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+# LEARNING FROM NOISY DATA WITH ROBUST REPRESENTATION LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Learning from noisy data has attracted much attention, where most methods focus on label noise. In this work, we propose a new framework which simultaneously addresses three types of noise commonly seen in real-world data: label noise, outof-distribution input, and input corruption. In contrast to most existing methods, we combat noise by learning robust representation. Specifically, we embed images into a low-dimensional subspace by training an autoencoder on the deep features. We regularize the geometric structure of the subspace with robust contrastive learning, which includes an unsupervised consistency loss and a supervised mixup prototypical loss. Furthermore, we leverage the structure of the learned subspace for noise cleaning, by aggregating information from neighboring samples. Experiments on multiple benchmarks demonstrate state-of-the-art performance of our method and robustness of the learned representation. Our code will be released1.
+
+# 1 INTRODUCTION
+
+Data in real life is noisy. However, deep models with remarkable performance are mostly trained on clean datasets with high-quality human annotations. Manual data cleaning and labeling is an expensive process that is difficult to scale. On the other hand, there exists almost infinite amount of noisy data online. It is crucial that deep neural networks (DNNs) could harvest noisy training data. However, it has been shown that DNNs are susceptible to overfitting to noise (Zhang et al., 2017).
+
+As shown in Figure $\mathbb { L } ,$ a real-world noisy image dataset often consists of multiple types of noise. Label noise refers to samples that are wrongly labeled as another class (e.g. flower labeled as orange). Out-of-distribution input refers to samples that do not belong to any known classes. Input corruption refers to image-level distortion (e.g. low brightness) that causes data shift between training and test.
+
+Most of the methods in literature focus on addressing the more detrimental label noise. Two dominant approaches include: (1) find clean samples as those with smaller loss and assign larger weights to them (Han et al., 2018; Yu et al., 2019; Shen & Sanghavi, 2019; Arazo et al., 2019); (2) relabel noisy samples using model’s predictions (Reed et al., 2015; Ma et al., 2018; Tanaka et al., 2018; Yi & Wu, $\dot { \overline { { 2 0 1 9 } } } )$ . The recently proposed DivideMix (Li et al., 2020a) integrates both approaches in a co-training framework, but it also increases computation cost. Previous methods that focus on addressing label noise do not consider out-of-distribution input or input corruption, which limits their performance in real-world scenarios. Furthermore, using a model’s own prediction to relabel samples could cause confirmation bias, where the prediction error accumulates and harms performance.
+
+We propose a new direction for effective learning from noisy data. Our method embeds images into noise-robust low-dimensional representations, and regularizes the geometric structure of the representations with contrastive learning. Specifically, our algorithmic contributions include:
+
+• We propose noise-robust contrastive learning, which introduces two contrastive losses. The first is an unsupervised consistency contrastive loss. It enforces inputs with perturbations to have similar normalized embeddings, which helps learn robust and discriminative representation.
+
+• Our second contrastive loss is a weakly-supervised mixup prototypical loss. We compute class prototypes as normalized mean embeddings, and enforces each sample’s embedding to be closer to its class prototype. Inspired by Mixup (Zhang et al., 2018), we construct virtual training samples as linear interpolation of inputs, and encourage the same linear relationship $w . r . t$ the class prototypes.
+
+![](images/638afbb1829c0fd505bc08ee611be1d2461a4ffc9d0e4631f4710188e435bbd6.jpg)  
+Figure 1: Google search images from WebVision (Li et al., 2017) dataset with keyword “orange”.
+
+• We train a linear autoencoder to reconstruct the high-dimensional features using low-dimensional embeddings. The autoendoer enables the high-dimensional features to maximally preserve the robustness of the low-dimensional embeddings, thus regularizing the classifier.
+
+• We propose a new noise cleaning method which exploits the structure of the learned representations. For each sample, we aggregate information from its top- $k$ neighbors to create a pseudo-label. A subset of training samples with confident pseudo-labels are selected to compute the weaklysupervised losses. This process can effectively clean both label noise and out-of-distribution (OOD) noise.
+
+# Our experimental contributions include:
+
+• We experimentally show that our method is robust to label noise, OOD input, and input corruption. Experiments are performed on multiple datasets with controlled noise and real-world noise, where our method achieves state-of-the-art performance.
+
+• We demonstrate that the proposed noise cleaning method can effectively clean a majority of label noise. It also learns a curriculum that gradually leverages more samples to compute the weakly-supervised losses as the pseudo-labels become more accurate.
+
+We validate the robustness of the learned low-dimensional representation by showing (1) $k$ -nearest neighbor classification outperforms the softmax classifier. (2) OOD samples can be separated from in-distribution samples. The efficacy of the proposed autoencoder is also verified.
+
+# 2 RELATED WORK
+
+Label noise learning. Learning from noisy labels have been extensively studied in the literature. While some methods require access to a small set of clean samples (Xiao et al., 2015; Vahdat, 2017; Veit et al., 2017; Lee et al., 2018; Hendrycks et al., 2018), most methods focus on the more challenging scenario where no clean labels are available. These methods can be categorized into two major types. The first type performs label correction using predictions from the network (Reed et al., 2015; Ma et al., 2018; Tanaka et al., 2018; Yi & Wu, 2019). The second type tries to separate clean samples from corrupted samples, and trains the model on clean samples (Han et al., 2018; Arazo et al., 2019; Jiang et al., 2018; 2020; Wang et al., 2018; Chen et al., 2019; Lyu & Tsang, 2020). The recently proposed DivideMix (Li et al., 2020a) effectively combines label correction and sample selection with the Mixup (Zhang et al., 2018) data augmentation under a co-training framework. However, it cost $2 \times$ the computational resource of our method.
+
+Different from existing methods, our method combats noise by learning noise-robust low-dimensional representations. We propose a more effective noise cleaning method by leveraging the structure of the learned representations. Furthermore, our model is robust not only to label noise, but also to out-of-distribution and corrupted input. A previous work has studied open-set noisy labels (Wang et al., 2018), but their method does not enjoy the same level of robustness as ours.
+
+Contrastive learning. Contrastive learning is at the core of recent self-supervised representation learning methods (Chen et al., 2020; He et al., 2019; Oord et al., 2018; Wu et al., 2018). In selfsupervised contrastive learning, two randomly augmented images are generated for each input image. Then a contrastive loss is applied to pull embeddings from the same source image closer, while pushing embeddings from different source images apart. Recently, prototypical contrastive learning (PCL) $\left. \underline { { \mathrm { L i } \mathrm { e t a l . } } } , \underline { { \mathrm { 2 0 2 0 b } } } \right.$ has been proposed, which uses cluster centroids as prototypes, and trains the network by pulling an image embedding closer to its assigned prototypes.
+
+![](images/094bf26750724724011e913dd594ec0ad21b0b1476e88ca023c596fd434b01b1.jpg)  
+Figure 2: Our proposed framework for noise-robust contrastive learning. We project images into a lowdimensional subspace, and regularize the geometric structure of the subspace with $( 1 ) \mathcal { L } _ { \mathrm { c c } }$ a consistency contrastive loss which enforces images with perturbations to have similar embeddings; $( 2 ) \mathcal { L } _ { \mathrm { p c . m i x } }$ : a prototypical contrastive loss augmented with mixup, which encourages the embedding for a linearly-interpolated input to have the same linear relationship w.r.t the class prototypes. The low-dimensional embeddings are also trained to reconstruct the high-dimensional features, which preserves the learned information and regularizes the classifier.
+
+Different from previous methods, our method performs contrastive learning in the principal subspace of the high-dimensional feature space, by training a linear autoencoder. Furthermore, our supervised contrastive loss improves PCL (Li et al., 2020b) with Mixup $\mathrm { ( } \mathbb { Z } \mathrm { h a n g ~ e t ~ a l . } \mathrm { ) } \mathbb { Z } \mathrm { 0 } 1 8 \mathrm { ) }$ . Different from the original Mixup where learning happens at the classification layer, our learning takes places in the low-dimensional subspace.
+
+# 3 METHOD
+
+Given a noisy training dataset $\mathbf { \mathcal { D } } = \{ ( \mathbf { \mathfrak { x } } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ , where $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ is an image and $y _ { i } \in \{ 1 , . . . , C \}$ is its class label. We aim to train a network that is robust to the noise in training data (i.e. label noise, OOD input, input corruption) and achieves high accuracy on a clean test set. The proposed network consists of three components: (1) a deep encoder (a convolutional neural network) that encodes an image $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ to a high-dimensional feature ${ \mathbf { } } v _ { i }$ ; (2) a classifier (a fully-connected layer followed by softmax) that receives ${ \mathbf { } } v _ { i }$ as input and outputs class predictions; (3) a linear autoencoder that projects ${ \bf \nabla } _ { v _ { i } }$ into a low-dimensional embedding $\dot { z } _ { i } \in \mathbb { R } ^ { d }$ . We show an illustration of our method in Figure $^ { 2 , }$ and a pseudo-code in appendix B. Next, we delineate its details.
+
+# 3.1 CONTRASTIVE LEARNING IN ROBUST LOW-DIMENSIONAL SUBSPACE
+
+Let $z _ { i } = \mathbf { W } _ { \mathrm { e } } { v } _ { i }$ be the linear projection from high-dimensional features to low-dimensional embeddings, and $\hat { z } _ { i } = z _ { i } / \left\| z _ { i } \right\| _ { 2 }$ be the normalized embeddings. We aim to learn robust embeddings with two contrastive losses: unsupervised consistency loss and weakly-supervised mixup prototypical loss.
+
+Unsupervised consistency contrastive loss. Following the NT-Xent (Chen et al., 2020) loss for selfsupervised representation learning, our consistency contrastive loss enforces images with semanticpreserving perturbations to have similar embeddings. Specifically, given a miniwe apply weak-augmentation and strong-augmentation to each image, and obtain tch of input $b$ $2 b$ $\{ \pmb { x } _ { i } \} _ { i = 1 } ^ { \breve { 2 } b }$ Weak augmentation is a standard flip-and-shift augmentation strategy, while strong augmentation consists of color and brightness changes with details given in Section 4.
+
+We project the inputs into the low-dimensional space to obtain their normalized embeddings $\{ \hat { z } _ { i } \} _ { i = 1 } ^ { 2 b }$ Let be the index of a weakly-augmented input, and $j ( i )$ be the index of the strongaugmented input from the same source image, the consistency contrastive loss is defined as:
+
+$$
+\mathcal { L } _ { \mathrm { c c } } = \sum _ { i = 1 } ^ { b } - \log \frac { \exp ( \hat { z } _ { i } \cdot \hat { z } _ { j ( i ) } / \tau ) } { \sum _ { k = 1 } ^ { 2 b } \mathbb { 1 } _ { i \neq k } \exp ( \hat { z } _ { i } \cdot \hat { z } _ { k } / \tau ) } ,
+$$
+
+where $\tau$ is a scalar temperature parameter. The consistency contrastive loss maximizes the inner product between the pair of positive embeddings $\hat { z } _ { i }$ and $\hat { z } _ { j ( i ) }$ , while minimizing the inner product between $2 ( b - 1 )$ pairs of negative embeddings. By mapping different views (augmentations) of the same image to neighboring embeddings, the consistency contrastive loss encourages the network to learn discriminative representation that is robust to low-level image corruption.
+
+Weakly-supervised mixup prototypical contrastive loss. Our second contrastive loss injects structural knowledge of classes into the embedding space. Let $\mathcal { T } _ { c }$ denote indices for the subset of images in $\mathcal { D }$ labeled with class $c$ , we calculate the class prototype as the normalized mean embedding:
+
+$$
+z ^ { c } = \frac { 1 } { \left| \mathbb { Z } _ { c } \right| } \sum _ { i \in \mathbb { Z } _ { c } } \hat { z } _ { i } , \hat { z } ^ { c } = \frac { z ^ { c } } { \left\| z ^ { c } \right\| _ { 2 } } ,
+$$
+
+where $\hat { z } _ { i }$ is the embedding of a center-cropped image, and the class prototypes are calculated at the beginning of each epoch.
+
+The prototypical contrastive loss enforces an image embedding $\hat { z } _ { i }$ to be more similar to its corresponding class prototype $\hat { z } ^ { y _ { i } }$ , in contrast to other class prototypes:
+
+$$
+\mathcal { L } _ { \mathrm { p c } } ( \hat { z } _ { i } , y _ { i } ) = - \log \frac { \exp ( \hat { z } _ { i } \cdot \hat { z } ^ { y _ { i } } / \tau ) } { \sum _ { c = 1 } ^ { C } \exp ( \hat { z } _ { i } \cdot \hat { z } ^ { c } / \tau ) } .
+$$
+
+Since the label $y _ { i }$ is noisy, we would like to regularize the encoder from memorizing training labels. Mixup (Zhang et al., $\boxed { 2 0 1 8 }$ has been shown to be an effective method against label noise (Arazo et al., $\mathord { \left. \kern - delimiterspace \right)} \frac { 2 0 1 9 } { \mathrm { ~ L i ~ e t ~ a l . } } \mathord { \left[ 2 0 2 0 \mathrm { a } \right] } $ . Inspired by it, we create virtual training samples by linearly interpolating a sample (indexed by $i$ ) with another sample (indexed by $m ( i ) .$ ) randomly chosen from the same minibatch:
+
+$$
+\begin{array} { r } { \pmb { x } _ { i } ^ { m } = \lambda \pmb { x } _ { i } + ( 1 - \lambda ) \pmb { x } _ { m ( i ) } , } \end{array}
+$$
+
+where $\lambda \sim \operatorname { B e t a } ( \alpha , \alpha )$
+
+Let $\hat { z } _ { i } ^ { m }$ be the normalized embedding for $\pmb { x } _ { i } ^ { m }$ , the mixup version of the prototypical contrastive loss is defined as a weighted combination of the two ${ \mathcal { L } } _ { \mathrm { p c } }$ w.r.t class $y _ { i }$ and $y _ { m ( i ) }$ . It enforces the embedding for the interpolated input to have the same linear relationship $w . r . t .$ . the class prototypes.
+
+$$
+\mathcal { L } _ { \mathrm { p c . m i x } } = \sum _ { i = 1 } ^ { 2 b } \lambda \mathcal { L } _ { \mathrm { p c } } ( \hat { z } _ { i } ^ { m } , y _ { i } ) + ( 1 - \lambda ) \mathcal { L } _ { \mathrm { p c } } ( \hat { z } _ { i } ^ { m } , y _ { m ( i ) } ) .
+$$
+
+Reconstruction loss. We also train a linear decoder $\mathbf { W } _ { \mathrm { d } }$ to reconstruct the high-dimensional feature ${ \mathbf { } } v _ { i }$ based on $z _ { i }$ . The reconstruction loss is defined as:
+
+$$
+\mathcal { L } _ { \mathrm { r e c o n } } = \sum _ { i = 1 } ^ { 2 b } \left\| \pmb { v } _ { i } - \mathbf { W } _ { \mathrm { d } } \pmb { z } _ { i } \right\| _ { 2 } ^ { 2 } .
+$$
+
+There are several benefits for training the autoencoder. First, with an optimal linear autoencoder, ${ \bf W } _ { \mathrm { e } }$ will project ${ \mathbf { } } v _ { i }$ into its low-dimensional principal subspace and can be understood as applying PCA $\mathrm { ( \mathbb { B } a l d i \ \& \ H o r n i k | , \frac { } { | 9 8 9 ) } }$ Thus the low-dimensional representation $z _ { i }$ is intrinsically robust to input noise. Second, minimizing the reconstruction error is maximizing a lower bound of the mutual information between ${ \mathbf { } } v _ { i }$ and $z _ { i }$ (Vincent et al., $\boxed { 2 0 1 0 }$ . Therefore, knowledge learned from the proposed contrastive losses can be maximally preserved in the high-dimensional representation, which helps regularize the classifier.
+
+Classification loss. Given the softmax output from the classifier, ${ \pmb p } ( { \pmb y } ; { \pmb x } _ { i } )$ , we define the classification loss as the cross-entropy loss. Note that it is only applied to the weakly-augmented inputs.
+
+$$
+\mathcal { L } _ { \mathrm { c e } } = - \sum _ { i = 1 } ^ { b } \log p ( y _ { i } ; \pmb { x } _ { i } ) .
+$$
+
+![](images/d37d4c61b89760d47df7abc4d1404e044ee37a5e9c3d002a73d196c2e4369eb5.jpg)  
+Figure 3: Curriculum learned by the proposed label correction method for training on CIFAR datasets with $50 \%$ sym. noise. (a) Accuracy of pseudo-labels w.r.t to clean training labels. (b) Number of samples in the weakly-supervised subset $\mathcal { D } _ { \mathrm { s u p } } ^ { t }$ . (c) Label noise ratio in the weakly-supervised subset.
+
+The overall training objective is to minimize a weighted sum of all losses:
+
+$$
+\mathcal { L } = \mathcal { L } _ { \mathrm { c e } } + \omega _ { \mathrm { c c } } \mathcal { L } _ { \mathrm { c c } } + \omega _ { \mathrm { p c } } \mathcal { L } _ { \mathrm { p c . m i x } } + \omega _ { \mathrm { r e c o n } } \mathcal { L } _ { \mathrm { r e c o n } }
+$$
+
+For all experiments, we fix $\omega _ { \mathrm { c c } } = 1$ , $\omega _ { \mathrm { r e c o n } } = 1$ , and change $\omega _ { \mathrm { p c } }$ only across datasets.
+
+# 3.2 NOISE CLEANING WITH SMOOTH NEIGHBORS
+
+After warming-up the model by training with the noisy labels $\{ y _ { i } \} _ { i = 1 } ^ { n }$ for $t _ { 0 }$ epochs, we aim to clean the noise by generating a soft pseudo-label $\pmb q _ { i }$ for each training sample. Different from previous methods that perform label correction purely using the model’s softmax prediction, our method exploits the structure of the low-dimensional subspace by aggregating information from top- $k$ neighboring samples, which helps alleviate the confirmation bias problem.
+
+At the $t$ -th epoch, for each sample $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ , let $\mathbf { \Delta } _ { p _ { i } ^ { t } } ^ { t }$ be the classifier’s softmax prediction, let $\pmb q _ { i } ^ { t - 1 }$ be its soft label from the previous epoch, we calculate the soft label for the current epoch as:
+
+$$
+\pmb { q } _ { i } ^ { t } = \frac { 1 } { 2 } \pmb { p } _ { i } ^ { t } + \frac { 1 } { 2 } \sum _ { j = 1 } ^ { k } w _ { i j } ^ { t } \pmb { q } _ { j } ^ { t - 1 } ,
+$$
+
+where $w _ { i j } ^ { t }$ represents the normalized affinity between a sample and its neighbor and is defined as $\begin{array} { r } { w _ { i j } ^ { t } = \frac { \mathrm { { ' } } \exp ( \hat { z } _ { i } ^ { t } \cdot \hat { z } _ { j } ^ { t } / \tau ) } { \sum _ { j = 1 } ^ { k } \exp ( \hat { z } _ { i } ^ { t } \cdot \hat { z } _ { j } ^ { t } / \tau ) } } \end{array}$ . We set $k = 2 0 0$ in all experiments.
+
+The soft label defined by eqn. $( 9 )$ is the minimizer of the following quadratic loss function:
+
+$$
+J ( { \pmb q } _ { i } ^ { t } ) = \sum _ { j = 1 } ^ { k } w _ { i j } ^ { t } \left\| { \pmb q } _ { i } ^ { t } - { \pmb q } _ { j } ^ { t - 1 } \right\| _ { 2 } ^ { 2 } + \left\| { \pmb q } _ { i } ^ { t } - { \pmb p } _ { i } ^ { t } \right\| _ { 2 } ^ { 2 } .
+$$
+
+The first term is a smoothness constraint which encourages the soft label to take a similar value as its neighbors’ labels, whereas the second term attempts to maintain the model’s class prediction.
+
+We construct a weakly-supervised subset which contains (1) clean sample whose soft label score for the original class $y _ { i }$ is higher than a threshold $\eta _ { 0 }$ , (2) pseudo-labeled sample whose maximum soft label score exceeds a threshold $\eta _ { 1 }$ . For pseudo-labeled samples, we convert their soft labels into hard labels by taking the class with the maximum score.
+
+$$
+\mathcal { D } _ { \operatorname* { s u p } } ^ { t } = \{ \alpha _ { i } , y _ { i } \mid q _ { i } ^ { t } ( y _ { i } ) > \eta _ { 0 } \} \cup \{ x _ { i } , \hat { y } _ { i } ^ { t } = \arg \operatorname* { m a x } _ { c } q _ { i } ^ { t } ( c ) \mid \forall \operatorname* { m a x } _ { c } q _ { i } ^ { t } ( c ) > \eta _ { 1 } , c \in \{ 1 , . . , C \} \}
+$$
+
+Given the weakly-supervised subset, we modify the classification loss $\mathcal { L } _ { \mathrm { c e } }$ , the mixup prototypical contrastive loss $\mathcal { L } _ { \mathrm { p c . m i x } }$ , and the calculation of prototypes $\hat { z } ^ { c }$ , such that they only use samples from $\mathcal { D } _ { \mathrm { s u p } } ^ { t }$ . The unsupervised losses (i.e. ${ \mathcal { L } } _ { \mathrm { c c } }$ and $\mathcal { L } _ { \mathrm { r e c o n } }$ ) still operate on all training samples.
+
+Learning curriculum. Our iterative noise cleaning method learns an effective training curriculum, which gradually increases the size of $\mathcal { D } _ { \mathrm { s u p } } ^ { t }$ as the pseudo-labels become more accurate. To demonstrate such curriculum, we analyse the noise cleaning statistics for training our model on CIFAR-10 and CIFAR-100 datasets with $50 \%$ label noise (experimental details explained in the next section). In Figure $\bigtriangledown$ (a), we show the accuracy of the soft pseudo-labels $w . r . t$ to clean training labels (only used for analysis purpose). Our method can significantly reduce the ratio of label noise from $50 \%$ to $5 \%$ (for CIFAR-10) and $17 \%$ (for CIFAR-100). Figure $\dot { 3 }$ (b) shows the size of $\mathcal { D } _ { \mathrm { s u p } } ^ { t }$ as a percentage of the total number of training samples, and Figure $\boxed { 3 }$ (c) shows the effective label noise ratio within the weakly-supervised subset $\mathcal { D } _ { \mathrm { s u p } } ^ { t }$ . Our method maintains a low noise ratio in the weakly-supervised subset, while gradually increasing its size to utilize more samples for the weakly-supervised losses.
+
+Table 1: Comparison with state-of-the-art methods on CIFAR datasets with label noise. Numbers indicate average test accuracy $( \% )$ over last 10 epochs. We report results over 3 independent runs with randomly-generated label noise. Results for previous methods are copied from Arazo et al. $\boxed { 2 0 1 9 }$ ; Li et al. $\textcircled { 2 0 2 0 2 }$ We re-run DivideMix (without ensemble) using the publicly available code on the same noisy data as ours.   
+
+<table><tr><td rowspan=1 colspan=6>Dataset                                          CIFAR-10                      CIFAR-100Noise type                         Sym 20% Sym 50%  Asym 40%  Sym 20% Sym 50%</td></tr><tr><td rowspan=9 colspan=1>Cross-Entropy (Li et al. 2020a)ForwardPatrini et al.12017Co-teaching+ (Yu et al.12019Mixup(Zhang et al.2018) P-correcuon (Yi&amp; Wu [2019)MLNTLi et al2019)M-correcion (Arazo et al]2019)DivideMix (Li et al.2020a)DivideMix (reproduced)</td><td rowspan=9 colspan=1>82.783.188.292.392.092.093.895.095.1±0.1</td><td rowspan=1 colspan=1>57.9</td><td rowspan=1 colspan=1>72.3</td><td rowspan=1 colspan=1>61.8</td><td rowspan=1 colspan=1>37.3</td></tr><tr><td rowspan=1 colspan=1>59.4</td><td rowspan=1 colspan=1>83.1</td><td rowspan=1 colspan=1>61.4</td><td rowspan=1 colspan=1>37.3</td></tr><tr><td rowspan=1 colspan=1>84.1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>64.1</td><td rowspan=1 colspan=1>45.3</td></tr><tr><td rowspan=1 colspan=1>77.6</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>66.0</td><td rowspan=1 colspan=1>46.6</td></tr><tr><td rowspan=1 colspan=1>88.7</td><td rowspan=1 colspan=1>88.1</td><td rowspan=1 colspan=1>68.1</td><td rowspan=1 colspan=1>56.4</td></tr><tr><td rowspan=1 colspan=1>88.8</td><td rowspan=1 colspan=1>88.6</td><td rowspan=1 colspan=1>67.7</td><td rowspan=1 colspan=1>58.0</td></tr><tr><td rowspan=3 colspan=1>91.993.793.6±0.2</td><td rowspan=1 colspan=1>86.3</td><td rowspan=1 colspan=1>73.4</td><td rowspan=1 colspan=1>65.4</td></tr><tr><td rowspan=1 colspan=1>91.4</td><td rowspan=1 colspan=1>74.8</td><td rowspan=1 colspan=1>72.1</td></tr><tr><td rowspan=1 colspan=1>91.3±0.8</td><td rowspan=1 colspan=1>75.1±0.2</td><td rowspan=1 colspan=1>72.1±0.3</td></tr><tr><td rowspan=1 colspan=1>Ours (classifier)Ours (knn)</td><td rowspan=1 colspan=1>95.8±0.195.9±0.1</td><td rowspan=1 colspan=1>94.3±0.294.5±0.1</td><td rowspan=1 colspan=1>91.9±0.892.4±0.9</td><td rowspan=1 colspan=1>79.1±0.179.4±0.1</td><td rowspan=1 colspan=1>74.8±0.475.0±0.4</td></tr></table>
+
+# 4 EXPERIMENT
+
+In this section, we validate the proposed method on multiple benchmarks with controlled noise and real-world noise. Our method achieves state-of-the-art performance across all benchmarks. For fair comparison, we compare with DivideMix $\mathbb { ( L i ~ e t ~ a l . ) } \index { [ 2 0 2 0 4 ) }$ without ensemble. In appendix A, we report the result of our method with co-training and ensemble, which further improves performance.
+
+# 4.1 EXPERIMENTS ON CONTROLLED NOISY LABELS
+
+Dataset. Following Tanaka et al. (2018); Li et al. $\textcircled { 1 2 0 2 0 } \textcircled { 2 }$ we corrupt the training data of CIFAR10 and CIFAR-100 (Krizhevsky & Hinton, $\textcircled { 2 0 0 9 }$ with two types of label noise: symmetric and asymmetric. Symmetric noise is injected by randomly selecting a percentage of samples and changing their labels to random labels. Asymmetric noise is class-dependant, where labels are only changed to similar classes (e.g. dog cat, deer horse). We experiment with multiple noise ratios: sym $20 \%$ , sym $50 \%$ , and asym $40 \%$ (see results for sym $80 \%$ and $90 \%$ in appendix $\mathbf { A } )$ . Note that asymmetric noise ratio cannot exceed $50 \%$ because certain classes would become theoretically indistinguishable.
+
+Implementation details. Same as previous works $\left( \mathrm { A r a z o ~ e t ~ a l . } \right) \left[ 2 0 1 9 \right. \mathrm { [ L i ~ e t ~ a l . ] } \left. 2 0 2 0 \mathrm { a } \right.$ we use PreAct ResNet-18 (He et al., $\boxed { 2 0 1 6 }$ as our encoder model. We set the dimensionality of the bottleneck layer as $d = 5 0$ . Our model is trained using SGD with a momentum of 0.9, a weight decay of 0.0005, and a batch size of 128. The network is trained for 200 epochs. We set the initial learning rate as 0.02 and use a cosine decay schedule. We apply standard crop and horizontal flip as the weak augmentation. For strong augmentation, we use AugMix $( \mathrm { \overline { { H e n d r y c k s \ e t { a l . } } } } , \mathrm { \overline { { 2 0 2 0 } } } )$ , though other methods (e.g. SimAug (Chen et al., $\boxed { 2 0 2 0 }$ ) work equally well. For all CIFAR experiments, we fix the hyper-parameters as $\omega _ { \mathrm { c c } } = 1 , \omega _ { \mathrm { p c } } = 5$ , $\omega _ { \mathrm { r e c o n } } = 1 , \tau = 0 . 3 , \alpha = 8 , \eta _ { 1 } = 0 . 9$ . For CIFAR-10, we activate noise cleaning at epoch $t _ { 0 } = 5$ , and set $\eta _ { 0 } = 0 . 1$ (sym.) or 0.4 (asym.). For CIFAR-100, we activate noise cleaning at epoch $t _ { 0 } = 1 5$ , and set $\eta _ { 0 } = 0 . 0 2$ . We use faiss-gpu (Johnson et al., 2017) for efficient knn search in the low-dimensional subspace, which finishes within 1 second.
+
+Results. Table $^ 1$ shows the comparison with existing methods. Our method outperforms previous methods across all label noise settings. On the more challenging CIFAR-100, we achieve $3- 4 \%$ accuracy improvement compared to the second-best method DivideMix. Moreover, our method is more computational efficient than DivideMix, which needs co-training for noise filtering.
+
+<table><tr><td rowspan="2">CIFAR-10 50% sym. noise</td><td rowspan="2">CE</td><td>Iterative</td><td>GCE</td><td rowspan="2">2018</td><td rowspan="2">DivideMix (Li et al. 2020a</td><td rowspan="2">Ours (cls.)</td><td rowspan="2">Ours (knn)</td></tr><tr><td>Wang et al. 2018</td><td>Zhang &amp; Sabuncu</td></tr><tr><td>+ CIFAR-100 20k</td><td>53.6</td><td>87.2</td><td>87.3</td><td></td><td>89.0</td><td>91.5</td><td>93.1±0.3</td></tr><tr><td>+ SVHN 20k</td><td>58.1</td><td>88.6</td><td>88.8</td><td></td><td>91.9</td><td>93.3</td><td>93.9±0.2</td></tr><tr><td>+ Image Corruption</td><td>53.8</td><td>87.7</td><td>87.9</td><td></td><td>89.8</td><td>91.4</td><td>91.6±0.2</td></tr></table>
+
+Table 2: Comparison with state-of-the-art methods on datasets with label noise and input noise. Numbers indicate average test accuracy $( \% )$ over last 10 epochs. We report results over 3 independent runs with randomlygenerated noise. We re-run previous methods using publicly available code with the same noisy data and model architecture as ours.
+
+In order to demonstrate the advantage of the proposed low-dimensional embeddings, we perform $k$ - nearest neighbor (knn) classification ( $k = 2 0 0$ ), by projecting test images into normalized embeddings. Compared to the trained classifier, knn achieves higher accuracy, which verifies the robustness of the learned low-dimensional representations.
+
+# 4.2 EXPERIMENTS ON CONTROLLED NOISY LABELS WITH NOISY IMAGES
+
+Dataset. We further corrupt a noisy CIFAR-10 dataset (sym. $50 \%$ ) by injecting two types of input noise: out-of-distribution (OOD) images and input corruption. For OOD noise, we follow Wang et al. (2018) and add $2 0 k$ images from either one of the two other datasets: CIFAR-100 and SVHN (Netzer $\boxed { \mathrm { e t ~ a l . } , \boxed { 2 0 1 1 } }$ , enlarging the training set to $7 0 k$ . A random CIFAR-10 label is assigned to each OOD image. For input corruption, we follow Hendrycks & Dietterich $\textcircled { 2 0 1 9 }$ and corrupt each image in CIFAR-10 with a noise randomly chosen from the following four types: Fog, Snow, Motion blur and Gaussian noise. Examples of both types of input noise are shown in Figure 4. We follow the same implementation details as the CIFAR-10 experiments described in Section 4.1.
+
+![](images/ac822823753a2fac58f6037c832534dcce1705ad8d63b701e0143ca0a0dbf8a7.jpg)  
+Figure 4: Examples of input noise injected to CIFAR-10.
+
+Results. Table 2 shows the results, where our method consistently outperforms existing methods by a substantial margin. We observe that OOD images from a similar domain (CIFAR-100) are more harmful than OOD images from a more different domain (SVHN). This is because noisy images that are closer to the test data distribution are more likely to distort the decision boundary in a way that negatively affects test performance. Nevertheless, performing knn classification using the learned embeddings demonstrates high robustness to input noise.
+
+In Figure $5 ,$ we show the t-SNE (Maaten & Hinton, 2008) visualization of the low-dimensional embeddings for all training samples. As training progresses, our model learns to separate OOD samples (represented as gray points) from in-distribution samples, and cluster samples of the same class together despite their noisy labels.
+
+![](images/7fcaf6998097aa975e3eac8c47e1d176c9a4ea41a30d28612dbc18b3e6873bec.jpg)  
+Figure 5: t-SNE visualization of low-dimensional embeddings for CIFAR-10 images (color represents the true class) $+ \mathrm { O O D }$ images (gray points) from CIFAR-100 or SVHN. The model is trained on noisy CIFAR-10 $5 0 k$ images with $50 \%$ label noise) and $2 0 k$ OOD images with random labels. Our method can effectively learn to (1) cluster CIFAR-10 images according to their true class, despite their noisy labels; (2) separate OOD samples from in-distribution samples, such that their harm is reduced.
+
+Table 3: Comparison with state-of-the-art methods trained on WebVision (mini).   
+
+<table><tr><td colspan="2">Test dataset</td><td colspan="2">WebVision</td><td colspan="2">ILSVRC12</td></tr><tr><td colspan="2">Accuracy (%)</td><td>top1</td><td>top5</td><td>top1</td><td>top5</td></tr><tr><td colspan="2"> Forward (Patrini et al.2017)</td><td>61.1</td><td>82.7</td><td>57.4</td><td>82.4</td></tr><tr><td colspan="2">Decoupling (Malach &amp; Shalev-Shwartz 12017</td><td>62.5</td><td>84.7</td><td>58.3</td><td>82.3</td></tr><tr><td colspan="2">D2L (Ma et al12018)</td><td>62.7</td><td>84.0</td><td>57.8</td><td>81.4</td></tr><tr><td colspan="2">MentorNet (Jiang et al. 2018</td><td>63.0</td><td>81.4</td><td>57.8</td><td>79.9</td></tr><tr><td colspan="2">Co-teaching (Han et al. 2018)</td><td>63.6</td><td>85.2</td><td>61.5</td><td>84.7</td></tr><tr><td colspan="2">INCV (Chen et al.,2019)</td><td>65.2</td><td>85.3</td><td>61.0</td><td>85.0</td></tr><tr><td colspan="2">DivideMix (Li et al. 12020a)</td><td>75.9</td><td>90.1</td><td>73.3</td><td>89.2</td></tr><tr><td colspan="2">Ours (w/o noise cleaning)</td><td>75.5</td><td>90.2</td><td>72.0</td><td>90.0</td></tr><tr><td colspan="2">Ours (classifier)</td><td>76.3</td><td>91.5</td><td>73.3</td><td>91.2</td></tr><tr><td colspan="2">Ours (knn)</td><td>77.8</td><td>91.3</td><td>74.4</td><td>90.9</td></tr></table>
+
+Table 4: Comparison with state-of-the-art methods on Clothing1M dataset.   
+
+<table><tr><td>Method</td><td>CE</td><td>Forward</td><td>Joint-Opt</td><td>MLNT</td><td>MentorMix</td><td>SL</td><td>DivideMix</td><td>Ours (cls.）</td><td>Ours (knn)</td></tr><tr><td>Accuracy</td><td>69.21</td><td>69.84</td><td>72.16</td><td>73.47</td><td>74.30</td><td>74.45</td><td>74.48</td><td>74.84</td><td>74.97</td></tr></table>
+
+# 4.3 EXPERIMENTS ON REAL-WORLD NOISY DATA
+
+Dataset and implementation details. We verify our method on two real-word noisy datasets: WebVision $\mathrm { ( } \mathbf { L i } \mathrm { e t a l . } \mathrm { , } \overline { { 2 0 1 7 } } \mathrm { ) }$ and Clothing1M (Xiao et al., 2015). Webvision contains images crawled from the web using the same concepts from ImageNet ILSVRC12 $( \mathbb { D e n g \ e t \ a l . } ) . 2 0 0 9 )$ . Following previous works (Chen et al., 2019; Li et al., 2020a), we perform experiments on the first 50 classes of the Google image subset. Clothing1M consists of images collected from online shopping websites where labels were generated from surrounding texts. Note that we do not use the additional clean set for training. For both experiments, we use the same model architecture as previous methods. More implementation details are given in the appendix.
+
+Results. We report the results for WebVision in Table $\textcircled { 3 }$ and Clothing1M in Table $4 ,$ where we achieve state-of-the-art performance on both datasets. Our method achieves competitive performance on WebVision even without performing noise cleaning, which demonstrates the robustness of the learned representation. Appendix D shows examples of noisy images that are cleaned by our method.
+
+# 4.4 ABLATION STUDY
+
+Effect of the proposed components. In order to study the effect of the proposed components, we remove each of them and report accuracy of the classifier (knn) across four benchmarks. As shown in Table $\boxed { 5 }$ the mixup prototypical contrastive loss $( \mathcal { L } _ { \mathrm { p c . m i x } } )$ is most crucial to the model’s performance. The consistency contrastive loss $( \mathcal { L } _ { \mathrm { c c } } )$ has a stronger effect with corrupted input or larger number of classes. We also experiment with removing mixup and using the standard prototypical contrastive loss, and using standard data augmentation (crop and horizontal flip) instead of AugMix. The proposed method still achieves state-of-the-art result with standard data augmentation.
+
+<table><tr><td></td><td>| CIFAR-10 Sym 50%</td><td>6|+CIFAR-10020k</td><td>+ Image Corruption</td><td>CIFAR-100 Sym 50%</td></tr><tr><td>w/o Lpc_mix</td><td>85.9 (86.1)</td><td>79.7 (81.5)</td><td>81.6 (81.7)</td><td>65.6 (65.9)</td></tr><tr><td>w/o Lcc</td><td>93.7 (93.8)</td><td>91.3 (91.5)</td><td>89.4 (89.5)</td><td>71.9 (71.8)</td></tr><tr><td>w/o Lrecon</td><td>93.3 (94.0)</td><td>90.7 (92.9)</td><td>90.2 (91.0)</td><td>73.2 (73.9)</td></tr><tr><td>w/o mixup</td><td>89.5 (89.9)</td><td>85.4 (87.0)</td><td>84.7 (84.9)</td><td>69.3 (69.7)</td></tr><tr><td> w/ standard aug.</td><td>94.1 (94.3)</td><td>90.8 (92.9)</td><td>90.5 (90.7)</td><td>74.5 (75.0)</td></tr><tr><td>DivideMix</td><td>93.6</td><td>89.0</td><td>89.8</td><td>72.1</td></tr><tr><td>Ours</td><td>94.3 (94.5)</td><td>91.5 (93.1)</td><td>91.4 (91.6)</td><td>74.8 (75.0)</td></tr></table>
+
+Table 5: Effect of the proposed components. We show the accuracy of the classifier (knn) on four benchmarks with different noise. Note that DivideMix (Li et al., 2020a) also performs mixup.
+
+Effect of bottleneck dimension. We vary the dimensionality of the bottleneck layer, $d$ , and examine the performance change in Table $\boxed { 6 }$ Our model is in general not very sensitive to the change of $d$ .   
+Table 6: Classifier’s test accuracy $( \% )$ with different low-dimensions.   
+
+<table><tr><td>bottleneck dimension |d= 25|d= 50丨d=100|d = 200</td><td></td><td></td><td></td><td></td></tr><tr><td>CIFAR-10 Sym 50%</td><td>93.4</td><td>94.3</td><td>94.2</td><td>93.7</td></tr><tr><td>CIFAR-100 Sym 50%</td><td>73.8</td><td>74.8</td><td>74.4</td><td>73.8</td></tr></table>
+
+# 5 CONCLUSION
+
+This paper proposes noise-robust contrastive learning, a new method to combat noise in training data by learning robust representation. We demonstrate our model’s state-of-the-art performance with extensive experiments on multiple noisy datasets. For future work, we are interested in adapting our method to other domains such as NLP or speech. We would also like to explore the potential of our method for learning transferable representations that could be useful for down-stream tasks.
