Fast Initializer entails a Slow Solver that learns to refine the “fast” keypoints. By slow, it is only relatively speaking. The Slow Keypointer is actually fast with near real-time speed. ",
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+        "text": "60 To realize the Slow Keypointer, as illustrated in Fig. 1   \n61 and Fig. 3, this paper presents a method of learning   \n62 LOcal-GlObal Contextual Adaptation for fully end-to  \n63 end and fast bottom-up human Pose estimation, dubbed   \n64 as LOGO-CAP. To quantitatively motivate the proposed   \n65 method, we first present a surprisingly strong observa  \n66 tion for a vanilla center-offset regression method (Ta  \n67 ble 1) in the fully-annotated subset of the COCO val-2017   \n68 dataset.Specifically, the vanilla regression method utilizes   \n69 the HRNet-W32 [27] as the feature backbone to directly   \n70 predict keypoints center heatmap and the offset vectors.   \n71 This vanilla center-offset model obtains 60.1 average pre  \n72 cision (AP), which is not great, but reasonably good. It   \n73 clearly shows that the pose keypoints center and the offset vectors can be learned reasonably well.   \n74 Instead of directly utilizing the learned offset vectors for human pose estimation, we treat them as   \n75 human pose keypoint initialization and do a local window search to compute the empirical upper  \n76 bound of performance. More detailed, based on the predicted human poses, by introducing a local   \n77 window (e.g., $1 1 \\times 1 1$ ) centered at each detected key point and by computing the single keypoint   \n78 similarity with the ground-truth keypoint, an empirical upper-bound of 88.9 AP is obtained, which   \n79 is significantly higher than the state of the art and shows the potential of improving the vanilla   \n80 center-offset regression paradigm.   \n81 Motivated by the above observation, a straightforward way is just to learn a local heatmap (e.g.,   \n82 $1 1 \\times 1 1$ ) for each human pose keypoint based on the learned center and offset vectors, and then to   \n83 compute the refined keypoints by taking arg max within the local heatmap. Although appealing,   \n84 this does not work as observed during our development of the LOGO-CAP. The underlying reason   \n85 is easy to understand: if this can work, the original offset vector regression should work at the   \n86 first place since no additional information is introduced through learning the local heatmap. We   \n87 hypothesize that on the one hand, on top of the local heatmap, the structural relationship between   \n88 different keypoints of a human pose needs to be taken into account, and on the other hand, the   \n89 intrinsic uncertainty of the local information in a local heatmap needs to be resolved. The former   \n90 is the key challenge of structured output prediction problems. Many message passing algorithms   \n91 have been developed in the literature. The latter can not be addressed by simply increasing the local   \n92 window size. It entails learning stronger local-global information interaction and adaptation,.   \n93 Along with the two hypotheses, the proposed LOGO  \n94 CAP lifts the initial keypoints via the center-offset pre  \n95 diction to keypoint expansion maps (KEMs) to counter   \n96 their lack of localization accuracy in two modules (Sec  \n97 tion 3.2). The KEMs extend the star-structured repre  \n98 sentation of the center-offset formulation to the pictorial   \n99 structure representation [10, 8]. The first module com  \n100 putes local KEMs and learns to account for the struc  \n101 tured output prediction nature of the human pose esti  \n102 mation problem, leading to the keypoint attraction maps   \n103 (KAMs). The second computes global KEMs and learns   \n104 to refine the global KEMs by leveraging the KAMs.   \n105 Our LOGO-CAP is a fully end-to-end bottom-up hu  \n106 man pose estimation method with near real-time infer  \n107 ence speed. It obtains $7 0 . 0 \\mathrm { A P }$ in the fully-annotated sub  \n108 set of the COCO val-2017 dataset, which is an absolute   \n109 increase of $9 . 9 \\mathrm { \\ A P }$ compared to the vanilla center-offset   \n110 method, making a significant step forward. Fig. 1 shows a   \n111 pose estimation example. Fig. 2 shows the advantage of the proposed LOGO-CAP in terms of over  \n112 all speed-accuracy comparisons between our LOGO-CAP and prior arts. Meanwhile, we should   \n113 notice that there is also a significant gap compared to the empirical upper bound (Table 1), which   \n114 encourages more work to be investigated. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/9a20fdc114c92a40bef045fe1f8b21f76cecb09a6c5e960030f6827b5e45b677.jpg",
+        "table_caption": [
+            "Table 1: The performance of a vanilla center-offset regression approach, its empirical upper bound, and the performance of our proposed LOGO-CAP using HRNet-W32 [27] as the feature backbone. See text for detail. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Baseline</td><td>Emp.Bound</td><td>LOGO-CAP</td></tr><tr><td>AP</td><td>60.1</td><td>88.9</td><td>70.0</td></tr><tr><td>Ap50</td><td>85.2</td><td>93.1</td><td>88.2</td></tr><tr><td>AP75</td><td>66.7</td><td>90.6</td><td>76.4</td></tr><tr><td>APM</td><td>53.7</td><td>87.7</td><td>64.4</td></tr><tr><td>APL</td><td>71.5</td><td>90.2</td><td>78.4</td></tr></table>",
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+            "Figure 2: Speed-accuracy comparisons between our LOGO-CAP and prior arts on the COCO val-2017 dataset. $\\mathrm { W } x .$ - $Y$ (e.g. W32-384) means that a model uses the backbone HRNet- $. \\mathrm { W } x$ (HRNetW32) and is tested with the image resolution $Y$ in the short side. "
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+        "text": "2 Related Works and Our Contributions ",
