diff --git a/parse/train/519VBzfEaKW/519VBzfEaKW.md b/parse/train/519VBzfEaKW/519VBzfEaKW.md new file mode 100644 index 0000000000000000000000000000000000000000..6eb61184d974de6035616edc61d30ad615722ca9 --- /dev/null +++ b/parse/train/519VBzfEaKW/519VBzfEaKW.md @@ -0,0 +1,219 @@ +# InfoGCL: Information-Aware Graph Contrastive Learning + +Dongkuan $\mathbf { X } \mathbf { u } ^ { 1 }$ Wei Cheng2 Dongsheng Luo1 Haifeng Chen2 Xiang Zhang1 1The Pennsylvania State University 2NEC Labs America 1{dux19,dul262,xzz89}@psu.edu 2{weicheng,haifeng}@nec-labs.com + +# Abstract + +Various graph contrastive learning models have been proposed to improve the performance of learning tasks on graph datasets in recent years. While effective and prevalent, these models are usually carefully customized. In particular, although all recent researches create two contrastive views, they differ greatly in view augmentations, architectures, and objectives. It remains an open question how to build your graph contrastive learning model from scratch for particular graph learning tasks and datasets. In this work, we aim to fill this gap by studying how graph information is transformed and transferred during the contrastive learning process and proposing an information-aware graph contrastive learning framework called InfoGCL. The key point of this framework is to follow the Information Bottleneck principle to reduce the mutual information between contrastive parts while keeping task-relevant information intact at both the levels of the individual module and the entire framework so that the information loss during graph representation learning can be minimized. We show for the first time that all recent graph contrastive learning methods can be unified by our framework. We empirically validate our theoretical analysis on both node and graph classification benchmark datasets, and demonstrate that our algorithm significantly outperforms the state-of-the-arts. + +# 1 Introduction + +Inspired by their success in the vision and language domains, contrastive learning methods have been wildly adopted by recent progress in graph learning to improve the performance of a variety of tasks [42, 13, 22]. In a nutshell, these methods typically learn representations by creating two augmented views of a graph and maximizing the feature consistency between the two views. Inheriting the advantages of self-supervised learning, contrastive learning relieves graph representation learning from its reliance on label information in graph domain, where label information can be very costly or even impossible to collect while unlabeled/partially labeled data is common, such as chemical graph data [28]. Graph contrastive learning methods have achieved similar (and even better) performance as compared to the equivalent methods trained with labels on benchmark graph datasets [42, 13, 6]. + +Despite being effective and prevalent, existing graph contrastive learning models differ mostly in augmented view design, encoding architecture, and contrastive objective (refer to Table 1 in Appendix for more comparisons). For a learning task, it usually requires a substantial degree of domain expertise to carefully design and customize these modules for the specific dataset. For example, while both DGI [33] and InfoGraph [28] seek to obtain graph representations by maximizing the mutual information between patch-level and graph-level representations, they adopt different graph encoders, GCN [15] and GIN [38] respectively. mvgrl [13] applies graph diffusion convolution to construct the augmented view, while GCC [22] and GRACE [45] adopt subgraph sampling and graph perturbation, respectively. + +![](images/31affb98767cd1b70acccd0192dfa07ba19c7212a4703b9817ace38dcc8ae1d7.jpg) +Figure 1: Graph contrastive learning approaches consist of three stages: 1) a graph $\mathcal { G }$ undergoes view augmentation, $q _ { i } ( \cdot ) , q _ { j } ( \cdot )$ , to obtain two semantically similar views, $\mathbf { v } _ { i }$ , $\mathbf { v } _ { j }$ . 2) the two views are fed into view encoder networks, $f _ { i } ( \cdot )$ , $f _ { j } ( \cdot )$ , to extract latent representations, ${ \bf z } _ { i } , { \bf z } _ { j }$ . 3) the feature consistency between representations is maximized to optimize the objective function based on contrastive mode $( c _ { i } ( \cdot ) , c _ { j } ( \cdot ) )$ , where $c _ { i } ( \cdot ) , c _ { j } ( \cdot )$ are aggregation operations applied to representations. + +The main question this paper attempts to answer is: how to perform contrastive learning for your learning tasks on specific graph datasets? However, answering this question is challenging. First, contrastive learning consists of multiple components, such as view augmentation and information encoding. For each of them, there are various choices. Numerous variations make it difficult to design models that are both robust and efficient. Existing graph contrastive learning approaches are carefully designed for different learning tasks on different datasets, however, none of them studies the guiding principles for choosing the best components. Second, graph data has unique properties that distinguish it from other types of data, such as rich structural information and highly diverse distribution [33, 28, 11, 10]. Thus, it is desirable to design the contrastive learning model that fits your graph data properties, even without any domain knowledge of the data. + +We propose to address these challenges via Information Bottleneck (IB) [31], which provides a crucial principle for representation learning. Specifically, IB encourages the representations to be maximally informative about the target in the downstream task, which helps keep task-relevant information. Concurrently, IB discourages the representation learning from acquiring the task-irrelevant information from the input data, which is related to the idea of minimal sufficient statistics [27]. However, different from the typical representation learning, there are two information flows involved in the two augmented views in contrastive learning. Therefore, we extend the previous IB work [37, 43] and propose InfoGCL, an information-aware contrastive learning framework for graph data. + +To study how information is transformed and transferred, we decouple a typical graph contrastive learning model into three sequential modules (as shown in Figure 1): view augmentation, view encoding, and representation contrasting. We further formalize how to find the optimal of the three modules into three optimization problems. To build the optimal graph contrastive learning model for the particular dataset and task, we argue that it is necessary and sufficient to minimize the mutual information between contrastive representations while maximizing task-relevant information at the levels of both individual module and entire framework. Our work is also motivated by the InfoMin theory [30], which suggests that a good set of views for contrastive learning in the vision domain should share the minimal information necessary to perform well at the downstream task. Beyond view selection, our work extends InfoMin to suggest principles of selecting view encodings and contrastive modes for graph learning considering the unique properties of graph data. + +We suggest practically feasible principles to find the optimal modules in graph contrastive learning and show that all recent graph contrastive learning methods can be unified by these principles: i) the augmented views should contain as much task-relevant information as possible, while they should share as little information as possible; ii) the view encoder should be task-relevant and simple as much as possible; iii) the contrastive mode should keep task-relevant information as much as possible after contrasting. Besides, we also investigate the role of negative samples in graph contrastive learning and argue that negative samples are not necessarily required, especially when graph data is not extremely sparse. Our proposed method, InfoGCL, is validated on a rich set of benchmark datasets for both node-level and graph-level tasks, where we analyze its ability to capture the unique structural properties of the graph data. The results demonstrate that our algorithm achieves highly competitive performance with up to $5 . 2 \%$ relative improvement in accuracy on graph classification task and competitive results on node classification task over the state-of-the-art unsupervised methods. + +# 2 Related Work + +Graph Contrastive Learning. Some recent research efforts in graph domain have been attracted by the success of contrastive learning in vision and language domains [3, 8, 4]. A number of graph contrastive learning approaches have been proposed [28, 22, 42, 13]. Despite all of them creating two views and targeting at maximizing the feature disagreement between the two views, these methods are carefully designed and differ in various aspects. Deep graph Infomax (DGI) [33] applied the InfoMax principle [18] to graph data by contrasting the representations of node-level and graph-level for node classification tasks. Different from DGI, InfoGraph [28] aims at node classification tasks and it contrasts the representations of graph-level and substructure-level of different granularity. In addition, DGI and InfoGraph use different graph encoders to extract latent representations. mvgrl [13] studies both node and graph classification tasks. It transforms the adjacency matrix to a diffusion matrix and treat the two matrices as two congruent views. However, in GCC [22] and GRACE [45], subgraph sampling and graph perturbation are used to create the augmented views. GraphCL [42] explores the view augmentations approaches for graph contrastive learning. Specifically, it studies the approaches of node dropping, edge perturbation, attribute masking, and subgraph sampling. A recent work [44] also studies graph data augmentations. However, it focuses on graph neural networks for node classification and does not study the contrastive learning framework. Our method differs from them. We aim to answer the question how to perform contrastive learning for your graph data and tasks. Instead of carefully designing the architectures, we decouple typical graph contrastive learning into three stages and provide our InfoGCL principles to analyze the optimality theoretically and practically. + +Information Bottleneck. Our method is related to the Information Bottleneck (IB) theory [31], which aims to find the best trade-off between accuracy and complexity when summarizing a random variable. IB has been recently used to study the deep learning approaches [26, 37, 43]. Specifically, IB expresses the trade-off between the mutual information measures $I ( \mathbf { D } , \mathbf { Z } )$ and $I ( \mathbf { Z } , \mathbf { y } )$ as + +$$ +\operatorname* { m a x } \mathbf { I } \mathbf { B } _ { \beta } = - I ( \mathbf { D } ; \mathbf { Z } ) + \beta I ( \mathbf { Z } ; \mathbf { y } ) , +$$ + +where D, Z, y are the input, the latent representation and the task label, respectively. $\theta$ is a hyperparameter. In other words, IB aims to learn representation $\mathbf { Z }$ that is maximally expressive about y, while being minimally expressive about $\mathbf { D }$ . More recently, there are some efforts applying the IB theory to graph representation learning. [37] aims to generate both expressive and robust graph representations, while [43] studies the subgraph recognition problem. In contrast, our work focuses on graph contrastive learning, where the two augmented views make the optimization objective (information trade-off) different. Our work is also related to the idea of minimal sufficient statistics [27], which has been recently studied in the vision domain [30], claiming that a good set of image views should share the minimal information necessary to perform well at the downstream task. Different from [30], we focus on the graph domain and propose three stages considering the unique properties of graph data. + +# 3 Preliminaries and Notations + +Graph Representation Learning. A graph is denoted by $\mathcal { G } = \left( \mathbf { A } , \mathbf { X } \right)$ . $\mathbf { A } \in \mathbb { R } ^ { n \times n }$ is the adjacency matrix. $\mathbf { X } \in \mathbb { R } ^ { n \times d }$ is the node attribute matrix, where $d$ is the attribute dimension. In this work, we focus on both node-level and graph-level tasks. For node-level task, given graph $\mathcal { G }$ and the labels of a subset of nodes, denoted by ${ \bf Y } _ { v }$ , the goal is to learn the latent representation $\mathbf { z } _ { v }$ for each node $v$ such that $\mathbf { z } _ { v }$ preserves both network structures and node attributes, which can be further used to predict ${ \bf Y } _ { v }$ . For graph-level task, given a set of graphs $\mathbb { G } = \{ \mathcal { G } ^ { 1 } , \mathcal { G } ^ { 2 } , \cdot \cdot \cdot \mathrm ~ \} \}$ and the labels of some graphs, denoted by $\mathbf { Y } _ { g }$ , the goal is to learn the latent representation $\mathbf { z } _ { g }$ for each graph such that $\mathbf { z } _ { g }$ can be used to predict $\mathbf { Y } _ { g }$ . Typically, the graph data is fed into graph neural networks (GNNs) to generate the representations, such as $\mathbf { z } _ { g } = \mathbf { G } \mathbf { N } \mathbf { N } \mathbf { s } ( \mathcal { G } )$ . + +![](images/31bbe816886e04d01d4fbcb07da4238bee69f393597ea0505da4fb5f0de51ece.jpg) +Figure 2: Illustration of optimal views. (Left) The relationships between graph $\mathcal { G }$ , two views, $\mathbf { v } _ { i }$ , $\mathbf { v } _ { j }$ , and task $y$ in terms of information entropy. A, B, C, D, E are overlapping areas. Two views are contained by graph because views are functions of graph. (Middle) A, B, E become null when Eqs. (4)-(5) hold, which indicates the views and graph share the same amount of task-relevant information. (Right) C further becomes null when Eq. (3) holds, which indicates all the shared information between views is task-relevant, i.e., the views are optimal. + +Graph Contrastive Learning. Given an input graph, graph contrastive learning aims to learn the representations of graph or nodes (for graph-level or node-level tasks respectively) through maximizing the feature consistency between two augmented views of the input graph via contrastive loss in the latent space. We decouple a typical graph contrastive learning model into three sequential modules. + +(i) View augmentation. Graph $\mathcal { G }$ undergoes data augmentation $q ( \cdot )$ to obtain two views $\mathbf { v } _ { i }$ , $\mathbf { v } _ { j }$ , i.e., $\mathbf { v } _ { i } \sim q _ { i } ( \mathcal { G } )$ and $\mathbf { v } _ { j } \sim q _ { j } ( { \mathcal { G } } )$ . A view is represented as graph data, such as $\mathbf { v } _ { i } = ( \mathbf { A } _ { v _ { i } } , \mathbf { X } _ { v _ { i } } )$ , where $\mathbf { A } _ { v _ { i } } \in \mathbb { R } ^ { n \times n }$ and $\mathbf { X } _ { v _ { i } } \in \mathbb { R } ^ { n \times d }$ . In practice, view augmentation approaches include node dropping, edge perturbation, subgraph sampling, etc. + +(ii) View encoding. Graph-level or node-level latent representation is extracted from views $\mathbf { v } _ { i } , \mathbf { v } _ { j }$ by using the view encoder networks $f ( \cdot )$ (a GNN backbone plus a projection MLP), i.e., $\mathbf { z } _ { i } \sim f _ { i } ( \mathbf { v } _ { i } )$ and $\mathbf { z } _ { j } \sim f _ { j } ( \mathbf { v } _ { j } )$ . The two encoders might or might not share parameters depending on whether they are from the same domain. + +(iii) Representation contrasting. Given the latent representations, a contrastive loss is optimized to score the positive pairs $\mathbf { z } _ { i } , \mathbf { z } _ { j }$ higher compared to other negative pairs. Typically, the negative pairs are constructed from the augmented views of other graphs in the same minibatch. The InfoNCE loss [21] has been adopted as one of popular contrastive losses, which is defined as: + +$$ +\mathcal { L } _ { N C E } = - \mathbb { E } \left[ l o g \frac { e x p ( h ( \mathbf { z } _ { i , n } , \mathbf { z } _ { j , n } ) ) } { \sum _ { n ^ { \prime } = 1 } ^ { N } e x p ( h ( \mathbf { z } _ { i , n } , \mathbf { z } _ { j , n ^ { \prime } } ) ) } \right] , +$$ + +where $h ( \cdot )$ is a contrasting operation to score the agreement between two representations. Theoretically, minimizing the InfoNCE loss equivalently maximizes a lower bound on the mutual information between the views of positive pairs. In other words, $I ( \mathbf { z } _ { i } , \mathbf { z } _ { j } ) \geqslant l o g ( N ) - \mathcal { L } _ { N C E }$ , where $I ( \cdot )$ measures the mutual information. + +# 4 Information-Aware Graph Contrastive Learning + +In this paper, we study how to perform contrastive learning for specific graph tasks and datasets. In particular, we attempt to answer the following questions for graph contrastive learning: (i) What is the optimal augmented views? (ii) What is the optimal view encoder? (iii) What is the optimal contrastive mode? + +# 4.1 View Augmentation + +The goal of view augmentation is to create realistically rational data via the transformation approaches that do not affect the semantic label. Compared to the augmentation in other domains, graph view augmentation needs to consider the structural information of graph data, such as the node, the edge, and the subgraph. There are various graph view augmentation methods proposed recently. We follow a similar definition used in [42] to categorize four kinds of view augmentation approaches for graph data. Node dropping discards a certain part of nodes along with their edges in the input graph to create a new graph view. Edge perturbation perturbs the connectivity in the graph via adding or dropping partial edges. Attribute masking masks part of node attributes and assumes that the missing attributes can be well predicted by the remaining ones. Subgraph sampling samples a subgraph from the input graph. The rationale behind these approaches is that the semantic meaning of graph has certain robustness to graph perturbation. + +The augmented views generated in the graph contrastive framework are typically used in a separate downstream task. To characterize what views are optimal for a downstream task, we define the optimality of views. The main motivation is: the optimal augmented views should contain the most task-relevant information, and the information shared between views should only be task-relevant. + +Corollary 1. (Optimal Augmented Views) For a downstream task $T$ whose goal is to predict a semantic label $y$ , the optimal views, $\mathbf { v } _ { i } ^ { * } , \mathbf { v } _ { j } ^ { * } ,$ , generated from the input graph $\mathcal { G }$ are the solutions to the following optimization problem : + +$$ +\begin{array} { c } { { ( \mathbf { v } _ { i } ^ { * } , \mathbf { v } _ { j } ^ { * } ) = \underset { \mathbf { v } _ { i } , \mathbf { v } _ { j } } { \arg \operatorname* { m i n } I ( \mathbf { v } _ { i } ; \mathbf { v } _ { j } ) } } } \\ { { s . t . ~ I ( \mathbf { v } _ { i } ; y ) = I ( \mathbf { v } _ { j } ; y ) } } \\ { { I ( \mathbf { v } _ { i } ; y ) = I ( \mathcal { G } ; y ) } } \end{array} +$$ + +This says that for the optimal graph views, the amount of information shared between them is minimized (Eq. (3)), while the two views contain the same amount of information with respect to $y$ (Eq. (4)), which is also the amount of information that the input gprah contains about the task (Eq. (5)). The illustration of the optimal views is shown in Figure 2 and the proof is in the Appendix. + +# 4.2 View Encoding + +View encoding aims to extract the latent representations of nodes or graphs via feeding the data of two views into view encoder networks such that the generated representations preserve both structure and attribute information in the views. The view encoders are quite flexible in graph contrastive learning and typically they are GCN [15], GAT [32], or GIN [38], etc. + +The representations extracted via view encoding are further utilized to optimize the objective function of contrastive learning. After well trained, the view encoders are used to generate the graph/node representations for a downstream task. To characterize what encoders are optimal, we define the optimality of view encoders for graph contrastive learning. The main motivation is: the representation generated by the optimal encoder for a view should keep all the shared information by the two contrastive views, meanwhile the kept information is all task-relevant. + +Corollary 2. (Optimal View Encoder) Given the optimal views, $\mathbf { v } _ { i } ^ { * } , \mathbf { v } _ { j } ^ { * }$ , for a downstream task $T$ whose goal is to predict a semantic label $y$ , the optimal view encoder for view $\mathbf { v } _ { i } ^ { * }$ is the solution to the following optimization problem : + +$$ +\boldsymbol f _ { i } ^ { * } = \arg \operatorname* { m i n } _ { \boldsymbol f _ { i } } I ( f _ { i } ( \mathbf { v } _ { i } ^ { * } ) ; \mathbf { v } _ { i } ^ { * } ) +$$ + +$$ +s . t . \ I ( f _ { i } ( \mathbf { v } _ { i } ^ { * } ) ; \mathbf { v } _ { j } ^ { * } ) = I ( \mathbf { v } _ { i } ^ { * } ; \mathbf { v } _ { j } ^ { * } ) +$$ + +It indicates that for the optimal view encoder, the amount of information shared between the optimal view and the extracted representation is minimized (Eq. (6)), while the information shared between the two optimal views is kept after the encoding process of one view (Eq. (7)). The illustration of the optimal encoder is shown in Figure 3 and the proof is illustrated in the Appendix. + +# 4.3 Representation Contrasting + +To allow flexible contrasting for graph data, we consider contrastive modes similar to [13]. A contrastive mode is denoted by $( c _ { i } ( \cdot ) , c _ { j } ( \cdot ) )$ , where $c _ { i } ( \cdot )$ , $c _ { j } ( \cdot )$ are the aggregation operations applied to the representations extracted by view encoders, The contrastive modes are unique to graph data because of the structural information inside a graph. Specifically, we consider five contrastive modes. In global-global mode, the graph representations from two views are contrasted. Thus, $c _ { i } ( \cdot )$ , $c _ { j } ( \cdot )$ are averaging aggregation operations in this mode. In local-global mode, we contrast the node representations from one view with the graph representations from the other view. Thus, $c _ { i } ( \cdot )$ , $c _ { j } ( \cdot )$ are the identical transformation and averaging aggregation operations, respectively. In local-local mode, the node representations from two views are contrasted. In multi-scale mode, we contrast graph representation of one view with the intermediate representation from the other. In hybrid mode, both global-global and local-global are applied. + +![](images/a7fd63d27d26cca071ead706f3fd728ce0211916cd89519025b1dee6ce661218.jpg) +Figure 3: Illustration of optimal view encoding. (Left) The relationships between two (optimal) views, $\mathbf { v } _ { i } ^ { * }$ , $\mathbf { v } _ { j } ^ { * }$ , task $y$ , and representation $\mathbf { z } _ { i }$ in terms of information entropy. A, B, C are null because the two views are optimal here. $\mathbf { z } _ { i }$ is contained by view $\mathbf { v } _ { i } ^ { * }$ because representations are functions of views. (Middle) $\mathbf { z } _ { i }$ covers $\mathbf { D }$ when Eq. (7) holds, which indicates the shared information between views is kept after encoding. $( R i g h t )$ View $\mathbf { z } _ { i }$ further exactly covers $\mathbf { D }$ and the view encoding becomes optimal, i.e., $\mathbf { z } _ { i } ^ { * }$ , which indicates all the information shared between view $\mathbf { v } _ { i } ^ { * }$ and representation $\mathbf { z } _ { i } ^ { * }$ is task-relevant. + +To characterize which mode is optimal, we define the optimality of contrastive mode for graph contrastive learning. The main motivation is: the optimal contrastive mode keeps the most taskrelevant information after the representations are aggregated. The proof is included in the Appendix. + +Corollary 3. (Optimal Contrastive Mode) Given the latent representations, $\mathbf { z } _ { i } ^ { * }$ , $\mathbf { z } _ { j } ^ { \ast }$ , extracted by the optimal view encoders, i.e., $\mathbf { z } _ { i } ^ { * } = f _ { i } ^ { * } ( \mathbf { v } _ { i } ^ { * } )$ , $\mathbf { z } _ { j _ { - } } ^ { * } = f _ { j _ { - } } ^ { * } ( \mathbf { v } _ { j } ^ { * } )$ , and a downstream task $T$ with label $y _ { \mathrm { { i } } }$ , the optimal contrastive mode is the solution to the following optimization problem, where $c _ { i }$ , $c _ { j }$ are the aggregation operations applied to the latent representations: + +$$ +\displaystyle ( c _ { i } ^ { * } , c _ { j } ^ { * } ) = \arg \operatorname* { m i n } _ { ( c _ { i } , c _ { j } ) } - I ( c _ { i } ( \mathbf { z } _ { i } ^ { * } ) ; c _ { j } ( \mathbf { z } _ { j } ^ { * } ) ) . +$$ + +# 4.4 InfoGCL Principle + +According to our proposed corollaries, we can theoretically design the optimal contrastive learning approach for our specific graph data and task. However, in real-world scenarios, the conditions to meet the exact optimality of contrastive learning is hard or even not practically possible to reach because of data noise and limited model capability. Therefore, we propose to achieve the optimal for each stage independently and practically, which is an approximation to achieve the original optimality. Specifically, we make the following propositions to address the questions of the optimal views, optimal view encoder, and optimal contrastive mode. + +Proposition 1. For a task $T$ with label $y _ { ; }$ , given a bunch of graph view augmentation methods, $\{ q _ { 1 } ( \cdot ) _ { : }$ , $q _ { 2 } ( \cdot ) , \cdot \cdot \cdot \}$ , that create two views $\mathbf { v } _ { i }$ , $\mathbf { v } _ { j }$ , the recommended augmentation methods are the ones, $q _ { i } ( \cdot )$ , $q _ { j } ( \cdot )$ , that maximize $I ( \mathbf { v } _ { i } ; y ) + I ( \mathbf { v } _ { j } ; y ) - I ( \mathbf { v } _ { i } ; \mathbf { v } _ { j } )$ , i.e., the area of $A { + } B { + } D$ in Figure 2. + +Proposition 2. Given a task $T$ with label $y$ and a set of view encoders, $\{ f _ { i } ^ { 1 } ( \cdot ) , f _ { i } ^ { 2 } ( \cdot ) , \cdot \cdot \cdot \ \} ,$ , that generate representation $\mathbf { z } _ { i }$ via taking view $\mathbf { v } _ { i }$ as input, the recommended view encoder is the one that maximizes the mutual information between $\mathbf { v } _ { i }$ , $\mathbf { z } _ { i }$ and $y$ . Symmetrically the same for view $\mathbf { v } _ { j }$ . + +Proposition 3. Given a task $T$ with label $y$ , the extracted representations, $z _ { i } , \ z _ { j }$ , and a set of aggregation operations, $\{ c _ { 1 } ( \cdot ) , c _ { 2 } ( \cdot ) , \cdot \cdot \cdot \ \} ,$ , the recommended contrastive mode is the one, $( c _ { i } , c _ { j } )$ , that has the largest amount of mutual information between $c _ { i } ( \mathbf { z } _ { i } )$ , $c _ { j } ( \mathbf { z } _ { j } )$ and $y$ . + +The qualitative and quantitative evaluation of these propositions are shown in Section 5.3. + +# 4.5 Role of Negative Samples + +Current graph contrastive learning approaches heavily depend on negative samples. However, recent progresses of contrastive learning in vision domain indicate that negative samples are not necessarily required [8, 4], of which the main benefit is to avoid careful treatment to retrieve the negative pairs. To study the influence of negative samples on graph contrastive learning, we follow the framework of SimSiam [4] to revise the loss function as Eq.(9). A very recent work [29] also studies graph contrastive learning without negative samples. Different from it, we focus on both node and graph classification tasks. + +Table 1: Dataset statistics. + +
Graph Task DatasetsNode Task Datasets
MUTAGPTC-MRIMDB-BIMDB-MNCI1COLLABCoraCiteseerPubmed
#Graphs1883441000150041105000111
#Nodes17.914.319.813.029.974.53327270819717
#Edges19.814.7193.165.91.133.04732542944338
#Classes222223673
+ +$$ +\mathcal { L } = - \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \frac { \mathbf { z } _ { i , n } } { \| \mathbf { z } _ { i , n } \| } \cdot \frac { \mathbf { z } _ { j , n } } { \| \mathbf { z } _ { j , n } \| } , +$$ + +# 5 Experiments + +In this section, we evaluate our InfoGCL with a number of experiments. We first describe datasets, evaluation protocol, and experimental setup. Then, we present the experimental results on both node and graph classification. Last, we analyze our proposed principles via ablation study. + +# 5.1 Setup + +We use both graph classification and node classification benchmark datasets that are widely used in the existing graph contrastive learning approaches. The graph classification datasets include MUTAG [17], PTC-MR [17], IMDB-B [40], IMDB-M [40], NCI1 [34], and COLLAB [40]. MUTAG is a collection of nitroaromatic compounds represented as graphs, where vertices stand for atoms and edges represent bonds between atoms. PTC-MR is a collection of 344 chemical compounds which report the carcinogenicity for rats. IMDB-B and IMDB-M are two movie collaboration datasets, where nodes represent actors/actress and there is an edge between them if they appear in the same movie. In NCI1, graphs are the representation of chemical compounds, where vertices stand for atoms and edges represent bonds between atoms. COLLAB is a collaboration dataset, where researchers are nodes and an edge indicates collaboration between two researchers. The node classification datasets include Citeseer, Cora, and Pubmed [23]. All of them are citation networks, where nodes are documents and edges are citation links. These datasets are summarized in Table 1. + +We closely follow the evaluation protocol of previous state-of-the-art graph contrastive learning approaches. For graph classification, we report the mean 10-fold cross validation accuracy after 5 runs followed by a linear SVM. The linear SVM is trained by applying cross validation on training data folds and the best mean accuracy is reported. For node classification, we report the mean accuracy on test set after 50 runs of training followed by a linear neural network model. To make comparison fair, we adopt the basic setting of InfoGraph for graph classification. We conduct experiment with the values of the number of GNN layers, the number of epochs, batch size, the parameter C of SVM in the sets {2, 4, 8, 12}, {10, 20, 40, 100}, {32, 64, 128, 256} and $\{ 1 0 ^ { - 3 } , \ : \mathrm { { \dot { 1 } 0 ^ { - 2 } } } , . . . , \ : 1 0 ^ { 2 } , 1 0 ^ { 3 } \ : \ : \}$ , respectively. We adopt the basic setting of DGI for node classification. Specifically, we set the number of GNN layers to 1 and experiment with the batch size in the set {2, 4, 8}. The hidden dimension of representations is set to 512. We also apply the early stopping strategy. + +# 5.2 Experimental Results + +To evaluate our method InfoGCL on graph classification, we use thhree categories of baselines. The kernel approaches include shortest path kernel (SP) [2], Graphlet kernel (GK) [25], WeisfeilerLehman sub-tree kernel (WL) [24], deep graph kernels (DGK) [39], and multi-scale Laplacian kernel (MLG) [16]. The supervised baselines include GraphSAGE [12], GCN [15], GIN [38], GAT [32]. We also compare with the unsupervised approaches, including RandomWalk [7], node2vec [9], sub2vec [14], graph2vec [20], InfoGraph [28], GraphCL [42], and mvgrl [13]. Table 2 shows the graph classification results. We observe that our approach achieves the best results compared to other unsupervised approaches. Our approach also outperforms or matches the best kernel approaches across the datasets. Even compared with the supervised ones, our approach achieves the best in 2 out of 6 datasets and the results of our approach on other 4 dataset are among the top. + +Table 2: Graph classification results $( \% )$ + +
MethodMUTAGPTC-MRIMDB-BIMDB-MNCI1COLLAB
Kernel Approaches
SP85.2 ± 2.458.2 ± 2.455.6± 0.238.0±0.373.5 ± 0.1
GK81.7 ± 2.157.3 ± 1.465.9 ± 1.043.9 ± 0.466.0 ± 0.172.8± 0.3
WL80.7 ± 3.058.0 ± 0.572.3 ± 3.447.0 ± 0.580.0± 0.578.9 ± 1.9
DGK87.4 ± 2.760.1 ± 2.667.0 ± 0.644.6 ± 0.580.3 ± 0.573.1 ± 0.3
MLG87.9 ± 1.663.3 ± 1.566.6 ± 0.341.2 ± 0.080.8 ± 1.3
Supervised Approaches
GraphSAGE85.1 ± 7.6 63.9 ± 7.772.3 ± 5.350.9 ± 2.277.7 ± 1.568.3 ± 4.2
GCN85.6 ± 5.864.2 ± 4.374.0± 3.451.9 ± 3.880.2 ± 2.079.0 ± 1.8
GIN-089.4 ± 5.664.6 ± 7.075.1 ± 5.152.3 ± 2.882.7 ± 1.780.2 ± 1.9
GIN-e89.0 ± 6.063.7 ±8.274.3 ± 5.152.1 ± 3.682.7 ± 1.680.1 ± 1.9
GAT89.4 ± 6.166.7 ± 5.170.5 ± 2.347.8 ± 3.166.6 ± 2.267.4 ± 2.9
Unsupervised Approaches
RandomWalk83.7 ± 1.557.9 ± 1.350.7±0.334.7 ± 0.264.3 ± 0.3
node2vec72.6 ±10.258.6±8.050.2 ± 0.936.0 ± 0.754.9 ± 1.656.1 ± 0.2
sub2vec61.1 ± 15.860.0 ± 6.455.3 ± 1.536.7 ± 0.852.8 ±1.5-
graph2vec83.2 ±9.660.2 ± 6.971.1 ± 0.550.4 ± 0.975.4 ±1.2
InfoGraph89.0 ± 1.161.7 ± 1.473.0± 0.949.7 ± 0.576.2 ±1.470.7 ± 1.1
GraphCL86.8 ± 1.361.3 ± 2.171.1 ± 0.449.2 ± 0.677.9 ± 0.471.4 ± 1.2
mvgrl89.7 ± 1.162.5 ± 1.774.2 ± 0.751.2 ± 0.577.0 ± 0.876.0 ± 1.2
51.4 ± 0.880.2 ± 0.6
InfoGCL91.2 ± 1.363.5 ± 1.575.1 ± 0.980.0 ± 1.3
+ +Table 3: Node classification results $( \% )$ . + +
MethodCoraCiteseerPubmed
Supervised Approaches
MLP55.146.571.4
ICA75.169.173.9
LP68.045.363.0
ManiReg59.560.170.7
SemiEmb59.059.671.7
Planetoid75.764.777.2
Chebyshev81.269.874.4
GCN81.570.379.0
JKNet82.7 ± 0.473.0 ± 0.577.9 ± 0.4
GAT83.0 ± 0.772.5± 0.779.0 ± 0.3
Unsupervised Approaches
Linear47.9 ± 0.449.3 ± 0.269.1 ± 0.3
DeepWalk70.7 ± 0.651.4 ± 0.574.3 ± 0.9
GAE71.5 ± 0.465.8 ±0.472.1 ± 0.5
VERSE72.5 ± 0.355.5± 0.4
DGI83.8 ± 0.572.0 ± 0.677.9 ± 0.3
GraphCL82.5 ± 0.173.1 ± 0.2
mvgrl86.8 ± 0.573.3 ± 0.580.1 ± 0.7
InfoGCL83.5 ± 0.373.5 ± 0.479.1 ± 0.2
+ +For node classification tasks, we compare InfoGCL with some supervised approaches and unsupervised approaches. The supervised baselines include a simple MLP model, iterative classification algorithm (ICA) [19], manifold regularization (ManiReg) [1], semi-supervised embedding (SemiEmb) [35], Planetoid [41], Chebyshev [5], GCN, JKNet [36], GAT. Table 3 shows the node classification results. It is observed that our approach achieves the state-of-the-art results and competes the best one with respect to the existing unsupervised approaches. Compared to supervised baselines, our approach outperforms all the baselines. + +
MethodMUTAGIMDB-BCOLLABCoraCiteseerPubmed
InfoGCL (w/o neg)91.0 ± 1.475.1± 0.580.2 ±1.078.6 ± 0.470.4 ± 0.677.4±0.7
InfoGCL (w/ neg)91.2 ± 1.375.1 ± 0.980.0± 1.383.5±0.373.5 ± 0.479.1±0.2
+ +Table 4: Comparison between InfoGCL with negative samples and without negative samples. + +# 5.3 Evaluation of InfoGCL Principle + +We can unify the existing graph contrastive learning methods through the perspective of InfoGCL principle: all recent graph contrastive learning methods can be decoupled into three stages that implicitly follow the InfoGCL principle, though being different in model architecture design and optimization strategies. Below, we analyze some observations from several recent work. Because of the limited space, please refer to the Appendix for more results of the quantitative analysis. + +Obs. i. Composing a graph and its augmentation benefits downstream performance [42]. Compared to composing a graph and the graph itself, augmentation leads to smaller $I ( \mathbf { v } _ { i } ; \mathbf { v } _ { j } )$ (Proposition 1). + +Obs. ii. Composing different augmentations benefits more [42]. Compared to composing a graph and its augmentations, two augmentations further decrease $I ( \mathbf { v } _ { i } ; \mathbf { v } _ { j } )$ (Proposition 1). + +Obs. iii. Node dropping and subgraph sampling are generally beneficial across datasets [42]. When compared to attribute masking and edge perturbation, they change the semantic meaning of the graph relatively slightly, which leads to higher $I ( \mathbf { v } _ { i } ; y )$ , $I ( \mathbf { v } _ { j } ; \bar { y } )$ (Proposition 1). + +Obs. iv. Edge perturbation benefits social networks but hurts some biochemical molecules [42]. The semantic meaning of social networks are robust to edge perturbation. However, the semantic meaning of some biochemical molecules are determined by local connection pattern, where edge perturbation decreases $I ( \mathbf { v } _ { i } ; y )$ (Proposition 1). + +Obs. v. Contrasting node and graph representations consistently performs better than other contrastive modes across benchmarks [13]. Compared to other contrastive modes, node-graph (i.e., local-global) mode generally extracts more graph structure information, which benefits predicting task label $y$ (Proposition 3). + +# 5.4 Effect of Negative Samples + +To study the effect of negative samples on graph contrastive learning, we follow SimSiam [4] and design the objective as Eq. (9). We conduct experiments on three graph task and three node task datasets. The results are reported in Table 4. It is observed that the negative samples show little influence on the three graph task datasets, while performance drops on the three node task datasets, especially the Cora dataset. + +According to the dataset statistics summarized in Table 1, we see the networks of Cora, Citeseer, Pubmed are much sparser (in terms of network topology). Furthermore, we know the node features of these three datasets are also much sparser (one-hot encoding with high dimensionality). We speculate this because the contrastive learning models tends to collapse easier if negative samples are not used, especially when data is too sparse. Therefore,we make the hypothesis: negative samples benefit graph modeling, especially when i) network topology, and ii) node features are extremely sparse. + +# 6 Conclusion and Limitations + +We propose InfoGCL, an information-aware graph contrastive learning framework for graph contrastive learning. Existing graph contrastive learning approaches are usually carefully designed. We aim to answer how to perform contrastive learning for your learning tasks on specific graph data. Our method decouples the typical contrastive learning approaches into three sequential modules and provides the theoretical analysis for reaching the optimality. To address the questions of optimality in a practical way, we propose the InfoGCL principle, which is implicitly followed by all recent graph contrastive learning approaches. In addition, we explore the role of negative samples in graph contrastive learning and find negative samples are not necessarily required. Experiments on both node and graph benchmark datasets demonstrate the effectiveness of our method. Note that our method is not without limitations. We can further improve our method by designing better practical approximations to the theoretical optimality of graph contrastive learning. + +# Funding Transparency Statement + +This project was partially supported by NSF projects IIS-1707548 and CBET-1638320. + +# References + +[1] Mikhail Belkin, Partha Niyogi, and Vikas Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. J. Mach. Learn. Res, 7, 2006. [2] Karsten M. Borgwardt and Hans-Peter Kriegel. Shortest-path kernels on graphs. In ICDM, 2005. [3] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey E. Hinton. A simple framework for contrastive learning of visual representations. 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Graph contrastive learning with augmentations. In NeurIPS, volume 33, 2020. +[43] Junchi Yu, Tingyang Xu, Yu Rong, Yatao Bian, Junzhou Huang, and Ran He. Graph information bottleneck for subgraph recognition. ICLR, 2021. +[44] Tong Zhao, Yozen Liu, Leonardo Neves, Oliver Woodford, Meng Jiang, and Neil Shah. Data augmentation for graph neural networks. arXiv preprint arXiv:2006.06830, 2020. +[45] Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Deep Graph Contrastive Representation Learning. In ICML Workshop, 2020. \ No newline at end of file diff --git a/parse/train/519VBzfEaKW/519VBzfEaKW_content_list.json b/parse/train/519VBzfEaKW/519VBzfEaKW_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..546aeb0c38f77ac83d6ec3355e15cd48a26d29c8 --- /dev/null +++ b/parse/train/519VBzfEaKW/519VBzfEaKW_content_list.json @@ -0,0 +1,1088 @@ +[ + { + "type": "text", + "text": "InfoGCL: Information-Aware Graph Contrastive Learning ", + "text_level": 1, + "bbox": [ + 200, + 122, + 799, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Dongkuan $\\mathbf { X } \\mathbf { u } ^ { 1 }$ Wei Cheng2 Dongsheng Luo1 Haifeng Chen2 Xiang Zhang1 1The Pennsylvania State University 2NEC Labs America 1{dux19,dul262,xzz89}@psu.edu 2{weicheng,haifeng}@nec-labs.com ", + "bbox": [ + 202, + 224, + 784, + 299 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 333, + 535, + 349 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Various graph contrastive learning models have been proposed to improve the performance of learning tasks on graph datasets in recent years. While effective and prevalent, these models are usually carefully customized. In particular, although all recent researches create two contrastive views, they differ greatly in view augmentations, architectures, and objectives. It remains an open question how to build your graph contrastive learning model from scratch for particular graph learning tasks and datasets. In this work, we aim to fill this gap by studying how graph information is transformed and transferred during the contrastive learning process and proposing an information-aware graph contrastive learning framework called InfoGCL. The key point of this framework is to follow the Information Bottleneck principle to reduce the mutual information between contrastive parts while keeping task-relevant information intact at both the levels of the individual module and the entire framework so that the information loss during graph representation learning can be minimized. We show for the first time that all recent graph contrastive learning methods can be unified by our framework. We empirically validate our theoretical analysis on both node and graph classification benchmark datasets, and demonstrate that our algorithm significantly outperforms the state-of-the-arts. ", + "bbox": [ + 233, + 367, + 766, + 602 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 635, + 310, + 651 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Inspired by their success in the vision and language domains, contrastive learning methods have been wildly adopted by recent progress in graph learning to improve the performance of a variety of tasks [42, 13, 22]. In a nutshell, these methods typically learn representations by creating two augmented views of a graph and maximizing the feature consistency between the two views. Inheriting the advantages of self-supervised learning, contrastive learning relieves graph representation learning from its reliance on label information in graph domain, where label information can be very costly or even impossible to collect while unlabeled/partially labeled data is common, such as chemical graph data [28]. Graph contrastive learning methods have achieved similar (and even better) performance as compared to the equivalent methods trained with labels on benchmark graph datasets [42, 13, 6]. ", + "bbox": [ + 174, + 669, + 825, + 794 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite being effective and prevalent, existing graph contrastive learning models differ mostly in augmented view design, encoding architecture, and contrastive objective (refer to Table 1 in Appendix for more comparisons). For a learning task, it usually requires a substantial degree of domain expertise to carefully design and customize these modules for the specific dataset. For example, while both DGI [33] and InfoGraph [28] seek to obtain graph representations by maximizing the mutual information between patch-level and graph-level representations, they adopt different graph encoders, GCN [15] and GIN [38] respectively. mvgrl [13] applies graph diffusion convolution to construct the augmented view, while GCC [22] and GRACE [45] adopt subgraph sampling and graph perturbation, respectively. ", + "bbox": [ + 174, + 800, + 825, + 897 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/31affb98767cd1b70acccd0192dfa07ba19c7212a4703b9817ace38dcc8ae1d7.jpg", + "image_caption": [ + "Figure 1: Graph contrastive learning approaches consist of three stages: 1) a graph $\\mathcal { G }$ undergoes view augmentation, $q _ { i } ( \\cdot ) , q _ { j } ( \\cdot )$ , to obtain two semantically similar views, $\\mathbf { v } _ { i }$ , $\\mathbf { v } _ { j }$ . 2) the two views are fed into view encoder networks, $f _ { i } ( \\cdot )$ , $f _ { j } ( \\cdot )$ , to extract latent representations, ${ \\bf z } _ { i } , { \\bf z } _ { j }$ . 3) the feature consistency between representations is maximized to optimize the objective function based on contrastive mode $( c _ { i } ( \\cdot ) , c _ { j } ( \\cdot ) )$ , where $c _ { i } ( \\cdot ) , c _ { j } ( \\cdot )$ are aggregation operations applied to representations. " + ], + "image_footnote": [], + "bbox": [ + 199, + 80, + 794, + 262 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 359, + 823, + 388 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main question this paper attempts to answer is: how to perform contrastive learning for your learning tasks on specific graph datasets? However, answering this question is challenging. First, contrastive learning consists of multiple components, such as view augmentation and information encoding. For each of them, there are various choices. Numerous variations make it difficult to design models that are both robust and efficient. Existing graph contrastive learning approaches are carefully designed for different learning tasks on different datasets, however, none of them studies the guiding principles for choosing the best components. Second, graph data has unique properties that distinguish it from other types of data, such as rich structural information and highly diverse distribution [33, 28, 11, 10]. Thus, it is desirable to design the contrastive learning model that fits your graph data properties, even without any domain knowledge of the data. ", + "bbox": [ + 174, + 393, + 825, + 532 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We propose to address these challenges via Information Bottleneck (IB) [31], which provides a crucial principle for representation learning. Specifically, IB encourages the representations to be maximally informative about the target in the downstream task, which helps keep task-relevant information. Concurrently, IB discourages the representation learning from acquiring the task-irrelevant information from the input data, which is related to the idea of minimal sufficient statistics [27]. However, different from the typical representation learning, there are two information flows involved in the two augmented views in contrastive learning. Therefore, we extend the previous IB work [37, 43] and propose InfoGCL, an information-aware contrastive learning framework for graph data. ", + "bbox": [ + 174, + 537, + 825, + 650 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To study how information is transformed and transferred, we decouple a typical graph contrastive learning model into three sequential modules (as shown in Figure 1): view augmentation, view encoding, and representation contrasting. We further formalize how to find the optimal of the three modules into three optimization problems. To build the optimal graph contrastive learning model for the particular dataset and task, we argue that it is necessary and sufficient to minimize the mutual information between contrastive representations while maximizing task-relevant information at the levels of both individual module and entire framework. Our work is also motivated by the InfoMin theory [30], which suggests that a good set of views for contrastive learning in the vision domain should share the minimal information necessary to perform well at the downstream task. Beyond view selection, our work extends InfoMin to suggest principles of selecting view encodings and contrastive modes for graph learning considering the unique properties of graph data. ", + "bbox": [ + 174, + 656, + 825, + 809 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We suggest practically feasible principles to find the optimal modules in graph contrastive learning and show that all recent graph contrastive learning methods can be unified by these principles: i) the augmented views should contain as much task-relevant information as possible, while they should share as little information as possible; ii) the view encoder should be task-relevant and simple as much as possible; iii) the contrastive mode should keep task-relevant information as much as possible after contrasting. Besides, we also investigate the role of negative samples in graph contrastive learning and argue that negative samples are not necessarily required, especially when graph data is not extremely sparse. Our proposed method, InfoGCL, is validated on a rich set of benchmark datasets for both node-level and graph-level tasks, where we analyze its ability to capture the unique structural properties of the graph data. The results demonstrate that our algorithm achieves highly competitive performance with up to $5 . 