+
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+        "text": "LEARNING FROM NOISY DATA WITH ROBUST REPRESENTATION LEARNING ",
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+        "text": "ABSTRACT ",
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+        "text": "Learning from noisy data has attracted much attention, where most methods focus on label noise. In this work, we propose a new framework which simultaneously addresses three types of noise commonly seen in real-world data: label noise, outof-distribution input, and input corruption. In contrast to most existing methods, we combat noise by learning robust representation. Specifically, we embed images into a low-dimensional subspace by training an autoencoder on the deep features. We regularize the geometric structure of the subspace with robust contrastive learning, which includes an unsupervised consistency loss and a supervised mixup prototypical loss. Furthermore, we leverage the structure of the learned subspace for noise cleaning, by aggregating information from neighboring samples. Experiments on multiple benchmarks demonstrate state-of-the-art performance of our method and robustness of the learned representation. Our code will be released1. ",
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+        "text": "1 INTRODUCTION ",
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+        "text": "Data in real life is noisy. However, deep models with remarkable performance are mostly trained on clean datasets with high-quality human annotations. Manual data cleaning and labeling is an expensive process that is difficult to scale. On the other hand, there exists almost infinite amount of noisy data online. It is crucial that deep neural networks (DNNs) could harvest noisy training data. However, it has been shown that DNNs are susceptible to overfitting to noise (Zhang et al., 2017). ",
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+        "text": "As shown in Figure $\\mathbb { L } ,$ a real-world noisy image dataset often consists of multiple types of noise. Label noise refers to samples that are wrongly labeled as another class (e.g. flower labeled as orange). Out-of-distribution input refers to samples that do not belong to any known classes. Input corruption refers to image-level distortion (e.g. low brightness) that causes data shift between training and test. ",
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+        "text": "Most of the methods in literature focus on addressing the more detrimental label noise. Two dominant approaches include: (1) find clean samples as those with smaller loss and assign larger weights to them (Han et al., 2018; Yu et al., 2019; Shen & Sanghavi, 2019; Arazo et al., 2019); (2) relabel noisy samples using model’s predictions (Reed et al., 2015; Ma et al., 2018; Tanaka et al., 2018; Yi & Wu, $\\dot { \\overline { { 2 0 1 9 } } } )$ . The recently proposed DivideMix (Li et al., 2020a) integrates both approaches in a co-training framework, but it also increases computation cost. Previous methods that focus on addressing label noise do not consider out-of-distribution input or input corruption, which limits their performance in real-world scenarios. Furthermore, using a model’s own prediction to relabel samples could cause confirmation bias, where the prediction error accumulates and harms performance. ",
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+        "text": "We propose a new direction for effective learning from noisy data. Our method embeds images into noise-robust low-dimensional representations, and regularizes the geometric structure of the representations with contrastive learning. Specifically, our algorithmic contributions include: ",
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+        "text": "• We propose noise-robust contrastive learning, which introduces two contrastive losses. The first is an unsupervised consistency contrastive loss. It enforces inputs with perturbations to have similar normalized embeddings, which helps learn robust and discriminative representation. ",
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+        "text": "• Our second contrastive loss is a weakly-supervised mixup prototypical loss. We compute class prototypes as normalized mean embeddings, and enforces each sample’s embedding to be closer to its class prototype. Inspired by Mixup (Zhang et al., 2018), we construct virtual training samples as linear interpolation of inputs, and encourage the same linear relationship $w . r . t$ the class prototypes. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We train a linear autoencoder to reconstruct the high-dimensional features using low-dimensional embeddings. The autoendoer enables the high-dimensional features to maximally preserve the robustness of the low-dimensional embeddings, thus regularizing the classifier. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a new noise cleaning method which exploits the structure of the learned representations. For each sample, we aggregate information from its top- $k$ neighbors to create a pseudo-label. A subset of training samples with confident pseudo-labels are selected to compute the weaklysupervised losses. This process can effectively clean both label noise and out-of-distribution (OOD) noise. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our experimental contributions include: ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We experimentally show that our method is robust to label noise, OOD input, and input corruption. Experiments are performed on multiple datasets with controlled noise and real-world noise, where our method achieves state-of-the-art performance. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We demonstrate that the proposed noise cleaning method can effectively clean a majority of label noise. It also learns a curriculum that gradually leverages more samples to compute the weakly-supervised losses as the pseudo-labels become more accurate. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We validate the robustness of the learned low-dimensional representation by showing (1) $k$ -nearest neighbor classification outperforms the softmax classifier. (2) OOD samples can be separated from in-distribution samples. The efficacy of the proposed autoencoder is also verified. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Label noise learning. Learning from noisy labels have been extensively studied in the literature. While some methods require access to a small set of clean samples (Xiao et al., 2015; Vahdat, 2017; Veit et al., 2017; Lee et al., 2018; Hendrycks et al., 2018), most methods focus on the more challenging scenario where no clean labels are available. These methods can be categorized into two major types. The first type performs label correction using predictions from the network (Reed et al., 2015; Ma et al., 2018; Tanaka et al., 2018; Yi & Wu, 2019). The second type tries to separate clean samples from corrupted samples, and trains the model on clean samples (Han et al., 2018; Arazo et al., 2019; Jiang et al., 2018; 2020; Wang et al., 2018; Chen et al., 2019; Lyu & Tsang, 2020). The recently proposed DivideMix (Li et al., 2020a) effectively combines label correction and sample selection with the Mixup (Zhang et al., 2018) data augmentation under a co-training framework. However, it cost $2 \\times$ the computational resource of our method. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Different from existing methods, our method combats noise by learning noise-robust low-dimensional representations. We propose a more effective noise cleaning method by leveraging the structure of the learned representations. Furthermore, our model is robust not only to label noise, but also to out-of-distribution and corrupted input. A previous work has studied open-set noisy labels (Wang et al., 2018), but their method does not enjoy the same level of robustness as ours. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contrastive learning. Contrastive learning is at the core of recent self-supervised representation learning methods (Chen et al., 2020; He et al., 2019; Oord et al., 2018; Wu et al., 2018). In selfsupervised contrastive learning, two randomly augmented images are generated for each input image. Then a contrastive loss is applied to pull embeddings from the same source image closer, while pushing embeddings from different source images apart. Recently, prototypical contrastive learning (PCL) $\\left. \\underline { { \\mathrm { L i } \\mathrm { e t a l . } } } , \\underline { { \\mathrm { 2 0 2 0 b } } } \\right.$ has been proposed, which uses cluster centroids as prototypes, and trains the network by pulling an image embedding closer to its assigned prototypes. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/094bf26750724724011e913dd594ec0ad21b0b1476e88ca023c596fd434b01b1.jpg",
+        "image_caption": [
+            "Figure 2: Our proposed framework for noise-robust contrastive learning. We project images into a lowdimensional subspace, and regularize the geometric structure of the subspace with $( 1 ) \\mathcal { L } _ { \\mathrm { c c } }$ a consistency contrastive loss which enforces images with perturbations to have similar embeddings; $( 2 ) \\mathcal { L } _ { \\mathrm { p c . m i x } }$ : a prototypical contrastive loss augmented with mixup, which encourages the embedding for a linearly-interpolated input to have the same linear relationship w.r.t the class prototypes. The low-dimensional embeddings are also trained to reconstruct the high-dimensional features, which preserves the learned information and regularizes the classifier. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            101,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Different from previous methods, our method performs contrastive learning in the principal subspace of the high-dimensional feature space, by training a linear autoencoder. Furthermore, our supervised contrastive loss improves PCL (Li et al., 2020b) with Mixup $\\mathrm { ( } \\mathbb { Z } \\mathrm { h a n g ~ e t ~ a l . } \\mathrm { ) } \\mathbb { Z } \\mathrm { 0 } 1 8 \\mathrm { ) }$ . Different from the original Mixup where learning happens at the classification layer, our learning takes places in the low-dimensional subspace. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a noisy training dataset $\\mathbf { \\mathcal { D } } = \\{ ( \\mathbf { \\mathfrak { x } } _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ , where $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ is an image and $y _ { i } \\in \\{ 1 , . . . , C \\}$ is its class label. We aim to train a network that is robust to the noise in training data (i.e. label noise, OOD input, input corruption) and achieves high accuracy on a clean test set. The proposed network consists of three components: (1) a deep encoder (a convolutional neural network) that encodes an image $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ to a high-dimensional feature ${ \\mathbf { } } v _ { i }$ ; (2) a classifier (a fully-connected layer followed by softmax) that receives ${ \\mathbf { } } v _ { i }$ as input and outputs class predictions; (3) a linear autoencoder that projects ${ \\bf \\nabla } _ { v _ { i } }$ into a low-dimensional embedding $\\dot { z } _ { i } \\in \\mathbb { R } ^ { d }$ . We show an illustration of our method in Figure $^ { 2 , }$ and a pseudo-code in appendix B. Next, we delineate its details. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 CONTRASTIVE LEARNING IN ROBUST LOW-DIMENSIONAL SUBSPACE ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $z _ { i } = \\mathbf { W } _ { \\mathrm { e } } { v } _ { i }$ be the linear projection from high-dimensional features to low-dimensional embeddings, and $\\hat { z } _ { i } = z _ { i } / \\left\\| z _ { i } \\right\\| _ { 2 }$ be the normalized embeddings. We aim to learn robust embeddings with two contrastive losses: unsupervised consistency loss and weakly-supervised mixup prototypical loss. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised consistency contrastive loss. Following the NT-Xent (Chen et al., 2020) loss for selfsupervised representation learning, our consistency contrastive loss enforces images with semanticpreserving perturbations to have similar embeddings. Specifically, given a miniwe apply weak-augmentation and strong-augmentation to each image, and obtain tch of input $b$ $2 b$ $\\{ \\pmb { x } _ { i } \\} _ { i = 1 } ^ { \\breve { 2 } b }$ Weak augmentation is a standard flip-and-shift augmentation strategy, while strong augmentation consists of color and brightness changes with details given in Section 4. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We project the inputs into the low-dimensional space to obtain their normalized embeddings $\\{ \\hat { z } _ { i } \\} _ { i = 1 } ^ { 2 b }$ Let be the index of a weakly-augmented input, and $j ( i )$ be the index of the strongaugmented input from the same source image, the consistency contrastive loss is defined as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            103,
+            774,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f7eb55fb1b2d86706c41435b72a94fae783caa0f25ecee2c1ad8db9b4069043.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { c c } } = \\sum _ { i = 1 } ^ { b } - \\log \\frac { \\exp ( \\hat { z } _ { i } \\cdot \\hat { z } _ { j ( i ) } / \\tau ) } { \\sum _ { k = 1 } ^ { 2 b } \\mathbb { 1 } _ { i \\neq k } \\exp ( \\hat { z } _ { i } \\cdot \\hat { z } _ { k } / \\tau ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            122,
+            648,
+            166
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ is a scalar temperature parameter. The consistency contrastive loss maximizes the inner product between the pair of positive embeddings $\\hat { z } _ { i }$ and $\\hat { z } _ { j ( i ) }$ , while minimizing the inner product between $2 ( b - 1 )$ pairs of negative embeddings. By mapping different views (augmentations) of the same image to neighboring embeddings, the consistency contrastive loss encourages the network to learn discriminative representation that is robust to low-level image corruption. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Weakly-supervised mixup prototypical contrastive loss. Our second contrastive loss injects structural knowledge of classes into the embedding space. Let $\\mathcal { T } _ { c }$ denote indices for the subset of images in $\\mathcal { D }$ labeled with class $c$ , we calculate the class prototype as the normalized mean embedding: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/10b306aed60e4511feb3b9104f8a39914a1c1b856e433dfc7100d89a3f0f78aa.jpg",
+        "text": "$$\nz ^ { c } = \\frac { 1 } { \\left| \\mathbb { Z } _ { c } \\right| } \\sum _ { i \\in \\mathbb { Z } _ { c } } \\hat { z } _ { i } , \\hat { z } ^ { c } = \\frac { z ^ { c } } { \\left\\| z ^ { c } \\right\\| _ { 2 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            292,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { z } _ { i }$ is the embedding of a center-cropped image, and the class prototypes are calculated at the beginning of each epoch. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The prototypical contrastive loss enforces an image embedding $\\hat { z } _ { i }$ to be more similar to its corresponding class prototype $\\hat { z } ^ { y _ { i } }$ , in contrast to other class prototypes: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a80b7782b2f036098044d74087f47f4c8faaed8b925f2fdcd9d3f696fd86006.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { p c } } ( \\hat { z } _ { i } , y _ { i } ) = - \\log \\frac { \\exp ( \\hat { z } _ { i } \\cdot \\hat { z } ^ { y _ { i } } / \\tau ) } { \\sum _ { c = 1 } ^ { C } \\exp ( \\hat { z } _ { i } \\cdot \\hat { z } ^ { c } / \\tau ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            404,
+            642,
+            441
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since the label $y _ { i }$ is noisy, we would like to regularize the encoder from memorizing training labels. Mixup (Zhang et al., $\\boxed { 2 0 1 8 }$ has been shown to be an effective method against label noise (Arazo et al., $\\mathord { \\left. \\kern - delimiterspace \\right)} \\frac { 2 0 1 9 } { \\mathrm { ~ L i ~ e t ~ a l . } } \\mathord { \\left[ 2 0 2 0 \\mathrm { a } \\right] } $ . Inspired by it, we create virtual training samples by linearly interpolating a sample (indexed by $i$ ) with another sample (indexed by $m ( i ) .$ ) randomly chosen from the same minibatch: ",
+        "bbox": [
+            173,
+            450,
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+            520
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8348dc8f0d599f75944d6f2437abf3cbddb975623be1a7079eb875982c1c237.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { x } _ { i } ^ { m } = \\lambda \\pmb { x } _ { i } + ( 1 - \\lambda ) \\pmb { x } _ { m ( i ) } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            520,
+            591,
+            537
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda \\sim \\operatorname { B e t a } ( \\alpha , \\alpha )$ ",
+        "bbox": [
+            174,
+            539,
+            328,
+            554
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let $\\hat { z } _ { i } ^ { m }$ be the normalized embedding for $\\pmb { x } _ { i } ^ { m }$ , the mixup version of the prototypical contrastive loss is defined as a weighted combination of the two ${ \\mathcal { L } } _ { \\mathrm { p c } }$ w.r.t class $y _ { i }$ and $y _ { m ( i ) }$ . It enforces the embedding for the interpolated input to have the same linear relationship $w . r . t .$ . the class prototypes. ",
+        "bbox": [
+            174,
+            560,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/375d9036a21353e296d4c1470bdab269fdeb1b654baa48527d3ec5a44f2277fc.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { p c . m i x } } = \\sum _ { i = 1 } ^ { 2 b } \\lambda \\mathcal { L } _ { \\mathrm { p c } } ( \\hat { z } _ { i } ^ { m } , y _ { i } ) + ( 1 - \\lambda ) \\mathcal { L } _ { \\mathrm { p c } } ( \\hat { z } _ { i } ^ { m } , y _ { m ( i ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            607,
+            684,
+            651
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Reconstruction loss. We also train a linear decoder $\\mathbf { W } _ { \\mathrm { d } }$ to reconstruct the high-dimensional feature ${ \\mathbf { } } v _ { i }$ based on $z _ { i }$ . The reconstruction loss is defined as: ",
+        "bbox": [
+            171,
+            661,
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+            689
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a0b1c7df2ccdb44a545abfc79bfe00fe1eae93d9f33b4be3a3e404b8169b2dfa.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { r e c o n } } = \\sum _ { i = 1 } ^ { 2 b } \\left\\| \\pmb { v } _ { i } - \\mathbf { W } _ { \\mathrm { d } } \\pmb { z } _ { i } \\right\\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            693,
+            599,
+            737
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "There are several benefits for training the autoencoder. First, with an optimal linear autoencoder, ${ \\bf W } _ { \\mathrm { e } }$ will project ${ \\mathbf { } } v _ { i }$ into its low-dimensional principal subspace and can be understood as applying PCA $\\mathrm { ( \\mathbb { B } a l d i \\ \\& \\ H o r n i k | , \\frac { } { | 9 8 9 ) } }$ Thus the low-dimensional representation $z _ { i }$ is intrinsically robust to input noise. Second, minimizing the reconstruction error is maximizing a lower bound of the mutual information between ${ \\mathbf { } } v _ { i }$ and $z _ { i }$ (Vincent et al., $\\boxed { 2 0 1 0 }$ . Therefore, knowledge learned from the proposed contrastive losses can be maximally preserved in the high-dimensional representation, which helps regularize the classifier. ",
+        "bbox": [
+            173,
+            746,
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+            845
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Classification loss. Given the softmax output from the classifier, ${ \\pmb p } ( { \\pmb y } ; { \\pmb x } _ { i } )$ , we define the classification loss as the cross-entropy loss. Note that it is only applied to the weakly-augmented inputs. ",
+        "bbox": [
+            171,
+            852,
+            823,
+            881
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1b8646e30548293f6065dfedb26b690d9949fc7e349c1a63cbde5875cfe0e99e.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { c e } } = - \\sum _ { i = 1 } ^ { b } \\log p ( y _ { i } ; \\pmb { x } _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            410,
+            885,
+            588,
+            928
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/d37d4c61b89760d47df7abc4d1404e044ee37a5e9c3d002a73d196c2e4369eb5.jpg",
+        "image_caption": [
+            "Figure 3: Curriculum learned by the proposed label correction method for training on CIFAR datasets with $50 \\%$ sym. noise. (a) Accuracy of pseudo-labels w.r.t to clean training labels. (b) Number of samples in the weakly-supervised subset $\\mathcal { D } _ { \\mathrm { s u p } } ^ { t }$ . (c) Label noise ratio in the weakly-supervised subset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            102,
+            821,
+            256
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The overall training objective is to minimize a weighted sum of all losses: ",
+        "bbox": [
+            176,
+            315,
+            656,
+            330
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/680bbf7c3e98744943bc2d460cf6354d60da7d257db7ffd79ea09e49f0e20207.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { \\mathrm { c e } } + \\omega _ { \\mathrm { c c } } \\mathcal { L } _ { \\mathrm { c c } } + \\omega _ { \\mathrm { p c } } \\mathcal { L } _ { \\mathrm { p c . m i x } } + \\omega _ { \\mathrm { r e c o n } } \\mathcal { L } _ { \\mathrm { r e c o n } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            337,
+            661,
+            356
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For all experiments, we fix $\\omega _ { \\mathrm { c c } } = 1$ , $\\omega _ { \\mathrm { r e c o n } } = 1$ , and change $\\omega _ { \\mathrm { p c } }$ only across datasets. ",
+        "bbox": [
+            176,
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+            385
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 NOISE CLEANING WITH SMOOTH NEIGHBORS",
+        "text_level": 1,
+        "bbox": [
+            174,
+            400,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "After warming-up the model by training with the noisy labels $\\{ y _ { i } \\} _ { i = 1 } ^ { n }$ for $t _ { 0 }$ epochs, we aim to clean the noise by generating a soft pseudo-label $\\pmb q _ { i }$ for each training sample. Different from previous methods that perform label correction purely using the model’s softmax prediction, our method exploits the structure of the low-dimensional subspace by aggregating information from top- $k$ neighboring samples, which helps alleviate the confirmation bias problem. ",
+        "bbox": [
+            173,
+            425,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "At the $t$ -th epoch, for each sample $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ , let $\\mathbf { \\Delta } _ { p _ { i } ^ { t } } ^ { t }$ be the classifier’s softmax prediction, let $\\pmb q _ { i } ^ { t - 1 }$ be its soft label from the previous epoch, we calculate the soft label for the current epoch as: ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3322cd0af1a38a3ce114078ea6eb186ea0b07a20a22a3395f7e5be470fe4d9c4.jpg",
+        "text": "$$\n\\pmb { q } _ { i } ^ { t } = \\frac { 1 } { 2 } \\pmb { p } _ { i } ^ { t } + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { k } w _ { i j } ^ { t } \\pmb { q } _ { j } ^ { t - 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            535,
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+            580
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { i j } ^ { t }$ represents the normalized affinity between a sample and its neighbor and is defined as $\\begin{array} { r } { w _ { i j } ^ { t } = \\frac { \\mathrm { { ' } } \\exp ( \\hat { z } _ { i } ^ { t } \\cdot \\hat { z } _ { j } ^ { t } / \\tau ) } { \\sum _ { j = 1 } ^ { k } \\exp ( \\hat { z } _ { i } ^ { t } \\cdot \\hat { z } _ { j } ^ { t } / \\tau ) } } \\end{array}$ . We set $k = 2 0 0$ in all experiments. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The soft label defined by eqn. $( 9 )$ is the minimizer of the following quadratic loss function: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/615e008559a184f8c3c2432b1502894b4bd7542f7124e04c3890d7ee4785d0c7.jpg",
+        "text": "$$\nJ ( { \\pmb q } _ { i } ^ { t } ) = \\sum _ { j = 1 } ^ { k } w _ { i j } ^ { t } \\left\\| { \\pmb q } _ { i } ^ { t } - { \\pmb q } _ { j } ^ { t - 1 } \\right\\| _ { 2 } ^ { 2 } + \\left\\| { \\pmb q } _ { i } ^ { t } - { \\pmb p } _ { i } ^ { t } \\right\\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            705
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The first term is a smoothness constraint which encourages the soft label to take a similar value as its neighbors’ labels, whereas the second term attempts to maintain the model’s class prediction. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We construct a weakly-supervised subset which contains (1) clean sample whose soft label score for the original class $y _ { i }$ is higher than a threshold $\\eta _ { 0 }$ , (2) pseudo-labeled sample whose maximum soft label score exceeds a threshold $\\eta _ { 1 }$ . For pseudo-labeled samples, we convert their soft labels into hard labels by taking the class with the maximum score. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cd6f9d3dd779b00d937d58b34b838797ca8ff3f171a9a3329912bf47f3e09436.jpg",
+        "text": "$$\n\\mathcal { D } _ { \\operatorname* { s u p } } ^ { t } = \\{ \\alpha _ { i } , y _ { i } \\mid q _ { i } ^ { t } ( y _ { i } ) > \\eta _ { 0 } \\} \\cup \\{ x _ { i } , \\hat { y } _ { i } ^ { t } = \\arg \\operatorname* { m a x } _ { c } q _ { i } ^ { t } ( c ) \\mid \\forall \\operatorname* { m a x } _ { c } q _ { i } ^ { t } ( c ) > \\eta _ { 1 } , c \\in \\{ 1 , . . , C \\} \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given the weakly-supervised subset, we modify the classification loss $\\mathcal { L } _ { \\mathrm { c e } }$ , the mixup prototypical contrastive loss $\\mathcal { L } _ { \\mathrm { p c . m i x } }$ , and the calculation of prototypes $\\hat { z } ^ { c }$ , such that they only use samples from $\\mathcal { D } _ { \\mathrm { s u p } } ^ { t }$ . The unsupervised losses (i.e. ${ \\mathcal { L } } _ { \\mathrm { c c } }$ and $\\mathcal { L } _ { \\mathrm { r e c o n } }$ ) still operate on all training samples. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Learning curriculum. Our iterative noise cleaning method learns an effective training curriculum, which gradually increases the size of $\\mathcal { D } _ { \\mathrm { s u p } } ^ { t }$ as the pseudo-labels become more accurate. To demonstrate such curriculum, we analyse the noise cleaning statistics for training our model on CIFAR-10 and CIFAR-100 datasets with $50 \\%$ label noise (experimental details explained in the next section). In Figure $\\bigtriangledown$ (a), we show the accuracy of the soft pseudo-labels $w . r . t$ to clean training labels (only used for analysis purpose). Our method can significantly reduce the ratio of label noise from $50 \\%$ to $5 \\%$ (for CIFAR-10) and $17 \\%$ (for CIFAR-100). Figure $\\dot { 3 }$ (b) shows the size of $\\mathcal { D } _ { \\mathrm { s u p } } ^ { t }$ as a percentage of the total number of training samples, and Figure $\\boxed { 3 }$ (c) shows the effective label noise ratio within the weakly-supervised subset $\\mathcal { D } _ { \\mathrm { s u p } } ^ { t }$ . Our method maintains a low noise ratio in the weakly-supervised subset, while gradually increasing its size to utilize more samples for the weakly-supervised losses. ",
+        "bbox": [
+            173,
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f8a4d188940af3f9de9d2c06aa0eb042229452bdef87bae96045d29702b74c82.jpg",
+        "table_caption": [
+            "Table 1: Comparison with state-of-the-art methods on CIFAR datasets with label noise. Numbers indicate average test accuracy $( \\% )$ over last 10 epochs. We report results over 3 independent runs with randomly-generated label noise. Results for previous methods are copied from Arazo et al. $\\boxed { 2 0 1 9 }$ ; Li et al. $\\textcircled { 2 0 2 0 2 }$ We re-run DivideMix (without ensemble) using the publicly available code on the same noisy data as ours. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=6>Dataset                                          CIFAR-10                      CIFAR-100Noise type                         Sym 20% Sym 50%  Asym 40%  Sym 20% Sym 50%</td></tr><tr><td rowspan=9 colspan=1>Cross-Entropy (Li et al. 2020a)ForwardPatrini et al.12017Co-teaching+ (Yu et al.12019Mixup(Zhang et al.2018) P-correcuon (Yi&amp; Wu [2019)MLNTLi et al2019)M-correcion (Arazo et al]2019)DivideMix (Li et al.2020a)DivideMix (reproduced)</td><td rowspan=9 colspan=1>82.783.188.292.392.092.093.895.095.1±0.1</td><td rowspan=1 colspan=1>57.9</td><td rowspan=1 colspan=1>72.3</td><td rowspan=1 colspan=1>61.8</td><td rowspan=1 colspan=1>37.3</td></tr><tr><td rowspan=1 colspan=1>59.4</td><td rowspan=1 colspan=1>83.1</td><td rowspan=1 colspan=1>61.4</td><td rowspan=1 colspan=1>37.3</td></tr><tr><td rowspan=1 colspan=1>84.1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>64.1</td><td rowspan=1 colspan=1>45.3</td></tr><tr><td rowspan=1 colspan=1>77.6</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>66.0</td><td rowspan=1 colspan=1>46.6</td></tr><tr><td rowspan=1 colspan=1>88.7</td><td rowspan=1 colspan=1>88.1</td><td rowspan=1 colspan=1>68.1</td><td rowspan=1 colspan=1>56.4</td></tr><tr><td rowspan=1 colspan=1>88.8</td><td rowspan=1 colspan=1>88.6</td><td rowspan=1 colspan=1>67.7</td><td rowspan=1 colspan=1>58.0</td></tr><tr><td rowspan=3 colspan=1>91.993.793.6±0.2</td><td rowspan=1 colspan=1>86.3</td><td rowspan=1 colspan=1>73.4</td><td rowspan=1 colspan=1>65.4</td></tr><tr><td rowspan=1 colspan=1>91.4</td><td rowspan=1 colspan=1>74.8</td><td rowspan=1 colspan=1>72.1</td></tr><tr><td rowspan=1 colspan=1>91.3±0.8</td><td rowspan=1 colspan=1>75.1±0.2</td><td rowspan=1 colspan=1>72.1±0.3</td></tr><tr><td rowspan=1 colspan=1>Ours (classifier)Ours (knn)</td><td rowspan=1 colspan=1>95.8±0.195.9±0.1</td><td rowspan=1 colspan=1>94.3±0.294.5±0.1</td><td rowspan=1 colspan=1>91.9±0.892.4±0.9</td><td rowspan=1 colspan=1>79.1±0.179.4±0.1</td><td rowspan=1 colspan=1>74.8±0.475.0±0.4</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we validate the proposed method on multiple benchmarks with controlled noise and real-world noise. Our method achieves state-of-the-art performance across all benchmarks. For fair comparison, we compare with DivideMix $\\mathbb { ( L i ~ e t ~ a l . ) } \\index { [ 2 0 2 0 4 ) }$ without ensemble. In appendix A, we report the result of our method with co-training and ensemble, which further improves performance. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTS ON CONTROLLED NOISY LABELS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Dataset. Following Tanaka et al. (2018); Li et al. $\\textcircled { 1 2 0 2 0 } \\textcircled { 2 }$ we corrupt the training data of CIFAR10 and CIFAR-100 (Krizhevsky & Hinton, $\\textcircled { 2 0 0 9 }$ with two types of label noise: symmetric and asymmetric. Symmetric noise is injected by randomly selecting a percentage of samples and changing their labels to random labels. Asymmetric noise is class-dependant, where labels are only changed to similar classes (e.g. dog cat, deer horse). We experiment with multiple noise ratios: sym $20 \\%$ , sym $50 \\%$ , and asym $40 \\%$ (see results for sym $80 \\%$ and $90 \\%$ in appendix $\\mathbf { A } )$ . Note that asymmetric noise ratio cannot exceed $50 \\%$ because certain classes would become theoretically indistinguishable. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Implementation details. Same as previous works $\\left( \\mathrm { A r a z o ~ e t ~ a l . } \\right) \\left[ 2 0 1 9 \\right. \\mathrm { [ L i ~ e t ~ a l . ] } \\left. 2 0 2 0 \\mathrm { a } \\right.$ we use PreAct ResNet-18 (He et al., $\\boxed { 2 0 1 6 }$ as our encoder model. We set the dimensionality of the bottleneck layer as $d = 5 0$ . Our model is trained using SGD with a momentum of 0.9, a weight decay of 0.0005, and a batch size of 128. The network is trained for 200 epochs. We set the initial learning rate as 0.02 and use a cosine decay schedule. We apply standard crop and horizontal flip as the weak augmentation. For strong augmentation, we use AugMix $( \\mathrm { \\overline { { H e n d r y c k s \\ e t { a l . } } } } , \\mathrm { \\overline { { 2 0 2 0 } } } )$ , though other methods (e.g. SimAug (Chen et al., $\\boxed { 2 0 2 0 }$ ) work equally well. For all CIFAR experiments, we fix the hyper-parameters as $\\omega _ { \\mathrm { c c } } = 1 , \\omega _ { \\mathrm { p c } } = 5$ , $\\omega _ { \\mathrm { r e c o n } } = 1 , \\tau = 0 . 3 , \\alpha = 8 , \\eta _ { 1 } = 0 . 9$ . For CIFAR-10, we activate noise cleaning at epoch $t _ { 0 } = 5$ , and set $\\eta _ { 0 } = 0 . 1$ (sym.) or 0.4 (asym.). For CIFAR-100, we activate noise cleaning at epoch $t _ { 0 } = 1 5$ , and set $\\eta _ { 0 } = 0 . 0 2$ . We use faiss-gpu (Johnson et al., 2017) for efficient knn search in the low-dimensional subspace, which finishes within 1 second. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Results. Table $^ 1$ shows the comparison with existing methods. Our method outperforms previous methods across all label noise settings. On the more challenging CIFAR-100, we achieve $3- 4 \\%$ accuracy improvement compared to the second-best method DivideMix. Moreover, our method is more computational efficient than DivideMix, which needs co-training for noise filtering. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/0251f502128246396fbc2f30b3b68d174004c1413f584f6403b329828842819e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">CIFAR-10 50% sym. noise</td><td rowspan=\"2\">CE</td><td>Iterative</td><td>GCE</td><td rowspan=\"2\">2018</td><td rowspan=\"2\">DivideMix (Li et al. 2020a</td><td rowspan=\"2\">Ours (cls.)</td><td rowspan=\"2\">Ours (knn)</td></tr><tr><td>Wang et al. 2018</td><td>Zhang &amp; Sabuncu</td></tr><tr><td>+ CIFAR-100 20k</td><td>53.6</td><td>87.2</td><td>87.3</td><td></td><td>89.0</td><td>91.5</td><td>93.1±0.3</td></tr><tr><td>+ SVHN 20k</td><td>58.1</td><td>88.6</td><td>88.8</td><td></td><td>91.9</td><td>93.3</td><td>93.9±0.2</td></tr><tr><td>+ Image Corruption</td><td>53.8</td><td>87.7</td><td>87.9</td><td></td><td>89.8</td><td>91.4</td><td>91.6±0.2</td></tr></table>",
+        "bbox": [
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+            101,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Comparison with state-of-the-art methods on datasets with label noise and input noise. Numbers indicate average test accuracy $( \\% )$ over last 10 epochs. We report results over 3 independent runs with randomlygenerated noise. We re-run previous methods using publicly available code with the same noisy data and model architecture as ours. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In order to demonstrate the advantage of the proposed low-dimensional embeddings, we perform $k$ - nearest neighbor (knn) classification ( $k = 2 0 0$ ), by projecting test images into normalized embeddings. Compared to the trained classifier, knn achieves higher accuracy, which verifies the robustness of the learned low-dimensional representations. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTS ON CONTROLLED NOISY LABELS WITH NOISY IMAGES ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Dataset. We further corrupt a noisy CIFAR-10 dataset (sym. $50 \\%$ ) by injecting two types of input noise: out-of-distribution (OOD) images and input corruption. For OOD noise, we follow Wang et al. (2018) and add $2 0 k$ images from either one of the two other datasets: CIFAR-100 and SVHN (Netzer $\\boxed { \\mathrm { e t ~ a l . } , \\boxed { 2 0 1 1 } }$ , enlarging the training set to $7 0 k$ . A random CIFAR-10 label is assigned to each OOD image. For input corruption, we follow Hendrycks & Dietterich $\\textcircled { 2 0 1 9 }$ and corrupt each image in CIFAR-10 with a noise randomly chosen from the following four types: Fog, Snow, Motion blur and Gaussian noise. Examples of both types of input noise are shown in Figure 4. We follow the same implementation details as the CIFAR-10 experiments described in Section 4.1. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/ac822823753a2fac58f6037c832534dcce1705ad8d63b701e0143ca0a0dbf8a7.jpg",
+        "image_caption": [
+            "Figure 4: Examples of input noise injected to CIFAR-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results. Table 2 shows the results, where our method consistently outperforms existing methods by a substantial margin. We observe that OOD images from a similar domain (CIFAR-100) are more harmful than OOD images from a more different domain (SVHN). This is because noisy images that are closer to the test data distribution are more likely to distort the decision boundary in a way that negatively affects test performance. Nevertheless, performing knn classification using the learned embeddings demonstrates high robustness to input noise. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Figure $5 ,$ we show the t-SNE (Maaten & Hinton, 2008) visualization of the low-dimensional embeddings for all training samples. As training progresses, our model learns to separate OOD samples (represented as gray points) from in-distribution samples, and cluster samples of the same class together despite their noisy labels. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/7fcaf6998097aa975e3eac8c47e1d176c9a4ea41a30d28612dbc18b3e6873bec.jpg",
+        "image_caption": [
+            "Figure 5: t-SNE visualization of low-dimensional embeddings for CIFAR-10 images (color represents the true class) $+ \\mathrm { O O D }$ images (gray points) from CIFAR-100 or SVHN. The model is trained on noisy CIFAR-10 $5 0 k$ images with $50 \\%$ label noise) and $2 0 k$ OOD images with random labels. Our method can effectively learn to (1) cluster CIFAR-10 images according to their true class, despite their noisy labels; (2) separate OOD samples from in-distribution samples, such that their harm is reduced. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/2c35f29fc3f32546e2d2ca435e23290f87290ba8ead775c4998ba9a5aa65cadc.jpg",
+        "table_caption": [
+            "Table 3: Comparison with state-of-the-art methods trained on WebVision (mini). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Test dataset</td><td colspan=\"2\">WebVision</td><td colspan=\"2\">ILSVRC12</td></tr><tr><td colspan=\"2\">Accuracy (%)</td><td>top1</td><td>top5</td><td>top1</td><td>top5</td></tr><tr><td colspan=\"2\"> Forward (Patrini et al.2017)</td><td>61.1</td><td>82.7</td><td>57.4</td><td>82.4</td></tr><tr><td colspan=\"2\">Decoupling (Malach &amp; Shalev-Shwartz 12017</td><td>62.5</td><td>84.7</td><td>58.3</td><td>82.3</td></tr><tr><td colspan=\"2\">D2L (Ma et al12018)</td><td>62.7</td><td>84.0</td><td>57.8</td><td>81.4</td></tr><tr><td colspan=\"2\">MentorNet (Jiang et al. 2018</td><td>63.0</td><td>81.4</td><td>57.8</td><td>79.9</td></tr><tr><td colspan=\"2\">Co-teaching (Han et al. 2018)</td><td>63.6</td><td>85.2</td><td>61.5</td><td>84.7</td></tr><tr><td colspan=\"2\">INCV (Chen et al.,2019)</td><td>65.2</td><td>85.3</td><td>61.0</td><td>85.0</td></tr><tr><td colspan=\"2\">DivideMix (Li et al. 12020a)</td><td>75.9</td><td>90.1</td><td>73.3</td><td>89.2</td></tr><tr><td colspan=\"2\">Ours (w/o noise cleaning)</td><td>75.5</td><td>90.2</td><td>72.0</td><td>90.0</td></tr><tr><td colspan=\"2\">Ours (classifier)</td><td>76.3</td><td>91.5</td><td>73.3</td><td>91.2</td></tr><tr><td colspan=\"2\">Ours (knn)</td><td>77.8</td><td>91.3</td><td>74.4</td><td>90.9</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4c59545cb2f0a54f1cda14b14c5f9abd004a624f95e412d4af14317acca6a416.jpg",
+        "table_caption": [
+            "Table 4: Comparison with state-of-the-art methods on Clothing1M dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>CE</td><td>Forward</td><td>Joint-Opt</td><td>MLNT</td><td>MentorMix</td><td>SL</td><td>DivideMix</td><td>Ours (cls.）</td><td>Ours (knn)</td></tr><tr><td>Accuracy</td><td>69.21</td><td>69.84</td><td>72.16</td><td>73.47</td><td>74.30</td><td>74.45</td><td>74.48</td><td>74.84</td><td>74.97</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 EXPERIMENTS ON REAL-WORLD NOISY DATA ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            414,
+            529,
+            428
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Dataset and implementation details. We verify our method on two real-word noisy datasets: WebVision $\\mathrm { ( } \\mathbf { L i } \\mathrm { e t a l . } \\mathrm { , } \\overline { { 2 0 1 7 } } \\mathrm { ) }$ and Clothing1M (Xiao et al., 2015). Webvision contains images crawled from the web using the same concepts from ImageNet ILSVRC12 $( \\mathbb { D e n g \\ e t \\ a l . } ) . 2 0 0 9 )$ . Following previous works (Chen et al., 2019; Li et al., 2020a), we perform experiments on the first 50 classes of the Google image subset. Clothing1M consists of images collected from online shopping websites where labels were generated from surrounding texts. Note that we do not use the additional clean set for training. For both experiments, we use the same model architecture as previous methods. More implementation details are given in the appendix. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results. We report the results for WebVision in Table $\\textcircled { 3 }$ and Clothing1M in Table $4 ,$ where we achieve state-of-the-art performance on both datasets. Our method achieves competitive performance on WebVision even without performing noise cleaning, which demonstrates the robustness of the learned representation. Appendix D shows examples of noisy images that are cleaned by our method. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            627,
+            339,
+            641
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Effect of the proposed components. In order to study the effect of the proposed components, we remove each of them and report accuracy of the classifier (knn) across four benchmarks. As shown in Table $\\boxed { 5 }$ the mixup prototypical contrastive loss $( \\mathcal { L } _ { \\mathrm { p c . m i x } } )$ is most crucial to the model’s performance. The consistency contrastive loss $( \\mathcal { L } _ { \\mathrm { c c } } )$ has a stronger effect with corrupted input or larger number of classes. We also experiment with removing mixup and using the standard prototypical contrastive loss, and using standard data augmentation (crop and horizontal flip) instead of AugMix. The proposed method still achieves state-of-the-art result with standard data augmentation. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4ceeab743d30b1b04118d2ab693327fed8fedc05a44164228804334107bd0b1f.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 5: Effect of the proposed components. We show the accuracy of the classifier (knn) on four benchmarks with different noise. Note that DivideMix (Li et al., 2020a) also performs mixup. "
+        ],
+        "table_body": "<table><tr><td></td><td>| CIFAR-10 Sym 50%</td><td>6|+CIFAR-10020k</td><td>+ Image Corruption</td><td>CIFAR-100 Sym 50%</td></tr><tr><td>w/o Lpc_mix</td><td>85.9 (86.1)</td><td>79.7 (81.5)</td><td>81.6 (81.7)</td><td>65.6 (65.9)</td></tr><tr><td>w/o Lcc</td><td>93.7 (93.8)</td><td>91.3 (91.5)</td><td>89.4 (89.5)</td><td>71.9 (71.8)</td></tr><tr><td>w/o Lrecon</td><td>93.3 (94.0)</td><td>90.7 (92.9)</td><td>90.2 (91.0)</td><td>73.2 (73.9)</td></tr><tr><td>w/o mixup</td><td>89.5 (89.9)</td><td>85.4 (87.0)</td><td>84.7 (84.9)</td><td>69.3 (69.7)</td></tr><tr><td> w/ standard aug.</td><td>94.1 (94.3)</td><td>90.8 (92.9)</td><td>90.5 (90.7)</td><td>74.5 (75.0)</td></tr><tr><td>DivideMix</td><td>93.6</td><td>89.0</td><td>89.8</td><td>72.1</td></tr><tr><td>Ours</td><td>94.3 (94.5)</td><td>91.5 (93.1)</td><td>91.4 (91.6)</td><td>74.8 (75.0)</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/e1e815f2b57f32da5fb5e656e68a68f69bdeecf860fb3fa5907c1d9d43f8f672.jpg",
+        "table_caption": [
+            "Effect of bottleneck dimension. We vary the dimensionality of the bottleneck layer, $d$ , and examine the performance change in Table $\\boxed { 6 }$ Our model is in general not very sensitive to the change of $d$ . ",
+            "Table 6: Classifier’s test accuracy $( \\% )$ with different low-dimensions. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>bottleneck dimension |d= 25|d= 50丨d=100|d = 200</td><td></td><td></td><td></td><td></td></tr><tr><td>CIFAR-10 Sym 50%</td><td>93.4</td><td>94.3</td><td>94.2</td><td>93.7</td></tr><tr><td>CIFAR-100 Sym 50%</td><td>73.8</td><td>74.8</td><td>74.4</td><td>73.8</td></tr></table>",
+        "bbox": [
+            292,
+            145,
+            699,
+            198
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+            174,
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+            320,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper proposes noise-robust contrastive learning, a new method to combat noise in training data by learning robust representation. We demonstrate our model’s state-of-the-art performance with extensive experiments on multiple noisy datasets. For future work, we are interested in adapting our method to other domains such as NLP or speech. We would also like to explore the potential of our method for learning transferable representations that could be useful for down-stream tasks. ",
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+        "page_idx": 8
+    },
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parse/train/HyxFF34FPr/HyxFF34FPr.md

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+# FOVEABOX: BEYOUND ANCHOR-BASED OBJECT DETECTION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We present FoveaBox, an accurate, flexible, and completely anchor-free framework for object detection. While almost all state-of-the-art object detectors utilize predefined anchors to enumerate possible locations, scales and aspect ratios for the search of the objects, their performance and generalization ability are also limited to the design of anchors. Instead, FoveaBox directly learns the object existing possibility and the bounding box coordinates without anchor reference. This is achieved by: (a) predicting category-sensitive semantic maps for the object existing possibility, and (b) producing category-agnostic bounding box for each position that potentially contains an object. The scales of target boxes are naturally associated with feature pyramid representations. We demonstrate its effectiveness on standard benchmarks and report extensive experimental analysis. Without bells and whistles, FoveaBox achieves state-of-the-art single model performance on the standard COCO detection benchmark. More importantly, FoveaBox avoids all computation and hyper-parameters related to anchor boxes, which are often sensitive to the final detection performance. We believe the simple and effective approach will serve as a solid baseline and help ease future research for object detection.
+
+# 1 INTRODUCTION
+
+Object detection requires the solution of two main tasks: recognition and localization. Given an arbitrary image, an object detection system needs to determine whether there are any instances of semantic objects from predefined categories and, if present, to return the spatial location and extent. To add the localization functionality to generic object detection systems, sliding window approaches have been the method of choice for many years (Lampert et al., 2008; Felzenszwalb et al., 2010; Liu et al., 2018).
+
+Recently, deep learning techniques have emerged as powerful methods for learning feature representations automatically from data (Simonyan & Zisserman, 2014; He et al., 2016; Huang et al., 2017a). For object detection, the anchor-based Region Proposal Networks (Ren et al., 2015) are widely used to serve as a common component for searching possible regions of interest for modern object detection frameworks (Liu et al., 2016; He et al., 2017; Lin et al., 2018). In short, anchor method suggests dividing the box space into discrete bins and refining the object box in the corresponding bin. Most state-of-the-art detectors rely on anchors to enumerate the possible locations, scales, and aspect ratios for target objects (Liu et al., 2018). Anchors are regression references and classification candidates to predict proposals for two-stage detectors or final bounding boxes for single-stage detectors. Nevertheless, anchors can be regarded as a feature-sharing sliding window scheme to cover the possible locations of objects.
+
+![](images/ee0090c10ecb98878b6e27f6bf8f893fea6ad40f1a694c27deeeb73a6acadbf6.jpg)  
+Figure 1: The anchor-based object detection frameworks need to (a) design anchors according to the ground-truth box distributions; (b) match anchors with ground-truth boxes to generate training target (anchor classification and refinement); and (c) utilize the target generated by (b) for training.
+
+However, anchors must be carefully designed and used in object detection frameworks. (a) One of the most important factors in designing anchors is how densely it covers the instance location space. To achieve a good recall rate, anchors are carefully designed based on the statistics computed from the training/validation set (Lin et al., 2018). (b) One design choice based on a particular dataset is not always applicable to other applications, which harms the generality (Yang et al., 2018). (c) At training phase, anchor-methods rely on the intersection-over-union (IoU) to define the positive/negative samples, which introduces additional computation and hyper-parameters for an object detection system (Wang et al., 2019).
+
+In contrast, our human vision system can recognize the instance in space and predict the boundary given the visual cortex map, without any pre-defined shape template (Bear et al., 2007). In other words, we human naturally recognize the object in the visual scene without enumerating the candidate boxes. Inspired by this, an intuitive question to ask is, is the anchor scheme the optimal way to guide the search of objects? And further, could we design an accurate object detection framework without anchors or candidate boxes? Without anchors, one may expect a complex method is required to achieve comparable performance. However, we show that a surprisingly simple and flexible system can match, even surpass the prior state-of-the-art object detection results without any requirement of candidate boxes.
+
+![](images/8bc10160a13d4b5beae54d063f3736f43c31f1b08a0bcf27e4fd650d2e4b7115.jpg)  
+Figure 2: FoveaBox object detector. For each output spacial position that potentially presents an object, FoveaBox directly predicts the confidences for all target categories and the bounding box.
+
+To this end, we present FoveaBox, a completely anchor-free framework for object detection. FoveaBox is motivated from the fovea of human eyes: the center of the vision field is with the highest visual acuity (Fig.2 left), which is necessary for activities where visual detail is of primary importance (Iwasaki & Inomata, 1986). FoveaBox jointly predicts the locations where the object’s center area is likely to exist as well as the bounding box at each valid location.
+
+In FoveaBox, each target object is predicted by category scores at center area, associated with 4-d bounding box, as shown in Fig.2 right. At training phase, we do not need to utilize anchors, or IoU matching to generate training target. Instead, the training target is directly generated by ground-truth boxes.
+
+To demonstrate the effectiveness of the proposed detection scheme, we combine the recent progress of feature pyramid networks and our detection head to form the framework of FoveaBox. Without bells and whistles, FoveaBox gets state-of-the-art single-model results on the COCO object detection task. Compared with the anchor-based RetinaNet, FoveaBox gets 2.2 AP gains, which also surpasses most of previously published anchor based single-model results. We believe the simple training/inference manner of FoveaBox, together with the flexibility and accuracy, will benefit future research on object detection and relevant topics.
+
+# 2 FOVEABOX
+
+FoveaBox is conceptually simple: It contains a backbone network and a fovea head network. The backbone is responsible for computing a convolutional feature map over an entire input image and is an off-the-shelf convolutional network. The fovea head is composed of two sub-branches, the first branch performs per pixel classification on the backbone’s output; the second branch performs box prediction for each position that potentially covered by an object.
+
+# 2.1 REVIEW OF FPN AND ANCHORS:
+
+We begin by briefly reviewing the Feature Pyramid Network (FPN) used for object detection (Lin et al., 2017). In general, FPN uses a top-down architecture with lateral connections to build an in-network feature pyramid from a single-scale input. FPN is independent of a specific task. For object detection, each level of the pyramid in FPN is used for detecting objects at a specific scale. On each feature pyramid, anchor-based methods uniformly place $A$ anchors on each of the $H \times W$ spacial position. After computing the IoU overlap between all anchors and the ground-truth boxes, the anchor-based methods can define training targets. Finally, the pyramid features are utilized to optimize the targets.
+
+![](images/dfe718192bcb34f1770531c855ee5fe6c07a0cef4ffd8a59e987bfc67478e369.jpg)  
+Figure 3: Anchor-based object detection v.s. FoveaBox object detection. left: The anchor-based method uniformly places $A$ ( $A = 3$ in this example) anchors on each output spacial position, and utilizes IoU to define the positive/negative anchors; right: FoveaBox directly define positive/negative samples for each output spacial position by ground-truth boxes, and predicts the box boundaries from the corresponding position.
+
+# 2.2 FOVEABOX
+
+FoveaBox directly predicts the object existing possibility and the corresponding boundary for each position potential contained by an instance. In this section, we introduce the key components stepby-step.