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+        "text": "116 There is a vast body of literature for human pose estimation. Many elegant representation schema   \n117 have been developed for modeling articulated human pose in the traditional approaches such as the   \n118 well-known pictorial structure model [10, 8] and its many variants [24, 1, 23, 33, 25]. Most of them   \n119 focused on single person pose estimation. They perform inference over a combination of local ob  \n120 servations on body parts (i.e., the data term) and the spatial dependencies between them (i.e., the   \n121 spring or clique term). The spatial dependencies are captured either using directed and acyclic struc  \n122 tures that facilitate the global optimization by dynamic programming [2, 9], or using structures with   \n123 loop introduced (for high-order part relationship modeling) which resort to approximate inference   \n124 by loopy belief propagation [19]. The bottleneck of the traditional methods lies in the data term   \n125 which is often based on hand-crafted features. With the resurgence of DNNs and the end-to-end   \n126 learning, the data term has been largely improved. We briefly review the recent deep learning based   \n127 approaches for bottom-up human pose estimation.   \n128 Limb-based Grouping Approaches have been extensively developed due to the naturalness of   \n129 modeling limbs based on keypoints. Given a predefined limb configuration (e.g., the COCO person   \n130 skeleton template consisting of 19 limbs based on 17 keypoints), the grouping can be addressed by   \n131 Part affinity field (PAF) [4, 3], Associative Embedding (AE) [20], mid-range offset fields in Person  \n132 Lab [22] and the fields of Part Intensity and Association [15]. Typically, sophisticated designs are   \n133 entailed to achieve good performance. For example, a bipartite graph matching is used in Open  \n134 Pose [3]. In addition to be computationally expensive, another drawback of these methods is not   \n135 fully end-to-end trainable. More recently, the differentiability issue was studied by the Hierarchical   \n136 Graph Clustering (HGG) method [13], which utilizes graph convolution networks to repeatedly de  \n137 lineate pose parameters of multiple persons from a keypoint graph. HGG improves the performance   \n138 compared to its baseline, the Associative Embedding method [20] at the expense of significantly   \n39 increased computational cost. In contrast to thoses approaches, our proposed LOGO-CAP is fully   \n40 end-to-end trainable and achieves near real-time inference speed.   \n141 Direct Regression based Approaches have attracted much attention due to their conceptually sim  \n142 ple formulation [6, 28, 26, 11, 30]. These center-offset based formulation are inspired by the re  \n143 cent remarkable success of direct bounding box regression in object detection such as the FCOS   \n144 method [29] and CenterNets [35, 6]. As aforementioned, one main challenge is the difficulty of ac  \n145 curately regress the offset vectors, especially for the long-range keypoints with respect to the center.   \n146 Sophisticated post-processing schema are often entailed to improve the performance. For example,   \n147 a method of matching the directly regressed poses to the nearest keypoints that are extracted from   \n148 the global keypiont heatmaps is used in [35]. Although being simple, the performance of this line of   \n149 work is usually inferior to the limb-based approaches. The mixture regression network [30] allevi  \n150 ated the issue of regression quality to some extent, but still remained an indispensable performance   \n151 gap comparing with the grouping-based approaches. Most recently, Geng et al. presented the first   \n152 competitive direct method, DEKR [11] with a novel pose-specific neural architecture for disentan  \n153 gled keypoint regression. To improve the performance, the DEKR method utilizes a lightweight   \n154 rescoring network to recalibrate the pose scores that are computed based on the keypoint heatmaps.   \n155 Despite good performance, the DEKR method entails the additional rescoring stage in both training   \n156 and testing, and thus is not fully end-to-end. The proposed LOGO-CAP retains the simplicity of the   \n157 vanilla center-offset formulation and enjoys fully end-to-end training and fast inference speed.   \n158 Our Contributions. The proposed LOGO-CAP makes three main contributions to the field of   \n159 bottom-up human pose estimation: (i) It addresses the drawback of the vanilla center-offset for  \n160 mulation while retaining its efficiency. It proposes the key idea of lifting a keypoint to a keypoint   \n161 expansion map to counter the lack of localization accuracy. To our knowledge, it is the first fully   \n162 end-to-end trainable method that achieves state-of-the-art performance. (ii) It presents a novel local  \n163 global contextual adaptation formulation that accounts for the nature of structured output predic  \n164 tion in human pose estimation and harnesses local-global structural information integration. (iii) It   \n165 obtains state-of-the-art performance in the COCO val-2017 and test-2017 datasets. It also shows   \n166 state-of-the-art transferability performance in the OCHuman dataset. ",
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+        "text": "3 Approach ",
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+        "text": "3.1 Problem Formulation ",
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+        "text": "169 We follow the COCO protocol of defining the human pose. It consists of 17 human pose keypoints:   \n170 8 pairs of symmetric keypoints (hips, ankles, knees, shoulders, elbows, wrists, ears and eyes) and   \n171 the nose keypoint. Let $P = \\{ 1 , \\cdots 1 7 \\}$ be the set of keypoint indexes using a predefined order.   \n172 Let $\\Lambda$ be an image lattice of the spatial size $H \\times W$ (e.g., $5 1 2 \\times 5 1 2$ ), and $I$ be an image defined   \n173 on $\\Lambda$ . Let $P _ { I } ^ { n }$ be the set of keypoint indexes for a human pose instance $n$ in an image $I$ and we   \n174 have $P _ { I } ^ { n } \\subseteq { \\bar { P } }$ . For example, in COCO, we typically have $1 \\leq n \\leq 3 0$ , and different human pose   \n175 instances have different number of visible keypoints due to occlusion and/or truncation. Denote by   \n176 $L _ { I } ^ { n } = \\{ ( x _ { i } , y _ { i } ) ; i \\in P _ { I } ^ { n } \\}$ the keypoint locations of a human pose instance $n$ in an image $I$ , where   \n177 $( x _ { i } , y _ { i } ) \\in \\Lambda$ . In the center-offset formulation, we introduce the keypoints center (i.e., the anchor),   \n178 $( x _ { c } , y _ { c } )$ based on a given $L _ { I } ^ { n }$ and we have, ",