2 \\%$ relative improvement in accuracy on graph classification task and competitive results on node classification task over the state-of-the-art unsupervised methods. ", + "bbox": [ + 174, + 814, + 823, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 180, + 321, + 196 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Graph Contrastive Learning. Some recent research efforts in graph domain have been attracted by the success of contrastive learning in vision and language domains [3, 8, 4]. A number of graph contrastive learning approaches have been proposed [28, 22, 42, 13]. Despite all of them creating two views and targeting at maximizing the feature disagreement between the two views, these methods are carefully designed and differ in various aspects. Deep graph Infomax (DGI) [33] applied the InfoMax principle [18] to graph data by contrasting the representations of node-level and graph-level for node classification tasks. Different from DGI, InfoGraph [28] aims at node classification tasks and it contrasts the representations of graph-level and substructure-level of different granularity. In addition, DGI and InfoGraph use different graph encoders to extract latent representations. mvgrl [13] studies both node and graph classification tasks. It transforms the adjacency matrix to a diffusion matrix and treat the two matrices as two congruent views. However, in GCC [22] and GRACE [45], subgraph sampling and graph perturbation are used to create the augmented views. GraphCL [42] explores the view augmentations approaches for graph contrastive learning. Specifically, it studies the approaches of node dropping, edge perturbation, attribute masking, and subgraph sampling. A recent work [44] also studies graph data augmentations. However, it focuses on graph neural networks for node classification and does not study the contrastive learning framework. Our method differs from them. We aim to answer the question how to perform contrastive learning for your graph data and tasks. Instead of carefully designing the architectures, we decouple typical graph contrastive learning into three stages and provide our InfoGCL principles to analyze the optimality theoretically and practically. ", + "bbox": [ + 174, + 210, + 825, + 488 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Information Bottleneck. Our method is related to the Information Bottleneck (IB) theory [31], which aims to find the best trade-off between accuracy and complexity when summarizing a random variable. IB has been recently used to study the deep learning approaches [26, 37, 43]. Specifically, IB expresses the trade-off between the mutual information measures $I ( \\mathbf { D } , \\mathbf { Z } )$ and $I ( \\mathbf { Z } , \\mathbf { y } )$ as ", + "bbox": [ + 174, + 494, + 825, + 550 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/7f84f97e8d7cee3894f71d06c6fa2677f0ffb3b47533ae9a9b4fe1e1c4da0508.jpg", + "text": "$$\n\\operatorname* { m a x } \\mathbf { I } \\mathbf { B } _ { \\beta } = - I ( \\mathbf { D } ; \\mathbf { Z } ) + \\beta I ( \\mathbf { Z } ; \\mathbf { y } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 558, + 619, + 575 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where D, Z, y are the input, the latent representation and the task label, respectively. $\\theta$ is a hyperparameter. In other words, IB aims to learn representation $\\mathbf { Z }$ that is maximally expressive about y, while being minimally expressive about $\\mathbf { D }$ . More recently, there are some efforts applying the IB theory to graph representation learning. [37] aims to generate both expressive and robust graph representations, while [43] studies the subgraph recognition problem. In contrast, our work focuses on graph contrastive learning, where the two augmented views make the optimization objective (information trade-off) different. Our work is also related to the idea of minimal sufficient statistics [27], which has been recently studied in the vision domain [30], claiming that a good set of image views should share the minimal information necessary to perform well at the downstream task. Different from [30], we focus on the graph domain and propose three stages considering the unique properties of graph data. ", + "bbox": [ + 173, + 582, + 825, + 734 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Preliminaries and Notations ", + "text_level": 1, + "bbox": [ + 176, + 753, + 439, + 770 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Graph Representation Learning. A graph is denoted by $\\mathcal { G } = \\left( \\mathbf { A } , \\mathbf { X } \\right)$ . $\\mathbf { A } \\in \\mathbb { R } ^ { n \\times n }$ is the adjacency matrix. $\\mathbf { X } \\in \\mathbb { R } ^ { n \\times d }$ is the node attribute matrix, where $d$ is the attribute dimension. In this work, we focus on both node-level and graph-level tasks. For node-level task, given graph $\\mathcal { G }$ and the labels of a subset of nodes, denoted by ${ \\bf Y } _ { v }$ , the goal is to learn the latent representation $\\mathbf { z } _ { v }$ for each node $v$ such that $\\mathbf { z } _ { v }$ preserves both network structures and node attributes, which can be further used to predict ${ \\bf Y } _ { v }$ . For graph-level task, given a set of graphs $\\mathbb { G } = \\{ \\mathcal { G } ^ { 1 } , \\mathcal { G } ^ { 2 } , \\cdot \\cdot \\cdot \\mathrm ~ \\} \\}$ and the labels of some graphs, denoted by $\\mathbf { Y } _ { g }$ , the goal is to learn the latent representation $\\mathbf { z } _ { g }$ for each graph such that $\\mathbf { z } _ { g }$ can be used to predict $\\mathbf { Y } _ { g }$ . Typically, the graph data is fed into graph neural networks (GNNs) to generate the representations, such as $\\mathbf { z } _ { g } = \\mathbf { G } \\mathbf { N } \\mathbf { N } \\mathbf { s } ( \\mathcal { G } )$ . ", + "bbox": [ + 174, + 785, + 825, + 912 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/31bbe816886e04d01d4fbcb07da4238bee69f393597ea0505da4fb5f0de51ece.jpg", + "image_caption": [ + "Figure 2: Illustration of optimal views. (Left) The relationships between graph $\\mathcal { G }$ , two views, $\\mathbf { v } _ { i }$ , $\\mathbf { v } _ { j }$ , and task $y$ in terms of information entropy. A, B, C, D, E are overlapping areas. Two views are contained by graph because views are functions of graph. (Middle) A, B, E become null when Eqs. (4)-(5) hold, which indicates the views and graph share the same amount of task-relevant information. (Right) C further becomes null when Eq. (3) holds, which indicates all the shared information between views is task-relevant, i.e., the views are optimal. " + ], + "image_footnote": [], + "bbox": [ + 207, + 71, + 794, + 217 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Graph Contrastive Learning. Given an input graph, graph contrastive learning aims to learn the representations of graph or nodes (for graph-level or node-level tasks respectively) through maximizing the feature consistency between two augmented views of the input graph via contrastive loss in the latent space. We decouple a typical graph contrastive learning model into three sequential modules. ", + "bbox": [ + 173, + 340, + 825, + 410 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(i) View augmentation. Graph $\\mathcal { G }$ undergoes data augmentation $q ( \\cdot )$ to obtain two views $\\mathbf { v } _ { i }$ , $\\mathbf { v } _ { j }$ , i.e., $\\mathbf { v } _ { i } \\sim q _ { i } ( \\mathcal { G } )$ and $\\mathbf { v } _ { j } \\sim q _ { j } ( { \\mathcal { G } } )$ . A view is represented as graph data, such as $\\mathbf { v } _ { i } = ( \\mathbf { A } _ { v _ { i } } , \\mathbf { X } _ { v _ { i } } )$ , where $\\mathbf { A } _ { v _ { i } } \\in \\mathbb { R } ^ { n \\times n }$ and $\\mathbf { X } _ { v _ { i } } \\in \\mathbb { R } ^ { n \\times d }$ . In practice, view augmentation approaches include node dropping, edge perturbation, subgraph sampling, etc. ", + "bbox": [ + 173, + 415, + 826, + 474 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(ii) View encoding. Graph-level or node-level latent representation is extracted from views $\\mathbf { v } _ { i } , \\mathbf { v } _ { j }$ by using the view encoder networks $f ( \\cdot )$ (a GNN backbone plus a projection MLP), i.e., $\\mathbf { z } _ { i } \\sim f _ { i } ( \\mathbf { v } _ { i } )$ and $\\mathbf { z } _ { j } \\sim f _ { j } ( \\mathbf { v } _ { j } )$ . The two encoders might or might not share parameters depending on whether they are from the same domain. ", + "bbox": [ + 173, + 479, + 825, + 536 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(iii) Representation contrasting. Given the latent representations, a contrastive loss is optimized to score the positive pairs $\\mathbf { z } _ { i } , \\mathbf { z } _ { j }$ higher compared to other negative pairs. Typically, the negative pairs are constructed from the augmented views of other graphs in the same minibatch. The InfoNCE loss [21] has been adopted as one of popular contrastive losses, which is defined as: ", + "bbox": [ + 173, + 541, + 825, + 598 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d05cd4c4ab1aa019c5cbd7a3e1158215a8e8279c058e4b2ea6f80dbb348a503c.jpg", + "text": "$$\n\\mathcal { L } _ { N C E } = - \\mathbb { E } \\left[ l o g \\frac { e x p ( h ( \\mathbf { z } _ { i , n } , \\mathbf { z } _ { j , n } ) ) } { \\sum _ { n ^ { \\prime } = 1 } ^ { N } e x p ( h ( \\mathbf { z } _ { i , n } , \\mathbf { z } _ { j , n ^ { \\prime } } ) ) } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 338, + 603, + 656, + 646 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $h ( \\cdot )$ is a contrasting operation to score the agreement between two representations. Theoretically, minimizing the InfoNCE loss equivalently maximizes a lower bound on the mutual information between the views of positive pairs. In other words, $I ( \\mathbf { z } _ { i } , \\mathbf { z } _ { j } ) \\geqslant l o g ( N ) - \\mathcal { L } _ { N C E }$ , where $I ( \\cdot )$ measures the mutual information. ", + "bbox": [ + 174, + 652, + 825, + 708 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Information-Aware Graph Contrastive Learning ", + "text_level": 1, + "bbox": [ + 173, + 727, + 612, + 744 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this paper, we study how to perform contrastive learning for specific graph tasks and datasets. In particular, we attempt to answer the following questions for graph contrastive learning: (i) What is the optimal augmented views? (ii) What is the optimal view encoder? (iii) What is the optimal contrastive mode? ", + "bbox": [ + 174, + 757, + 825, + 814 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 View Augmentation ", + "text_level": 1, + "bbox": [ + 174, + 830, + 349, + 845 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The goal of view augmentation is to create realistically rational data via the transformation approaches that do not affect the semantic label. Compared to the augmentation in other domains, graph view augmentation needs to consider the structural information of graph data, such as the node, the edge, and the subgraph. There are various graph view augmentation methods proposed recently. We follow a similar definition used in [42] to categorize four kinds of view augmentation approaches for graph data. Node dropping discards a certain part of nodes along with their edges in the input graph to create a new graph view. Edge perturbation perturbs the connectivity in the graph via adding or dropping partial edges. Attribute masking masks part of node attributes and assumes that the missing attributes can be well predicted by the remaining ones. Subgraph sampling samples a subgraph from the input graph. The rationale behind these approaches is that the semantic meaning of graph has certain robustness to graph perturbation. ", + "bbox": [ + 174, + 856, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 825, + 189 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The augmented views generated in the graph contrastive framework are typically used in a separate downstream task. To characterize what views are optimal for a downstream task, we define the optimality of views. The main motivation is: the optimal augmented views should contain the most task-relevant information, and the information shared between views should only be task-relevant. ", + "bbox": [ + 174, + 194, + 825, + 251 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Corollary 1. (Optimal Augmented Views) For a downstream task $T$ whose goal is to predict a semantic label $y$ , the optimal views, $\\mathbf { v } _ { i } ^ { * } , \\mathbf { v } _ { j } ^ { * } ,$ , generated from the input graph $\\mathcal { G }$ are the solutions to the following optimization problem : ", + "bbox": [ + 173, + 256, + 825, + 299 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/125154646e54d112b4e2f0b360da0088cfe4aa1ee53731aad8f5655118279355.jpg", + "text": "$$\n\\begin{array} { c } { { ( \\mathbf { v } _ { i } ^ { * } , \\mathbf { v } _ { j } ^ { * } ) = \\underset { \\mathbf { v } _ { i } , \\mathbf { v } _ { j } } { \\arg \\operatorname* { m i n } I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } ) } } } \\\\ { { s . t . ~ I ( \\mathbf { v } _ { i } ; y ) = I ( \\mathbf { v } _ { j } ; y ) } } \\\\ { { I ( \\mathbf { v } _ { i } ; y ) = I ( \\mathcal { G } ; y ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 400, + 303, + 598, + 369 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This says that for the optimal graph views, the amount of information shared between them is minimized (Eq. (3)), while the two views contain the same amount of information with respect to $y$ (Eq. (4)), which is also the amount of information that the input gprah contains about the task (Eq. (5)). The illustration of the optimal views is shown in Figure 2 and the proof is in the Appendix. ", + "bbox": [ + 173, + 375, + 825, + 431 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 View Encoding ", + "text_level": 1, + "bbox": [ + 174, + 446, + 318, + 462 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "View encoding aims to extract the latent representations of nodes or graphs via feeding the data of two views into view encoder networks such that the generated representations preserve both structure and attribute information in the views. The view encoders are quite flexible in graph contrastive learning and typically they are GCN [15], GAT [32], or GIN [38], etc. ", + "bbox": [ + 174, + 472, + 825, + 529 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The representations extracted via view encoding are further utilized to optimize the objective function of contrastive learning. After well trained, the view encoders are used to generate the graph/node representations for a downstream task. To characterize what encoders are optimal, we define the optimality of view encoders for graph contrastive learning. The main motivation is: the representation generated by the optimal encoder for a view should keep all the shared information by the two contrastive views, meanwhile the kept information is all task-relevant. ", + "bbox": [ + 174, + 534, + 825, + 617 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Corollary 2. (Optimal View Encoder) Given the optimal views, $\\mathbf { v } _ { i } ^ { * } , \\mathbf { v } _ { j } ^ { * }$ , for a downstream task $T$ whose goal is to predict a semantic label $y$ , the optimal view encoder for view $\\mathbf { v } _ { i } ^ { * }$ is the solution to the following optimization problem : ", + "bbox": [ + 174, + 623, + 825, + 665 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/5d79d772c3dbbfc6cadd034a3c8060d85b65cdbbf856c42f70ed5b69a5848e92.jpg", + "text": "$$\n\\boldsymbol f _ { i } ^ { * } = \\arg \\operatorname* { m i n } _ { \\boldsymbol f _ { i } } I ( f _ { i } ( \\mathbf { v } _ { i } ^ { * } ) ; \\mathbf { v } _ { i } ^ { * } )\n$$", + "text_format": "latex", + "bbox": [ + 406, + 671, + 598, + 698 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0d136a4cfea519c254df35fe0488265f4797a827eba594b8a1635cf351fa73ae.jpg", + "text": "$$\ns . t . \\ I ( f _ { i } ( \\mathbf { v } _ { i } ^ { * } ) ; \\mathbf { v } _ { j } ^ { * } ) = I ( \\mathbf { v } _ { i } ^ { * } ; \\mathbf { v } _ { j } ^ { * } )\n$$", + "text_format": "latex", + "bbox": [ + 393, + 700, + 602, + 719 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "It indicates that for the optimal view encoder, the amount of information shared between the optimal view and the extracted representation is minimized (Eq. (6)), while the information shared between the two optimal views is kept after the encoding process of one view (Eq. (7)). The illustration of the optimal encoder is shown in Figure 3 and the proof is illustrated in the Appendix. ", + "bbox": [ + 174, + 729, + 825, + 786 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.3 Representation Contrasting ", + "text_level": 1, + "bbox": [ + 176, + 803, + 406, + 818 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To allow flexible contrasting for graph data, we consider contrastive modes similar to [13]. A contrastive mode is denoted by $( c _ { i } ( \\cdot ) , c _ { j } ( \\cdot ) )$ , where $c _ { i } ( \\cdot )$ , $c _ { j } ( \\cdot )$ are the aggregation operations applied to the representations extracted by view encoders, The contrastive modes are unique to graph data because of the structural information inside a graph. Specifically, we consider five contrastive modes. In global-global mode, the graph representations from two views are contrasted. Thus, $c _ { i } ( \\cdot )$ , $c _ { j } ( \\cdot )$ are averaging aggregation operations in this mode. In local-global mode, we contrast the node representations from one view with the graph representations from the other view. Thus, $c _ { i } ( \\cdot )$ , $c _ { j } ( \\cdot )$ are the identical transformation and averaging aggregation operations, respectively. In local-local mode, the node representations from two views are contrasted. In multi-scale mode, we contrast graph representation of one view with the intermediate representation from the other. In hybrid mode, both global-global and local-global are applied. ", + "bbox": [ + 174, + 827, + 825, + 912 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/a7fd63d27d26cca071ead706f3fd728ce0211916cd89519025b1dee6ce661218.jpg", + "image_caption": [ + "Figure 3: Illustration of optimal view encoding. (Left) The relationships between two (optimal) views, $\\mathbf { v } _ { i } ^ { * }$ , $\\mathbf { v } _ { j } ^ { * }$ , task $y$ , and representation $\\mathbf { z } _ { i }$ in terms of information entropy. A, B, C are null because the two views are optimal here. $\\mathbf { z } _ { i }$ is contained by view $\\mathbf { v } _ { i } ^ { * }$ because representations are functions of views. (Middle) $\\mathbf { z } _ { i }$ covers $\\mathbf { D }$ when Eq. (7) holds, which indicates the shared information between views is kept after encoding. $( R i g h t )$ View $\\mathbf { z } _ { i }$ further exactly covers $\\mathbf { D }$ and the view encoding becomes optimal, i.e., $\\mathbf { z } _ { i } ^ { * }$ , which indicates all the information shared between view $\\mathbf { v } _ { i } ^ { * }$ and representation $\\mathbf { z } _ { i } ^ { * }$ is task-relevant. " + ], + "image_footnote": [], + "bbox": [ + 207, + 73, + 792, + 184 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 321, + 825, + 393 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To characterize which mode is optimal, we define the optimality of contrastive mode for graph contrastive learning. The main motivation is: the optimal contrastive mode keeps the most taskrelevant information after the representations are aggregated. The proof is included in the Appendix. ", + "bbox": [ + 173, + 398, + 825, + 441 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Corollary 3. (Optimal Contrastive Mode) Given the latent representations, $\\mathbf { z } _ { i } ^ { * }$ , $\\mathbf { z } _ { j } ^ { \\ast }$ , extracted by the optimal view encoders, i.e., $\\mathbf { z } _ { i } ^ { * } = f _ { i } ^ { * } ( \\mathbf { v } _ { i } ^ { * } )$ , $\\mathbf { z } _ { j _ { - } } ^ { * } = f _ { j _ { - } } ^ { * } ( \\mathbf { v } _ { j } ^ { * } )$ , and a downstream task $T$ with label $y _ { \\mathrm { { i } } }$ , the optimal contrastive mode is the solution to the following optimization problem, where $c _ { i }$ , $c _ { j }$ are the aggregation operations applied to the latent representations: ", + "bbox": [ + 173, + 446, + 825, + 505 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2c3af1818fa8af72de7349137d1173eb8200bc8d576eacace9f582cdf15289e2.jpg", + "text": "$$\n\\displaystyle ( c _ { i } ^ { * } , c _ { j } ^ { * } ) = \\arg \\operatorname* { m i n } _ { ( c _ { i } , c _ { j } ) } - I ( c _ { i } ( \\mathbf { z } _ { i } ^ { * } ) ; c _ { j } ( \\mathbf { z } _ { j } ^ { * } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 510, + 629, + 537 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.4 InfoGCL Principle ", + "text_level": 1, + "bbox": [ + 174, + 553, + 344, + 568 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "According to our proposed corollaries, we can theoretically design the optimal contrastive learning approach for our specific graph data and task. However, in real-world scenarios, the conditions to meet the exact optimality of contrastive learning is hard or even not practically possible to reach because of data noise and limited model capability. Therefore, we propose to achieve the optimal for each stage independently and practically, which is an approximation to achieve the original optimality. Specifically, we make the following propositions to address the questions of the optimal views, optimal view encoder, and optimal contrastive mode. ", + "bbox": [ + 173, + 577, + 825, + 675 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 1. For a task $T$ with label $y _ { ; }$ , given a bunch of graph view augmentation methods, $\\{ q _ { 1 } ( \\cdot ) _ { : }$ , $q _ { 2 } ( \\cdot ) , \\cdot \\cdot \\cdot \\}$ , that create two views $\\mathbf { v } _ { i }$ , $\\mathbf { v } _ { j }$ , the recommended augmentation methods are the ones, $q _ { i } ( \\cdot )$ , $q _ { j } ( \\cdot )$ , that maximize $I ( \\mathbf { v } _ { i } ; y ) + I ( \\mathbf { v } _ { j } ; y ) - I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } )$ , i.e., the area of $A { + } B { + } D$ in Figure 2. ", + "bbox": [ + 174, + 681, + 826, + 724 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 2. Given a task $T$ with label $y$ and a set of view encoders, $\\{ f _ { i } ^ { 1 } ( \\cdot ) , f _ { i } ^ { 2 } ( \\cdot ) , \\cdot \\cdot \\cdot \\ \\} ,$ , that generate representation $\\mathbf { z } _ { i }$ via taking view $\\mathbf { v } _ { i }$ as input, the recommended view encoder is the one that maximizes the mutual information between $\\mathbf { v } _ { i }$ , $\\mathbf { z } _ { i }$ and $y$ . Symmetrically the same for view $\\mathbf { v } _ { j }$ . ", + "bbox": [ + 174, + 729, + 826, + 773 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 3. Given a task $T$ with label $y$ , the extracted representations, $z _ { i } , \\ z _ { j }$ , and a set of aggregation operations, $\\{ c _ { 1 } ( \\cdot ) , c _ { 2 } ( \\cdot ) , \\cdot \\cdot \\cdot \\ \\} ,$ , the recommended contrastive mode is the one, $( c _ { i } , c _ { j } )$ , that has the largest amount of mutual information between $c _ { i } ( \\mathbf { z } _ { i } )$ , $c _ { j } ( \\mathbf { z } _ { j } )$ and $y$ . ", + "bbox": [ + 173, + 779, + 825, + 821 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The qualitative and quantitative evaluation of these propositions are shown in Section 5.3. ", + "bbox": [ + 173, + 827, + 759, + 842 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.5 Role of Negative Samples ", + "text_level": 1, + "bbox": [ + 174, + 857, + 388, + 872 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Current graph contrastive learning approaches heavily depend on negative samples. However, recent progresses of contrastive learning in vision domain indicate that negative samples are not necessarily required [8, 4], of which the main benefit is to avoid careful treatment to retrieve the negative pairs. To study the influence of negative samples on graph contrastive learning, we follow the framework of SimSiam [4] to revise the loss function as Eq.(9). A very recent work [29] also studies graph contrastive learning without negative samples. Different from it, we focus on both node and graph classification tasks. ", + "bbox": [ + 174, + 882, + 823, + 911 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/d27d31742747e95258dc95f8006e5345c23f4824d93edd3c724ac3922de15242.jpg", + "table_caption": [ + "Table 1: Dataset statistics. " + ], + "table_footnote": [], + "table_body": "
Graph Task DatasetsNode Task Datasets
MUTAGPTC-MRIMDB-BIMDB-MNCI1COLLABCoraCiteseerPubmed
#Graphs1883441000150041105000111
#Nodes17.914.319.813.029.974.53327270819717
#Edges19.814.7193.165.91.133.04732542944338
#Classes222223673
", + "bbox": [ + 179, + 88, + 818, + 174 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 217, + 825, + 286 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/43d11dbf627226ec2239c08ec4341ee94a4da08f41197bcd575b09df6f98e01b.jpg", + "text": "$$\n\\mathcal { L } = - \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\frac { \\mathbf { z } _ { i , n } } { \\| \\mathbf { z } _ { i , n } \\| } \\cdot \\frac { \\mathbf { z } _ { j , n } } { \\| \\mathbf { z } _ { j , n } \\| } ,\n$$", + "text_format": "latex", + "bbox": [ + 395, + 290, + 601, + 333 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 Experiments ", + "text_level": 1, + "bbox": [ + 173, + 347, + 312, + 364 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we evaluate our InfoGCL with a number of experiments. We first describe datasets, evaluation protocol, and experimental setup. Then, we present the experimental results on both node and graph classification. Last, we analyze our proposed principles via ablation study. ", + "bbox": [ + 176, + 378, + 825, + 421 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 Setup", + "text_level": 1, + "bbox": [ + 174, + 436, + 253, + 452 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use both graph classification and node classification benchmark datasets that are widely used in the existing graph contrastive learning approaches. The graph classification datasets include MUTAG [17], PTC-MR [17], IMDB-B [40], IMDB-M [40], NCI1 [34], and COLLAB [40]. MUTAG is a collection of nitroaromatic compounds represented as graphs, where vertices stand for atoms and edges represent bonds between atoms. PTC-MR is a collection of 344 chemical compounds which report the carcinogenicity for rats. IMDB-B and IMDB-M are two movie collaboration datasets, where nodes represent actors/actress and there is an edge between them if they appear in the same movie. In NCI1, graphs are the representation of chemical compounds, where vertices stand for atoms and edges represent bonds between atoms. COLLAB is a collaboration dataset, where researchers are nodes and an edge indicates collaboration between two researchers. The node classification datasets include Citeseer, Cora, and Pubmed [23]. All of them are citation networks, where nodes are documents and edges are citation links. These datasets are summarized in Table 1. ", + "bbox": [ + 174, + 462, + 825, + 627 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We closely follow the evaluation protocol of previous state-of-the-art graph contrastive learning approaches. For graph classification, we report the mean 10-fold cross validation accuracy after 5 runs followed by a linear SVM. The linear SVM is trained by applying cross validation on training data folds and the best mean accuracy is reported. For node classification, we report the mean accuracy on test set after 50 runs of training followed by a linear neural network model. To make comparison fair, we adopt the basic setting of InfoGraph for graph classification. We conduct experiment with the values of the number of GNN layers, the number of epochs, batch size, the parameter C of SVM in the sets {2, 4, 8, 12}, {10, 20, 40, 100}, {32, 64, 128, 256} and $\\{ 1 0 ^ { - 3 } , \\ : \\mathrm { { \\dot { 1 } 0 ^ { - 2 } } } , . . . , \\ : 1 0 ^ { 2 } , 1 0 ^ { 3 } \\ : \\ : \\}$ , respectively. We adopt the basic setting of DGI for node classification. Specifically, we set the number of GNN layers to 1 and experiment with the batch size in the set {2, 4, 8}. The hidden dimension of representations is set to 512. We also apply the early stopping strategy. ", + "bbox": [ + 173, + 633, + 825, + 787 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 Experimental Results ", + "text_level": 1, + "bbox": [ + 174, + 803, + 362, + 818 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To evaluate our method InfoGCL on graph classification, we use thhree categories of baselines. The kernel approaches include shortest path kernel (SP) [2], Graphlet kernel (GK) [25], WeisfeilerLehman sub-tree kernel (WL) [24], deep graph kernels (DGK) [39], and multi-scale Laplacian kernel (MLG) [16]. The supervised baselines include GraphSAGE [12], GCN [15], GIN [38], GAT [32]. We also compare with the unsupervised approaches, including RandomWalk [7], node2vec [9], sub2vec [14], graph2vec [20], InfoGraph [28], GraphCL [42], and mvgrl [13]. Table 2 shows the graph classification results. We observe that our approach achieves the best results compared to other unsupervised approaches. Our approach also outperforms or matches the best kernel approaches across the datasets. Even compared with the supervised ones, our approach achieves the best in 2 out of 6 datasets and the results of our approach on other 4 dataset are among the top. ", + "bbox": [ + 174, + 827, + 825, + 912 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/100341ee754bc7ccbfbd5552183bc1ec5e1b1be1a782962292628b2341f3673b.jpg", + "table_caption": [ + "Table 2: Graph classification results $( \\% )$ " + ], + "table_footnote": [], + "table_body": "
MethodMUTAGPTC-MRIMDB-BIMDB-MNCI1COLLAB
Kernel Approaches
SP85.2 ± 2.458.2 ± 2.455.6± 0.238.0±0.373.5 ± 0.1
GK81.7 ± 2.157.3 ± 1.465.9 ± 1.043.9 ± 0.466.0 ± 0.172.8± 0.3
WL80.7 ± 3.058.0 ± 0.572.3 ± 3.447.0 ± 0.580.0± 0.578.9 ± 1.9
DGK87.4 ± 2.760.1 ± 2.667.0 ± 0.644.6 ± 0.580.3 ± 0.573.1 ± 0.3
MLG87.9 ± 1.663.3 ± 1.566.6 ± 0.341.2 ± 0.080.8 ± 1.3
Supervised Approaches
GraphSAGE85.1 ± 7.6 63.9 ± 7.772.3 ± 5.350.9 ± 2.277.7 ± 1.568.3 ± 4.2
GCN85.6 ± 5.864.2 ± 4.374.0± 3.451.9 ± 3.880.2 ± 2.079.0 ± 1.8
GIN-089.4 ± 5.664.6 ± 7.075.1 ± 5.152.3 ± 2.882.7 ± 1.780.2 ± 1.9
GIN-e89.0 ± 6.063.7 ±8.274.3 ± 5.152.1 ± 3.682.7 ± 1.680.1 ± 1.9
GAT89.4 ± 6.166.7 ± 5.170.5 ± 2.347.8 ± 3.166.6 ± 2.267.4 ± 2.9
Unsupervised Approaches
RandomWalk83.7 ± 1.557.9 ± 1.350.7±0.334.7 ± 0.264.3 ± 0.3
node2vec72.6 ±10.258.6±8.050.2 ± 0.936.0 ± 0.754.9 ± 1.656.1 ± 0.2
sub2vec61.1 ± 15.860.0 ± 6.455.3 ± 1.536.7 ± 0.852.8 ±1.5-
graph2vec83.2 ±9.660.2 ± 6.971.1 ± 0.550.4 ± 0.975.4 ±1.2
InfoGraph89.0 ± 1.161.7 ± 1.473.0± 0.949.7 ± 0.576.2 ±1.470.7 ± 1.1
GraphCL86.8 ± 1.361.3 ± 2.171.1 ± 0.449.2 ± 0.677.9 ± 0.471.4 ± 1.2
mvgrl89.7 ± 1.162.5 ± 1.774.2 ± 0.751.2 ± 0.577.0 ± 0.876.0 ± 1.2
51.4 ± 0.880.2 ± 0.6
InfoGCL91.2 ± 1.363.5 ± 1.575.1 ± 0.980.0 ± 1.3
", + "bbox": [ + 183, + 87, + 812, + 367 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/bb3604b9cde8633c8adb22a2b93c182327925e26e7eabc91176a409025cf1d38.jpg", + "table_caption": [ + "Table 3: Node classification results $( \\% )$ . " + ], + "table_footnote": [], + "table_body": "
MethodCoraCiteseerPubmed
Supervised Approaches
MLP55.146.571.4
ICA75.169.173.9
LP68.045.363.0
ManiReg59.560.170.7
SemiEmb59.059.671.7
Planetoid75.764.777.2
Chebyshev81.269.874.4
GCN81.570.379.0
JKNet82.7 ± 0.473.0 ± 0.577.9 ± 0.4
GAT83.0 ± 0.772.5± 0.779.0 ± 0.3
Unsupervised Approaches
Linear47.9 ± 0.449.3 ± 0.269.1 ± 0.3
DeepWalk70.7 ± 0.651.4 ± 0.574.3 ± 0.9
GAE71.5 ± 0.465.8 ±0.472.1 ± 0.5
VERSE72.5 ± 0.355.5± 0.4
DGI83.8 ± 0.572.0 ± 0.677.9 ± 0.3
GraphCL82.5 ± 0.173.1 ± 0.2
mvgrl86.8 ± 0.573.3 ± 0.580.1 ± 0.7
InfoGCL83.5 ± 0.373.5 ± 0.479.1 ± 0.2
", + "bbox": [ + 297, + 422, + 699, + 684 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 752, + 825, + 808 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For node classification tasks, we compare InfoGCL with some supervised approaches and unsupervised approaches. The supervised baselines include a simple MLP model, iterative classification algorithm (ICA) [19], manifold regularization (ManiReg) [1], semi-supervised embedding (SemiEmb) [35], Planetoid [41], Chebyshev [5], GCN, JKNet [36], GAT. Table 3 shows the node classification results. It is observed that our approach achieves the state-of-the-art results and competes the best one with respect to the existing unsupervised approaches. Compared to supervised baselines, our approach outperforms all the baselines. ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/f0477851d52bef85626a9194a114b45dcbb4525569d2ce3f3528abe6efd26a28.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MethodMUTAGIMDB-BCOLLABCoraCiteseerPubmed
InfoGCL (w/o neg)91.0 ± 1.475.1± 0.580.2 ±1.078.6 ± 0.470.4 ± 0.677.4±0.7
InfoGCL (w/ neg)91.2 ± 1.375.1 ± 0.980.0± 1.383.5±0.373.5 ± 0.479.1±0.2
", + "bbox": [ + 184, + 74, + 813, + 125 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 4: Comparison between InfoGCL with negative samples and without negative samples. ", + "bbox": [ + 183, + 132, + 800, + 146 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.3 Evaluation of InfoGCL Principle ", + "text_level": 1, + "bbox": [ + 174, + 164, + 442, + 179 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We can unify the existing graph contrastive learning methods through the perspective of InfoGCL principle: all recent graph contrastive learning methods can be decoupled into three stages that implicitly follow the InfoGCL principle, though being different in model architecture design and optimization strategies. Below, we analyze some observations from several recent work. Because of the limited space, please refer to the Appendix for more results of the quantitative analysis. ", + "bbox": [ + 173, + 190, + 825, + 260 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Obs. i. Composing a graph and its augmentation benefits downstream performance [42]. Compared to composing a graph and the graph itself, augmentation leads to smaller $I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } )$ (Proposition 1). ", + "bbox": [ + 173, + 266, + 821, + 295 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Obs. ii. Composing different augmentations benefits more [42]. Compared to composing a graph and its augmentations, two augmentations further decrease $I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } )$ (Proposition 1). ", + "bbox": [ + 173, + 299, + 821, + 329 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Obs. iii. Node dropping and subgraph sampling are generally beneficial across datasets [42]. When compared to attribute masking and edge perturbation, they change the semantic meaning of the graph relatively slightly, which leads to higher $I ( \\mathbf { v } _ { i } ; y )$ , $I ( \\mathbf { v } _ { j } ; \\bar { y } )$ (Proposition 1). ", + "bbox": [ + 174, + 334, + 825, + 377 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Obs. iv. Edge perturbation benefits social networks but hurts some biochemical molecules [42]. The semantic meaning of social networks are robust to edge perturbation. However, the semantic meaning of some biochemical molecules are determined by local connection pattern, where edge perturbation decreases $I ( \\mathbf { v } _ { i } ; y )$ (Proposition 1). ", + "bbox": [ + 174, + 382, + 825, + 439 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Obs. v. Contrasting node and graph representations consistently performs better than other contrastive modes across benchmarks [13]. Compared to other contrastive modes, node-graph (i.e., local-global) mode generally extracts more graph structure information, which benefits predicting task label $y$ (Proposition 3). ", + "bbox": [ + 174, + 444, + 825, + 501 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.4 Effect of Negative Samples ", + "text_level": 1, + "bbox": [ + 174, + 517, + 398, + 531 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To study the effect of negative samples on graph contrastive learning, we follow SimSiam [4] and design the objective as Eq. (9). We conduct experiments on three graph task and three node task datasets. The results are reported in Table 4. It is observed that the negative samples show little influence on the three graph task datasets, while performance drops on the three node task datasets, especially the Cora dataset. ", + "bbox": [ + 174, + 542, + 825, + 611 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "According to the dataset statistics summarized in Table 1, we see the networks of Cora, Citeseer, Pubmed are much sparser (in terms of network topology). Furthermore, we know the node features of these three datasets are also much sparser (one-hot encoding with high dimensionality). We speculate this because the contrastive learning models tends to collapse easier if negative samples are not used, especially when data is too sparse. Therefore,we make the hypothesis: negative samples benefit graph modeling, especially when i) network topology, and ii) node features are extremely sparse. ", + "bbox": [ + 174, + 618, + 825, + 702 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Conclusion and Limitations ", + "text_level": 1, + "bbox": [ + 176, + 719, + 437, + 738 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We propose InfoGCL, an information-aware graph contrastive learning framework for graph contrastive learning. Existing graph contrastive learning approaches are usually carefully designed. We aim to answer how to perform contrastive learning for your learning tasks on specific graph data. Our method decouples the typical contrastive learning approaches into three sequential modules and provides the theoretical analysis for reaching the optimality. To address the questions of optimality in a practical way, we propose the InfoGCL principle, which is implicitly followed by all recent graph contrastive learning approaches. In addition, we explore the role of negative samples in graph contrastive learning and find negative samples are not necessarily required. Experiments on both node and graph benchmark datasets demonstrate the effectiveness of our method. Note that our method is not without limitations. We can further improve our method by designing better practical approximations to the theoretical optimality of graph contrastive learning. ", + "bbox": [ + 174, + 751, + 825, + 904 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Funding Transparency Statement ", + "text_level": 1, + "bbox": [ + 174, + 89, + 455, + 107 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This project was partially supported by NSF projects IIS-1707548 and CBET-1638320. 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However, answering this question is challenging. First,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "contrastive learning consists of multiple components, such as view augmentation and information", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "encoding. For each of them, there are various choices. Numerous variations make it difficult to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "design models that are both robust and efficient. Existing graph contrastive learning approaches are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "carefully designed for different learning tasks on different datasets, however, none of them studies", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "the guiding principles for choosing the best components. Second, graph data has unique properties", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "that distinguish it from other types of data, such as rich structural information and highly diverse", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "distribution [33, 28, 11, 10]. Thus, it is desirable to design the contrastive learning model that fits", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 410, + 412, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 412, + 424 + ], + "score": 1.0, + "content": "your graph data properties, even without any domain knowledge of the data.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "We propose to address these challenges via Information Bottleneck (IB) [31], which provides a crucial", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "principle for representation learning. Specifically, IB encourages the representations to be maximally", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 450, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 506, + 461 + ], + "score": 1.0, + "content": "informative about the target in the downstream task, which helps keep task-relevant information.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 461, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 471 + ], + "score": 1.0, + "content": "Concurrently, IB discourages the representation learning from acquiring the task-irrelevant informa-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "tion from the input data, which is related to the idea of minimal sufficient statistics [27]. However,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "different from the typical representation learning, there are two information flows involved in the two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 504 + ], + "score": 1.0, + "content": "augmented views in contrastive learning. Therefore, we extend the previous IB work [37, 43] and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 458, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 458, + 516 + ], + "score": 1.0, + "content": "propose InfoGCL, an information-aware contrastive learning framework for graph data.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "To study how information is transformed and transferred, we decouple a typical graph contrastive", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "learning model into three sequential modules (as shown in Figure 1): view augmentation, view", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "encoding, and representation contrasting. 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) , c _ { j } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 261, + 507, + 274 + ], + "score": 1.0, + "content": "are aggregation operations applied to representations.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 504, + 308 + ], + "lines": [], + "index": 8.5, + "bbox_fs": [ + 105, + 284, + 506, + 310 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 312, + 505, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "The main question this paper attempts to answer is: how to perform contrastive learning for your", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "learning tasks on specific graph datasets? However, answering this question is challenging. First,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "contrastive learning consists of multiple components, such as view augmentation and information", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "encoding. For each of them, there are various choices. Numerous variations make it difficult to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "design models that are both robust and efficient. Existing graph contrastive learning approaches are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "carefully designed for different learning tasks on different datasets, however, none of them studies", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "the guiding principles for choosing the best components. Second, graph data has unique properties", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "that distinguish it from other types of data, such as rich structural information and highly diverse", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "distribution [33, 28, 11, 10]. Thus, it is desirable to design the contrastive learning model that fits", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 410, + 412, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 412, + 424 + ], + "score": 1.0, + "content": "your graph data properties, even without any domain knowledge of the data.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 312, + 506, + 424 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "We propose to address these challenges via Information Bottleneck (IB) [31], which provides a crucial", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "principle for representation learning. Specifically, IB encourages the representations to be maximally", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 450, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 506, + 461 + ], + "score": 1.0, + "content": "informative about the target in the downstream task, which helps keep task-relevant information.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 461, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 471 + ], + "score": 1.0, + "content": "Concurrently, IB discourages the representation learning from acquiring the task-irrelevant informa-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "tion from the input data, which is related to the idea of minimal sufficient statistics [27]. However,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "different from the typical representation learning, there are two information flows involved in the two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 504 + ], + "score": 1.0, + "content": "augmented views in contrastive learning. Therefore, we extend the previous IB work [37, 43] and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 458, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 458, + 516 + ], + "score": 1.0, + "content": "propose InfoGCL, an information-aware contrastive learning framework for graph data.