+
+# 2.2.1 OBJECT OCCURRENCE POSSIBILITY
+
+Given a valid ground-truth box denoted as $( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } )$ . We first map the box into the target feature pyramid $P _ { l }$
+
+$$
+\begin{array} { c } { { x _ { 1 } ^ { ' } = \frac { x _ { 1 } } { s _ { l } } , y _ { 1 } ^ { ' } = \frac { y _ { 1 } } { s _ { l } } , x _ { 2 } ^ { ' } = \frac { x _ { 2 } } { s _ { l } } , y _ { 2 } ^ { ' } = \frac { y _ { 2 } } { s _ { l } } , } } \\ { { { } } } \\ { { c _ { x } ^ { ' } = 0 . 5 ( x _ { 2 } ^ { ' } + x _ { 1 } ^ { ' } ) , c _ { y } ^ { ' } = 0 . 5 ( y _ { 2 } ^ { ' } + y _ { 1 } ^ { ' } ) , w ^ { ' } = x _ { 2 } ^ { ' } - x _ { 1 } ^ { ' } , h ^ { ' } = y _ { 2 } ^ { ' } - y _ { 1 } ^ { ' } , } } \end{array}
+$$
+
+where $s _ { l }$ is the down-sample factor. The positive area $R ^ { p o s }$ on the score map is designed to be roughly a shrunk version of the original one (Fig.3 right):
+
+$$
+\begin{array} { r } { x _ { 1 } ^ { p o s } = c _ { x } ^ { ' } - 0 . 5 \sigma w ^ { ' } , y _ { 1 } ^ { p o s } = c _ { y } ^ { ' } - 0 . 5 \sigma h ^ { ' } , } \\ { x _ { 2 } ^ { p o s } = c _ { x } ^ { ' } + 0 . 5 \sigma w ^ { ' } , y _ { 2 } ^ { p o s } = c _ { y } ^ { ' } + 0 . 5 \sigma h ^ { ' } , } \end{array}
+$$
+
+where $\sigma$ is the shrunk factor. At training phase, each cell inside the positive area is annotated with the corresponding target class label. The negative area is the whole feature map excluding area in $R ^ { p o s }$ . For predicting, each output set of pyramidal heat-map has $C$ channels, where $C$ is the number of categories, and is of size $H \times W$ . Each channel is a binary mask indicating the possibility for a class, like FCNs in semantic segmentation (Long et al., 2015). The positive area usually accounts for a small portion of the whole feature map, so we adopt Focal Loss (Lin et al., 2018) to train this branch.
+
+# 2.2.2 SCALE ASSIGNMENT
+
+While our goal is to predict the boundary of the target objects, directly predicting these numbers is not stable, due to the large scale variations of the objects. Instead, we divide the scales of objects into several bins, according to the number of feature pyramidal levels. Each pyramid has a basic scale $r _ { l }$ ranging from 32 to 512 on pyramid levels $P _ { 3 }$ to $P _ { 7 }$ , respectively. The valid scale range of the target boxes for pyramid level $l$ is computed as
+
+$$
+[ r _ { l } / \eta , r _ { l } \cdot \eta ] ,
+$$
+
+where $\eta$ is set empirically to control the scale range for each pyramid. Target objects not in the corresponding scale range are ignored during training. Note that an object may be detected by multiple pyramids of the networks, which is different from previous practice that maps objects to only one feature pyramid (He et al., 2017).
+
+# 2.2.3 BOX PREDICTION
+
+Each ground-truth bounding box is specified in the way $G \ : = \ : ( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } )$ . Starting from a positive point $( x , y )$ in $R ^ { p o s }$ , FoveaBox directly computes the normalized offset between $( x , y )$ and four boundaries:
+
+$$
+\begin{array} { l l } { { t _ { x } } _ { 1 } = \log \frac { s _ { l } ( x + 0 . 5 ) - x _ { 1 } } { r _ { l } } , t _ { y _ { 1 } } = \log \frac { s _ { l } ( y + 0 . 5 ) - y _ { 1 } } { r _ { l } } , } \\ { { t _ { x } } _ { 2 } = \log \frac { x _ { 2 } - s _ { l } ( x + 0 . 5 ) } { r _ { l } } , t _ { y _ { 2 } } = \log \frac { y _ { 2 } - s _ { l } ( y + 0 . 5 ) } { r _ { l } } . } \end{array}
+$$
+
+This function first maps the coordinate $( x , y )$ to the input image, then computes the normalized offset between the projected coordinate and $G$ . Finally the targets are regularized with the log-space function. $r _ { l }$ is the basic scale defined in section 2.2.2.
+
+For simplicity, we adopt the widely used Smooth $L _ { 1 }$ loss (Ren et al., 2015) to train the box prediction $L _ { b o x }$ . After targets being optimized, we can generate the box boundary for each cell $( x , y )$ on the output feature maps1. In box branch, each output set of pyramidal heatmap has 4 channels, for jointly prediction of $( t _ { x _ { 1 } } , t _ { y _ { 1 } } , t _ { x _ { 2 } } , t _ { y _ { 2 } } )$ .
+
+# 2.2.4 NETWORK ARCHITECTURE
+
+To demonstrate the generality of our approach, we instantiate FoveaBox with multiple architectures. For clarity, we differentiate between: (i) the convolutional backbone architecture used for feature extraction over an entire image, and (ii) the network head for computing the final results.
+
+![](images/862b43123997d3662af060569300731534469cd4e3d82efd5890e75b12ee4bee.jpg)  
+Figure 4: On each FPN feature level, FoveaBox attaches two subnetworks, one for classifying the corresponding cells and one for predict the $( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } )$ of ground-truth object box. Right is the score output map with their corresponding predicted boxes before feeding into non-maximum suppression (NMS). The score probability in each position is denoted by the color density. More examples are shown in Fig.5.
+
+Most of the experiments are based on the head architecture as shown in Fig.4. We also utilize different head variants to further study the generality. More complex designs have the potential to improve performance but are not the focus of this work.
+
+# 2.2.5 IMPLEMENTATIONS
+
+We adopt the widely used FPN networks for fair comparison. Concretely, we construct a pyramid with levels $\{ P _ { l } \} , l \ = \ 3 , 4 , \cdots , 7$ , where $l$ indicates pyramid level. $P _ { l }$ has $1 / 2 ^ { l }$ resolution of the input. All pyramid levels have $C = 2 5 6$ channels. Fovea head is attached on each pyramid level. Parameters are shared across all pyramid levels.
+
+FoveaBox is trained with stochastic gradient descent (SGD). We use synchronized SGD over 4 GPUs with a total of 16 images per minibatch (4 images per GPU). Unless otherwise specified, all models are trained for 12 epochs with an initial learning rate of 0.01, which is then divided by 10 at 8th and again at 11th epochs. Weight decay of 0.0001 and momentum of 0.9 are used. Only standard horizontal image flipping is used for data augmentation. During inference, we first use a confidence threshold of 0.05 to filter out predictions with low confidence. Then, we select the top 1000 scoring boxes from each prediction layer. Next, NMS with threshold 0.5 is applied for each class separately. Finally, the top-100 scoring predictions are selected for each image. Although there are more intelligent ways to perform post-processing, such as bbox voting (Gidaris & Komodakis, 2015), Soft-NMS (Bodla et al., 2017) or test-time image augmentations, in order to keep simplicity and to fairly compare against the baseline models, we do not use those tricks here.
+
+# 3 EXPERIMENTS
+
+We present experimental results on the bounding box detection track of the MS COCO benchmark. All models are trained on MS COCO trainval35k. If not specified, ResNet-50-FPN backbone and a 600 pixel train and test image scale are used to do the ablation study. We report lesion and sensitivity studies by evaluating on the minival split. For our main results, we report COCO AP on the test-dev split, which has no public labels and requires use of the evaluation server.
+
+# 3.1 ABLATION STUDY
+
+Various anchor densities and FoveaBox: One of the most important design factors in an anchorbased detection system is how densely it covers the space of possible objects. As anchor-based detectors use a fixed sampling grid, a popular approach for achieving high coverage of boxes is to use multiple anchors at each spatial position. One may expect that we can always get better performance when attaching denser anchors on each position. To verify this assumption, we sweep over the number of scale and aspect ratio anchors used at each spatial position and each pyramid level in RetinaNet, including a single square anchor at each location to 12 anchors per location (Table.1(a)). Increasing beyond 6-9 anchors does not show further gains. The saturation of performance w.r.t. density implies the handcrafted, over-density anchors do not offer an advantage.
+
+Over-density anchors not only increase the foreground-background optimization difficulty, but also likely to cause the ambiguous position definition problem. For each output spatial location, there are $A$ anchors whose labels are defined by the IoU with the ground-truth. Among them, some of the anchors are defined as positive samples, while others are negatives. However they are sharing the same input features. The classifier needs to not only distinguish the samples from different positions, but also different anchors at the same position.
+
+In contrast, FoveaBox explicitly predicts one target at each position and gets no worse performance than the best anchor-based model. Compare with the anchor based scheme, FoveaBox enjoys several advantages. (a) Since we only predict one target at each position, the output space has been reduced to $1 / A$ of the anchor-based method. (b) There is no ambiguous problem and the optimization target is more straightforward. (c) FoveaBox has fewer hyper-parameters, and is more flexible, since we do not need to extensively design anchors to see a relatively better choice.
+
+FoveaBox is more robust to box distribution: One of the major benefits of FoveaBox is the robust prediction of bounding boxes. To verify this, we divide the boxes in the validation set into three groups according to the ground-truth aspect ratios $\begin{array} { r } { u = \operatorname* { m a x } ( \frac { h } { w } , \frac { w } { h } ) } \end{array}$ . We compare FoveaBox and
+
+Table 1: Ablation experiments for FoveaBox. All models are trained on trainval35k, test on minival. If not specified, default values are $\eta = 2 . 0$ and $\sigma = 0 . 4$ . (a) Our anchor-free FoveaBox get $0 . 9 \mathrm { \ A P }$ gains compared with the best model of anchor-based RetinaNet; (b) FoveaBox is more robust to bounding box distributions; (c) FoveaBox could also generate high-quality region proposals; (d) Accuracy of FoveaBox for various network depths and image scales; (e) and (f): FoveaBox gets best performance with $\eta = 2 . 0$ and $\sigma = 0 . 4$ . See Section 3.1 for details.
+
+(a) Varying anchor density and FoveaBox.   
+
+<table><tr><td>method</td><td>#sc</td><td>#ar</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>RetinaNet</td><td>1</td><td>1</td><td>30.2</td><td>49.0</td><td>31.7</td></tr><tr><td>RetinaNet</td><td>2</td><td></td><td>31.9</td><td>50.0</td><td>34.1</td></tr><tr><td>RetinaNet</td><td>3</td><td></td><td>31.9</td><td>49.4</td><td>33.8</td></tr><tr><td>RetinaNet</td><td>2</td><td>１１3</td><td>34.2</td><td>53.1</td><td>36.5</td></tr><tr><td>RetinaNet</td><td>3</td><td>33</td><td>34.2</td><td>53.2</td><td>36.9</td></tr><tr><td>RetinaNet</td><td>4</td><td></td><td>33.9</td><td>52.1</td><td>36.2</td></tr><tr><td>FoveaBox</td><td>-</td><td>-</td><td>35.1</td><td>54.3</td><td>37.1</td></tr></table>
+
+(b) Detection with different aspect ratios   
+
+<table><tr><td>method</td><td>AP</td><td>APu&lt;3</td><td>AP3≤u&lt;5</td><td>APu&gt;5</td></tr><tr><td>RetinaNet</td><td>34.2</td><td>36.5</td><td>24.5</td><td>10.2</td></tr><tr><td>FoveaBox</td><td>35.1</td><td>36.8</td><td>26.8</td><td>16.4</td></tr></table>
+
+(c) Region proposal performance.   
+
+<table><tr><td>method</td><td>backbone</td><td>AR100</td><td>AR300</td><td>AR1000</td></tr><tr><td>RPN</td><td>ResNet-50</td><td>44.5</td><td>51.1</td><td>56.6</td></tr><tr><td>FoveaBox</td><td>ResNet-50</td><td>52.9</td><td>57.3</td><td>61.5</td></tr></table>
+
+(d) Different input resolutions and models.   
+(e) Varying η $( \sigma = 0 . 4 )$ . (f) Varying σ $( \eta = 2 . 0 )$ ).   
+
+<table><tr><td>net-depth-scale</td><td>AP</td><td>AP50</td><td>AP75</td><td>m</td><td>AP</td><td>AP50</td><td>AP75</td><td>0</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>FoveaBox-50-400</td><td>31.9+1.4</td><td>49.6</td><td>33.8</td><td>1.0</td><td>32.0</td><td>50.4</td><td>31.8</td><td>0.2</td><td>34.1</td><td>53.2</td><td>36.0</td></tr><tr><td>FoveaBox-50-600</td><td>35.1+0.9</td><td>54.3</td><td>37.1</td><td>1.5</td><td>34.1</td><td>53.3</td><td>36.0</td><td>0.3</td><td>34.8</td><td>54.0</td><td>36.7</td></tr><tr><td>FoveaBox-50-800</td><td>36.4+0.9</td><td>56.2</td><td>38.7</td><td>2.0</td><td>35.1</td><td>54.4</td><td>37.0</td><td>0.4</td><td>35.1</td><td>54.4</td><td>37.0</td></tr><tr><td>FoveaBox-101-400</td><td>33.3+1.4</td><td>51.0</td><td>35.0</td><td>2.5</td><td>35.0</td><td>54.2</td><td>36.8</td><td>0.5</td><td>34.8</td><td>53.9</td><td>36.6</td></tr><tr><td>FoveaBox-101-600</td><td>37.0+1.0</td><td>56.4</td><td>39.3</td><td>3.0</td><td>34.3</td><td>53.3</td><td>36.5</td><td>0.6</td><td>34.1</td><td>53.1</td><td>36.0</td></tr><tr><td>FoveaBox-101-800</td><td>38.6+0.9</td><td>58.0</td><td>41.2</td><td>4.0</td><td>32.8</td><td>51.0</td><td>34.5</td><td>0.7</td><td>33.3</td><td>52.5</td><td>34.9</td></tr></table>
+
+RetinaNet at different aspect ratio thresholds, as shown in Table 1(b). We see that both methods get best performance when $u$ is low. Although FoveaBox also suffers performance decrease when $u$ increases, it is much better than the baseline model.
+
+Generating high-quality region proposals: Changing the classification target to class-agnostic head is straightforward and could generate region proposals. We compare the proposal performance against FPN-based RPN (Lin et al., 2017) and evaluate average recalls (AR) with different numbers of proposals on minival set, as shown in Table 1(c). Surprisingly, our method outperforms the RPN baseline by a large margin, among all criteria. Specifically, with top 100 region proposals, FoveaBox gets 52.9 AR, outperforming RPN by 8.4 points. This validates that our model’s capacity in generating high quality region proposals.
+
+Across model depth and scale: Table 1(d) shows FoveaBox utilizing different backbone networks and input resolutions. The train/inference settings are exactly the same as the baseline method (Lin et al., 2018). Under the same settings, FoveaBox consistently gets $0 . 9 { \sim } 1 . 4$ higher AP. When comparing the inference speed, we find that FoveaBox models are about $1 . 1 { \sim } 1 . 3$ times faster than the RetinaNet counterparts.
+
+Analysis of $\eta$ and $\sigma$ : In Eq.(3), $\eta$ controls the scale assignment extent for each pyramid. As $\eta$ increases, each pyramid will response to more scales of objects. Table 1(e) shows the impact of $\eta$ on the final detection performance. Another important hyper-parameter is the shrunk factor $\sigma$ which controls the positive/negative samples. Table 1(f) shows the model performance with respect to $\sigma$ changes. In this paper, $\sigma = 0 . 4$ and $\eta = 2$ are used in other experiments.
+
+IoU-based assignment v.s. fovea area: Another choice of defining the positive/negative samples is firstly gets the predicted box from the box branch, and then assign the target labels based on the IoU between the predicted boxes and ground-truth boxes. As shown in Table 2, the shrunk version gets better performance $( + 0 . 4$ AP) than the IoU-based assignment process.
+
+Better head and feature alignment: The most recent works (Chen et al., 2019; Yang et al., 2019) suggest to align the features in one-stage object detection frameworks with anchors. In FoveaBox, we adopt deformable convolution (Dai et al., 2017) based on the box offset learned by Eq.(4) to refine the classification branch2. FoveaBox works well when adding such techniques. Specifically, when we change the classification branch to a heavier head, together with feature alignment and GN, FoveaBox gets 40.1 AP using ResNet-50 as backbone! This experiment demonstrates the generality of our approach to the network design (Table 3).
+
+Table 2: Label assignment strategy (ResNet-50, 800 scale).   
+
+<table><tr><td>assign method</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>IoU (0.5/0.4) IoU (0.6/0.5)</td><td>35.6 35.9</td><td>54.7 54.7</td><td>37.7 38.4</td></tr><tr><td>IoU (0.5/0.5)</td><td>36.0</td><td>54.9</td><td>38.4</td></tr><tr><td>Fovea (σ = 0.4)</td><td>36.4</td><td>56.2</td><td>38.7</td></tr></table>
+
+Table 3: Feature alignment and group normalization (ResNet-50, 800 scale).   
+
+<table><tr><td>cls branch</td><td>alignment</td><td>GN</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>256(3×3) -→256(3×3)-→256(3×3)-&gt;256(3×3)</td><td></td><td></td><td>36.4</td><td>56.2</td><td>38.7</td></tr><tr><td>256(3×3)→&gt;256(3×3)-&gt;256(3×3)-→256(3×3)</td><td>√</td><td></td><td>36.8</td><td>56.5</td><td>38.9</td></tr><tr><td>256(3×3)-→256(3×3)-→256(3×3)-&gt;256(3×3)</td><td>√</td><td>一</td><td>37.1</td><td>56.7</td><td>39.2</td></tr><tr><td>1024(3×3)-→1024(1×1)</td><td></td><td></td><td>36.7</td><td>57.0</td><td>39.1</td></tr><tr><td>1024(3×3)-→&gt;1024(1×1)</td><td>√</td><td></td><td>37.2</td><td>57.4</td><td>39.4</td></tr><tr><td>1024(3×3) -→1024(1×1)</td><td></td><td></td><td>37.5</td><td>58.2</td><td>39.5</td></tr><tr><td>1024(3×3) -→1024(1×1),2× epochs</td><td></td><td></td><td>37.9</td><td>58.4</td><td>40.4</td></tr><tr><td>1024(3×3)-→&gt;1024(1×1),2× epochs,mstrain</td><td></td><td></td><td>40.1</td><td>60.8</td><td>42.5</td></tr></table>
+
+# 3.2 MAIN RESULTS
+
+We compare FoveaBox to the state-of-the-art methods in Table 4. All instantiations of our model outperform baseline variants of previous state-of-the-art models. The first group of detectors on Table 4 are two-stage detectors, the second group one-stage detectors, and the last group the FoveaBox detector. FoveaBox outperforms all single-stage detectors under ResNet-101 backbone, under all evaluation metrics. This includes the recent one-stage CornerNet and ExtremeNet (Law & Deng, 2018; Zhou et al., 2019b). FoveaBox also outperforms most of two-stage detectors, including FPN (Lin et al., 2017), Mask R-CNN (He et al., 2017) and IoU-Net (Jiang et al., 2018).
+
+Two-stage detectors rely on region-wise sub-networks to further classify the sparse region proposals. Since FoveaBox could also generate region proposals by changing the model head to class agnostic scheme (Table 1(c)), we believe it could further improve the performance of two-stage detectors, which beyond the focus of this paper.
+
+# 4 IN CONTEXT OF RELATED WORK
+
+Our work is related to previous works in different aspects. Before closing, we discuss the relations and differences in details.
+
+Anchor-based Object Detection: The anchor-based object detection frameworks can be generally grouped into two factions: two-stage, proposal driven detectors and one-stage, proposal free methods. Anchors are regression references and classification candidates to predict proposals for two-stage detectors(Ren et al., 2015; Lin et al., 2017; He et al., 2017; Cai & Vasconcelos, 2018) or final bounding boxes for single-stage detectors (Liu et al., 2016; Lin et al., 2018; Redmon & Farhadi, 2017). Most top one-stage detectors rely on the anchor boxes to enumerate the possible locations of target objects.
+
+Anchor-Free Explorations: There are also some prior works trying to remove the dependence of anchors. Due to the absence of anchors or region proposals, usually they lack the ability to deal with complex scenes and cases (Huang et al., 2015; Redmon et al., 2016). In text detection, the score mask technique has been used due to the arbitrary shape of target text (Zhang et al., 2016; Hu et al., 2017; Zhou et al., 2017). Such works usually utilize the fully convolutional networks to predict the existence of target scene text and the quadrilateral shapes. Guided-Anchoring (Wang et al., 2019) jointly predicts the locations where the center of objects are likely to exist as well as the scales and aspect ratios centered at the corresponding locations. Guided-Anchoring still relies on predefined anchors to optimize the object shape, and utilizes the center points to give the best predictions. In contrast, FoveaBox predicts the (left, top, right, bottom) boundaries of the object for each foreground position.
+
+Table 4: Object detection single-model results v.s. state-of-the-arts on COCO test-dev. We show results for our FoveaBox models with 800 input scale. FoveaBox-align indicates utilizing feature alignment discussed in Section 3.1.   
+
+<table><tr><td></td><td>backbone</td><td>AP</td><td>AP50</td><td>AP75</td><td>APs</td><td>APM</td><td>APL</td></tr><tr><td>two-stage methods Faster R-CNN w FPN (Lin et al., 2017)</td><td>ResNet-101</td><td rowspan="8">36.2 38.2 34.7</td><td rowspan="8">59.1</td><td rowspan="8">39.0 41.7</td><td rowspan="8">18.2 20.1</td><td rowspan="8"></td><td rowspan="8"></td><td>48.2 50.2 52.0</td></tr><tr><td>Mask R-CNN(He et al.,2017) Faster R-CNN by G-RMI(Huang et al.,2017b)</td><td>ResNet-101 Inception-ResNet-v2 Inception-ResNet-v2</td><td>60.3 55.5 57.7</td><td>36.7 13.5</td><td>39.0 41.1 38.1</td></tr><tr><td>Faster R-CNN w TDM(Shrivastava et al.,2016)</td></tr><tr><td>Relation Network (Hu et al.,2018)</td><td>36.8 DCN-101 39.0</td><td>39.2 58.6 42.9 59.0</td><td>16.2 39.8 - -</td></tr><tr><td>IoU-Net (Jiang et al.,2018)</td><td>ResNet-101 40.6 ResNet-101 42.8 62.1</td><td>■ ■ 46.3 23.7 45.5</td></tr><tr><td>CascadeR-CNN(Cai&amp; Vasconcelos,2018)</td></tr><tr><td>one-stagemethods YOLOv2 (Redmon et al., 2016)</td></tr><tr><td>YOLOv3 (Redmon &amp; Farhadi,2018) SSD513 (Fu et al.,2017)</td></tr><tr><td>DSSD513 (Fu et al., 2017) RetinaNet (Lin et al.,2018) RetinaNet (Lin et al., 2018) RPDet (Yang et al.,2019) FCOS (Tian et al.,2019)</td><td>Darknet-53 ResNet-101 ResNet-101 ResNet-101 ResNeXt-101 ResNeXt-101</td><td colspan="9">33.0 57.9 31.2 50.4 33.2 53.3 39.1 59.1 40.8</td></tr><tr><td></td><td>ResNeXt-101 Hourglass-104</td><td>41.0 42.1</td><td>61.1 62.9 62.1</td><td>44.1 44.3 45.2</td><td>24.1 23.6 25.6</td><td>44.2 44.1 44.9 42.7</td><td>51.2 51.7 52.0 53.9</td></tr><tr><td>CornerNet (Law&amp; Deng,2018) ExtremeNet (Zhou et al.,2019b)</td><td>Hourglass-104</td><td>40.5 40.1</td><td>56.5 55.3</td><td>43.1 43.2</td><td>19.4 20.3</td><td>43.2</td><td></td></tr><tr><td>CenterNet (Duan et al.,2019)</td><td>Hourglass-104</td><td>42.1</td><td>61.1</td><td>45.9</td><td>24.1</td><td>45.5</td><td>53.1 52.8</td></tr><tr><td>ours</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FoveaBox</td><td>ResNet-101</td><td>40.8</td><td>61.4</td><td>44.0</td><td>24.1</td><td></td><td>45.3</td><td>53.2</td></tr><tr><td>FoveaBox</td><td>ResNeXt-101</td><td>42.3</td><td>62.9</td><td>45.4</td><td></td><td>25.3</td><td></td><td></td></tr><tr><td>FoveaBox-align</td><td>ResNet-101</td><td>42.1</td><td></td><td></td><td></td><td></td><td>46.8</td><td>55.0</td></tr><tr><td></td><td>ResNeXt-101</td><td>43.9</td><td>62.7</td><td>45.5</td><td></td><td>25.2</td><td>46.6</td><td>54.5</td></tr><tr><td>FoveaBox-align</td><td></td><td></td><td>63.5</td><td>47.7</td><td></td><td>26.8</td><td>46.9</td><td>55.6</td></tr></table>
+
+Contemporary Works: Also there are contemporary works (Tian et al., 2019; Zhou et al., 2019a) similar to the idea of FoveaBox. FCOS relies on the proposed centerness map for better learning of the instance. Instead, FoveaBox directly predict the final class probability without centerness voting, which is more simple. The CenterNet (Zhou et al., 2019a) represent each instance by its features at the center point, which could also get comparable performance when adopting heavier networks (Law & Deng, 2018). Compare with a single point, our fovea-based positive sample definition process is more reasonable (The performance drops dramatically when we decrease $\sigma$ ). Again, we note that FoveaBox, CenterNet and FCOS are concurrent works.
+
+Bottom-up Methods: In CornerNet (Law & Deng, 2018), the authors propose to detect an object bounding box as a pair of key-points, the top-left corner and the bottom-right corner. CornerNet adopts the Associative Embedding (Newell et al., 2017) technique to separate different instances. Also there are some following works in bottom-up grouping manner (Zhou et al., 2019b; Duan et al., 2019). It should be noted that the bottom-up methods also do not need anchors during training and inference.
+
+# 5 CONCLUSION
+
+We have presented FoveaBox, a simple, effective, and completely anchor-free framework for generic object detection. By simultaneously predict the object position and the corresponding boundary, FoveaBox gives a clean solution for detecting objects without prior candidate boxes. We demonstrate its effectiveness on standard benchmarks and report extensive experimental analysis. We believe the simple and effective approach will serve as a solid baseline and help ease future research for object detection.
+
+# REFERENCES
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+Joseph Redmon, Santosh Divvala, Ross Girshick, and Ali Farhadi. You only look once: Unified, real-time object detection. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 779–788, 2016.
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+Abhinav Shrivastava, Rahul Sukthankar, Jitendra Malik, and Abhinav Gupta. Beyond skip connections: Top-down modulation for object detection. arXiv preprint arXiv:1612.06851, 2016.
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+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.
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+Zhi Tian, Chunhua Shen, Hao Chen, and Tong He. Fcos: Fully convolutional one-stage object detection. arXiv preprint arXiv:1904.01355, 2019.
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+Jiaqi Wang, Kai Chen, Shuo Yang, Chen Change Loy, and Dahua Lin. Region proposal by guided anchoring. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2965–2974, 2019.
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+Tong Yang, Xiangyu Zhang, Zeming Li, Wenqiang Zhang, and Jian Sun. Metaanchor: Learning to detect objects with customized anchors. In Advances in Neural Information Processing Systems, pp. 318–328, 2018.
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+Xingyi Zhou, Dequan Wang, and Philipp Krahenb ¨ uhl. Objects as points. ¨ arXiv preprint arXiv:1904.07850, 2019a.
+
+Xingyi Zhou, Jiacheng Zhuo, and Philipp Krahenbuhl. Bottom-up object detection by grouping extreme and center points. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 850–859, 2019b.
+
+Xinyu Zhou, Cong Yao, He Wen, Yuzhi Wang, Shuchang Zhou, Weiran He, and Jiajun Liang. East: an efficient and accurate scene text detector. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp. 5551–5560, 2017.
+
+# A APPENDIX
+
+# A.1 QUALITATIVE RESULTS
+
+Fig.5 shows the detection outputs of FoveaBox. Points and boxes with class probability larger than 0.5 are shown (before feeding into NMS). For each object, though there are several active points, the predicted boxes are very close to the ground-truth. These figures demonstrate that FoveaBox could directly generate accurate, robust box predictions, without the requirement of candidate anchors.
+
+![](images/daa123700845237aff0c1602e0f807a2721b2b0ae599350fe6ae893b0fa790be.jpg)  
+Figure 5: FoveaBox results on the COCO minival set. These results are based on ResNet-101, achieving a single model box AP of 38.6. For each pair, left is the detection results with bounding box, category, and confidence. Right is the score output map with their corresponding bounding boxes before feeding into non-maximum suppression (NMS). The score probability in each position is denoted by the color density.
+
+# A.2 PER-CLASS DIFFERENCE:
+
+Fig. 6 shows per-class AP difference of FoveaBox and RetinaNet. Both of them are with ResNet-50- FPN backbone and 800 input scale. The vertical axis shows $\mathbf { A P } _ { F o v e a B o x }$ -APRetinaNet. FoveaBox shows improvement in most of the classes.
+
+![](images/9d75c5fbf37f13dc1c4aa1335506f3078e85b21adfca94ba3a80c4050cb16a64.jpg)  
+Figure 6: AP difference of FoveaNet and RetinaNet on COCO dataset. Both models use ResNetFPN-50 as backbone and 800 input scales.
+
+# A.3 MORE TRAINING TIME:
+
+Table 5 shows the detection performance under $2 \mathbf { x }$ setting (24 epochs training). FoveaBox also outperforms the anchor-based baseline method.
+
+Table $5 \colon 2 \mathrm { x }$ epochs training (800 scale).   
+
+<table><tr><td>method</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>RetinaNet-50</td><td>36.4</td><td>56.3</td><td>39.4</td></tr><tr><td>RetinaNet-101</td><td>38.1</td><td>57.8</td><td>41.1</td></tr><tr><td>FoveaBox-50</td><td>37.1</td><td>56.7</td><td>39.3</td></tr><tr><td>FoveaBox-101</td><td>38.7</td><td>58.4</td><td>41.1</td></tr></table>
+
+# A.4 FEATURE ALIGNMENT
+
+For feature alignment, we adopt a $3 \times 3$ deformable convolutional layer to implement the transformation, as shown in Fig.7. The offset input to the deformable convolutional layer is $( { \hat { t } } _ { x _ { 1 } } , { \hat { t } } _ { y _ { 1 } } , { \hat { t } } _ { x _ { 2 } } , { \hat { t } } _ { y _ { 2 } } )$ of the bbox output.
+
+![](images/7d0edf63e7fd123b34f67edf62dac10ba3ba1ed7dbc2ab6e42647f9f47a867a9.jpg)  
+Figure 7: Feature alignment process.

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+        "type": "text",
+        "text": "FOVEABOX: BEYOUND ANCHOR-BASED OBJECT DETECTION ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+    },
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+        "text": "We present FoveaBox, an accurate, flexible, and completely anchor-free framework for object detection. While almost all state-of-the-art object detectors utilize predefined anchors to enumerate possible locations, scales and aspect ratios for the search of the objects, their performance and generalization ability are also limited to the design of anchors. Instead, FoveaBox directly learns the object existing possibility and the bounding box coordinates without anchor reference. This is achieved by: (a) predicting category-sensitive semantic maps for the object existing possibility, and (b) producing category-agnostic bounding box for each position that potentially contains an object. The scales of target boxes are naturally associated with feature pyramid representations. We demonstrate its effectiveness on standard benchmarks and report extensive experimental analysis. Without bells and whistles, FoveaBox achieves state-of-the-art single model performance on the standard COCO detection benchmark. More importantly, FoveaBox avoids all computation and hyper-parameters related to anchor boxes, which are often sensitive to the final detection performance. We believe the simple and effective approach will serve as a solid baseline and help ease future research for object detection. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Object detection requires the solution of two main tasks: recognition and localization. Given an arbitrary image, an object detection system needs to determine whether there are any instances of semantic objects from predefined categories and, if present, to return the spatial location and extent. To add the localization functionality to generic object detection systems, sliding window approaches have been the method of choice for many years (Lampert et al., 2008; Felzenszwalb et al., 2010; Liu et al., 2018). ",
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+        "text": "Recently, deep learning techniques have emerged as powerful methods for learning feature representations automatically from data (Simonyan & Zisserman, 2014; He et al., 2016; Huang et al., 2017a). For object detection, the anchor-based Region Proposal Networks (Ren et al., 2015) are widely used to serve as a common component for searching possible regions of interest for modern object detection frameworks (Liu et al., 2016; He et al., 2017; Lin et al., 2018). In short, anchor method suggests dividing the box space into discrete bins and refining the object box in the corresponding bin. Most state-of-the-art detectors rely on anchors to enumerate the possible locations, scales, and aspect ratios for target objects (Liu et al., 2018). Anchors are regression references and classification candidates to predict proposals for two-stage detectors or final bounding boxes for single-stage detectors. Nevertheless, anchors can be regarded as a feature-sharing sliding window scheme to cover the possible locations of objects. ",
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+        "image_caption": [
+            "Figure 1: The anchor-based object detection frameworks need to (a) design anchors according to the ground-truth box distributions; (b) match anchors with ground-truth boxes to generate training target (anchor classification and refinement); and (c) utilize the target generated by (b) for training. "
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+        "text": "However, anchors must be carefully designed and used in object detection frameworks. (a) One of the most important factors in designing anchors is how densely it covers the instance location space. To achieve a good recall rate, anchors are carefully designed based on the statistics computed from the training/validation set (Lin et al., 2018). (b) One design choice based on a particular dataset is not always applicable to other applications, which harms the generality (Yang et al., 2018). (c) At training phase, anchor-methods rely on the intersection-over-union (IoU) to define the positive/negative samples, which introduces additional computation and hyper-parameters for an object detection system (Wang et al., 2019). ",
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+        "type": "text",
+        "text": "In contrast, our human vision system can recognize the instance in space and predict the boundary given the visual cortex map, without any pre-defined shape template (Bear et al., 2007). In other words, we human naturally recognize the object in the visual scene without enumerating the candidate boxes. Inspired by this, an intuitive question to ask is, is the anchor scheme the optimal way to guide the search of objects? And further, could we design an accurate object detection framework without anchors or candidate boxes? Without anchors, one may expect a complex method is required to achieve comparable performance. However, we show that a surprisingly simple and flexible system can match, even surpass the prior state-of-the-art object detection results without any requirement of candidate boxes. ",
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+        "image_caption": [
+            "Figure 2: FoveaBox object detector. For each output spacial position that potentially presents an object, FoveaBox directly predicts the confidences for all target categories and the bounding box. "
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+    {
+        "type": "text",
+        "text": "To this end, we present FoveaBox, a completely anchor-free framework for object detection. FoveaBox is motivated from the fovea of human eyes: the center of the vision field is with the highest visual acuity (Fig.2 left), which is necessary for activities where visual detail is of primary importance (Iwasaki & Inomata, 1986). FoveaBox jointly predicts the locations where the object’s center area is likely to exist as well as the bounding box at each valid location. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In FoveaBox, each target object is predicted by category scores at center area, associated with 4-d bounding box, as shown in Fig.2 right. At training phase, we do not need to utilize anchors, or IoU matching to generate training target. Instead, the training target is directly generated by ground-truth boxes. ",
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+    {
+        "type": "text",
+        "text": "To demonstrate the effectiveness of the proposed detection scheme, we combine the recent progress of feature pyramid networks and our detection head to form the framework of FoveaBox. Without bells and whistles, FoveaBox gets state-of-the-art single-model results on the COCO object detection task. Compared with the anchor-based RetinaNet, FoveaBox gets 2.2 AP gains, which also surpasses most of previously published anchor based single-model results. We believe the simple training/inference manner of FoveaBox, together with the flexibility and accuracy, will benefit future research on object detection and relevant topics. ",
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+    {
+        "type": "text",
+        "text": "2 FOVEABOX ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "FoveaBox is conceptually simple: It contains a backbone network and a fovea head network. The backbone is responsible for computing a convolutional feature map over an entire input image and is an off-the-shelf convolutional network. The fovea head is composed of two sub-branches, the first branch performs per pixel classification on the backbone’s output; the second branch performs box prediction for each position that potentially covered by an object. ",
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+    {
+        "type": "text",
+        "text": "2.1 REVIEW OF FPN AND ANCHORS: ",
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+    {
+        "type": "text",
+        "text": "We begin by briefly reviewing the Feature Pyramid Network (FPN) used for object detection (Lin et al., 2017). In general, FPN uses a top-down architecture with lateral connections to build an in-network feature pyramid from a single-scale input. FPN is independent of a specific task. For object detection, each level of the pyramid in FPN is used for detecting objects at a specific scale. On each feature pyramid, anchor-based methods uniformly place $A$ anchors on each of the $H \\times W$ spacial position. After computing the IoU overlap between all anchors and the ground-truth boxes, the anchor-based methods can define training targets. Finally, the pyramid features are utilized to optimize the targets. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/dfe718192bcb34f1770531c855ee5fe6c07a0cef4ffd8a59e987bfc67478e369.jpg",
+        "image_caption": [
+            "Figure 3: Anchor-based object detection v.s. FoveaBox object detection. left: The anchor-based method uniformly places $A$ ( $A = 3$ in this example) anchors on each output spacial position, and utilizes IoU to define the positive/negative anchors; right: FoveaBox directly define positive/negative samples for each output spacial position by ground-truth boxes, and predicts the box boundaries from the corresponding position. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "2.2 FOVEABOX ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "FoveaBox directly predicts the object existing possibility and the corresponding boundary for each position potential contained by an instance. In this section, we introduce the key components stepby-step. ",
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+    {
+        "type": "text",
+        "text": "2.2.1 OBJECT OCCURRENCE POSSIBILITY",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Given a valid ground-truth box denoted as $( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } )$ . We first map the box into the target feature pyramid $P _ { l }$ ",
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+    {
+        "type": "equation",
+        "img_path": "images/7ebf4c68a8b5ee3bfb72961570bc9e7549665659c202895100017fe3e39e2d3f.jpg",
+        "text": "$$\n\\begin{array} { c } { { x _ { 1 } ^ { ' } = \\frac { x _ { 1 } } { s _ { l } } , y _ { 1 } ^ { ' } = \\frac { y _ { 1 } } { s _ { l } } , x _ { 2 } ^ { ' } = \\frac { x _ { 2 } } { s _ { l } } , y _ { 2 } ^ { ' } = \\frac { y _ { 2 } } { s _ { l } } , } } \\\\ { { { } } } \\\\ { { c _ { x } ^ { ' } = 0 . 5 ( x _ { 2 } ^ { ' } + x _ { 1 } ^ { ' } ) , c _ { y } ^ { ' } = 0 . 5 ( y _ { 2 } ^ { ' } + y _ { 1 } ^ { ' } ) , w ^ { ' } = x _ { 2 } ^ { ' } - x _ { 1 } ^ { ' } , h ^ { ' } = y _ { 2 } ^ { ' } - y _ { 1 } ^ { ' } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $s _ { l }$ is the down-sample factor. The positive area $R ^ { p o s }$ on the score map is designed to be roughly a shrunk version of the original one (Fig.3 right): ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/67d428eb0fdb6dc2f4aa6384586680d763d3eaad349141bf5be83f752fa8b3e5.jpg",
+        "text": "$$\n\\begin{array} { r } { x _ { 1 } ^ { p o s } = c _ { x } ^ { ' } - 0 . 5 \\sigma w ^ { ' } , y _ { 1 } ^ { p o s } = c _ { y } ^ { ' } - 0 . 5 \\sigma h ^ { ' } , } \\\\ { x _ { 2 } ^ { p o s } = c _ { x } ^ { ' } + 0 . 5 \\sigma w ^ { ' } , y _ { 2 } ^ { p o s } = c _ { y } ^ { ' } + 0 . 5 \\sigma h ^ { ' } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ is the shrunk factor. At training phase, each cell inside the positive area is annotated with the corresponding target class label. The negative area is the whole feature map excluding area in $R ^ { p o s }$ . For predicting, each output set of pyramidal heat-map has $C$ channels, where $C$ is the number of categories, and is of size $H \\times W$ . Each channel is a binary mask indicating the possibility for a class, like FCNs in semantic segmentation (Long et al., 2015). The positive area usually accounts for a small portion of the whole feature map, so we adopt Focal Loss (Lin et al., 2018) to train this branch. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2.2 SCALE ASSIGNMENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "While our goal is to predict the boundary of the target objects, directly predicting these numbers is not stable, due to the large scale variations of the objects. Instead, we divide the scales of objects into several bins, according to the number of feature pyramidal levels. Each pyramid has a basic scale $r _ { l }$ ranging from 32 to 512 on pyramid levels $P _ { 3 }$ to $P _ { 7 }$ , respectively. The valid scale range of the target boxes for pyramid level $l$ is computed as ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/273cf265c0c9bca815459f17984b01507b2a37ce49c804f81a57ff6bfb71e8c2.jpg",
+        "text": "$$\n[ r _ { l } / \\eta , r _ { l } \\cdot \\eta ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\eta$ is set empirically to control the scale range for each pyramid. Target objects not in the corresponding scale range are ignored during training. Note that an object may be detected by multiple pyramids of the networks, which is different from previous practice that maps objects to only one feature pyramid (He et al., 2017). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2.3 BOX PREDICTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Each ground-truth bounding box is specified in the way $G \\ : = \\ : ( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } )$ . Starting from a positive point $( x , y )$ in $R ^ { p o s }$ , FoveaBox directly computes the normalized offset between $( x , y )$ and four boundaries: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ea84541e51ed9d6021015c7df1c20600c4f6b2b0d334d52a02f9a138b4228cb.jpg",
+        "text": "$$\n\\begin{array} { l l } { { t _ { x } } _ { 1 } = \\log \\frac { s _ { l } ( x + 0 . 5 ) - x _ { 1 } } { r _ { l } } , t _ { y _ { 1 } } = \\log \\frac { s _ { l } ( y + 0 . 5 ) - y _ { 1 } } { r _ { l } } , } \\\\ { { t _ { x } } _ { 2 } = \\log \\frac { x _ { 2 } - s _ { l } ( x + 0 . 5 ) } { r _ { l } } , t _ { y _ { 2 } } = \\log \\frac { y _ { 2 } - s _ { l } ( y + 0 . 5 ) } { r _ { l } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This function first maps the coordinate $( x , y )$ to the input image, then computes the normalized offset between the projected coordinate and $G$ . Finally the targets are regularized with the log-space function. $r _ { l }$ is the basic scale defined in section 2.2.2. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For simplicity, we adopt the widely used Smooth $L _ { 1 }$ loss (Ren et al., 2015) to train the box prediction $L _ { b o x }$ . After targets being optimized, we can generate the box boundary for each cell $( x , y )$ on the output feature maps1. In box branch, each output set of pyramidal heatmap has 4 channels, for jointly prediction of $( t _ { x _ { 1 } } , t _ { y _ { 1 } } , t _ { x _ { 2 } } , t _ { y _ { 2 } } )$ . ",
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2.4 NETWORK ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            578,
+            416,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the generality of our approach, we instantiate FoveaBox with multiple architectures. For clarity, we differentiate between: (i) the convolutional backbone architecture used for feature extraction over an entire image, and (ii) the network head for computing the final results. ",
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/862b43123997d3662af060569300731534469cd4e3d82efd5890e75b12ee4bee.jpg",
+        "image_caption": [
+            "Figure 4: On each FPN feature level, FoveaBox attaches two subnetworks, one for classifying the corresponding cells and one for predict the $( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } )$ of ground-truth object box. Right is the score output map with their corresponding predicted boxes before feeding into non-maximum suppression (NMS). The score probability in each position is denoted by the color density. More examples are shown in Fig.5. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Most of the experiments are based on the head architecture as shown in Fig.4. We also utilize different head variants to further study the generality. More complex designs have the potential to improve performance but are not the focus of this work. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.2.5 IMPLEMENTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            161,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We adopt the widely used FPN networks for fair comparison. Concretely, we construct a pyramid with levels $\\{ P _ { l } \\} , l \\ = \\ 3 , 4 , \\cdots , 7$ , where $l$ indicates pyramid level. $P _ { l }$ has $1 / 2 ^ { l }$ resolution of the input. All pyramid levels have $C = 2 5 6$ channels. Fovea head is attached on each pyramid level. Parameters are shared across all pyramid levels. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "FoveaBox is trained with stochastic gradient descent (SGD). We use synchronized SGD over 4 GPUs with a total of 16 images per minibatch (4 images per GPU). Unless otherwise specified, all models are trained for 12 epochs with an initial learning rate of 0.01, which is then divided by 10 at 8th and again at 11th epochs. Weight decay of 0.0001 and momentum of 0.9 are used. Only standard horizontal image flipping is used for data augmentation. During inference, we first use a confidence threshold of 0.05 to filter out predictions with low confidence. Then, we select the top 1000 scoring boxes from each prediction layer. Next, NMS with threshold 0.5 is applied for each class separately. Finally, the top-100 scoring predictions are selected for each image. Although there are more intelligent ways to perform post-processing, such as bbox voting (Gidaris & Komodakis, 2015), Soft-NMS (Bodla et al., 2017) or test-time image augmentations, in order to keep simplicity and to fairly compare against the baseline models, we do not use those tricks here. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We present experimental results on the bounding box detection track of the MS COCO benchmark. All models are trained on MS COCO trainval35k. If not specified, ResNet-50-FPN backbone and a 600 pixel train and test image scale are used to do the ablation study. We report lesion and sensitivity studies by evaluating on the minival split. For our main results, we report COCO AP on the test-dev split, which has no public labels and requires use of the evaluation server. ",