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+        "text": "$$\n\\bar { x _ { c } } = 1 / | L _ { I } ^ { n } | \\cdot \\sum _ { i \\in P _ { I } ^ { n } } x _ { i } , \\quad y _ { c } = 1 / | L _ { I } ^ { n } | \\cdot \\sum _ { i \\in P _ { I } ^ { n } } y _ { i } .\n$$",
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+        "text": "179 With the anchor, a keypoint $( x _ { i } , y _ { i } )$ is equivalently defined by its offset/displacement, denoted by   \n180 $( \\Delta x _ { i } , \\Delta y _ { i } )$ with $\\Delta x _ { i } = x _ { i } - x _ { c }$ and $\\Delta y _ { i } = y _ { i } - y _ { c }$ . So, $L _ { I } ^ { n }$ can also be equivalently expressed as   \n181 $L _ { I } ^ { n } = \\{ ( x _ { c } , y _ { c } ) , ( \\Delta x _ { i } , \\Delta y _ { i } ) ; i \\in P _ { I } ^ { n } \\}$ .   \n182 The objective of human pose estimation is to recover $L _ { I } ^ { n } = \\{ ( x _ { i } , y _ { i } ) ; i \\in P _ { I } ^ { n } \\}$ for all human pose   \n183 instances in an image. Denote by $\\hat { L } _ { I } ^ { n } = \\{ ( \\hat { x } _ { i } , \\hat { y } _ { i } ) ; i \\in P _ { I } ^ { n } \\}$ the estimated human pose. Following   \n184 the COCO protocol, the object keypoint similarity (OKS) is used to evaluate the accuracy, ",
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+        "text": "$$\n\\ell _ { O K S } ( \\hat { L } _ { I } ^ { n } , L _ { I } ^ { n } ) = 1 / | P _ { I } ^ { n } | \\cdot \\sum _ { i \\in P _ { I } ^ { n } } \\exp { ( - d _ { i } ^ { 2 } / 2 s ^ { 2 } \\kappa _ { i } ^ { 2 } ) } ,\n$$",
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+        "text": "185 where $d _ { i }$ is the Euclidian distance between the ground-truth keypoint $( x _ { i } , y _ { i } )$ and the predicted one   \n186 $( \\hat { x } _ { i } , \\hat { y } _ { i } )$ . $s$ is the square root of the human segment area, and $\\kappa$ per-keypoint constant that controls   \n187 fall-off in evaluation. We have $\\ell _ { O K S } ( \\hat { L } _ { I } ^ { n } , L _ { I } ^ { n } ) \\in [ 0 , 1 ]$ . The OKS metric is to evaluate the distance   \n188 between predicted keypoints and ground-truth keypoints normalized by the scale of the person with   \n189 the importance of keypoints equalized. In benchmarking different methods, the average precision ",
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+        "image_caption": [
+            "Figure 3: Illustration of the network and algorithmic flow of the proposed LOGO-CAP for bottomup human pose estimation. See text for detail. "
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+        "text": "(AP) at190 $\\mathrm { O K S } { = } 0 . 5 0 : 0 . 0 5 : 0 . 9 5$ is used as the primary metric, together with $A P ^ { 5 0 }$ at $\\mathrm { O K S = 0 . 5 0 }$ , 191 $\\dot { A } P ^ { 7 5 }$ at $\\mathrm { O K S = 0 . 7 5 }$ , and AP across medium and large scales, $A { \\tilde { P } } ^ { M }$ and $A P ^ { L }$ respectively. ",
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+        "text": "3.2 The Proposed LOGO-CAP ",
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+        "text": "We first present the network and the inference of LOGO-CAP, and then give details of the training. We keep different modules of the proposed LOGO-CAP simple, which in turn highlights the effectiveness of the proposed representation and algorithmic flow. ",
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+        "text": "3.2.1 The Network and the Inference ",
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+        "text": "As illustrated in Fig. 3, the proposed LOGO-CAP consists of four components as follows. ",
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+        "text": "i) A convolution neural network feature backbone. Given an input image $I$ , the output of the feature backbone is a $C$ -dim feature map, denoted by $F \\in R ^ { C \\times h \\times w }$ , where $C$ is the feature dimension of the last convolutional layer in the feature backbone, and the spatial size $h \\times w$ depends on the total stride in the feature backbone. We use off-the-shelf HRNets [27] in our experiments. ",
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+        "text": "202 ii) A parallel keypoint-offset regression module. Given the feature map $F$ , the output of keypoint   \n203 regression is an 18-dim feature map (i.e., heatmaps) for the 17 keypoints and the keypoints center   \n204 respectively. Denote by $\\mathcal { H } \\in \\mathcal { R } ^ { 1 8 \\times h \\times w }$ the heatmaps, and by $\\bar { \\mathcal { H } } ^ { \\dagger ^ { \\star } } \\in \\mathit { R } ^ { 1 8 \\times H \\times W }$ the up-sampled   \n205 heatmaps (using bi-linear interpolation in our experiments). The output of offset regression is a   \n206 34-dim feature map (i.e., the offset vector fields) for the 17 keypoints. Denote by $O \\in R ^ { 3 4 \\times h \\times w }$   \n207 the offset fields. We adopt a minimally-simple design in realizing the regression modules using a   \n208 channel-wise multi-layer perceptron (MLP). In implementation, we first apply dimension reduction   \n209 to the feature map $F$ using a $1 \\times 1$ convolution followed by a Batch Normalization (BN) and a Rec  \n210 tified Linear Unit (ReLU). Then, the output is computed by a $1 \\times 1$ convolution. More specifically,   \n211 we have the two parallel branches as follows, ",
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+        "text": "$$\n\\begin{array} { r } { F _ { C \\times h \\times w } \\xrightarrow [ C \\times 1 \\times 1 \\times C _ { 1 } ] { C o n v + B N + R e L U } F _ { C _ { 1 } \\times h \\times w } ^ { \\mathcal { H } } \\xrightarrow [ C _ { 1 } \\times 1 \\times 1 \\times 1 \\mathrm { S } ] { C o n v } \\mathcal { H } _ { 1 8 \\times h \\times w } \\xrightarrow [ ] { U p S a m p l i n g } \\mathcal { H } _ { 1 8 \\times H \\times W } ^ { \\uparrow } , } \\end{array}\n$$",