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 427, + 506, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "To study how information is transformed and transferred, we decouple a typical graph contrastive", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "learning model into three sequential modules (as shown in Figure 1): view augmentation, view", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "encoding, and representation contrasting. We further formalize how to find the optimal of the three", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "modules into three optimization problems. To build the optimal graph contrastive learning model for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "the particular dataset and task, we argue that it is necessary and sufficient to minimize the mutual", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "information between contrastive representations while maximizing task-relevant information at the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 586, + 504, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 504, + 597 + ], + "score": 1.0, + "content": "levels of both individual module and entire framework. Our work is also motivated by the InfoMin", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "theory [30], which suggests that a good set of views for contrastive learning in the vision domain", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "should share the minimal information necessary to perform well at the downstream task. Beyond", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "view selection, our work extends InfoMin to suggest principles of selecting view encodings and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 629, + 448, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 448, + 642 + ], + "score": 1.0, + "content": "contrastive modes for graph learning considering the unique properties of graph data.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 519, + 506, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 659 + ], + "score": 1.0, + "content": "We suggest practically feasible principles to find the optimal modules in graph contrastive learning", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 656, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 669 + ], + "score": 1.0, + "content": "and show that all recent graph contrastive learning methods can be unified by these principles: i) the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 667, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 679 + ], + "score": 1.0, + "content": "augmented views should contain as much task-relevant information as possible, while they should", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "share as little information as possible; ii) the view encoder should be task-relevant and simple as", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 689, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 700 + ], + "score": 1.0, + "content": "much as possible; iii) the contrastive mode should keep task-relevant information as much as possible", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "after contrasting. Besides, we also investigate the role of negative samples in graph contrastive", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "learning and argue that negative samples are not necessarily required, especially when graph data", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "is not extremely sparse. Our proposed method, InfoGCL, is validated on a rich set of benchmark", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "datasets for both node-level and graph-level tasks, where we analyze its ability to capture the unique", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "score": 1.0, + "content": "structural properties of the graph data. The results demonstrate that our algorithm achieves highly", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 252, + 118 + ], + "score": 1.0, + "content": "competitive performance with up to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 252, + 106, + 275, + 116 + ], + "score": 0.87, + "content": "5 . 2 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 275, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "relative improvement in accuracy on graph classification", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "task and competitive results on node classification task over the state-of-the-art unsupervised methods.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 644, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "is not extremely sparse. Our proposed method, InfoGCL, is validated on a rich set of benchmark", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "datasets for both node-level and graph-level tasks, where we analyze its ability to capture the unique", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "score": 1.0, + "content": "structural properties of the graph data. The results demonstrate that our algorithm achieves highly", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 252, + 118 + ], + "score": 1.0, + "content": "competitive performance with up to", + "type": "text" + }, + { + "bbox": [ + 252, + 106, + 275, + 116 + ], + "score": 0.87, + "content": "5 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "relative improvement in accuracy on graph classification", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "task and competitive results on node classification task over the state-of-the-art unsupervised methods.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 143, + 197, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 198, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 198, + 158 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "Graph Contrastive Learning. Some recent research efforts in graph domain have been attracted", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 180, + 504, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 504, + 191 + ], + "score": 1.0, + "content": "by the success of contrastive learning in vision and language domains [3, 8, 4]. A number of graph", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 202 + ], + "score": 1.0, + "content": "contrastive learning approaches have been proposed [28, 22, 42, 13]. Despite all of them creating two", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 202, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 505, + 213 + ], + "score": 1.0, + "content": "views and targeting at maximizing the feature disagreement between the two views, these methods", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "are carefully designed and differ in various aspects. Deep graph Infomax (DGI) [33] applied the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "InfoMax principle [18] to graph data by contrasting the representations of node-level and graph-level", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 245 + ], + "score": 1.0, + "content": "for node classification tasks. Different from DGI, InfoGraph [28] aims at node classification tasks", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "score": 1.0, + "content": "and it contrasts the representations of graph-level and substructure-level of different granularity. In", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "addition, DGI and InfoGraph use different graph encoders to extract latent representations. mvgrl [13]", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "studies both node and graph classification tasks. It transforms the adjacency matrix to a diffusion", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 507, + 291 + ], + "score": 1.0, + "content": "matrix and treat the two matrices as two congruent views. However, in GCC [22] and GRACE [45],", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "score": 1.0, + "content": "subgraph sampling and graph perturbation are used to create the augmented views. GraphCL [42]", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "explores the view augmentations approaches for graph contrastive learning. Specifically, it studies the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 310, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 323 + ], + "score": 1.0, + "content": "approaches of node dropping, edge perturbation, attribute masking, and subgraph sampling. A recent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "work [44] also studies graph data augmentations. However, it focuses on graph neural networks for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "node classification and does not study the contrastive learning framework. Our method differs from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "them. We aim to answer the question how to perform contrastive learning for your graph data and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 368 + ], + "score": 1.0, + "content": "tasks. Instead of carefully designing the architectures, we decouple typical graph contrastive learning", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "into three stages and provide our InfoGCL principles to analyze the optimality theoretically and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 376, + 154, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 154, + 388 + ], + "score": 1.0, + "content": "practically.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 507, + 405 + ], + "score": 1.0, + "content": "Information Bottleneck. Our method is related to the Information Bottleneck (IB) theory [31],", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "score": 1.0, + "content": "which aims to find the best trade-off between accuracy and complexity when summarizing a random", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "variable. IB has been recently used to study the deep learning approaches [26, 37, 43]. Specifically,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 424, + 479, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 380, + 438 + ], + "score": 1.0, + "content": "IB expresses the trade-off between the mutual information measures", + "type": "text" + }, + { + "bbox": [ + 381, + 425, + 416, + 437 + ], + "score": 0.93, + "content": "I ( \\mathbf { D } , \\mathbf { Z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 424, + 434, + 438 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 434, + 425, + 466, + 437 + ], + "score": 0.93, + "content": "I ( \\mathbf { Z } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 424, + 479, + 438 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 442, + 379, + 456 + ], + "lines": [ + { + "bbox": [ + 230, + 442, + 379, + 456 + ], + "spans": [ + { + "bbox": [ + 230, + 442, + 379, + 456 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } \\mathbf { I } \\mathbf { B } _ { \\beta } = - I ( \\mathbf { D } ; \\mathbf { Z } ) + \\beta I ( \\mathbf { Z } ; \\mathbf { y } ) ,", + "type": "interline_equation", + "image_path": "7f84f97e8d7cee3894f71d06c6fa2677f0ffb3b47533ae9a9b4fe1e1c4da0508.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 230, + 442, + 379, + 456 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 478, + 474 + ], + "score": 1.0, + "content": "where D, Z, y are the input, the latent representation and the task label, respectively.", + "type": "text" + }, + { + "bbox": [ + 478, + 462, + 485, + 471 + ], + "score": 0.75, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 375, + 484 + ], + "score": 1.0, + "content": "hyperparameter. In other words, IB aims to learn representation", + "type": "text" + }, + { + "bbox": [ + 375, + 472, + 383, + 482 + ], + "score": 0.53, + "content": "\\mathbf { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "that is maximally expressive", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 303, + 497 + ], + "score": 1.0, + "content": "about y, while being minimally expressive about", + "type": "text" + }, + { + "bbox": [ + 304, + 483, + 314, + 493 + ], + "score": 0.3, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 482, + 506, + 497 + ], + "score": 1.0, + "content": ". More recently, there are some efforts applying", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "the IB theory to graph representation learning. [37] aims to generate both expressive and robust", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "graph representations, while [43] studies the subgraph recognition problem. In contrast, our work", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "focuses on graph contrastive learning, where the two augmented views make the optimization", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "objective (information trade-off) different. Our work is also related to the idea of minimal sufficient", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "statistics [27], which has been recently studied in the vision domain [30], claiming that a good set of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "image views should share the minimal information necessary to perform well at the downstream task.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "Different from [30], we focus on the graph domain and propose three stages considering the unique", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 571, + 206, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 206, + 582 + ], + "score": 1.0, + "content": "properties of graph data.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 597, + 269, + 610 + ], + "lines": [ + { + "bbox": [ + 104, + 596, + 270, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 270, + 612 + ], + "score": 1.0, + "content": "3 Preliminaries and Notations", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 339, + 636 + ], + "score": 1.0, + "content": "Graph Representation Learning. A graph is denoted by", + "type": "text" + }, + { + "bbox": [ + 340, + 622, + 388, + 634 + ], + "score": 0.87, + "content": "\\mathcal { G } = \\left( \\mathbf { A } , \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 620, + 393, + 636 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 393, + 622, + 438, + 633 + ], + "score": 0.86, + "content": "\\mathbf { A } \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 620, + 506, + 636 + ], + "score": 1.0, + "content": "is the adjacency", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 632, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 138, + 648 + ], + "score": 1.0, + "content": "matrix.", + "type": "text" + }, + { + "bbox": [ + 138, + 633, + 183, + 645 + ], + "score": 0.92, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 632, + 323, + 648 + ], + "score": 1.0, + "content": "is the node attribute matrix, where", + "type": "text" + }, + { + "bbox": [ + 324, + 635, + 330, + 644 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 632, + 506, + 648 + ], + "score": 1.0, + "content": "is the attribute dimension. In this work, we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 644, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 422, + 658 + ], + "score": 1.0, + "content": "focus on both node-level and graph-level tasks. For node-level task, given graph", + "type": "text" + }, + { + "bbox": [ + 422, + 646, + 430, + 656 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 644, + 505, + 658 + ], + "score": 1.0, + "content": "and the labels of a", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 218, + 669 + ], + "score": 1.0, + "content": "subset of nodes, denoted by", + "type": "text" + }, + { + "bbox": [ + 218, + 657, + 232, + 667 + ], + "score": 0.89, + "content": "{ \\bf Y } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 655, + 408, + 669 + ], + "score": 1.0, + "content": ", the goal is to learn the latent representation", + "type": "text" + }, + { + "bbox": [ + 408, + 658, + 419, + 667 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 655, + 477, + 669 + ], + "score": 1.0, + "content": "for each node", + "type": "text" + }, + { + "bbox": [ + 477, + 658, + 483, + 666 + ], + "score": 0.71, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 124, + 679 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 668, + 135, + 678 + ], + "score": 0.87, + "content": "\\mathbf { z } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "preserves both network structures and node attributes, which can be further used to predict", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 120, + 689 + ], + "score": 0.88, + "content": "{ \\bf Y } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 676, + 299, + 691 + ], + "score": 1.0, + "content": ". For graph-level task, given a set of graphs", + "type": "text" + }, + { + "bbox": [ + 300, + 677, + 378, + 690 + ], + "score": 0.92, + "content": "\\mathbb { G } = \\{ \\mathcal { G } ^ { 1 } , \\mathcal { G } ^ { 2 } , \\cdot \\cdot \\cdot \\mathrm ~ \\} \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "and the labels of some graphs,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 154, + 702 + ], + "score": 1.0, + "content": "denoted by", + "type": "text" + }, + { + "bbox": [ + 154, + 689, + 168, + 701 + ], + "score": 0.89, + "content": "\\mathbf { Y } _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 688, + 351, + 702 + ], + "score": 1.0, + "content": ", the goal is to learn the latent representation", + "type": "text" + }, + { + "bbox": [ + 351, + 691, + 362, + 702 + ], + "score": 0.85, + "content": "\\mathbf { z } _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 688, + 463, + 702 + ], + "score": 1.0, + "content": "for each graph such that", + "type": "text" + }, + { + "bbox": [ + 464, + 690, + 475, + 702 + ], + "score": 0.87, + "content": "\\mathbf { z } _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 168, + 714 + ], + "score": 1.0, + "content": "used to predict", + "type": "text" + }, + { + "bbox": [ + 168, + 700, + 182, + 712 + ], + "score": 0.89, + "content": "\\mathbf { Y } _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 699, + 506, + 714 + ], + "score": 1.0, + "content": ". Typically, the graph data is fed into graph neural networks (GNNs) to generate", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 284, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 218, + 725 + ], + "score": 1.0, + "content": "the representations, such as", + "type": "text" + }, + { + "bbox": [ + 218, + 711, + 279, + 723 + ], + "score": 0.92, + "content": "\\mathbf { z } _ { g } = \\mathbf { G } \\mathbf { N } \\mathbf { N } \\mathbf { s } ( \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 709, + 284, + 725 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 127 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 73, + 506, + 129 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 143, + 197, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 198, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 198, + 158 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "Graph Contrastive Learning. Some recent research efforts in graph domain have been attracted", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 180, + 504, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 504, + 191 + ], + "score": 1.0, + "content": "by the success of contrastive learning in vision and language domains [3, 8, 4]. A number of graph", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 202 + ], + "score": 1.0, + "content": "contrastive learning approaches have been proposed [28, 22, 42, 13]. Despite all of them creating two", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 202, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 505, + 213 + ], + "score": 1.0, + "content": "views and targeting at maximizing the feature disagreement between the two views, these methods", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "are carefully designed and differ in various aspects. Deep graph Infomax (DGI) [33] applied the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "InfoMax principle [18] to graph data by contrasting the representations of node-level and graph-level", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 245 + ], + "score": 1.0, + "content": "for node classification tasks. Different from DGI, InfoGraph [28] aims at node classification tasks", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "score": 1.0, + "content": "and it contrasts the representations of graph-level and substructure-level of different granularity. In", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "addition, DGI and InfoGraph use different graph encoders to extract latent representations. mvgrl [13]", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "studies both node and graph classification tasks. It transforms the adjacency matrix to a diffusion", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 507, + 291 + ], + "score": 1.0, + "content": "matrix and treat the two matrices as two congruent views. However, in GCC [22] and GRACE [45],", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "score": 1.0, + "content": "subgraph sampling and graph perturbation are used to create the augmented views. GraphCL [42]", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "explores the view augmentations approaches for graph contrastive learning. Specifically, it studies the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 310, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 323 + ], + "score": 1.0, + "content": "approaches of node dropping, edge perturbation, attribute masking, and subgraph sampling. A recent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "work [44] also studies graph data augmentations. However, it focuses on graph neural networks for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "node classification and does not study the contrastive learning framework. Our method differs from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "them. We aim to answer the question how to perform contrastive learning for your graph data and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 368 + ], + "score": 1.0, + "content": "tasks. Instead of carefully designing the architectures, we decouple typical graph contrastive learning", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "into three stages and provide our InfoGCL principles to analyze the optimality theoretically and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 376, + 154, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 154, + 388 + ], + "score": 1.0, + "content": "practically.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 168, + 507, + 388 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 507, + 405 + ], + "score": 1.0, + "content": "Information Bottleneck. Our method is related to the Information Bottleneck (IB) theory [31],", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "score": 1.0, + "content": "which aims to find the best trade-off between accuracy and complexity when summarizing a random", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "variable. IB has been recently used to study the deep learning approaches [26, 37, 43]. Specifically,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 424, + 479, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 380, + 438 + ], + "score": 1.0, + "content": "IB expresses the trade-off between the mutual information measures", + "type": "text" + }, + { + "bbox": [ + 381, + 425, + 416, + 437 + ], + "score": 0.93, + "content": "I ( \\mathbf { D } , \\mathbf { Z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 424, + 434, + 438 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 434, + 425, + 466, + 437 + ], + "score": 0.93, + "content": "I ( \\mathbf { Z } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 424, + 479, + 438 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 390, + 507, + 438 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 442, + 379, + 456 + ], + "lines": [ + { + "bbox": [ + 230, + 442, + 379, + 456 + ], + "spans": [ + { + "bbox": [ + 230, + 442, + 379, + 456 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } \\mathbf { I } \\mathbf { B } _ { \\beta } = - I ( \\mathbf { D } ; \\mathbf { Z } ) + \\beta I ( \\mathbf { Z } ; \\mathbf { y } ) ,", + "type": "interline_equation", + "image_path": "7f84f97e8d7cee3894f71d06c6fa2677f0ffb3b47533ae9a9b4fe1e1c4da0508.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 230, + 442, + 379, + 456 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 478, + 474 + ], + "score": 1.0, + "content": "where D, Z, y are the input, the latent representation and the task label, respectively.", + "type": "text" + }, + { + "bbox": [ + 478, + 462, + 485, + 471 + ], + "score": 0.75, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 375, + 484 + ], + "score": 1.0, + "content": "hyperparameter. 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A graph is denoted by", + "type": "text" + }, + { + "bbox": [ + 340, + 622, + 388, + 634 + ], + "score": 0.87, + "content": "\\mathcal { G } = \\left( \\mathbf { A } , \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 620, + 393, + 636 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 393, + 622, + 438, + 633 + ], + "score": 0.86, + "content": "\\mathbf { A } \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 620, + 506, + 636 + ], + "score": 1.0, + "content": "is the adjacency", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 632, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 138, + 648 + ], + "score": 1.0, + "content": "matrix.", + "type": "text" + }, + { + "bbox": [ + 138, + 633, + 183, + 645 + ], + "score": 0.92, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 632, + 323, + 648 + ], + "score": 1.0, + "content": "is the node attribute matrix, where", + "type": "text" + }, + { + "bbox": [ + 324, + 635, + 330, + 644 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 632, + 506, + 648 + ], + "score": 1.0, + "content": "is the attribute dimension. 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(Left) The relationships between graph", + "type": "text" + }, + { + "bbox": [ + 433, + 181, + 441, + 191 + ], + "score": 0.78, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 178, + 491, + 194 + ], + "score": 1.0, + "content": ", two views,", + "type": "text" + }, + { + "bbox": [ + 492, + 182, + 502, + 191 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 178, + 507, + 194 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 190, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 117, + 203 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 190, + 159, + 204 + ], + "score": 1.0, + "content": ", and task", + "type": "text" + }, + { + "bbox": [ + 159, + 193, + 166, + 203 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 190, + 506, + 204 + ], + "score": 1.0, + "content": "in terms of information entropy. A, B, C, D, E are overlapping areas. Two views", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "are contained by graph because views are functions of graph. (Middle) A, B, E become null when", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "Eqs. (4)-(5) hold, which indicates the views and graph share the same amount of task-relevant", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "information. (Right) C further becomes null when Eq. (3) holds, which indicates all the shared", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 234, + 389, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 389, + 246 + ], + "score": 1.0, + "content": "information between views is task-relevant, i.e., the views are optimal.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "Graph Contrastive Learning. 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We decouple a typical graph contrastive learning model into three sequential", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 313, + 145, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 145, + 326 + ], + "score": 1.0, + "content": "modules.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 506, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 233, + 343 + ], + "score": 1.0, + "content": "(i) View augmentation. 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A view is represented as graph data, such as", + "type": "text" + }, + { + "bbox": [ + 406, + 341, + 473, + 353 + ], + "score": 0.91, + "content": "\\mathbf { v } _ { i } = ( \\mathbf { A } _ { v _ { i } } , \\mathbf { X } _ { v _ { i } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 340, + 506, + 354 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 350, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 159, + 365 + ], + "score": 0.9, + "content": "\\mathbf { A } _ { v _ { i } } \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 350, + 178, + 367 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 353, + 230, + 365 + ], + "score": 0.91, + "content": "\\mathbf { X } _ { v _ { i } } \\in \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 350, + 507, + 367 + ], + "score": 1.0, + "content": ". In practice, view augmentation approaches include node dropping,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 364, + 279, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 279, + 378 + ], + "score": 1.0, + "content": "edge perturbation, subgraph sampling, etc.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 465, + 393 + ], + "score": 1.0, + "content": "(ii) View encoding. 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The two encoders might or might not share parameters depending on whether they", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 413, + 215, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 215, + 425 + ], + "score": 1.0, + "content": "are from the same domain.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "(iii) Representation contrasting. 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In", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "score": 1.0, + "content": "particular, we attempt to answer the following questions for graph contrastive learning: (i) What", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "is the optimal augmented views? (ii) What is the optimal view encoder? 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(Left) The relationships between graph", + "type": "text" + }, + { + "bbox": [ + 433, + 181, + 441, + 191 + ], + "score": 0.78, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 178, + 491, + 194 + ], + "score": 1.0, + "content": ", two views,", + "type": "text" + }, + { + "bbox": [ + 492, + 182, + 502, + 191 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 178, + 507, + 194 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 190, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 117, + 203 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 190, + 159, + 204 + ], + "score": 1.0, + "content": ", and task", + "type": "text" + }, + { + "bbox": [ + 159, + 193, + 166, + 203 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 190, + 506, + 204 + ], + "score": 1.0, + "content": "in terms of information entropy. 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(3) holds, which indicates all the shared", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 234, + 389, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 389, + 246 + ], + "score": 1.0, + "content": "information between views is task-relevant, i.e., the views are optimal.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "Graph Contrastive Learning. 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Graph", + "type": "text" + }, + { + "bbox": [ + 233, + 331, + 241, + 341 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 330, + 361, + 343 + ], + "score": 1.0, + "content": "undergoes data augmentation", + "type": "text" + }, + { + "bbox": [ + 361, + 330, + 378, + 342 + ], + "score": 0.93, + "content": "q ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 330, + 459, + 343 + ], + "score": 1.0, + "content": "to obtain two views", + "type": "text" + }, + { + "bbox": [ + 460, + 333, + 470, + 341 + ], + "score": 0.67, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 330, + 473, + 343 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 474, + 332, + 485, + 342 + ], + "score": 0.72, + "content": "\\mathbf { v } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 330, + 506, + 343 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 152, + 353 + ], + "score": 0.91, + "content": "\\mathbf { v } _ { i } \\sim q _ { i } ( \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 340, + 171, + 354 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 171, + 341, + 218, + 353 + ], + "score": 0.91, + "content": "\\mathbf { v } _ { j } \\sim q _ { j } ( { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 340, + 406, + 354 + ], + "score": 1.0, + "content": ". A view is represented as graph data, such as", + "type": "text" + }, + { + "bbox": [ + 406, + 341, + 473, + 353 + ], + "score": 0.91, + "content": "\\mathbf { v } _ { i } = ( \\mathbf { A } _ { v _ { i } } , \\mathbf { X } _ { v _ { i } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 340, + 506, + 354 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 350, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 159, + 365 + ], + "score": 0.9, + "content": "\\mathbf { A } _ { v _ { i } } \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 350, + 178, + 367 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 353, + 230, + 365 + ], + "score": 0.91, + "content": "\\mathbf { X } _ { v _ { i } } \\in \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 350, + 507, + 367 + ], + "score": 1.0, + "content": ". In practice, view augmentation approaches include node dropping,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 364, + 279, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 279, + 378 + ], + "score": 1.0, + "content": "edge perturbation, subgraph sampling, etc.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 330, + 507, + 378 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 465, + 393 + ], + "score": 1.0, + "content": "(ii) View encoding. 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The two encoders might or might not share parameters depending on whether they", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 413, + 215, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 215, + 425 + ], + "score": 1.0, + "content": "are from the same domain.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 380, + 506, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "(iii) Representation contrasting. Given the latent representations, a contrastive loss is optimized", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 213, + 453 + ], + "score": 1.0, + "content": "to score the positive pairs", + "type": "text" + }, + { + "bbox": [ + 214, + 442, + 239, + 453 + ], + "score": 0.3, + "content": "\\mathbf { z } _ { i } , \\mathbf { z } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "higher compared to other negative pairs. Typically, the negative", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "pairs are constructed from the augmented views of other graphs in the same minibatch. The InfoNCE", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 463, + 442, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 442, + 474 + ], + "score": 1.0, + "content": "loss [21] has been adopted as one of popular contrastive losses, which is defined as:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 430, + 505, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 478, + 402, + 512 + ], + "lines": [ + { + "bbox": [ + 207, + 478, + 402, + 512 + ], + "spans": [ + { + "bbox": [ + 207, + 478, + 402, + 512 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { N C E } = - \\mathbb { E } \\left[ l o g \\frac { e x p ( h ( \\mathbf { z } _ { i , n } , \\mathbf { z } _ { j , n } ) ) } { \\sum _ { n ^ { \\prime } = 1 } ^ { N } e x p ( h ( \\mathbf { z } _ { i , n } , \\mathbf { z } _ { j , n ^ { \\prime } } ) ) } \\right] ,", + "type": "interline_equation", + "image_path": "d05cd4c4ab1aa019c5cbd7a3e1158215a8e8279c058e4b2ea6f80dbb348a503c.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 478, + 402, + 495.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 207, + 495.0, + 402, + 512.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 133, + 530 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 517, + 150, + 529 + ], + "score": 0.9, + "content": "h ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "is a contrasting operation to score the agreement between two representations. Theoreti-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "cally, minimizing the InfoNCE loss equivalently maximizes a lower bound on the mutual information", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 307, + 552 + ], + "score": 1.0, + "content": "between the views of positive pairs. In other words,", + "type": "text" + }, + { + "bbox": [ + 307, + 538, + 419, + 551 + ], + "score": 0.92, + "content": "I ( \\mathbf { z } _ { i } , \\mathbf { z } _ { j } ) \\geqslant l o g ( N ) - \\mathcal { L } _ { N C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 539, + 448, + 552 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 449, + 539, + 465, + 551 + ], + "score": 0.9, + "content": "I ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "measures", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 203, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 203, + 561 + ], + "score": 1.0, + "content": "the mutual information.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 517, + 506, + 561 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 576, + 375, + 590 + ], + "lines": [ + { + "bbox": [ + 104, + 573, + 376, + 593 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 376, + 593 + ], + "score": 1.0, + "content": "4 Information-Aware Graph Contrastive Learning", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "In this paper, we study how to perform contrastive learning for specific graph tasks and datasets. In", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "score": 1.0, + "content": "particular, we attempt to answer the following questions for graph contrastive learning: (i) What", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "is the optimal augmented views? (ii) What is the optimal view encoder? (iii) What is the optimal", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 635, + 181, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 181, + 644 + ], + "score": 1.0, + "content": "contrastive mode?", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 601, + 506, + 644 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 658, + 214, + 670 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 216, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 216, + 671 + ], + "score": 1.0, + "content": "4.1 View Augmentation", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "The goal of view augmentation is to create realistically rational data via the transformation approaches", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "that do not affect the semantic label. Compared to the augmentation in other domains, graph view", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "augmentation needs to consider the structural information of graph data, such as the node, the edge,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 712, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 505, + 723 + ], + "score": 1.0, + "content": "and the subgraph. There are various graph view augmentation methods proposed recently. We follow", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "a similar definition used in [42] to categorize four kinds of view augmentation approaches for graph", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "data. Node dropping discards a certain part of nodes along with their edges in the input graph", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "to create a new graph view. Edge perturbation perturbs the connectivity in the graph via adding", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "or dropping partial edges. Attribute masking masks part of node attributes and assumes that the", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "missing attributes can be well predicted by the remaining ones. Subgraph sampling samples a", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "score": 1.0, + "content": "subgraph from the input graph. The rationale behind these approaches is that the semantic meaning", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 321, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 321, + 151 + ], + "score": 1.0, + "content": "of graph has certain robustness to graph perturbation.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 678, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "a similar definition used in [42] to categorize four kinds of view augmentation approaches for graph", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "data. Node dropping discards a certain part of nodes along with their edges in the input graph", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "to create a new graph view. Edge perturbation perturbs the connectivity in the graph via adding", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "or dropping partial edges. Attribute masking masks part of node attributes and assumes that the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "missing attributes can be well predicted by the remaining ones. 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The rationale behind these approaches is that the semantic meaning", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 321, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 321, + 151 + ], + "score": 1.0, + "content": "of graph has certain robustness to graph perturbation.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "The augmented views generated in the graph contrastive framework are typically used in a separate", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "score": 1.0, + "content": "downstream task. To characterize what views are optimal for a downstream task, we define the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "optimality of views. The main motivation is: the optimal augmented views should contain the most", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 500, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 500, + 200 + ], + "score": 1.0, + "content": "task-relevant information, and the information shared between views should only be task-relevant.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 383, + 216 + ], + "score": 1.0, + "content": "Corollary 1. 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The illustration of the optimal views is shown in Figure 2 and the proof is in the Appendix.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 354, + 195, + 366 + ], + "lines": [ + { + "bbox": [ + 104, + 351, + 197, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 197, + 369 + ], + "score": 1.0, + "content": "4.2 View Encoding", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "View encoding aims to extract the latent representations of nodes or graphs via feeding the data of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "two views into view encoder networks such that the generated representations preserve both structure", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "and attribute information in the views. The view encoders are quite flexible in graph contrastive", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 388, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 388, + 420 + ], + "score": 1.0, + "content": "learning and typically they are GCN [15], GAT [32], or GIN [38], etc.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "The representations extracted via view encoding are further utilized to optimize the objective function", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "score": 1.0, + "content": "of contrastive learning. After well trained, the view encoders are used to generate the graph/node", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "representations for a downstream task. To characterize what encoders are optimal, we define the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "optimality of view encoders for graph contrastive learning. The main motivation is: the representation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "generated by the optimal encoder for a view should keep all the shared information by the two", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 388, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 388, + 489 + ], + "score": 1.0, + "content": "contrastive views, meanwhile the kept information is all task-relevant.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 504, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 368, + 507 + ], + "score": 1.0, + "content": "Corollary 2. (Optimal View Encoder) Given the optimal views,", + "type": "text" + }, + { + "bbox": [ + 368, + 495, + 397, + 509 + ], + "score": 0.25, + "content": "\\mathbf { v } _ { i } ^ { * } , \\mathbf { v } _ { j } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 493, + 495, + 507 + ], + "score": 1.0, + "content": ", for a downstream task", + "type": "text" + }, + { + "bbox": [ + 495, + 495, + 504, + 504 + ], + "score": 0.75, + "content": "T", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 273, + 517 + ], + "score": 1.0, + "content": "whose goal is to predict a semantic label", + "type": "text" + }, + { + "bbox": [ + 273, + 507, + 280, + 517 + ], + "score": 0.52, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 505, + 422, + 517 + ], + "score": 1.0, + "content": ", the optimal view encoder for view", + "type": "text" + }, + { + "bbox": [ + 423, + 506, + 434, + 517 + ], + "score": 0.88, + "content": "\\mathbf { v } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "is the solution to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 516, + 255, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 255, + 528 + ], + "score": 1.0, + "content": "the following optimization problem :", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 532, + 366, + 553 + ], + "lines": [ + { + "bbox": [ + 249, + 532, + 366, + 553 + ], + "spans": [ + { + "bbox": [ + 249, + 532, + 366, + 553 + ], + "score": 0.57, + "content": "\\boldsymbol f _ { i } ^ { * } = \\arg \\operatorname* { m i n } _ { \\boldsymbol f _ { i } } I ( f _ { i } ( \\mathbf { v } _ { i } ^ { * } ) ; \\mathbf { v } _ { i } ^ { * } )", + "type": "interline_equation", + "image_path": "5d79d772c3dbbfc6cadd034a3c8060d85b65cdbbf856c42f70ed5b69a5848e92.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 249, + 532, + 366, + 553 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 555, + 369, + 570 + ], + "lines": [ + { + "bbox": [ + 241, + 555, + 369, + 570 + ], + "spans": [ + { + "bbox": [ + 241, + 555, + 369, + 570 + ], + "score": 0.47, + "content": "s . t . \\ I ( f _ { i } ( \\mathbf { v } _ { i } ^ { * } ) ; \\mathbf { v } _ { j } ^ { * } ) = I ( \\mathbf { v } _ { i } ^ { * } ; \\mathbf { v } _ { j } ^ { * } )", + "type": "interline_equation", + "image_path": "0d136a4cfea519c254df35fe0488265f4797a827eba594b8a1635cf351fa73ae.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 241, + 555, + 369, + 570 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "It indicates that for the optimal view encoder, the amount of information shared between the optimal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "view and the extracted representation is minimized (Eq. (6)), while the information shared between", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "the two optimal views is kept after the encoding process of one view (Eq. (7)). The illustration of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 612, + 434, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 434, + 624 + ], + "score": 1.0, + "content": "optimal encoder is shown in Figure 3 and the proof is illustrated in the Appendix.