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+    {
+        "type": "text",
+        "text": "3.1 ABLATION STUDY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Various anchor densities and FoveaBox: One of the most important design factors in an anchorbased detection system is how densely it covers the space of possible objects. As anchor-based detectors use a fixed sampling grid, a popular approach for achieving high coverage of boxes is to use multiple anchors at each spatial position. One may expect that we can always get better performance when attaching denser anchors on each position. To verify this assumption, we sweep over the number of scale and aspect ratio anchors used at each spatial position and each pyramid level in RetinaNet, including a single square anchor at each location to 12 anchors per location (Table.1(a)). Increasing beyond 6-9 anchors does not show further gains. The saturation of performance w.r.t. density implies the handcrafted, over-density anchors do not offer an advantage. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Over-density anchors not only increase the foreground-background optimization difficulty, but also likely to cause the ambiguous position definition problem. For each output spatial location, there are $A$ anchors whose labels are defined by the IoU with the ground-truth. Among them, some of the anchors are defined as positive samples, while others are negatives. However they are sharing the same input features. The classifier needs to not only distinguish the samples from different positions, but also different anchors at the same position. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrast, FoveaBox explicitly predicts one target at each position and gets no worse performance than the best anchor-based model. Compare with the anchor based scheme, FoveaBox enjoys several advantages. (a) Since we only predict one target at each position, the output space has been reduced to $1 / A$ of the anchor-based method. (b) There is no ambiguous problem and the optimization target is more straightforward. (c) FoveaBox has fewer hyper-parameters, and is more flexible, since we do not need to extensively design anchors to see a relatively better choice. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "FoveaBox is more robust to box distribution: One of the major benefits of FoveaBox is the robust prediction of bounding boxes. To verify this, we divide the boxes in the validation set into three groups according to the ground-truth aspect ratios $\\begin{array} { r } { u = \\operatorname* { m a x } ( \\frac { h } { w } , \\frac { w } { h } ) } \\end{array}$ . We compare FoveaBox and ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Ablation experiments for FoveaBox. All models are trained on trainval35k, test on minival. If not specified, default values are $\\eta = 2 . 0$ and $\\sigma = 0 . 4$ . (a) Our anchor-free FoveaBox get $0 . 9 \\mathrm { \\ A P }$ gains compared with the best model of anchor-based RetinaNet; (b) FoveaBox is more robust to bounding box distributions; (c) FoveaBox could also generate high-quality region proposals; (d) Accuracy of FoveaBox for various network depths and image scales; (e) and (f): FoveaBox gets best performance with $\\eta = 2 . 0$ and $\\sigma = 0 . 4$ . See Section 3.1 for details. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/0c7597d7090d477001c173bda208e1b3eb08672a06c91ca2cf658b4d40794144.jpg",
+        "table_caption": [
+            "(a) Varying anchor density and FoveaBox. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>method</td><td>#sc</td><td>#ar</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>RetinaNet</td><td>1</td><td>1</td><td>30.2</td><td>49.0</td><td>31.7</td></tr><tr><td>RetinaNet</td><td>2</td><td></td><td>31.9</td><td>50.0</td><td>34.1</td></tr><tr><td>RetinaNet</td><td>3</td><td></td><td>31.9</td><td>49.4</td><td>33.8</td></tr><tr><td>RetinaNet</td><td>2</td><td>１１3</td><td>34.2</td><td>53.1</td><td>36.5</td></tr><tr><td>RetinaNet</td><td>3</td><td>33</td><td>34.2</td><td>53.2</td><td>36.9</td></tr><tr><td>RetinaNet</td><td>4</td><td></td><td>33.9</td><td>52.1</td><td>36.2</td></tr><tr><td>FoveaBox</td><td>-</td><td>-</td><td>35.1</td><td>54.3</td><td>37.1</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/9ba1f5c7952a0d18aedc1bf41fc92be0d9cd9d63875d9300fe6ad6ddfda91c3d.jpg",
+        "table_caption": [
+            "(b) Detection with different aspect ratios "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>method</td><td>AP</td><td>APu&lt;3</td><td>AP3≤u&lt;5</td><td>APu&gt;5</td></tr><tr><td>RetinaNet</td><td>34.2</td><td>36.5</td><td>24.5</td><td>10.2</td></tr><tr><td>FoveaBox</td><td>35.1</td><td>36.8</td><td>26.8</td><td>16.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/3b28dbf72cf972ae345f4a12eed9f3a8a579d7dff6a1d51b3a1629c1c0e4f1e8.jpg",
+        "table_caption": [
+            "(c) Region proposal performance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>method</td><td>backbone</td><td>AR100</td><td>AR300</td><td>AR1000</td></tr><tr><td>RPN</td><td>ResNet-50</td><td>44.5</td><td>51.1</td><td>56.6</td></tr><tr><td>FoveaBox</td><td>ResNet-50</td><td>52.9</td><td>57.3</td><td>61.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/83ce28fe93cfe2d18ba777fb5a4c14792b5e7fe109119a6f8665f4280aef945a.jpg",
+        "table_caption": [
+            "(d) Different input resolutions and models. ",
+            "(e) Varying η $( \\sigma = 0 . 4 )$ . (f) Varying σ $( \\eta = 2 . 0 )$ ). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>net-depth-scale</td><td>AP</td><td>AP50</td><td>AP75</td><td>m</td><td>AP</td><td>AP50</td><td>AP75</td><td>0</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>FoveaBox-50-400</td><td>31.9+1.4</td><td>49.6</td><td>33.8</td><td>1.0</td><td>32.0</td><td>50.4</td><td>31.8</td><td>0.2</td><td>34.1</td><td>53.2</td><td>36.0</td></tr><tr><td>FoveaBox-50-600</td><td>35.1+0.9</td><td>54.3</td><td>37.1</td><td>1.5</td><td>34.1</td><td>53.3</td><td>36.0</td><td>0.3</td><td>34.8</td><td>54.0</td><td>36.7</td></tr><tr><td>FoveaBox-50-800</td><td>36.4+0.9</td><td>56.2</td><td>38.7</td><td>2.0</td><td>35.1</td><td>54.4</td><td>37.0</td><td>0.4</td><td>35.1</td><td>54.4</td><td>37.0</td></tr><tr><td>FoveaBox-101-400</td><td>33.3+1.4</td><td>51.0</td><td>35.0</td><td>2.5</td><td>35.0</td><td>54.2</td><td>36.8</td><td>0.5</td><td>34.8</td><td>53.9</td><td>36.6</td></tr><tr><td>FoveaBox-101-600</td><td>37.0+1.0</td><td>56.4</td><td>39.3</td><td>3.0</td><td>34.3</td><td>53.3</td><td>36.5</td><td>0.6</td><td>34.1</td><td>53.1</td><td>36.0</td></tr><tr><td>FoveaBox-101-800</td><td>38.6+0.9</td><td>58.0</td><td>41.2</td><td>4.0</td><td>32.8</td><td>51.0</td><td>34.5</td><td>0.7</td><td>33.3</td><td>52.5</td><td>34.9</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "RetinaNet at different aspect ratio thresholds, as shown in Table 1(b). We see that both methods get best performance when $u$ is low. Although FoveaBox also suffers performance decrease when $u$ increases, it is much better than the baseline model. ",
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+    {
+        "type": "text",
+        "text": "Generating high-quality region proposals: Changing the classification target to class-agnostic head is straightforward and could generate region proposals. We compare the proposal performance against FPN-based RPN (Lin et al., 2017) and evaluate average recalls (AR) with different numbers of proposals on minival set, as shown in Table 1(c). Surprisingly, our method outperforms the RPN baseline by a large margin, among all criteria. Specifically, with top 100 region proposals, FoveaBox gets 52.9 AR, outperforming RPN by 8.4 points. This validates that our model’s capacity in generating high quality region proposals. ",
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+    {
+        "type": "text",
+        "text": "Across model depth and scale: Table 1(d) shows FoveaBox utilizing different backbone networks and input resolutions. The train/inference settings are exactly the same as the baseline method (Lin et al., 2018). Under the same settings, FoveaBox consistently gets $0 . 9 { \\sim } 1 . 4$ higher AP. When comparing the inference speed, we find that FoveaBox models are about $1 . 1 { \\sim } 1 . 3$ times faster than the RetinaNet counterparts. ",
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+    {
+        "type": "text",
+        "text": "Analysis of $\\eta$ and $\\sigma$ : In Eq.(3), $\\eta$ controls the scale assignment extent for each pyramid. As $\\eta$ increases, each pyramid will response to more scales of objects. Table 1(e) shows the impact of $\\eta$ on the final detection performance. Another important hyper-parameter is the shrunk factor $\\sigma$ which controls the positive/negative samples. Table 1(f) shows the model performance with respect to $\\sigma$ changes. In this paper, $\\sigma = 0 . 4$ and $\\eta = 2$ are used in other experiments. ",
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+    {
+        "type": "text",
+        "text": "IoU-based assignment v.s. fovea area: Another choice of defining the positive/negative samples is firstly gets the predicted box from the box branch, and then assign the target labels based on the IoU between the predicted boxes and ground-truth boxes. As shown in Table 2, the shrunk version gets better performance $( + 0 . 4$ AP) than the IoU-based assignment process. ",
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+    {
+        "type": "text",
+        "text": "Better head and feature alignment: The most recent works (Chen et al., 2019; Yang et al., 2019) suggest to align the features in one-stage object detection frameworks with anchors. In FoveaBox, we adopt deformable convolution (Dai et al., 2017) based on the box offset learned by Eq.(4) to refine the classification branch2. FoveaBox works well when adding such techniques. Specifically, when we change the classification branch to a heavier head, together with feature alignment and GN, FoveaBox gets 40.1 AP using ResNet-50 as backbone! This experiment demonstrates the generality of our approach to the network design (Table 3). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6d45abd1a9953b76d52fc03ad3cbf4b242fb209a0261fcf24184623f32083ff0.jpg",
+        "table_caption": [
+            "Table 2: Label assignment strategy (ResNet-50, 800 scale). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>assign method</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>IoU (0.5/0.4) IoU (0.6/0.5)</td><td>35.6 35.9</td><td>54.7 54.7</td><td>37.7 38.4</td></tr><tr><td>IoU (0.5/0.5)</td><td>36.0</td><td>54.9</td><td>38.4</td></tr><tr><td>Fovea (σ = 0.4)</td><td>36.4</td><td>56.2</td><td>38.7</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/46b8093bc1aa0d0c0afb7a29fe0cee96200502baf586d2c8e65b948dfca075bd.jpg",
+        "table_caption": [
+            "Table 3: Feature alignment and group normalization (ResNet-50, 800 scale). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>cls branch</td><td>alignment</td><td>GN</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>256(3×3) -→256(3×3)-→256(3×3)-&gt;256(3×3)</td><td></td><td></td><td>36.4</td><td>56.2</td><td>38.7</td></tr><tr><td>256(3×3)→&gt;256(3×3)-&gt;256(3×3)-→256(3×3)</td><td>√</td><td></td><td>36.8</td><td>56.5</td><td>38.9</td></tr><tr><td>256(3×3)-→256(3×3)-→256(3×3)-&gt;256(3×3)</td><td>√</td><td>一</td><td>37.1</td><td>56.7</td><td>39.2</td></tr><tr><td>1024(3×3)-→1024(1×1)</td><td></td><td></td><td>36.7</td><td>57.0</td><td>39.1</td></tr><tr><td>1024(3×3)-→&gt;1024(1×1)</td><td>√</td><td></td><td>37.2</td><td>57.4</td><td>39.4</td></tr><tr><td>1024(3×3) -→1024(1×1)</td><td></td><td></td><td>37.5</td><td>58.2</td><td>39.5</td></tr><tr><td>1024(3×3) -→1024(1×1),2× epochs</td><td></td><td></td><td>37.9</td><td>58.4</td><td>40.4</td></tr><tr><td>1024(3×3)-→&gt;1024(1×1),2× epochs,mstrain</td><td></td><td></td><td>40.1</td><td>60.8</td><td>42.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.2 MAIN RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We compare FoveaBox to the state-of-the-art methods in Table 4. All instantiations of our model outperform baseline variants of previous state-of-the-art models. The first group of detectors on Table 4 are two-stage detectors, the second group one-stage detectors, and the last group the FoveaBox detector. FoveaBox outperforms all single-stage detectors under ResNet-101 backbone, under all evaluation metrics. This includes the recent one-stage CornerNet and ExtremeNet (Law & Deng, 2018; Zhou et al., 2019b). FoveaBox also outperforms most of two-stage detectors, including FPN (Lin et al., 2017), Mask R-CNN (He et al., 2017) and IoU-Net (Jiang et al., 2018). ",
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+        "type": "text",
+        "text": "Two-stage detectors rely on region-wise sub-networks to further classify the sparse region proposals. Since FoveaBox could also generate region proposals by changing the model head to class agnostic scheme (Table 1(c)), we believe it could further improve the performance of two-stage detectors, which beyond the focus of this paper. ",
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+    {
+        "type": "text",
+        "text": "4 IN CONTEXT OF RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our work is related to previous works in different aspects. Before closing, we discuss the relations and differences in details. ",
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+    {
+        "type": "text",
+        "text": "Anchor-based Object Detection: The anchor-based object detection frameworks can be generally grouped into two factions: two-stage, proposal driven detectors and one-stage, proposal free methods. Anchors are regression references and classification candidates to predict proposals for two-stage detectors(Ren et al., 2015; Lin et al., 2017; He et al., 2017; Cai & Vasconcelos, 2018) or final bounding boxes for single-stage detectors (Liu et al., 2016; Lin et al., 2018; Redmon & Farhadi, 2017). Most top one-stage detectors rely on the anchor boxes to enumerate the possible locations of target objects. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Anchor-Free Explorations: There are also some prior works trying to remove the dependence of anchors. Due to the absence of anchors or region proposals, usually they lack the ability to deal with complex scenes and cases (Huang et al., 2015; Redmon et al., 2016). In text detection, the score mask technique has been used due to the arbitrary shape of target text (Zhang et al., 2016; Hu et al., 2017; Zhou et al., 2017). Such works usually utilize the fully convolutional networks to predict the existence of target scene text and the quadrilateral shapes. Guided-Anchoring (Wang et al., 2019) jointly predicts the locations where the center of objects are likely to exist as well as the scales and aspect ratios centered at the corresponding locations. Guided-Anchoring still relies on predefined anchors to optimize the object shape, and utilizes the center points to give the best predictions. In contrast, FoveaBox predicts the (left, top, right, bottom) boundaries of the object for each foreground position. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e84260c2da6b571f845c3ca8b81417be7fb4979d0c1e221bc66b6ef5d8b4b9c3.jpg",
+        "table_caption": [
+            "Table 4: Object detection single-model results v.s. state-of-the-arts on COCO test-dev. We show results for our FoveaBox models with 800 input scale. FoveaBox-align indicates utilizing feature alignment discussed in Section 3.1. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>backbone</td><td>AP</td><td>AP50</td><td>AP75</td><td>APs</td><td>APM</td><td>APL</td></tr><tr><td>two-stage methods Faster R-CNN w FPN (Lin et al., 2017)</td><td>ResNet-101</td><td rowspan=\"8\">36.2 38.2 34.7</td><td rowspan=\"8\">59.1</td><td rowspan=\"8\">39.0 41.7</td><td rowspan=\"8\">18.2 20.1</td><td rowspan=\"8\"></td><td rowspan=\"8\"></td><td>48.2 50.2 52.0</td></tr><tr><td>Mask R-CNN(He et al.,2017) Faster R-CNN by G-RMI(Huang et al.,2017b)</td><td>ResNet-101 Inception-ResNet-v2 Inception-ResNet-v2</td><td>60.3 55.5 57.7</td><td>36.7 13.5</td><td>39.0 41.1 38.1</td></tr><tr><td>Faster R-CNN w TDM(Shrivastava et al.,2016)</td></tr><tr><td>Relation Network (Hu et al.,2018)</td><td>36.8 DCN-101 39.0</td><td>39.2 58.6 42.9 59.0</td><td>16.2 39.8 - -</td></tr><tr><td>IoU-Net (Jiang et al.,2018)</td><td>ResNet-101 40.6 ResNet-101 42.8 62.1</td><td>■ ■ 46.3 23.7 45.5</td></tr><tr><td>CascadeR-CNN(Cai&amp; Vasconcelos,2018)</td></tr><tr><td>one-stagemethods YOLOv2 (Redmon et al., 2016)</td></tr><tr><td>YOLOv3 (Redmon &amp; Farhadi,2018) SSD513 (Fu et al.,2017)</td></tr><tr><td>DSSD513 (Fu et al., 2017) RetinaNet (Lin et al.,2018) RetinaNet (Lin et al., 2018) RPDet (Yang et al.,2019) FCOS (Tian et al.,2019)</td><td>Darknet-53 ResNet-101 ResNet-101 ResNet-101 ResNeXt-101 ResNeXt-101</td><td colspan=\"9\">33.0 57.9 31.2 50.4 33.2 53.3 39.1 59.1 40.8</td></tr><tr><td></td><td>ResNeXt-101 Hourglass-104</td><td>41.0 42.1</td><td>61.1 62.9 62.1</td><td>44.1 44.3 45.2</td><td>24.1 23.6 25.6</td><td>44.2 44.1 44.9 42.7</td><td>51.2 51.7 52.0 53.9</td></tr><tr><td>CornerNet (Law&amp; Deng,2018) ExtremeNet (Zhou et al.,2019b)</td><td>Hourglass-104</td><td>40.5 40.1</td><td>56.5 55.3</td><td>43.1 43.2</td><td>19.4 20.3</td><td>43.2</td><td></td></tr><tr><td>CenterNet (Duan et al.,2019)</td><td>Hourglass-104</td><td>42.1</td><td>61.1</td><td>45.9</td><td>24.1</td><td>45.5</td><td>53.1 52.8</td></tr><tr><td>ours</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FoveaBox</td><td>ResNet-101</td><td>40.8</td><td>61.4</td><td>44.0</td><td>24.1</td><td></td><td>45.3</td><td>53.2</td></tr><tr><td>FoveaBox</td><td>ResNeXt-101</td><td>42.3</td><td>62.9</td><td>45.4</td><td></td><td>25.3</td><td></td><td></td></tr><tr><td>FoveaBox-align</td><td>ResNet-101</td><td>42.1</td><td></td><td></td><td></td><td></td><td>46.8</td><td>55.0</td></tr><tr><td></td><td>ResNeXt-101</td><td>43.9</td><td>62.7</td><td>45.5</td><td></td><td>25.2</td><td>46.6</td><td>54.5</td></tr><tr><td>FoveaBox-align</td><td></td><td></td><td>63.5</td><td>47.7</td><td></td><td>26.8</td><td>46.9</td><td>55.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Contemporary Works: Also there are contemporary works (Tian et al., 2019; Zhou et al., 2019a) similar to the idea of FoveaBox. FCOS relies on the proposed centerness map for better learning of the instance. Instead, FoveaBox directly predict the final class probability without centerness voting, which is more simple. The CenterNet (Zhou et al., 2019a) represent each instance by its features at the center point, which could also get comparable performance when adopting heavier networks (Law & Deng, 2018). Compare with a single point, our fovea-based positive sample definition process is more reasonable (The performance drops dramatically when we decrease $\\sigma$ ). Again, we note that FoveaBox, CenterNet and FCOS are concurrent works. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Bottom-up Methods: In CornerNet (Law & Deng, 2018), the authors propose to detect an object bounding box as a pair of key-points, the top-left corner and the bottom-right corner. CornerNet adopts the Associative Embedding (Newell et al., 2017) technique to separate different instances. Also there are some following works in bottom-up grouping manner (Zhou et al., 2019b; Duan et al., 2019). It should be noted that the bottom-up methods also do not need anchors during training and inference. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We have presented FoveaBox, a simple, effective, and completely anchor-free framework for generic object detection. By simultaneously predict the object position and the corresponding boundary, FoveaBox gives a clean solution for detecting objects without prior candidate boxes. We demonstrate its effectiveness on standard benchmarks and report extensive experimental analysis. We believe the simple and effective approach will serve as a solid baseline and help ease future research for object detection. ",
+        "bbox": [
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+    {
+        "type": "text",
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+    {
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+        "text": "Xinyu Zhou, Cong Yao, He Wen, Yuzhi Wang, Shuchang Zhou, Weiran He, and Jiajun Liang. East: an efficient and accurate scene text detector. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp. 5551–5560, 2017. ",
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+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            428,
+            297,
+            444
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 QUALITATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            379,
+            476
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Fig.5 shows the detection outputs of FoveaBox. Points and boxes with class probability larger than 0.5 are shown (before feeding into NMS). For each object, though there are several active points, the predicted boxes are very close to the ground-truth. These figures demonstrate that FoveaBox could directly generate accurate, robust box predictions, without the requirement of candidate anchors. ",
+        "bbox": [
+            173,
+            488,
+            825,
+            544
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/daa123700845237aff0c1602e0f807a2721b2b0ae599350fe6ae893b0fa790be.jpg",
+        "image_caption": [
+            "Figure 5: FoveaBox results on the COCO minival set. These results are based on ResNet-101, achieving a single model box AP of 38.6. For each pair, left is the detection results with bounding box, category, and confidence. Right is the score output map with their corresponding bounding boxes before feeding into non-maximum suppression (NMS). The score probability in each position is denoted by the color density. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            559,
+            816,
+            819
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 PER-CLASS DIFFERENCE: ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            392,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Fig. 6 shows per-class AP difference of FoveaBox and RetinaNet. Both of them are with ResNet-50- FPN backbone and 800 input scale. The vertical axis shows $\\mathbf { A P } _ { F o v e a B o x }$ -APRetinaNet. FoveaBox shows improvement in most of the classes. ",
+        "bbox": [
+            174,
+            128,
+            823,
+            171
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/9d75c5fbf37f13dc1c4aa1335506f3078e85b21adfca94ba3a80c4050cb16a64.jpg",
+        "image_caption": [
+            "Figure 6: AP difference of FoveaNet and RetinaNet on COCO dataset. Both models use ResNetFPN-50 as backbone and 800 input scales. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            183,
+            818,
+            358
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 MORE TRAINING TIME: ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            433,
+            380,
+            446
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 5 shows the detection performance under $2 \\mathbf { x }$ setting (24 epochs training). FoveaBox also outperforms the anchor-based baseline method. ",
+        "bbox": [
+            173,
+            458,
+            823,
+            487
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/ee366a1d89befcca3152aa7a0da0daab7e6c993fa379672c26f64d97cc324079.jpg",
+        "table_caption": [
+            "Table $5 \\colon 2 \\mathrm { x }$ epochs training (800 scale). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>method</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>RetinaNet-50</td><td>36.4</td><td>56.3</td><td>39.4</td></tr><tr><td>RetinaNet-101</td><td>38.1</td><td>57.8</td><td>41.1</td></tr><tr><td>FoveaBox-50</td><td>37.1</td><td>56.7</td><td>39.3</td></tr><tr><td>FoveaBox-101</td><td>38.7</td><td>58.4</td><td>41.1</td></tr></table>",
+        "bbox": [
+            359,
+            529,
+            633,
+            607
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.4 FEATURE ALIGNMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            631,
+            372,
+            645
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For feature alignment, we adopt a $3 \\times 3$ deformable convolutional layer to implement the transformation, as shown in Fig.7. The offset input to the deformable convolutional layer is $( { \\hat { t } } _ { x _ { 1 } } , { \\hat { t } } _ { y _ { 1 } } , { \\hat { t } } _ { x _ { 2 } } , { \\hat { t } } _ { y _ { 2 } } )$ of the bbox output. ",
+        "bbox": [
+            174,
+            656,
+            825,
+            699
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/7d0edf63e7fd123b34f67edf62dac10ba3ba1ed7dbc2ab6e42647f9f47a867a9.jpg",
+        "image_caption": [
+            "Figure 7: Feature alignment process. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            709,
+            787,
+            881
+        ],
+        "page_idx": 11
+    }
+]

parse/train/U_mat0b9iv/U_mat0b9iv.md

@@ -0,0 +1,715 @@
+# Multi-Prize Lottery Ticket Hypothesis: FINDING ACCURATE BINARY NEURAL NETWORKS BY PRUNING A RANDOMLY WEIGHTED NETWORK
+
+James Diffenderfer & Bhavya Kailkhura
+
+Center for Applied Scientific Computing Lawrence Livermore National Laboratory Livermore, CA 94550, USA {diffenderfer2,kailkhura1}@llnl.gov
+
+# ABSTRACT
+
+Recently, Frankle & Carbin (2019) demonstrated that randomly-initialized dense networks contain subnetworks that once found can be trained to reach test accuracy comparable to the trained dense network. However, finding these high performing trainable subnetworks is expensive, requiring iterative process of training and pruning weights. In this paper, we propose (and prove) a stronger Multi-Prize Lottery Ticket Hypothesis:
+
+A sufficiently over-parameterized neural network with random weights contains several subnetworks (winning tickets) that (a) have comparable accuracy to a dense target network with learned weights (prize 1), (b) do not require any further training to achieve prize 1 (prize 2), and (c) is robust to extreme forms of quantization (i.e., binary weights and/or activation) (prize 3).
+
+This provides a new paradigm for learning compact yet highly accurate binary neural networks simply by pruning and quantizing randomly weighted full precision neural networks. We also propose an algorithm for finding multi-prize tickets (MPTs) and test it by performing a series of experiments on CIFAR-10 and ImageNet datasets. Empirical results indicate that as models grow deeper and wider, multi-prize tickets start to reach similar (and sometimes even higher) test accuracy compared to their significantly larger and full-precision counterparts that have been weight-trained. Without ever updating the weight values, our MPTs-1/32 not only set new binary weight network state-of-the-art (SOTA) Top-1 accuracy – $9 4 . 8 \%$ on CIFAR-10 and $7 4 . 0 3 \%$ on ImageNet – but also outperform their full-precision counterparts by $1 . 7 8 \%$ and $0 . 7 6 \%$ , respectively. Further, our MPT-1/1 achieves SOTA Top-1 accuracy $( 9 1 . 9 \% )$ for binary neural networks on CIFAR-10. Code and pre-trained models are available at: https: //github.com/chrundle/biprop.
+
+# 1 INTRODUCTION
+
+Deep learning (DL) has made a significant breakthroughs in a wide range of applications (Goodfellow et al., 2016). These performance improvements can be attributed to the significant growth in the model size and the availability of massive computational resources to train such models. Therefore, these gains have come at the cost of large memory consumption, high inference time, and increased power consumption. This not only limits the potential applications where DL can make an impact but also have some serious consequences, such as, (a) generating huge carbon footprint, and (b) creating roadblocks to the democratization of AI. Note that significant parameter redundancy and a large number of floating-point operations are key factors incurring the these costs. Thus, for discarding the redundancy from DNNs, one can either (a) Prune: remove non-essential connections from an existing dense network, or (b) Quantize: constrain the full-precision (FP) weight and activation values to a set of discrete values which allows them to be represented using fewer bits. Further, one can exploit the complementary nature of pruning and quantization to combine their strengths.
+
+Figure 1: Multi-Prize Ticket Performance: Multi-prize tickets, obtained only by pruning and binarizing random networks, outperforms trained full precision and SOTA binary weight networks.   
+
+<table><tr><td></td><td>Configuration</td><td>Model</td><td>Weights</td><td>Memory Savings</td><td>Comp. Savings</td><td>Params</td><td>Accuracy</td></tr><tr><td rowspan="3">CTITAPIT</td><td>Dense (32/32)</td><td>ResNet18</td><td>Learned</td><td>1×</td><td>1×</td><td>11.2 M</td><td>93.02%</td></tr><tr><td>ProxQuant (1/32)</td><td>ResNet56</td><td>Learned</td><td>~32×</td><td>~2x</td><td>0.85 M</td><td>92.3%</td></tr><tr><td>MPT (1/32)</td><td>ResNet18</td><td>Pruned</td><td>~32x</td><td>~2x</td><td>2.2 M</td><td>94.8%</td></tr><tr><td rowspan="3">raeenee</td><td>Dense (32/32)</td><td>ResNet34</td><td>Learned</td><td>1×</td><td>1×</td><td>21.8M</td><td>73.27%</td></tr><tr><td>Quant-Net (1/32)</td><td>ResNet50</td><td>Learned</td><td>~32x</td><td>~2x</td><td>25.6M</td><td>72.8%</td></tr><tr><td>MPT (1/32)</td><td>WRN-50</td><td>Pruned</td><td>~32x</td><td>~2x</td><td>13.7 M</td><td>74.03%</td></tr></table>
+
+Although pruning and quantization1 are typical approaches used for compressing DNNs (Neill, 2020), it is not clear under what conditions and to what extent compression can be achieved without sacrificing the accuracy. The most extreme form of quanitization is binarization, where weights and/or activations can only have two possible values, namely $- 1 ( 0 )$ or $+ 1$ (the interest of this paper). In addition to saving memory, binarization results in more power efficient networks with significant computation acceleration since expensive multiply-accumulate operations (MACs) can be replaced by cheap XNOR and bit-counting operations (Qin et al., 2020a). In light of these benefits, it is of interest to question if conditions exists such that a binarized DNN can be pruned to achieve accuracy comparable to the dense FP DNN. More importantly, even if these favourable conditions are met then how do we find these extremely compressed (or compact) and highly accurate subnetworks?
+
+Traditional pruning schemes have shown that a pretrained DNN can be pruned without a significant loss in the performance. Recently, (Frankle & Carbin, 2019) made a breakthrough by showing that dense network contain sparse subnetworks that can match the performance of the original network when trained from scratch with weights being reset to their initialization (Lottery Ticket Hypothesis). Although the original approach to find these subnetworks still required training the dense network, some efforts (Wang et al., 2020b; You et al., 2019; Wang et al., 2020a) have been carried out to overcome this limitation. Recently a more intriguing phenomenon has been reported – a dense network with random initialization contains subnetworks that achieve high accuracy, without any further training (Zhou et al., 2019; Ramanujan et al., 2020; Malach et al., 2020; Orseau et al., 2020). These trends highlight good progress being made towards efficiently and accurately pruning DNNs.
+
+In contrast to these positive developments for pruning, results on binarizing DNNs have been mostly negative. To the best of our knowledge, post-training schemes have not been successful in binarizing pretrained models without retraining. Even with training binary neural networks (BNNs) from scratch (though inefficient), the community has not been able to make BNNs achieve comparable results to their full precision counterparts. The main reason being that network structures and weight optimization techniques are predominantly developed for full precision DNNs and may not be suitable for training BNNs. Thus, closing the gap in accuracy between the full precision and the binarized version may require a paradigm shift. Furthermore, this also makes one wonder if efficiently and accurately binarizing DNNs similar to the recent trends in pruning is ever feasible.
+
+In this paper, we show that a randomly initialized dense network contains extremely sparse binary subnetworks that without any weight training (i.e., efficient) have comparable performance to their trained dense and full-precision counterparts (i.e., accurate). Based on this, we state our hypothesis:
+
+Multi-Prize Lottery Ticket Hypothesis. A sufficiently over-parameterized neural network with random weights contains several subnetworks (winning tickets) that (a) have comparable accuracy to a dense target network with learned weights (prize 1), (b) do not require any further training to achieve prize 1 (prize 2), and (c) is robust to extreme forms of quantization (i.e., binary weights and/or activation) (prize 3).
+
+Contributions. First, we propose the multi-prize lottery ticket hypothesis as a new perspective on finding neural networks with drastically reduced memory size, much faster test-time inference and lower power consumption compared to their dense and full-precision counterparts. Next, we provide theoretical evidence of the existence of highly accurate binary subnetworks within a randomly weighted DNN (i.e., proving the multi-prize lottery ticket hypothesis). Specifically, we mathematically prove that we can find an $\varepsilon$ -approximation of a fully-connected ReLU DNN with width $n$ and depth $\ell$ using a sparse binary-weight DNN of sufficient width. Our proof indicates that this can be accomplished by pruning and binarizing the weights of a randomly weighted neural network that is a factor $O ( n ^ { 3 / 2 } \ell / \varepsilon )$ wider and $2 \ell$ deeper. To the best of our knowledge, this is the first theoretical work proving the existence of highly accurate binary subnetworks within a sufficiently overparameterized randomly initialized neural network. Finally, we provide biprop (binarize-prune optimizer) in Algorithm 1 to identify MPTs within randomly weighted DNNs and empirically test our hypothesis. This provides a completely new way to learn BNNs without relying on weight-optimization.
+
+Results. We explore two variants of multi-prize tickets – one with binary weights (MPT-1/32) and other with binary weights and activation (MPT-1/1) where $x / y$ denotes $x$ and $y$ bits to represent weights and activation, respectively. MPTs we find have $6 0 \mathrm { ~ - ~ } 8 0 \%$ fewer parameters than the original network. We perform a series of experiments on on small and large scale datasets for image recognition, namely CIFAR-10 (Krizhevsky et al., 2009) and ImageNet (Deng et al., 2009). On CIFAR-10, we test the performance of multi-prize tickets against the trend of making the model deeper and wider. We found that as models grow deeper and wider, both variants of multi-prize tickets start to reach similar (and sometimes even higher) test accuracy compared to the dense and full precision original network with learned weights. In other words, the performance of multiprize tickets improves with the amount of redundancy in the original network. We also carry out experiments with state-of-the-art (SOTA) architectures on CIFAR-10 and ImageNet datasets with an aim to investigate their redundancy. We find that within most randomly weighted SOTA DNNs reside extremely compact (i.e., sparse and binary) subnetworks which are smaller than, but match the performance of trained target dense and full precision networks. Furthermore, with minimal hyperparameter tuning, our MPTs achieve Top-1 accuracy comparable to (or higher than) SOTA BNNs. The performance of MPTs is further improved by allowing the parameters in BatchNorm layer to be learned. Finally, on both CIFAR-10 and ImageNet, MPT-1/32 subnetworks outperform their significantly larger and full-precision counterparts that have been weight-trained.
+
+# 2 MULTI-PRIZE LOTTERY TICKETS: THEORY AND ALGORITHMS
+
+We first prove the existence of MPTs in an overparameterized randomly weighted DNN. For ease of presentation, we state an informal version of Theorem 2 which can be found in Appendix B. We then explore two variants of tickets (MPT-1/32 and MPT-1/1) and provide an algorithm to find them.
+
+2.1 PROVING THE MULTI-PRIZE LOTTERY TICKETS HYPOTHESIS
+
+In this section we seek to answer the following question: What is the required amount of overparameterization such that a randomly weighted neural network can be compressed to a sparse binary subnetwork that approximates a dense trained target network?
+
+Theorem 1. (Informal Statement of Theorem 2) Let $\varepsilon , \delta > 0$ . For every fully-connected $( F C )$ target network with ReLU activations of depth $\ell$ and width $n$ with bounded weights, a random binary $F C$ network with ReLU activations of depth $2 \ell$ and width $\mathcal { O } \left( ( \ell n ^ { 3 / 2 } / \varepsilon ) + \ell n \log ( \ell n / \delta ) \right)$ contains with probability $( 1 - \delta )$ a binary subnetwork that approximates the target network with error at most $\varepsilon$ .
+
+Sketch of Proof. Consider a FC ReLU network ${ \cal F } ( { \pmb x } ) = { \cal W } ^ { ( \ell ) } \sigma ( { \pmb W } ^ { ( \ell - 1 ) } \cdot \cdot \cdot \sigma ( { \pmb W } ^ { ( 1 ) } { \pmb x } ) )$ , where $\sigma ( x ) = \operatorname* { m a x } \{ 0 , x \}$ , $\pmb { x } \in \mathbb { R } ^ { d }$ , $\boldsymbol { W } ^ { ( i ) } \in \mathbb { R } ^ { k _ { i } \times k _ { i - 1 } }$ , $k _ { 0 } = d$ , and $i \in [ \ell ]$ . Additionally, consider a FC network with binary weights given by $G ( \pmb { x } ) = B ^ { ( \ell ^ { \prime } ) } \sigma ( \pmb { B } ^ { ( \ell ^ { \prime } - 1 ) } \cdot \cdot \cdot \sigma ( \pmb { B } ^ { ( 1 ) } \pmb { x } ) )$ , where $B ^ { ( i ) } \in$ $\{ - 1 , + 1 \} ^ { k _ { i } ^ { \prime } \times k _ { i - 1 } ^ { \prime } }$ , $k _ { 0 } ^ { \prime } = d$ , and $i \in [ \ell ^ { \prime } ]$ . Our goal is to determine a lower bound on the depth, $\ell ^ { \prime }$ , and the widths, $\{ k _ { i } ^ { \prime } \} _ { i = 1 } ^ { \ell ^ { \prime } }$ , such that with probability $( 1 - \delta )$ the network $G ( \pmb { x } )$ contains a subnetwork $\tilde { G } ( { \pmb x } )$ satisfying $\| { \tilde { G } } ( { \pmb x } ) - F ( { \pmb x } ) \| \le \varepsilon$ , for any $\varepsilon > 0$ and $\delta \in \mathsf { \Gamma } ( 0 , 1 )$ . We first establish lower bounds on the width of a network of the form ${ \pmb g } ( { \pmb x } ) = { \pmb B } ^ { ( 2 ) } \sigma ( { \pmb B } ^ { ( 1 ) } { \pmb x } )$ such that with probability $( 1 - \delta ^ { \prime } )$ there exists a subnetwork $\tilde { \pmb g } ( \pmb x )$ of $\pmb { g } ( \pmb { x } )$ s.t. $\| \tilde { \pmb g } ( \pmb x ) - \sigma ( \pmb W \pmb x ) \| \le \varepsilon ^ { \prime }$ , for any $\bar { \varepsilon } ^ { \prime } > 0$ and $\delta ^ { \prime } \in ( 0 , 1 )$ . This process is carried out in detail in Lemmas 1, 2, and 3 in Appendix B. We have now approximated a single layer FC real-valued network using a subnetwork of a two-layer FC binary network. Hence, we can take $\ell ^ { \prime } = 2 \ell$ and Lemma 3 provides lower bounds on the width of each intermediate layer such that with probability $( 1 - \delta )$ there exists a subnetwork $\tilde { G } ( { \pmb x } )$ of $G ( \pmb { x } )$ satisfying $\| { \tilde { G } } ( { \pmb x } ) - F ( { \pmb x } ) \| \leq \varepsilon .$ . This is accomplished in Theorem 2 in Appendix B. □
+
+To the best of our knowledge this is the first theoretical result proving that a sparse binary-weight DNN that can approximate a real-valued target DNN. As it has been established that real-valued DNNs are universal approximators (Scarselli & Tsoi, 1998), our result carries the implication that sparse binary-weight DNNs are also universal approximators. In relation to the first result establishing the existence of real-valued subnetworks in a randomly weighted DNN approximating a realvalued target DNN (Malach et al., 2020), the lower bound on the width established in Theorem 2 is better than their lower bound of $O \left( \ell ^ { 2 } n ^ { 2 } \log ( \ell n / \delta ) / \varepsilon ^ { 2 } \right)$ .
+
+# 2.2 FINDING MULTI-PRIZE WINNING TICKETS
+
+Given the existence of multi-prize winning tickets from Theorem 2, a natural question arises – How should we find them? In this section, we answer this question by introducing an algorithm for finding multi-prize tickets.2 Specifically, we explore two variants of multi-prize tickets in this paper – 1) MPT-1/32 where weights are quantized to 1-bit with activations being real valued (i.e., 32-bits) and 2) MPT-1/1 where both weights and activations are quantized to 1-bit. We first outline a generic process for identifying MPTs along with some theoretical motivation for our approach.
+
+Given a neural network $g ( \pmb { x } ; \pmb { W } )$ with weights $W \in \mathbb { R } ^ { m }$ , we can express a subnetwork of $g$ using a binary mask $M \in \{ 0 , 1 \} ^ { m }$ as ${ \overset { \cdot } { g } } ( { \pmb x } ; M \odot { \pmb W } )$ , where $\odot$ denotes the Hadamard product. Hence, a binary subnetwork can be expressed as $g ( { \pmb x } ; M \odot B )$ , where $B \in \{ - 1 , + 1 \} ^ { m }$ . Lemma 1 in Appendix B indicates that rescaling the binary weights to $\{ - \alpha , \alpha \}$ using a gain term $\alpha \in \mathbb { R }$ is necessary to achieve good performance of the resulting subnetwork. We note that the use of gain terms is common in binary neural networks (Qin et al., 2020a; Martinez et al., 2020; Bulat & Tzimiropoulos, 2019). Combining all this allows us to represent a binary subnetwork as $g ( { \pmb x } ; \alpha ( M \odot { \pmb B } ) )$ .
+
+Now we focus on how to update $M , B$ , and $\alpha$ . Suppose $f ( x ; W ^ { * } )$ is a target network with optimized weights $W ^ { * }$ that we wish to approximate. Assuming ${ \dot { g } } ( { \pmb x } ; { \cdot } )$ is $\kappa$ -Lipschitz continuous yields
+
+$$
+\underbrace { \| g \left( x ; \alpha ( M \odot B ) \right) - f ( x ; W ^ { * } ) \| } _ { \mathrm { M p r e r o r } } \leq \kappa \underbrace { \| M \odot ( W - \alpha B ) \| } _ { \mathrm { B i n a r i z a t o n e r o r } } + \underbrace { \| g ( x ; M \odot W ) - f ( x ; W ^ { * } ) \| } _ { \mathrm { S u b n e t w o t k e r o r } } .
+$$
+
+Hence, the MPT error is bounded above by the error of the subnetwork of $g$ with the original weights and the error from binarizing the current subnetwork. This informs our approach for identifying MPTs: 1) Update a pruning mask $M$ that reduces the subnetwork error (lines $7 - 9$ in Algorithm 1), and 2) apply binarization with a gain term that minimizes the binarization error (lines 4 and 10).
+
+We first discuss how to update $M$ . While we could search for $M$ by minimizing the subnetwork error in (1), this would require the use of a pretrained target network (i.e., $f ( \pmb { x } ; \pmb { W } ^ { * } ) )$ . To avoid requiring a target network in our method we instead aim to minimize the training loss w.r.t. $M$ in the current binary subnetwork. Directly optimizing over the pruning mask is a combinatorial problem. So to update the pruning mask efficiently we optimize over a set of scores $\pmb { S } \in \mathbb { R } ^ { m }$ corresponding to each randomly initialized weight in the network. In this approach, each component of the randomly initialized weights is assigned a pruning score. The pruning scores are updated via backpropagation by computing the gradient of the loss function over minibatches with respect to the pruning scores (line 7). Then the magnitude of the scores in absolute value are used to identify the $P$ percent of weights in each layer that are least important to the success of the binary subnetwork (line 8). The components of the pruning mask corresponding to these indices are set to 0 and the remaining components are set to 1 (line 9). To avoid unintentionally pruning an entire layer of the network, we use a pruning mask for each layer that prunes $P$ percent of the weights in that layer. The choice to use pruning scores to update the mask $M$ was due to the fact that it is computationally efficient. The use of pruning scores is a well-established optimization technique used in a range of applications (Joshi & Boyd, 2009; Ramanujan et al., 2020).
+
+Algorithm 1 biprop: Finding multi-prize tickets in a randomly weighted neural network   
+1: Input: Neural network $g ( { \pmb x } ; \cdot )$ with $1 \mathrm { - }$ or 32-bit activations; Network depth $\ell$ ; Layer widths   
+$\{ k _ { j } \} _ { j = 1 } ^ { \ell }$ ; Loss function $L$ ; Training data $\{ ( \pmb { x } ^ { ( i ) } , \pmb { y } ^ { ( i ) } ) \} _ { i = 1 } ^ { N }$ ; Pruning percentage $P$ .   
+2: Randomly Initialize FP Parameters: Network weights $\{ W ^ { ( j ) } \} _ { j = 1 } ^ { \ell }$ ; Pruning scores $\{ S ^ { ( j ) } \} _ { j = 1 } ^ { \ell }$ .   
+3: Initialize Layerwise Pruning Masks: $\{ M ^ { ( j ) } \} _ { j = 1 } ^ { \ell }$ each to 1.   
+4: Initialize Binary Subnetwork Weights: $\{ B ^ { ( j ) } \} _ { j = 1 } ^ { \ell }  \{ \mathrm { s i g n } ( W ^ { ( j ) } ) \} _ { j = 1 } ^ { \ell }$ .   
+5: Initialize Layerwise Gain Terms: $\begin{array} { r } { \{ \alpha ^ { ( j ) } \} _ { j = 1 } ^ { \ell }  \{ \| M ^ { ( j ) } \odot W ^ { ( j ) } \| _ { 1 } / \| M ^ { ( j ) } \| _ { 1 } \} _ { j = 1 } ^ { \ell } . } \end{array}$ .   