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+        "img_path": "images/7939d79af294e684a600a40812d96890a6e07216eefeb0d42dfdf87e8f575dd6.jpg",
+        "text": "$$\n\\begin{array} { r } { F _ { C \\times h \\times w } \\xrightarrow [ { C \\times 1 \\times 1 \\times C _ { 2 } } ] { C o n v + B N + R e L U } F _ { C _ { 2 } \\times h \\times w } ^ { \\mathcal { O } } \\xrightarrow [ { C _ { 2 } \\times 1 \\times 1 \\times 3 \\mathrm { { \\ell } } ] { C o n v } } { \\mathcal { O } } _ { 3 4 \\times h \\times w } , } \\end{array}\n$$",
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+        "text": "212 where $C _ { 1 }$ and $C _ { 2 }$ are predefined (e.g., $C _ { 1 } = 3 2$ and $C _ { 2 } = 2 5 6$ are typically used). ",
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+        "text": "213 Initial pose estimation via the center-offset approach. Based on the computed keypoints center   \n214 heatmap H↑(18) and offset fields $\\mathcal { O }$ , a predefined maximum number of pose candidates is computed   \n215 as done in the vanilla center-offset approach. A non-maximum suppression (NMS) with a $3 \\times 3$   \n216 window is applied in $\\mathcal { H } _ { ( 1 8 ) } ^ { \\uparrow }$ and then the top- $. N$ keypoints centers are selected (e.g., $N = 3 0$ in our   \n217 experiments). The $N$ pose instances are computed by retrieving their offset vectors in $\\mathcal { O }$ based on   \n218 the selected $N$ keypoints centers. The $N$ pose instances are further pruned by thresholding their   \n219 confidence scores in $\\mathcal { H } _ { ( 1 8 ) } ^ { \\uparrow }$ with a predefined threshold (e.g., 0.01 used in our experiments). Without   \n220 confusion in the context, we still use $N$ to denote the number of poses instances by this initial pose   \n221 estimation step. We obtain the set of estimated keypoints centers, denoted by $\\mathcal { C } _ { N \\times 3 }$ each row of   \n222 which represents the position coordinates and the confidence score.   \n223 Lifting a keypoint to a keypoint expansion map (KEM) by imposing a mesh. For each of the $N$   \n224 pose instances, each of the 17 keypoints are placed in a local geometric mesh (e.g., $1 1 \\times 1 1 )$ ) with the   \n225 estimated location as the mesh center, capturing the uncertainty of the center-offset pose estimation   \n226 as aforementioned in the introduction. This mesh can thus be interpreted as keypoint expansion   \n227 map (KEM), accounting for competency-aware representations. The entire mesh is denoted by   \n228 $\\mathcal { M } _ { N \\times 1 7 \\times 1 1 \\times 1 1 \\times 2 }$ , which is used in computing the empirical upper bound in Table 1. We have, ",
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+        "text": "$$\n\\{ \\mathcal { H } _ { ( 1 8 ) } ^ { \\uparrow } , \\mathcal { O } _ { 3 4 \\times h \\times w } \\} \\xrightarrow { \\mathrm { i n i t i a l p o s e ~ e s t i m a t i o n } } \\{ \\mathcal { C } _ { N \\times 3 } , \\mathcal { M } _ { N \\times 1 7 \\times 1 1 \\times 1 1 \\times 2 } \\}\n$$",
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+        "text": "229 iii) A convolution message passing module. We first encode the geometric mesh $\\mathcal { M } _ { N \\times 1 7 \\times 1 1 \\times 1 1 \\times 2 }$   \n230 in a latent space with the dimensionality $C _ { 3 }$ (e.g., 64 in our experiments), computed based on the   \n231 feature backbone output. Then, a keypoint is represented by a $C _ { 3 } \\times 1 1 \\times 1 1$ local feature map. A   \n232 pose instance is represented by concatenating all the 17 keypoints. We have, ",
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+        "text": "$$\n\\begin{array} { r } { F _ { C \\times h \\times w } \\xrightarrow [ C \\times 1 \\times 1 \\times C _ { 3 } ] { C o n v + B N + R e L U } F _ { C _ { 3 } \\times h \\times w } ^ { M } \\xrightarrow [ ] { \\overset { M _ { N \\times 1 7 \\times 1 1 \\times 1 1 \\times 2 } } { \\underbrace { \\mathrm { b i } \\cdot \\mathrm { l i n e a r } } } } \\mathcal { K } _ { N \\times ( 1 7 \\times C _ { 3 } ) \\times 1 1 \\times 1 1 } , } \\end{array}\n$$",
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+        "text": "233 where the bi-linear interpolation is used due to the sub-pixel based locations in the mesh and for   \n234 better feature alignment.   \n235 To facilitate the structural information flow between different latent codes of the keypoints of a pose   \n236 instance, we propose a simple convolutional message passing (CMP) module with three layers of   \n237 Conv $\\mathbf { + B N + }$ ReLU operations, ",
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+        "text": "$$\n\\begin{array} { r } { K _ { N \\times ( 1 7 \\times C _ { 3 } ) \\times 1 1 \\times 1 1 } \\Rightarrow [ \\frac { C o n v + B N + R e L U } { C _ { i n } \\times 3 \\times 3 \\times C _ { o u t } } ] _ { \\times 3 } \\Rightarrow \\cdot \\xrightarrow [ C _ { 6 } \\times 1 \\times 1 \\times 1 7 ] { C o n v } K _ { N \\times 1 7 \\times 1 1 \\times 1 1 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            369,
+            758,
+            396
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "238 where $C _ { i n } \\in \\{ ( 1 7 \\times C _ { 3 } ) , C _ { 4 } , C _ { 5 } \\}$ and $C _ { o u t } \\in \\{ C _ { 4 } , C _ { 5 } , C _ { 6 } \\}$ (e.g., $C _ { 4 } = 5 1 2 , C _ { 5 } = 2 5 6 , C _ { 6 } = 1 2 8$   \n239 in our experiments). The resulting $K _ { N \\times 1 7 \\times 1 1 \\times 1 1 }$ can be interpreted as keypoint attraction maps   \n240 (KAMs) which are “re-focused” based on the KEMs by the CMP. To account for the specificity of   \n241 different pose instances in the CMP, we adopt the Attention Normalization [17] to replace the BN in   \n242 the second Conv+BN+ReLU layer, which further improves the performance in our experiments. ",
+        "bbox": [
+            140,
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+            467