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 108, + 636, + 249, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 250, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 250, + 651 + ], + "score": 1.0, + "content": "4.3 Representation Contrasting", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "To allow flexible contrasting for graph data, we consider contrastive modes similar to [13]. A", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 230, + 680 + ], + "score": 1.0, + "content": "contrastive mode is denoted by", + "type": "text" + }, + { + "bbox": [ + 230, + 667, + 280, + 680 + ], + "score": 0.92, + "content": "( c _ { i } ( \\cdot ) , c _ { j } ( \\cdot ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 666, + 309, + 680 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 310, + 667, + 329, + 679 + ], + "score": 0.64, + "content": "c _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 666, + 332, + 680 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 332, + 667, + 353, + 680 + ], + "score": 0.58, + "content": "c _ { j } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 666, + 505, + 680 + ], + "score": 1.0, + "content": "are the aggregation operations applied", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "to the representations extracted by view encoders, The contrastive modes are unique to graph data", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "because of the structural information inside a graph. Specifically, we consider five contrastive modes.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 461, + 713 + ], + "score": 1.0, + "content": "In global-global mode, the graph representations from two views are contrasted. Thus,", + "type": "text" + }, + { + "bbox": [ + 461, + 700, + 480, + 712 + ], + "score": 0.55, + "content": "c _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 700, + 484, + 713 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 484, + 699, + 505, + 712 + ], + "score": 0.59, + "content": "c _ { j } ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "are averaging aggregation operations in this mode. 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To characterize what views are optimal for a downstream task, we define the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "optimality of views. The main motivation is: the optimal augmented views should contain the most", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 500, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 500, + 200 + ], + "score": 1.0, + "content": "task-relevant information, and the information shared between views should only be task-relevant.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 154, + 506, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 383, + 216 + ], + "score": 1.0, + "content": "Corollary 1. (Optimal Augmented Views) For a downstream task", + "type": "text" + }, + { + "bbox": [ + 383, + 204, + 392, + 213 + ], + "score": 0.75, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "whose goal is to predict a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 213, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 104, + 213, + 165, + 228 + ], + "score": 1.0, + "content": "semantic label", + "type": "text" + }, + { + "bbox": [ + 165, + 216, + 172, + 226 + ], + "score": 0.55, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 213, + 248, + 228 + ], + "score": 1.0, + "content": ", the optimal views,", + "type": "text" + }, + { + "bbox": [ + 248, + 215, + 277, + 228 + ], + "score": 0.25, + "content": "\\mathbf { v } _ { i } ^ { * } , \\mathbf { v } _ { j } ^ { * } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 213, + 404, + 228 + ], + "score": 1.0, + "content": ", generated from the input graph", + "type": "text" + }, + { + "bbox": [ + 405, + 215, + 412, + 225 + ], + "score": 0.8, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 213, + 505, + 228 + ], + "score": 1.0, + "content": "are the solutions to the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 241, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 241, + 239 + ], + "score": 1.0, + "content": "following optimization problem :", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 203, + 505, + 239 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 240, + 366, + 293 + ], + "lines": [ + { + "bbox": [ + 245, + 240, + 364, + 292 + ], + "spans": [ + { + "bbox": [ + 245, + 240, + 364, + 292 + ], + "score": 0.44, + "content": "\\begin{array} { c } { { ( \\mathbf { v } _ { i } ^ { * } , \\mathbf { v } _ { j } ^ { * } ) = \\underset { \\mathbf { v } _ { i } , \\mathbf { v } _ { j } } { \\arg \\operatorname* { m i n } I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } ) } } } \\\\ { { s . t . ~ I ( \\mathbf { v } _ { i } ; y ) = I ( \\mathbf { v } _ { j } ; y ) } } \\\\ { { I ( \\mathbf { v } _ { i } ; y ) = I ( \\mathcal { G } ; y ) } } \\end{array}", + "type": "interline_equation", + "image_path": "125154646e54d112b4e2f0b360da0088cfe4aa1ee53731aad8f5655118279355.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 240, + 366, + 266.5 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 245, + 266.5, + 366, + 293.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "This says that for the optimal graph views, the amount of information shared between them is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 308, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 506, + 320 + ], + "score": 1.0, + "content": "minimized (Eq. 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The illustration of the optimal views is shown in Figure 2 and the proof is in the Appendix.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 296, + 507, + 343 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 354, + 195, + 366 + ], + "lines": [ + { + "bbox": [ + 104, + 351, + 197, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 197, + 369 + ], + "score": 1.0, + "content": "4.2 View Encoding", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "View encoding aims to extract the latent representations of nodes or graphs via feeding the data of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "two views into view encoder networks such that the generated representations preserve both structure", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "and attribute information in the views. The view encoders are quite flexible in graph contrastive", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 388, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 388, + 420 + ], + "score": 1.0, + "content": "learning and typically they are GCN [15], GAT [32], or GIN [38], etc.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 374, + 506, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "The representations extracted via view encoding are further utilized to optimize the objective function", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "score": 1.0, + "content": "of contrastive learning. After well trained, the view encoders are used to generate the graph/node", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "representations for a downstream task. To characterize what encoders are optimal, we define the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "optimality of view encoders for graph contrastive learning. The main motivation is: the representation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "generated by the optimal encoder for a view should keep all the shared information by the two", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 388, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 388, + 489 + ], + "score": 1.0, + "content": "contrastive views, meanwhile the kept information is all task-relevant.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 423, + 506, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 504, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 368, + 507 + ], + "score": 1.0, + "content": "Corollary 2. (Optimal View Encoder) Given the optimal views,", + "type": "text" + }, + { + "bbox": [ + 368, + 495, + 397, + 509 + ], + "score": 0.25, + "content": "\\mathbf { v } _ { i } ^ { * } , \\mathbf { v } _ { j } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 493, + 495, + 507 + ], + "score": 1.0, + "content": ", for a downstream task", + "type": "text" + }, + { + "bbox": [ + 495, + 495, + 504, + 504 + ], + "score": 0.75, + "content": "T", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 273, + 517 + ], + "score": 1.0, + "content": "whose goal is to predict a semantic label", + "type": "text" + }, + { + "bbox": [ + 273, + 507, + 280, + 517 + ], + "score": 0.52, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 505, + 422, + 517 + ], + "score": 1.0, + "content": ", the optimal view encoder for view", + "type": "text" + }, + { + "bbox": [ + 423, + 506, + 434, + 517 + ], + "score": 0.88, + "content": "\\mathbf { v } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "is the solution to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 516, + 255, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 255, + 528 + ], + "score": 1.0, + "content": "the following optimization problem :", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 493, + 505, + 528 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 532, + 366, + 553 + ], + "lines": [ + { + "bbox": [ + 249, + 532, + 366, + 553 + ], + "spans": [ + { + "bbox": [ + 249, + 532, + 366, + 553 + ], + "score": 0.57, + "content": "\\boldsymbol f _ { i } ^ { * } = \\arg \\operatorname* { m i n } _ { \\boldsymbol f _ { i } } I ( f _ { i } ( \\mathbf { v } _ { i } ^ { * } ) ; \\mathbf { v } _ { i } ^ { * } )", + "type": "interline_equation", + "image_path": "5d79d772c3dbbfc6cadd034a3c8060d85b65cdbbf856c42f70ed5b69a5848e92.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 249, + 532, + 366, + 553 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 555, + 369, + 570 + ], + "lines": [ + { + "bbox": [ + 241, + 555, + 369, + 570 + ], + "spans": [ + { + "bbox": [ + 241, + 555, + 369, + 570 + ], + "score": 0.47, + "content": "s . t . \\ I ( f _ { i } ( \\mathbf { v } _ { i } ^ { * } ) ; \\mathbf { v } _ { j } ^ { * } ) = I ( \\mathbf { v } _ { i } ^ { * } ; \\mathbf { v } _ { j } ^ { * } )", + "type": "interline_equation", + "image_path": "0d136a4cfea519c254df35fe0488265f4797a827eba594b8a1635cf351fa73ae.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 241, + 555, + 369, + 570 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "It indicates that for the optimal view encoder, the amount of information shared between the optimal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "view and the extracted representation is minimized (Eq. (6)), while the information shared between", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "the two optimal views is kept after the encoding process of one view (Eq. (7)). The illustration of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 612, + 434, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 434, + 624 + ], + "score": 1.0, + "content": "optimal encoder is shown in Figure 3 and the proof is illustrated in the Appendix.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 579, + 505, + 624 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 636, + 249, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 250, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 250, + 651 + ], + "score": 1.0, + "content": "4.3 Representation Contrasting", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "To allow flexible contrasting for graph data, we consider contrastive modes similar to [13]. A", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 230, + 680 + ], + "score": 1.0, + "content": "contrastive mode is denoted by", + "type": "text" + }, + { + "bbox": [ + 230, + 667, + 280, + 680 + ], + "score": 0.92, + "content": "( c _ { i } ( \\cdot ) , c _ { j } ( \\cdot ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 666, + 309, + 680 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 310, + 667, + 329, + 679 + ], + "score": 0.64, + "content": "c _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 666, + 332, + 680 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 332, + 667, + 353, + 680 + ], + "score": 0.58, + "content": "c _ { j } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 666, + 505, + 680 + ], + "score": 1.0, + "content": "are the aggregation operations applied", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "to the representations extracted by view encoders, The contrastive modes are unique to graph data", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "because of the structural information inside a graph. Specifically, we consider five contrastive modes.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 461, + 713 + ], + "score": 1.0, + "content": "In global-global mode, the graph representations from two views are contrasted. Thus,", + "type": "text" + }, + { + "bbox": [ + 461, + 700, + 480, + 712 + ], + "score": 0.55, + "content": "c _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 700, + 484, + 713 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 484, + 699, + 505, + 712 + ], + "score": 0.59, + "content": "c _ { j } ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "are averaging aggregation operations in this mode. In local-global mode, we contrast the node", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 461, + 270 + ], + "score": 1.0, + "content": "representations from one view with the graph representations from the other view. Thus,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 461, + 256, + 481, + 268 + ], + "score": 0.5, + "content": "c _ { i } ( \\cdot )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 481, + 256, + 484, + 270 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 484, + 256, + 505, + 269 + ], + "score": 0.57, + "content": "c _ { j } ( \\cdot )", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "are the identical transformation and averaging aggregation operations, respectively. 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(Left) The relationships between two (optimal) views,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 167, + 507, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 118, + 180 + ], + "score": 0.77, + "content": "\\mathbf { v } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 167, + 121, + 181 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 122, + 168, + 133, + 181 + ], + "score": 0.82, + "content": "\\mathbf { v } _ { j } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 167, + 155, + 181 + ], + "score": 1.0, + "content": ", task", + "type": "text" + }, + { + "bbox": [ + 155, + 169, + 162, + 179 + ], + "score": 0.78, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 167, + 239, + 181 + ], + "score": 1.0, + "content": ", and representation", + "type": "text" + }, + { + "bbox": [ + 239, + 169, + 249, + 178 + ], + "score": 0.85, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 167, + 507, + 181 + ], + "score": 1.0, + "content": "in terms of information entropy. A, B, C are null because the two", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 179, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 203, + 190 + ], + "score": 1.0, + "content": "views are optimal here.", + "type": "text" + }, + { + "bbox": [ + 203, + 180, + 213, + 189 + ], + "score": 0.85, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 179, + 300, + 190 + ], + "score": 1.0, + "content": "is contained by view", + "type": "text" + }, + { + "bbox": [ + 300, + 179, + 312, + 190 + ], + "score": 0.89, + "content": "\\mathbf { v } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 179, + 506, + 190 + ], + "score": 1.0, + "content": "because representations are functions of views.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 144, + 201 + ], + "score": 1.0, + "content": "(Middle)", + "type": "text" + }, + { + "bbox": [ + 144, + 190, + 154, + 200 + ], + "score": 0.85, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 189, + 185, + 201 + ], + "score": 1.0, + "content": "covers", + "type": "text" + }, + { + "bbox": [ + 185, + 190, + 194, + 199 + ], + "score": 0.38, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 189, + 506, + 201 + ], + "score": 1.0, + "content": "when Eq. 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Thus,", + "type": "text" + }, + { + "bbox": [ + 461, + 256, + 481, + 268 + ], + "score": 0.5, + "content": "c _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 256, + 484, + 270 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 484, + 256, + 505, + 269 + ], + "score": 0.57, + "content": "c _ { j } ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "are the identical transformation and averaging aggregation operations, respectively. In local-local", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "mode, the node representations from two views are contrasted. In multi-scale mode, we contrast", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "score": 1.0, + "content": "graph representation of one view with the intermediate representation from the other. In hybrid", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 299, + 325, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 325, + 312 + ], + "score": 1.0, + "content": "mode, both global-global and local-global are applied.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "To characterize which mode is optimal, we define the optimality of contrastive mode for graph", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 327, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 339 + ], + "score": 1.0, + "content": "contrastive learning. The main motivation is: the optimal contrastive mode keeps the most task-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "relevant information after the representations are aggregated. The proof is included in the Appendix.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 409, + 368 + ], + "score": 1.0, + "content": "Corollary 3. (Optimal Contrastive Mode) Given the latent representations,", + "type": "text" + }, + { + "bbox": [ + 410, + 356, + 420, + 367 + ], + "score": 0.72, + "content": "\\mathbf { z } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 354, + 424, + 368 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 424, + 355, + 435, + 368 + ], + "score": 0.78, + "content": "\\mathbf { z } _ { j } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 354, + 505, + 368 + ], + "score": 1.0, + "content": ", extracted by the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 366, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 218, + 381 + ], + "score": 1.0, + "content": "optimal view encoders, i.e.,", + "type": "text" + }, + { + "bbox": [ + 219, + 367, + 271, + 379 + ], + "score": 0.81, + "content": "\\mathbf { z } _ { i } ^ { * } = f _ { i } ^ { * } ( \\mathbf { v } _ { i } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 366, + 274, + 381 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 275, + 367, + 326, + 380 + ], + "score": 0.76, + "content": "\\mathbf { z } _ { j _ { - } } ^ { * } = f _ { j _ { - } } ^ { * } ( \\mathbf { v } _ { j } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 366, + 428, + 381 + ], + "score": 1.0, + "content": ", and a downstream task", + "type": "text" + }, + { + "bbox": [ + 428, + 368, + 437, + 377 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 366, + 480, + 381 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 480, + 369, + 487, + 379 + ], + "score": 0.49, + "content": "y _ { \\mathrm { { i } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 366, + 506, + 381 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 451, + 391 + ], + "score": 1.0, + "content": "optimal contrastive mode is the solution to the following optimization problem, where", + "type": "text" + }, + { + "bbox": [ + 451, + 380, + 460, + 389 + ], + "score": 0.68, + "content": "c _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 379, + 464, + 391 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 464, + 380, + 473, + 390 + ], + "score": 0.73, + "content": "c _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "are the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 389, + 351, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 351, + 402 + ], + "score": 1.0, + "content": "aggregation operations applied to the latent representations:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 404, + 385, + 426 + ], + "lines": [ + { + "bbox": [ + 226, + 404, + 385, + 426 + ], + "spans": [ + { + "bbox": [ + 226, + 404, + 385, + 426 + ], + "score": 0.95, + "content": "\\displaystyle ( c _ { i } ^ { * } , c _ { j } ^ { * } ) = \\arg \\operatorname* { m i n } _ { ( c _ { i } , c _ { j } ) } - I ( c _ { i } ( \\mathbf { z } _ { i } ^ { * } ) ; c _ { j } ( \\mathbf { z } _ { j } ^ { * } ) ) .", + "type": "interline_equation", + "image_path": "2c3af1818fa8af72de7349137d1173eb8200bc8d576eacace9f582cdf15289e2.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 226, + 404, + 385, + 426 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 438, + 211, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 212, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 212, + 452 + ], + "score": 1.0, + "content": "4.4 InfoGCL Principle", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 471 + ], + "score": 1.0, + "content": "According to our proposed corollaries, we can theoretically design the optimal contrastive learning", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "approach for our specific graph data and task. However, in real-world scenarios, the conditions to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "meet the exact optimality of contrastive learning is hard or even not practically possible to reach", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "because of data noise and limited model capability. Therefore, we propose to achieve the optimal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "for each stage independently and practically, which is an approximation to achieve the original", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "optimality. Specifically, we make the following propositions to address the questions of the optimal", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 524, + 347, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 347, + 536 + ], + "score": 1.0, + "content": "views, optimal view encoder, and optimal contrastive mode.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 506, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 210, + 553 + ], + "score": 1.0, + "content": "Proposition 1. For a task", + "type": "text" + }, + { + "bbox": [ + 210, + 541, + 219, + 550 + ], + "score": 0.72, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 538, + 261, + 553 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 261, + 542, + 268, + 552 + ], + "score": 0.59, + "content": "y _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 538, + 477, + 553 + ], + "score": 1.0, + "content": ", given a bunch of graph view augmentation methods,", + "type": "text" + }, + { + "bbox": [ + 478, + 540, + 503, + 552 + ], + "score": 0.89, + "content": "\\{ q _ { 1 } ( \\cdot ) _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 538, + 506, + 553 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 550, + 508, + 565 + ], + "spans": [ + { + "bbox": [ + 107, + 551, + 152, + 563 + ], + "score": 0.88, + "content": "q _ { 2 } ( \\cdot ) , \\cdot \\cdot \\cdot \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 550, + 240, + 565 + ], + "score": 1.0, + "content": ", that create two views", + "type": "text" + }, + { + "bbox": [ + 240, + 553, + 250, + 561 + ], + "score": 0.75, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 550, + 254, + 565 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 254, + 553, + 266, + 562 + ], + "score": 0.78, + "content": "\\mathbf { v } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 550, + 483, + 565 + ], + "score": 1.0, + "content": ", the recommended augmentation methods are the ones,", + "type": "text" + }, + { + "bbox": [ + 484, + 552, + 503, + 563 + ], + "score": 0.89, + "content": "q _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 550, + 508, + 565 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 561, + 468, + 575 + ], + "spans": [ + { + "bbox": [ + 107, + 563, + 127, + 574 + ], + "score": 0.89, + "content": "q _ { j } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 561, + 189, + 575 + ], + "score": 1.0, + "content": ", that maximize", + "type": "text" + }, + { + "bbox": [ + 189, + 562, + 312, + 574 + ], + "score": 0.9, + "content": "I ( \\mathbf { v } _ { i } ; y ) + I ( \\mathbf { v } _ { j } ; y ) - I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 561, + 378, + 575 + ], + "score": 1.0, + "content": ", i.e., the area of", + "type": "text" + }, + { + "bbox": [ + 378, + 562, + 416, + 572 + ], + "score": 0.89, + "content": "A { + } B { + } D", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 561, + 468, + 575 + ], + "score": 1.0, + "content": "in Figure 2.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 506, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 225, + 593 + ], + "score": 1.0, + "content": "Proposition 2. Given a task", + "type": "text" + }, + { + "bbox": [ + 225, + 580, + 234, + 589 + ], + "score": 0.74, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 578, + 279, + 593 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 279, + 581, + 286, + 591 + ], + "score": 0.33, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 578, + 402, + 593 + ], + "score": 1.0, + "content": "and a set of view encoders,", + "type": "text" + }, + { + "bbox": [ + 402, + 578, + 483, + 591 + ], + "score": 0.91, + "content": "\\{ f _ { i } ^ { 1 } ( \\cdot ) , f _ { i } ^ { 2 } ( \\cdot ) , \\cdot \\cdot \\cdot \\ \\} ,", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 578, + 506, + 593 + ], + "score": 1.0, + "content": ", that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 591, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 201, + 602 + ], + "score": 1.0, + "content": "generate representation", + "type": "text" + }, + { + "bbox": [ + 201, + 592, + 211, + 601 + ], + "score": 0.84, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 591, + 273, + 602 + ], + "score": 1.0, + "content": "via taking view", + "type": "text" + }, + { + "bbox": [ + 274, + 592, + 284, + 601 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 591, + 506, + 602 + ], + "score": 1.0, + "content": "as input, the recommended view encoder is the one that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 481, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 280, + 615 + ], + "score": 1.0, + "content": "maximizes the mutual information between", + "type": "text" + }, + { + "bbox": [ + 280, + 603, + 290, + 612 + ], + "score": 0.79, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 599, + 294, + 615 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 294, + 603, + 304, + 612 + ], + "score": 0.8, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 599, + 322, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 323, + 603, + 329, + 613 + ], + "score": 0.5, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 599, + 465, + 615 + ], + "score": 1.0, + "content": ". Symmetrically the same for view", + "type": "text" + }, + { + "bbox": [ + 465, + 603, + 476, + 614 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 599, + 481, + 615 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 229, + 630 + ], + "score": 1.0, + "content": "Proposition 3. Given a task", + "type": "text" + }, + { + "bbox": [ + 229, + 618, + 238, + 627 + ], + "score": 0.73, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 616, + 285, + 630 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 285, + 619, + 291, + 629 + ], + "score": 0.74, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 616, + 422, + 630 + ], + "score": 1.0, + "content": ", the extracted representations,", + "type": "text" + }, + { + "bbox": [ + 422, + 620, + 447, + 630 + ], + "score": 0.52, + "content": "z _ { i } , \\ z _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 616, + 506, + 630 + ], + "score": 1.0, + "content": ", and a set of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 205, + 642 + ], + "score": 1.0, + "content": "aggregation operations,", + "type": "text" + }, + { + "bbox": [ + 206, + 628, + 281, + 640 + ], + "score": 0.9, + "content": "\\{ c _ { 1 } ( \\cdot ) , c _ { 2 } ( \\cdot ) , \\cdot \\cdot \\cdot \\ \\} ,", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 628, + 474, + 642 + ], + "score": 1.0, + "content": ", the recommended contrastive mode is the one,", + "type": "text" + }, + { + "bbox": [ + 474, + 628, + 502, + 641 + ], + "score": 0.92, + "content": "( c _ { i } , c _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 628, + 506, + 642 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 639, + 427, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 342, + 652 + ], + "score": 1.0, + "content": "that has the largest amount of mutual information between", + "type": "text" + }, + { + "bbox": [ + 343, + 639, + 367, + 651 + ], + "score": 0.85, + "content": "c _ { i } ( \\mathbf { z } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 639, + 371, + 652 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 371, + 639, + 398, + 652 + ], + "score": 0.8, + "content": "c _ { j } ( \\mathbf { z } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 639, + 417, + 652 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 417, + 641, + 423, + 651 + ], + "score": 0.65, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 639, + 427, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 465, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 466, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 466, + 669 + ], + "score": 1.0, + "content": "The qualitative and quantitative evaluation of these propositions are shown in Section 5.3.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 107, + 679, + 238, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 239, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 239, + 694 + ], + "score": 1.0, + "content": "4.5 Role of Negative Samples", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "Current graph contrastive learning approaches heavily depend on negative samples. 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A, B, C are null because the two", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 179, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 203, + 190 + ], + "score": 1.0, + "content": "views are optimal here.", + "type": "text" + }, + { + "bbox": [ + 203, + 180, + 213, + 189 + ], + "score": 0.85, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 179, + 300, + 190 + ], + "score": 1.0, + "content": "is contained by view", + "type": "text" + }, + { + "bbox": [ + 300, + 179, + 312, + 190 + ], + "score": 0.89, + "content": "\\mathbf { v } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 179, + 506, + 190 + ], + "score": 1.0, + "content": "because representations are functions of views.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 144, + 201 + ], + "score": 1.0, + "content": "(Middle)", + "type": "text" + }, + { + "bbox": [ + 144, + 190, + 154, + 200 + ], + "score": 0.85, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 189, + 185, + 201 + ], + "score": 1.0, + "content": "covers", + "type": "text" + }, + { + "bbox": [ + 185, + 190, + 194, + 199 + ], + "score": 0.38, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 189, + 506, + 201 + ], + "score": 1.0, + "content": "when Eq. 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The main motivation is: the optimal contrastive mode keeps the most task-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "relevant information after the representations are aggregated. The proof is included in the Appendix.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 316, + 506, + 351 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 409, + 368 + ], + "score": 1.0, + "content": "Corollary 3. (Optimal Contrastive Mode) Given the latent representations,", + "type": "text" + }, + { + "bbox": [ + 410, + 356, + 420, + 367 + ], + "score": 0.72, + "content": "\\mathbf { z } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 354, + 424, + 368 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 424, + 355, + 435, + 368 + ], + "score": 0.78, + "content": "\\mathbf { z } _ { j } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 354, + 505, + 368 + ], + "score": 1.0, + "content": ", extracted by the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 366, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 218, + 381 + ], + "score": 1.0, + "content": "optimal view encoders, i.e.,", + "type": "text" + }, + { + "bbox": [ + 219, + 367, + 271, + 379 + ], + "score": 0.81, + "content": "\\mathbf { z } _ { i } ^ { * } = f _ { i } ^ { * } ( \\mathbf { v } _ { i } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 366, + 274, + 381 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 275, + 367, + 326, + 380 + ], + "score": 0.76, + "content": "\\mathbf { z } _ { j _ { - } } ^ { * } = f _ { j _ { - } } ^ { * } ( \\mathbf { v } _ { j } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 366, + 428, + 381 + ], + "score": 1.0, + "content": ", and a downstream task", + "type": "text" + }, + { + "bbox": [ + 428, + 368, + 437, + 377 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 366, + 480, + 381 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 480, + 369, + 487, + 379 + ], + "score": 0.49, + "content": "y _ { \\mathrm { { i } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 366, + 506, + 381 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 451, + 391 + ], + "score": 1.0, + "content": "optimal contrastive mode is the solution to the following optimization problem, where", + "type": "text" + }, + { + "bbox": [ + 451, + 380, + 460, + 389 + ], + "score": 0.68, + "content": "c _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 379, + 464, + 391 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 464, + 380, + 473, + 390 + ], + "score": 0.73, + "content": "c _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "are the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 389, + 351, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 351, + 402 + ], + "score": 1.0, + "content": "aggregation operations applied to the latent representations:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 354, + 506, + 402 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 404, + 385, + 426 + ], + "lines": [ + { + "bbox": [ + 226, + 404, + 385, + 426 + ], + "spans": [ + { + "bbox": [ + 226, + 404, + 385, + 426 + ], + "score": 0.95, + "content": "\\displaystyle ( c _ { i } ^ { * } , c _ { j } ^ { * } ) = \\arg \\operatorname* { m i n } _ { ( c _ { i } , c _ { j } ) } - I ( c _ { i } ( \\mathbf { z } _ { i } ^ { * } ) ; c _ { j } ( \\mathbf { z } _ { j } ^ { * } ) ) .", + "type": "interline_equation", + "image_path": "2c3af1818fa8af72de7349137d1173eb8200bc8d576eacace9f582cdf15289e2.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 226, + 404, + 385, + 426 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 438, + 211, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 212, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 212, + 452 + ], + "score": 1.0, + "content": "4.4 InfoGCL Principle", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 471 + ], + "score": 1.0, + "content": "According to our proposed corollaries, we can theoretically design the optimal contrastive learning", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "approach for our specific graph data and task. However, in real-world scenarios, the conditions to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "meet the exact optimality of contrastive learning is hard or even not practically possible to reach", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "because of data noise and limited model capability. Therefore, we propose to achieve the optimal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "for each stage independently and practically, which is an approximation to achieve the original", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "optimality. Specifically, we make the following propositions to address the questions of the optimal", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 524, + 347, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 347, + 536 + ], + "score": 1.0, + "content": "views, optimal view encoder, and optimal contrastive mode.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 456, + 506, + 536 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 506, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 210, + 553 + ], + "score": 1.0, + "content": "Proposition 1. For a task", + "type": "text" + }, + { + "bbox": [ + 210, + 541, + 219, + 550 + ], + "score": 0.72, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 538, + 261, + 553 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 261, + 542, + 268, + 552 + ], + "score": 0.59, + "content": "y _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 538, + 477, + 553 + ], + "score": 1.0, + "content": ", given a bunch of graph view augmentation methods,", + "type": "text" + }, + { + "bbox": [ + 478, + 540, + 503, + 552 + ], + "score": 0.89, + "content": "\\{ q _ { 1 } ( \\cdot ) _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 538, + 506, + 553 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 550, + 508, + 565 + ], + "spans": [ + { + "bbox": [ + 107, + 551, + 152, + 563 + ], + "score": 0.88, + "content": "q _ { 2 } ( \\cdot ) , \\cdot \\cdot \\cdot \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 550, + 240, + 565 + ], + "score": 1.0, + "content": ", that create two views", + "type": "text" + }, + { + "bbox": [ + 240, + 553, + 250, + 561 + ], + "score": 0.75, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 550, + 254, + 565 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 254, + 553, + 266, + 562 + ], + "score": 0.78, + "content": "\\mathbf { v } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 550, + 483, + 565 + ], + "score": 1.0, + "content": ", the recommended augmentation methods are the ones,", + "type": "text" + }, + { + "bbox": [ + 484, + 552, + 503, + 563 + ], + "score": 0.89, + "content": "q _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 550, + 508, + 565 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 561, + 468, + 575 + ], + "spans": [ + { + "bbox": [ + 107, + 563, + 127, + 574 + ], + "score": 0.89, + "content": "q _ { j } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 561, + 189, + 575 + ], + "score": 1.0, + "content": ", that maximize", + "type": "text" + }, + { + "bbox": [ + 189, + 562, + 312, + 574 + ], + "score": 0.9, + "content": "I ( \\mathbf { v } _ { i } ; y ) + I ( \\mathbf { v } _ { j } ; y ) - I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 561, + 378, + 575 + ], + "score": 1.0, + "content": ", i.e., the area of", + "type": "text" + }, + { + "bbox": [ + 378, + 562, + 416, + 572 + ], + "score": 0.89, + "content": "A { + } B { + } D", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 561, + 468, + 575 + ], + "score": 1.0, + "content": "in Figure 2.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 538, + 508, + 575 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 506, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 225, + 593 + ], + "score": 1.0, + "content": "Proposition 2. Given a task", + "type": "text" + }, + { + "bbox": [ + 225, + 580, + 234, + 589 + ], + "score": 0.74, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 578, + 279, + 593 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 279, + 581, + 286, + 591 + ], + "score": 0.33, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 578, + 402, + 593 + ], + "score": 1.0, + "content": "and a set of view encoders,", + "type": "text" + }, + { + "bbox": [ + 402, + 578, + 483, + 591 + ], + "score": 0.91, + "content": "\\{ f _ { i } ^ { 1 } ( \\cdot ) , f _ { i } ^ { 2 } ( \\cdot ) , \\cdot \\cdot \\cdot \\ \\} ,", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 578, + 506, + 593 + ], + "score": 1.0, + "content": ", that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 591, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 201, + 602 + ], + "score": 1.0, + "content": "generate representation", + "type": "text" + }, + { + "bbox": [ + 201, + 592, + 211, + 601 + ], + "score": 0.84, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 591, + 273, + 602 + ], + "score": 1.0, + "content": "via taking view", + "type": "text" + }, + { + "bbox": [ + 274, + 592, + 284, + 601 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 591, + 506, + 602 + ], + "score": 1.0, + "content": "as input, the recommended view encoder is the one that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 481, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 280, + 615 + ], + "score": 1.0, + "content": "maximizes the mutual information between", + "type": "text" + }, + { + "bbox": [ + 280, + 603, + 290, + 612 + ], + "score": 0.79, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 599, + 294, + 615 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 294, + 603, + 304, + 612 + ], + "score": 0.8, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 599, + 322, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 323, + 603, + 329, + 613 + ], + "score": 0.5, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 599, + 465, + 615 + ], + "score": 1.0, + "content": ". Symmetrically the same for view", + "type": "text" + }, + { + "bbox": [ + 465, + 603, + 476, + 614 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 599, + 481, + 615 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 578, + 506, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 229, + 630 + ], + "score": 1.0, + "content": "Proposition 3. Given a task", + "type": "text" + }, + { + "bbox": [ + 229, + 618, + 238, + 627 + ], + "score": 0.73, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 616, + 285, + 630 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 285, + 619, + 291, + 629 + ], + "score": 0.74, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 616, + 422, + 630 + ], + "score": 1.0, + "content": ", the extracted representations,", + "type": "text" + }, + { + "bbox": [ + 422, + 620, + 447, + 630 + ], + "score": 0.52, + "content": "z _ { i } , \\ z _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 616, + 506, + 630 + ], + "score": 1.0, + "content": ", and a set of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 205, + 642 + ], + "score": 1.0, + "content": "aggregation operations,", + "type": "text" + }, + { + "bbox": [ + 206, + 628, + 281, + 640 + ], + "score": 0.9, + "content": "\\{ c _ { 1 } ( \\cdot ) , c _ { 2 } ( \\cdot ) , \\cdot \\cdot \\cdot \\ \\} ,", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 628, + 474, + 642 + ], + "score": 1.0, + "content": ", the recommended contrastive mode is the one,", + "type": "text" + }, + { + "bbox": [ + 474, + 628, + 502, + 641 + ], + "score": 0.92, + "content": "( c _ { i } , c _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 628, + 506, + 642 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 639, + 427, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 342, + 652 + ], + "score": 1.0, + "content": "that has the largest amount of mutual information between", + "type": "text" + }, + { + "bbox": [ + 343, + 639, + 367, + 651 + ], + "score": 0.85, + "content": "c _ { i } ( \\mathbf { z } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 639, + 371, + 652 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 371, + 639, + 398, + 652 + ], + "score": 0.8, + "content": "c _ { j } ( \\mathbf { z } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 639, + 417, + 652 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 417, + 641, + 423, + 651 + ], + "score": 0.65, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 639, + 427, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 616, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 465, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 466, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 466, + 669 + ], + "score": 1.0, + "content": "The qualitative and quantitative evaluation of these propositions are shown in Section 5.3.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 654, + 466, + 669 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 679, + 238, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 239, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 239, + 694 + ], + "score": 1.0, + "content": "4.5 Role of Negative Samples", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "Current graph contrastive learning approaches heavily depend on negative samples. However, recent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "score": 1.0, + "content": "progresses of contrastive learning in vision domain indicate that negative samples are not necessarily", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 172, + 507, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 507, + 185 + ], + "score": 1.0, + "content": "required [8, 4], of which the main benefit is to avoid careful treatment to retrieve the negative pairs.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "To study the influence of negative samples on graph contrastive learning, we follow the framework", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "of SimSiam [4] to revise the loss function as Eq.(9). A very recent work [29] also studies graph", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 204, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 218 + ], + "score": 1.0, + "content": "contrastive learning without negative samples. 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Last, we analyze our proposed principles via ablation study.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 346, + 155, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 157, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 157, + 361 + ], + "score": 1.0, + "content": "5.1 Setup", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "We use both graph classification and node classification benchmark datasets that are widely used", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "in the existing graph contrastive learning approaches. The graph classification datasets include", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "MUTAG [17], PTC-MR [17], IMDB-B [40], IMDB-M [40], NCI1 [34], and COLLAB [40]. MUTAG", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "is a collection of nitroaromatic compounds represented as graphs, where vertices stand for atoms and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "edges represent bonds between atoms. PTC-MR is a collection of 344 chemical compounds which", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "report the carcinogenicity for rats. IMDB-B and IMDB-M are two movie collaboration datasets,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "where nodes represent actors/actress and there is an edge between them if they appear in the same", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "movie. In NCI1, graphs are the representation of chemical compounds, where vertices stand for atoms", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "and edges represent bonds between atoms. COLLAB is a collaboration dataset, where researchers", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 465, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 504, + 476 + ], + "score": 1.0, + "content": "are nodes and an edge indicates collaboration between two researchers. The node classification", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 476, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 504, + 487 + ], + "score": 1.0, + "content": "datasets include Citeseer, Cora, and Pubmed [23]. All of them are citation networks, where nodes are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 438, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 438, + 498 + ], + "score": 1.0, + "content": "documents and edges are citation links. These datasets are summarized in Table 1.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "score": 1.0, + "content": "We closely follow the evaluation protocol of previous state-of-the-art graph contrastive learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "approaches. For graph classification, we report the mean 10-fold cross validation accuracy after 5", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 523, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 523, + 506, + 538 + ], + "score": 1.0, + "content": "runs followed by a linear SVM. The linear SVM is trained by applying cross validation on training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "data folds and the best mean accuracy is reported. For node classification, we report the mean accuracy", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "on test set after 50 runs of training followed by a linear neural network model. To make comparison", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "fair, we adopt the basic setting of InfoGraph for graph classification. We conduct experiment with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "the values of the number of GNN layers, the number of epochs, batch size, the parameter C of SVM", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 577, + 507, + 593 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 388, + 593 + ], + "score": 1.0, + "content": "in the sets {2, 4, 8, 12}, {10, 20, 40, 100}, {32, 64, 128, 256} and", + "type": "text" + }, + { + "bbox": [ + 388, + 578, + 501, + 591 + ], + "score": 0.4, + "content": "\\{ 1 0 ^ { - 3 } , \\ : \\mathrm { { \\dot { 1 } 0 ^ { - 2 } } } , . . . , \\ : 1 0 ^ { 2 } , 1 0 ^ { 3 } \\ : \\ : \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 577, + 507, + 593 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "respectively. We adopt the basic setting of DGI for node classification. Specifically, we set the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "number of GNN layers to 1 and experiment with the batch size in the set {2, 4, 8}. The hidden", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 447, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 447, + 626 + ], + "score": 1.0, + "content": "dimension of representations is set to 512. We also apply the early stopping strategy.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 222, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 223, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 223, + 650 + ], + "score": 1.0, + "content": "5.2 Experimental Results", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "To evaluate our method InfoGCL on graph classification, we use thhree categories of baselines.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "The kernel approaches include shortest path kernel (SP) [2], Graphlet kernel (GK) [25], Weisfeiler-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "Lehman sub-tree kernel (WL) [24], deep graph kernels (DGK) [39], and multi-scale Laplacian kernel", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "(MLG) [16]. The supervised baselines include GraphSAGE [12], GCN [15], GIN [38], GAT [32].", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "score": 1.0, + "content": "We also compare with the unsupervised approaches, including RandomWalk [7], node2vec [9],", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 724 + ], + "score": 1.0, + "content": "sub2vec [14], graph2vec [20], InfoGraph [28], GraphCL [42], and mvgrl [13]. 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Graph Task DatasetsNode Task Datasets