+6: for $k = 1$ to $N _ { e p o c h s }$ do   
+7: $S ^ { ( j ) } \gets S ^ { ( j ) } - \eta \nabla _ { S ^ { ( j ) } } L \big ( \{ \alpha ^ { ( j ) } ( M ^ { ( j ) } \odot B ^ { ( j ) } ) \} _ { j = 1 } ^ { \ell } \big )$ Update pruning scores at layer $j$   
+8: $\{ \tau ( i ) \} _ { i = 1 } ^ { k _ { j } } \gets$ Sorting of indices $\{ i \} _ { i = 1 } ^ { k _ { j } }$ s.t. $| S _ { \tau ( i ) } ^ { ( j ) } | \le | S _ { \tau ( i + 1 ) } ^ { ( j ) } |$ Index sort over values $| S ^ { ( j ) } |$   
+9: $M _ { i } ^ { ( j ) } \gets \mathbb { 1 } _ { \{ \tau ( i ) \geq \lceil k _ { j } P / 1 0 0 \rceil \} } ( i )$ Update pruning mask at layer $j$   
+10: $\alpha ^ { ( j ) }  \Vert M ^ { ( j ) } \odot W ^ { ( j ) } \Vert _ { 1 } / \Vert M ^ { ( j ) } \Vert _ { 1 }$ Update gain term at layer $j$   
+11: Output: Return Binarized Subnetwork $g ( \pmb { x } ; \{ \alpha ^ { ( j ) } ( \pmb { M } ^ { ( j ) } \odot \pmb { B } ^ { ( j ) } ) \} _ { j = 1 } ^ { \ell } )$ .
+
+We now consider how to update $\textbf {  { B } }$ and $\alpha$ . By keeping $M$ fixed, we can derive the following closed form expressions that minimize the binarization error in (1): $B ^ { * } = \mathrm { s i g n } ( W )$ and $\alpha ^ { * } =$ $\| M \odot W \| _ { 1 } / \| M \| _ { 1 }$ . These closed form expressions indicate that only the gain term needs to be recomputed after each update to $M$ . Hence, $B = \mathrm { s i g n } ( W )$ throughout our entire approach (line 4). We update a gain term for each layer of the subnetwork in our approach based on the formula for $\alpha ^ { * }$ (line 10). More details on the derivation of $B ^ { * }$ and $\alpha ^ { * }$ are provided in Appendix C.
+
+Pseudocode for our method biprop (binarize-prune optimizer) is provided in Algorithm 1 and crossentropy loss is used in our experiments. Note that the process for identifying MPT-1/32 and MPT-1/1 differs only in computation of the gradient. Next, we explain how these gradients can be computed.
+
+2.2.1 UPDATING PRUNING SCORES FOR BINARY-WEIGHT TICKETS (MPT-1/32)
+
+As an example, for a FC network where the state at each layer is defined recursively by ${ \cal U } ^ { ( 1 ) } \ = \ \alpha ^ { ( 1 ) } ( B ^ { ( 1 ) } \odot M ^ { ( 1 ) } ) x$ and ${ \cal U } ^ { ( j ) } \ : = \ : \alpha ^ { ( j ) } \big ( { \cal B } ^ { ( j ) } \odot { \cal M } ^ { ( j ) } \big ) \sigma \big ( { \cal U } ^ { ( j - 1 ) } \big )$ we have $\begin{array} { r } { \frac { \partial \hat { L ^ { ( j ) } } } { \partial S _ { p , q } ^ { ( j ) } } = } \end{array}$ $\frac { \partial \cal L } { \partial U _ { q } ^ { ( j ) } } \frac { \partial U _ { q } ^ { ( j ) } } { \partial M _ { p , q } ^ { ( j ) } } \frac { \partial M _ { p , q } ^ { ( j ) } } { \partial S _ { p , q } ^ { ( j ) } }$ . We use the straight-through estimator (Bengio et al., 2013) for $\frac { \partial M _ { p , q } ^ { ( j ) } } { \partial S _ { p , q } ^ { ( j ) } }$ which yields $\begin{array} { r } { \frac { \partial L } { \partial S _ { p , q } ^ { ( j ) } } = \frac { \partial L } { \partial U _ { q } ^ { ( j ) } } \alpha ^ { ( j ) } B _ { p , q } ^ { ( j ) } \sigma \left( U _ { p } ^ { ( j - 1 ) } \right) } \end{array}$ , where $\frac { \partial L } { \partial U _ { q } ^ { ( j ) } }$ is computed via backpropagation.
+
+# 2.2.2 UPDATING PRUNING SCORES FOR BINARY-ACTIVATION TICKETS (MPT-1/1)
+
+Note that MPT-1/1 uses the sign activation function. From Section 2.2.1, it immediately follows that $\begin{array} { r } { \frac { \partial L } { \partial S _ { p , q } ^ { ( j ) } } = \frac { \partial L } { \partial U _ { q } ^ { ( j ) } } \alpha ^ { ( j ) } B _ { p , q } ^ { ( j ) } \ \mathrm { s i g n } \left( U _ { p } ^ { ( j - 1 ) } \right) } \end{array}$ However, updating $\frac { \partial L } { \partial U _ { q } ^ { ( j ) } }$ via backpropagation requires a gradient estimator for the sign activation function. To motivate our choice of estimator note that we can approximate the sign function using a quadratic spline parameterized by some $t > 0$ :
+
+$$
+s _ { t } ( x ) = \left\{ \begin{array} { c c c } { - 1 } & { : } & { x < - t } \\ { q _ { 1 } ( x ) } & { : } & { x \in [ - t , 0 ) } \\ { q _ { 2 } ( x ) } & { : } & { x \in [ 0 , t ) } \\ { 1 } & { : } & { x \geq t } \end{array} \right. .
+$$
+
+In (2), $q _ { i } ( x ) = a _ { i } x ^ { 2 } + b _ { i } x + c _ { i }$ and suitable values for the coefficients are derived using the following zero- and first-order constraints: $q _ { 1 } ( - t ) = - 1 , q _ { 1 } ( 0 ) = 0 , q _ { 2 } ( 0 ) = 0$ , $q _ { 2 } ( t ) \ \stackrel { - } { = } \ 1$ , $q _ { 1 } ^ { \prime } ( - t ) ~ = ~ 0 , ~ q _ { 1 } ^ { \prime } ( 0 ) ~ = ~ q _ { 2 } ^ { \prime } ( 0 )$ , and $q _ { 2 } ^ { \prime } ( t ) \ = \ 0$ . This yields $q _ { 1 } ( x ) ~ = ~ ( x / t ) ^ { 2 } + 2 ( x / t )$ and $q _ { 2 } ( x ) ~ = ~ - ( x / t ) ^ { 2 } + 2 ( x / t )$ . As $s _ { t } ( x )$ approximates $\mathrm { s i g n } ( x )$ , we can use $s _ { t } ^ { \prime } ( x )$ as our gradient estimator. Since $\begin{array} { r } { q _ { 1 } ^ { \prime } ( x ) \ = \ \frac { 2 } { t } ( 1 + \frac { x } { t } ) } \end{array}$ and $\begin{array} { r } { q _ { 2 } ^ { \prime } ( x ) ~ = ~ \frac { 2 } { t } ( 1 - \frac { x } { t } ) } \end{array}$ it follows that $s _ { t } ^ { \prime } ( x ) \ =$ $\begin{array} { r } { \left[ \frac { 2 } { t } \left( 1 - \frac { | x | } { t } \right) \right] \mathbb { 1 } _ { \{ x \in [ - t , t ] \} } ( x ) } \end{array}$ . The choice to approximate sign using a quadratic spline insteada gradient estimator that can be implemented efficiently in PyTorch as torch.clamp( $^ { 2 \star }$ (1-torch.abs $( \mathrm { x } ) / \mathrm { t } ) / \mathrm { t } , \mathrm { m i n } { = } 0 . 0 )$ ). We note that $\begin{array} { r l } { \operatorname* { l i m } _ { t \to 0 } s _ { t } ( x ) = } \end{array}$ $\mathrm { s i g n } ( x )$ , which suggests that smaller values of $t$ yield more suitable approximations. Our experiments use $s _ { 1 } ^ { \prime } ( x )$ as the gradient estimator since we found it to work well in practice. Finally, we note that taking $t = 1$ in our gradient estimator yields the same value as the gradient estimator in (Liu et al., 2018a), however, our implementation in PyTorch is $6 \times$ more memory efficient.
+
+# 3 EXPERIMENTAL RESULTS
+
+The primary goal of the experiments in Section 3.1 is to empirically verify our Multi-Prize Lottery Ticket Hypothesis. As a secondary objective, we would like to determine tunable factors that make randomly-initialized networks amenable to containing readily identifiable Multi-Prize Tickets (MPTs). Thus, we test our hypothesis against the general trend of increasing the model size (depth and width) and monitor the accuracy of the identified MPTs. After verifying our Multi-Prize Lottery Ticket Hypothesis, we consider the performance of MPTs compared to state-of-the-arts in binary neural networks and their dense counterparts on CIFAR-10 and ImageNet datasets in Section 3.2. Building upon edge-popup (Ramanujan et al., 2020), we implement Algorithm 1 to identify MPTs.3
+
+# 3.1 WHERE CAN WE EXPECT TO FIND MULTI-PRIZE TICKETS?
+
+In this section, we empirically test the effect of overparameterization on the performance of MPTs.   
+We overparameterize networks by making them (a) deeper (Sec. 3.1.1) and (b) wider (Sec. 3.1.2).
+
+![](images/341865cf4b701935aea67288a65087db658f204ca615ec157c3b2a759fbb061c.jpg)  
+Figure 2: Effect of Varying Depth and Pruning Rate: Comparing the Top-1 accuracy of small and binary MPTs to a large, full-precision, and weight-optimized network on CIFAR-10.
+
+We use VGG (Simonyan & Zisserman, 2014) variants as our network architectures for searching for MPTs. In each randomly weighted network, we find winning tickets MPT-1/32 and MPT-1/1 for different pruning rates using Algorithm 1. We choose our baselines as dense full-precision models with learned weights. In all experiments, we use three independent initializations and report the average of Top-1 accuracy with with error bars extending to the lowest and highest Top-1 accuracy. Additional experiment configuration details are provided in Appendix A.
+
+# 3.1.1 DO WINNING TICKETS EXIST IN DEEP NETWORKS?
+
+In this experiment, we empirically test the following hypothesis: As a network grows deeper, the performance of multi-prize tickets in the randomly initialized network will approach the performance of the same network with learned weights. We are further interested in exploring the required network depth for our hypothesis to be true.
+
+In Figure 2, we vary the depth of VGG architectures ( $\dot { d } = 2$ to 8) and compare the Top-1 accuracy of MPTs (at different pruning rates) with weight-trained dense network. We notice that there exist a range of pruning rates where the performance of MPTs are very similar, and beyond this range the performance drops quickly. Interestingly, as the network depth increases, more parameters can be pruned without hurting the performance of MPTs. For example, MPT-1/32 can match the performance of trained Conv-8 while having only $\sim 2 0 \%$ of its parameter count. Interestingly, the performance gap between MPT-1/32 and MPT-1/1 does not change much with depth across different pruning rates. We further note that the performance of MPTs improve when increasing the depth and both start to approach the performance of the dense model with learned weights. This gain starts to plateau beyond a certain depth, suggesting that the MPTs might be approaching the limit of their achievable accuracy. Surprisingly, MPT-1/32 performs equally good (or better) than the weight-trained model regardless of having $5 0 - 8 0 \%$ lesser parameters and weights being binarized.
+
+![](images/b344eab1cd1b7353193c60ebc7ab6acd5091b0c3a9879954ced47e1e4090962b.jpg)  
+Figure 3: Effect of Varying Width on MPT-1/32: Comparing the Top-1 accuracy of sparse and binary MPT-1/32 to dense, full-precision, and weight-optimized network on CIFAR-10.
+
+# 3.1.2 DO WINNING TICKETS EXIST IN WIDE NETWORKS?
+
+![](images/aa982870a628a7f2a08d4d8c829db951831b0a58f653ebf4b0d53ad5ab73a45f.jpg)  
+Figure 4: Effect of Varying Width on MPT-1/1: Comparing the Top-1 accuracy of sparse and binary MPT-1/1 to dense, full-precision, and weight-optimized network on CIFAR-10.
+
+Similar to the previous experiment, in this experiment, we empirically test the following hypothesis: As a network grows wider, the performance of multi-prize tickets in the randomly initialized network will approach the performance of the same network with learned weights. We are further interested in exploring the required layer width for our hypothesis to be true.
+
+In Figures 3 and 4, we vary the width of different VGG architectures and compare the Top-1 accuracy of MPT-1/32 and MPT-1/1 tickets (at different pruning rates) with weight-trained dense network. A width multiplier of value 1 corresponds to the models in Figure 2. Performance of all the models improves when increasing the width and the performance of both MPT-1/32 and MPT-1/1 start to approach the performance of the dense model with learned weights. Although, this gain starts to plateau beyond a certain width. For both MPT-1/32 and MPT-1/1, as the width and depth increase the performance at different pruning rates approach the same value. This observed phenomenon yields a more significant gain in the performance for MPTs with higher pruning rates. Similar to the previous experiment, the performance of MPT-1/32 matches (or exceeds) the performance of dense models for a large range of pruning rates. Furthermore, in the high width regime, a large number of weights $( \sim 9 0 \%$ ) can be pruned without having a noticeable impact on the performance of MPTs. We also notice that the performance gap between MPT-1/32 and MPT-1/1 decreases significantly with an increase the width which is in sharp contrast with the with the depth experiments where the performance gap between MPT-1/32 and MPT-1/1 appeared to be largely independent of the depth.
+
+Key Takeaways. Our experiments verify Multi-Prize Lottery Ticket Hypothesis and additionally convey the significance of choosing appropriate network depth and layer width for a given pruning rate. In particular, we find that a network with a large width can be pruned more aggressively without sacrificing much accuracy, while the accuracy of a network with smaller widths suffers when pruning a large percentage of the weights. Similar patterns hold for the depth of the networks as well. The amount of overparametrization needed to approach the performance of dense networks seems to differ for MPT variants – MPT-1/1 requires higher depth and width compared to MPT-1/32.
+
+3.2 HOW REDUNDANT ARE STATE-OF-THE-ART DEEP NEURAL NETWORKS?
+
+Having shown that MPTs can perform equally good (or better) than overparameterized networks, this experiment aims to answer: Are state-of-the-art weight-trained DNNs overparametrized enough that significantly smaller multi-prize tickets can match (or beat) their performance?
+
+Experimental Configuration. Instead of focusing on extremely large DNNs, we experiment with small to moderate size DNNs. Specifically, we analyze the redundancy of following backbone models: (1) VGG-Small and ResNet-18 on CIFAR-10, and (2) WideResNet-34 and WideResNet50 on ImageNet. As we will show later that even these models are highly redundant, thus, our finding automatically extends to larger models. In this process, we also perform a comprehensive comparison of the performance of our multi-prize winning tickets with state-of-the-art in binary neural networks (BNNs). Details on the experimental configuration are provided in Appendix A.
+
+This experiment uses Algorithm 1 to find MPTs within randomly initialized backbone networks. We compare the Top-1 accuracy and number of non-zero parameters for our MPT-1/32 and MPT1/1 tickets with selected baselines in BNNs (Qin et al., 2020a). Results for CIFAR-10 and ImageNet are shown in Tables 1, 2 and Tables 3, 4, respectively. Next to each MPT method we include the percentage of weights pruned in parentheses. Motivated by (Frankle et al., 2020), we also include models in which the BatchNorm parameters are learned when identifying the random subnetwork using biprop, indicated by $+ \mathrm { B N }$ . A more comprehensive comparison can be found in Appendix D.
+
+<table><tr><td>Method</td><td>Model</td><td>Top-1</td><td>Params</td></tr><tr><td>BinaryConnect</td><td>VGG-Small</td><td>91.7</td><td>4.6M</td></tr><tr><td>ProxQuant</td><td>ResNet-56</td><td>92.3</td><td>0.85M</td></tr><tr><td>DSQ</td><td>ResNet-20</td><td>90.2</td><td>0.27M</td></tr><tr><td>IR-Net</td><td>ResNet-20</td><td>90.8</td><td>0.27M</td></tr><tr><td>Full-Precision</td><td>ResNet-18</td><td>93.02</td><td>11.2M</td></tr><tr><td>MPT (80)</td><td>ResNet-18</td><td>94.66</td><td>2.2M</td></tr><tr><td>MPT(80) +BN</td><td>ResNet-18</td><td>94.8</td><td>2.2M</td></tr></table>
+
+Table 1: Comparison of MPT-1/32 with trained binary-1/32 networks on CIFAR-10.
+
+<table><tr><td>Method</td><td>Model</td><td>Top-1</td><td>Params</td></tr><tr><td>BNN</td><td>VGG-Small</td><td>89.9</td><td>4.6M</td></tr><tr><td>XNOR-Net</td><td>VGG-Small</td><td>89.8</td><td>4.6M</td></tr><tr><td>DSQ</td><td>VGG-Small</td><td>91.7</td><td>4.6M</td></tr><tr><td>IR-Net</td><td>ResNet-18</td><td>91.5</td><td>11.2 M</td></tr><tr><td>Full-Precision</td><td>VGG-Small</td><td>93.6</td><td>4.6M</td></tr><tr><td>MPT (75)</td><td>VGG-Small</td><td>88.52</td><td>1.44 M</td></tr><tr><td>MPT(75) +BN</td><td>VGG-Small</td><td>91.9</td><td>1.44 M</td></tr></table>
+
+Table 2: Comparison of MPT-1/1 with trained binary-1/1 networks on CIFAR-10.
+
+<table><tr><td>Method</td><td>Model</td><td>Top-1</td><td>Params</td></tr><tr><td>ABC-Net</td><td>ResNet-18</td><td>62.8</td><td>11.2 M</td></tr><tr><td>BWN</td><td>ResNet-18</td><td>60.8</td><td>11.2M</td></tr><tr><td>IR-Net</td><td>ResNet-34</td><td>70.4</td><td>21.8M</td></tr><tr><td>Quant-Net</td><td>ResNet-50</td><td>72.8</td><td>25.6M</td></tr><tr><td>Full-Precision</td><td>ResNet-34</td><td>73.27</td><td>21.8M</td></tr><tr><td>MPT (80)</td><td>WRN-50</td><td>72.67</td><td>13.7M</td></tr><tr><td>MPT(80)+BN</td><td>WRN-50</td><td>74.03</td><td>13.7M</td></tr></table>
+
+Table 3: Comparison of MPT-1/32 with trained binary-1/32 networks on ImageNet.
+
+<table><tr><td>Method</td><td>Model</td><td>Top-1</td><td>Params</td></tr><tr><td>BNN</td><td>AlexNet</td><td>27.9</td><td>62.3M</td></tr><tr><td>XNOR-Net</td><td>AlexNet</td><td>44.2</td><td>62.3M</td></tr><tr><td>ABC-Net</td><td>ResNet-34</td><td>52.4</td><td>21.8M</td></tr><tr><td>IR-Net</td><td>ResNet-34</td><td>62.9</td><td>21.8M</td></tr><tr><td>Full-Precision</td><td>ResNet-34</td><td>73.27</td><td>21.8M</td></tr><tr><td>MPT (60)</td><td>WRN-34</td><td>45.06</td><td>19.3M</td></tr><tr><td>MPT(60)+BN</td><td>WRN-34</td><td>52.07</td><td>19.3M</td></tr></table>
+
+Table 4: Comparison of MPT-1/1 with trained binary-1/1 networks on ImageNet.
+
+Our results highlight that SOTA DNN models are extremely redundant. For similar parameter count, our binary MPT-1/32 models outperform even full-precision models with learned weights. When compared to state-of-the-art in BNNs, with minimal hyperparameter tuning our multi-prize tickets achieve comparable (or higher) Top-1 accuracy. Specifically, our MPT-1/32 outperform trained binary weight networks on CIFAR-10 and ImageNet and our MPT-1/1 outperforms trained binary weight and activation networks on CIFAR-10. Further, on CIFAR-10 and ImageNet, MPT-1/32 networks with significantly reduced parameter counts outperform dense and full precision networks with learned weights. Searches for MPT-1/1 in BNN-specific architectures (Kim et al., 2020; Bulat et al., 2020a) and adopting other commonly used tricks to improve model & representation capacities (Bulat et al., 2020b; Yang et al., 2020; Lin et al., 2020; 2021) are likely to yield MPT-1/1 networks with improved performance. For example, up to a $7 \%$ gain in the MPT-1/1 accuracy was achieved by simply allowing BatchNorm parameters to be updated. Additionally, alternative approaches for updating the pruning mask in biprop could alleviate issues with back-propagating gradients through binary activation networks.
+
+# 4 DISCUSSION AND IMPLICATIONS
+
+Existing compression approaches (e.g., pruning and binarization) typically rely on some form of weight-training. This paper showed that a sufficiently overparametrized randomly weighted network contains binary subnetworks that achieve high accuracy (comparable to dense and full precision original network with learned weights) without any training. We referred to this finding as the Multi-Prize Lottery Ticket Hypothesis. We also proved the existence of such winning tickets and presented a generic procedure to find them. Our comparison with state-of-the-art neural networks corroborated our hypothesis. With minimal hyperparameter tuning, our binary weight multi-prize tickets outperformed current state-of-the-art in BNNs and proved its practical importance. Our work has several important practical and theoretical implications.
+
+Algorithmic. Our biprop framework enjoys certain advantages over traditional weightoptimization. First, contemporary experience suggests that sparse BNN training from scratch is challenging. Both sparseness and binarization bring their own challenges for gradient-based weight training – getting stuck at bad local minima in the sparse regime, incompatibility of backpropagation due to discontinuity in activation function, etc. Although we used gradient-based approaches in this paper, biprop is flexible to accommodate different class of algorithms that might avoid the pitfalls of gradient-based weight training. Next, in contrast to weight-optimization that requires large model size and massive compute resources to achieve high performance, our hypothesis suggests that one can achieve similar performance without ever training the large model. Therefore, strategies such as fast ticket search (You et al., 2019) or forward ticket selection (Ye et al., 2020) can be developed to enable more efficient ways of finding–or even designing–MPTs. Finally, as opposed to weight-optimization, biprop by design achieves compact yet accurate models.
+
+Theoretical. MPTs achieve similar performance as the model with learned weights. First, this observation notes the benefit of overparameterization in the neural network learning and reinforces the idea that an important task of gradient descent (and learning in general) may be to effectively compress overparametrized models to find multi-prize tickets. Next, our results highlight the expressive power of MPTs – since we showed that compressed subnetworks can approximate any target neural network who are known to be universal approximators, our MPTs are also universal approximators. Finally, the multi-prize lottery ticket hypothesis also uncovers the generalization properties of DNNs. Generalization theory for DL is still in its infancy and its not clear what and how DNNs learn (Neyshabur et al., 2017). Multi-prize lottery ticket hypothesis may serve as a valuable tool for answering such questions as it indicates the dependence of generalization on the compressiblity.
+
+Practical. Huge storage and heavy computation requirements of state-of-the-art deep neural networks inevitably limit their applications in practice. Multi-prize tickets are significantly lighter, faster, and efficient while maintaining performance. This unlocks a range of potential applications DL could be applied to (e.g., applications with resource-constrained devices such as mobile phones, embedded devices, etc.). Our results also indicate that existing SOTA models might be spending far more compute and power than is needed to achieve a certain performance. In other words, SOTA DL models have terrible energy efficiency and significant carbon footprint (Strubell et al., 2019). In this regard, MPTs have the potential to enable environmentally friendly artificial intelligence.
+
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+
+# ACKNOWLEDGEMENTS
+
+The authors would like to thank Shreya Chaganti for her valuable contributions to the biprop open source code development and for her help on training MPT models for the final version of the paper.
+
+This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344, Lawrence Livermore National Security, LLC. This document was prepared as an account of the work sponsored by an agency of the United States Government. Neither the United States Government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or Lawrence Livermore National Security, LLC. The views and opinions of the authors expressed herein do not necessarily state or reflect those of the United States Government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes. This work was supported by LLNL Laboratory Directed Research and Development project 20-ER-014 and released with LLNL tracking number LLNL-CONF-815432.
+
+# A HYPERPARAMETER CONFIGURATIONS
+
+# A.1 HYPERPARAMETERS FOR SECTION 3.1
+
+Experimental Configuration. For MPT-1/32 tickets, the network structure is not modified from the original. For MPT-1/1 tickets, the network structure is modified by moving the max-pooling layer directly after the convolution layer and adding a batch-normalization layer before the binary activation function, as is common in many BNN architectures (Rastegari et al., 2016). We choose our baselines as dense full precision models with learned weights. The baselines were obtained by training backbone networks using the Adam optimizer with learning rate of 0.0003 for 100 epochs and with a batch size of 60. In each randomly weighted backbone network, we find winning tickets MPT1/32 and MPT-1/1 for different pruning rates using Algorithm 1. For both the weight-optimized and MPT networks, the weights are initialized using the Kaiming Normal distribution (He et al., 2015). All training routines make use of a cosine decay learning rate policy.
+
+Table 5: Hyperparameter Configurations for CIFAR-10 Experiments   
+
+<table><tr><td>Method</td><td>Model</td><td>Optimizer</td><td>LR</td><td>Momentum</td><td>Weight Decay</td><td>Batch</td><td>Epochs</td></tr><tr><td>MPT-1/32</td><td>Conv2/4/6/8</td><td>SGD</td><td>0.1</td><td>0.9</td><td>1e-4</td><td>128</td><td>250</td></tr><tr><td>MPT-1/1</td><td>Conv2/4/6/8</td><td>Adam</td><td>0.1</td><td></td><td>1e-4</td><td>128</td><td>250</td></tr></table>
+
+# A.2 HYPERPARAMETERS FOR SECTION 3.2
+
+In these experiments, the weights are initialized using the Kaiming Normal distribution (He et al., 2015) for all the models except for MPT-1/32 on ImageNet where we use the Signed Constant initialization (Ramanujan et al., 2020) as it yielded slightly better performance. All training routines make use of a cosine decay learning rate policy. For ImageNet training we used a label smoothing value of 0.1 and a learning rate warmup length of 5 epochs.
+
+Table 6: Hyperparameter Configurations for CIFAR-10 Experiments   
+
+<table><tr><td>Method</td><td>Model</td><td>Opt.</td><td>LR</td><td>Momentum</td><td>Weight Decay</td><td>Batch</td><td>Epochs</td></tr><tr><td>MPT-1/32</td><td>ResNet-18</td><td>SGD</td><td>0.1</td><td>0.9</td><td>5e-4</td><td>256</td><td>250</td></tr><tr><td>MPT-1/32+BN</td><td>ResNet-18</td><td>SGD</td><td>0.1</td><td>0.9</td><td>5e-4</td><td>256</td><td>250</td></tr><tr><td>MPT-1/1</td><td>VGG-Small</td><td>Adam</td><td>3.63e-3</td><td>=</td><td>17.335</td><td>128</td><td>600</td></tr><tr><td>MPT-1/1+BN</td><td>VGG-Small</td><td>Adam</td><td>3.63e-3</td><td>1</td><td>1e-4</td><td>128</td><td>600</td></tr></table>
+
+Table 7: Hyperparameter Configurations for ImageNet Experiments   
+
+<table><tr><td>Method</td><td>Model</td><td>Optimizer</td><td>LR</td><td>Momentum</td><td>Weight Decay</td><td>Batch</td><td>Epochs</td></tr><tr><td>MPT-1/32</td><td>WRN-50</td><td>SGD</td><td>0.256</td><td>0.875</td><td>3.051757812e-5</td><td>256</td><td>120</td></tr><tr><td>MPT-1/32+BN</td><td>WRN-50</td><td>SGD</td><td>0.256</td><td>0.875</td><td>3.051757812e-5</td><td>256</td><td>120</td></tr><tr><td>MPT-1/1</td><td>WRN-34</td><td>Adam</td><td>2.56e-4</td><td></td><td>3.051757812e-5</td><td>256</td><td>250</td></tr><tr><td>MPT-1/1 +BN</td><td>WRN-34</td><td>Adam</td><td>2.56e-4</td><td></td><td>3.051757812e-5</td><td>256</td><td>250</td></tr></table>
+
+# B EXISTENCE OF BINARY-WEIGHT SUBNETWORK APPROXIMATING TARGET NETWORK
+
+In the following analysis, note that we write $B i n ( \{ - 1 , + 1 \} ^ { m \times n } )$ to denote matrices of dimension $m \times n$ whose components are independently sampled from a binomial distribution with elements $\{ - 1 , + 1 \}$ and probability $p = 1 / 2$ .
+
+Lemma 1. Let $s \in [ d ]$ , $\alpha \in \left[ - \frac { 1 } { \sqrt { s } } , \frac { 1 } { \sqrt { s } } \right]$ , $i \in [ d ]$ , and $\varepsilon , \delta \geq 0$ be given. Let $B \in \{ - 1 , + 1 \} ^ { k \times d }$ be chosen randomly from $B i n ( \{ - 1 , 1 \} ^ { k \times d } )$ and $\textbf { \em u } \in \{ - 1 , + 1 \} ^ { k }$ be chosen randomly from
+
+$B i n ( \{ - 1 , + 1 \} ^ { k } )$ . If
+
+$$
+k \geq \frac { 1 6 } { \varepsilon \sqrt { s } } + 1 6 \log \left( \frac { 2 } { \delta } \right) ,
+$$
+
+then with probability at least $1 - \delta$ there exist masks $\tilde { m } \in \{ 0 , 1 \} ^ { k }$ and $M \in \{ 0 , 1 \} ^ { k \times d }$ such that the function $g : \mathbb { R } ^ { d } \overset { \cdot } {  } \mathbb { R }$ defined by
+
+$$
+g ( \pmb { x } ) = ( \tilde { \pmb { m } } \odot \pmb { u } ) ^ { \top } \sigma \left( \varepsilon ( \pmb { M } \odot \pmb { B } ) \pmb { x } \right) ,
+$$
+
+satisfies
+
+$$
+| g ( \pmb { x } ) - \alpha x _ { i } | \leq \varepsilon ,
+$$
+
+for all $\| \pmb { x } \| _ { \infty } \leq 1$ . Furthermore, $\begin{array} { r } { \| \tilde { \pmb { m } } \| _ { 0 } = \| \pmb { M } \| _ { 0 } \le \frac { 2 } { \varepsilon \sqrt { s } } } \end{array}$ , and $\begin{array} { r } { \operatorname* { m a x } _ { 1 \leq j \leq k } \| M _ { j , : } \| _ { 0 } \leq 1 . } \end{array}$
+
+Proof. If $| \alpha | \le \varepsilon$ then taking $M = \mathbf { 0 }$ yields the desired result. Suppose that $| \alpha | > \varepsilon$ . Then there exists a $c _ { i } \in \mathbb { N }$ such that
+
+$$
+c _ { i } \varepsilon \leq | \alpha | \leq ( c _ { i } + 1 ) \varepsilon \quad { \mathrm { a n d } } \quad | c _ { i } \varepsilon - | \alpha | | \leq \varepsilon .
+$$
+
+Hence, it follows that
+
+$$
+| c _ { i } \varepsilon \mathrm { s i g n } ( \alpha ) x _ { i } - \alpha x _ { i } | = | x _ { i } | | c _ { i } \varepsilon - | \alpha | | \leq \varepsilon ,
+$$
+
+where the final inequality follows from (6) and the hypothesis that $\| { \pmb x } \| _ { \infty } \le 1$ . Our goal now is to show that with probability $1 - \delta$ the random initialization of $\textbf { \em u }$ and $\textbf {  { B } }$ yield masks $\tilde { m }$ and $M$ such that $\begin{array} { r } { g ( \pmb { x } ) = c _ { i } \bar { \varepsilon } \mathrm { s i g n } ( \alpha ) x _ { i } } \end{array}$ .
+
+Now fix $i \in [ d ]$ and take $\begin{array} { r } { k ^ { \prime } = \frac { k } { 2 } } \end{array}$ . First, we consider the probability
+
+$$
+P \left( | \{ j \in [ k ^ { \prime } ] : u _ { j } = + 1 \mathrm { ~ a n d ~ } B _ { j , i } = \operatorname { s i g n } ( \alpha ) \} | < c _ { i } \right) .
+$$
+
+As $\textbf { \em u }$ and $B _ { : , i }$ are each sampled from a binomial distribution with $k ^ { \prime }$ trials, the distribution that the pair $( u _ { j } , B _ { j , i } )$ is sampled from is a multinomial distribution with four possible events each having a probability of $1 / 4$ . Since we are only interested in the event $( u _ { j } , B _ { j , i } ) = ( + 1 , \mathrm { s i g n } ( \alpha ) )$ occurring, we can instead consider a binomial distribution where $P ( ( u _ { j } , B _ { j , i } ) \ = \ ( + 1 , \mathrm { s i g n } ( \alpha ) ) \ = \ { \textstyle { \frac { 1 } { 4 } } }$ and $P ( ( u _ { j } , B _ { j , i } ) \neq ( + 1 , \mathrm { s i g n } ( \alpha ) ) = \frac { 3 } { 4 }$ . Hence, using Hoeffding’s inequality we have that
+
+$$
+\begin{array} { r l r } & { } & { P \left( | \{ j \in [ k ^ { \prime } ] : u _ { j } = + 1 \mathrm { ~ a n d ~ } B _ { j , i } = \mathrm { s i g n } ( \alpha ) \} | < c _ { i } \right) \leq \exp \left( - 2 k ^ { \prime } \left( \frac { 1 } { 4 } - \frac { c _ { i } } { k ^ { \prime } } \right) ^ { 2 } \right) } \\ & { } & { = \exp \left( - \displaystyle \frac { 1 } { 8 } k ^ { \prime } + c _ { i } - 2 \frac { c _ { i } ^ { 2 } } { k ^ { \prime } } \right) } \\ & { } & { < \exp \left( - \displaystyle \frac { 1 } { 8 } k ^ { \prime } + 2 c _ { i } \right) , } \end{array}
+$$
+
+where the final inequality follows since $\exp ( )$ is an increasing function and $- 2 \frac { c _ { i } ^ { 2 } } { k ^ { \prime } } < 0$ . From (6) and the fact that $\begin{array} { r } { | \alpha | \le \frac { 1 } { \sqrt { s } } } \end{array}$ , it follows that
+
+$$
+c _ { i } \leq { \frac { 1 } { \varepsilon { \sqrt { s } } } } .
+$$
+
+Combining our hypothesis in (3) with (12) yields that
+
+$$
+- \frac { 1 } { 8 } k ^ { \prime } + c _ { i } = - \frac { 1 } { 1 6 } k + c _ { i } \le - \frac { 1 } { 1 6 } \left( \frac { 1 6 } { \varepsilon \sqrt { s } } + 1 6 \log \left( \frac { 2 } { \delta } \right) \right) + \frac { 1 } { \varepsilon \sqrt { s } } = \log \left( \frac { \delta } { 2 } \right) .
+$$
+
+Substituting (13) into (11) yields
+
+$$
+P \left( | \{ j \in [ k ^ { \prime } ] : u _ { j } = + 1 \mathrm { ~ a n d ~ } B _ { j , i } = \mathrm { s i g n } ( \alpha ) \} | < c _ { i } \right) < \frac { \delta } { 2 } .
+$$
+
+Additionally, it follows from the same argument that
+
+$$
+P \left( | \{ k ^ { \prime } < j \leq k : u _ { j } = - 1 \mathrm { ~ a n d ~ } B _ { j , i } = - \mathrm { s i g n } ( \alpha ) \} | < c _ { i } \right) < \frac { \delta } { 2 } .
+$$
+
+From (14) and (15) it follows with probability at least $1 - \delta$ that there exist sets $S _ { + } ~ : = ~ \{ j ~ :$ $u _ { j } ~ = ~ + 1$ and $B _ { j , i } = \mathrm { s i g n } ( \alpha ) \}$ and $S _ { - } ~ : = ~ \{ j ~ : ~ u _ { j } = - 1$ and $B _ { j , i } = - \mathrm { s i g n } ( \alpha ) \}$ satisfying $| \bar { S } _ { + } | = | S _ { - } | = \bar { c _ { i } }$ and $S _ { + } \cap S _ { - } = \emptyset$ . Using these sets, we define the components of the mask $\tilde { m }$ and $M$ by
+
+$$
+\tilde { m } _ { j } = \left\{ \begin{array} { l l } { 1 } & { : \quad j \in S _ { + } \cup S _ { - } } \\ { 0 } & { : \quad \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+and
+
+$$
+M _ { j , \ell } = \left\{ \begin{array} { l l l } { 1 } & { : } & { j \in S _ { + } \cup S _ { - } \mathrm { ~ a n d ~ } \ell = i } \\ { 0 } & { : } & { \mathrm { o t h e r w i s e } } \end{array} \right. .
+$$
+
+Using the definition of $g ( { \pmb x } )$ in (4) we now have that
+
+$$
+\begin{array} { l } { { g ( { \pmb x } ) = \displaystyle \sum _ { i \in S _ { + } } \sigma \left( \varepsilon \mathrm { s i g n } ( \alpha ) x _ { i } \right) - \sum _ { i \in S _ { - } } \sigma \left( - \varepsilon \mathrm { s i g n } ( \alpha ) x _ { i } \right) } } \\ { { \nonumber = c _ { i } \sigma \left( \varepsilon \mathrm { s i g n } ( \alpha ) x _ { i } \right) - c _ { i } \sigma \left( - \varepsilon \mathrm { s i g n } ( \alpha ) x _ { i } \right) } } \\ { { \qquad = c _ { i } \varepsilon \mathrm { s i g n } ( \alpha ) x _ { i } , } } \end{array}
+$$
+
+where the final equality follows from the identity $\sigma ( a ) - \sigma ( - a ) = a$ , for all $a \in \mathbb { R }$ . This concludes the proof of (5).
+
+Lastly, by our choice of $\tilde { m }$ in (16), $M$ in (17), and (12), it follows that
+
+$$
+\| \tilde { \pmb { m } } \| _ { 0 } = \| \pmb { M } \| _ { 0 } = 2 c _ { i } \leq \frac { 2 } { \varepsilon \sqrt { s } } ,
+$$
+
+and
+
+$$
+\operatorname* { m a x } _ { 1 \leq j \leq k } \| M _ { j , : } \| _ { 0 } \leq 1 ,
+$$
+
+which concludes the proof.
+
+The next step is to consider an analogue for Lemma A.2 from (Malach et al., 2020) which we provide in Lemma 2.
+
+Lemma 2. Let $s \in [ d ]$ , $\pmb { w } ^ { * } \in \left[ - \frac { 1 } { \sqrt { s } } , \frac { 1 } { \sqrt { s } } \right] ^ { d }$ with $\| \pmb { w } ^ { * } \| _ { 0 } \leq s$ , and $\varepsilon , \delta > 0$ be given. Let $\textbf { \textit { B } } \in$ $\{ - 1 , + 1 \} ^ { k \times d }$ be chosen randomly from $\bar { B i n } ( \{ - 1 , 1 \} ^ { k \times d } )$ and $\pmb { u } \in \{ - 1 , + 1 \} ^ { k }$ be chosen randomly from $B i \bar { n } ( \{ - 1 , + 1 \} ^ { k } )$ . If
+
+$$
+k \geq s \cdot \left\lceil \frac { 1 6 \sqrt { s } } { \varepsilon } + 1 6 \log \left( \frac { 2 s } { \delta } \right) \right\rceil ,
+$$
+
+then with probability at least $1 - \delta$ there exist masks $\tilde { m } \in \{ 0 , 1 \} ^ { k }$ and $M \in \{ 0 , 1 \} ^ { k \times d }$ such that the function $g : \mathbb { R } ^ { d } \overset { \cdot } {  } \mathbb { R }$ defined by
+
+$$
+g ( \pmb { x } ) = ( \tilde { \pmb { m } } \odot \pmb { u } ) ^ { \top } \sigma \left( \varepsilon ( \pmb { M } \odot \pmb { B } ) \pmb { x } \right) ,
+$$
+
+satisfies
+
+$$
+| g ( \pmb { x } ) - \langle \pmb { w } ^ { * } , \pmb { x } \rangle | \leq \varepsilon , f o r a l l \| \pmb { x } \| _ { \infty } \leq 1 .
+$$
+
+Furthermore, $\begin{array} { r } { \| \tilde { m } \| _ { 0 } = \| M \| _ { 0 } \leq \frac { 2 s \sqrt { s } } { \varepsilon } a n d \operatorname* { m a x } _ { 1 \leq j \leq k } \| M _ { j , : } \| _ { 0 } \leq 1 . } \end{array}$
+
+$\begin{array} { r } {  { k } \ = \  { s } \cdot \left\lceil \frac { 1 6 \sqrt { s } } { \varepsilon } + 1 6 \log \left( \frac { 2 s } { \delta } \right) \right\rceil } \end{array}$ and set  be maske $\begin{array} { l l l } { k ^ { \prime } } & { = } & { { \frac { k } { s } } } \end{array}$ . Note that if  the desired value $k \_ s$ $\begin{array} { r } { \left\lceil \frac { 1 6 \sqrt { s } } { \varepsilon } + 1 6 \log \left( \frac { 2 s } { \delta } \right) \right\rceil } \end{array}$ $k$
+
+decompose $u , \tilde { m } , B$ , and $M$ into $s$ equal size submatrices by defining
+
+$$
+\begin{array} { r l } & { u ^ { ( i ) } : = \left[ u _ { k ^ { \prime } ( i - 1 ) + 1 } \quad \cdots \quad u _ { k ^ { \prime } i } \right] ^ { \top } \in \left\{ - 1 , + 1 \right\} ^ { k ^ { \prime } \times 1 } } \\ & { \tilde { m } ^ { ( i ) } : = \left[ \tilde { m } _ { k ^ { \prime } ( i - 1 ) + 1 } \quad \cdots \quad \tilde { m } _ { k ^ { \prime } i } \right] ^ { \top } \in \left\{ 0 , 1 \right\} ^ { k ^ { \prime } \times 1 } } \\ & { B ^ { ( i ) } : = \left[ \begin{array} { c c c } { b ( k ^ { \prime } ( i - 1 ) + 1 ) , 1 } & { \cdots } & { b ( k ^ { \prime } ( i - 1 ) + 1 ) , d } \\ { \vdots } & { \ddots } & { \vdots } \\ { b _ { k ^ { \prime } i , 1 } } & { \cdots } & { b _ { k ^ { \prime } i , d } } \end{array} \right] \in \left\{ - 1 , + 1 \right\} ^ { k ^ { \prime } \times d } } \\ & { M ^ { ( i ) } : = \left[ \begin{array} { c c c } { m _ { ( k ^ { \prime } ( i - 1 ) + 1 ) , 1 } } & { \cdots } & { m _ { ( k ^ { \prime } ( i - 1 ) + 1 ) , d } } \\ { \vdots } & { \ddots } & { \vdots } \\ { m _ { k ^ { \prime } i , 1 } } & { \cdots } & { m _ { k ^ { \prime } i , d } } \end{array} \right] \in \left\{ 0 , 1 \right\} ^ { k ^ { \prime } \times d } , } \end{array}
+$$
+
+for $i \in [ s ]$ . Note that these submatrices satisfy
+
+$$
+{ \pmb u } = \left[ \begin{array} { c } { { { \pmb u } ^ { ( 1 ) } } } \\ { { \vdots } } \\ { { { \pmb u } ^ { ( s ) } } } \end{array} \right] , \tilde { m } = \left[ \begin{array} { c } { { { \tilde { m } } ^ { ( 1 ) } } } \\ { { \vdots } } \\ { { { \tilde { m } } ^ { ( s ) } } } \end{array} \right] , B = \left[ \begin{array} { c } { { { \pmb B } ^ { ( 1 ) } } } \\ { { \vdots } } \\ { { { \pmb B } ^ { ( s ) } } } \end{array} \right] , M = \left[ \begin{array} { c } { { { \pmb U } ^ { ( 1 ) } } } \\ { { \vdots } } \\ { { { \pmb U } ^ { ( s ) } } } \end{array} \right] .
+$$
+
+Now let $\mathcal { T } : = \{ i \in [ d ] : w _ { i } ^ { * } \neq 0 \}$ . By our hypothesis that $\| \pmb { w } ^ { * } \| _ { 0 } \leq s$ , it follows that $| \mathcal { T } | \leq s$ WLOG, assume that $\mathcal { T } \subseteq [ s ]$ . Now fix $i \in [ s ]$ and define $g _ { i } : \mathbb { R } ^ { d }  \mathbb { R }$ by
+
+$$
+g _ { i } ( \pmb { x } ) : = \left( \tilde { \pmb { m } } ^ { ( i ) } \odot \pmb { u } ^ { ( i ) } \right) ^ { \top } \sigma \left( \varepsilon ( M ^ { ( i ) } \odot \pmb { B } ^ { ( i ) } ) \pmb { x } \right)
+$$
+
+By (23), taking $\varepsilon ^ { \prime } = \textstyle { \frac { \varepsilon } { s } }$ and $\begin{array} { r } { \delta ^ { \prime } = \frac { \delta } { s } } \end{array}$ yields that $\begin{array} { r } { k ^ { \prime } \ge \frac { 1 6 } { \varepsilon ^ { \prime } \sqrt { s } } + 1 6 \log \left( \frac { 2 } { \delta ^ { \prime } } \right) } \end{array}$ . Hence, it follows from Lemma 1 that with probability at least $1 - \delta ^ { \prime }$ there exist $\tilde { m } ^ { ( i ) } \in \{ 0 , 1 \} ^ { k ^ { \prime } }$ and $M ^ { ( i ) } \in \{ 0 , 1 \} ^ { k ^ { \prime } \times d }$ such that
+
+$$
+| g _ { i } ( \pmb { x } ) - w _ { i } ^ { * } x _ { i } | \leq \varepsilon ^ { \prime } = \frac { \varepsilon } { s } ,
+$$
+
+for every $\pmb { x } \in \mathbb { R } ^ { d }$ with $\| \pmb { x } \| _ { \infty } \leq 1$ , and
+
+$$
+\| \tilde { \pmb { m } } ^ { ( i ) } \| _ { 0 } = \| \pmb { M } ^ { ( i ) } \| _ { 0 } \le \frac { 2 } { \varepsilon ^ { \prime } \sqrt { s } } = \frac { 2 \sqrt { s } } { \varepsilon } \quad \mathrm { a n d } \quad \operatorname* { m a x } _ { \substack { k ^ { \prime } ( i - 1 ) + 1 \le j \le k ^ { \prime } i } } \| M _ { j , i } ^ { ( i ) } \| _ { 0 } \le 1 .