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Through the CMP, we obtain the dynamic (a.k.a., data-driven) kernels for the 17 keypoints in a pose instance-sensitive way, which are used to refine the global heatmaps $\\mathcal { H } ^ { \\uparrow }$ for the 17 keypoints. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "iv) A local-global contextual adaptation module. We first compute another geometric mesh with enlarged mesh window $a \\times a$ (e.g., $a = 9 7$ ) for each keypoint of the $N$ pose instances, and the entire mesh is denoted by $\\mathcal { M } _ { N \\times 1 7 \\times a \\times a \\times 2 } ^ { L }$ , as done in Eqn. 5. The mesh can be interpreted as the global KEM. It is then instantiated with appearance features extracted from the global heatmaps H↑(1:17), similar to Eqn. 6, and we have, ",
+        "bbox": [
+            165,
+            506,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8be70fbfbcc891631a7015ffd59e81b586b327e8ea00b955c51469aab1c163f1.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { H } _ { ( 1 : 1 7 ) } ^ { \\uparrow } \\xrightarrow { \\mathcal { M } _ { N \\times 1 7 \\times a \\times a \\times 2 } ^ { L } } \\mathbb { H } _ { N \\times 1 7 \\times a \\times a } \\xrightarrow [ \\mathrm { r e w e i g h i n g } ] { \\mathcal { G } _ { a \\times a } ( 0 , \\sigma ) } \\bar { \\mathbb { H } } _ { N \\times 1 7 \\times a \\times a } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            583,
+            700,
+            614
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where to encode the Gaussian prior of keypoint heatmaps, the resulting pose-guided heatmaps $\\mathbb { H }$ is reweighed by a Gaussian kernel $\\begin{array} { r } { \\mathcal { G } _ { a \\times a } ( 0 , \\dot { \\sigma } = \\frac { a - 1 } { 2 \\times 3 } ) } \\end{array}$ (e.g., $\\sigma = 1 6$ when $a = 9 7$ ) in an element-wise way. By doing so, it means that the enlarged mesh follows the $3 \\sigma$ principle. ",
+        "bbox": [
+            161,
+            616,
+            825,
+            660
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Then, we apply the learned keypoint $1 1 \\times 1 1$ kernels $K _ { n , i }$ ’s (Eqn. 7) to convolve the reweighed $a \\times a$ heatmap $\\bar { \\mathbb H } _ { n , i }$ (Eqn. 8) in a pose instance-sensitive and keypoint-specific way, leading to LOcal-GlObal Contextual Adaptation, ",
+        "bbox": [
+            165,
+            664,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0978d54b3399ad656cbf297a9354a7c01328553ac66082bede228097c9c42490.jpg",
+        "text": "$$\n\\mathbb { \\tilde { H } } _ { N \\times 1 7 \\times a \\times a } \\xrightarrow [ \\mathrm { L O G O - C A } ] { K _ { N \\times 1 7 \\times 1 1 \\times 1 1 } } \\mathbb { \\tilde { H } } _ { N \\times 1 7 \\times a \\times a } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            709,
+            633,
+            734
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "256 which represents the refined heatmaps for the 17 human pose keypoints. ",
+        "bbox": [
+            150,
+            736,
+            645,
+            750
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The Pose Estimation Output. With the local-global contextually adpated heatmaps $\\tilde { \\mathbb { H } } _ { N \\times 1 7 \\times a \\times a }$ , we maintain the top-2 locations for each keypoint within the $a \\times a$ heatmap, and then utilize a convex average of the top-2 locations as the final predicted offset vectors (i.e. $( \\Delta x _ { i } ^ { \\prime } , \\Delta y _ { i } ^ { \\prime } )$ ’s in Fig. 3), and of their confidence scores as the prediction score, with a predefined weight $\\lambda$ for the top-1 location (0.75 in our experiments). Together with the predicted keypoints centers $\\mathcal { C } _ { N \\times 3 }$ (Eqn. 5), the final prediction score for each keypoint is the product between the convex average confidence score and the center confidence score. We keep the keypoints whose final scores are greater than 0. We have, ",
+        "bbox": [
+            166,
+            757,
+            825,
+            854
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/30d709fb2337a1350f915e060c088bf16a8e7cd80996b9b97654515b37a035f9.jpg",
+        "text": "$$\n\\left\\{ \\mathcal C _ { N \\times 3 } , \\tilde { \\mathbb H } _ { N \\times 1 7 \\times a \\times a } \\right\\} \\xrightarrow [ \\mathrm { S c o r e ~ t h r e s h o l d i n g } ] { \\mathrm { O u t p u t } } \\left\\{ \\hat { L } _ { I } ^ { n } ; n = 1 , \\cdots N ^ { \\prime } \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            856,
+            691,
+            882
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where 264 $N ^ { \\prime }$ is the number of the final predicted pose instances in an image $I$ . ",
+        "bbox": [
+            140,
+            885,
+            666,
+            900
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2.2 Loss Functions in Training ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            413,
+            106
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the fully end-to-end training, we need to define loss functions for the global heatmap $\\mathcal { H }$ (Eqn. 3), the refined local heatmap $\\tilde { \\mathbb { H } }$ (Eqn. 9), the offset field $\\mathcal { O }$ (Eqn. 4), and the keypoint kernels (Eqn. 7). ",
+        "bbox": [
+            165,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "$\\mathcal { H } _ { 1 8 \\times h \\times w } ^ { G T }$ map Loss. The widely adopted mean squared error (MSE) loss is used. Denoted bythe ground truth heatmaps in which each keypoint (including the center) is modeled by a 2-D Gaussian with dataset-provided mean and variance. Let $\\mathbf { p } = ( i , \\mathbf { x } )$ be the index of the domain $D$ of dimensions $1 8 \\times h \\times w$ . For the predicted heatmaps $\\mathcal { H } _ { 1 8 \\times h \\times w }$ , the MSE loss is defined by, ",
+        "bbox": [
+            169,
+            145,
+            825,
+            203
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/413b0fa8feeab03d56de8e0f19c9101b1d4783b642fc90da22758f4e70142378.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathcal { H } } = 1 / | D | \\cdot \\sum _ { \\mathbf { p } \\in D } \\| w ( \\mathbf { x } ) ( \\mathcal { H } ( \\mathbf { p } ) - \\mathcal { \\hat { H } } ( \\mathbf { p } ) ) \\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            203,