MUTAGPTC-MRIMDB-BIMDB-MNCI1COLLABCoraCiteseerPubmed
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We first describe datasets,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "evaluation protocol, and experimental setup. Then, we present the experimental results on both node", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 322, + 447, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 447, + 335 + ], + "score": 1.0, + "content": "and graph classification. Last, we analyze our proposed principles via ablation study.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 300, + 506, + 335 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 346, + 155, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 157, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 157, + 361 + ], + "score": 1.0, + "content": "5.1 Setup", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "We use both graph classification and node classification benchmark datasets that are widely used", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "in the existing graph contrastive learning approaches. The graph classification datasets include", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "MUTAG [17], PTC-MR [17], IMDB-B [40], IMDB-M [40], NCI1 [34], and COLLAB [40]. MUTAG", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "is a collection of nitroaromatic compounds represented as graphs, where vertices stand for atoms and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "edges represent bonds between atoms. PTC-MR is a collection of 344 chemical compounds which", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "report the carcinogenicity for rats. IMDB-B and IMDB-M are two movie collaboration datasets,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "where nodes represent actors/actress and there is an edge between them if they appear in the same", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "movie. In NCI1, graphs are the representation of chemical compounds, where vertices stand for atoms", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "and edges represent bonds between atoms. COLLAB is a collaboration dataset, where researchers", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 465, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 504, + 476 + ], + "score": 1.0, + "content": "are nodes and an edge indicates collaboration between two researchers. The node classification", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 476, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 504, + 487 + ], + "score": 1.0, + "content": "datasets include Citeseer, Cora, and Pubmed [23]. All of them are citation networks, where nodes are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 438, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 438, + 498 + ], + "score": 1.0, + "content": "documents and edges are citation links. These datasets are summarized in Table 1.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 366, + 506, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "score": 1.0, + "content": "We closely follow the evaluation protocol of previous state-of-the-art graph contrastive learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "approaches. For graph classification, we report the mean 10-fold cross validation accuracy after 5", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 523, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 523, + 506, + 538 + ], + "score": 1.0, + "content": "runs followed by a linear SVM. The linear SVM is trained by applying cross validation on training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "data folds and the best mean accuracy is reported. For node classification, we report the mean accuracy", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "on test set after 50 runs of training followed by a linear neural network model. To make comparison", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "fair, we adopt the basic setting of InfoGraph for graph classification. We conduct experiment with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "the values of the number of GNN layers, the number of epochs, batch size, the parameter C of SVM", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 577, + 507, + 593 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 388, + 593 + ], + "score": 1.0, + "content": "in the sets {2, 4, 8, 12}, {10, 20, 40, 100}, {32, 64, 128, 256} and", + "type": "text" + }, + { + "bbox": [ + 388, + 578, + 501, + 591 + ], + "score": 0.4, + "content": "\\{ 1 0 ^ { - 3 } , \\ : \\mathrm { { \\dot { 1 } 0 ^ { - 2 } } } , . . . , \\ : 1 0 ^ { 2 } , 1 0 ^ { 3 } \\ : \\ : \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 577, + 507, + 593 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "respectively. We adopt the basic setting of DGI for node classification. Specifically, we set the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "number of GNN layers to 1 and experiment with the batch size in the set {2, 4, 8}. The hidden", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 447, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 447, + 626 + ], + "score": 1.0, + "content": "dimension of representations is set to 512. We also apply the early stopping strategy.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 501, + 507, + 626 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 222, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 223, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 223, + 650 + ], + "score": 1.0, + "content": "5.2 Experimental Results", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "To evaluate our method InfoGCL on graph classification, we use thhree categories of baselines.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "The kernel approaches include shortest path kernel (SP) [2], Graphlet kernel (GK) [25], Weisfeiler-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "Lehman sub-tree kernel (WL) [24], deep graph kernels (DGK) [39], and multi-scale Laplacian kernel", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "(MLG) [16]. The supervised baselines include GraphSAGE [12], GCN [15], GIN [38], GAT [32].", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "score": 1.0, + "content": "We also compare with the unsupervised approaches, including RandomWalk [7], node2vec [9],", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 724 + ], + "score": 1.0, + "content": "sub2vec [14], graph2vec [20], InfoGraph [28], GraphCL [42], and mvgrl [13]. Table 2 shows the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "score": 1.0, + "content": "graph classification results. We observe that our approach achieves the best results compared to other", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "unsupervised approaches. Our approach also outperforms or matches the best kernel approaches", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "across the datasets. Even compared with the supervised ones, our approach achieves the best in 2 out", + "type": "text", + "cross_page": true + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 628, + 434, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 434, + 642 + ], + "score": 1.0, + "content": "of 6 datasets and the results of our approach on other 4 dataset are among the top.", + "type": "text", + "cross_page": true + } + ], + "index": 23 + } + ], + "index": 42.5, + "bbox_fs": [ + 104, + 655, + 507, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 112, + 69, + 497, + 291 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 112, + 69, + 497, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 69, + 497, + 291 + ], + "spans": [ + { + "bbox": [ + 112, + 69, + 497, + 291 + ], + "score": 0.982, + "html": "
MethodMUTAGPTC-MRIMDB-BIMDB-MNCI1COLLAB
Kernel Approaches
SP85.2 ± 2.458.2 ± 2.455.6± 0.238.0±0.373.5 ± 0.1
GK81.7 ± 2.157.3 ± 1.465.9 ± 1.043.9 ± 0.466.0 ± 0.172.8± 0.3
WL80.7 ± 3.058.0 ± 0.572.3 ± 3.447.0 ± 0.580.0± 0.578.9 ± 1.9
DGK87.4 ± 2.760.1 ± 2.667.0 ± 0.644.6 ± 0.580.3 ± 0.573.1 ± 0.3
MLG87.9 ± 1.663.3 ± 1.566.6 ± 0.341.2 ± 0.080.8 ± 1.3
Supervised Approaches
GraphSAGE85.1 ± 7.6 63.9 ± 7.772.3 ± 5.350.9 ± 2.277.7 ± 1.568.3 ± 4.2
GCN85.6 ± 5.864.2 ± 4.374.0± 3.451.9 ± 3.880.2 ± 2.079.0 ± 1.8
GIN-089.4 ± 5.664.6 ± 7.075.1 ± 5.152.3 ± 2.882.7 ± 1.780.2 ± 1.9
GIN-e89.0 ± 6.063.7 ±8.274.3 ± 5.152.1 ± 3.682.7 ± 1.680.1 ± 1.9
GAT89.4 ± 6.166.7 ± 5.170.5 ± 2.347.8 ± 3.166.6 ± 2.267.4 ± 2.9
Unsupervised Approaches
RandomWalk83.7 ± 1.557.9 ± 1.350.7±0.334.7 ± 0.264.3 ± 0.3
node2vec72.6 ±10.258.6±8.050.2 ± 0.936.0 ± 0.754.9 ± 1.656.1 ± 0.2
sub2vec61.1 ± 15.860.0 ± 6.455.3 ± 1.536.7 ± 0.852.8 ±1.5-
graph2vec83.2 ±9.660.2 ± 6.971.1 ± 0.550.4 ± 0.975.4 ±1.2
InfoGraph89.0 ± 1.161.7 ± 1.473.0± 0.949.7 ± 0.576.2 ±1.470.7 ± 1.1
GraphCL86.8 ± 1.361.3 ± 2.171.1 ± 0.449.2 ± 0.677.9 ± 0.471.4 ± 1.2
mvgrl89.7 ± 1.162.5 ± 1.774.2 ± 0.751.2 ± 0.577.0 ± 0.876.0 ± 1.2
51.4 ± 0.880.2 ± 0.6
InfoGCL91.2 ± 1.363.5 ± 1.575.1 ± 0.980.0 ± 1.3
", + "type": "table", + "image_path": "100341ee754bc7ccbfbd5552183bc1ec5e1b1be1a782962292628b2341f3673b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 69, + 497, + 143.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 143.0, + 497, + 217.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 217.0, + 497, + 291.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 222, + 297, + 385, + 308 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 220, + 295, + 383, + 310 + ], + "spans": [ + { + "bbox": [ + 220, + 295, + 367, + 310 + ], + "score": 1.0, + "content": "Table 2: Graph classification results", + "type": "text" + }, + { + "bbox": [ + 367, + 297, + 383, + 308 + ], + "score": 0.71, + "content": "( \\% )", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "table", + "bbox": [ + 182, + 335, + 428, + 542 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 182, + 335, + 428, + 542 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 182, + 335, + 428, + 542 + ], + "spans": [ + { + "bbox": [ + 182, + 335, + 428, + 542 + ], + "score": 0.977, + "html": "
MethodCoraCiteseerPubmed
Supervised Approaches
MLP55.146.571.4
ICA75.169.173.9
LP68.045.363.0
ManiReg59.560.170.7
SemiEmb59.059.671.7
Planetoid75.764.777.2
Chebyshev81.269.874.4
GCN81.570.379.0
JKNet82.7 ± 0.473.0 ± 0.577.9 ± 0.4
GAT83.0 ± 0.772.5± 0.779.0 ± 0.3
Unsupervised Approaches
Linear47.9 ± 0.449.3 ± 0.269.1 ± 0.3
DeepWalk70.7 ± 0.651.4 ± 0.574.3 ± 0.9
GAE71.5 ± 0.465.8 ±0.472.1 ± 0.5
VERSE72.5 ± 0.355.5± 0.4
DGI83.8 ± 0.572.0 ± 0.677.9 ± 0.3
GraphCL82.5 ± 0.173.1 ± 0.2
mvgrl86.8 ± 0.573.3 ± 0.580.1 ± 0.7
InfoGCL83.5 ± 0.373.5 ± 0.479.1 ± 0.2
", + "type": "table", + "image_path": "bb3604b9cde8633c8adb22a2b93c182327925e26e7eabc91176a409025cf1d38.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 182, + 335, + 428, + 348.8 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 182, + 348.8, + 428, + 362.6 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 182, + 362.6, + 428, + 376.40000000000003 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 182, + 376.40000000000003, + 428, + 390.20000000000005 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 182, + 390.20000000000005, + 428, + 404.00000000000006 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 182, + 404.00000000000006, + 428, + 417.80000000000007 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 182, + 417.80000000000007, + 428, + 431.6000000000001 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 182, + 431.6000000000001, + 428, + 445.4000000000001 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 182, + 445.4000000000001, + 428, + 459.2000000000001 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 182, + 459.2000000000001, + 428, + 473.0000000000001 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 182, + 473.0000000000001, + 428, + 486.8000000000001 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 182, + 486.8000000000001, + 428, + 500.60000000000014 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 182, + 500.60000000000014, + 428, + 514.4000000000001 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 182, + 514.4000000000001, + 428, + 528.2 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 182, + 528.2, + 428, + 542.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 224, + 547, + 384, + 559 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 222, + 545, + 385, + 561 + ], + "spans": [ + { + "bbox": [ + 222, + 545, + 365, + 561 + ], + "score": 1.0, + "content": "Table 3: Node classification results", + "type": "text" + }, + { + "bbox": [ + 366, + 548, + 381, + 559 + ], + "score": 0.67, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 545, + 385, + 561 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + } + ], + "index": 15.0 + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "score": 1.0, + "content": "graph classification results. We observe that our approach achieves the best results compared to other", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "unsupervised approaches. Our approach also outperforms or matches the best kernel approaches", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "across the datasets. Even compared with the supervised ones, our approach achieves the best in 2 out", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 628, + 434, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 434, + 642 + ], + "score": 1.0, + "content": "of 6 datasets and the results of our approach on other 4 dataset are among the top.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "For node classification tasks, we compare InfoGCL with some supervised approaches and unsu-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 656, + 507, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 507, + 668 + ], + "score": 1.0, + "content": "pervised approaches. The supervised baselines include a simple MLP model, iterative classifi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "cation algorithm (ICA) [19], manifold regularization (ManiReg) [1], semi-supervised embedding", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "(SemiEmb) [35], Planetoid [41], Chebyshev [5], GCN, JKNet [36], GAT. Table 3 shows the node", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "classification results. It is observed that our approach achieves the state-of-the-art results and com-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "petes the best one with respect to the existing unsupervised approaches. Compared to supervised", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 711, + 323, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 323, + 724 + ], + "score": 1.0, + "content": "baselines, our approach outperforms all the baselines.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 112, + 69, + 497, + 291 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 112, + 69, + 497, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 69, + 497, + 291 + ], + "spans": [ + { + "bbox": [ + 112, + 69, + 497, + 291 + ], + "score": 0.982, + "html": "
MethodMUTAGPTC-MRIMDB-BIMDB-MNCI1COLLAB
Kernel Approaches
SP85.2 ± 2.458.2 ± 2.455.6± 0.238.0±0.373.5 ± 0.1
GK81.7 ± 2.157.3 ± 1.465.9 ± 1.043.9 ± 0.466.0 ± 0.172.8± 0.3
WL80.7 ± 3.058.0 ± 0.572.3 ± 3.447.0 ± 0.580.0± 0.578.9 ± 1.9
DGK87.4 ± 2.760.1 ± 2.667.0 ± 0.644.6 ± 0.580.3 ± 0.573.1 ± 0.3
MLG87.9 ± 1.663.3 ± 1.566.6 ± 0.341.2 ± 0.080.8 ± 1.3
Supervised Approaches
GraphSAGE85.1 ± 7.6 63.9 ± 7.772.3 ± 5.350.9 ± 2.277.7 ± 1.568.3 ± 4.2
GCN85.6 ± 5.864.2 ± 4.374.0± 3.451.9 ± 3.880.2 ± 2.079.0 ± 1.8
GIN-089.4 ± 5.664.6 ± 7.075.1 ± 5.152.3 ± 2.882.7 ± 1.780.2 ± 1.9
GIN-e89.0 ± 6.063.7 ±8.274.3 ± 5.152.1 ± 3.682.7 ± 1.680.1 ± 1.9
GAT89.4 ± 6.166.7 ± 5.170.5 ± 2.347.8 ± 3.166.6 ± 2.267.4 ± 2.9
Unsupervised Approaches
RandomWalk83.7 ± 1.557.9 ± 1.350.7±0.334.7 ± 0.264.3 ± 0.3
node2vec72.6 ±10.258.6±8.050.2 ± 0.936.0 ± 0.754.9 ± 1.656.1 ± 0.2
sub2vec61.1 ± 15.860.0 ± 6.455.3 ± 1.536.7 ± 0.852.8 ±1.5-
graph2vec83.2 ±9.660.2 ± 6.971.1 ± 0.550.4 ± 0.975.4 ±1.2
InfoGraph89.0 ± 1.161.7 ± 1.473.0± 0.949.7 ± 0.576.2 ±1.470.7 ± 1.1
GraphCL86.8 ± 1.361.3 ± 2.171.1 ± 0.449.2 ± 0.677.9 ± 0.471.4 ± 1.2
mvgrl89.7 ± 1.162.5 ± 1.774.2 ± 0.751.2 ± 0.577.0 ± 0.876.0 ± 1.2
51.4 ± 0.880.2 ± 0.6
InfoGCL91.2 ± 1.363.5 ± 1.575.1 ± 0.980.0 ± 1.3
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MethodCoraCiteseerPubmed
Supervised Approaches
MLP55.146.571.4
ICA75.169.173.9
LP68.045.363.0
ManiReg59.560.170.7
SemiEmb59.059.671.7
Planetoid75.764.777.2
Chebyshev81.269.874.4
GCN81.570.379.0
JKNet82.7 ± 0.473.0 ± 0.577.9 ± 0.4
GAT83.0 ± 0.772.5± 0.779.0 ± 0.3
Unsupervised Approaches
Linear47.9 ± 0.449.3 ± 0.269.1 ± 0.3
DeepWalk70.7 ± 0.651.4 ± 0.574.3 ± 0.9
GAE71.5 ± 0.465.8 ±0.472.1 ± 0.5
VERSE72.5 ± 0.355.5± 0.4
DGI83.8 ± 0.572.0 ± 0.677.9 ± 0.3
GraphCL82.5 ± 0.173.1 ± 0.2
mvgrl86.8 ± 0.573.3 ± 0.580.1 ± 0.7
InfoGCL83.5 ± 0.373.5 ± 0.479.1 ± 0.2
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MethodMUTAGIMDB-BCOLLABCoraCiteseerPubmed
InfoGCL (w/o neg)91.0 ± 1.475.1± 0.580.2 ±1.078.6 ± 0.470.4 ± 0.677.4±0.7
InfoGCL (w/ neg)91.2 ± 1.375.1 ± 0.980.0± 1.383.5±0.373.5 ± 0.479.1±0.2
", + "type": "table", + "image_path": "f0477851d52bef85626a9194a114b45dcbb4525569d2ce3f3528abe6efd26a28.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 59, + 498, + 72.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 72.33333333333333, + 498, + 85.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 85.66666666666666, + 498, + 98.99999999999999 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 112, + 105, + 490, + 116 + ], + "lines": [ + { + "bbox": [ + 116, + 102, + 491, + 118 + ], + "spans": [ + { + "bbox": [ + 116, + 102, + 491, + 118 + ], + "score": 1.0, + "content": "Table 4: Comparison between InfoGCL with negative samples and without negative samples.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 107, + 130, + 271, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 271, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 271, + 144 + ], + "score": 1.0, + "content": "5.3 Evaluation of InfoGCL Principle", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 151, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "We can unify the existing graph contrastive learning methods through the perspective of InfoGCL", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 505, + 174 + ], + "score": 1.0, + "content": "principle: all recent graph contrastive learning methods can be decoupled into three stages that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "implicitly follow the InfoGCL principle, though being different in model architecture design and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 184, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 506, + 196 + ], + "score": 1.0, + "content": "optimization strategies. Below, we analyze some observations from several recent work. Because of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 471, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 471, + 208 + ], + "score": 1.0, + "content": "the limited space, please refer to the Appendix for more results of the quantitative analysis.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 503, + 234 + ], + "lines": [ + { + "bbox": [ + 106, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "Obs. i. Composing a graph and its augmentation benefits downstream performance [42]. Compared", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 222, + 503, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 398, + 235 + ], + "score": 1.0, + "content": "to composing a graph and the graph itself, augmentation leads to smaller", + "type": "text" + }, + { + "bbox": [ + 399, + 222, + 436, + 234 + ], + "score": 0.93, + "content": "I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 222, + 503, + 235 + ], + "score": 1.0, + "content": "(Proposition 1).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 503, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 252 + ], + "score": 1.0, + "content": "Obs. ii. Composing different augmentations benefits more [42]. Compared to composing a graph", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 249, + 446, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 343, + 262 + ], + "score": 1.0, + "content": "and its augmentations, two augmentations further decrease", + "type": "text" + }, + { + "bbox": [ + 343, + 249, + 380, + 262 + ], + "score": 0.92, + "content": "I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 249, + 446, + 262 + ], + "score": 1.0, + "content": "(Proposition 1).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "Obs. iii. Node dropping and subgraph sampling are generally beneficial across datasets [42]. When", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "compared to attribute masking and edge perturbation, they change the semantic meaning of the graph", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 406, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 268, + 301 + ], + "score": 1.0, + "content": "relatively slightly, which leads to higher", + "type": "text" + }, + { + "bbox": [ + 269, + 288, + 302, + 299 + ], + "score": 0.61, + "content": "I ( \\mathbf { v } _ { i } ; y )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 287, + 305, + 301 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 306, + 287, + 339, + 300 + ], + "score": 0.55, + "content": "I ( \\mathbf { v } _ { j } ; \\bar { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 287, + 406, + 301 + ], + "score": 1.0, + "content": "(Proposition 1).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 504, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 504, + 315 + ], + "score": 1.0, + "content": "Obs. iv. Edge perturbation benefits social networks but hurts some biochemical molecules [42]. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 328 + ], + "score": 1.0, + "content": "semantic meaning of social networks are robust to edge perturbation. However, the semantic meaning", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "of some biochemical molecules are determined by local connection pattern, where edge perturbation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 247, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 147, + 348 + ], + "score": 1.0, + "content": "decreases", + "type": "text" + }, + { + "bbox": [ + 147, + 336, + 180, + 348 + ], + "score": 0.92, + "content": "I ( \\mathbf { v } _ { i } ; y )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 337, + 247, + 348 + ], + "score": 1.0, + "content": "(Proposition 1).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "Obs. v. Contrasting node and graph representations consistently performs better than other contrastive", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "modes across benchmarks [13]. Compared to other contrastive modes, node-graph (i.e., local-global)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 375, + 504, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 497, + 388 + ], + "score": 1.0, + "content": "mode generally extracts more graph structure information, which benefits predicting task label", + "type": "text" + }, + { + "bbox": [ + 497, + 376, + 504, + 386 + ], + "score": 0.73, + "content": "y", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 170, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 170, + 398 + ], + "score": 1.0, + "content": "(Proposition 3).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 410, + 244, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 245, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 245, + 424 + ], + "score": 1.0, + "content": "5.4 Effect of Negative Samples", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "To study the effect of negative samples on graph contrastive learning, we follow SimSiam [4] and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "design the objective as Eq. (9). We conduct experiments on three graph task and three node task", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "datasets. The results are reported in Table 4. It is observed that the negative samples show little", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 461, + 507, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 507, + 477 + ], + "score": 1.0, + "content": "influence on the three graph task datasets, while performance drops on the three node task datasets,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 218, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 218, + 486 + ], + "score": 1.0, + "content": "especially the Cora dataset.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 490, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 507, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 507, + 502 + ], + "score": 1.0, + "content": "According to the dataset statistics summarized in Table 1, we see the networks of Cora, Citeseer,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Pubmed are much sparser (in terms of network topology). Furthermore, we know the node features of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "these three datasets are also much sparser (one-hot encoding with high dimensionality). We speculate", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "this because the contrastive learning models tends to collapse easier if negative samples are not used,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "especially when data is too sparse. Therefore,we make the hypothesis: negative samples benefit graph", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 544, + 470, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 470, + 558 + ], + "score": 1.0, + "content": "modeling, especially when i) network topology, and ii) node features are extremely sparse.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 268, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 268, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 268, + 587 + ], + "score": 1.0, + "content": "6 Conclusion and Limitations", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "We propose InfoGCL, an information-aware graph contrastive learning framework for graph con-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "trastive learning. Existing graph contrastive learning approaches are usually carefully designed. We", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "aim to answer how to perform contrastive learning for your learning tasks on specific graph data.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "Our method decouples the typical contrastive learning approaches into three sequential modules and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "provides the theoretical analysis for reaching the optimality. To address the questions of optimality", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 651, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 506, + 663 + ], + "score": 1.0, + "content": "in a practical way, we propose the InfoGCL principle, which is implicitly followed by all recent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "score": 1.0, + "content": "graph contrastive learning approaches. In addition, we explore the role of negative samples in graph", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "contrastive learning and find negative samples are not necessarily required. Experiments on both", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 683, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 505, + 696 + ], + "score": 1.0, + "content": "node and graph benchmark datasets demonstrate the effectiveness of our method. Note that our", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 695, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 707 + ], + "score": 1.0, + "content": "method is not without limitations. We can further improve our method by designing better practical", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 705, + 403, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 705, + 403, + 719 + ], + "score": 1.0, + "content": "approximations to the theoretical optimality of graph contrastive learning.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 113, + 59, + 498, + 99 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 113, + 59, + 498, + 99 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 59, + 498, + 99 + ], + "spans": [ + { + "bbox": [ + 113, + 59, + 498, + 99 + ], + "score": 0.974, + "html": "
MethodMUTAGIMDB-BCOLLABCoraCiteseerPubmed
InfoGCL (w/o neg)91.0 ± 1.475.1± 0.580.2 ±1.078.6 ± 0.470.4 ± 0.677.4±0.7
InfoGCL (w/ neg)91.2 ± 1.375.1 ± 0.980.0± 1.383.5±0.373.5 ± 0.479.1±0.2
", + "type": "table", + "image_path": "f0477851d52bef85626a9194a114b45dcbb4525569d2ce3f3528abe6efd26a28.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 59, + 498, + 72.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 72.33333333333333, + 498, + 85.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 85.66666666666666, + 498, + 98.99999999999999 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 112, + 105, + 490, + 116 + ], + "lines": [ + { + "bbox": [ + 116, + 102, + 491, + 118 + ], + "spans": [ + { + "bbox": [ + 116, + 102, + 491, + 118 + ], + "score": 1.0, + "content": "Table 4: Comparison between InfoGCL with negative samples and without negative samples.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 116, + 102, + 491, + 118 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 130, + 271, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 271, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 271, + 144 + ], + "score": 1.0, + "content": "5.3 Evaluation of InfoGCL Principle", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 151, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "We can unify the existing graph contrastive learning methods through the perspective of InfoGCL", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 505, + 174 + ], + "score": 1.0, + "content": "principle: all recent graph contrastive learning methods can be decoupled into three stages that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "implicitly follow the InfoGCL principle, though being different in model architecture design and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 184, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 506, + 196 + ], + "score": 1.0, + "content": "optimization strategies. Below, we analyze some observations from several recent work. Because of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 471, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 471, + 208 + ], + "score": 1.0, + "content": "the limited space, please refer to the Appendix for more results of the quantitative analysis.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 150, + 506, + 208 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 503, + 234 + ], + "lines": [ + { + "bbox": [ + 106, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "Obs. i. Composing a graph and its augmentation benefits downstream performance [42]. Compared", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 222, + 503, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 398, + 235 + ], + "score": 1.0, + "content": "to composing a graph and the graph itself, augmentation leads to smaller", + "type": "text" + }, + { + "bbox": [ + 399, + 222, + 436, + 234 + ], + "score": 0.93, + "content": "I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 222, + 503, + 235 + ], + "score": 1.0, + "content": "(Proposition 1).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 106, + 211, + 505, + 235 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 503, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 252 + ], + "score": 1.0, + "content": "Obs. ii. Composing different augmentations benefits more [42]. Compared to composing a graph", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 249, + 446, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 343, + 262 + ], + "score": 1.0, + "content": "and its augmentations, two augmentations further decrease", + "type": "text" + }, + { + "bbox": [ + 343, + 249, + 380, + 262 + ], + "score": 0.92, + "content": "I ( \\mathbf { v } _ { i } ; \\mathbf { v } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 249, + 446, + 262 + ], + "score": 1.0, + "content": "(Proposition 1).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 236, + 505, + 262 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "Obs. iii. Node dropping and subgraph sampling are generally beneficial across datasets [42]. When", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "compared to attribute masking and edge perturbation, they change the semantic meaning of the graph", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 406, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 268, + 301 + ], + "score": 1.0, + "content": "relatively slightly, which leads to higher", + "type": "text" + }, + { + "bbox": [ + 269, + 288, + 302, + 299 + ], + "score": 0.61, + "content": "I ( \\mathbf { v } _ { i } ; y )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 287, + 305, + 301 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 306, + 287, + 339, + 300 + ], + "score": 0.55, + "content": "I ( \\mathbf { v } _ { j } ; \\bar { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 287, + 406, + 301 + ], + "score": 1.0, + "content": "(Proposition 1).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 265, + 505, + 301 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 504, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 504, + 315 + ], + "score": 1.0, + "content": "Obs. iv. Edge perturbation benefits social networks but hurts some biochemical molecules [42]. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 328 + ], + "score": 1.0, + "content": "semantic meaning of social networks are robust to edge perturbation. However, the semantic meaning", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "of some biochemical molecules are determined by local connection pattern, where edge perturbation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 247, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 147, + 348 + ], + "score": 1.0, + "content": "decreases", + "type": "text" + }, + { + "bbox": [ + 147, + 336, + 180, + 348 + ], + "score": 0.92, + "content": "I ( \\mathbf { v } _ { i } ; y )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 337, + 247, + 348 + ], + "score": 1.0, + "content": "(Proposition 1).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 303, + 505, + 348 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "Obs. v. Contrasting node and graph representations consistently performs better than other contrastive", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "modes across benchmarks [13]. Compared to other contrastive modes, node-graph (i.e., local-global)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 375, + 504, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 497, + 388 + ], + "score": 1.0, + "content": "mode generally extracts more graph structure information, which benefits predicting task label", + "type": "text" + }, + { + "bbox": [ + 497, + 376, + 504, + 386 + ], + "score": 0.73, + "content": "y", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 170, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 170, + 398 + ], + "score": 1.0, + "content": "(Proposition 3).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 352, + 505, + 398 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 410, + 244, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 245, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 245, + 424 + ], + "score": 1.0, + "content": "5.4 Effect of Negative Samples", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "To study the effect of negative samples on graph contrastive learning, we follow SimSiam [4] and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "design the objective as Eq. (9). We conduct experiments on three graph task and three node task", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "datasets. The results are reported in Table 4. It is observed that the negative samples show little", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 461, + 507, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 507, + 477 + ], + "score": 1.0, + "content": "influence on the three graph task datasets, while performance drops on the three node task datasets,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 218, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 218, + 486 + ], + "score": 1.0, + "content": "especially the Cora dataset.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 429, + 507, + 486 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 490, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 507, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 507, + 502 + ], + "score": 1.0, + "content": "According to the dataset statistics summarized in Table 1, we see the networks of Cora, Citeseer,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Pubmed are much sparser (in terms of network topology). Furthermore, we know the node features of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "these three datasets are also much sparser (one-hot encoding with high dimensionality). We speculate", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "this because the contrastive learning models tends to collapse easier if negative samples are not used,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "especially when data is too sparse. Therefore,we make the hypothesis: negative samples benefit graph", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 544, + 470, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 470, + 558 + ], + "score": 1.0, + "content": "modeling, especially when i) network topology, and ii) node features are extremely sparse.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 489, + 507, + 558 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 268, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 268, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 268, + 587 + ], + "score": 1.0, + "content": "6 Conclusion and Limitations", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "We propose InfoGCL, an information-aware graph contrastive learning framework for graph con-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "trastive learning. Existing graph contrastive learning approaches are usually carefully designed. We", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "aim to answer how to perform contrastive learning for your learning tasks on specific graph data.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "Our method decouples the typical contrastive learning approaches into three sequential modules and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "provides the theoretical analysis for reaching the optimality. To address the questions of optimality", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 651, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 506, + 663 + ], + "score": 1.0, + "content": "in a practical way, we propose the InfoGCL principle, which is implicitly followed by all recent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "score": 1.0, + "content": "graph contrastive learning approaches. In addition, we explore the role of negative samples in graph", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "contrastive learning and find negative samples are not necessarily required. Experiments on both", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 683, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 505, + 696 + ], + "score": 1.0, + "content": "node and graph benchmark datasets demonstrate the effectiveness of our method. Note that our", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 695, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 707 + ], + "score": 1.0, + "content": "method is not without limitations. We can further improve our method by designing better practical", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 705, + 403, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 705, + 403, + 719 + ], + "score": 1.0, + "content": "approximations to the theoretical optimality of graph contrastive learning.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 596, + 506, + 719 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 279, + 85 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 281, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 281, + 88 + ], + "score": 1.0, + "content": "Funding Transparency Statement", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 455, + 108 + ], + "lines": [ + { + "bbox": [ + 106, + 95, + 456, + 110 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 456, + 110 + ], + "score": 1.0, + "content": "This project was partially supported by NSF projects IIS-1707548 and CBET-1638320.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 124, + 163, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 123, + 165, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 165, + 138 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 507, + 727 + ], + "lines": [ + { + "bbox": [ + 111, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 111, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "[1] Mikhail Belkin, Partha Niyogi, and Vikas Sindhwani. 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Graph Task DatasetsNode Task Datasets
MUTAGPTC-MRIMDB-BIMDB-MNCI1COLLABCoraCiteseerPubmed
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#Edges19.814.7193.165.91.133.04732542944338
#Classes222223673
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MethodMUTAGPTC-MRIMDB-BIMDB-MNCI1COLLAB
Kernel Approaches
SP85.2 ± 2.458.2 ± 2.455.6± 0.238.0±0.373.5 ± 0.1
GK81.7 ± 2.157.3 ± 1.465.9 ± 1.043.9 ± 0.466.0 ± 0.172.8± 0.3
WL80.7 ± 3.058.0 ± 0.572.3 ± 3.447.0 ± 0.580.0± 0.578.9 ± 1.9
DGK87.4 ± 2.760.1 ± 2.667.0 ± 0.644.6 ± 0.580.3 ± 0.573.1 ± 0.3
MLG87.9 ± 1.663.3 ± 1.566.6 ± 0.341.2 ± 0.080.8 ± 1.3
Supervised Approaches
GraphSAGE85.1 ± 7.6 63.9 ± 7.772.3 ± 5.350.9 ± 2.277.7 ± 1.568.3 ± 4.2
GCN85.6 ± 5.864.2 ± 4.374.0± 3.451.9 ± 3.880.2 ± 2.079.0 ± 1.8
GIN-089.4 ± 5.664.6 ± 7.075.1 ± 5.152.3 ± 2.882.7 ± 1.780.2 ± 1.9
GIN-e89.0 ± 6.063.7 ±8.274.3 ± 5.152.1 ± 3.682.7 ± 1.680.1 ± 1.9
GAT89.4 ± 6.166.7 ± 5.170.5 ± 2.347.8 ± 3.166.6 ± 2.267.4 ± 2.9
Unsupervised Approaches
RandomWalk83.7 ± 1.557.9 ± 1.350.7±0.334.7 ± 0.264.3 ± 0.3
node2vec72.6 ±10.258.6±8.050.2 ± 0.936.0 ± 0.754.9 ± 1.656.1 ± 0.2
sub2vec61.1 ± 15.860.0 ± 6.455.3 ± 1.536.7 ± 0.852.8 ±1.5-
graph2vec83.2 ±9.660.2 ± 6.971.1 ± 0.550.4 ± 0.975.4 ±1.2
InfoGraph89.0 ± 1.161.7 ± 1.473.0± 0.949.7 ± 0.576.2 ±1.470.7 ± 1.1
GraphCL86.8 ± 1.361.3 ± 2.171.1 ± 0.449.2 ± 0.677.9 ± 0.471.4 ± 1.2
mvgrl89.7 ± 1.162.5 ± 1.774.2 ± 0.751.2 ± 0.577.0 ± 0.876.0 ± 1.2
51.4 ± 0.880.2 ± 0.6
InfoGCL91.2 ± 1.363.5 ± 1.575.1 ± 0.980.0 ± 1.3
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MethodCoraCiteseerPubmed
Supervised Approaches
MLP55.146.571.4
ICA75.169.173.9
LP68.045.363.0
ManiReg59.560.170.7
SemiEmb59.059.671.7
Planetoid75.764.777.2
Chebyshev81.269.874.4
GCN81.570.379.0
JKNet82.7 ± 0.473.0 ± 0.577.9 ± 0.4
GAT83.0 ± 0.772.5± 0.779.0 ± 0.3
Unsupervised Approaches
Linear47.9 ± 0.449.3 ± 0.269.1 ± 0.3
DeepWalk70.7 ± 0.651.4 ± 0.574.3 ± 0.9
GAE71.5 ± 0.465.8 ±0.472.1 ± 0.5
VERSE72.5 ± 0.355.5± 0.4
DGI83.8 ± 0.572.0 ± 0.677.9 ± 0.3
GraphCL82.5 ± 0.173.1 ± 0.2
mvgrl86.8 ± 0.573.3 ± 0.580.1 ± 0.7
InfoGCL83.5 ± 0.373.5 ± 0.479.1 ± 0.2
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MethodMUTAGIMDB-BCOLLABCoraCiteseerPubmed
InfoGCL (w/o neg)91.0 ± 1.475.1± 0.580.2 ±1.078.6 ± 0.470.4 ± 0.677.4±0.7
InfoGCL (w/ neg)91.2 ± 1.375.1 ± 0.980.0± 1.383.5±0.373.5 ± 0.479.1±0.2