+$$
+
+By the definition of $g ( { \pmb x } )$ in (24), using (30) yields
+
+$$
+g ( \pmb { x } ) = \left( \tilde { m } \odot \pmb { u } \right) ^ { \top } \sigma \left( \varepsilon ( M \odot B ) \pmb { x } \right) = \sum _ { i = 1 } ^ { s } \left( \tilde { m } ^ { ( i ) } \odot \pmb { u } ^ { ( i ) } \right) ^ { \top } \sigma \left( \varepsilon ( M ^ { ( i ) } \odot \pmb { B } ^ { ( i ) } ) \pmb { x } \right) = \sum _ { i = 1 } ^ { s } g _ { i } ( \pmb { x } ) .
+$$
+
+Hence, combining (32) for all $i \in [ s ]$ , it follows that with probability at least $1 - \delta$ we have
+
+$$
+| g ( \pmb { x } ) - \langle \pmb { w } ^ { * } , \pmb { x } \rangle | = \left| \sum _ { i = 1 } ^ { s } g _ { i } ( \pmb { x } ) - \sum _ { i = 1 } ^ { s } w _ { i } ^ { * } \pmb { x } _ { i } \right| \leq \sum _ { i = 1 } ^ { s } | g _ { i } ( \pmb { x } ) - w _ { i } ^ { * } \pmb { x } _ { i } | \leq \varepsilon .
+$$
+
+Finally, it follows from (30) and (33) that
+
+$$
+\| \tilde { \ b { m } } \| _ { 0 } = \| \ b { M } \| _ { 0 } \leq \frac { 2 s \sqrt { s } } { \varepsilon } \quad \mathrm { a n d } \quad \operatorname* { m a x } _ { 1 \leq j \leq k } \| \ b { M } _ { j , : } \| _ { 0 } \leq 1 ,
+$$
+
+which concludes the proof.
+
+We now state and prove an analogue to Lemma A.5 in (Malach et al., 2020) which is the last lemma we will need to establish the desired result.
+
+Lemma 3. Let $s \in [ d ]$ , $\begin{array} { r } { W ^ { * } ~ \in ~ \left[ - \frac { 1 } { \sqrt { s } } , \frac { 1 } { \sqrt { s } } \right] ^ { n \times d } } \end{array}$ with $\| W ^ { * } \| _ { 0 } \leq s , F : \mathbb { R } ^ { d } \to \mathbb { R } ^ { n }$ defined by $F _ { i } ( \pmb { x } ) = \sigma ( \langle \pmb { w } _ { i } ^ { * } , \pmb { x } \rangle )$ , and $\varepsilon , \delta > 0$ be given. Let $B \in \{ - 1 , + 1 \} ^ { k \times d }$ be chosen randomly from $B i n ( \{ - 1 , 1 \} ^ { k \times d } )$ and $U \in \{ - 1 , + 1 \} ^ { k \times n }$ be chosen randomly from $B i n ( \{ - 1 , + 1 \} ^ { k \times n } )$ . If
+
+$$
+k \geq n s \cdot \left\lceil \frac { 1 6 \sqrt { n s } } { \varepsilon } + 1 6 \log \left( \frac { 2 n s } { \delta } \right) \right\rceil ,
+$$
+
+then with probability at least $1 - \delta$ there exist masks $\tilde { M } \in \{ 0 , 1 \} ^ { k \times n }$ and $M \in \{ 0 , 1 \} ^ { k \times d }$ such that the function $G : \mathbb { R } ^ { d }  \mathbb { R } ^ { n }$ defined by
+
+$$
+G ( \pmb { x } ) = \sigma \left( ( \tilde { M } \odot \pmb { U } ) ^ { \top } \sigma \left( \varepsilon ( M \odot \pmb { B } ) \pmb { x } \right) \right) ,
+$$
+
+satisfies
+
+$$
+\| G ( \pmb { x } ) - F ( \pmb { x } ) \| _ { 2 } \leq \varepsilon , f o r a l l \| \pmb { x } \| _ { \infty } \leq 1 .
+$$
+
+Furthermore, $\begin{array} { r } { \| \tilde { M } \| _ { 0 } = \| M \| _ { 0 } \leq \frac { 2 n s \sqrt { n s } } { \varepsilon } } \end{array}$ .
+
+Proof. Assume $\begin{array} { r } { k \ = \ n s \ \cdot \ \left\lceil \frac { 1 6 \sqrt { n s } } { \varepsilon } + 1 6 \log \left( \frac { 2 n s } { \delta } \right) \right\rceil } \end{array}$ and set $\begin{array} { l } { { k ^ { \prime } } } \end{array} = \begin{array} { l } { { \frac { k } { n } } } \end{array}$ . Note that if $k \ > \ n s \ .$ $\begin{array} { r } { \left\lceil { \frac { 1 6 \sqrt { n s } } { \varepsilon } } + 1 6 \log \left( \frac { 2 n s } { \delta } \right) \right\rceil } \end{array}$ then excess neurons can be masked to yield the desired value for $k$ . As in the proof of Lemma 2, we can split $U , { \tilde { M } } ,$ B, and $M$ into $n$ submatrices, denoted $U ^ { ( i ) } \in \{ - 1 , + 1 \} ^ { k ^ { \prime } \times n }$ , $\tilde { M } ^ { ( i ) } \in \{ - 1 , + 1 \} ^ { k ^ { \prime } \times n }$ , $B ^ { ( i ) } \in \{ - 1 , + 1 \} ^ { k ^ { \prime } \times d }$ , and $M ^ { ( i ) } \in \{ - 1 , + 1 \} ^ { k ^ { \prime } \times d }$ for $i \in [ n ]$ , such that
+
+$$
+\boldsymbol { U } = \left[ \begin{array} { c } { \boldsymbol { U } ^ { ( 1 ) } } \\ { \vdots } \\ { \boldsymbol { U } ^ { ( n ) } } \end{array} \right] , \tilde { \boldsymbol { M } } = \left[ \begin{array} { c } { \boldsymbol { \tilde { M } } ^ { ( 1 ) } } \\ { \vdots } \\ { \boldsymbol { \tilde { M } } ^ { ( n ) } } \end{array} \right] , \boldsymbol { B } = \left[ \begin{array} { c } { \boldsymbol { B } ^ { ( 1 ) } } \\ { \vdots } \\ { \boldsymbol { B } ^ { ( n ) } } \end{array} \right] , \mathrm { a n d } \boldsymbol { M } = \left[ \begin{array} { c } { \boldsymbol { M } ^ { ( 1 ) } } \\ { \vdots } \\ { \boldsymbol { M } ^ { ( n ) } } \end{array} \right] .
+$$
+
+To simplify notation in the following definition, we define the vectors $\tilde { m } ^ { ( i ) } : = \tilde { M } _ { : , i } ^ { ( i ) }$ and $\tilde { \mathbf { \boldsymbol { u } } } ^ { ( i ) } : =$ $\tilde { U } _ { : , i } ^ { ( i ) }$ . Now we define the functions $g _ { i } : \mathbb { R } ^ { d }  \mathbb { R }$ by
+
+$$
+g _ { i } ( \pmb { x } ) = \left( \tilde { \pmb { m } } ^ { ( i ) } \odot \pmb { u } ^ { ( i ) } \right) ^ { \top } \sigma \left( \beta ( \pmb { M } ^ { ( i ) } \odot \pmb { B } ^ { ( i ) } ) \pmb { x } \right) ,
+$$
+
+for each $\textit { i } \in \ [ n ]$ . Taking $\begin{array} { l l l } { \varepsilon ^ { \prime } } & { = } & { { \frac { \varepsilon } { \sqrt { n } } } } \end{array}$ and $\begin{array} { l l l } { \delta ^ { \prime } } & { = } & { { \frac { \delta } { n } } } \end{array}$ , it follows from (37) that $k ^ { \prime } \geq s \mathrm { ~ . ~ }$ $\begin{array} { r } { \left\lceil { \frac { 1 6 \sqrt { s } } { \varepsilon ^ { \prime } } } + 1 6 \log \left( \frac { 2 s } { \delta ^ { \prime } } \right) \right\rceil } \end{array}$ . As the hypotheses of Lemma 2 are satisfied, with probability at least $\textstyle 1 - { \frac { \delta } { n } }$ there exist masks $\tilde { m } ^ { ( i ) }$ and $M ^ { ( i ) }$ with
+
+$$
+\| \tilde { \pmb { m } } ^ { ( i ) } \| _ { 0 } = \| \pmb { M } ^ { ( i ) } \| _ { 0 } \leq \frac { 2 s \sqrt { s } } { \varepsilon ^ { \prime } } = \frac { 2 s \sqrt { n s } } { \varepsilon }
+$$
+
+such that
+
+$$
+| g _ { i } ( \pmb { x } ) - \langle \pmb { W } _ { i } ^ { * } , \pmb { x } \rangle | \leq \frac { \varepsilon } { \sqrt { n } } , \mathrm { ~ f o r ~ a l l ~ } \| \pmb { x } \| _ { \infty } \leq 1 .
+$$
+
+For each $i \in [ n ]$ , note that this results in choosing the columns of the mask $\tilde { M } ^ { ( i ) }$ by
+
+$$
+\tilde { M } _ { : , \ell } ^ { ( i ) } = \left\{ \begin{array} { l l } { \tilde { m } ^ { ( i ) } } & { : \quad \ell = i } \\ { { \bf 0 } } & { : \quad \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+Combining this choice with (40) yields
+
+$$
+( \tilde { M } \odot U ) ^ { \top } \sigma \left( \beta ( M \odot B ) \pmb { x } \right) = \left[ \begin{array} { l } { g _ { 1 } ( \pmb { x } ) } \\ { \vdots } \\ { g _ { n } ( \pmb { x } ) } \end{array} \right] .
+$$
+
+By the definition of $G ( \pmb { x } )$ in (38), it follows from (45) that
+
+$$
+G ( \pmb { x } ) = \left[ \begin{array} { c } { \sigma ( g _ { 1 } ( \pmb { x } ) ) } \\ { \vdots } \\ { \sigma ( g _ { n } ( \pmb { x } ) ) } \end{array} \right] .
+$$
+
+Combining (43) and (46), we have with probability at least $1 - \delta$ that
+
+$$
+\| G ( \pmb { x } ) - F ( \pmb { x } ) \| _ { 2 } ^ { 2 } = \sum _ { i = 1 } ^ { n } \left( \sigma ( g _ { i } ( \pmb { x } ) ) - \sigma ( \langle \pmb { w } _ { i } ^ { * } , \pmb { x } \rangle ) \right) ^ { 2 } \leq \sum _ { i = 1 } ^ { n } \left( g _ { i } ( \pmb { x } ) - \langle \pmb { W } _ { i } ^ { * } , \pmb { x } \rangle \right) ^ { 2 } \leq \varepsilon ^ { 2 } .
+$$
+
+Finally, it follows from (42) and (44) that
+
+$$
+\| \tilde { M } \| _ { 0 } = \| M \| _ { 0 } \leq \frac { 2 n s \sqrt { n s } } { \varepsilon }
+$$
+
+which concludes the proof.
+
+We are now ready to prove the main result in Theorem 2.
+
+Theorem 2. Let $\begin{array} { r l r } { \ell , n , s } & { \in \mathrm {  ~ { \cal ~ N } ~ } , \ W ^ { ( 1 ) * } } & { \in } & { \Big [ - \frac { 1 } { \sqrt { s } } , \frac { 1 } { \sqrt { s } } \Big ] ^ { d \times n } , \ \{ \pmb { W } } ^ { ( i ) * } \} _ { i = 2 } ^ { \ell - 1 } \in \Big [ - \frac { 1 } { \sqrt { n } } , \frac { 1 } { \sqrt { n } } \Big ] ^ { n \times n } ,  \end{array}$ and W ( \` ) ∗ ∈ h − √1n , √1n i . Assume that for each $i \in [ \ell ]$ we have $\| \mathbf { W } ^ { ( i ) * } \| _ { 2 } ~ \le ~ 1$ and $\operatorname* { m a x } _ { j } \| \pmb { W } _ { j } ^ { ( i ) * } \| _ { 0 } \leq s .$ . Define $F ( x ) : = F ^ { ( \ell ) } \circ \cdot \cdot \cdot \circ F ^ { ( 1 ) } ( { \pmb x } )$ where $F ^ { ( i ) } ( { \pmb x } ) = \sigma ( { \pmb W } ^ { ( i ) * } { \pmb x } )$ for $i \in [ \ell - 1 ]$ and $F ^ { ( \ell ) } ( { \pmb x } ) = { \pmb W } ^ { ( \ell ) * } { \pmb x } . F i x \varepsilon , \delta \in ( 0 , 1 ) .$ .
+
+Let $B ^ { ( 1 ) } \ \in \ \{ - 1 , + 1 \} ^ { k \times d }$ be sampled from $B i n ( \{ - 1 , + 1 \} ^ { k \times d } )$ , $\{ B ^ { ( i ) } \} _ { i = 2 } ^ { \ell } ~ \in ~ \{ - 1 , + 1 \} ^ { k \times n }$ be sampled from Bin({−1, +1}k×n), {U (i)}\`−1i=1 $\begin{array} { r l r } { \{ U ^ { ( i ) } \} _ { i = 1 } ^ { \ell - 1 } } & { { } \in } & { \{ - 1 , + 1 \} ^ { k \times n } } \end{array}$ be sampled from $B i n ( \{ - 1 , + 1 \} ^ { k \times n } )$ and $U ^ { ( \ell ) } \in \{ - 1 , + 1 \} ^ { k \times 1 }$ sampled from $B i n ( \{ - 1 , + 1 \} ^ { k \times 1 } )$ . If
+
+$$
+k \geq n s \cdot \left\lceil \frac { 3 2 \ell \sqrt { n s } } { \varepsilon } + 1 6 \log \left( \frac { 2 n s \ell } { \delta } \right) \right\rceil ,
+$$
+
+then with probability at least $1 - \delta$ there exist binary masks $\{ M ^ { ( i ) } \} _ { i = 1 } ^ { \ell }$ and $\{ \tilde { M } ^ { ( i ) } \} _ { i = 1 } ^ { \ell }$ for $\{ B ^ { ( i ) } \} _ { i = 1 } ^ { \ell }$ and $\{ \pmb { U } ^ { ( i ) } \} _ { i = 1 } ^ { \ell }$ , respectively, such that the function $G : \mathbb { R } ^ { d }  \mathbb { R }$ defined by
+
+$$
+G ( \pmb { x } ) : = G ^ { ( \ell ) } \circ \cdot \cdot \cdot \circ G ^ { ( 1 ) } ( \pmb { x } ) ,
+$$
+
+where
+
+$$
+\begin{array} { r l } & { G ^ { ( i ) } ( \pmb { x } ) : = \sigma \left( ( \tilde { M } ^ { ( i ) } \odot \pmb { U } ^ { ( i ) } ) ^ { \top } \sigma ( \varepsilon ( M ^ { ( i ) } \odot \pmb { B } ^ { ( i ) } ) \pmb { x } ) \right) , f o r i \in [ \ell - 1 ] } \\ & { G ^ { ( \ell ) } ( \pmb { x } ) : = ( \tilde { M } ^ { ( i ) } \odot \pmb { U } ^ { ( i ) } ) ^ { \top } \sigma ( \varepsilon ( M ^ { ( i ) } \odot \pmb { B } ^ { ( i ) } ) \pmb { x } ) , } \end{array}
+$$
+
+satisfies
+
+$$
+| G ( \pmb { x } ) - F ( \pmb { x } ) | \leq \varepsilon , f o r a l l \| \pmb { x } \| _ { 2 } .
+$$
+
+Additionally, kM˜ k0 = kM k0 ≤ 4ns\`2 nsε .
+
+Proof. Let $i \in [ \ell - 1 ]$ . Using Lemma 3 with $\begin{array} { r } { \varepsilon ^ { \prime } = \frac { \varepsilon } { 2 \ell } } \end{array}$ and $\begin{array} { r } { \delta ^ { \prime } = \frac { \delta } { \ell } } \end{array}$ , with probability at least $\textstyle 1 - { \frac { \delta } { \ell } }$ there exist $M ^ { ( i ) }$ and $\tilde { M } ^ { ( i ) }$ such that
+
+$$
+\| G ^ { ( i ) } ( { \pmb x } ) - F ^ { ( i ) } ( { \pmb x } ) \| _ { 2 } \leq \frac { \varepsilon } { 2 \ell } , \mathrm { f o r ~ a l l ~ } \| { \pmb x } \| _ { \infty } \leq 1
+$$
+
+and
+
+$$
+\| \tilde { \boldsymbol { M } } ^ { ( i ) } \| _ { 0 } = \| \boldsymbol { M } ^ { ( i ) } \| _ { 0 } \leq \frac { 2 n s \sqrt { n s } } { \varepsilon ^ { \prime } } = \frac { 4 n s \ell \sqrt { n s } } { \varepsilon } .
+$$
+
+The remainder of the proof follows from applying the same argument as in the proof of Theorem A.6 from (Malach et al., 2020). □
+
+# C MOTIVATION FOR FRAMEWORK TO IDENTIFY MPTS
+
+Suppose that $f ( { \pmb x } ; { \pmb W } ^ { * } )$ with optimized weights $W ^ { * }$ is a target network that we wish to approximate. Let $g ( \pmb { x } ; \pmb { W } )$ denote the network in which we want to identify a MPT-1/32 that is an $\varepsilon$ - approximation of $f ( { \pmb x } ; { \pmb W } ^ { * } )$ , for some $\varepsilon > 0$ .
+
+Now assume that $g ( { \pmb x } ; \cdot )$ is Lipschitz continuous with constant $\kappa$ , $B \in \{ - 1 , + 1 \} ^ { m }$ are binary parameters for $g$ , and $\alpha \in \mathbb { R }$ is gain term. It follows that
+
+$$
+\begin{array} { r l } { \| g \left( \pmb { x } ; \alpha ( M \odot \pmb { B } ) \right) - f ( \pmb { x } ; \pmb { W } ^ { * } ) \| \leq \| g \left( \pmb { x } ; \alpha ( M \odot \pmb { B } ) - g ( \pmb { x } ; M \odot \pmb { B } ) \right\| } & { } \\ { + \| g ( \pmb { x } ; M \odot \pmb { W } ) - f ( \pmb { x } ; \pmb { W } ^ { * } ) \| } & { } \\ { < \kappa \| \big ( M \odot \pmb { W } \big ) - \alpha \big ( M \odot \pmb { B } \big ) \| } & { } \\ { + \| g ( \pmb { x } ; M \odot \pmb { W } ) - f ( \pmb { x } ; \pmb { W } ^ { * } ) \| . } \end{array}
+$$
+
+If we take $M$ to be a fixed binary mask, we can minimize the error of binarizing the subnetwork parameters $M \odot W$ by solving the optimization problem
+
+$$
+\begin{array} { r l } { \underset { \alpha , B } { \operatorname* { m i n } } } & { \| ( M \odot W ) - \alpha ( M \odot B ) \| ^ { 2 } } \\ { \mathrm { s . t . } } & { \quad \alpha \in \mathbb { R } , B \in \{ - 1 , 1 \} ^ { n } } \end{array}
+$$
+
+where $M , W$ , and $\textbf {  { B } }$ are stacked into vectors of some length, say $n$ . As the pruning mask $M$ is applied to both $W$ and $\textbf {  { B } }$ , solving problem (57) is equivalent to solving problem (2) in (Rastegari et al., 2016) with a different dimension. Hence, it immediately follows that one closed form solution for $\textbf {  { B } }$ in problem (57) is
+
+$$
+B ^ { * } = \mathrm { s i g n } ( W ) .
+$$
+
+Taking the derivative of the cost function in (57) with respect to $\alpha$ and setting it equal to zero yields
+
+$$
+\alpha ( M \odot B ^ { * } ) ^ { \top } ( M \odot B ^ { * } ) - ( M \odot W ) ^ { \top } ( M \odot B ^ { * } ) = 0 .
+$$
+
+Recalling that $M \in \{ 0 , 1 \} ^ { n }$ and using (58), we have
+
+$$
+( M \odot B ^ { * } ) ^ { \top } ( M \odot B ^ { * } ) = \sum _ { i = 1 } ^ { n } ( M _ { i } B _ { i } ^ { * } ) ^ { 2 } = \sum _ { i = 1 } ^ { n } M _ { i } ^ { 2 } ( \mathrm { s i g n } ( W _ { i } ) ) ^ { 2 } = \sum _ { i = 1 } ^ { n } M _ { i } = \| M \| _ { 1 } = \frac { \partial M ^ { * } } { \partial \bigstar }
+$$
+
+and
+
+$$
+( M \odot W ) ^ { \top } ( M \odot B ^ { * } ) = \sum _ { i = 1 } ^ { n } M _ { i } ^ { 2 } W _ { i } \operatorname { s i g n } ( W _ { i } ) = \sum _ { i = 1 } ^ { n } M _ { i } | W _ { i } | = \| M \odot W \| _ { 1 } .
+$$
+
+Substituting (60) and (61) into (59) and solving for $\alpha$ yields the closed form solution
+
+$$
+\alpha ^ { * } = \frac { \| M \odot W \| _ { 1 } } { \| M \| _ { 1 } } .
+$$
+
+Hence, $\alpha ^ { * }$ and $B ^ { * }$ minimize the right hand side of (56) and, consequently, reduce the approximation error of the MPT-1/32. So when the binarization error, $\left\| \left( M \odot W \right) - \alpha ( M \odot \mathrm { s i g n } ( W ) ) \right\|$ , and the subnetwork error, $\| g ( \pmb { x } ; M \odot W ) - f ( \pmb { x } ; \pmb { W } ^ { * } ) \|$ , are sufficiently small then the binarized subnetwork $g \left( \pmb { x } ; \alpha ( M \odot \mathrm { s i g n } ( \pmb { W } ) ) \right)$ serves as a good approximation to the target network.
+
+These closed form expressions for the gain term and the binarized weights are the updates used for the gain term and binary subnetwork weights in biprop after updating the binary pruning mask.
+
+# D COMPARISON OF MPTS WITH BINARY NEURAL NETWORK SOTA
+
+Here we provide a more exhaustive comparison of MPT–1/32 and MPT–1/1 on CIFAR-10 and ImageNet to SOTA methods – BinaryConnect (Courbariaux et al., 2015), BNN (Courbariaux et al., 2016), DoReFa-Net (Zhou et al., 2016), LQ-Nets (Zhang et al., 2018), BWN and XNOR-Net (Rastegari et al., 2016), ABC-Net (Lin et al., 2017), IR-Net (Qin et al., 2020b), LAB (Hou et al., 2016), ProxQuant (Bai et al., 2018), DSQ (Gong et al., 2019), and BBG (Shen et al., 2020). Results for CIFAR-10 can be found in Tables 8 and 9 and results for ImageNet can be found in Tables 10 and 11. Next to the MPT method we include the percentage of weights pruned and the layer width multiplier (if larger than 1) in parentheses.
+
+Table 8: Comparison of MPT-1/32 with Trained Binary (1/32) Networks on CIFAR-10   
+
+<table><tr><td>Method</td><td>Model</td><td>Top-1 Params</td></tr><tr><td>BinaryConnect VGG-Small</td><td>91.7</td><td>4.6M</td></tr><tr><td>BWN VGG-Small</td><td>90.1</td><td>4.6M</td></tr><tr><td>DoReFa-Net ResNet-20</td><td>90.0</td><td>0.27M</td></tr><tr><td>LQ-Nets ResNet-20</td><td>90.1</td><td>0.27M</td></tr><tr><td>LAB VGG-Small</td><td>89.5</td><td>4.6M</td></tr><tr><td>ProxQuant ResNet-56</td><td>92.3</td><td>0.85M</td></tr><tr><td>DSQ ResNet-20</td><td>90.2</td><td>0.27M</td></tr><tr><td>IR-Net ResNet-20</td><td>90.8</td><td>0.27M</td></tr><tr><td>Full-Precision ResNet-18</td><td>93.02</td><td>11.2M</td></tr><tr><td>MPT-1/32 (95) VGG-Small</td><td>91.48</td><td>0.23M</td></tr><tr><td>MPT(80) ResNet-18</td><td>94.66</td><td>2.2M</td></tr><tr><td>MPT(80)+BN ResNet-18</td><td>94.8</td><td>2.2M</td></tr></table>
+
+Table 9: Comparison of MPT-1/1 with Trained Binary (1/1) Networks on CIFAR-10   
+
+<table><tr><td>Method</td><td>Model</td><td>Top-1</td><td>Params</td></tr><tr><td>BNN</td><td>VGG-Small</td><td>89.9</td><td>4.6M</td></tr><tr><td>XNOR-Net</td><td>VGG-Small</td><td>89.8</td><td>4.6M</td></tr><tr><td>DoReFa-Net</td><td>ResNet-20</td><td>79.3</td><td>0.27M</td></tr><tr><td>BBG</td><td>ResNet-20</td><td>85.3</td><td>0.27M</td></tr><tr><td>LAB</td><td>VGG-Small</td><td>87.7</td><td>4.6M</td></tr><tr><td>DSQ</td><td>VGG-Small</td><td>91.7</td><td>4.6M</td></tr><tr><td>IR-Net</td><td>ResNet-18</td><td>91.5</td><td>4.6M</td></tr><tr><td>Full-Precision</td><td>VGG-Small</td><td>93.6</td><td>4.6M</td></tr><tr><td>MPT(75,1.25x)</td><td>VGG-Small</td><td>88.49</td><td>1.44 M</td></tr><tr><td>MPT(75,1.25x) +BN</td><td>VGG-Small</td><td>91.9</td><td>1.44 M</td></tr></table>
+
+Table 10: Comparison of MPT-1/32 with Trained Binary (1/32) Networks on ImageNet   
+
+<table><tr><td>Method</td><td>Model</td><td>Top-1</td><td>Params</td></tr><tr><td>ABC-Net</td><td>ResNet-18</td><td>62.8</td><td>11.2 M</td></tr><tr><td>BWN</td><td>ResNet-18</td><td>60.8</td><td>11.2M</td></tr><tr><td>BWNH</td><td>ResNet-18</td><td>64.3</td><td>11.2M</td></tr><tr><td>PACT</td><td>ResNet-18</td><td>65.8</td><td>11.2 M</td></tr><tr><td>IR-Net</td><td>ResNet-34</td><td>70.4</td><td>21.8M</td></tr><tr><td>Quantization-Networks</td><td>ResNet-18</td><td>66.5</td><td>11.2M</td></tr><tr><td>Quantization-Networks</td><td>ResNet-50</td><td>72.8</td><td>25.6M</td></tr><tr><td>Full-Precision</td><td>ResNet-34</td><td>73.27</td><td>21.8M</td></tr><tr><td>MPT (80)</td><td>WRN-50</td><td>72.67</td><td>13.7 M</td></tr><tr><td>MPT(80)+BN</td><td>WRN-50</td><td>74.03</td><td>13.7 M</td></tr></table>
+
+# E COMPARISON TO EDGEPOPUP FOR MPT-1/32
+
+Note that binarization step of biprop can be avoided while finding MPT-1/32 – by initializing (and pruning) our backbone neural network with binary initialization (e.g., edgepopup with Signed Constant initialization (Ramanujan et al., 2020)). In this specific instance, biprop boils down to edgepopup with proper scaling. Next, we compare the performance of MPT-1/32 networks identified using these two approaches. Both networks presented below use the same hyperparameter configurations and are trained for 250 epochs on the CIFAR-10 dataset. We initialize the networks identified with edgepopup using the Signed Constant initialization as it yielded their best performance. MPT-1/32 networks identified using biprop are initialized using the Kaiming Normal initialization. We plot the average over three experiments for each pruning percentage and bars extending to the minimum and maximum accuracy for each pruning percentage. Additionally, for each network we include the Top-1 accuracy of a dense model with learned weights. These plots can be found in
+
+Table 11: Comparison of MPT-1/1 with Trained Binary (1/1) Networks on ImageNet   
+
+<table><tr><td>Method</td><td>Model</td><td>Top-1</td><td>Params</td></tr><tr><td>BNN</td><td>AlexNet</td><td>27.9</td><td>62.3M</td></tr><tr><td>XNOR-Net</td><td>AlexNet</td><td>44.2</td><td>62.3M</td></tr><tr><td>ABC-Net</td><td>ResNet-18</td><td>42.7</td><td>11.2M</td></tr><tr><td>ABC-Net</td><td>ResNet-34</td><td>52.4</td><td>21.8M</td></tr><tr><td>TSQ</td><td>AlexNet</td><td>58.0</td><td>62.3M</td></tr><tr><td>WRPN</td><td>ResNet-34</td><td>60.5</td><td>21.8M</td></tr><tr><td>HWGQ</td><td>AlexNet</td><td>52.7</td><td>62.3M</td></tr><tr><td>IR-Net</td><td>ResNet-18</td><td>58.1</td><td>11.2M</td></tr><tr><td>IR-Net</td><td>ResNet-34</td><td>62.9</td><td>21.8M</td></tr><tr><td>Full-Precision</td><td>ResNet-34</td><td>73.27</td><td>21.8M</td></tr><tr><td>MPT (60)</td><td>WRN-34</td><td>45.06</td><td>19.3M</td></tr><tr><td>MPT(60)+BN</td><td>WRN-34</td><td>52.07</td><td>19.3M</td></tr></table>
+
+![](images/46bbb794069082037cdbf1b33091b58b956ba38e019c3946bf7ee4e3a9066281.jpg)  
+Figure 5. We find that the performance of MPT-1/32 identified with biprop outperforms networks identified using edgepopup. This highlights the benefit of binarization (in conjunction with pruning) as a learning strategy.   
+Figure 5: Comparing biprop and edgepopup: Comparing the Top-1 accuracy of MPT-1/32 to binary weight networks of the same size identified using edgepopup on CIFAR-10.
+
+# F RELATED WORK
+
+F.1 PRUNING
+
+We categorize pruning methods based on whether a model is pruned either after the training or before the training (see (Neill, 2020) for a comprehensive review).
+
+Post-Training Pruning. The traditional pruning methods leverage a three-stage pipeline – pretraining (a large model), pruning, and fine-tuning. The main distinction lies among these approaches is what type of criteria is used for pruning. One of the most popular approach is the magnitude-based pruning where the weights with the magnitude below a certain threshold are discarded (Hagiwara, 1993). Further, certain penalty term (e.g., $l _ { 1 } , l _ { 2 }$ or lasso weight regularization) can be used during training to encourage a model to learn certain smaller magnitude weights and removing them posttraining (Weigend et al., 1991). Models can also be pruned by measuring the importance of weights by computing the sensitivity of the loss function when weights are removed and prune those which cause the smallest change in the loss (LeCun et al., 1990).
+
+Pruning Before Training. Thus far, we have have discussed methods for pruning pretrained DNNs.
+
+Recently, (Frankle & Carbin, 2019) proposed the Lottery Ticket Hypothesis and showed that randomly-initialized neural networks contain sparse subnetworks that can be effectively trained from scratch when reset to their initialization. Further, (Liu et al., 2018b) showed that the training an over-parameterized model is often not necessary to obtain an efficient final model and network architecture itself is more important than the remaining weights after pruning pretrained networks. These findings has revived interest in finding approaches for searching sparse and trainable subnetworks. For example, (Lee et al., 2018; Wang et al., 2020b; You et al., 2019; Wang et al., 2020a) explored efficient approaches to search for these sparse and trainable subnetworks. Along this line of work, a striking finding was reported by (Zhou et al., 2019; Ramanujan et al., 2020) showing that randomly-initialized neural networks contain sparse subnetworks that achieve good performance without any training. (Malach et al., 2020; Pensia et al., 2020) provided theoretical evidences for this phenomenon and showed that one can approximate any target neural network, by pruning a sufficiently over-parameterized network of random weights.
+
+# F.2 BINARIZATION
+
+Similar to pruning, we categorize binarization methods based on whether a model is binarized either after the training or during the training (see (Qin et al., 2020a) for a comprehensive review).
+
+Post-Training Binarization. To the best of our knowledge, none of the post-training schemes have been successful in binarizing pretrained models with or without retraining to achieve reasonable test accuracy. Most existing works (Han et al., 2015; Zhou et al., 2017) are limited to ternary weight quantization.
+
+Training-Aware Binarization. There are several efforts to improve the performance of BNN training. This is a challenging problem as binarization introduces discontinuities which makes differentiation during backpropogation difficult. Binaryconnect (Courbariaux et al., 2015) established how to train networks with binary weights within the familiar back-propagation paradigm. BinaryNet (Courbariaux et al., 2016) further quantize both the weights and the activations to 1-bit values. Unfortunately, these early schemes resulted in a staggering drop in the accuracy compared to their full precision counterparts. In an attempt to improve the performance, XNOR-Net (Rastegari et al., 2016) proposed to add a real-valued channel-wise scaling factor. Dorefa-Net (Zhou et al., 2016) extends XNOR-Net to accelerate the training process using quantized gradients. ABC-Net (Lin et al., 2017) improved the performance by using more weight bases and activation bases at the cost of increase in memory and computation. There have also been efforts in making modifications to the network architectures to make them amenable for the binary neural network training. For example, Bireal-Net (Liu et al., 2018a) added layer-wise identity short-cut, and AutoBNN (Shen et al., 2020) proposed to widen or squeeze the channels in an automatic manner. (Han et al., 2020) proposed to learn to binarize neurons with noisy supervision. Some efforts also have been carried out to designing gradient estimators extending straight-through estimator (STE) (Bengio et al., 2013) for accurate gradient back-propagation. DSQ (Gong et al., 2019) used differentiable soft quantization to have accurate gradients in backward propagation. On the other hand, PCNN Gu et al. (2019) proposed a new discrete back-propagation via projection algorithm to build BNNs.
+
+# F.3 OTHER RELATED DIRECTIONS
+
+Gaier & Ha (2019) proposed a search method for neural network architectures that can already perform a task without any explicit weight training, i.e., each weight in the network has the same shared value. Recent work in randomly wired neural networks (Xie et al., 2019) showed that constructing neural networks with random graph algorithms often outperforms a manually engineered architecture. As opposed to fixed wirings in (Xie et al., 2019), (Wortsman et al., 2019) learned the network parameters as well as the structure. This show that finding a good architecture is akin to finding a sparse subnetwork of the complete graph.

parse/train/nWSZ30wrEw3/nWSZ30wrEw3.md

@@ -0,0 +1,288 @@
+# Adaptive Machine Unlearning
+
+Varun Gupta1, Christopher Jung1, Seth Neel2, Aaron Roth1, Saeed Sharifi-Malvajerdi1, and Chris Waites3
+
+1University of Pennsylvania 2Harvard University 3Stanford University
+
+# Abstract
+
+Data deletion algorithms aim to remove the influence of deleted data points from trained models at a cheaper computational cost than fully retraining those models. However, for sequences of deletions, most prior work in the non-convex setting gives valid guarantees only for sequences that are chosen independently of the models that are published. If people choose to delete their data as a function of the published models (because they don’t like what the models reveal about them, for example), then the update sequence is adaptive. In this paper, we give a general reduction from deletion guarantees against adaptive sequences to deletion guarantees against non-adaptive sequences, using differential privacy and its connection to max information. Combined with ideas from prior work which give guarantees for non-adaptive deletion sequences, this leads to extremely flexible algorithms able to handle arbitrary model classes and training methodologies, giving strong provable deletion guarantees for adaptive deletion sequences. We show in theory how prior work for non-convex models fails against adaptive deletion sequences, and use this intuition to design a practical attack against the SISA algorithm of Bourtoule et al. [2021] on CIFAR-10, MNIST, Fashion-MNIST.
+
+# 1 Introduction
+
+Businesses like Facebook and Google depend on training sophisticated models on user data. Increasingly—in part because of regulations like the European Union’s General Data Protection Act and the California Consumer Privacy Act—these organizations are receiving requests to delete the data of particular users. But what should that mean? It is straightforward to delete a customer’s data from a database and stop using it to train future models. But what about models that have already been trained using an individual’s data? These are not necessarily safe; it is known that individual training data can be exfiltrated from models trained in standard ways via model inversion attacks [Shokri et al., 2017, Veale et al., 2018, Fredrikson et al., 2015]. Regulators are still grappling with when a trained model should be considered to contain personal data of individuals in the training set and the potential legal implications. In 2020 draft guidance, the U.K.’s Information Commissioner’s Office addressed how to comply with data deletion requests as they pertain to ML models:
+
+If the request is for rectification or erasure of the data, this may not be possible without re-training the model...or deleting the model altogether [ICO, 2020].
+
+Fully retraining the model every time a deletion request is received can be prohibitive in terms of both time and money—especially for large models and frequent deletion requests. The problem of data deletion (also known as machine unlearning) is to find an algorithmic middle ground between the compliant but impractical baseline of retraining, and the potentially illegal standard of doing nothing. We iteratively update models as deletion requests come in, with the twin goals of having computational cost that is substantially less than the cost of full retraining, and the guarantee that the models we produce are (almost) indistinguishable from the models that would have resulted from full retraining.
+
+After an initial model is deployed deletion requests arrive over time as users make decisions about whether to delete their data. It is easy to see how these decisions may be adaptive with respect to the models. For example, security researchers may publish a new model inversion attack that identifies a specific subset of people in the training data, thus leading to increased deletion requests for people in that subset. In this paper we give the first machine unlearning algorithms that both have rigorous deletion guarantees against these kind of adaptive deletion sequence, and can accommodate arbitrary non-convex models like deep neural networks without requiring pretraining on non-user data.
+
+# 1.1 Main Results
+
+The deletion guarantees proven for several prior methods crucially rely on the implicit assumption that the points that are deleted are independent of the randomness used to train the models. However this assumption fails unless the sequence of deletion requests is chosen independently of the information that the model provider has made public. This is a very strong assumption, because users may wish to delete their data exactly because of what deployed models reveal about them.
+
+We give a generic reduction. We show that if:
+
+1. A data deletion algorithm $\mathcal { R } _ { A }$ for a learning algorithm $\mathcal { A }$ has deletion guarantees for oblivious sequences of deletion requests (as those from past work do), and 2. Information about the internal randomness of $\mathcal { R } _ { A }$ is revealed only in a manner that satisfies differential privacy, then
+
+$( \mathcal { A } , \mathcal { R } _ { \mathcal { A } } )$ also satisfies data deletion guarantees against an adaptive sequence of deletion requests, that can depend in arbitrary ways on the information that the model provider has made public.
+
+In Section 3, we motivate our main result with a theoretical example which illustrates that past method’s lack of guarantees for adaptive sequences is not simply a failure of analysis, but an actual failure of these methods to satisfy deletion guarantees for adaptive deletion sequences. As an exemplar, we use a variant of SISA from Bourtoule et al. [2021] that satisfies perfect deletion guarantees for non-adaptive deletion sequences and exhibit adaptive deletion sequences that strongly separate the resulting distribution on models compared to the retraining baseline.
+
+The generic reduction found in Section 4 can be used to give adaptive data deletion mechanisms for a wide variety of problems by leveraging past work on deletion algorithms for non-adaptive sequences, and a line of work on differentially private aggregation [Papernot et al., 2018, Dwork and Feldman, 2018]. Since prior deletion algorithms themselves tend to use existing learning algorithms in a black-box way, the entire pipeline is modular and easy to bolt-on to existing methods. In Section 5, we show how this can be accomplished by using a variant of the SISA framework of Bourtoule et al. [2021] together with a differentially private aggregation method.
+
+In Section 6, we complement our main result with a set of experimental results on CIFAR-10, MNIST, and Fashion-MNIST that demonstrate differential privacy may be useful in giving adaptive guarantees beyond the statement of our theorems. Specifically we show that small amounts of noise addition (insufficient for our theorems to apply) already serve to break the adaptive deletion strategies that we use to falsify the adaptive deletion guarantees in our experiments described in Section 3 and do so at minimal expense in model accuracy.
+
+# 1.2 Related Work
+
+Data deletion was introduced by Cao and Yang [2015]; we adopt the randomized formulation of Ginart et al. [2019]. Ginart et al. [2019] anticipate the problem of deletion requests that might be correlated with internal state of the algorithm, and define (and propose as a study for future work) robust data deletion which is a data deletion guarantee that holds for adversaries with knowledge of the internal state. Our insight is that we can provide deletion guarantees against adaptive sequences by instead obscuring the internal state of the algorithm using techniques from differential privacy.
+
+We are the first to explicitly consider the problem of adaptive sequences of deletion requests, but some techniques from past work do have deletion guarantees that extend to adaptive sequences. Deterministic methods and methods that depend only on randomness that is sampled after the deletion request are already robust to adaptive deletion. This includes techniques that find an approximately optimal solution to a strongly convex problem and then perturb the solution to obscure the optimizer within a small radius e.g. Guo et al. [2019], Neel et al. [2021], Sekhari et al. [2021]. It also includes the approach of Golatkar et al. [2020a,b] which pre-trains a nonconvex model on data that will never be deleted and then does convex fine-tuning on user data on top of that. Techniques whose deletion guarantees depend on randomness sampled at training in general do not have guarantees against adaptive deletions. This includes algorithms given in Ginart et al. [2019], Bourtoule et al. [2021], Neel et al. [2021] — the SISA framework of Bourtoule et al. [2021] being of particular interest as it is agnostic to the class of models and training methodology, and so is extremely flexible.
+
+Differential privacy has been used as a mitigation for adaptivity since the work of Dwork et al. [2015c,a]. In machine learning, it has been used to mitigate the bias of adaptive data gathering strategies as used in bandit learning algorithms [Neel and Roth, 2018]. The application that is most similar to our work is Hassidim et al. [2020], which uses differential privacy of the internal randomness of an algorithm (as we do) to reduce streaming algorithms with guarantees against adaptive adversarial streams to streaming algorithms with guarantees against oblivious adversaries. Our techniques differ; while Hassidim et al. [2020] reduce to the so-called “transfer theorem for linear and low sensitivity queries” developed over a series of works Dwork et al. [2015c], Bassily et al. [2021], Jung et al. [2020], we use a more general connection between differential privacy and “max-information” established in Dwork et al. [2015b], Rogers et al. [2016].
+
+# 2 Preliminaries
+
+Let $\mathcal { Z }$ be the data domain. A dataset $D$ is a multi-set of elements from $\mathcal { Z }$ . We consider update requests of two types: deletion and addition. These update requests are formally defined below, similar to how they are defined in [Neel et al., 2021].
+
+Definition 2.1 (Update Operations and Sequences). An update $u$ is a pair $( z , \bullet )$ where $z \in { \mathcal { Z } }$ is $a$ datapoint and $\bullet \in \mathcal { T } = \{ ^ { \prime } \mathbf { a d d } ^ { \prime } , ^ { \prime } \mathbf { d e l e t e } ^ { \prime } \}$ determines the type of the update. An update sequence $U$ is a sequence $( u ^ { 1 } , u ^ { 2 } , \ldots )$ where $u ^ { t } \in \mathcal { Z } \times \mathcal { T }$ for all $t$ . Given a dataset $D$ and an update $u = ( z , \bullet )$ , the update operation is defined as:
+
+$$
+D \circ u \triangleq { \left\{ { D \cup \{ z \} } \quad i f \bullet = { ' } { \mathsf { a d d } } ^ { \prime } \right.} _ { D \setminus \{ z \} }   _ { i f \bullet = { ' } { \mathsf { d e l e t e } } ^ { \prime } }
+$$
+
+Given an update sequence $U = ( u ^ { 1 } , u ^ { 2 } , \ldots ) ;$ , we have $D \circ U \triangleq ( ( ( D \circ u ^ { 1 } ) \circ u ^ { 2 } ) \circ \ldots ) .$ .
+
+We use $\Theta$ to denote the space of models. A learning or training algorithm is a mapping $\mathcal { A } : \mathcal { Z } ^ { * } \to \Theta ^ { * }$ that maps a dataset $D \in { \mathcal { Z } } ^ { * }$ to a collection of models $\theta \in \Theta ^ { * }$ . An unlearning or update algorithm for $\mathcal { A }$ is a mapping $\mathcal { R } _ { A } : \mathcal { Z } ^ { * } \times ( \mathcal { Z } \times \mathcal { T } ) \times \mathcal { S } \to \Theta ^ { * }$ which takes in a data set $D \in { \mathcal { Z } } ^ { * }$ , an update request $u \in \mathcal { Z } \times \mathcal { T }$ , and some current state for the algorithms $s \in S$ (the domain $s$ can be arbitrary), and outputs an updated collection of models $\theta ^ { \prime } \in \Theta ^ { * }$ . In this paper we consider a setting in which a stream of update requests arrive in sequence. We note that in this sequential framework, the update algorithm $\mathcal { R } _ { A }$ also updates the state of the algorithm after each update request is processed; however, for notational economy, we do not explicitly write the updated state as an output of the algorithm.