+            647,
+            237
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "272 where $w ( \\mathbf { x } )$ represents the weight for the foreground and the background pixels. The foreground   \n273 mask is provided by the dataset annotation. In our experiment, we set $w ( \\mathbf { x } ) = 1$ for a foreground   \n274 pixel and $w ( \\mathbf { x } ) = 0 . 1$ for a background pixel. ",
+        "bbox": [
+            142,
+            237,
+            825,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In defining the loss function $\\mathcal { L } _ { \\tilde { \\mathbb { H } } }$ for the refined local heatmap $\\tilde { \\mathbb { H } }$ (Eqn. 9), the ground-truth heatmap $\\tilde { \\mathbb { H } } ^ { G T }$ is generated on-the-fly based on the mesh s using a Gaussian model with mean bein $\\mathcal { M } _ { N \\times 1 7 \\times a \\times a } ^ { L }$ (Eqn. 8) and the ground-truth key-ment between the current predicted keypoints and the ground-truth ones, and variance $\\sigma$ (i.e., the standard deviation of the reweighing Gaussion prior model in Eqn. 8). ",
+        "bbox": [
+            171,
+            286,
+            825,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The Offset Field Loss. The widely adopted SmoothL1 loss $[ ]$ is used. Let $\\mathcal { O } _ { 3 4 \\times h \\times w } ^ { G T }$ be the groundtruth offset field, and be the non-empty set of ground-truth keypoints centers (Eqn. 1). For the predicted offset field $\\mathcal { O } _ { 3 4 \\times h \\times w }$ (Eqn. 4), we have, ",
+        "bbox": [
+            165,
+            364,
+            825,
+            411
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7bb1cb1f1c2f3f440b786df5bddb8db2205455a76c9a84acd8379e0d2c3184bb.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathcal { O } } = 1 / | \\mathcal { C } ^ { G T } | \\cdot \\sum _ { \\mathbf { p } \\in \\mathcal { C } ^ { G T } } \\mathcal { A } ( \\mathbf { p } ) \\cdot \\mathrm { S m o o t h L 1 } \\left( \\mathcal { O } ( \\cdot , \\mathbf { p } ) , \\mathcal { O } ^ { G T } ( \\cdot , \\mathbf { p } ) ; \\beta \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            277,
+            411,
+            718,
+            445
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\scriptstyle A ( \\mathbf { p } )$ is the area of the person centered at the pixel $\\mathbf { p }$ , and $\\beta$ the cutting-off threshold (e.g., $\\frac { 1 } { 9 }$ in our experiments), and SmoothL1 $( a , b ; \\beta ) = 0 . 5 \\times | a - b | ^ { 2 } / \\beta$ if $| a - b | \\le \\beta$ , otherwise $| a - b | { - } 0 . 5 { \\times } \\beta$ . ",
+        "bbox": [
+            163,
+            446,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The OKS Loss for the Keyoint Kernels. Consider a single predicted pose instance, learning the keypoint kernels, $K _ { 1 7 \\times 1 1 \\times 1 1 }$ (Eqn. 7) is the key to facilitate the local-global contextual adaptation. To that end, the figure of merits of the KEF, $\\mathcal { M } _ { 1 7 \\times 1 1 \\times 1 1 \\times 2 }$ (Eqn. 5) needs to directly reflect the task loss, i.e., the OKS loss (Eqn. 2). With respect to the $N ^ { G T }$ ground-truth pose instances in an image, we can compute the similarity score per keypoint candidate in the KEF, and obtain the score tensor $S _ { 1 7 \\times 1 1 \\times 1 1 \\times N ^ { G T } }$ . The score tensor is further clamped with a threshold 0.5, i.e., $S _ { 1 7 \\times 1 1 \\times 1 1 \\times N ^ { G T } } =$ $\\operatorname* { m a x } ( S _ { 1 7 \\times 1 1 \\times 1 1 \\times N ^ { G T } } , 0 . 5 )$ . A mean reduction is applied to the first three dimensions of the clamped score tensor to compute the matching score for each of the $N ^ { G T }$ pose instance. Then, the best ground-truth pose instance indexed by $n ^ { * }$ is selected in terms of the matching score, and its matching score is denoted by $s _ { n ^ { * } }$ . Based on the selected ground-truth pose instance, we compute the perkeypoint similarity score for the predicted pose instance at hand, denoted by $s _ { k }$ ( $k \\in [ 1 , 1 7 ] ,$ . Then, the loss function fo the keypoint kernels are defined by, ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/23a89bfe3baae393a08ac386fdab942a9db739f0ec58c0038cd0a2cbcbda1778.jpg",
+        "text": "$$\n\\mathcal { L } _ { K } = s _ { n ^ { * } } \\cdot \\sum _ { k , i , j } s _ { k } \\cdot | K _ { k , i , j } - S _ { k , i , j , n ^ { * } } | ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "97 The Total Loss is then defined by $\\mathcal { L } = \\mathcal { L } _ { \\mathcal { H } } + \\mathcal { L } _ { \\tilde { \\mathbb { H } } } + \\lambda \\cdot \\left( \\mathcal { L } _ { \\mathcal { O } } + \\mathcal { L } _ { K } \\right)$ , where the trade-off parameter   \n98 $\\lambda$ is used to balance the different loss items $\\lambda = 0 . 0 1$ in our experiments). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present detailed experimental results and analyses of the proposed LOGO-CAP.   \nOur PyTorch source code will be released for reproducibility. ",
+        "bbox": [
+            160,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Datasets. We use two datasets in our experiments: The COCO dataset [18] is the most popular testbed for human pose estimation. It consists of $6 5 \\mathrm { k }$ , 5k and $2 0 \\mathrm { k }$ images with human pose well-annotated in the training, validation and testing datasets respectively. In all experiments, the proposed LOGO-CAP is trained using the 65k training images. The OCHuman dataset [34] is one popular testing-only dataset for evaluating human pose estimation under the occlusion scenarios. It consists of a total number of 4713 images with 8110 detailed annotated human pose instances using the COCO keypoint configuration. All the annotated 8110 human pose instances have occlusions with the maxIOU $\\geq 0 . 5$ . Furthermore, $3 2 \\%$ instances are more challenging with the maxIOU $\\geq 0 . 7 5$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/0e72cc55f2e31a42274abfd31b7efb1e906ecac81785af11a053ab3d68f99595.jpg",
+        "image_caption": [