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0000000000000000000000000000000000000000..6518adfa8fa241108d453be488f9ea024eabf3b1 --- /dev/null +++ b/parse/train/S1X7nhsxl/S1X7nhsxl.md @@ -0,0 +1,262 @@ +# IMPROVING GENERATIVE ADVERSARIAL NETWORKS WITH DENOISING FEATURE MATCHING + +David Warde-Farley & Yoshua Bengio? +Montreal Institute for Learning Algorithms, $\star$ CIFAR Senior Fellow +Universite de Montr´ eal´ +Montreal, Quebec, Canada +{david.warde-farley,yoshua.bengio}@umontreal.ca + +# ABSTRACT + +We propose an augmented training procedure for generative adversarial networks designed to address shortcomings of the original by directing the generator towards probable configurations of abstract discriminator features. We estimate and track the distribution of these features, as computed from data, with a denoising auto-encoder, and use it to propose high-level targets for the generator. We combine this new loss with the original and evaluate the hybrid criterion on the task of unsupervised image synthesis from datasets comprising a diverse set of visual categories, noting a qualitative and quantitative improvement in the “objectness” of the resulting samples. + +# 1 INTRODUCTION + +Generative adversarial networks (Goodfellow et al., 2014a) (GANs) have become well known for their strength at realistic image synthesis. The objective function for the generative network is an implicit function of a learned discriminator network, estimated in parallel with the generator, which aims to tell apart real data from synthesized. Ideally, the discriminator learns to capture distinguishing features of real data, which the generator learns to imitate, and the process iterates until real data and synthesized data are indistinguishable. + +In practice, GANs are well known for being quite challenging to train effectively. The relative model capacities of the generator and discriminator must be carefully balanced in order for the generator to effectively learn. Compounding the problem is the lack of an unambiguous and computable convergence criterion. Nevertheless, particularly when trained on image collections from relatively narrow domains such as bedroom scenes (Yu et al., 2015) and human faces (Liu et al., 2015), GANs have been shown to produce very compelling results. + +For diverse image collections comprising a wider variety of the visual world, the results have generally been less impressive. For example, samples from models trained on ImageNet (Russakovsky et al., 2014) roughly match the local and global statistics of natural images but yield few recognizable objects. Recent work (Salimans et al., 2016) has sought to address this problem by training the discriminator in a semi-supervised fashion, granting the discriminator’s internal representations knowledge of the class structure of (some fraction of) the training data it is presented. This technique markedly increases sample quality, but is unsatisfying from the perspective of GANs as a tool for unsupervised learning. + +We propose to augment the generator’s training criterion with a second training objective which guides the generator towards samples more like those in the training set by explicitly modeling the data density in addition to the adversarial discriminator. Rather than deploy a second computationally expensive convolutional network for this task, the additional objective is computed in the space of features learned by the discriminator. In that space, we train a denoising auto-encoder, a family of models which is known to estimate the energy gradient of the data on which it is trained. We evaluate the denoising auto-encoder on samples drawn from the generator, and use the “denoised” features as targets – nearby feature configurations which are more likely than those of the generated sample, according to the distribution estimated by the denoiser. + +We show that this yields generators which consistently produce recognizable objects on the CIFAR10 dataset without the use of label information as in Salimans et al. (2016). The criterion appears to improve stability and possesses a degree of natural robustness to the well known “collapse” pathology. We further investigate the criterion’s performance on two larger and more diverse collections of images, and validate our qualitative observations quantitatively with the Inception score proposed in Salimans et al. (2016). + +# 2 BACKGROUND + +# 2.1 GENERATIVE ADVERSARIAL NETWORKS + +The generative adversarial networks paradigm (Goodfellow et al., 2014a) estimates generative samplers by means of a training procedure which pits a generator $G$ against a discriminator $D$ . $D$ is trained to tell apart training examples from samples produced by $G$ , while $G$ is trained to increase the probability of its samples being incorrectly classified as data. In the original formulation, the training procedure defines a continuous minimax game + +$$ +\underset { G } { \arg \operatorname* { m i n } } \ \underset { D } { \arg \operatorname* { m a x } } \mathbb { E } _ { \mathbf { x } \sim \mathcal { D } } \log D ( \mathbf { x } ) + \mathbb { E } _ { \mathbf { z } \sim p ( \mathbf { z } ) } \log \left( 1 - D \left( G ( \mathbf { z } ) \right) \right) +$$ + +where $\mathcal { D }$ is a data distribution on $\mathbb { R } ^ { n }$ , $D$ is a function that maps $\mathbb { R } ^ { n }$ to the unit interval, and $G$ is a function that maps a noise vector $\mathbf { z } \in \mathbb { R } ^ { m }$ , drawn from a simple distribution $p ( \mathbf { z } )$ , to the ambient space of the training data, $\mathbb { R } ^ { n }$ . The idealized algorithm can be shown to converge and to minimize the Jensen-Shannon divergence between the data generating distribution and the distribution parameterized by $G$ . + +Goodfellow et al. (2014a) found that in practice, minimizing (1) with respect to the parameters of $G$ proved difficult, and elected instead to optimize an alternate objective, + +$$ +\underset { G } { \arg \operatorname* { m a x } } \mathbb { E } _ { \mathbf { z } \sim p ( \mathbf { z } ) } \log D \left( G ( z ) \right) +$$ + +at the same time as $D$ is optimized as above. $\log D ( G ( \mathbf { z } ) )$ yields more favourably scaled persample gradients for $G$ when $D$ confidently identifies a sample as counterfeit, avoiding the vanishing gradients arising in that case with the $- \log ( 1 - D ( G ( \mathbf { z } ) ) )$ objective. + +Subsequent authors have investigated applications and extensions of GANs; for a review of this body of literature, see Warde-Farley $\&$ Goodfellow (2016). Of particular note for our purposes is Radford et al. (2015), who provide a set of general guidelines for the successful training of generative adversarial networks, and Salimans et al. (2016), who build upon these techniques with a number of useful heuristics and explore a variant in which the discriminator $D$ is trained to correctly classify labeled training data, resulting in gradients with respect to the discriminator evidently containing a great deal of information relevant to generating “object-like” samples. + +# 2.2 CHALLENGES AND LIMITATIONS OF GANS + +While Goodfellow et al. (2014a) provides a theoretical basis for the GAN criterion, the theory relies on certain assumptions that are not satisfied in practice. Proofs demonstrate convergence of the GAN criterion in the unconstrained space of arbitrary functions; in practice, finitely parameterized families of functions such as neural networks are employed. As a consequence, the “inner loop” of the idealized algorithm – maximizing (1) with respect to (the parameters of) $D$ , is infeasible to perform exactly, and in practice only one or a few gradient steps stand in for this maximization. This results in a de facto criterion for $G$ which minimizes a lower bound on the correct objective (Goodfellow, 2014). + +A commonly observed failure mode is that of full or partial collapse, where $G$ maps a large fraction of probable regions under $p ( \mathbf { z } )$ to only a few, low-volume regions of $\mathbb { R } ^ { n }$ ; in the case of images, this manifests as the appearance of many near-duplicate images in independent draws from $G$ , as well as a lower diversity of samples and modes than what is observed in the dataset. As $G$ and $D$ are typically trained via mini-batch stochastic gradient descent, several authors have proposed heuristics that penalize such duplication within each mini-batch (Salimans et al., 2016; Zhao et al., 2016). + +GANs represent a departure from traditional probabilistic models based on maximum likelihood and its approximations in that they parameterize a sampler directly and lack a closed form for the likelihood. This makes objective, quantitative evaluation difficult. While previous results in the literature have reported approximate likelihoods based on Parzen window estimates, Theis et al. (2015) has convincingly argued that these estimates can be quite misleading for high-dimensional data. In this work, we adopt the Inception score proposed by Salimans et al. (2016), which uses a reference Inception convolutional neural network (Szegedy et al., 2015) to compute + +$$ +I ( \{ \mathbf { x } \} _ { 1 } ^ { N } ) = \exp { ( \mathbb { E } \left[ D _ { K L } ( p ( y | \mathbf { x } ) \| p ( y ) \right] ) ) } +$$ + +where $p ( y | \mathbf { x } )$ is provided by the output of the Inception network and $\begin{array} { r } { p ( y ) = \int _ { \mathbf { x } } p ( \mathbf { x } ) p ( y | \mathbf { x } ) d \mathbf { x } \approx } \end{array}$ $\begin{array} { r } { \frac { 1 } { N } \sum p ( y | \mathbf { x } _ { i } ) } \end{array}$ . Note that this score can be made larger by a low-entropy per-sample posterior (i.e. the Inception network classifies a given sample with greater certainty) as well as a higher entropy aggregate posterior (i.e. the Inception network identifies a wide variety of classes among the samples presented to it). Salimans et al. (2016) found this score correlated well with human evaluations of samplers trained on CIFAR-10; we therefore employ the Inception score here as a quantitative measure of visual fidelity of the samples, following the previous work’s protocol of evaluating the average Inception score over 10 independent groups of 5,000 samples each. Error estimates correspond to standard deviations, in keeping with previously reported results. + +# 3 IMPROVING UNSUPERVISED GAN TRAINING ON DIVERSE DATASETS + +In this work, we focus on the apparent difficulty of training GANs to produce “object-like” samples when trained on diverse collections of natural images. While Salimans et al. (2016) make progress on this problem by employing labeled data and training the discriminator, here we aim to make progress on the unsupervised case. Nevertheless, our methods would be readily applicable to supervised, semi-supervised or (with slight modifications) conditional setting. + +We begin from the slightly subtle observation that in realistic manifestations of the GAN training procedure, the discriminator’s (negative) gradient with respect to a sample points in a direction of (infinitesimal) local improvement with respect to the discriminator’s estimate of the sample being data; it does not necessarily point in the direction of a draw from the data distribution. Indeed, the literature is replete with instances of gradient descent with respect to the input of a classification model, particularly wide-domain natural image classifiers, producing ghostly approximations to a particular class exemplar (Le et al., 2012; Erhan et al., 2009; Yosinski et al., 2015) when this procedure is carried out without additional guidance, to say nothing of the problems posed by adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2014b) and fooling examples (Nguyen et al., 2015). + +While the gradient of the loss function defined by the discriminator may be a source of information mostly relevant to very local improvements, the discriminator itself is a potentially valuable source of compact descriptors of the training data. Many authors have noted the remarkable versatility of high-level features learned by convolutional networks (Donahue et al., 2014; Yosinski et al., 2014) and the degree to which high-level semantics can be reconstructed from even the deepest layers of a network (Dosovitskiy & Brox, 2016). Although non-stationary, the distribution of the highlevel activations of the discriminator when evaluated on data is ripe for exploitation as an additional source of knowledge about salient aspects of the data distribution. + +We propose in this work to track this distribution with a denoising auto-encoder $r ( \cdot )$ trained on the discriminator’s hidden states when evaluated on training data. Alain & Bengio (2014) showed that a denoising auto-encoder trained on data from a distribution $q ( \mathbf { h } )$ estimates via $r ( \mathbf { h } ) - \mathbf { h }$ the gradient of the true log-dtransformed training data $\frac { \partial \log q ( \mathbf { h } ) } { \partial \mathbf { h } }$ ce, if, then isingwith er on theindicates $\mathbf { h } = \Phi ( \mathbf { x } )$ $\mathbf { x } \sim \mathcal { D }$ $r ( \Phi ( \mathbf { x } ^ { \prime } ) ) - \Phi ( \mathbf { x } ^ { \prime } )$ $\mathbf { x } ^ { \prime } = G ( \mathbf { z } )$ in which direction $\mathbf { x } ^ { \prime }$ should be changed in order to make $\mathbf { h } = \Phi ( \mathbf { x } ^ { \prime } )$ more like those features seen with the data. Minimizing $| | r ( \Phi ( \mathbf { x } ^ { \prime } ) ) - \Phi ( \mathbf { x } ^ { \prime } ) | | ^ { 2 }$ with respect to $\mathbf { x } ^ { \prime }$ would thus push $\mathbf { x } ^ { \prime }$ towards higher probability configurations according to the data distribution in the feature space $\Phi ( \mathbf { x } )$ . We thus evaluate the discriminator features $\Phi ( \mathbf { x } )$ , and the denoising auto-encoder, on samples from the generator, and treat the denoiser’s output reconstruction as a fixed target for the generator. We refer to this procedure as denoising feature matching, and employ it as a learning signal for the generator in addition to the traditional GAN generator objective. + +Formally, let $G$ be the generator parameterized by $\theta _ { G }$ , and $D = d { \circ } \Phi$ be our discriminator composing feature extractor $\Phi ( \cdot ) \ : \mathbb { R } ^ { n } \ \bar { \mathbb { R } } ^ { k }$ and a classifier $d ( \cdot ) : \mathbb { R } ^ { k } \to [ 0 , 1 ]$ . Let $C ( \cdot ) : \mathbb { R } ^ { k } \to \mathbb { R } ^ { \bar { k } }$ be a corruption function to be applied at the input of the denoising auto-encoder when it is trained to denoise. The parameters of the discriminator $D$ , comprising the parameters of both $d$ and $\Phi$ , is trained as in Goodfellow et al. (2014a), while the generator is trained according to + +$$ +\underset { \theta _ { G } } { \arg \operatorname* { m i n } } \mathbb { E } _ { \mathbf { z } \sim p ( \mathbf { z } ) } \left[ \lambda _ { \mathrm { d e n o i s e } } \| \Phi ( G ( \mathbf { z } ) ) - r ( \Phi ( G ( \mathbf { z } ) ) ) \| | ^ { 2 } - \lambda _ { \mathrm { a d v } } \log D ( G ( z ) ) \right] +$$ + +where $r ( G ( \mathbf { z } ) )$ is treated as constant with respect to gradient computations. Simultaneously, the denoiser $r ( \cdot )$ is trained according to the objective + +$$ +\underset { \theta _ { r } } { \arg \operatorname* { m i n } } \mathop { { \mathbb { E } } } _ { { \mathbf { x } } \sim { \mathcal { D } } } \| \Phi ( { \mathbf { x } } ) - r ( C ( \Phi ( { \mathbf { x } } ) ) ) \| ^ { 2 } +$$ + +# 3.1 EFFECT OF $\Phi$ + +The theory surrounding denoising auto-encoders applies when estimating a denoising function from a data distribution $p ( v e c x )$ . Here, we propose to estimate the denoising auto-encoder in the space of discriminator features, giving rise to a distribution $q ( \Phi ( \mathbf { x } ) )$ . A natural question is what effect this has on the gradient being backpropagated. This is difficult to analyze in general, as for most choices the mapping $\Phi$ will not be invertible, though it is instructive to examine the invertible case. Assuming an invertible $\Phi : \mathbb { R } ^ { n } \mathbb { R } ^ { n }$ , let $\begin{array} { r } { J = \frac { \partial \Phi ( \mathbf { \bar { x } } ) } { \partial \mathbf { x } } } \end{array}$ be the Jacobian of $\Phi$ , and $q ( \Phi ( \mathbf { x } ) ) = p ( \mathbf { x } ) | J |$ . By the inverse function theorem, $J$ is also invertible (and is in fact the Jacobian of the inverse $\Phi ^ { - 1 }$ ). Applying the chain rule and re-arranging terms, taking advantage of the invertibility of $J$ , we arrive at a straightforward relationship between the score of $q$ and the score of $p$ : + +$$ +\begin{array} { l } { \displaystyle \frac { \partial \log q ( \Phi ( \mathbf { x } ) ) } { \partial \Phi ( \mathbf { x } ) } = \frac { \partial \log p ( \mathbf { \boldsymbol { p } } ( \mathbf { \mathbf { x } } ) \mid J \mid ] } { \partial \Phi ( \mathbf { x } ) } } \\ { \displaystyle ~ = \frac { \partial \log p ( \mathbf { \boldsymbol { x } } ) } { \partial \Phi ( \mathbf { x } ) } + \frac { \partial \log \left| \frac { \partial \Phi ( \mathbf { \boldsymbol { x } } ) } { \partial \mathbf { x } } \right| } { \partial \Phi ( \mathbf { x } ) } } \\ { \displaystyle ~ = \left( \frac { \partial \log p ( \mathbf { \boldsymbol { x } } ) } { \partial \mathbf { x } } + \frac { \partial \log \left| J \right| } { \partial \mathbf { x } } \right) J ^ { - 1 } } \end{array} +$$ + +where + +$$ +{ \frac { \partial \log | J | } { \partial x _ { k } } } = \operatorname { T r } \left( J ^ { - 1 } { \frac { d J } { d x _ { k } } } \right) +$$ + +and $\scriptstyle { \frac { d J } { d x _ { k } } }$ is a matrix of scalar derivatives of elements of $\mathbf { J }$ with respect to $x _ { k }$ . Thus, we see that the gradient backpropagated to the generator in an ideal setting is the gradient of the data distribution $p ( \mathbf { x } )$ along with an additive term which accounts for the changes in the rate of volume expansion/contraction in $\Phi$ locally around $\mathbf { x }$ . In practice, $\Phi$ is not invertible, but the added benefit of the denoiser-targeted gradient appears to reduce underfitting to the modes of $p$ in the generator, irrespective of any distortions $\Phi$ may introduce. + +# 4 RELATED WORK + +Denoising feature matching was originally inspired by feature matching introduced by Salimans et al. (2016) as an alternative training criterion for GAN generators, namely (in our notation) + +$$ +\underset { \theta _ { G } } { \arg \operatorname* { m i n } } \left. \left. \mathbb { E } _ { \mathbf { x } \sim \mathcal { D } } \left[ \Phi ( \mathbf { x } ) \right] - \mathbb { E } _ { \mathbf { z } \sim p ( z ) } \left[ \Phi ( G ( \mathbf { z } ) ) \right] \right. \right. ^ { 2 } +$$ + +Feature matching is equivalent to linear maximum mean discrepancy (Gretton et al., 2006), employing linear first moment matching in the space of discriminator features $\Phi ( \cdot )$ rather than the more familiar kernelized formulation. When performed on features in the penultimate layer, Salimans et al. (2016) found that the feature matching criterion was useful for the purpose of improving results on semi-supervised classification, using classification of samples from the generator as a sophisticated form of data augmentation. Feature matching was, however, less successful at producing samples with high visual fidelity. This is somewhat unsurprising given that the criterion is insensitive to higher-order statistics of the respective feature distributions. Indeed, a degenerate $G$ which deterministically reproduces a single sample $\hat { \mathbf { m } }$ such that $\Phi ( \hat { \mathbf { m } } ) = \mathbb { E } _ { \mathbf { x } \in \mathcal { D } } \Phi ( \mathbf { x } )$ trivially minimizes (10); in practice the joint training dynamics of $D$ and $G$ do not appear to yield such degenerate solutions. + +Rather than aiming to merely reduce linear separability between data and samples in the feature space defined by $\Phi ( \cdot )$ , denoising feature matching selects a more probable (according to the feature distribution implied by the data, as captured by the denoiser) feature space target for each sample produced by $G$ and regresses $G$ towards it. While an early loss of entropy in $G$ could result in the generator locking on to one or a few attractors in the denoiser’s energy landscape, we observe that this does not happen when used in conjunction with the traditional GAN objective, and in fact that the combination of the two objectives is notably robust to the collapses often observed in GAN training, even without taking additional measures to prevent them. + +This work also draws inspiration from Alain & Bengio (2014), which showed that a suitably trained denoiser learns an operator which locally maps a sample towards regions of high probability under the data distribution. They further showed that a suitably trained1 reconstruction function $r ( \cdot )$ behaves such that + +$$ +r ( \mathbf { x } ) - \mathbf { x } \propto \frac { \partial \log p ( \mathbf { x } ) } { \partial \mathbf { x } } +$$ + +That is, $r ( \mathbf { x } ) - \mathbf { x }$ estimates the score of the data generating distribution, up to a multiplicative constant. Our use of denoising auto-encoders necessarily departs from idealized conditions in that the denoiser is estimated online from an ever-changing distribution of features. + +Several approaches to GAN-like models have cast the problem in terms of learning an energy function. Kim & Bengio (2016) extends GANs by modeling the data distribution simultaneously with an energy function parameterized by a deep neural network (playing the role of the discriminator) and the traditional generator, carrying out learning with a learning rule resembling that of the Boltzmann machine (Ackley et al., 1985), where the “negative phase” gradient is estimated from samples from the generator. The energy-based GAN formulation of Zhao et al. (2016) resembles our work in their use of an auto-encoder which is trained to faithfully reconstruct (in our case, a corrupted, function of) the training data. The energy-based GAN replaces the discriminator with an auto-encoder, which is trained to assign low energy $L _ { 2 }$ reconstruction error) to training data and higher energy to samples from $G$ . To discourage generator collapses, a “pull-away term” penalizes the normalized dot product in a feature space defined by the auto-encoder’s internal representation. In this work, we preserve the discriminator, trained in the usual discriminative fashion, and in fact preserve the traditional generator loss, instead augmenting it with a source of complementary information provided by targets obtained from the denoiser. The energy-based GAN can be viewed as training the generator to seek fixed points of the autoencoding function (i.e. by backpropagating through the decoder and encoder in order to decrease reconstruction error), whereas we treat the output of $r ( \cdot )$ as constant with respect to the optimization as in Lee et al. (2015). That is to say, rather than using backpropagation to steer the dynamics of the autoencoder, we instead employ our denoising autoencoder to augment the gradient information obtained by ordinary backpropagation. + +Closest to our own approach, concurrent work on model-based super-resolution by Sønderby et al. (2016) trains a denoising auto-encoder on high-resolution ground truth and evaluates it on synthesized super-resolution images, using the difference between the original synthesized image and the denoiser’s output as an additional training signal for refining the output of the super-resolution network. Both Sønderby et al. (2016) and our own work are motivated by the results of Alain & Bengio (2014) discussed above. Aside from addressing a different application area, our denoiser is learned on-the-fly from a high-level feature representation which is itself learned. + +# 5 EXPERIMENTS + +We evaluate denoising feature matching on learning synthesis models from three datasets of increasing diversity and size: CIFAR-10, STL-10, and ImageNet. Although several authors have described GAN-based image synthesis models operating at $1 2 8 \times 1 2 8$ (Salimans et al., 2016; Zhao et al., 2016) and $2 5 6 \times 2 5 6$ (Zhao et al., 2016) resolution, we carry out our investigations at relatively low resolutions, both for computational ease and because we believe that the problem of unconditional modeling of diverse image collections is not well solved even at low resolutions; making progress in this regime is likely to yield insights that apply to the higher-resolution case. + +In all experiments, we employ isotropic Gaussian corruption noise with $\sigma = 1$ . Although we experimented with annealing $\sigma$ towards 0 (as also performed in Sønderby et al. (2016)), an annealing schedule which consistently outperformed fixed noise remained elusive. We experimented with convolutional denoisers, but our best results to date were obtained with deep, fully-connected denoisers using the ReLU nonlinearity on the penultimate layer of the discriminator. The number of hidden units was fixed to the same value in all denoiser layers, and the procedure is apparently robust to this hyperparameter choice, as long as it is greater than or equal to the input dimensionality. + +Our generator and discriminator architectures follow the methods outlined in Radford et al. (2015). Accordingly, batch normalization (Ioffe & Szegedy, 2015) was used in the generator and discriminator in the same manner as Radford et al. (2015), and in all layers of the denoiser except the output layer. In particular, as in Radford et al. (2015), we separately batch normalize data and generator samples for the discriminator and denoiser with respect to each source’s statistics. We calculate updates with respect to all losses with the parameters of all three networks fixed, and update all parameters simultaneously. + +All networks were trained with the Adam optimizer Kingma & Ba (2014) with a learning rate of $1 0 ^ { - 4 }$ and $\beta _ { 1 } = 0 . 5$ . The Adam optimizer is scale invariant, and so it suffices to e.g. tune $\lambda _ { \mathrm { d e n o i s e } }$ and fix $\lambda _ { \mathrm { a d v } }$ to 1. In our experiments, we set $\lambda _ { d e n o i s e }$ to $0 . 0 3 / n _ { h }$ , where $n _ { h }$ is the number of discriminator hidden units fed as input to the denoiser; this division decouples the scale of the first term of (4) from the dimensionality of the representation used, reducing the need to adjust this hyperparameter simply because we altered the architecture of the discriminator. + +# 5.1 CIFAR-10 + +CIFAR-10 (Krizhevsky & Hinton, 2009) is a small, well-studied dataset consisting of $5 0 , 0 0 0 3 2 \times 3 2$ pixel RGB training images and 10,000 test images from 10 classes: airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck. + +Samples from our model trained on CIFAR-10 are shown in Figure 1, and Inception scores for several methods, including those reported in Salimans et al. (2016) and scores computed from samples generated from a model presented in Dumoulin et al. (2016), are presented in Table 1. We achieve a mean Inception score of 7.72, falling slightly short of Salimans et al. (2016), which employed a supervised discriminator network (the same work reports a score of $4 . 3 6 \pm . 0 4$ when labels are omitted from their training procedure). Qualitatively, the samples include recognizable cars, boats and various animals. The best performing generator network consisted of the $3 2 \times 3 2$ ImageNet architecture from Radford et al. (2015) with half the number of parameters at each layer, and less than $40 \%$ of the parameters of the CIFAR-10 generator presented in Salimans et al. (2016). + +$$ +\frac { \mathrm { R e a l ~ d a t a ^ { \star } ~ } } { 1 1 . 2 4 \pm . 1 2 } \left| \begin{array} { c } { { \mathrm { S e m i - s u p e r v i s e d } } } \\ { { \mathrm { I m p r o v e d ~ G A N ~ ( S a l i m a n s ~ } e t a l ) ^ { \star } } } \\ { { \mathrm { 8 . 0 9 \pm . 0 7 } } } \end{array} \right| \left| \begin{array} { c } { { \mathrm { U n s u p e r v i s e d } } } \\ { { \mathrm { A L I ~ ( D u m o u l i n ~ } e t a l ) ^ { \dagger } } } \\ { { 5 . 3 4 \pm 0 . 0 5 } } \end{array} \right| \left. \begin{array} { c } { { \mathrm { 0 u r s ~ } } } \\ { { \mathrm { 7 . 7 2 \pm 0 . 1 3 } } } \end{array} \right| +$$ + +Table 1: Inception scores for models of CIFAR-10. ? as reported in Salimans et al. (2016); semisupervised † computed from samples drawn using author-provided model parameters and implementation. + +![](images/2920d907092d5dbf8c4c4f921094af220f27cdd9f81d1d9481a0c8268052aabc.jpg) +Figure 1: Samples generated from a model trained with denoising feature matching on CIFAR10. + +# 5.2 STL-10 + +STL-10 (Coates et al., 2011) is a dataset consisting of a small labeled set and larger (100,000) unlabeled set of $9 6 \times 9 6$ RGB images. The unlabeled set is a subset of ImageNet that is more diverse than CIFAR-10 (or the labeled set of STL-10), but less diverse than full ImageNet. We downsample by a factor of 2 on each dimension and train our networks at $4 8 \times 4 8$ . Inception scores for our model and a baseline, consisting of the same architecture trained without denoising feature matching (both trained for 50 epochs), are shown in Table 2. Samples are displayed in Figure 2. + +Table 2: Inception scores for models of the unlabeled set of STL-10. + +
Real dataOursGANBaseline
26.08±.268.51 ± 0.137.84 ± .07
+ +# 5.3 IMAGENET + +The ImageNet database (Russakovsky et al., 2014) is a large-scale database of natural images. We train on the designated training set of the most widely used release, the 2012 ImageNet Large Scale Visual Recognition Challenge (ILSVRC2012), consisting of a highly unbalanced split among 1,000 object classes. We preprocess the dataset as rescaled central crops following the procedure of Krizhevsky et al. (2012), except at $3 2 \times 3 2$ resolution to facilitate comparison with Radford et al. (2015). + +ImageNet poses a particular challenge for unsupervised GANs due to its high level of diversity and class skew. With a generator and discriminator architecture identical to that used for the same dataset in Radford et al. (2015), we achieve a higher Inception score using denoising feature matching, using denoiser with 10 hidden layers of 2,048 rectified linear units each. Both fall far short of the score assigned to real data at this resolution; there is still plenty of room for improvement. Samples are displayed in Figure 3. + +# 6 DISCUSSION AND FUTURE DIRECTIONS + +We have shown that training a denoising model on high-level discriminator activations in a GAN, and using the denoiser to propose high-level feature targets for the generator, can usefully improve + +![](images/29920cf699d2fec30f17d655c1dfae487045c1563aaf102ee3b2620d354b1c93.jpg) +Figure 2: Samples from a model trained with denoising feature matching on the unlabeled portion of the STL-10 dataset. + +![](images/be1a200c0eb6f3b8c857de32f4dd2d30763756f36937a68393656c0630cafdd7.jpg) +Figure 3: Samples from our model of ILSVRC2012 at $3 2 \times 3 2$ resolution. + +
Real dataRadford et al*Ours
25.78 ± .478.83 ± 0.149.18± .13
+ +Table 3: Inception scores for models of ILSVRC 2012 at $3 2 \times 3 2$ resolution. ? computed from samples drawn using author-provided model parameters and implementation. + +GAN image models. Higher Inception scores, as well as visual inspection, suggest that the procedure captures class-specific features of the training data in a manner superior to the original adversarial objective alone. That being said, we do not believe we are yet making optimal use of the paradigm. The non-stationarity of the feature distribution on which the denoiser is trained could be limiting the ability of the denoiser to obtain a good fit, and the information backpropagated to the generator is always slightly stale. Steps to reduce this non-stationarity may be fruitful; we experimented briefly with historical averaging as explored in Salimans et al. (2016) but did not observe a clear benefit thus far. Structured denoisers, including denoisers that learn an energy function for multiple hidden layers at once, could conceivably aid in obtaining a better fit. Learning a partially stochastic transition operator rather than a deterministic denoiser could conceivably capture interesting multimodalities that are “blurred” by a unimodal denoising function. + +Our method is orthogonal and could conceivably be used in combination with several other GAN extensions. For example, methods incorporating an encoder component (Donahue et al., 2016; Dumoulin et al., 2016), various existing conditional architectures (Mirza & Osindero, 2014; Denton et al., 2015; Reed et al., 2016), or the semi-supervised variant employed in Salimans et al. (2016), could all be trained with an additional denoising feature matching objective. + +We have proposed a useful heuristic, but a better theoretical grounding regarding how GANs are trained in practice is a necessary direction for future work, including grounded criteria for assessing mode coverage and mass misassignment, and principled criteria for assessing convergence or performing early stopping. + +# ACKNOWLEDGMENTS + +We thank Ian Goodfellow, Laurent Dinh, Yaroslav Ganin and Kyle Kastner for helpful discussions. We thank Vincent Dumoulin and Ishmael Belghazi for making available code and model parameters used in comparison to ALI, as well as Alec Radford for making available the code and model parameters for his ImageNet model. We would like to thank Antonia Creswell and Hiroyuki Yamazaki for pointing out an error in the initial version of this manuscript, and anonymous reviewers for valuable feedback. 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Energy-based generative adversarial network. arXiv preprint arXiv:1609.03126, 2016. \ No newline at end of file diff --git a/parse/train/S1X7nhsxl/S1X7nhsxl_content_list.json b/parse/train/S1X7nhsxl/S1X7nhsxl_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..24f245ba53a11526ed9b180e069a6c8a456df937 --- /dev/null +++ b/parse/train/S1X7nhsxl/S1X7nhsxl_content_list.json @@ -0,0 +1,1354 @@ +[ + { + "type": "text", + "text": "IMPROVING GENERATIVE ADVERSARIAL NETWORKS WITH DENOISING FEATURE MATCHING ", + "text_level": 1, + "bbox": [ + 174, + 98, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "David Warde-Farley & Yoshua Bengio? \nMontreal Institute for Learning Algorithms, $\\star$ CIFAR Senior Fellow \nUniversite de Montr´ eal´ \nMontreal, Quebec, Canada \n{david.warde-farley,yoshua.bengio}@umontreal.ca ", + "bbox": [ + 184, + 170, + 638, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 276, + 544, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose an augmented training procedure for generative adversarial networks designed to address shortcomings of the original by directing the generator towards probable configurations of abstract discriminator features. We estimate and track the distribution of these features, as computed from data, with a denoising auto-encoder, and use it to propose high-level targets for the generator. We combine this new loss with the original and evaluate the hybrid criterion on the task of unsupervised image synthesis from datasets comprising a diverse set of visual categories, noting a qualitative and quantitative improvement in the “objectness” of the resulting samples. ", + "bbox": [ + 233, + 309, + 764, + 435 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 465, + 336, + 482 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Generative adversarial networks (Goodfellow et al., 2014a) (GANs) have become well known for their strength at realistic image synthesis. The objective function for the generative network is an implicit function of a learned discriminator network, estimated in parallel with the generator, which aims to tell apart real data from synthesized. Ideally, the discriminator learns to capture distinguishing features of real data, which the generator learns to imitate, and the process iterates until real data and synthesized data are indistinguishable. ", + "bbox": [ + 173, + 498, + 825, + 583 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In practice, GANs are well known for being quite challenging to train effectively. The relative model capacities of the generator and discriminator must be carefully balanced in order for the generator to effectively learn. Compounding the problem is the lack of an unambiguous and computable convergence criterion. Nevertheless, particularly when trained on image collections from relatively narrow domains such as bedroom scenes (Yu et al., 2015) and human faces (Liu et al., 2015), GANs have been shown to produce very compelling results. ", + "bbox": [ + 174, + 589, + 825, + 674 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "For diverse image collections comprising a wider variety of the visual world, the results have generally been less impressive. For example, samples from models trained on ImageNet (Russakovsky et al., 2014) roughly match the local and global statistics of natural images but yield few recognizable objects. Recent work (Salimans et al., 2016) has sought to address this problem by training the discriminator in a semi-supervised fashion, granting the discriminator’s internal representations knowledge of the class structure of (some fraction of) the training data it is presented. This technique markedly increases sample quality, but is unsatisfying from the perspective of GANs as a tool for unsupervised learning. ", + "bbox": [ + 173, + 680, + 825, + 791 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose to augment the generator’s training criterion with a second training objective which guides the generator towards samples more like those in the training set by explicitly modeling the data density in addition to the adversarial discriminator. Rather than deploy a second computationally expensive convolutional network for this task, the additional objective is computed in the space of features learned by the discriminator. In that space, we train a denoising auto-encoder, a family of models which is known to estimate the energy gradient of the data on which it is trained. We evaluate the denoising auto-encoder on samples drawn from the generator, and use the “denoised” features as targets – nearby feature configurations which are more likely than those of the generated sample, according to the distribution estimated by the denoiser. ", + "bbox": [ + 173, + 799, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We show that this yields generators which consistently produce recognizable objects on the CIFAR10 dataset without the use of label information as in Salimans et al. (2016). The criterion appears to improve stability and possesses a degree of natural robustness to the well known “collapse” pathology. We further investigate the criterion’s performance on two larger and more diverse collections of images, and validate our qualitative observations quantitatively with the Inception score proposed in Salimans et al. (2016). ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 209, + 326, + 226 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 GENERATIVE ADVERSARIAL NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 242, + 493, + 257 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The generative adversarial networks paradigm (Goodfellow et al., 2014a) estimates generative samplers by means of a training procedure which pits a generator $G$ against a discriminator $D$ . $D$ is trained to tell apart training examples from samples produced by $G$ , while $G$ is trained to increase the probability of its samples being incorrectly classified as data. In the original formulation, the training procedure defines a continuous minimax game ", + "bbox": [ + 174, + 268, + 825, + 339 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/4bae875f072788494d334e95f81a93789efac2bd43a37360555bc3df3b0f4562.jpg", + "text": "$$\n\\underset { G } { \\arg \\operatorname* { m i n } } \\ \\underset { D } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { \\mathbf { x } \\sim \\mathcal { D } } \\log D ( \\mathbf { x } ) + \\mathbb { E } _ { \\mathbf { z } \\sim p ( \\mathbf { z } ) } \\log \\left( 1 - D \\left( G ( \\mathbf { z } ) \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 281, + 347, + 715, + 373 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mathcal { D }$ is a data distribution on $\\mathbb { R } ^ { n }$ , $D$ is a function that maps $\\mathbb { R } ^ { n }$ to the unit interval, and $G$ is a function that maps a noise vector $\\mathbf { z } \\in \\mathbb { R } ^ { m }$ , drawn from a simple distribution $p ( \\mathbf { z } )$ , to the ambient space of the training data, $\\mathbb { R } ^ { n }$ . The idealized algorithm can be shown to converge and to minimize the Jensen-Shannon divergence between the data generating distribution and the distribution parameterized by $G$ . ", + "bbox": [ + 174, + 381, + 825, + 452 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Goodfellow et al. (2014a) found that in practice, minimizing (1) with respect to the parameters of $G$ proved difficult, and elected instead to optimize an alternate objective, ", + "bbox": [ + 176, + 458, + 820, + 486 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/9d439296646e30dd2efcc2fbc50bc1a5ca743a51ef1bd8486d61cb34072d5669.jpg", + "text": "$$\n\\underset { G } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { \\mathbf { z } \\sim p ( \\mathbf { z } ) } \\log D \\left( G ( z ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 395, + 494, + 601, + 520 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "at the same time as $D$ is optimized as above. $\\log D ( G ( \\mathbf { z } ) )$ yields more favourably scaled persample gradients for $G$ when $D$ confidently identifies a sample as counterfeit, avoiding the vanishing gradients arising in that case with the $- \\log ( 1 - D ( G ( \\mathbf { z } ) ) )$ objective. ", + "bbox": [ + 174, + 529, + 825, + 571 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Subsequent authors have investigated applications and extensions of GANs; for a review of this body of literature, see Warde-Farley $\\&$ Goodfellow (2016). Of particular note for our purposes is Radford et al. (2015), who provide a set of general guidelines for the successful training of generative adversarial networks, and Salimans et al. (2016), who build upon these techniques with a number of useful heuristics and explore a variant in which the discriminator $D$ is trained to correctly classify labeled training data, resulting in gradients with respect to the discriminator evidently containing a great deal of information relevant to generating “object-like” samples. ", + "bbox": [ + 173, + 578, + 825, + 676 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 CHALLENGES AND LIMITATIONS OF GANS ", + "text_level": 1, + "bbox": [ + 174, + 695, + 514, + 709 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While Goodfellow et al. (2014a) provides a theoretical basis for the GAN criterion, the theory relies on certain assumptions that are not satisfied in practice. Proofs demonstrate convergence of the GAN criterion in the unconstrained space of arbitrary functions; in practice, finitely parameterized families of functions such as neural networks are employed. As a consequence, the “inner loop” of the idealized algorithm – maximizing (1) with respect to (the parameters of) $D$ , is infeasible to perform exactly, and in practice only one or a few gradient steps stand in for this maximization. This results in a de facto criterion for $G$ which minimizes a lower bound on the correct objective (Goodfellow, 2014). ", + "bbox": [ + 173, + 720, + 825, + 833 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A commonly observed failure mode is that of full or partial collapse, where $G$ maps a large fraction of probable regions under $p ( \\mathbf { z } )$ to only a few, low-volume regions of $\\mathbb { R } ^ { n }$ ; in the case of images, this manifests as the appearance of many near-duplicate images in independent draws from $G$ , as well as a lower diversity of samples and modes than what is observed in the dataset. As $G$ and $D$ are typically trained via mini-batch stochastic gradient descent, several authors have proposed heuristics that penalize such duplication within each mini-batch (Salimans et al., 2016; Zhao et al., 2016). ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "GANs represent a departure from traditional probabilistic models based on maximum likelihood and its approximations in that they parameterize a sampler directly and lack a closed form for the likelihood. This makes objective, quantitative evaluation difficult. While previous results in the literature have reported approximate likelihoods based on Parzen window estimates, Theis et al. (2015) has convincingly argued that these estimates can be quite misleading for high-dimensional data. In this work, we adopt the Inception score proposed by Salimans et al. (2016), which uses a reference Inception convolutional neural network (Szegedy et al., 2015) to compute ", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/ee40ce6c4f641053150c9044c4c60954540ba3c3751f8800cd2121b953dddd2c.jpg", + "text": "$$\nI ( \\{ \\mathbf { x } \\} _ { 1 } ^ { N } ) = \\exp { ( \\mathbb { E } \\left[ D _ { K L } ( p ( y | \\mathbf { x } ) \\| p ( y ) \\right] ) ) }\n$$", + "text_format": "latex", + "bbox": [ + 359, + 207, + 637, + 227 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $p ( y | \\mathbf { x } )$ is provided by the output of the Inception network and $\\begin{array} { r } { p ( y ) = \\int _ { \\mathbf { x } } p ( \\mathbf { x } ) p ( y | \\mathbf { x } ) d \\mathbf { x } \\approx } \\end{array}$ $\\begin{array} { r } { \\frac { 1 } { N } \\sum p ( y | \\mathbf { x } _ { i } ) } \\end{array}$ . Note that this score can be made larger by a low-entropy per-sample posterior (i.e. the Inception network classifies a given sample with greater certainty) as well as a higher entropy aggregate posterior (i.e. the Inception network identifies a wide variety of classes among the samples presented to it). Salimans et al. (2016) found this score correlated well with human evaluations of samplers trained on CIFAR-10; we therefore employ the Inception score here as a quantitative measure of visual fidelity of the samples, following the previous work’s protocol of evaluating the average Inception score over 10 independent groups of 5,000 samples each. Error estimates correspond to standard deviations, in keeping with previously reported results. ", + "bbox": [ + 174, + 233, + 825, + 362 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 IMPROVING UNSUPERVISED GAN TRAINING ON DIVERSE DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 381, + 782, + 397 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work, we focus on the apparent difficulty of training GANs to produce “object-like” samples when trained on diverse collections of natural images. While Salimans et al. (2016) make progress on this problem by employing labeled data and training the discriminator, here we aim to make progress on the unsupervised case. Nevertheless, our methods would be readily applicable to supervised, semi-supervised or (with slight modifications) conditional setting. ", + "bbox": [ + 174, + 412, + 825, + 483 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We begin from the slightly subtle observation that in realistic manifestations of the GAN training procedure, the discriminator’s (negative) gradient with respect to a sample points in a direction of (infinitesimal) local improvement with respect to the discriminator’s estimate of the sample being data; it does not necessarily point in the direction of a draw from the data distribution. Indeed, the literature is replete with instances of gradient descent with respect to the input of a classification model, particularly wide-domain natural image classifiers, producing ghostly approximations to a particular class exemplar (Le et al., 2012; Erhan et al., 2009; Yosinski et al., 2015) when this procedure is carried out without additional guidance, to say nothing of the problems posed by adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2014b) and fooling examples (Nguyen et al., 2015). ", + "bbox": [ + 174, + 489, + 825, + 628 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "While the gradient of the loss function defined by the discriminator may be a source of information mostly relevant to very local improvements, the discriminator itself is a potentially valuable source of compact descriptors of the training data. Many authors have noted the remarkable versatility of high-level features learned by convolutional networks (Donahue et al., 2014; Yosinski et al., 2014) and the degree to which high-level semantics can be reconstructed from even the deepest layers of a network (Dosovitskiy & Brox, 2016). Although non-stationary, the distribution of the highlevel activations of the discriminator when evaluated on data is ripe for exploitation as an additional source of knowledge about salient aspects of the data distribution. ", + "bbox": [ + 173, + 635, + 825, + 747 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We propose in this work to track this distribution with a denoising auto-encoder $r ( \\cdot )$ trained on the discriminator’s hidden states when evaluated on training data. Alain & Bengio (2014) showed that a denoising auto-encoder trained on data from a distribution $q ( \\mathbf { h } )$ estimates via $r ( \\mathbf { h } ) - \\mathbf { h }$ the gradient of the true log-dtransformed training data $\\frac { \\partial \\log q ( \\mathbf { h } ) } { \\partial \\mathbf { h } }$ ce, if, then isingwith er on theindicates $\\mathbf { h } = \\Phi ( \\mathbf { x } )$ $\\mathbf { x } \\sim \\mathcal { D }$ $r ( \\Phi ( \\mathbf { x } ^ { \\prime } ) ) - \\Phi ( \\mathbf { x } ^ { \\prime } )$ $\\mathbf { x } ^ { \\prime } = G ( \\mathbf { z } )$ in which direction $\\mathbf { x } ^ { \\prime }$ should be changed in order to make $\\mathbf { h } = \\Phi ( \\mathbf { x } ^ { \\prime } )$ more like those features seen with the data. Minimizing $| | r ( \\Phi ( \\mathbf { x } ^ { \\prime } ) ) - \\Phi ( \\mathbf { x } ^ { \\prime } ) | | ^ { 2 }$ with respect to $\\mathbf { x } ^ { \\prime }$ would thus push $\\mathbf { x } ^ { \\prime }$ towards higher probability configurations according to the data distribution in the feature space $\\Phi ( \\mathbf { x } )$ . We thus evaluate the discriminator features $\\Phi ( \\mathbf { x } )$ , and the denoising auto-encoder, on samples from the generator, and treat the denoiser’s output reconstruction as a fixed target for the generator. We refer to this procedure as denoising feature matching, and employ it as a learning signal for the generator in addition to the traditional GAN generator objective. ", + "bbox": [ + 173, + 753, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Formally, let $G$ be the generator parameterized by $\\theta _ { G }$ , and $D = d { \\circ } \\Phi$ be our discriminator composing feature extractor $\\Phi ( \\cdot ) \\ : \\mathbb { R } ^ { n } \\ \\bar { \\mathbb { R } } ^ { k }$ and a classifier $d ( \\cdot ) : \\mathbb { R } ^ { k } \\to [ 0 , 1 ]$ . Let $C ( \\cdot ) : \\mathbb { R } ^ { k } \\to \\mathbb { R } ^ { \\bar { k } }$ be a corruption function to be applied at the input of the denoising auto-encoder when it is trained to denoise. The parameters of the discriminator $D$ , comprising the parameters of both $d$ and $\\Phi$ , is trained as in Goodfellow et al. (2014a), while the generator is trained according to ", + "bbox": [ + 173, + 102, + 825, + 174 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9fb33ca769730051bdce5ac575ddc010ceaf5f6af105d6e45b004e40d243d7f9.jpg", + "text": "$$\n\\underset { \\theta _ { G } } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { \\mathbf { z } \\sim p ( \\mathbf { z } ) } \\left[ \\lambda _ { \\mathrm { d e n o i s e } } \\| \\Phi ( G ( \\mathbf { z } ) ) - r ( \\Phi ( G ( \\mathbf { z } ) ) ) \\| | ^ { 2 } - \\lambda _ { \\mathrm { a d v } } \\log D ( G ( z ) ) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 245, + 179, + 750, + 208 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $r ( G ( \\mathbf { z } ) )$ is treated as constant with respect to gradient computations. Simultaneously, the denoiser $r ( \\cdot )$ is trained according to the objective ", + "bbox": [ + 176, + 214, + 823, + 243 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/58e69619f4024842b8268a99064c7f064bdf50c5191a4b3292f99e470b719851.jpg", + "text": "$$\n\\underset { \\theta _ { r } } { \\arg \\operatorname* { m i n } } \\mathop { { \\mathbb { E } } } _ { { \\mathbf { x } } \\sim { \\mathcal { D } } } \\| \\Phi ( { \\mathbf { x } } ) - r ( C ( \\Phi ( { \\mathbf { x } } ) ) ) \\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 250, + 629, + 279 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 EFFECT OF $\\Phi$ ", + "text_level": 1, + "bbox": [ + 174, + 294, + 307, + 308 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The theory surrounding denoising auto-encoders applies when estimating a denoising function from a data distribution $p ( v e c x )$ . Here, we propose to estimate the denoising auto-encoder in the space of discriminator features, giving rise to a distribution $q ( \\Phi ( \\mathbf { x } ) )$ . A natural question is what effect this has on the gradient being backpropagated. This is difficult to analyze in general, as for most choices the mapping $\\Phi$ will not be invertible, though it is instructive to examine the invertible case. Assuming an invertible $\\Phi : \\mathbb { R } ^ { n } \\mathbb { R } ^ { n }$ , let $\\begin{array} { r } { J = \\frac { \\partial \\Phi ( \\mathbf { \\bar { x } } ) } { \\partial \\mathbf { x } } } \\end{array}$ be the Jacobian of $\\Phi$ , and $q ( \\Phi ( \\mathbf { x } ) ) = p ( \\mathbf { x } ) | J |$ . By the inverse function theorem, $J$ is also invertible (and is in fact the Jacobian of the inverse $\\Phi ^ { - 1 }$ ). Applying the chain rule and re-arranging terms, taking advantage of the invertibility of $J$ , we arrive at a straightforward relationship between the score of $q$ and the score of $p$ : ", + "bbox": [ + 173, + 319, + 826, + 452 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/45b54f0b1f4bfc178d0457b8d54649a988ec8d755d452103517aed884c257747.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { \\partial \\log q ( \\Phi ( \\mathbf { x } ) ) } { \\partial \\Phi ( \\mathbf { x } ) } = \\frac { \\partial \\log p ( \\mathbf { \\boldsymbol { p } } ( \\mathbf { \\mathbf { x } } ) \\mid J \\mid ] } { \\partial \\Phi ( \\mathbf { x } ) } } \\\\ { \\displaystyle ~ = \\frac { \\partial \\log p ( \\mathbf { \\boldsymbol { x } } ) } { \\partial \\Phi ( \\mathbf { x } ) } + \\frac { \\partial \\log \\left| \\frac { \\partial \\Phi ( \\mathbf { \\boldsymbol { x } } ) } { \\partial \\mathbf { x } } \\right| } { \\partial \\Phi ( \\mathbf { x } ) } } \\\\ { \\displaystyle ~ = \\left( \\frac { \\partial \\log p ( \\mathbf { \\boldsymbol { x } } ) } { \\partial \\mathbf { x } } + \\frac { \\partial \\log \\left| J \\right| } { \\partial \\mathbf { x } } \\right) J ^ { - 1 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 336, + 473, + 661, + 592 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 174, + 595, + 217, + 609 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/761c94ce4ea062d65fbead458c5b7ce1def1e2107844bd595dde46564edf0bd5.jpg", + "text": "$$\n{ \\frac { \\partial \\log | J | } { \\partial x _ { k } } } = \\operatorname { T r } \\left( J ^ { - 1 } { \\frac { d J } { d x _ { k } } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 405, + 614, + 591, + 650 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "and $\\scriptstyle { \\frac { d J } { d x _ { k } } }$ is a matrix of scalar derivatives of elements of $\\mathbf { J }$ with respect to $x _ { k }$ . Thus, we see that the gradient backpropagated to the generator in an ideal setting is the gradient of the data distribution $p ( \\mathbf { x } )$ along with an additive term which accounts for the changes in the rate of volume expansion/contraction in $\\Phi$ locally around $\\mathbf { x }$ . In practice, $\\Phi$ is not invertible, but the added benefit of the denoiser-targeted gradient appears to reduce underfitting to the modes of $p$ in the generator, irrespective of any distortions $\\Phi$ may introduce. ", + "bbox": [ + 173, + 656, + 825, + 743 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 763, + 341, + 780 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Denoising feature matching was originally inspired by feature matching introduced by Salimans et al. (2016) as an alternative training criterion for GAN generators, namely (in our notation) ", + "bbox": [ + 173, + 795, + 823, + 824 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d1bdaaf6fad4b07cb9daeaa3ba4c997736cb68bde4b7d2710f49d419afd9a5a3.jpg", + "text": "$$\n\\underset { \\theta _ { G } } { \\arg \\operatorname* { m i n } } \\left. \\left. \\mathbb { E } _ { \\mathbf { x } \\sim \\mathcal { D } } \\left[ \\Phi ( \\mathbf { x } ) \\right] - \\mathbb { E } _ { \\mathbf { z } \\sim p ( z ) } \\left[ \\Phi ( G ( \\mathbf { z } ) ) \\right] \\right. \\right. ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 334, + 830, + 661, + 861 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Feature matching is equivalent to linear maximum mean discrepancy (Gretton et al., 2006), employing linear first moment matching in the space of discriminator features $\\Phi ( \\cdot )$ rather than the more familiar kernelized formulation. When performed on features in the penultimate layer, Salimans et al. (2016) found that the feature matching criterion was useful for the purpose of improving results on semi-supervised classification, using classification of samples from the generator as a sophisticated form of data augmentation. Feature matching was, however, less successful at producing samples with high visual fidelity. This is somewhat unsurprising given that the criterion is insensitive to higher-order statistics of the respective feature distributions. Indeed, a degenerate $G$ which deterministically reproduces a single sample $\\hat { \\mathbf { m } }$ such that $\\Phi ( \\hat { \\mathbf { m } } ) = \\mathbb { E } _ { \\mathbf { x } \\in \\mathcal { D } } \\Phi ( \\mathbf { x } )$ trivially minimizes (10); in practice the joint training dynamics of $D$ and $G$ do not appear to yield such degenerate solutions. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 200 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Rather than aiming to merely reduce linear separability between data and samples in the feature space defined by $\\Phi ( \\cdot )$ , denoising feature matching selects a more probable (according to the feature distribution implied by the data, as captured by the denoiser) feature space target for each sample produced by $G$ and regresses $G$ towards it. While an early loss of entropy in $G$ could result in the generator locking on to one or a few attractors in the denoiser’s energy landscape, we observe that this does not happen when used in conjunction with the traditional GAN objective, and in fact that the combination of the two objectives is notably robust to the collapses often observed in GAN training, even without taking additional measures to prevent them. ", + "bbox": [ + 174, + 208, + 823, + 319 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This work also draws inspiration from Alain & Bengio (2014), which showed that a suitably trained denoiser learns an operator which locally maps a sample towards regions of high probability under the data distribution. They further showed that a suitably trained1 reconstruction function $r ( \\cdot )$ behaves such that ", + "bbox": [ + 174, + 325, + 823, + 382 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/acd946045e8303d1508b052cdcef4cb65cd32b7b0e084d0fbb155da7484d98b9.jpg", + "text": "$$\nr ( \\mathbf { x } ) - \\mathbf { x } \\propto \\frac { \\partial \\log p ( \\mathbf { x } ) } { \\partial \\mathbf { x } }\n$$", + "text_format": "latex", + "bbox": [ + 419, + 401, + 576, + 433 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "That is, $r ( \\mathbf { x } ) - \\mathbf { x }$ estimates the score of the data generating distribution, up to a multiplicative constant. Our use of denoising auto-encoders necessarily departs from idealized conditions in that the denoiser is estimated online from an ever-changing distribution of features. ", + "bbox": [ + 174, + 453, + 823, + 496 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Several approaches to GAN-like models have cast the problem in terms of learning an energy function. Kim & Bengio (2016) extends GANs by modeling the data distribution simultaneously with an energy function parameterized by a deep neural network (playing the role of the discriminator) and the traditional generator, carrying out learning with a learning rule resembling that of the Boltzmann machine (Ackley et al., 1985), where the “negative phase” gradient is estimated from samples from the generator. The energy-based GAN formulation of Zhao et al. (2016) resembles our work in their use of an auto-encoder which is trained to faithfully reconstruct (in our case, a corrupted, function of) the training data. The energy-based GAN replaces the discriminator with an auto-encoder, which is trained to assign low energy $L _ { 2 }$ reconstruction error) to training data and higher energy to samples from $G$ . To discourage generator collapses, a “pull-away term” penalizes the normalized dot product in a feature space defined by the auto-encoder’s internal representation. In this work, we preserve the discriminator, trained in the usual discriminative fashion, and in fact preserve the traditional generator loss, instead augmenting it with a source of complementary information provided by targets obtained from the denoiser. The energy-based GAN can be viewed as training the generator to seek fixed points of the autoencoding function (i.e. by backpropagating through the decoder and encoder in order to decrease reconstruction error), whereas we treat the output of $r ( \\cdot )$ as constant with respect to the optimization as in Lee et al. (2015). That is to say, rather than using backpropagation to steer the dynamics of the autoencoder, we instead employ our denoising autoencoder to augment the gradient information obtained by ordinary backpropagation. ", + "bbox": [ + 174, + 502, + 825, + 766 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Closest to our own approach, concurrent work on model-based super-resolution by Sønderby et al. (2016) trains a denoising auto-encoder on high-resolution ground truth and evaluates it on synthesized super-resolution images, using the difference between the original synthesized image and the denoiser’s output as an additional training signal for refining the output of the super-resolution network. Both Sønderby et al. (2016) and our own work are motivated by the results of Alain & Bengio (2014) discussed above. Aside from addressing a different application area, our denoiser is learned on-the-fly from a high-level feature representation which is itself learned. ", + "bbox": [ + 174, + 772, + 823, + 871 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 326, + 117 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate denoising feature matching on learning synthesis models from three datasets of increasing diversity and size: CIFAR-10, STL-10, and ImageNet. Although several authors have described GAN-based image synthesis models operating at $1 2 8 \\times 1 2 8$ (Salimans et al., 2016; Zhao et al., 2016) and $2 5 6 \\times 2 5 6$ (Zhao et al., 2016) resolution, we carry out our investigations at relatively low resolutions, both for computational ease and because we believe that the problem of unconditional modeling of diverse image collections is not well solved even at low resolutions; making progress in this regime is likely to yield insights that apply to the higher-resolution case. ", + "bbox": [ + 174, + 142, + 825, + 241 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In all experiments, we employ isotropic Gaussian corruption noise with $\\sigma = 1$ . Although we experimented with annealing $\\sigma$ towards 0 (as also performed in Sønderby et al. (2016)), an annealing schedule which consistently outperformed fixed noise remained elusive. We experimented with convolutional denoisers, but our best results to date were obtained with deep, fully-connected denoisers using the ReLU nonlinearity on the penultimate layer of the discriminator. The number of hidden units was fixed to the same value in all denoiser layers, and the procedure is apparently robust to this hyperparameter choice, as long as it is greater than or equal to the input dimensionality. ", + "bbox": [ + 174, + 247, + 825, + 344 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our generator and discriminator architectures follow the methods outlined in Radford et al. (2015). Accordingly, batch normalization (Ioffe & Szegedy, 2015) was used in the generator and discriminator in the same manner as Radford et al. (2015), and in all layers of the denoiser except the output layer. In particular, as in Radford et al. (2015), we separately batch normalize data and generator samples for the discriminator and denoiser with respect to each source’s statistics. We calculate updates with respect to all losses with the parameters of all three networks fixed, and update all parameters simultaneously. ", + "bbox": [ + 174, + 351, + 825, + 449 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "All networks were trained with the Adam optimizer Kingma & Ba (2014) with a learning rate of $1 0 ^ { - 4 }$ and $\\beta _ { 1 } = 0 . 5$ . The Adam optimizer is scale invariant, and so it suffices to e.g. tune $\\lambda _ { \\mathrm { d e n o i s e } }$ and fix $\\lambda _ { \\mathrm { a d v } }$ to 1. In our experiments, we set $\\lambda _ { d e n o i s e }$ to $0 . 0 3 / n _ { h }$ , where $n _ { h }$ is the number of discriminator hidden units fed as input to the denoiser; this division decouples the scale of the first term of (4) from the dimensionality of the representation used, reducing the need to adjust this hyperparameter simply because we altered the architecture of the discriminator. ", + "bbox": [ + 174, + 455, + 825, + 540 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 CIFAR-10 ", + "text_level": 1, + "bbox": [ + 174, + 571, + 290, + 585 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "CIFAR-10 (Krizhevsky & Hinton, 2009) is a small, well-studied dataset consisting of $5 0 , 0 0 0 3 2 \\times 3 2$ pixel RGB training images and 10,000 test images from 10 classes: airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck. ", + "bbox": [ + 174, + 603, + 823, + 646 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Samples from our model trained on CIFAR-10 are shown in Figure 1, and Inception scores for several methods, including those reported in Salimans et al. (2016) and scores computed from samples generated from a model presented in Dumoulin et al. (2016), are presented in Table 1. We achieve a mean Inception score of 7.72, falling slightly short of Salimans et al. (2016), which employed a supervised discriminator network (the same work reports a score of $4 . 3 6 \\pm . 0 4$ when labels are omitted from their training procedure). Qualitatively, the samples include recognizable cars, boats and various animals. The best performing generator network consisted of the $3 2 \\times 3 2$ ImageNet architecture from Radford et al. (2015) with half the number of parameters at each layer, and less than $40 \\%$ of the parameters of the CIFAR-10 generator presented in Salimans et al. (2016). ", + "bbox": [ + 174, + 652, + 825, + 779 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/8b94c8eb78cf54076859c0560bdd30258661f60b5deadf38471e18f76219f8d5.jpg", + "text": "$$\n\\frac { \\mathrm { R e a l ~ d a t a ^ { \\star } ~ } } { 1 1 . 2 4 \\pm . 1 2 } \\left| \\begin{array} { c } { { \\mathrm { S e m i - s u p e r v i s e d } } } \\\\ { { \\mathrm { I m p r o v e d ~ G A N ~ ( S a l i m a n s ~ } e t a l ) ^ { \\star } } } \\\\ { { \\mathrm { 8 . 0 9 \\pm . 0 7 } } } \\end{array} \\right| \\left| \\begin{array} { c } { { \\mathrm { U n s u p e r v i s e d } } } \\\\ { { \\mathrm { A L I ~ ( D u m o u l i n ~ } e t a l ) ^ { \\dagger } } } \\\\ { { 5 . 3 4 \\pm 0 . 0 5 } } \\end{array} \\right| \\left. \\begin{array} { c } { { \\mathrm { 0 u r s ~ } } } \\\\ { { \\mathrm { 7 . 7 2 \\pm 0 . 1 3 } } } \\end{array} \\right|\n$$", + "text_format": "latex", + "bbox": [ + 192, + 803, + 803, + 852 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "", + "table_caption": [ + "Table 1: Inception scores for models of CIFAR-10. ? as reported in Salimans et al. (2016); semisupervised † computed from samples drawn using author-provided model parameters and implementation. " + ], + "table_footnote": [], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/2920d907092d5dbf8c4c4f921094af220f27cdd9f81d1d9481a0c8268052aabc.jpg", + "image_caption": [ + "Figure 1: Samples generated from a model trained with denoising feature matching on CIFAR10. " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 823, + 387 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 STL-10 ", + "text_level": 1, + "bbox": [ + 174, + 441, + 271, + 457 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "STL-10 (Coates et al., 2011) is a dataset consisting of a small labeled set and larger (100,000) unlabeled set of $9 6 \\times 9 6$ RGB images. The unlabeled set is a subset of ImageNet that is more diverse than CIFAR-10 (or the labeled set of STL-10), but less diverse than full ImageNet. We downsample by a factor of 2 on each dimension and train our networks at $4 8 \\times 4 8$ . Inception scores for our model and a baseline, consisting of the same architecture trained without denoising feature matching (both trained for 50 epochs), are shown in Table 2. Samples are displayed in Figure 2. ", + "bbox": [ + 174, + 468, + 825, + 553 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/c38dc87ca763335a4636a98017817abffe08c22e36a3c5a8372d04f8f7a18e56.jpg", + "table_caption": [ + "Table 2: Inception scores for models of the unlabeled set of STL-10. " + ], + "table_footnote": [], + "table_body": "
Real dataOursGANBaseline
26.08±.268.51 ± 0.137.84 ± .07
", + "bbox": [ + 341, + 564, + 656, + 595 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 IMAGENET ", + "text_level": 1, + "bbox": [ + 174, + 643, + 292, + 657 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The ImageNet database (Russakovsky et al., 2014) is a large-scale database of natural images. We train on the designated training set of the most widely used release, the 2012 ImageNet Large Scale Visual Recognition Challenge (ILSVRC2012), consisting of a highly unbalanced split among 1,000 object classes. We preprocess the dataset as rescaled central crops following the procedure of Krizhevsky et al. (2012), except at $3 2 \\times 3 2$ resolution to facilitate comparison with Radford et al. (2015). ", + "bbox": [ + 173, + 669, + 825, + 753 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "ImageNet poses a particular challenge for unsupervised GANs due to its high level of diversity and class skew. With a generator and discriminator architecture identical to that used for the same dataset in Radford et al. (2015), we achieve a higher Inception score using denoising feature matching, using denoiser with 10 hidden layers of 2,048 rectified linear units each. Both fall far short of the score assigned to real data at this resolution; there is still plenty of room for improvement. Samples are displayed in Figure 3. ", + "bbox": [ + 174, + 760, + 825, + 844 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 DISCUSSION AND FUTURE DIRECTIONS ", + "text_level": 1, + "bbox": [ + 174, + 864, + 537, + 881 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We have shown that training a denoising model on high-level discriminator activations in a GAN, and using the denoiser to propose high-level feature targets for the generator, can usefully improve ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/29920cf699d2fec30f17d655c1dfae487045c1563aaf102ee3b2620d354b1c93.jpg", + "image_caption": [ + "Figure 2: Samples from a model trained with denoising feature matching on the unlabeled portion of the STL-10 dataset. " + ], + "image_footnote": [], + "bbox": [ + 174, + 152, + 823, + 420 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/be1a200c0eb6f3b8c857de32f4dd2d30763756f36937a68393656c0630cafdd7.jpg", + "image_caption": [ + "Figure 3: Samples from our model of ILSVRC2012 at $3 2 \\times 3 2$ resolution. " + ], + "image_footnote": [], + "bbox": [ + 174, + 571, + 825, + 840 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/76af805e34a814df92a346b3fe98cf47a9e0dbb7f53cc191e57e04cc6402a5bd.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Real dataRadford et al*Ours
25.78 ± .478.83 ± 0.149.18± .13
", + "bbox": [ + 348, + 99, + 648, + 131 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 3: Inception scores for models of ILSVRC 2012 at $3 2 \\times 3 2$ resolution. ? computed from samples drawn using author-provided model parameters and implementation. ", + "bbox": [ + 171, + 141, + 823, + 170 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "GAN image models. Higher Inception scores, as well as visual inspection, suggest that the procedure captures class-specific features of the training data in a manner superior to the original adversarial objective alone. That being said, we do not believe we are yet making optimal use of the paradigm. The non-stationarity of the feature distribution on which the denoiser is trained could be limiting the ability of the denoiser to obtain a good fit, and the information backpropagated to the generator is always slightly stale. Steps to reduce this non-stationarity may be fruitful; we experimented briefly with historical averaging as explored in Salimans et al. (2016) but did not observe a clear benefit thus far. Structured denoisers, including denoisers that learn an energy function for multiple hidden layers at once, could conceivably aid in obtaining a better fit. Learning a partially stochastic transition operator rather than a deterministic denoiser could conceivably capture interesting multimodalities that are “blurred” by a unimodal denoising function. ", + "bbox": [ + 174, + 195, + 825, + 349 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Our method is orthogonal and could conceivably be used in combination with several other GAN extensions. For example, methods incorporating an encoder component (Donahue et al., 2016; Dumoulin et al., 2016), various existing conditional architectures (Mirza & Osindero, 2014; Denton et al., 2015; Reed et al., 2016), or the semi-supervised variant employed in Salimans et al. (2016), could all be trained with an additional denoising feature matching objective. ", + "bbox": [ + 174, + 356, + 825, + 426 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have proposed a useful heuristic, but a better theoretical grounding regarding how GANs are trained in practice is a necessary direction for future work, including grounded criteria for assessing mode coverage and mass misassignment, and principled criteria for assessing convergence or performing early stopping. ", + "bbox": [ + 174, + 433, + 823, + 489 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 506, + 326, + 518 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We thank Ian Goodfellow, Laurent Dinh, Yaroslav Ganin and Kyle Kastner for helpful discussions. We thank Vincent Dumoulin and Ishmael Belghazi for making available code and model parameters used in comparison to ALI, as well as Alec Radford for making available the code and model parameters for his ImageNet model. We would like to thank Antonia Creswell and Hiroyuki Yamazaki for pointing out an error in the initial version of this manuscript, and anonymous reviewers for valuable feedback. We thank the University of Montreal and Compute Canada for the computational resources used for this investigation, as well as the authors of Theano (Al-Rfou et al., 2016), Blocks and Fuel (van Merrienboer et al., 2015). 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We com-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "score": 1.0, + "content": "bine this new loss with the original and evaluate the hybrid criterion on the task", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 312, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 142, + 312, + 470, + 325 + ], + "score": 1.0, + "content": "of unsupervised image synthesis from datasets comprising a diverse set of visual", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 323, + 469, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 469, + 335 + ], + "score": 1.0, + "content": "categories, noting a qualitative and quantitative improvement in the “objectness”", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 334, + 242, + 346 + ], + "spans": [ + { + "bbox": [ + 142, + 334, + 242, + 346 + ], + "score": 1.0, + "content": "of the resulting samples.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 246, + 470, + 346 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 369, + 206, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 368, + 208, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 208, + 385 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "score": 1.0, + "content": "Generative adversarial networks (Goodfellow et al., 2014a) (GANs) have become well known for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "their strength at realistic image synthesis. The objective function for the generative network is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "an implicit function of a learned discriminator network, estimated in parallel with the generator,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "which aims to tell apart real data from synthesized. Ideally, the discriminator learns to capture", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "distinguishing features of real data, which the generator learns to imitate, and the process iterates", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 450, + 336, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 336, + 462 + ], + "score": 1.0, + "content": "until real data and synthesized data are indistinguishable.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 396, + 505, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "In practice, GANs are well known for being quite challenging to train effectively. The relative model", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "capacities of the generator and discriminator must be carefully balanced in order for the generator", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "to effectively learn. Compounding the problem is the lack of an unambiguous and computable", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "convergence criterion. 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(2016), who build upon these techniques with a number of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 371, + 515 + ], + "score": 1.0, + "content": "useful heuristics and explore a variant in which the discriminator", + "type": "text" + }, + { + "bbox": [ + 372, + 503, + 381, + 512 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "is trained to correctly classify", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "labeled training data, resulting in gradients with respect to the discriminator evidently containing a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 525, + 388, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 388, + 537 + ], + "score": 1.0, + "content": "great deal of information relevant to generating “object-like” samples.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 458, + 506, + 537 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 551, + 315, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 316, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 316, + 564 + ], + "score": 1.0, + "content": "2.2 CHALLENGES AND LIMITATIONS OF GANS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "While Goodfellow et al. 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As", + "type": "text" + }, + { + "bbox": [ + 450, + 699, + 459, + 709 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 699, + 479, + 711 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 479, + 699, + 488, + 709 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "typically trained via mini-batch stochastic gradient descent, several authors have proposed heuristics", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 489, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 489, + 733 + ], + "score": 1.0, + "content": "that penalize such duplication within each mini-batch (Salimans et al., 2016; Zhao et al., 2016).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 665, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "GANs represent a departure from traditional probabilistic models based on maximum likelihood", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "and its approximations in that they parameterize a sampler directly and lack a closed form for the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "likelihood. This makes objective, quantitative evaluation difficult. While previous results in the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 128 + ], + "score": 1.0, + "content": "literature have reported approximate likelihoods based on Parzen window estimates, Theis et al.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "(2015) has convincingly argued that these estimates can be quite misleading for high-dimensional", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 507, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 507, + 150 + ], + "score": 1.0, + "content": "data. In this work, we adopt the Inception score proposed by Salimans et al. 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Note that this score can be made larger by a low-entropy per-sample posterior (i.e.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "score": 1.0, + "content": "the Inception network classifies a given sample with greater certainty) as well as a higher entropy", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "aggregate posterior (i.e. the Inception network identifies a wide variety of classes among the sam-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "ples presented to it). Salimans et al. 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Error estimates corre-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 400, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 400, + 287 + ], + "score": 1.0, + "content": "spond to standard deviations, in keeping with previously reported results.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 302, + 479, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 481, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 481, + 317 + ], + "score": 1.0, + "content": "3 IMPROVING UNSUPERVISED GAN TRAINING ON DIVERSE DATASETS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "In this work, we focus on the apparent difficulty of training GANs to produce “object-like” sam-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "ples when trained on diverse collections of natural images. While Salimans et al. (2016) make", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "progress on this problem by employing labeled data and training the discriminator, here we aim to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "make progress on the unsupervised case. Nevertheless, our methods would be readily applicable to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 370, + 420, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 420, + 385 + ], + "score": 1.0, + "content": "supervised, semi-supervised or (with slight modifications) conditional setting.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "score": 1.0, + "content": "We begin from the slightly subtle observation that in realistic manifestations of the GAN training", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 399, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 411 + ], + "score": 1.0, + "content": "procedure, the discriminator’s (negative) gradient with respect to a sample points in a direction of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "(infinitesimal) local improvement with respect to the discriminator’s estimate of the sample being", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 422, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 433 + ], + "score": 1.0, + "content": "data; it does not necessarily point in the direction of a draw from the data distribution. Indeed, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "literature is replete with instances of gradient descent with respect to the input of a classification", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "model, particularly wide-domain natural image classifiers, producing ghostly approximations to a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "particular class exemplar (Le et al., 2012; Erhan et al., 2009; Yosinski et al., 2015) when this proce-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "dure is carried out without additional guidance, to say nothing of the problems posed by adversarial", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "examples (Szegedy et al., 2013; Goodfellow et al., 2014b) and fooling examples (Nguyen et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 485, + 136, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 136, + 500 + ], + "score": 1.0, + "content": "2015).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "While the gradient of the loss function defined by the discriminator may be a source of information", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 515, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 526 + ], + "score": 1.0, + "content": "mostly relevant to very local improvements, the discriminator itself is a potentially valuable source", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "score": 1.0, + "content": "of compact descriptors of the training data. Many authors have noted the remarkable versatility of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "high-level features learned by convolutional networks (Donahue et al., 2014; Yosinski et al., 2014)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "and the degree to which high-level semantics can be reconstructed from even the deepest layers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "of a network (Dosovitskiy & Brox, 2016). Although non-stationary, the distribution of the high-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "level activations of the discriminator when evaluated on data is ripe for exploitation as an additional", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 581, + 372, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 372, + 592 + ], + "score": 1.0, + "content": "source of knowledge about salient aspects of the data distribution.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 597, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 442, + 609 + ], + "score": 1.0, + "content": "We propose in this work to track this distribution with a denoising auto-encoder", + "type": "text" + }, + { + "bbox": [ + 442, + 597, + 459, + 609 + ], + "score": 0.93, + "content": "r ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "trained on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "the discriminator’s hidden states when evaluated on training data. 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This makes objective, quantitative evaluation difficult. While previous results in the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 128 + ], + "score": 1.0, + "content": "literature have reported approximate likelihoods based on Parzen window estimates, Theis et al.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "(2015) has convincingly argued that these estimates can be quite misleading for high-dimensional", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 507, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 507, + 150 + ], + "score": 1.0, + "content": "data. In this work, we adopt the Inception score proposed by Salimans et al. 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Note that this score can be made larger by a low-entropy per-sample posterior (i.e.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "score": 1.0, + "content": "the Inception network classifies a given sample with greater certainty) as well as a higher entropy", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "aggregate posterior (i.e. the Inception network identifies a wide variety of classes among the sam-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "ples presented to it). Salimans et al. 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Error estimates corre-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 400, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 400, + 287 + ], + "score": 1.0, + "content": "spond to standard deviations, in keeping with previously reported results.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 184, + 506, + 287 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 302, + 479, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 481, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 481, + 317 + ], + "score": 1.0, + "content": "3 IMPROVING UNSUPERVISED GAN TRAINING ON DIVERSE DATASETS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "In this work, we focus on the apparent difficulty of training GANs to produce “object-like” sam-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "ples when trained on diverse collections of natural images. While Salimans et al. (2016) make", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "progress on this problem by employing labeled data and training the discriminator, here we aim to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "make progress on the unsupervised case. 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Indeed, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "literature is replete with instances of gradient descent with respect to the input of a classification", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "model, particularly wide-domain natural image classifiers, producing ghostly approximations to a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "particular class exemplar (Le et al., 2012; Erhan et al., 2009; Yosinski et al., 2015) when this proce-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "dure is carried out without additional guidance, to say nothing of the problems posed by adversarial", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "examples (Szegedy et al., 2013; Goodfellow et al., 2014b) and fooling examples (Nguyen et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 485, + 136, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 136, + 500 + ], + "score": 1.0, + "content": "2015).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 387, + 506, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "While the gradient of the loss function defined by the discriminator may be a source of information", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 515, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 526 + ], + "score": 1.0, + "content": "mostly relevant to very local improvements, the discriminator itself is a potentially valuable source", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "score": 1.0, + "content": "of compact descriptors of the training data. Many authors have noted the remarkable versatility of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "high-level features learned by convolutional networks (Donahue et al., 2014; Yosinski et al., 2014)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "and the degree to which high-level semantics can be reconstructed from even the deepest layers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "of a network (Dosovitskiy & Brox, 2016). 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Here, we propose to estimate the denoising auto-encoder in the space", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 327, + 288 + ], + "score": 1.0, + "content": "of discriminator features, giving rise to a distribution", + "type": "text" + }, + { + "bbox": [ + 327, + 276, + 361, + 288 + ], + "score": 0.92, + "content": "q ( \\Phi ( \\mathbf { x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 276, + 505, + 288 + ], + "score": 1.0, + "content": ". A natural question is what effect", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "score": 1.0, + "content": "this has on the gradient being backpropagated. This is difficult to analyze in general, as for most", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 191, + 310 + ], + "score": 1.0, + "content": "choices the mapping", + "type": "text" + }, + { + "bbox": [ + 191, + 298, + 200, + 308 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "will not be invertible, though it is instructive to examine the invertible case.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 307, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 201, + 325 + ], + "score": 1.0, + "content": "Assuming an invertible", + "type": "text" + }, + { + "bbox": [ + 201, + 311, + 258, + 322 + ], + "score": 0.9, + "content": "\\Phi : \\mathbb { R } ^ { n } \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 307, + 274, + 325 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 275, + 308, + 318, + 324 + ], + "score": 0.94, + "content": "\\begin{array} { r } { J = \\frac { \\partial \\Phi ( \\mathbf { \\bar { x } } ) } { \\partial \\mathbf { x } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 307, + 394, + 325 + ], + "score": 1.0, + "content": "be the Jacobian of", + "type": "text" + }, + { + "bbox": [ + 394, + 311, + 402, + 321 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 307, + 422, + 325 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 423, + 310, + 501, + 323 + ], + "score": 0.92, + "content": "q ( \\Phi ( \\mathbf { x } ) ) = p ( \\mathbf { x } ) | J |", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 307, + 505, + 325 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 239, + 335 + ], + "score": 1.0, + "content": "By the inverse function theorem,", + "type": "text" + }, + { + "bbox": [ + 239, + 324, + 246, + 334 + ], + "score": 0.81, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 324, + 479, + 335 + ], + "score": 1.0, + "content": "is also invertible (and is in fact the Jacobian of the inverse", + "type": "text" + }, + { + "bbox": [ + 479, + 324, + 498, + 334 + ], + "score": 0.87, + "content": "\\Phi ^ { - 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In practice,", + "type": "text" + }, + { + "bbox": [ + 340, + 556, + 348, + 566 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "is not invertible, but the added benefit", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 429, + 580 + ], + "score": 1.0, + "content": "of the denoiser-targeted gradient appears to reduce underfitting to the modes of", + "type": "text" + }, + { + "bbox": [ + 429, + 569, + 435, + 578 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 566, + 505, + 580 + ], + "score": 1.0, + "content": "in the generator,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 578, + 299, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 227, + 590 + ], + "score": 1.0, + "content": "irrespective of any distortions", + "type": "text" + }, + { + "bbox": [ + 227, + 578, + 236, + 587 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 578, + 299, + 590 + ], + "score": 1.0, + "content": "may introduce.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 605, + 209, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 210, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 210, + 619 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 630, + 504, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 630, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 643 + ], + "score": 1.0, + "content": "Denoising feature matching was originally inspired by feature matching introduced by Salimans", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 641, + 477, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 477, + 653 + ], + "score": 1.0, + "content": "et al. 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Our use of denoising auto-encoders necessarily departs from idealized conditions in that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 381, + 423, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 423, + 394 + ], + "score": 1.0, + "content": "the denoiser is estimated online from an ever-changing distribution of features.