+
+At each round, we provide access to the models through a mapping $f _ { \mathrm { p u b l i s h } } ^ { t } : \Theta ^ { * } \to \Psi$ that takes in the collection of models and outputs some object $\psi \in \Psi$ . A published object $\psi \in \Psi$ can, for instance, be the aggregate predictions of the learned models on a data set, or, some aggregation of the models. To model adaptively chosen update sequences, we define an arbitrary “update requester” who interacts with the learning and unlearning algorithms $( \mathcal { A } , \mathcal { R } _ { \mathcal { A } } )$ through the publishing function $f _ { \mathrm { p u b l i s h } }$ in rounds to generate a sequence of updates. The update requester is denoted by UpdReq and defined in Definition 2.2, and the interaction between the algorithms and the update requester is described in Algorithm 1.
+
+Throughout we will use $u ^ { t }$ to denote the update request at round $t$ . We will use $D ^ { t }$ to denote the data set at round $t$ : $D ^ { 0 }$ is the initial training data set and for all $t \geq 1$ , $D ^ { t } = D ^ { t - 1 } \circ u ^ { t }$ . We will use $\theta ^ { t }$ to denote the learned models at round $t$ : $\theta ^ { 0 }$ is generated by the initial training algorithm $\mathcal { A }$ , and $\theta ^ { t }$ for $t \geq 1$ denotes the updated models at round $t$ generated by the update algorithm $\mathcal { R } _ { A }$ . $\psi ^ { t }$ denotes the published object at round $t$ : $\psi ^ { t } = f _ { \mathrm { p u b l i s h } } ^ { t } ( \theta ^ { t } )$ .
+
+Algorithm 1: Interaction between $( \mathcal { A } , \mathcal { R } _ { \mathcal { A } } )$ and UpdReq   
+
+<table><tr><td></td><td>1: Input: Data set D</td></tr><tr><td>2:</td><td>Let D°← D.</td></tr><tr><td>3:</td><td>Train 0° ← A(D).</td></tr><tr><td>4:</td><td>Publish y0←pubish(00).</td></tr><tr><td>5:</td><td> Save the initial state so.</td></tr><tr><td>6:</td><td>for t = 1,2,... do</td></tr><tr><td>7:</td><td> The update requester requests a new update, given the history of interaction:</td></tr><tr><td>8:</td><td>ut←UpdReq(o,u¹,1,u², ,ut-1,γt-1).</td></tr><tr><td>9:</td><td>The algorithms update, given ut:</td></tr><tr><td>10:</td><td>Update the models 0t ← RA (Dt-1,ut,st-1).</td></tr><tr><td>11:</td><td>Publish bt ← fpubish (0t).</td></tr><tr><td>12:</td><td>Save the updated state st .</td></tr><tr><td>13:</td><td>Update the data set Dt ← Dt-1 o ut .</td></tr></table>
+
+Definition 2.2 (Update Requester (UpdReq)). The update sequence is generated by an update requester which is modeled by a (possibly randomized) mapping UpdReq : $\Psi ^ { * } \times ( \mathcal { Z } \times \mathcal { T } ) ^ { * }  ( \mathcal { Z } \times \mathcal { T } )$ that takes as input the history of interaction between herself and the algorithms, and outputs a new update for the current round. Given an update requester UpdReq, algorithms $( \mathcal { A } , \mathcal { R } _ { \mathcal { A } } )$ and publishing functions $\{ f _ { p u b l i s h } ^ { t } \} _ { t }$ , the update sequence $U = \{ u ^ { t } \} _ { t }$ can be written as
+
+$$
+\boldsymbol { u } ^ { 1 } = \mathrm { U p d R e q } \left( \boldsymbol { \psi } ^ { 0 } \right) , \boldsymbol { u } ^ { 2 } = \mathrm { U p d R e q } \left( \boldsymbol { \psi } ^ { 0 } , \boldsymbol { u } ^ { 1 } , \boldsymbol { \psi } ^ { 1 } \right) , \boldsymbol { \cdot } , \boldsymbol { \cdot } , \boldsymbol { u } ^ { t } = \mathrm { U p d R e q } \left( \boldsymbol { \psi } ^ { 0 } , \boldsymbol { u } ^ { 1 } , \boldsymbol { \psi } ^ { 1 } , \boldsymbol { \cdot } , \boldsymbol { \cdot } , \boldsymbol { u } ^ { t - 1 } , \boldsymbol { \psi } ^ { t - 1 } \right)
+$$
+
+We say an update requester UpdReq is nonadaptive if it is independent of the published objects, i.e., if there exists a mapping UpdR $\mathfrak { s q } ^ { \prime } : ( \mathcal { Z } \times \mathcal { T } ) ^ { \ast }  ( \mathcal { Z } \times \mathcal { T } )$ such that for all $t \geq 1$ ,
+
+$$
+u ^ { t } = \mathtt { U p d R e q } \left( \psi ^ { 0 } , u ^ { 1 } , \psi ^ { 1 } , u ^ { 2 } , \ldots , u ^ { t - 1 } , \psi ^ { t - 1 } \right) = \mathtt { U p d R e q } ^ { \prime } \left( u ^ { 1 } , u ^ { 2 } , \ldots , u ^ { t - 1 } \right)
+$$
+
+This is equivalent to saying that the update sequence is fixed before the interaction occurs.
+
+Following [Ginart et al., 2019], we propose the following definition for an unlearning algorithm in the sequential update setting ([Ginart et al., 2019] gives a definition for a single deletion request, whereas here we define a natural extension for an arbitrarily long sequence of deletions, as well as additions, that can be chosen adaptively.). Informally, we require that at every round, and for all possible update requesters, with high probability over the draw of the update sequence, no subset of models resulting from deletion occurs with substantially higher probability than it would have under full retraining.
+
+Definition 2.3 $( \alpha , \beta , \gamma )$ -unlearning). We say that $\mathcal { R } _ { A }$ is an $( \alpha , \beta , \gamma )$ -unlearning algorithm for $\mathcal { A }$ if for all datasets $D = D ^ { 0 }$ and all update requesters UpdReq, the following condition holds: For every update step $t \geq 1$ , with probability at least $1 - \gamma$ over the draw of the update sequence $u ^ { \le t } \overset { \cdot } { = } \hat { ( } u ^ { 1 } , \ldots , \overset { \cdot } { u } ^ { t } )$ from UpdReq,
+
+$$
+\begin{array} { r } { \forall E \subseteq \Theta ^ { * } : \quad \operatorname* { P r } \left[ \mathcal { R } _ { { \cal A } } \left( D ^ { t - 1 } , u ^ { t } , s ^ { t - 1 } \right) \in E \middle | u ^ { \leq t } \right] \leq e ^ { \alpha } \cdot \operatorname* { P r } \left[ { \cal A } \left( D ^ { t } \right) \in E \right] + \beta } \end{array}
+$$
+
+We say $\mathcal { R } _ { A }$ is a nonadaptive $( \alpha , \beta , \gamma )$ -unlearning algorithm for $\mathcal { A }$ if the above condition holds for any nonadaptive UpdReq.
+
+Remark 2.1. Our definition of unlearning is reminiscent of differential privacy, but following [Ginart et al., 2019], we ask only for $a$ one-sided guarantee: that the probability of any event under the unlearning scheme is not too much larger than the probability of the same event under full retraining, but not vice versa. The reason is that we do not want there to be events that can substantially increase an observer’s confidence that we did not engage in full retraining, but we do not object to observers who strongly update their beliefs that we did engage in full retraining. Our events $E$ are defined directly over the sets of models in $\Theta ^ { * }$ output by $\mathcal { A }$ and $\mathcal { R } _ { A }$ — note that because of information processing inequalities, this is only stronger than defining events $E$ over the observable outcome space $\Psi$ .
+
+# 2.1 Differential Privacy and Max-Information
+
+Differential privacy will be a key tool in our results. Let $\mathcal { X }$ denote an arbitrary data domain. We use $x \in \mathcal { X }$ to denote an individual element of $\mathcal { X }$ , and $X \in \mathcal { X } ^ { \ast }$ to denote a collection of elements from $\mathcal { X }$ — which we call a data set. We say two data sets $X , X ^ { \prime } \in { \mathcal { X } } ^ { * }$ are neighboring if they differ in at most one element. We say an algorithm $M : \mathcal { X } ^ { n }  \mathcal { O }$ is differentially private if its output distributions on neighboring data sets are close, formalized below.
+
+Definition 2.4 (Differential Privacy (DP) [Dwork et al., 2006b,a]). An algorithm $M : \mathcal { X } ^ { m }  \mathcal { O }$ is $( \epsilon , \delta )$ -differentially private, if for every neighboring $X$ and $X ^ { \prime }$ , and for every $O \subseteq { \mathcal { O } }$ , we have $\mathrm { P r } \left[ M ( X ) \in O \right] \leq e ^ { \epsilon } \mathrm { P r } \left[ M ( \dot { X } ^ { \prime } ) \in O \right] + \delta$ .
+
+We remark at the outset that the “datasets” to which we will eventually ask for differential privacy with respect to will not be the datasets on which our learning algorithms are trained, but will instead be collections of random bits parameterizing our randomized algorithms.
+
+Differentially private algorithms are robust to data-independent post-processing:
+
+Lemma 2.1 (Post-processing preserves DP [Dwork et al., 2006b]). If $M : \mathcal { X } ^ { m }  \mathcal { O }$ is $( \epsilon , \delta )$ - differentially private, then for all $f : \mathcal { O }  \mathcal { R }$ , we have $f \circ M : \mathcal { X } ^ { m }  \mathcal { R }$ defined by $f \circ M ( X ) =$ $f ( M ( X ) )$ is $( \epsilon , \delta )$ -differentially private.
+
+The max-information between two jointly distributed random variables measures how close their joint distribution is to the product of their corresponding marginal distributions.
+
+Definition 2.5 (Max-Information [Dwork et al., 2015b]). Let $X$ and $Y$ be jointly distributed random variables over the domain $( \mathcal { X } , \mathcal { Y } )$ . The $\beta$ -approximate max-information between $X$ and $Y$ is:
+
+$$
+I _ { \infty } ^ { \beta } ( X ; Y ) = \log \operatorname* { s u p } _ { \substack { E \subseteq ( \mathcal { X } , \mathcal { Y } ) , \operatorname* { P r } [ ( X , Y ) \in E ] > \beta } } \frac { \operatorname* { P r } [ ( X , Y ) \in E ] - \beta } { \operatorname* { P r } [ ( X \otimes Y ) \in E ] }
+$$
+
+where $( X \otimes Y )$ represents the product distribution of $X$ and $Y$ .
+
+The max-information of an algorithm $M$ that takes a dataset $X$ as input and outputs $M ( X )$ , is defined as the max-information between $X$ and $M ( X )$ for the worst case product distribution over $X$ :
+
+Definition 2.6 (Max-Information of an Algorithm [Dwork et al., 2015b]). Let $M : \mathcal { X } ^ { m }  \mathcal { O }$ be an Algorithm. We say $M$ has $\beta$ -approximate max-information of $k$ , written $T _ { \infty } ^ { \beta } ( M , m ) \leq k$ , if for every distribution $\mathcal { P }$ over $\mathcal { X }$ , we have $I _ { \infty } ^ { \beta } ( X ; M ( X ) ) \le k$ when $X \sim \mathcal { P } ^ { m }$ .
+
+In this paper, we will use the fact that differentially private algorithms have bounded max-information:
+
+Theorem 2.1 (DP implies bounded max-information [Rogers et al., 2016]). Let $M : \mathcal { X } ^ { m }  \mathcal { O }$ be an $( \epsilon , \delta )$ -differentially private algorithm for $0 < \epsilon \le 1 / 2$ and $0 < \delta < \epsilon$ . Then, $I _ { \infty } ^ { \beta } ( M , m ) =$ $O \left( \epsilon ^ { 2 } m + m \sqrt { \delta / \epsilon } \right) f o r \beta = e ^ { - \epsilon ^ { 2 } m } + O \left( m \sqrt { \delta / \epsilon } \right) .$ .
+
+# 3 Falsifying Unlearning Guarantees with Adaptivity
+
+In this section we demonstrate that the deletion guarantees of algorithms in the SISA framework [Bourtoule et al., 2021] fail for adaptive deletion sequences. We give a clean toy construction which shows algorithms in the SISA framework fail to have nontrivial adaptive deletion guarantees even in the black-box setting when the models within each shard are not made public, only aggregations of their classification outputs. In the Appendix we experimentally evaluate a more realistic instantiation of this construction.
+
+The setting we consider directly corresponds to the setting in which our final algorithms operate: what is made public is the aggregate predictions of the ensemble of models, but not the models themselves. For non-adaptive sequences of deletions, distributed algorithms of the sort described in Section 5 have perfect deletion guarantees. We demonstrate via a simple example that these guarantees dramatically fail for adaptive deletion sequences.
+
+Suppose we have a dataset consisting of real-valued points with binary labels $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { 2 n }$ , $x _ { i } \in \mathbb { R } ^ { d }$ $y _ { i } \ \stackrel { - } { \in } \ \{ 0 , 1 \}$ in which there are exactly two copies of each distinct training example. Consider a simplistic classification model, resembling a lookup table, which given a point $x _ { i }$ predicts the label $y _ { i }$ if the model has been trained on $( x _ { i } , y _ { i } )$ and a dummy prediction value " $" \perp "$ otherwise:
+
+$$
+f _ { \mathcal { D } } ( x _ { i } ) = \left\{ \begin{array} { l l } { y _ { i } } & { \mathrm { i f } \left( x _ { i } , y _ { i } \right) \in \mathcal { D } , } \\ { \perp } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+Consider what happens when the training algorithm randomly partitions this dataset into three pieces and trains such a model on each partition. This constructs an ensemble which, at query time, predicts the class with the majority vote. On this dataset, the ensemble will predict the labels of roughly $2 / 3$ of the training points correctly—that is, exactly those points for which the duplicates have fallen into distinct partitions, so that the ensemble gets the majority vote right.
+
+We construct an adaptive adversary who chooses to delete exactly those training points that the ensemble correctly classifies (which are those points for whom the duplicates have fallen into distinct shards). The result is that the model resulting from this deletion sequence will misclassify every remaining training point. Full retraining (because it would rerandomize the partition) would again lead to training accuracy of approximately $2 / 3$ . Recalling that our deletion notion requires that the probability of any event under the unlearning scheme is not much larger than the probability of the same event under full retraining, this demonstrates that there are algorithms in the SISA framework — even if the models are not directly exposed — that do not satisfy $( \alpha , \beta , \gamma )$ -deletion guarantees for any nontrivial value of $\alpha$ . We formalize this below:
+
+Theorem 3.1. There are learning and unlearning algorithms in the SISA framework $( \mathcal { A } , \mathcal { R } _ { \mathcal { A } } )$ such that for any $\alpha _ { i }$ , and any $\beta , \gamma < 1 / 4$ , $\mathcal { R } _ { A }$ is not an $( \alpha , \beta , \gamma )$ -unlearning algorithm for $\mathcal { A }$ .
+
+A proof of this theorem can be found in the appendix.
+
+# 4 A Reduction from Adaptive to Nonadaptive Update Requesters
+
+In our analysis we imagine without loss of generality that the learning algorithm $\mathcal { A }$ draws an i.i.d. sequence of random variables $r \sim \mathcal { P } ^ { m }$ (that encodes all the randomness to be used over the course of the updates) from some distribution $\mathcal { P }$ , and passes it to the unlearning algorithm $\mathcal { R } _ { A }$ . Note $r$ is drawn once in the initial training, and given $r$ , $\mathcal { A }$ and $\mathcal { R } _ { A }$ become deterministic mappings. We can also view the state $s ^ { t }$ as a deterministic mapping of $r$ , the update requests so far $u ^ { \le t } = ( \bar { u ^ { 1 } } , \dots , u ^ { t } )$ , and the original data set $D ^ { 0 }$ . We write $s ^ { t } = g ^ { t } \bar { ( } D ^ { 0 } , u ^ { \le t } , r \bar { ) }$ for some deterministic mapping $g ^ { t }$ . We can therefore summarize the trajectory of the algorithms $( \mathcal { A } , \mathcal { R } _ { \mathcal { A } } )$ as follows.
+
+In this view, the randomness $r$ used by the learning algorithm $\mathcal { A }$ and the subsequent invocations of the unlearning algorithm $\mathcal { R } _ { A }$ is represented as part of the internal state. Past analyses of unlearning algorithms have crucially assumed that $r$ is statistically independent of the updates $( \dot { u } ^ { 1 } , u ^ { 2 } , \dots )$ (which is the case for non-adaptive update requesters, but not for adaptive update requesters). In the following general theorem, we show that if a learning/unlearning pair satisfies unlearning guarantees against non-adaptive update requesters, and the publishing function is differentially private in the internal randomness $r$ , then the resulting algorithms also satisfy unlearning guarantees against adaptive update requesters. Note that what is important is that the publishing algorithms are differentially private in the internal randomness $r$ , not in the datapoints used for training.
+
+Theorem 4.1 (A General Theorem). Fix a pair of learning and unlearning algorithms $( \mathcal { A } , \mathcal { R } _ { \mathcal { A } } )$ and the publishing functions $\{ f _ { p u b l i s h } ^ { t } \} _ { t }$ . Suppose for every round $t$ , the sequence of publishing functions $\{ f _ { p u b l i s h } ^ { t ^ { \prime } } \} _ { t ^ { \prime } \leq t }$ is $( \epsilon , \delta )$ -differentially private in $r \sim \mathcal { P } ^ { m }$ , for $0 < \epsilon \le 1 / 2$ and $0 < \delta < \epsilon$ . Suppose $\mathcal { R } _ { A }$ is a non-adaptive $( \alpha , \beta , \gamma )$ -unlearning algorithm for $\mathcal { A }$ . Then $\mathcal { R } _ { A }$ is an $( \alpha ^ { \prime } , \beta ^ { \prime } , \gamma ^ { \prime } )$ -unlearning algorithm for $\mathcal { A }$ for $\alpha ^ { \prime } = \alpha + \epsilon ^ { \prime } , \beta ^ { \prime } = \beta e ^ { \epsilon ^ { \prime } } + \sqrt { \delta ^ { \prime } } , \gamma ^ { \prime } = \gamma + \sqrt { \delta ^ { \prime } }$ where $\epsilon ^ { \prime } = O \left( \epsilon ^ { 2 } m + m \sqrt { \delta / \epsilon } \right)$ and $\delta ^ { \prime } = e ^ { - \epsilon ^ { 2 } m } + O \left( m \sqrt { \delta / \epsilon } \right) .$ .
+
+The proof can be found in the Appendix, but at an intuitive level, it proceeds as follows. Because it does not change the joint distribution on update requests and internal state, we can imagine in our analysis that $r$ is redrawn after each update request from its conditional distribution, conditioned on the observed update sequence so far. Because the publishing function is differentially private in $r$ , by the fact that post-processing preserves differential privacy (Lemma 2.1), so is the update sequence. We may therefore apply the max-information bound (Theorem 2.1), which allows us to relate the conditional distribution on $r$ to its original (prior) distribution $\mathcal { P } ^ { m }$ . But resampling $r$ from $\mathcal { P } ^ { m }$ removes the dependence between $r$ and the update sequence, which places us in the non-adaptive case, and allows us to apply the hypothesized unlearning guarantees for nonadaptive update requesters.
+
+Algorithm 2: ${ \mathcal { A } } ^ { \mathrm { d i s t r } }$ : Distributed Learning Algorithm   
+
+<table><tr><td>Input: dataset D = D° of size n Draw the shards: D = Sampler(D°,p), for every i ∈ [k]. Train the models: 0 = Asingle(D),for every i∈ [k]. Save the state: s°= ({D&#x27;}i∈[k],{}iε[k]) // to be used for the 1st update.</td></tr></table>
+
+# 5 Distributed Algorithms
+
+In this section, we describe a general family of distributed learning and unlearning algorithms that are in the spirit of the “SISA” framework of Bourtoule et al. [2021] (with one crucial modification). At a high level, the SISA framework operates by first randomly dividing the data into $k$ “shards”, and separately training a model on each shard. When a new point is deleted, it is removed from the shards that contained it, and only the models corresponding to those shards are retrained. The flexibility of this methodology is that the models and training procedures used in each shard can be arbitrary, as can the aggregation done at the end to convert the resulting ensemble into predictions: however these choices are instantiated, this framework gives a $( 0 , 0 , 0 )$ -unlearning algorithm against any non-adaptive update requester (Lemma 5.1). Here we show that if the $k$ shards are selected independently of one another, then we can apply our reduction given in the previous section with $m = k$ and obtain algorithms that satisfy deletion guarantees against adaptive update requesters.
+
+A distributed learning algorithm $\mathcal { A } ^ { \mathrm { d i s t r } } : \mathcal { Z } ^ { * } \to \Theta ^ { * }$ is described by a single-shard learning algorithm $\mathcal { A } ^ { \mathrm { s i n g l e } } : \mathcal { Z } ^ { * } \to \Theta$ and a routine Sampler, used to select the points in a shard. Sampler, given a dataset $D$ and some probability $p \in [ 0 , 1 ]$ , includes each element of $D$ in the shard with probability $p$
+
+Distributed learning algorithm ${ \mathcal { A } } ^ { \mathrm { d i s t r } }$ creates $k$ independent shards from the dataset $D$ of size $n$ by running Sampler $k$ times and training a model with $\bar { \mathcal { A } } ^ { \mathrm { s i n g l e } }$ on each shard $i \in [ k ]$ to form an ensemble of $k$ models. To emphasize that the randomness across shards is independent, we will instantiate $k$ independent samplers $\mathtt { S a m p l e r } _ { i }$ and training algorithms $\mathcal { A } _ { i } ^ { \mathrm { s i n g l e } }$ for each shard $i \in [ k ]$ . We formally describe ${ \mathcal { A } } ^ { \mathrm { d i s t r } }$ in Algorithm 2.
+
+The state $s$ of the unlearning algorithm ${ \mathcal { R } } _ { A ^ { \mathrm { d i s t r } } }$ records the $k$ shards $\{ D _ { i } \} _ { i }$ and the ensemble of $k$ models $\{ \theta _ { i } \} _ { i }$ . Thus $\mathcal { S } = \{ \mathcal { Z } ^ { * } \} ^ { k } \times \Theta ^ { k }$ . As an update request $u$ is received, the update function removes the data point from every shard that contains it (for deletion) or adds the new point to each shard with probability $p$ (for addition). In either case, only the models corresponding to shards that have been updated are retrained using $\mathcal { A } ^ { \mathrm { s i n g l e } }$ . We formally describe ${ \mathcal { R } } _ { A ^ { \mathrm { d i s t r } } }$ in Algorithm 3.
+
+First, we show that if the update requester is non-adaptive, ${ \mathcal { R } } _ { A ^ { \mathrm { d i s t r } } }$ is a $( 0 , 0 , 0 )$ -unlearning algorithm: Lemma 5.1. $\mathcal { R } _ { \mathcal { A } ^ { d i s t r } }$ is a non-adaptive $( 0 , 0 , 0 )$ -unlearning algorithm for $\mathcal { A } ^ { d i s t r }$ .
+
+Now, by combining Lemma 5.1 and our general Theorem 4.1, we can show the following:
+
+Theorem 5.1 (Unlearning Guarantees). If for every round $t$ , the sequence of publishing functions $\{ f _ { p u b l i s h } ^ { t ^ { \prime } } \} _ { t ^ { \prime } \leq t }$ is $( \epsilon , \delta )$ -differentially private in the random seeds $r \sim \mathcal { P } ^ { k }$ of the algorithms for $0 < \epsilon \leq 1 / 2$ and $0 < \delta < \epsilon _ { \cdot }$ , then $\mathcal { R } _ { \mathcal { A } ^ { d i s t r } }$ is an $( \alpha , \beta , \gamma )$ -unlearning algorithm for $\mathcal { A } ^ { d i s t r }$ where
+
+$$
+\alpha = O \left( \epsilon ^ { 2 } k + k \sqrt { \delta / \epsilon } \right) , \quad \beta = \gamma = O \left( \sqrt { e ^ { - \epsilon ^ { 2 } k } + k \sqrt { \delta / \epsilon } } \right)
+$$
+
+Next, we bound the time complexity of our algorithms:
+
+Input: dataset $D ^ { t - 1 }$ , update $u ^ { t } = ( z ^ { t } , \bullet ^ { t } )$ , state $s ^ { t - 1 } = ( \{ D _ { i } ^ { t - 1 } \} _ { i \in [ k ] } , \{ \theta _ { i } ^ { t - 1 } \} _ { i \in [ k ] } )$   
+if $\bullet ^ { t } = { } ^ { \prime } \mathtt { d e l e t e } ^ { \prime }$ then $S = \left\{ i \in [ k ] : z ^ { t } \in D _ { i } ^ { t - 1 } \right\} / /$ the shards $z ^ { t }$ belongs to.   
+else $S = \{ i \in [ k ] : \mathtt { S a m p l e r } _ { i } ( \{ z ^ { t } \} , p ) \neq \{ \} \} / \prime$ the shards $z ^ { t }$ will be added to.   
+Update the shards: $D _ { i } ^ { t } = \left\{ { D _ { i } ^ { t - 1 } \circ u } \right.$ t if ot $i \in S$ se , for every $i \in [ k ]$ .   
+Update the models: $\theta _ { i } ^ { t } = \Big \{ \mathcal { A } _ { i } ^ { \mathrm { s i n g l e } } ( D _ { i } ^ { t } )$ if ot $i \in S$ se , for every $i \in [ k ]$ .   
+Update the state: $s ^ { t } = ( \{ D _ { i } ^ { t } \} _ { i \in [ k ] } , \{ \theta _ { i } ^ { t } \} _ { i \in [ k ] } ) / /$ to be used for the next update.   
+Output: {θti }i∈[k]
+
+Theorem 5.2 (Run-time Guarantees). Let $p = 1 / k$ . Suppose the publishing functions satisfy the differential privacy requirement of Theorem 5.1. Let $N ^ { t }$ denote the number of times $\mathcal { R } _ { \mathcal { A } } ^ { d i s t r }$ calls $\mathcal { A } ^ { s i n g l e }$ at round $t$ . We have that $N ^ { 0 } = k$ , and for every round $t \geq 1$ : 1) if the update requester is non-adaptive, for every $\xi$ , with probability at least $1 - \xi$ , $N ^ { t } \leq 1 + \sqrt { 2 \log { ( 1 / \xi ) } }$ . 2) if the update requester is adaptive, for every $\xi$ , with probability at least 1 − ξ, $N ^ { t } \leq 1 + \sqrt { 2 \log { ( ( n + t ) / \xi ) } }$ . Furthermore, for $\xi > \delta ^ { \prime }$ , with probability at least $1 - \xi$ , we have
+
+$$
+\begin{array} { c } { { N ^ { t } \leq 1 + \operatorname* { m i n } \left\{ \sqrt { 2 \log \left( 2 ( n + t ) / ( \xi - \delta ^ { \prime } ) \right) } , \sqrt { 2 \epsilon ^ { \prime } + 2 \log \left( 2 / ( \xi - \delta ^ { \prime } ) \right) } \right\} } } \\ { { { } } } \\ { { = O \left( \epsilon ^ { 2 } k + k \sqrt { \delta / \epsilon } \right) a n d \delta ^ { \prime } = e ^ { - \epsilon ^ { 2 } k } + O \left( k \sqrt { \delta / \epsilon } \right) } } \end{array}
+$$
+
+The proof can be found in the appendix, but at a high level it proceeds as follows. For a deletion request, we must retrain every shard that contains the point to be deleted. For a non-adaptive deletion request, we retrain one shard in expectation and we can obtain a high probability upper bound by using a Hoeffding bound. In the adaptive case, this may no longer be true, but there are two ways to obtain upper bounds that correspond to the two bounds in our Theorem. We can provide a worst-case upper bound on the number of shards that any of the √ $n$ data points belongs to, which incurs a cost of order $\sqrt { \log n }$ . Alternately, we can apply max-information bounds to reduce to the non-adaptive case, using an argument that is similar to our reduction for deletion guarantees.
+
+Remark 5.1. We note that there is an alternative algorithm that one might consider, resulting from group differential privacy. If a learning algorithm satisfies $\frac { \epsilon } { k }$ −differential privacy, a valid unlearning procedure is to do nothing for $k$ updates and then fully retrain on the $( k + 1 ) ^ { t h }$ update. This follows from the −differential privacy guarantee the algorithm will have for groups of size $k$ . Our algorithm substantially outperforms this alternative algorithm as well, namely because our privacy parameter degrades much slower than in this group privacy baseline. Our analysis leverages adaptive composition of privacy across the publishing functions which means that privacy degrades with the square root of the number of updates, while it degrades linearly with group privacy. Consequently the group privacy baseline would require a full retraining every $k$ updates, but our algorithm requires a full retraining only every $k ^ { 2 }$ updates.
+
+# 5.1 Private Aggregation
+
+We briefly describe how we serve prediction requests by privately aggregating the output of the ensemble of models such that the published predictions are differentially private in the random seeds $r$ . At each round $t$ , while $\mathcal { R } _ { \mathcal { A } } ^ { \mathrm { d i s t r } }$ is waiting for the next update request $\hat { u ^ { t + 1 } }$ , we receive prediction requests $x$ and serve predictions $\hat { y }$ . For each prediction request, we privately aggregate the predictions made by the ensemble of models $\{ \theta _ { i } ^ { t } \} _ { i }$ ; Dwork and Feldman [2018] show several ways to privately aggregate predictions (one simple technique is to use the exponential mechanism to approximate the majority vote). Suppose we aggregate the predictions made by the ensemble of models using PrivatePredi $\mathfrak { L } _ { \epsilon ^ { \prime } } ^ { k } : \bar { \Theta } ^ { k } \times \mathcal { X } \stackrel { \left. } { \right. } \mathcal { Y }$ , which takes in an ensemble of $k$ models and a data point, aggregates predictions from the ensemble models, and outputs a label that is $\epsilon ^ { \prime }$ -differentially private in the models. If we receive $l ^ { t }$ many prediction requests $( x _ { 1 } ^ { t } , \ldots , x _ { l ^ { t } } ^ { t } )$ before our next update request $\boldsymbol u ^ { t + 1 }$ , we can write $( \hat { y } _ { 1 } ^ { t } , \dots , \hat { y } _ { l ^ { t } } ^ { t } ) = f _ { \mathrm { p u b l i s h } } ^ { t } ( \{ \theta _ { i } ^ { t } \} _ { i } )$ where $\begin{array} { r } { \hat { y } _ { j } ^ { t } = \mathtt { P r i v a t e P r e d i c t } _ { \epsilon ^ { \prime } } ^ { k } ( \{ \theta _ { i } ^ { t } \} _ { i } , x _ { j } ^ { t } ) . } \end{array}$ .
+
+Theorem 5.1, tells us that desired unlearning parameters $( \alpha , \beta , \gamma )$ can be obtained by guaranteeing that the sequence of predictions is $( \epsilon , \delta )$ differentially private in the models (and hence $r$ ), for target parameters $\epsilon , \delta$ . As we serve prediction requests using PrivatePredict $\mathbf { \Sigma } _ { \epsilon ^ { \prime } } ^ { k }$ our privacy loss will accumulate and eventually exhaust our budget of $( \epsilon , \delta )$ -differential privacy. Hence we must track our accumulated privacy loss in the state of our unlearning algorithm, and when it is exhausted, fully retrain using $\bar { \mathcal { A } } ^ { \mathrm { d i s t r } }$ . This resamples $r$ and hence resets our privacy budget. Standard composition theorems (see Dwork and Roth [2014]) show that we exhaust our privacy budget (and need to fully retrain) every time the number of prediction requests made since the last full retraining exceeds $\left\lfloor { \frac { \epsilon ^ { 2 } } { 8 ( \epsilon ^ { \prime } ) ^ { 2 } \ln ( { \frac { 1 } { \delta } } ) } } \right\rfloor$ We formally describe this process denoted as PrivatePredictionInteraction $( \epsilon ^ { \prime } , \epsilon , \delta , k )$ in the appendix and state its unlearning guarantee in Theorem 5.3.
+
+Theorem 5.3. The models $\{ \{ \theta _ { i } ^ { t } \} _ { i } \} _ { t }$ in PrivatePredictionInteraction $( \epsilon ^ { \prime } , \epsilon , \delta , k )$ satisfy $( \alpha , \beta , \gamma )$ -unlearning guarantee for $\mathcal { A } ^ { d i s t r }$ where $\begin{array} { r l r } { \alpha } & { { } = } & { O \left( \epsilon ^ { 2 } k + k \sqrt { \delta / \epsilon } \right) } \end{array}$ and $\beta , \gamma \quad = \quad$ $O \left( \sqrt { e ^ { - \epsilon ^ { 2 } k } + k \sqrt { \delta / \epsilon } } \right)$ , $i f 0 < \epsilon \leq 1 / 2$ and $0 < \delta < \epsilon$ .
+
+# 6 Evaluation of Unlearning Guarantees
+
+In this section we consider the white-box setting in which the models in each shard are made public. SISA continues to have perfect deletion guarantees against non-adaptive deletion sequences in this setting. Experimental results on CIFAR-10 [Krizhevsky and Hinton, 2009], MNIST [Lecun et al., 1998], and Fashion-MNIST [Xiao et al., 2017] show both the failure of SISA to satisfy adaptive deletion guarantees, and give evidence that differential privacy can mitigate this problem well beyond the setting of our theorems while achieving accuracy only modestly worse than SISA. The code for our experiments can be found at https://github.com/ChrisWaites/adaptive-machine-unlearning.
+
+We train SISA with an ensemble of convolutional neural networks on several datasets of points with categorical labels. Given a new point at query time, each model in the ensemble votes on the most likely label and aggregates their votes. The models are exposed publicly. This scheme has perfect non-adaptive deletion guarantees.
+
+To construct an adaptive deletion sequence to falsify the hypothesis that the scheme has adaptive deletion guarantees, we exploit the observation that neural networks are often overconfident in the correct label for points on which they have been trained. For each training point, we guess that it falls into the shard corresponding to the model that has the highest confidence for the correct label. We then delete points for which we guess that they fall into the first $k / 2$ of the shards, and do not delete any others. After deleting the targeted points, we compute a test statistic: the indicator of whether the average accuracy of the models from the targeted shards is lower than the average accuracy of the models from the non-targeted shards. Under full retraining, by the symmetry of the random partition, the expectation of this test statistic is 0.5. Thus under the null hypothesis that the deletion algorithm satisfies perfect deletion guarantees, the test statistic also has expectation 0.5. Therefore, to the extent that the expectation of the indicator differs from 0.5, we falsify the null hypothesis that SISA has adaptive data deletion guarantees, and larger deviations from 0.5 falsify weaker deletion guarantees.
+
+We run this experiment on three datasets (CIFAR-10, MNIST, and Fashion-MNIST), and plot the results in Figure 1. We then repeat the experiment by adding various amounts of noise to the gradients in the model training process to guarantee finite levels of differential privacy (though much weaker privacy guarantees than would be needed to invoke our theorems). We observe that on each dataset, modest amounts of noise are sufficient to break our attack (i.e. $9 5 \%$ confidence intervals for the expectation of our indicator include 0.5, and hence fail to falsify the null hypothesis) while still approaching the accuracy of our models trained without differential privacy. This is also plotted in Figure 1. This gives evidence that differential privacy can improve deletion guarantees in the presence of adaptivity even in regimes beyond which our theory gives nontrivial guarantees.
+
+Full experimental details can be found in the appendix.
+
+![](images/44fba5964c6c783ca2ae42a63a10ae0552b68072bfa7bccf8697ae8c62cbba64.jpg)  
+Figure 1: The top row and bottom row show experiments with $k = 6$ and $k = 2$ shards respectively. The 3 columns report on 3 datasets. The $x$ axis denotes estimated expectation of our test statistic (the null hypothesis is expectation 0.5). The $y$ axis denotes the accuracy of the ensemble after deletion. Each point is annotated with the noise multiplier used in DP-SGD, the standard deviation of Gaussian noise applied to gradients during training. A label of 0.0 for a point represents the baseline case of no noise (original SISA algorithm). Points are affixed with $9 5 \%$ confidence intervals along both axes (over the randomness of repeating the training/deletion experiment). Horizontal confidence intervals that overlap the line denoting expectation 0.5 fail to reject the null hypothesis that the algorithm has adaptive data deletion guarantees at $p \leq 0 . 0 5$ . We get to this point with a level of noise addition that results in only a modest degradation in ensemble performance compared to SISA.
+
+# 7 Conclusion and Discussion
+
+We identify an important blindspot in the data deletion literature (the tenuous implicit assumption that deletion requests are independent of previously released models), and provide a very general methodology to reduce adaptive deletion guarantees to oblivious deletion guarantees. Through this reduction we get the first model and training algorithm agnostic methodology that allows for deletion of arbitrary sequences of adaptively chosen points while giving rigorous guarantees. The constants that our theorems inherit from the max information bounds of Rogers et al. [2016] are such that in most realistic settings they will not give useful parameters. But we hope that these constants will be improved in future work, and we give empirical evidence that differential privacy mitigates adaptive deletion “attacks” at very practical levels, beyond the promises of our theoretical results. We note that like for differential privacy, the $( \alpha , \beta , \gamma )$ -deletion guarantees we give in this paper are parameterized, and are not meaningful absent a specification of those parameters. There is a risk with such technologies that they will be used with large values of the parameters that give only very weak guarantees, but will be described publicly in a way that glosses over this issue. We therefore recommend that if adopted in deployed products, deletion guarantees always be discussed in public in a way that is precise about what they promise, including the relevant parameter settings.
+
+# Acknowledgements
+
+V.G., C.J., A.R., and S.S. were supported in part by NSF grants CCF-1934876 and AF-1763307, and a grant from the Simons Foundation.
+
+# References
+
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+
+Lucas Bourtoule, Varun Chandrasekaran, Christopher Choquette-Choo, Hengrui Jia, Adelin Travers, Baiwu Zhang, David Lie, and Nicolas Papernot. Machine unlearning. In Proceedings of the 42nd IEEE Symposium on Security and Privacy, San Francisco, CA., 2021.
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+Cynthia Dwork, Vitaly Feldman, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Aaron Roth. Generalization in adaptive data analysis and holdout reuse. In Proceedings of the 28th International Conference on Neural Information Processing Systems-Volume 2, pages 2350–2358, 2015b.
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+
+The U.K. Information Commissioner’s Office ICO. Guidance on the ai auditing framework. Draft Consultation, 2020. URL https://ico.org.uk/media/about-the-ico/consultations/ 2617219/guidance-on-the-ai-auditing-framework-draft-for-consultation.pdf.