+            "Figure 4: Examples of human pose estimation in the COCO val-2017 dataset by the proposed LOGO-CAP with the HRNet-W32 backbone. Top: The COCO skeleton template based visualization. Bottom: The close-up visualization and OKS comparisons between the initial center-offset estimation and the refined keypoints. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Evaluation results on the COCO-val-2017 and COCO-testdev-2017 dataset. For HGG [13] and SimplePose [16], the multi-scale inference† is applied on the testdev-2017 dataset. For DEKR [11] that uses an rescoring network to get the final predictions, we report both the performance with and without rescoring (which is the fair baseline for our LOGO-CAP). The numbers of SPM [21] and HGG [13] are extracted from their papers. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/d0928fd2b66bdccdea70da13ae5eec5ff65e038bb0d664b7c284312fcbf6d8ad.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Method</td><td rowspan=\"2\">Backbone</td><td colspan=\"5\">COCO-val-2017</td><td colspan=\"5\">COCO-testdev-2017</td></tr><tr><td>AP[%]</td><td>AP50[%]</td><td>|AP75[%]|APM[%]|</td><td></td><td>| AP𝐿 [%]</td><td>AP[%]</td><td>AP50[%]|AP75[%]</td><td></td><td>|APM[%]</td><td>APL[%]</td></tr><tr><td rowspan=\"5\">Cerrnonn</td><td>OpenPose [35]</td><td>VGG-19</td><td>61.0</td><td>84.9</td><td>67.5</td><td>56.3</td><td>69.3</td><td>61.8</td><td>84.9</td><td>67.5</td><td>57.1</td><td>68.2</td></tr><tr><td>PifPaf[15]</td><td>ResNet-152</td><td>67.4</td><td>86.9</td><td>73.8</td><td>63.1</td><td>74.1</td><td>66.7</td><td>87.8</td><td>73.6</td><td>62.4</td><td>72.9</td></tr><tr><td>PersonLab [22]</td><td>ResNet-152</td><td>66.5</td><td>86.2</td><td>71.9</td><td>62.3</td><td>73.2</td><td>66.5</td><td>88.0</td><td>72.6</td><td>62.4</td><td>72.3</td></tr><tr><td>AE [20, 5]</td><td>HrHRNet-W32</td><td>67.1</td><td>86.2</td><td>73.0</td><td>61.5</td><td>76.1</td><td>66.4</td><td>87.5</td><td>72.8</td><td>61.2</td><td>74.2</td></tr><tr><td>HGG [13]</td><td>HrHRNet-W48</td><td>69.9</td><td>87.2</td><td>76.1</td><td>65.4</td><td>76.4</td><td>68.4</td><td>88.2</td><td>75.1</td><td>64.4</td><td>74.2</td></tr><tr><td rowspan=\"8\"></td><td>SimplePose [16]</td><td>Hourglass IMHN</td><td>60.4 66.1</td><td>83.0 85.9</td><td>66.2 71.6</td><td>59.8</td><td>1 76.2</td><td>67.6† 68.5t</td><td>85.1 86.7†</td><td>73.7† 74.9†</td><td>62.7 66.4†</td><td>74.6t 71.9†</td></tr><tr><td>SPM[21]</td><td>Hourglass</td><td>一</td><td></td><td></td><td>一</td><td>一</td><td>66.9</td><td>88.5</td><td>72.9</td><td>62.6</td><td>0.731</td></tr><tr><td>CenterNet [35]</td><td>Hourglass</td><td>64.0</td><td>85.6</td><td>70.2</td><td>59.4</td><td>72.1</td><td>63.0</td><td>86.8</td><td>69.6</td><td>58.9</td><td>70.4</td></tr><tr><td>DEKR[11]</td><td>HRNet-W32</td><td>68.0</td><td>86.7</td><td>74.5</td><td>62.1</td><td>77.7</td><td>67.3</td><td>87.9</td><td>74.1</td><td>61.5</td><td>76.1</td></tr><tr><td>(w. Rescoring)</td><td>HRNet-W48</td><td>71.0</td><td>88.3</td><td>77.4</td><td>66.7</td><td>78.5</td><td>70.0</td><td>89.4</td><td>77.3</td><td>65.7</td><td>76.9</td></tr><tr><td>DEKR [11]</td><td>HRNet-W32</td><td>67.2</td><td>86.3</td><td>73.8</td><td>61.7</td><td>77.1</td><td>66.6</td><td>87.6</td><td>73.5</td><td>61.2</td><td>75.6</td></tr><tr><td>(w.o. Rescoring)</td><td>HRNet-W48</td><td>70.3</td><td>87.9</td><td>76.8</td><td>66.3</td><td>78.0</td><td>69.3</td><td>89.1</td><td>76.7</td><td>65.3</td><td>76.4</td></tr><tr><td>LOGO-CAP (Ours)</td><td>HRNet-W32 HRNet-W48</td><td>69.6 72.2</td><td>87.5 88.9</td><td>75.9 78.9</td><td>64.1 68.1</td><td>78.0 78.9</td><td>68.2 70.8</td><td>88.7 89.7</td><td>74.9 77.8</td><td>62.8 66.7</td><td>76.0 77.0</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "10 4.1 Results on the COCO dataset ",
+        "text_level": 1,
+        "bbox": [
+            151,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Fig. 4 shows some qualitative examples of human pose estimation by the proposed LOGO-CAP.   \nMore examples will be provided in the supplementary material. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The proposed LOGO-CAP is compared with prior arts including OpenPose [3], PifPaf [15], PersonLab [22], AE [20] and DEKR [11]. As reported in Table 2, the proposed LOGO-CAP outperforms all of them on both both validation and test-dev datasets. ",
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+        "text": "In comparisons to the best-performing grouping approach, AE [20] with a larger backbone HrHRNet-W48 [5], our LOGO-CAP obtains competitive performance with a smaller HRNet-32 backbone, and improves the AP score with HRNet-W48 backbone on the validation and testdev datasets by 2.3 and 2.5 points, respectively. For the fully differentiable grouping approach HGG [13], our LOGO-CAP achieves better performance by a significantly large margin, more than 9.2 points on the validation set under the single-scale testing. Although the performance of HGG is improved by the multi-scale testing on the test-dev set, the performance of our LOGO-CAP is still significantly better without using the multi-scale testing scheme. ",
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+        "text": "In comparisons to the direct regression based approaches, our LOGO-CAP obtains the best results without incurring either the matching scheme used in CenterNet [35] or the additional rescoring network used in DEKR [11]. When we disable the rescoring network for DEKR [11] for fair comparisons, our LOGO-CAP significantly improves the AP on the validation and testdev datasets by 2.4 points and 1.6 points respectively when HRNet-W32 is used as backbone. The larger backbone is beneficial for both DEKR and our method, which further improves the AP score of our LOGO-CAP to 72.2 and 70.8 on the validation and test-dev dataset respectively, outperforming DEKR by 1.9 and 1.5 respectively. ",
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+        "img_path": "images/c1fce16633ed01d6ff7a5c5c12e2c47466a1c9fc70a976049c17a15df932eca0.jpg",
+        "table_caption": [
+            "Table 3: Results on the OCHuman validation and testing datasets [34]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>Val.AP[%]</td><td rowspan=1 colspan=1>TestAP [%]</td></tr><tr><td rowspan=1 colspan=1>umop-doL</td><td rowspan=1 colspan=1>RMPE [7]SBL [32]SBL [32]</td><td rowspan=1 colspan=1>HourglassResNet-50ResNet-152</td><td rowspan=1 colspan=1>38.837.841.0</td><td rowspan=1 colspan=1>30.730.433.3</td></tr><tr><td rowspan=2 colspan=1>dn-uonog</td><td rowspan=1 colspan=1>AE [20]HGG [20]DEKR [11]</td><td rowspan=1 colspan=1>HourglassHourglassHRNet-W32HRNet-W48</td><td rowspan=1 colspan=1>32.135.637.938.8</td><td rowspan=1 colspan=1>29.534.836.538.2</td></tr><tr><td rowspan=1 colspan=1>LOGO-CAP(Ours)</td><td rowspan=1 colspan=1>HRNet-W32HRNet-W48</td><td rowspan=1 colspan=1>39.041.2</td><td rowspan=1 colspan=1>38.140.4</td></tr></table>",
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+        "table_caption": [
+            "Table 4: The single image inference speed comparison for bottom-up human pose estimation approaches. "
+        ],
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+        "table_body": "<table><tr><td>Method</td><td>AP [%]</td><td>Backbone</td><td>Time↓ [ms]</td><td>FPS个</td></tr><tr><td>PifPaf[15]</td><td>67.4</td><td>ResNet-152</td><td>213</td><td>4.68</td></tr><tr><td>AE[20,5]</td><td>67.1</td><td>HrHRNet-W32</td><td>560</td><td>1.78</td></tr><tr><td>CenterNet [35]</td><td>64.0</td><td>Hourglass</td><td>147</td><td>6.80</td></tr><tr><td>DEKR[11]</td><td>68.0</td><td>HRNet-W32</td><td>63</td><td>15.8</td></tr><tr><td>DEKR [11]</td><td>71.0</td><td>HRNet-W48</td><td>139</td><td>7.21</td></tr><tr><td>LOGO-CAP</td><td>69.6</td><td>HRNet-W32</td><td>48</td><td>20.7</td></tr><tr><td>LOGO-CAP</td><td>72.2</td><td>HRNet-W48</td><td>112</td><td>8.95</td></tr></table>",
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+        "type": "text",
+        "text": "4.2 Results on the OCHuman dataset ",
+        "text_level": 1,
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+        "text": "Table 3 shows that our LOGO-CAP achieves the best AP performance on both the validation and testing datasets by significant margins of 2.4 and 2.2 points in comparing with the bottom-up approaches. For the top-down approaches, although they obtain strong AP scores on the validation split, there exists a large performance gap between the validation and testing sets. In comparisons to DEKR [11] (with the rescoring network), our LOGO-CAP improves the performance from 37.9 to 39.0 and from 36.5 to 38.1 on the validation and testing splits with the same backbone HRNetW32, respectively. The similar improvement is observed when the HRNet-W48 backbone is used, outperforming both bottom-up and top-down approaches. ",
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+        "type": "text",
+        "text": "4.3 Inference Speed ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In comparing the inference speed, we test all the models on a single TITAN RTX GPU for its popularity in practice. The average inference speed, FPS (frames per second), over the 5000 images in COCO-val-2017 is used for the comparison. For DEKR [11], we re-implement their inference code with better speed obtained for fair comparisons at the algorithm level. For methods that have post-processing schema on CPU, only one thread is used. As shown in Table 4, our LOGO-CAP runs significantly faster than PifPaf [15] and AE [20]. The CenterNet [35] runs slower than DEKR and our LOCO-CAP as it requires a post-processing scheme to match the predicted offsets to the keypoints obtained from heatmaps. Comparing with DEKR, the speed improvement of our LOGOCAP is from the lightweight design of head modules since the same backbones are used. For the comparisons in Table 2, we run the models with different resolutions of testing images. ",
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+        "type": "text",
+        "text": "4.4 Potentials and Limitations of the Proposed LOGO-CAP ",
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+        "type": "text",
+        "text": "Consider the generic applicability of the center-offset formulation to many computer vision tasks as demonstrated in [35], we hypothesize that the proposed LOGO-CAP has a great potential to remedy the lack of sufficient accuracy using the vanilla center-offset method in those tasks. We also notice that the minimally-simple design in learning the “Slow Keypointer” can be relaxed for different accuracy-speed trade-offs in practice. For example, for the convolutional message passing module, an alternative method could be the Transformer model [31], which potentially will further improve the performance at the expense of inference speed. We leave these for future work. ",
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+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This paper focuses on deep learning based formulation for bottom-up human pose estimation. It presents a method of learning LOcal-GlObal Contextual Adaptation for Pose estimation, dubbed as LOGO-CAP. The proposed LOGO-CAP is built on the conceptually simple center-offset paradigm and addresses its drawback of lacking the capability of accurately localizing human pose keypoints. The key idea of our LOG-CAP is to lift the center-offset predicted keypoints to keypoint expansion maps (KEMs),which counters the inaccuracy and uncertainty of the initial keypoints. Two types of KEMs are introduced in two parallel modules on top of the feature backbone. Local KEMs are used to learn keypoint attraction maps (KAMs) via a convolutional message passing module that accounts for the structured output prediction nature of human pose estimation. Global KEMs are used to learn local-global contextual adaptation which convolves global KEMs using the KAMs as kernels. The refined global KEMs are used in computing the final human pose estimation. The proposed LOGO-CAP obtains state-of-the-art performance in COCO val-2017 and test-dev 2017 datasets for bottom-up human pose estimation. It also achieves state-of-the-art transferability performance in the OCHuman dataset with the COCO trained models. ",
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+        "type": "text",
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