+
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+        "text": "Varun Gupta1, Christopher Jung1, Seth Neel2, Aaron Roth1, Saeed Sharifi-Malvajerdi1, and Chris Waites3 ",
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+        "text": "Abstract ",
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+        ],
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+    },
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+        "text": "Data deletion algorithms aim to remove the influence of deleted data points from trained models at a cheaper computational cost than fully retraining those models. However, for sequences of deletions, most prior work in the non-convex setting gives valid guarantees only for sequences that are chosen independently of the models that are published. If people choose to delete their data as a function of the published models (because they don’t like what the models reveal about them, for example), then the update sequence is adaptive. In this paper, we give a general reduction from deletion guarantees against adaptive sequences to deletion guarantees against non-adaptive sequences, using differential privacy and its connection to max information. Combined with ideas from prior work which give guarantees for non-adaptive deletion sequences, this leads to extremely flexible algorithms able to handle arbitrary model classes and training methodologies, giving strong provable deletion guarantees for adaptive deletion sequences. We show in theory how prior work for non-convex models fails against adaptive deletion sequences, and use this intuition to design a practical attack against the SISA algorithm of Bourtoule et al. [2021] on CIFAR-10, MNIST, Fashion-MNIST. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Businesses like Facebook and Google depend on training sophisticated models on user data. Increasingly—in part because of regulations like the European Union’s General Data Protection Act and the California Consumer Privacy Act—these organizations are receiving requests to delete the data of particular users. But what should that mean? It is straightforward to delete a customer’s data from a database and stop using it to train future models. But what about models that have already been trained using an individual’s data? These are not necessarily safe; it is known that individual training data can be exfiltrated from models trained in standard ways via model inversion attacks [Shokri et al., 2017, Veale et al., 2018, Fredrikson et al., 2015]. Regulators are still grappling with when a trained model should be considered to contain personal data of individuals in the training set and the potential legal implications. In 2020 draft guidance, the U.K.’s Information Commissioner’s Office addressed how to comply with data deletion requests as they pertain to ML models: ",
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+        "type": "text",
+        "text": "If the request is for rectification or erasure of the data, this may not be possible without re-training the model...or deleting the model altogether [ICO, 2020]. ",
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+        "text": "Fully retraining the model every time a deletion request is received can be prohibitive in terms of both time and money—especially for large models and frequent deletion requests. The problem of data deletion (also known as machine unlearning) is to find an algorithmic middle ground between the compliant but impractical baseline of retraining, and the potentially illegal standard of doing nothing. We iteratively update models as deletion requests come in, with the twin goals of having computational cost that is substantially less than the cost of full retraining, and the guarantee that the models we produce are (almost) indistinguishable from the models that would have resulted from full retraining. ",
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+        "text": "After an initial model is deployed deletion requests arrive over time as users make decisions about whether to delete their data. It is easy to see how these decisions may be adaptive with respect to the models. For example, security researchers may publish a new model inversion attack that identifies a specific subset of people in the training data, thus leading to increased deletion requests for people in that subset. In this paper we give the first machine unlearning algorithms that both have rigorous deletion guarantees against these kind of adaptive deletion sequence, and can accommodate arbitrary non-convex models like deep neural networks without requiring pretraining on non-user data. ",
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+        "text": "1.1 Main Results ",
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+        "text": "The deletion guarantees proven for several prior methods crucially rely on the implicit assumption that the points that are deleted are independent of the randomness used to train the models. However this assumption fails unless the sequence of deletion requests is chosen independently of the information that the model provider has made public. This is a very strong assumption, because users may wish to delete their data exactly because of what deployed models reveal about them. ",
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+        "text": "1. A data deletion algorithm $\\mathcal { R } _ { A }$ for a learning algorithm $\\mathcal { A }$ has deletion guarantees for oblivious sequences of deletion requests (as those from past work do), and 2. Information about the internal randomness of $\\mathcal { R } _ { A }$ is revealed only in a manner that satisfies differential privacy, then ",
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+    {
+        "type": "text",
+        "text": "$( \\mathcal { A } , \\mathcal { R } _ { \\mathcal { A } } )$ also satisfies data deletion guarantees against an adaptive sequence of deletion requests, that can depend in arbitrary ways on the information that the model provider has made public. ",
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+        "text": "In Section 3, we motivate our main result with a theoretical example which illustrates that past method’s lack of guarantees for adaptive sequences is not simply a failure of analysis, but an actual failure of these methods to satisfy deletion guarantees for adaptive deletion sequences. As an exemplar, we use a variant of SISA from Bourtoule et al. [2021] that satisfies perfect deletion guarantees for non-adaptive deletion sequences and exhibit adaptive deletion sequences that strongly separate the resulting distribution on models compared to the retraining baseline. ",
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+        "type": "text",
+        "text": "The generic reduction found in Section 4 can be used to give adaptive data deletion mechanisms for a wide variety of problems by leveraging past work on deletion algorithms for non-adaptive sequences, and a line of work on differentially private aggregation [Papernot et al., 2018, Dwork and Feldman, 2018]. Since prior deletion algorithms themselves tend to use existing learning algorithms in a black-box way, the entire pipeline is modular and easy to bolt-on to existing methods. In Section 5, we show how this can be accomplished by using a variant of the SISA framework of Bourtoule et al. [2021] together with a differentially private aggregation method. ",
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+        "type": "text",
+        "text": "In Section 6, we complement our main result with a set of experimental results on CIFAR-10, MNIST, and Fashion-MNIST that demonstrate differential privacy may be useful in giving adaptive guarantees beyond the statement of our theorems. Specifically we show that small amounts of noise addition (insufficient for our theorems to apply) already serve to break the adaptive deletion strategies that we use to falsify the adaptive deletion guarantees in our experiments described in Section 3 and do so at minimal expense in model accuracy. ",
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+    {
+        "type": "text",
+        "text": "1.2 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Data deletion was introduced by Cao and Yang [2015]; we adopt the randomized formulation of Ginart et al. [2019]. Ginart et al. [2019] anticipate the problem of deletion requests that might be correlated with internal state of the algorithm, and define (and propose as a study for future work) robust data deletion which is a data deletion guarantee that holds for adversaries with knowledge of the internal state. Our insight is that we can provide deletion guarantees against adaptive sequences by instead obscuring the internal state of the algorithm using techniques from differential privacy. ",
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+        "type": "text",
+        "text": "We are the first to explicitly consider the problem of adaptive sequences of deletion requests, but some techniques from past work do have deletion guarantees that extend to adaptive sequences. Deterministic methods and methods that depend only on randomness that is sampled after the deletion request are already robust to adaptive deletion. This includes techniques that find an approximately optimal solution to a strongly convex problem and then perturb the solution to obscure the optimizer within a small radius e.g. Guo et al. [2019], Neel et al. [2021], Sekhari et al. [2021]. It also includes the approach of Golatkar et al. [2020a,b] which pre-trains a nonconvex model on data that will never be deleted and then does convex fine-tuning on user data on top of that. Techniques whose deletion guarantees depend on randomness sampled at training in general do not have guarantees against adaptive deletions. This includes algorithms given in Ginart et al. [2019], Bourtoule et al. [2021], Neel et al. [2021] — the SISA framework of Bourtoule et al. [2021] being of particular interest as it is agnostic to the class of models and training methodology, and so is extremely flexible. ",
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+        "type": "text",
+        "text": "Differential privacy has been used as a mitigation for adaptivity since the work of Dwork et al. [2015c,a]. In machine learning, it has been used to mitigate the bias of adaptive data gathering strategies as used in bandit learning algorithms [Neel and Roth, 2018]. The application that is most similar to our work is Hassidim et al. [2020], which uses differential privacy of the internal randomness of an algorithm (as we do) to reduce streaming algorithms with guarantees against adaptive adversarial streams to streaming algorithms with guarantees against oblivious adversaries. Our techniques differ; while Hassidim et al. [2020] reduce to the so-called “transfer theorem for linear and low sensitivity queries” developed over a series of works Dwork et al. [2015c], Bassily et al. [2021], Jung et al. [2020], we use a more general connection between differential privacy and “max-information” established in Dwork et al. [2015b], Rogers et al. [2016]. ",
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+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
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+        "text": "Let $\\mathcal { Z }$ be the data domain. A dataset $D$ is a multi-set of elements from $\\mathcal { Z }$ . We consider update requests of two types: deletion and addition. These update requests are formally defined below, similar to how they are defined in [Neel et al., 2021]. ",
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+        "text": "Definition 2.1 (Update Operations and Sequences). An update $u$ is a pair $( z , \\bullet )$ where $z \\in { \\mathcal { Z } }$ is $a$ datapoint and $\\bullet \\in \\mathcal { T } = \\{ ^ { \\prime } \\mathbf { a d d } ^ { \\prime } , ^ { \\prime } \\mathbf { d e l e t e } ^ { \\prime } \\}$ determines the type of the update. An update sequence $U$ is a sequence $( u ^ { 1 } , u ^ { 2 } , \\ldots )$ where $u ^ { t } \\in \\mathcal { Z } \\times \\mathcal { T }$ for all $t$ . Given a dataset $D$ and an update $u = ( z , \\bullet )$ , the update operation is defined as: ",
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+        "img_path": "images/96b4b5ec8998a0e7c67003ba2bfac3fc872c4ac9a59a896e8b6c404ede3d3cb3.jpg",
+        "text": "$$\nD \\circ u \\triangleq { \\left\\{ { D \\cup \\{ z \\} } \\quad i f \\bullet = { ' } { \\mathsf { a d d } } ^ { \\prime } \\right.} _ { D \\setminus \\{ z \\} }   _ { i f \\bullet = { ' } { \\mathsf { d e l e t e } } ^ { \\prime } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Given an update sequence $U = ( u ^ { 1 } , u ^ { 2 } , \\ldots ) ;$ , we have $D \\circ U \\triangleq ( ( ( D \\circ u ^ { 1 } ) \\circ u ^ { 2 } ) \\circ \\ldots ) .$ . ",
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+        "text": "We use $\\Theta$ to denote the space of models. A learning or training algorithm is a mapping $\\mathcal { A } : \\mathcal { Z } ^ { * } \\to \\Theta ^ { * }$ that maps a dataset $D \\in { \\mathcal { Z } } ^ { * }$ to a collection of models $\\theta \\in \\Theta ^ { * }$ . An unlearning or update algorithm for $\\mathcal { A }$ is a mapping $\\mathcal { R } _ { A } : \\mathcal { Z } ^ { * } \\times ( \\mathcal { Z } \\times \\mathcal { T } ) \\times \\mathcal { S } \\to \\Theta ^ { * }$ which takes in a data set $D \\in { \\mathcal { Z } } ^ { * }$ , an update request $u \\in \\mathcal { Z } \\times \\mathcal { T }$ , and some current state for the algorithms $s \\in S$ (the domain $s$ can be arbitrary), and outputs an updated collection of models $\\theta ^ { \\prime } \\in \\Theta ^ { * }$ . In this paper we consider a setting in which a stream of update requests arrive in sequence. We note that in this sequential framework, the update algorithm $\\mathcal { R } _ { A }$ also updates the state of the algorithm after each update request is processed; however, for notational economy, we do not explicitly write the updated state as an output of the algorithm. ",
+        "bbox": [
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+        "type": "text",
+        "text": "At each round, we provide access to the models through a mapping $f _ { \\mathrm { p u b l i s h } } ^ { t } : \\Theta ^ { * } \\to \\Psi$ that takes in the collection of models and outputs some object $\\psi \\in \\Psi$ . A published object $\\psi \\in \\Psi$ can, for instance, be the aggregate predictions of the learned models on a data set, or, some aggregation of the models. To model adaptively chosen update sequences, we define an arbitrary “update requester” who interacts with the learning and unlearning algorithms $( \\mathcal { A } , \\mathcal { R } _ { \\mathcal { A } } )$ through the publishing function $f _ { \\mathrm { p u b l i s h } }$ in rounds to generate a sequence of updates. The update requester is denoted by UpdReq and defined in Definition 2.2, and the interaction between the algorithms and the update requester is described in Algorithm 1. ",
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+        "type": "text",
+        "text": "Throughout we will use $u ^ { t }$ to denote the update request at round $t$ . We will use $D ^ { t }$ to denote the data set at round $t$ : $D ^ { 0 }$ is the initial training data set and for all $t \\geq 1$ , $D ^ { t } = D ^ { t - 1 } \\circ u ^ { t }$ . We will use $\\theta ^ { t }$ to denote the learned models at round $t$ : $\\theta ^ { 0 }$ is generated by the initial training algorithm $\\mathcal { A }$ , and $\\theta ^ { t }$ for $t \\geq 1$ denotes the updated models at round $t$ generated by the update algorithm $\\mathcal { R } _ { A }$ . $\\psi ^ { t }$ denotes the published object at round $t$ : $\\psi ^ { t } = f _ { \\mathrm { p u b l i s h } } ^ { t } ( \\theta ^ { t } )$ . ",
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+    {
+        "type": "table",
+        "img_path": "images/49af37bb3ca08e560a92e8b40ce7bbcad36a565b9bdae3ca4959afdb266bd33a.jpg",
+        "table_caption": [
+            "Algorithm 1: Interaction between $( \\mathcal { A } , \\mathcal { R } _ { \\mathcal { A } } )$ and UpdReq "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>1: Input: Data set D</td></tr><tr><td>2:</td><td>Let D°← D.</td></tr><tr><td>3:</td><td>Train 0° ← A(D).</td></tr><tr><td>4:</td><td>Publish y0←pubish(00).</td></tr><tr><td>5:</td><td> Save the initial state so.</td></tr><tr><td>6:</td><td>for t = 1,2,... do</td></tr><tr><td>7:</td><td> The update requester requests a new update, given the history of interaction:</td></tr><tr><td>8:</td><td>ut←UpdReq(o,u¹,1,u², ,ut-1,γt-1).</td></tr><tr><td>9:</td><td>The algorithms update, given ut:</td></tr><tr><td>10:</td><td>Update the models 0t ← RA (Dt-1,ut,st-1).</td></tr><tr><td>11:</td><td>Publish bt ← fpubish (0t).</td></tr><tr><td>12:</td><td>Save the updated state st .</td></tr><tr><td>13:</td><td>Update the data set Dt ← Dt-1 o ut .</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Definition 2.2 (Update Requester (UpdReq)). The update sequence is generated by an update requester which is modeled by a (possibly randomized) mapping UpdReq : $\\Psi ^ { * } \\times ( \\mathcal { Z } \\times \\mathcal { T } ) ^ { * }  ( \\mathcal { Z } \\times \\mathcal { T } )$ that takes as input the history of interaction between herself and the algorithms, and outputs a new update for the current round. Given an update requester UpdReq, algorithms $( \\mathcal { A } , \\mathcal { R } _ { \\mathcal { A } } )$ and publishing functions $\\{ f _ { p u b l i s h } ^ { t } \\} _ { t }$ , the update sequence $U = \\{ u ^ { t } \\} _ { t }$ can be written as ",
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+        "img_path": "images/4518acdec353d550f167a7e753c1d2dea3736017a727d5ca5e9755ddcf286abc.jpg",
+        "text": "$$\n\\boldsymbol { u } ^ { 1 } = \\mathrm { U p d R e q } \\left( \\boldsymbol { \\psi } ^ { 0 } \\right) , \\boldsymbol { u } ^ { 2 } = \\mathrm { U p d R e q } \\left( \\boldsymbol { \\psi } ^ { 0 } , \\boldsymbol { u } ^ { 1 } , \\boldsymbol { \\psi } ^ { 1 } \\right) , \\boldsymbol { \\cdot } , \\boldsymbol { \\cdot } , \\boldsymbol { u } ^ { t } = \\mathrm { U p d R e q } \\left( \\boldsymbol { \\psi } ^ { 0 } , \\boldsymbol { u } ^ { 1 } , \\boldsymbol { \\psi } ^ { 1 } , \\boldsymbol { \\cdot } , \\boldsymbol { \\cdot } , \\boldsymbol { u } ^ { t - 1 } , \\boldsymbol { \\psi } ^ { t - 1 } \\right)\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We say an update requester UpdReq is nonadaptive if it is independent of the published objects, i.e., if there exists a mapping UpdR $\\mathfrak { s q } ^ { \\prime } : ( \\mathcal { Z } \\times \\mathcal { T } ) ^ { \\ast }  ( \\mathcal { Z } \\times \\mathcal { T } )$ such that for all $t \\geq 1$ , ",
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+        "img_path": "images/787e0ccfabb01426b1d2ca8a285bb48e80098736481a4556a309978293362e34.jpg",
+        "text": "$$\nu ^ { t } = \\mathtt { U p d R e q } \\left( \\psi ^ { 0 } , u ^ { 1 } , \\psi ^ { 1 } , u ^ { 2 } , \\ldots , u ^ { t - 1 } , \\psi ^ { t - 1 } \\right) = \\mathtt { U p d R e q } ^ { \\prime } \\left( u ^ { 1 } , u ^ { 2 } , \\ldots , u ^ { t - 1 } \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This is equivalent to saying that the update sequence is fixed before the interaction occurs. ",
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+        "type": "text",
+        "text": "Following [Ginart et al., 2019], we propose the following definition for an unlearning algorithm in the sequential update setting ([Ginart et al., 2019] gives a definition for a single deletion request, whereas here we define a natural extension for an arbitrarily long sequence of deletions, as well as additions, that can be chosen adaptively.). Informally, we require that at every round, and for all possible update requesters, with high probability over the draw of the update sequence, no subset of models resulting from deletion occurs with substantially higher probability than it would have under full retraining. ",
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+        "type": "text",
+        "text": "Definition 2.3 $( \\alpha , \\beta , \\gamma )$ -unlearning). We say that $\\mathcal { R } _ { A }$ is an $( \\alpha , \\beta , \\gamma )$ -unlearning algorithm for $\\mathcal { A }$ if for all datasets $D = D ^ { 0 }$ and all update requesters UpdReq, the following condition holds: For every update step $t \\geq 1$ , with probability at least $1 - \\gamma$ over the draw of the update sequence $u ^ { \\le t } \\overset { \\cdot } { = } \\hat { ( } u ^ { 1 } , \\ldots , \\overset { \\cdot } { u } ^ { t } )$ from UpdReq, ",
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+        "text": "$$\n\\begin{array} { r } { \\forall E \\subseteq \\Theta ^ { * } : \\quad \\operatorname* { P r } \\left[ \\mathcal { R } _ { { \\cal A } } \\left( D ^ { t - 1 } , u ^ { t } , s ^ { t - 1 } \\right) \\in E \\middle | u ^ { \\leq t } \\right] \\leq e ^ { \\alpha } \\cdot \\operatorname* { P r } \\left[ { \\cal A } \\left( D ^ { t } \\right) \\in E \\right] + \\beta } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We say $\\mathcal { R } _ { A }$ is a nonadaptive $( \\alpha , \\beta , \\gamma )$ -unlearning algorithm for $\\mathcal { A }$ if the above condition holds for any nonadaptive UpdReq. ",
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+        "type": "text",
+        "text": "Remark 2.1. Our definition of unlearning is reminiscent of differential privacy, but following [Ginart et al., 2019], we ask only for $a$ one-sided guarantee: that the probability of any event under the unlearning scheme is not too much larger than the probability of the same event under full retraining, but not vice versa. The reason is that we do not want there to be events that can substantially increase an observer’s confidence that we did not engage in full retraining, but we do not object to observers who strongly update their beliefs that we did engage in full retraining. Our events $E$ are defined directly over the sets of models in $\\Theta ^ { * }$ output by $\\mathcal { A }$ and $\\mathcal { R } _ { A }$ — note that because of information processing inequalities, this is only stronger than defining events $E$ over the observable outcome space $\\Psi$ . ",
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+    {
+        "type": "text",
+        "text": "2.1 Differential Privacy and Max-Information ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Differential privacy will be a key tool in our results. Let $\\mathcal { X }$ denote an arbitrary data domain. We use $x \\in \\mathcal { X }$ to denote an individual element of $\\mathcal { X }$ , and $X \\in \\mathcal { X } ^ { \\ast }$ to denote a collection of elements from $\\mathcal { X }$ — which we call a data set. We say two data sets $X , X ^ { \\prime } \\in { \\mathcal { X } } ^ { * }$ are neighboring if they differ in at most one element. We say an algorithm $M : \\mathcal { X } ^ { n }  \\mathcal { O }$ is differentially private if its output distributions on neighboring data sets are close, formalized below. ",
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+        "type": "text",
+        "text": "Definition 2.4 (Differential Privacy (DP) [Dwork et al., 2006b,a]). An algorithm $M : \\mathcal { X } ^ { m }  \\mathcal { O }$ is $( \\epsilon , \\delta )$ -differentially private, if for every neighboring $X$ and $X ^ { \\prime }$ , and for every $O \\subseteq { \\mathcal { O } }$ , we have $\\mathrm { P r } \\left[ M ( X ) \\in O \\right] \\leq e ^ { \\epsilon } \\mathrm { P r } \\left[ M ( \\dot { X } ^ { \\prime } ) \\in O \\right] + \\delta$ . ",
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+        "type": "text",
+        "text": "We remark at the outset that the “datasets” to which we will eventually ask for differential privacy with respect to will not be the datasets on which our learning algorithms are trained, but will instead be collections of random bits parameterizing our randomized algorithms. ",
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+        "type": "text",
+        "text": "Differentially private algorithms are robust to data-independent post-processing: ",
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+        "text": "Lemma 2.1 (Post-processing preserves DP [Dwork et al., 2006b]). If $M : \\mathcal { X } ^ { m }  \\mathcal { O }$ is $( \\epsilon , \\delta )$ - differentially private, then for all $f : \\mathcal { O }  \\mathcal { R }$ , we have $f \\circ M : \\mathcal { X } ^ { m }  \\mathcal { R }$ defined by $f \\circ M ( X ) =$ $f ( M ( X ) )$ is $( \\epsilon , \\delta )$ -differentially private. ",
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+    {
+        "type": "text",
+        "text": "The max-information between two jointly distributed random variables measures how close their joint distribution is to the product of their corresponding marginal distributions. ",
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+        "type": "text",
+        "text": "Definition 2.5 (Max-Information [Dwork et al., 2015b]). Let $X$ and $Y$ be jointly distributed random variables over the domain $( \\mathcal { X } , \\mathcal { Y } )$ . The $\\beta$ -approximate max-information between $X$ and $Y$ is: ",
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+        "type": "equation",
+        "img_path": "images/c01dbe67c5f13bd3c0e5eb1ead4db719c15dd45a5707b961fc413e7c40663f11.jpg",
+        "text": "$$\nI _ { \\infty } ^ { \\beta } ( X ; Y ) = \\log \\operatorname* { s u p } _ { \\substack { E \\subseteq ( \\mathcal { X } , \\mathcal { Y } ) , \\operatorname* { P r } [ ( X , Y ) \\in E ] > \\beta } } \\frac { \\operatorname* { P r } [ ( X , Y ) \\in E ] - \\beta } { \\operatorname* { P r } [ ( X \\otimes Y ) \\in E ] }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $( X \\otimes Y )$ represents the product distribution of $X$ and $Y$ . ",
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+        "type": "text",
+        "text": "The max-information of an algorithm $M$ that takes a dataset $X$ as input and outputs $M ( X )$ , is defined as the max-information between $X$ and $M ( X )$ for the worst case product distribution over $X$ : ",
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+        "type": "text",
+        "text": "Definition 2.6 (Max-Information of an Algorithm [Dwork et al., 2015b]). Let $M : \\mathcal { X } ^ { m }  \\mathcal { O }$ be an Algorithm. We say $M$ has $\\beta$ -approximate max-information of $k$ , written $T _ { \\infty } ^ { \\beta } ( M , m ) \\leq k$ , if for every distribution $\\mathcal { P }$ over $\\mathcal { X }$ , we have $I _ { \\infty } ^ { \\beta } ( X ; M ( X ) ) \\le k$ when $X \\sim \\mathcal { P } ^ { m }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we will use the fact that differentially private algorithms have bounded max-information: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2.1 (DP implies bounded max-information [Rogers et al., 2016]). Let $M : \\mathcal { X } ^ { m }  \\mathcal { O }$ be an $( \\epsilon , \\delta )$ -differentially private algorithm for $0 < \\epsilon \\le 1 / 2$ and $0 < \\delta < \\epsilon$ . Then, $I _ { \\infty } ^ { \\beta } ( M , m ) =$ $O \\left( \\epsilon ^ { 2 } m + m \\sqrt { \\delta / \\epsilon } \\right) f o r \\beta = e ^ { - \\epsilon ^ { 2 } m } + O \\left( m \\sqrt { \\delta / \\epsilon } \\right) .$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Falsifying Unlearning Guarantees with Adaptivity ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section we demonstrate that the deletion guarantees of algorithms in the SISA framework [Bourtoule et al., 2021] fail for adaptive deletion sequences. We give a clean toy construction which shows algorithms in the SISA framework fail to have nontrivial adaptive deletion guarantees even in the black-box setting when the models within each shard are not made public, only aggregations of their classification outputs. In the Appendix we experimentally evaluate a more realistic instantiation of this construction. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The setting we consider directly corresponds to the setting in which our final algorithms operate: what is made public is the aggregate predictions of the ensemble of models, but not the models themselves. For non-adaptive sequences of deletions, distributed algorithms of the sort described in Section 5 have perfect deletion guarantees. We demonstrate via a simple example that these guarantees dramatically fail for adaptive deletion sequences. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Suppose we have a dataset consisting of real-valued points with binary labels $\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { 2 n }$ , $x _ { i } \\in \\mathbb { R } ^ { d }$ $y _ { i } \\ \\stackrel { - } { \\in } \\ \\{ 0 , 1 \\}$ in which there are exactly two copies of each distinct training example. Consider a simplistic classification model, resembling a lookup table, which given a point $x _ { i }$ predicts the label $y _ { i }$ if the model has been trained on $( x _ { i } , y _ { i } )$ and a dummy prediction value \" $\" \\perp \"$ otherwise: ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/2dad0327245c5c034647574ad26888aa1312e4c98faf0655406d013a697d07cc.jpg",
+        "text": "$$\nf _ { \\mathcal { D } } ( x _ { i } ) = \\left\\{ \\begin{array} { l l } { y _ { i } } & { \\mathrm { i f } \\left( x _ { i } , y _ { i } \\right) \\in \\mathcal { D } , } \\\\ { \\perp } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Consider what happens when the training algorithm randomly partitions this dataset into three pieces and trains such a model on each partition. This constructs an ensemble which, at query time, predicts the class with the majority vote. On this dataset, the ensemble will predict the labels of roughly $2 / 3$ of the training points correctly—that is, exactly those points for which the duplicates have fallen into distinct partitions, so that the ensemble gets the majority vote right. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We construct an adaptive adversary who chooses to delete exactly those training points that the ensemble correctly classifies (which are those points for whom the duplicates have fallen into distinct shards). The result is that the model resulting from this deletion sequence will misclassify every remaining training point. Full retraining (because it would rerandomize the partition) would again lead to training accuracy of approximately $2 / 3$ . Recalling that our deletion notion requires that the probability of any event under the unlearning scheme is not much larger than the probability of the same event under full retraining, this demonstrates that there are algorithms in the SISA framework — even if the models are not directly exposed — that do not satisfy $( \\alpha , \\beta , \\gamma )$ -deletion guarantees for any nontrivial value of $\\alpha$ . We formalize this below: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1. There are learning and unlearning algorithms in the SISA framework $( \\mathcal { A } , \\mathcal { R } _ { \\mathcal { A } } )$ such that for any $\\alpha _ { i }$ , and any $\\beta , \\gamma < 1 / 4$ , $\\mathcal { R } _ { A }$ is not an $( \\alpha , \\beta , \\gamma )$ -unlearning algorithm for $\\mathcal { A }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "A proof of this theorem can be found in the appendix. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 A Reduction from Adaptive to Nonadaptive Update Requesters ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our analysis we imagine without loss of generality that the learning algorithm $\\mathcal { A }$ draws an i.i.d. sequence of random variables $r \\sim \\mathcal { P } ^ { m }$ (that encodes all the randomness to be used over the course of the updates) from some distribution $\\mathcal { P }$ , and passes it to the unlearning algorithm $\\mathcal { R } _ { A }$ . Note $r$ is drawn once in the initial training, and given $r$ , $\\mathcal { A }$ and $\\mathcal { R } _ { A }$ become deterministic mappings. We can also view the state $s ^ { t }$ as a deterministic mapping of $r$ , the update requests so far $u ^ { \\le t } = ( \\bar { u ^ { 1 } } , \\dots , u ^ { t } )$ , and the original data set $D ^ { 0 }$ . We write $s ^ { t } = g ^ { t } \\bar { ( } D ^ { 0 } , u ^ { \\le t } , r \\bar { ) }$ for some deterministic mapping $g ^ { t }$ . We can therefore summarize the trajectory of the algorithms $( \\mathcal { A } , \\mathcal { R } _ { \\mathcal { A } } )$ as follows. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this view, the randomness $r$ used by the learning algorithm $\\mathcal { A }$ and the subsequent invocations of the unlearning algorithm $\\mathcal { R } _ { A }$ is represented as part of the internal state. Past analyses of unlearning algorithms have crucially assumed that $r$ is statistically independent of the updates $( \\dot { u } ^ { 1 } , u ^ { 2 } , \\dots )$ (which is the case for non-adaptive update requesters, but not for adaptive update requesters). In the following general theorem, we show that if a learning/unlearning pair satisfies unlearning guarantees against non-adaptive update requesters, and the publishing function is differentially private in the internal randomness $r$ , then the resulting algorithms also satisfy unlearning guarantees against adaptive update requesters. Note that what is important is that the publishing algorithms are differentially private in the internal randomness $r$ , not in the datapoints used for training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 4.1 (A General Theorem). Fix a pair of learning and unlearning algorithms $( \\mathcal { A } , \\mathcal { R } _ { \\mathcal { A } } )$ and the publishing functions $\\{ f _ { p u b l i s h } ^ { t } \\} _ { t }$ . Suppose for every round $t$ , the sequence of publishing functions $\\{ f _ { p u b l i s h } ^ { t ^ { \\prime } } \\} _ { t ^ { \\prime } \\leq t }$ is $( \\epsilon , \\delta )$ -differentially private in $r \\sim \\mathcal { P } ^ { m }$ , for $0 < \\epsilon \\le 1 / 2$ and $0 < \\delta < \\epsilon$ . Suppose $\\mathcal { R } _ { A }$ is a non-adaptive $( \\alpha , \\beta , \\gamma )$ -unlearning algorithm for $\\mathcal { A }$ . Then $\\mathcal { R } _ { A }$ is an $( \\alpha ^ { \\prime } , \\beta ^ { \\prime } , \\gamma ^ { \\prime } )$ -unlearning algorithm for $\\mathcal { A }$ for $\\alpha ^ { \\prime } = \\alpha + \\epsilon ^ { \\prime } , \\beta ^ { \\prime } = \\beta e ^ { \\epsilon ^ { \\prime } } + \\sqrt { \\delta ^ { \\prime } } , \\gamma ^ { \\prime } = \\gamma + \\sqrt { \\delta ^ { \\prime } }$ where $\\epsilon ^ { \\prime } = O \\left( \\epsilon ^ { 2 } m + m \\sqrt { \\delta / \\epsilon } \\right)$ and $\\delta ^ { \\prime } = e ^ { - \\epsilon ^ { 2 } m } + O \\left( m \\sqrt { \\delta / \\epsilon } \\right) .$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The proof can be found in the Appendix, but at an intuitive level, it proceeds as follows. Because it does not change the joint distribution on update requests and internal state, we can imagine in our analysis that $r$ is redrawn after each update request from its conditional distribution, conditioned on the observed update sequence so far. Because the publishing function is differentially private in $r$ , by the fact that post-processing preserves differential privacy (Lemma 2.1), so is the update sequence. We may therefore apply the max-information bound (Theorem 2.1), which allows us to relate the conditional distribution on $r$ to its original (prior) distribution $\\mathcal { P } ^ { m }$ . But resampling $r$ from $\\mathcal { P } ^ { m }$ removes the dependence between $r$ and the update sequence, which places us in the non-adaptive case, and allows us to apply the hypothesized unlearning guarantees for nonadaptive update requesters. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/75b45716a089d8a2549115650460c6df9fe917ad406518bdf6dd3055e3d50ee3.jpg",
+        "table_caption": [
+            "Algorithm 2: ${ \\mathcal { A } } ^ { \\mathrm { d i s t r } }$ : Distributed Learning Algorithm "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Input: dataset D = D° of size n Draw the shards: D = Sampler(D°,p), for every i ∈ [k]. Train the models: 0 = Asingle(D),for every i∈ [k]. Save the state: s°= ({D&#x27;}i∈[k],{}iε[k]) // to be used for the 1st update.</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 Distributed Algorithms ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe a general family of distributed learning and unlearning algorithms that are in the spirit of the “SISA” framework of Bourtoule et al. [2021] (with one crucial modification). At a high level, the SISA framework operates by first randomly dividing the data into $k$ “shards”, and separately training a model on each shard. When a new point is deleted, it is removed from the shards that contained it, and only the models corresponding to those shards are retrained. The flexibility of this methodology is that the models and training procedures used in each shard can be arbitrary, as can the aggregation done at the end to convert the resulting ensemble into predictions: however these choices are instantiated, this framework gives a $( 0 , 0 , 0 )$ -unlearning algorithm against any non-adaptive update requester (Lemma 5.1). Here we show that if the $k$ shards are selected independently of one another, then we can apply our reduction given in the previous section with $m = k$ and obtain algorithms that satisfy deletion guarantees against adaptive update requesters. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A distributed learning algorithm $\\mathcal { A } ^ { \\mathrm { d i s t r } } : \\mathcal { Z } ^ { * } \\to \\Theta ^ { * }$ is described by a single-shard learning algorithm $\\mathcal { A } ^ { \\mathrm { s i n g l e } } : \\mathcal { Z } ^ { * } \\to \\Theta$ and a routine Sampler, used to select the points in a shard. Sampler, given a dataset $D$ and some probability $p \\in [ 0 , 1 ]$ , includes each element of $D$ in the shard with probability $p$ ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Distributed learning algorithm ${ \\mathcal { A } } ^ { \\mathrm { d i s t r } }$ creates $k$ independent shards from the dataset $D$ of size $n$ by running Sampler $k$ times and training a model with $\\bar { \\mathcal { A } } ^ { \\mathrm { s i n g l e } }$ on each shard $i \\in [ k ]$ to form an ensemble of $k$ models. To emphasize that the randomness across shards is independent, we will instantiate $k$ independent samplers $\\mathtt { S a m p l e r } _ { i }$ and training algorithms $\\mathcal { A } _ { i } ^ { \\mathrm { s i n g l e } }$ for each shard $i \\in [ k ]$ . We formally describe ${ \\mathcal { A } } ^ { \\mathrm { d i s t r } }$ in Algorithm 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The state $s$ of the unlearning algorithm ${ \\mathcal { R } } _ { A ^ { \\mathrm { d i s t r } } }$ records the $k$ shards $\\{ D _ { i } \\} _ { i }$ and the ensemble of $k$ models $\\{ \\theta _ { i } \\} _ { i }$ . Thus $\\mathcal { S } = \\{ \\mathcal { Z } ^ { * } \\} ^ { k } \\times \\Theta ^ { k }$ . As an update request $u$ is received, the update function removes the data point from every shard that contains it (for deletion) or adds the new point to each shard with probability $p$ (for addition). In either case, only the models corresponding to shards that have been updated are retrained using $\\mathcal { A } ^ { \\mathrm { s i n g l e } }$ . We formally describe ${ \\mathcal { R } } _ { A ^ { \\mathrm { d i s t r } } }$ in Algorithm 3. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "First, we show that if the update requester is non-adaptive, ${ \\mathcal { R } } _ { A ^ { \\mathrm { d i s t r } } }$ is a $( 0 , 0 , 0 )$ -unlearning algorithm: Lemma 5.1. $\\mathcal { R } _ { \\mathcal { A } ^ { d i s t r } }$ is a non-adaptive $( 0 , 0 , 0 )$ -unlearning algorithm for $\\mathcal { A } ^ { d i s t r }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Now, by combining Lemma 5.1 and our general Theorem 4.1, we can show the following: ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5.1 (Unlearning Guarantees). If for every round $t$ , the sequence of publishing functions $\\{ f _ { p u b l i s h } ^ { t ^ { \\prime } } \\} _ { t ^ { \\prime } \\leq t }$ is $( \\epsilon , \\delta )$ -differentially private in the random seeds $r \\sim \\mathcal { P } ^ { k }$ of the algorithms for $0 < \\epsilon \\leq 1 / 2$ and $0 < \\delta < \\epsilon _ { \\cdot }$ , then $\\mathcal { R } _ { \\mathcal { A } ^ { d i s t r } }$ is an $( \\alpha , \\beta , \\gamma )$ -unlearning algorithm for $\\mathcal { A } ^ { d i s t r }$ where ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/129e8e9ed74c537bbd51cd4895d8c06221772667ff86984801778b047e7df81f.jpg",
+        "text": "$$\n\\alpha = O \\left( \\epsilon ^ { 2 } k + k \\sqrt { \\delta / \\epsilon } \\right) , \\quad \\beta = \\gamma = O \\left( \\sqrt { e ^ { - \\epsilon ^ { 2 } k } + k \\sqrt { \\delta / \\epsilon } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Next, we bound the time complexity of our algorithms: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Input: dataset $D ^ { t - 1 }$ , update $u ^ { t } = ( z ^ { t } , \\bullet ^ { t } )$ , state $s ^ { t - 1 } = ( \\{ D _ { i } ^ { t - 1 } \\} _ { i \\in [ k ] } , \\{ \\theta _ { i } ^ { t - 1 } \\} _ { i \\in [ k ] } )$   \nif $\\bullet ^ { t } = { } ^ { \\prime } \\mathtt { d e l e t e } ^ { \\prime }$ then $S = \\left\\{ i \\in [ k ] : z ^ { t } \\in D _ { i } ^ { t - 1 } \\right\\} / /$ the shards $z ^ { t }$ belongs to.   \nelse $S = \\{ i \\in [ k ] : \\mathtt { S a m p l e r } _ { i } ( \\{ z ^ { t } \\} , p ) \\neq \\{ \\} \\} / \\prime$ the shards $z ^ { t }$ will be added to.   \nUpdate the shards: $D _ { i } ^ { t } = \\left\\{ { D _ { i } ^ { t - 1 } \\circ u } \\right.$ t if ot $i \\in S$ se , for every $i \\in [ k ]$ .   \nUpdate the models: $\\theta _ { i } ^ { t } = \\Big \\{ \\mathcal { A } _ { i } ^ { \\mathrm { s i n g l e } } ( D _ { i } ^ { t } )$ if ot $i \\in S$ se , for every $i \\in [ k ]$ .   \nUpdate the state: $s ^ { t } = ( \\{ D _ { i } ^ { t } \\} _ { i \\in [ k ] } , \\{ \\theta _ { i } ^ { t } \\} _ { i \\in [ k ] } ) / /$ to be used for the next update.   \nOutput: {θti }i∈[k] ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5.2 (Run-time Guarantees). Let $p = 1 / k$ . Suppose the publishing functions satisfy the differential privacy requirement of Theorem 5.1. Let $N ^ { t }$ denote the number of times $\\mathcal { R } _ { \\mathcal { A } } ^ { d i s t r }$ calls $\\mathcal { A } ^ { s i n g l e }$ at round $t$ . We have that $N ^ { 0 } = k$ , and for every round $t \\geq 1$ : 1) if the update requester is non-adaptive, for every $\\xi$ , with probability at least $1 - \\xi$ , $N ^ { t } \\leq 1 + \\sqrt { 2 \\log { ( 1 / \\xi ) } }$ . 2) if the update requester is adaptive, for every $\\xi$ , with probability at least 1 − ξ, $N ^ { t } \\leq 1 + \\sqrt { 2 \\log { ( ( n + t ) / \\xi ) } }$ . Furthermore, for $\\xi > \\delta ^ { \\prime }$ , with probability at least $1 - \\xi$ , we have ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae1774c441c6ddb0054e6f3f295ae6b952de2af5475d4b11fb5793e00a93a87f.jpg",
+        "text": "$$\n\\begin{array} { c } { { N ^ { t } \\leq 1 + \\operatorname* { m i n } \\left\\{ \\sqrt { 2 \\log \\left( 2 ( n + t ) / ( \\xi - \\delta ^ { \\prime } ) \\right) } , \\sqrt { 2 \\epsilon ^ { \\prime } + 2 \\log \\left( 2 / ( \\xi - \\delta ^ { \\prime } ) \\right) } \\right\\} } } \\\\ { { { } } } \\\\ { { = O \\left( \\epsilon ^ { 2 } k + k \\sqrt { \\delta / \\epsilon } \\right) a n d \\delta ^ { \\prime } = e ^ { - \\epsilon ^ { 2 } k } + O \\left( k \\sqrt { \\delta / \\epsilon } \\right) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The proof can be found in the appendix, but at a high level it proceeds as follows. For a deletion request, we must retrain every shard that contains the point to be deleted. For a non-adaptive deletion request, we retrain one shard in expectation and we can obtain a high probability upper bound by using a Hoeffding bound. In the adaptive case, this may no longer be true, but there are two ways to obtain upper bounds that correspond to the two bounds in our Theorem. We can provide a worst-case upper bound on the number of shards that any of the √ $n$ data points belongs to, which incurs a cost of order $\\sqrt { \\log n }$ . Alternately, we can apply max-information bounds to reduce to the non-adaptive case, using an argument that is similar to our reduction for deletion guarantees. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Remark 5.1. We note that there is an alternative algorithm that one might consider, resulting from group differential privacy. If a learning algorithm satisfies $\\frac { \\epsilon } { k }$ −differential privacy, a valid unlearning procedure is to do nothing for $k$ updates and then fully retrain on the $( k + 1 ) ^ { t h }$ update. This follows from the \u000f−differential privacy guarantee the algorithm will have for groups of size $k$ . Our algorithm substantially outperforms this alternative algorithm as well, namely because our privacy parameter degrades much slower than in this group privacy baseline. Our analysis leverages adaptive composition of privacy across the publishing functions which means that privacy degrades with the square root of the number of updates, while it degrades linearly with group privacy. Consequently the group privacy baseline would require a full retraining every $k$ updates, but our algorithm requires a full retraining only every $k ^ { 2 }$ updates. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1 Private Aggregation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We briefly describe how we serve prediction requests by privately aggregating the output of the ensemble of models such that the published predictions are differentially private in the random seeds $r$ . At each round $t$ , while $\\mathcal { R } _ { \\mathcal { A } } ^ { \\mathrm { d i s t r } }$ is waiting for the next update request $\\hat { u ^ { t + 1 } }$ , we receive prediction requests $x$ and serve predictions $\\hat { y }$ . For each prediction request, we privately aggregate the predictions made by the ensemble of models $\\{ \\theta _ { i } ^ { t } \\} _ { i }$ ; Dwork and Feldman [2018] show several ways to privately aggregate predictions (one simple technique is to use the exponential mechanism to approximate the majority vote). Suppose we aggregate the predictions made by the ensemble of models using PrivatePredi $\\mathfrak { L } _ { \\epsilon ^ { \\prime } } ^ { k } : \\bar { \\Theta } ^ { k } \\times \\mathcal { X } \\stackrel { \\left. } { \\right. } \\mathcal { Y }$ , which takes in an ensemble of $k$ models and a data point, aggregates predictions from the ensemble models, and outputs a label that is $\\epsilon ^ { \\prime }$ -differentially private in the models. If we receive $l ^ { t }$ many prediction requests $( x _ { 1 } ^ { t } , \\ldots , x _ { l ^ { t } } ^ { t } )$ before our next update request $\\boldsymbol u ^ { t + 1 }$ , we can write $( \\hat { y } _ { 1 } ^ { t } , \\dots , \\hat { y } _ { l ^ { t } } ^ { t } ) = f _ { \\mathrm { p u b l i s h } } ^ { t } ( \\{ \\theta _ { i } ^ { t } \\} _ { i } )$ where $\\begin{array} { r } { \\hat { y } _ { j } ^ { t } = \\mathtt { P r i v a t e P r e d i c t } _ { \\epsilon ^ { \\prime } } ^ { k } ( \\{ \\theta _ { i } ^ { t } \\} _ { i } , x _ { j } ^ { t } ) . } \\end{array}$ . ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Theorem 5.1, tells us that desired unlearning parameters $( \\alpha , \\beta , \\gamma )$ can be obtained by guaranteeing that the sequence of predictions is $( \\epsilon , \\delta )$ differentially private in the models (and hence $r$ ), for target parameters $\\epsilon , \\delta$ . As we serve prediction requests using PrivatePredict $\\mathbf { \\Sigma } _ { \\epsilon ^ { \\prime } } ^ { k }$ our privacy loss will accumulate and eventually exhaust our budget of $( \\epsilon , \\delta )$ -differential privacy. Hence we must track our accumulated privacy loss in the state of our unlearning algorithm, and when it is exhausted, fully retrain using $\\bar { \\mathcal { A } } ^ { \\mathrm { d i s t r } }$ . This resamples $r$ and hence resets our privacy budget. Standard composition theorems (see Dwork and Roth [2014]) show that we exhaust our privacy budget (and need to fully retrain) every time the number of prediction requests made since the last full retraining exceeds $\\left\\lfloor { \\frac { \\epsilon ^ { 2 } } { 8 ( \\epsilon ^ { \\prime } ) ^ { 2 } \\ln ( { \\frac { 1 } { \\delta } } ) } } \\right\\rfloor$ We formally describe this process denoted as PrivatePredictionInteraction $( \\epsilon ^ { \\prime } , \\epsilon , \\delta , k )$ in the appendix and state its unlearning guarantee in Theorem 5.3. ",
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+        "type": "text",
+        "text": "Theorem 5.3. The models $\\{ \\{ \\theta _ { i } ^ { t } \\} _ { i } \\} _ { t }$ in PrivatePredictionInteraction $( \\epsilon ^ { \\prime } , \\epsilon , \\delta , k )$ satisfy $( \\alpha , \\beta , \\gamma )$ -unlearning guarantee for $\\mathcal { A } ^ { d i s t r }$ where $\\begin{array} { r l r } { \\alpha } & { { } = } & { O \\left( \\epsilon ^ { 2 } k + k \\sqrt { \\delta / \\epsilon } \\right) } \\end{array}$ and $\\beta , \\gamma \\quad = \\quad$ $O \\left( \\sqrt { e ^ { - \\epsilon ^ { 2 } k } + k \\sqrt { \\delta / \\epsilon } } \\right)$ , $i f 0 < \\epsilon \\leq 1 / 2$ and $0 < \\delta < \\epsilon$ . ",
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+    {
+        "type": "text",
+        "text": "6 Evaluation of Unlearning Guarantees ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we consider the white-box setting in which the models in each shard are made public. SISA continues to have perfect deletion guarantees against non-adaptive deletion sequences in this setting. Experimental results on CIFAR-10 [Krizhevsky and Hinton, 2009], MNIST [Lecun et al., 1998], and Fashion-MNIST [Xiao et al., 2017] show both the failure of SISA to satisfy adaptive deletion guarantees, and give evidence that differential privacy can mitigate this problem well beyond the setting of our theorems while achieving accuracy only modestly worse than SISA. The code for our experiments can be found at https://github.com/ChrisWaites/adaptive-machine-unlearning. ",
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+    {
+        "type": "text",
+        "text": "We train SISA with an ensemble of convolutional neural networks on several datasets of points with categorical labels. Given a new point at query time, each model in the ensemble votes on the most likely label and aggregates their votes. The models are exposed publicly. This scheme has perfect non-adaptive deletion guarantees. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To construct an adaptive deletion sequence to falsify the hypothesis that the scheme has adaptive deletion guarantees, we exploit the observation that neural networks are often overconfident in the correct label for points on which they have been trained. For each training point, we guess that it falls into the shard corresponding to the model that has the highest confidence for the correct label. We then delete points for which we guess that they fall into the first $k / 2$ of the shards, and do not delete any others. After deleting the targeted points, we compute a test statistic: the indicator of whether the average accuracy of the models from the targeted shards is lower than the average accuracy of the models from the non-targeted shards. Under full retraining, by the symmetry of the random partition, the expectation of this test statistic is 0.5. Thus under the null hypothesis that the deletion algorithm satisfies perfect deletion guarantees, the test statistic also has expectation 0.5. Therefore, to the extent that the expectation of the indicator differs from 0.5, we falsify the null hypothesis that SISA has adaptive data deletion guarantees, and larger deviations from 0.5 falsify weaker deletion guarantees. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We run this experiment on three datasets (CIFAR-10, MNIST, and Fashion-MNIST), and plot the results in Figure 1. We then repeat the experiment by adding various amounts of noise to the gradients in the model training process to guarantee finite levels of differential privacy (though much weaker privacy guarantees than would be needed to invoke our theorems). We observe that on each dataset, modest amounts of noise are sufficient to break our attack (i.e. $9 5 \\%$ confidence intervals for the expectation of our indicator include 0.5, and hence fail to falsify the null hypothesis) while still approaching the accuracy of our models trained without differential privacy. This is also plotted in Figure 1. This gives evidence that differential privacy can improve deletion guarantees in the presence of adaptivity even in regimes beyond which our theory gives nontrivial guarantees. ",
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+    {
+        "type": "text",
+        "text": "Full experimental details can be found in the appendix. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/44fba5964c6c783ca2ae42a63a10ae0552b68072bfa7bccf8697ae8c62cbba64.jpg",
+        "image_caption": [
+            "Figure 1: The top row and bottom row show experiments with $k = 6$ and $k = 2$ shards respectively. The 3 columns report on 3 datasets. The $x$ axis denotes estimated expectation of our test statistic (the null hypothesis is expectation 0.5). The $y$ axis denotes the accuracy of the ensemble after deletion. Each point is annotated with the noise multiplier used in DP-SGD, the standard deviation of Gaussian noise applied to gradients during training. A label of 0.0 for a point represents the baseline case of no noise (original SISA algorithm). Points are affixed with $9 5 \\%$ confidence intervals along both axes (over the randomness of repeating the training/deletion experiment). Horizontal confidence intervals that overlap the line denoting expectation 0.5 fail to reject the null hypothesis that the algorithm has adaptive data deletion guarantees at $p \\leq 0 . 0 5$ . We get to this point with a level of noise addition that results in only a modest degradation in ensemble performance compared to SISA. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion and Discussion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We identify an important blindspot in the data deletion literature (the tenuous implicit assumption that deletion requests are independent of previously released models), and provide a very general methodology to reduce adaptive deletion guarantees to oblivious deletion guarantees. Through this reduction we get the first model and training algorithm agnostic methodology that allows for deletion of arbitrary sequences of adaptively chosen points while giving rigorous guarantees. The constants that our theorems inherit from the max information bounds of Rogers et al. [2016] are such that in most realistic settings they will not give useful parameters. But we hope that these constants will be improved in future work, and we give empirical evidence that differential privacy mitigates adaptive deletion “attacks” at very practical levels, beyond the promises of our theoretical results. We note that like for differential privacy, the $( \\alpha , \\beta , \\gamma )$ -deletion guarantees we give in this paper are parameterized, and are not meaningful absent a specification of those parameters. There is a risk with such technologies that they will be used with large values of the parameters that give only very weak guarantees, but will be described publicly in a way that glosses over this issue. We therefore recommend that if adopted in deployed products, deletion guarantees always be discussed in public in a way that is precise about what they promise, including the relevant parameter settings. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "V.G., C.J., A.R., and S.S. were supported in part by NSF grants CCF-1934876 and AF-1763307, and a grant from the Simons Foundation. ",
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