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 412 + ], + "score": 1.0, + "content": "Several approaches to GAN-like models have cast the problem in terms of learning an energy func-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "tion. 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Our use of denoising auto-encoders necessarily departs from idealized conditions in that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 381, + 423, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 423, + 394 + ], + "score": 1.0, + "content": "the denoiser is estimated online from an ever-changing distribution of features.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 358, + 505, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 412 + ], + "score": 1.0, + "content": "Several approaches to GAN-like models have cast the problem in terms of learning an energy func-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "tion. 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The energy-based GAN replaces the discriminator with an auto-encoder,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 264, + 500 + ], + "score": 1.0, + "content": "which is trained to assign low energy", + "type": "text" + }, + { + "bbox": [ + 265, + 487, + 277, + 497 + ], + "score": 0.85, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 484, + 505, + 500 + ], + "score": 1.0, + "content": "reconstruction error) to training data and higher energy", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 175, + 510 + ], + "score": 1.0, + "content": "to samples from", + "type": "text" + }, + { + "bbox": [ + 176, + 497, + 185, + 507 + ], + "score": 0.66, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 497, + 505, + 510 + ], + "score": 1.0, + "content": ". To discourage generator collapses, a “pull-away term” penalizes the normal-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "ized dot product in a feature space defined by the auto-encoder’s internal representation. 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The energy-based GAN can be viewed as training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "the generator to seek fixed points of the autoencoding function (i.e. by backpropagating through", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "score": 1.0, + "content": "the decoder and encoder in order to decrease reconstruction error), whereas we treat the output of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 123, + 586 + ], + "score": 0.92, + "content": "r ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "as constant with respect to the optimization as in Lee et al. (2015). That is to say, rather than", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "score": 1.0, + "content": "using backpropagation to steer the dynamics of the autoencoder, we instead employ our denoising", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 596, + 461, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 461, + 609 + ], + "score": 1.0, + "content": "autoencoder to augment the gradient information obtained by ordinary backpropagation.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 397, + 506, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 612, + 504, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "Closest to our own approach, concurrent work on model-based super-resolution by Sønderby et al.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 624, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 635 + ], + "score": 1.0, + "content": "(2016) trains a denoising auto-encoder on high-resolution ground truth and evaluates it on synthe-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "sized super-resolution images, using the difference between the original synthesized image and the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "score": 1.0, + "content": "denoiser’s output as an additional training signal for refining the output of the super-resolution net-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "work. Both Sønderby et al. (2016) and our own work are motivated by the results of Alain & Bengio", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "score": 1.0, + "content": "(2014) discussed above. Aside from addressing a different application area, our denoiser is learned", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 679, + 401, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 401, + 690 + ], + "score": 1.0, + "content": "on-the-fly from a high-level feature representation which is itself learned.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 612, + 506, + 690 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 201, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 201, + 96 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 113, + 505, + 191 + ], + "lines": [ + { + "bbox": [ + 106, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "We evaluate denoising feature matching on learning synthesis models from three datasets of increas-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 124, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 505, + 137 + ], + "score": 1.0, + "content": "ing diversity and size: CIFAR-10, STL-10, and ImageNet. Although several authors have described", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 135, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 311, + 148 + ], + "score": 1.0, + "content": "GAN-based image synthesis models operating at", + "type": "text" + }, + { + "bbox": [ + 311, + 136, + 357, + 146 + ], + "score": 0.9, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 135, + 505, + 148 + ], + "score": 1.0, + "content": "(Salimans et al., 2016; Zhao et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 146, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 149, + 159 + ], + "score": 1.0, + "content": "2016) and", + "type": "text" + }, + { + "bbox": [ + 149, + 146, + 192, + 157 + ], + "score": 0.89, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 146, + 505, + 159 + ], + "score": 1.0, + "content": "(Zhao et al., 2016) resolution, we carry out our investigations at relatively low", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 158, + 504, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 504, + 169 + ], + "score": 1.0, + "content": "resolutions, both for computational ease and because we believe that the problem of unconditional", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "modeling of diverse image collections is not well solved even at low resolutions; making progress", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 424, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 424, + 192 + ], + "score": 1.0, + "content": "in this regime is likely to yield insights that apply to the higher-resolution case.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 400, + 209 + ], + "score": 1.0, + "content": "In all experiments, we employ isotropic Gaussian corruption noise with", + "type": "text" + }, + { + "bbox": [ + 400, + 197, + 428, + 207 + ], + "score": 0.89, + "content": "\\sigma = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 196, + 505, + 209 + ], + "score": 1.0, + "content": ". Although we ex-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 215, + 221 + ], + "score": 1.0, + "content": "perimented with annealing", + "type": "text" + }, + { + "bbox": [ + 216, + 209, + 223, + 217 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 206, + 505, + 221 + ], + "score": 1.0, + "content": "towards 0 (as also performed in Sønderby et al. (2016)), an annealing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 231 + ], + "score": 1.0, + "content": "schedule which consistently outperformed fixed noise remained elusive. We experimented with con-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "score": 1.0, + "content": "volutional denoisers, but our best results to date were obtained with deep, fully-connected denoisers", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "using the ReLU nonlinearity on the penultimate layer of the discriminator. The number of hidden", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "units was fixed to the same value in all denoiser layers, and the procedure is apparently robust to this", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 457, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 457, + 275 + ], + "score": 1.0, + "content": "hyperparameter choice, as long as it is greater than or equal to the input dimensionality.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "Our generator and discriminator architectures follow the methods outlined in Radford et al. (2015).", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "Accordingly, batch normalization (Ioffe & Szegedy, 2015) was used in the generator and discrimi-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "nator in the same manner as Radford et al. (2015), and in all layers of the denoiser except the output", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 313, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 324 + ], + "score": 1.0, + "content": "layer. In particular, as in Radford et al. 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We calculate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "updates with respect to all losses with the parameters of all three networks fixed, and update all", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 345, + 218, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 218, + 357 + ], + "score": 1.0, + "content": "parameters simultaneously.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "All networks were trained with the Adam optimizer Kingma & Ba (2014) with a learning rate of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 370, + 504, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 128, + 383 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 370, + 147, + 387 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 147, + 372, + 186, + 384 + ], + "score": 0.91, + "content": "\\beta _ { 1 } = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 370, + 471, + 387 + ], + "score": 1.0, + "content": ". 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In our experiments, we set", + "type": "text" + }, + { + "bbox": [ + 302, + 384, + 336, + 395 + ], + "score": 0.92, + "content": "\\lambda _ { d e n o i s e }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 383, + 350, + 397 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 351, + 383, + 385, + 395 + ], + "score": 0.91, + "content": "0 . 0 3 / n _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 383, + 418, + 397 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 419, + 385, + 431, + 394 + ], + "score": 0.86, + "content": "n _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "is the number of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "discriminator hidden units fed as input to the denoiser; this division decouples the scale of the first", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "term of (4) from the dimensionality of the representation used, reducing the need to adjust this", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 416, + 426, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 426, + 429 + ], + "score": 1.0, + "content": "hyperparameter simply because we altered the architecture of the discriminator.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 107, + 453, + 178, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 179, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 179, + 466 + ], + "score": 1.0, + "content": "5.1 CIFAR-10", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 504, + 512 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 444, + 491 + ], + "score": 1.0, + "content": "CIFAR-10 (Krizhevsky & Hinton, 2009) is a small, well-studied dataset consisting of", + "type": "text" + }, + { + "bbox": [ + 444, + 478, + 504, + 489 + ], + "score": 0.88, + "content": "5 0 , 0 0 0 3 2 \\times 3 2", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "pixel RGB training images and 10,000 test images from 10 classes: airplane, automobile, bird, cat,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 500, + 260, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 260, + 514 + ], + "score": 1.0, + "content": "deer, dog, frog, horse, ship, and truck.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "Samples from our model trained on CIFAR-10 are shown in Figure 1, and Inception scores for sev-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "eral methods, including those reported in Salimans et al. 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Qualitatively, the samples include recognizable cars, boats", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 426, + 596 + ], + "score": 1.0, + "content": "and various animals. The best performing generator network consisted of the", + "type": "text" + }, + { + "bbox": [ + 426, + 583, + 461, + 594 + ], + "score": 0.89, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "ImageNet", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "architecture from Radford et al. 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Although several authors have described", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 135, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 311, + 148 + ], + "score": 1.0, + "content": "GAN-based image synthesis models operating at", + "type": "text" + }, + { + "bbox": [ + 311, + 136, + 357, + 146 + ], + "score": 0.9, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 135, + 505, + 148 + ], + "score": 1.0, + "content": "(Salimans et al., 2016; Zhao et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 146, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 149, + 159 + ], + "score": 1.0, + "content": "2016) and", + "type": "text" + }, + { + "bbox": [ + 149, + 146, + 192, + 157 + ], + "score": 0.89, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 146, + 505, + 159 + ], + "score": 1.0, + "content": "(Zhao et al., 2016) resolution, we carry out our investigations at relatively low", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 158, + 504, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 504, + 169 + ], + "score": 1.0, + "content": "resolutions, both for computational ease and because we believe that the problem of unconditional", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "modeling of diverse image collections is not well solved even at low resolutions; making progress", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 424, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 424, + 192 + ], + "score": 1.0, + "content": "in this regime is likely to yield insights that apply to the higher-resolution case.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 113, + 505, + 192 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 400, + 209 + ], + "score": 1.0, + "content": "In all experiments, we employ isotropic Gaussian corruption noise with", + "type": "text" + }, + { + "bbox": [ + 400, + 197, + 428, + 207 + ], + "score": 0.89, + "content": "\\sigma = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 196, + 505, + 209 + ], + "score": 1.0, + "content": ". Although we ex-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 215, + 221 + ], + "score": 1.0, + "content": "perimented with annealing", + "type": "text" + }, + { + "bbox": [ + 216, + 209, + 223, + 217 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 206, + 505, + 221 + ], + "score": 1.0, + "content": "towards 0 (as also performed in Sønderby et al. (2016)), an annealing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 231 + ], + "score": 1.0, + "content": "schedule which consistently outperformed fixed noise remained elusive. We experimented with con-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "score": 1.0, + "content": "volutional denoisers, but our best results to date were obtained with deep, fully-connected denoisers", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "using the ReLU nonlinearity on the penultimate layer of the discriminator. The number of hidden", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "units was fixed to the same value in all denoiser layers, and the procedure is apparently robust to this", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 457, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 457, + 275 + ], + "score": 1.0, + "content": "hyperparameter choice, as long as it is greater than or equal to the input dimensionality.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 196, + 505, + 275 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "Our generator and discriminator architectures follow the methods outlined in Radford et al. 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Real dataRadford et al*Ours
25.78 ± .478.83 ± 0.149.18± .13
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Higher Inception scores, as well as visual inspection, suggest that the procedure", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "captures class-specific features of the training data in a manner superior to the original adversarial", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "score": 1.0, + "content": "objective alone. That being said, we do not believe we are yet making optimal use of the paradigm.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "The non-stationarity of the feature distribution on which the denoiser is trained could be limiting the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "ability of the denoiser to obtain a good fit, and the information backpropagated to the generator is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "always slightly stale. Steps to reduce this non-stationarity may be fruitful; we experimented briefly", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "with historical averaging as explored in Salimans et al. (2016) but did not observe a clear benefit thus", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "far. Structured denoisers, including denoisers that learn an energy function for multiple hidden layers", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 244, + 504, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 504, + 255 + ], + "score": 1.0, + "content": "at once, could conceivably aid in obtaining a better fit. Learning a partially stochastic transition", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "operator rather than a deterministic denoiser could conceivably capture interesting multimodalities", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 317, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 317, + 279 + ], + "score": 1.0, + "content": "that are “blurred” by a unimodal denoising function.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "Our method is orthogonal and could conceivably be used in combination with several other GAN", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "extensions. 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(2016),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 412, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 412, + 339 + ], + "score": 1.0, + "content": "could all be trained with an additional denoising feature matching objective.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "We have proposed a useful heuristic, but a better theoretical grounding regarding how GANs are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "trained in practice is a necessary direction for future work, including grounded criteria for assess-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "ing mode coverage and mass misassignment, and principled criteria for assessing convergence or", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 215, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 215, + 390 + ], + "score": 1.0, + "content": "performing early stopping.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 401, + 200, + 411 + ], + "lines": [ + { + "bbox": [ + 107, + 402, + 200, + 412 + ], + "spans": [ + { + "bbox": [ + 107, + 402, + 200, + 412 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "We thank Ian Goodfellow, Laurent Dinh, Yaroslav Ganin and Kyle Kastner for helpful discussions.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "We thank Vincent Dumoulin and Ishmael Belghazi for making available code and model parameters", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "used in comparison to ALI, as well as Alec Radford for making available the code and model pa-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "rameters for his ImageNet model. We would like to thank Antonia Creswell and Hiroyuki Yamazaki", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 504, + 476 + ], + "score": 1.0, + "content": "for pointing out an error in the initial version of this manuscript, and anonymous reviewers for valu-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "able feedback. We thank the University of Montreal and Compute Canada for the computational", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 504, + 497 + ], + "score": 1.0, + "content": "resources used for this investigation, as well as the authors of Theano (Al-Rfou et al., 2016), Blocks", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "and Fuel (van Merrienboer et al., 2015). We thank CIFAR, NSERC, Google, Samsung and Canada ¨", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 224, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 224, + 520 + ], + "score": 1.0, + "content": "Research Chairs for funding.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 535, + 175, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 176, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 176, + 549 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 504, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 568 + ], + "score": 1.0, + "content": "David H Ackley, Geoffrey E Hinton, and Terrence J Sejnowski. 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Real dataRadford et al*Ours
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That being said, we do not believe we are yet making optimal use of the paradigm.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "The non-stationarity of the feature distribution on which the denoiser is trained could be limiting the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "ability of the denoiser to obtain a good fit, and the information backpropagated to the generator is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "always slightly stale. Steps to reduce this non-stationarity may be fruitful; we experimented briefly", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "with historical averaging as explored in Salimans et al. (2016) but did not observe a clear benefit thus", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "far. Structured denoisers, including denoisers that learn an energy function for multiple hidden layers", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 244, + 504, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 504, + 255 + ], + "score": 1.0, + "content": "at once, could conceivably aid in obtaining a better fit. 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(2016),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 412, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 412, + 339 + ], + "score": 1.0, + "content": "could all be trained with an additional denoising feature matching objective.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 282, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "We have proposed a useful heuristic, but a better theoretical grounding regarding how GANs are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "trained in practice is a necessary direction for future work, including grounded criteria for assess-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "ing mode coverage and mass misassignment, and principled criteria for assessing convergence or", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 215, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 215, + 390 + ], + "score": 1.0, + "content": "performing early stopping.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 343, + 506, + 390 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 401, + 200, + 411 + ], + "lines": [ + { + "bbox": [ + 107, + 402, + 200, + 412 + ], + "spans": [ + { + "bbox": [ + 107, + 402, + 200, + 412 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "We thank Ian Goodfellow, Laurent Dinh, Yaroslav Ganin and Kyle Kastner for helpful discussions.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "We thank Vincent Dumoulin and Ishmael Belghazi for making available code and model parameters", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "used in comparison to ALI, as well as Alec Radford for making available the code and model pa-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "rameters for his ImageNet model. We would like to thank Antonia Creswell and Hiroyuki Yamazaki", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 504, + 476 + ], + "score": 1.0, + "content": "for pointing out an error in the initial version of this manuscript, and anonymous reviewers for valu-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "able feedback. We thank the University of Montreal and Compute Canada for the computational", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 504, + 497 + ], + "score": 1.0, + "content": "resources used for this investigation, as well as the authors of Theano (Al-Rfou et al., 2016), Blocks", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "and Fuel (van Merrienboer et al., 2015). 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Real dataRadford et al*Ours
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Having a radial basis function centred on each", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "score": 1.0, + "content": "training feature is made scalable by treating it as an approximate nearest neighbour", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "score": 1.0, + "content": "search problem. End-to-end learning of the network and solver is carried out,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 470, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 470, + 299 + ], + "score": 1.0, + "content": "mapping high dimensional features into clusters of the same class. This results in a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 470, + 312 + ], + "score": 1.0, + "content": "well formed embedding space, where semantically related instances are likely to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 470, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 470, + 323 + ], + "score": 1.0, + "content": "be located near one another, regardless of whether or not the network was trained", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "score": 1.0, + "content": "on those classes. The same loss function is used for both the metric learning and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 333, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 142, + 333, + 469, + 344 + ], + "score": 1.0, + "content": "classification problems. We show that our radial basis function solver outperforms", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 471, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 471, + 355 + ], + "score": 1.0, + "content": "state-of-the-art embedding approaches on the Stanford Cars196 and CUB-200-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 354, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 142, + 354, + 469, + 366 + ], + "score": 1.0, + "content": "2011 datasets. Additionally, we show that when used as a classifier, our method", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 366, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 469, + 376 + ], + "score": 1.0, + "content": "outperforms a conventional softmax classifier on the CUB-200-2011, Stanford", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 376, + 461, + 388 + ], + "spans": [ + { + "bbox": [ + 142, + 376, + 461, + 388 + ], + "score": 1.0, + "content": "Cars196, Oxford 102 Flowers and Leafsnap fine-grained classification datasets.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5, + "bbox_fs": [ + 141, + 212, + 471, + 388 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 409, + 206, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 208, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 208, + 425 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "score": 1.0, + "content": "The solver of a neural network is vital to its performance, as it defines the objective and drives the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 457 + ], + "score": 1.0, + "content": "learning. We define a solver as the layers of the network that are aware of the class labels of the data.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "In the domain of image classification, a softmax solver is conventionally used to transform activations", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "into a distribution across class labels (Krizhevsky et al., 2012; Simonyan & Zisserman, 2014; Szegedy", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "et al., 2015; He et al., 2016). 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These methods aim to form a single cluster per class. In contrast,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "our approach pushes a feature around the embedding space based only on the local neighbourhood", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "of that feature. This means that the current structure of the space is considered, allowing multiple", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "clusters to form for a single class, if that is appropriate. 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Radial basis functions have", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "been used as neural network solvers in the form of support vector machines. In one such formulation,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "a neural network is used as a fixed feature extractor and separate support vector machines are trained", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "to classify the features (Razavian et al., 2014; Donahue et al., 2014). 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The RBF solver by its very nature is a classifier, but learns the classification", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 426, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 426, + 287 + ], + "score": 1.0, + "content": "problem in the exact same way it learns the embedding space building problem.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 175, + 506, + 287 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 504, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "The main advantages of our novel radial basis function solver for neural networks can be summarised", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 153, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 153, + 315 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 291, + 505, + 315 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 324, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 133, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 133, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "• Our solver can be directly applied to two previously separate problems; classification and", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 334, + 251, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 251, + 348 + ], + "score": 1.0, + "content": "embedding space learning.", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 131, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "• End-to-end learning can be made scalable by leveraging fast approximate nearest neighbour", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 361, + 268, + 373 + ], + "spans": [ + { + "bbox": [ + 141, + 361, + 268, + 373 + ], + "score": 1.0, + "content": "search (as seen in Section 3.2).", + "type": "text" + } + ], + "index": 23, + "is_list_end_line": true + }, + { + "bbox": [ + 140, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 140, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "Our approach outperforms current state-of-the-art deep metric learning algorithms on the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 387, + 428, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 387, + 428, + 399 + ], + "score": 1.0, + "content": "Stanford Cars196 and CUB-200-2011 datasets (as seen in Section 4.1).", + "type": "text" + } + ], + "index": 25, + "is_list_end_line": true + }, + { + "bbox": [ + 137, + 403, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 137, + 403, + 505, + 414 + ], + "score": 1.0, + "content": "Finally, our radial basis function classifier outperforms a conventional softmax classifier", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 414, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 505, + 424 + ], + "score": 1.0, + "content": "on the fine-grained classification datasets CUB-200-2011, Stanford Cars196, Oxford 102", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 424, + 330, + 436 + ], + "spans": [ + { + "bbox": [ + 141, + 424, + 330, + 436 + ], + "score": 1.0, + "content": "Flowers and Leafsnap (as seen in Section 4.2).", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + } + ], + "index": 24, + "bbox_fs": [ + 131, + 324, + 505, + 436 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 452, + 211, + 465 + ], + "lines": [ + { + "bbox": [ + 104, + 451, + 213, + 468 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 213, + 468 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 490 + ], + "score": 1.0, + "content": "Radial Basis Functions in Neural Networks Radial basis function networks were introduced by", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 487, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 501 + ], + "score": 1.0, + "content": "Broomhead & Lowe (1988). The networks formulate activation functions as RBFs, resulting in an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 500, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 511 + ], + "score": 1.0, + "content": "output that is a sum of radial basis function values between the input and network parameters. In", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "contrast to these radial basis function networks, our approach uses RBFs in the solver of a deep", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 522, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 533 + ], + "score": 1.0, + "content": "convolutional neural network and our radial basis function centres are coupled to high dimensional", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "score": 1.0, + "content": "embeddings of training samples, rather than being network parameters. Radial basis functions have", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "been used as neural network solvers in the form of support vector machines. In one such formulation,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "a neural network is used as a fixed feature extractor and separate support vector machines are trained", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "to classify the features (Razavian et al., 2014; Donahue et al., 2014). No joint training occurs between", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "the solver (classifier) and network. Such an approach is often used for transfer learning, where the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "network is trained on vast amounts of data and the support vector machines are trained for problems", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "in which labelled training data is scarce. 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In this case, the solver and network are trained jointly, meaning the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 263, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 263, + 632 + ], + "score": 1.0, + "content": "loss that is minimised is margin based.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 476, + 506, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Metric Learning Early methods in the domain of metric learning include those that use Siamese", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "networks (Bromley et al., 1993) and contrastive loss (Hadsell et al., 2006; Chopra et al., 2005).", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "The objective of these approaches is to pull pairwise samples of the same class together and push", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "pairwise samples of different classes apart. 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For example, the probability that the feature embedding", + "type": "text" + }, + { + "bbox": [ + 459, + 687, + 466, + 695 + ], + "score": 0.61, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 685, + 505, + 697 + ], + "score": 1.0, + "content": "has class", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 694, + 151, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 128, + 708 + ], + "score": 1.0, + "content": "label", + "type": "text" + }, + { + "bbox": [ + 129, + 696, + 137, + 707 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 694, + 151, + 708 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 630, + 506, + 708 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 704, + 384, + 736 + ], + "lines": [ + { + "bbox": [ + 225, + 704, + 384, + 736 + ], + "spans": [ + { + "bbox": [ + 225, + 704, + 384, + 736 + ], + "score": 0.95, + "content": "P r ( \\mathbf { x } \\in \\operatorname { c l a s s } Q ) = \\frac { \\sum _ { i \\in Q } w _ { i } f ( \\mathbf { x } , \\mathbf { c _ { i } } ) } { \\sum _ { j = 1 } ^ { m } w _ { j } f ( \\mathbf { x } , \\mathbf { c _ { j } } ) } ,", + "type": "interline_equation", + "image_path": "c99b5f94e1bec4588b5d8120c9f4f6a0833844fafd8a9e96c24ba09352dbd282.jpg" + } + ] + } + ], + "index": 40.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 704, + 384, + 720.0 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 225, + 720.0, + 384, + 736.0 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 133, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 141, + 94 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 83, + 190, + 95 + ], + "score": 1.0, + "content": "is the RBF,", + "type": "text" + }, + { + "bbox": [ + 191, + 83, + 217, + 94 + ], + "score": 0.91, + "content": "i \\in Q", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 83, + 323, + 95 + ], + "score": 1.0, + "content": "are the centres with label", + "type": "text" + }, + { + "bbox": [ + 324, + 83, + 333, + 94 + ], + "score": 0.77, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 83, + 336, + 95 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 337, + 84, + 347, + 93 + ], + "score": 0.6, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "is the number of training samples and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 118, + 105 + ], + "score": 0.83, + "content": "w _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 93, + 273, + 106 + ], + "score": 1.0, + "content": "is a learnable weight for RBF centre", + "type": "text" + }, + { + "bbox": [ + 273, + 95, + 278, + 104 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ". Of course, if a sample is in the training set and has a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "corresponding RBF centre, the distance calculation to itself is omitted during the computation of the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 359, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 359, + 127 + ], + "score": 1.0, + "content": "classification distribution, the loss function and the derivatives.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 105, + 132, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "The loss function used for optimisation is simply the summed negative logarithm of the probabilities", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 465, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 288, + 155 + ], + "score": 1.0, + "content": "of the true class labels. For example, the loss", + "type": "text" + }, + { + "bbox": [ + 288, + 144, + 296, + 153 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 144, + 342, + 155 + ], + "score": 1.0, + "content": "for sample", + "type": "text" + }, + { + "bbox": [ + 342, + 145, + 349, + 153 + ], + "score": 0.51, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 144, + 443, + 155 + ], + "score": 1.0, + "content": "with ground truth label", + "type": "text" + }, + { + "bbox": [ + 444, + 144, + 453, + 153 + ], + "score": 0.83, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 144, + 465, + 155 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 157, + 373, + 171 + ], + "lines": [ + { + "bbox": [ + 237, + 157, + 373, + 171 + ], + "spans": [ + { + "bbox": [ + 237, + 157, + 373, + 171 + ], + "score": 0.93, + "content": "L ( \\mathbf { x } ) = - \\ln \\left( P r ( \\mathbf { x } \\in \\operatorname { c l a s s } R ) \\right) .", + "type": "interline_equation", + "image_path": "9bb32f330f211af7618a01c37cd98921b95b2bf14d564756505ef2e151c55f82.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 237, + 157, + 373, + 171 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 505, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "score": 1.0, + "content": "The same loss function is used regardless of whether the network is being trained for classification,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "as above, or for embedding space building (distance metric learning). This is possible since the RBF", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "classifier is directly computed from distances between features in the embedding space. This means", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "that a network trained for classification will result in features of the same class being located near", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "one another, and similarly a network trained for metric learning will result in an embedding space in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 306, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 306, + 240 + ], + "score": 1.0, + "content": "which features can be well classified using RBFs.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 251, + 287, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 251, + 288, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 288, + 264 + ], + "score": 1.0, + "content": "3.2 NEAREST NEIGHBOUR RBF SOLVER", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 505, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "In Equation 2, the distribution is calculated by summing over all RBF centres. However, since these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 296 + ], + "score": 1.0, + "content": "centres are attached to training samples, of which there could be any large number, computing that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "sum is both intractable and unnecessary. The majority of RBF values for a given feature embedding", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 504, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 504, + 317 + ], + "score": 1.0, + "content": "will be effectively zero, as the sample feature will lie only within a subset of the RBF centres’", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 317, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 328 + ], + "score": 1.0, + "content": "Gaussian windows. As such, only the local neighbourhood around a feature embedding should be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "considered. Operating on the set of the nearest RBF centres to a feature ensures that most of the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 488, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 488, + 351 + ], + "score": 1.0, + "content": "distance values computed are pertinent to the loss calculation. The classifier equation becomes:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 351, + 390, + 382 + ], + "lines": [ + { + "bbox": [ + 218, + 351, + 390, + 382 + ], + "spans": [ + { + "bbox": [ + 218, + 351, + 390, + 382 + ], + "score": 0.95, + "content": "P r ( \\mathbf x \\in \\mathrm { c l a s s } Q ) = \\frac { \\sum _ { i \\in Q \\cap \\mathcal { N } } w _ { i } f ( \\mathbf x , \\mathbf c _ { \\mathbf i } ) } { \\sum _ { j \\in \\mathcal { N } } w _ { j } f ( \\mathbf x , \\mathbf c _ { \\mathbf j } ) } ,", + "type": "interline_equation", + "image_path": "325b18d1e53eff7c6d216b7383e0c9df86986896d30805f7d46d60dbc51063fc.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 351, + 390, + 366.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 218, + 366.5, + 390, + 382.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 133, + 396 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 384, + 144, + 394 + ], + "score": 0.81, + "content": "\\mathcal { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 383, + 370, + 396 + ], + "score": 1.0, + "content": "is the set of approximate nearest neighbours for sample", + "type": "text" + }, + { + "bbox": [ + 371, + 386, + 378, + 394 + ], + "score": 0.73, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 383, + 434, + 396 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 434, + 384, + 480, + 395 + ], + "score": 0.92, + "content": "i \\in Q \\cap \\mathcal N", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 325, + 408 + ], + "score": 1.0, + "content": "set of approximate nearest neighbours that have label", + "type": "text" + }, + { + "bbox": [ + 326, + 395, + 335, + 406 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 394, + 505, + 408 + ], + "score": 1.0, + "content": ". Again, we note that training set samples", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 361, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 361, + 418 + ], + "score": 1.0, + "content": "exclude their own RBF centre from their nearest neighbour list.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "In the interest of providing a scalable solution, we use approximate nearest neighbour search to obtain", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "candidate nearest neighbour lists. This allows for a trade off between precision and computational", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "efficiency. Specifically, we use a Fast Approximate Nearest Neighbour Graph (FANNG) (Harwood &", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "Drummond, 2016), as it provides the most efficiency when needing a high probability of finding the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "true nearest neighbours of a query point. Importantly, FANNG provides scalability in terms of the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 477, + 343, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 343, + 490 + ], + "score": 1.0, + "content": "number of dimensions and the number of training samples.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 107, + 502, + 236, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 237, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 237, + 514 + ], + "score": 1.0, + "content": "3.3 END-TO-END LEARNING", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "The network and solver weights are learned end-to-end. As the weights are constantly being updated", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "score": 1.0, + "content": "during training, the locations of the RBF centres are changing. This leads to complications in the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "computation of the derivatives of the loss with respect to the embeddings. This calculation requires", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "dimension by dimension differences between the training embeddings and the RBF centres. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "centres are moving as the network is being updated, but computing the current RBF centre locations", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "online is intractable. For example, if considering 100 nearest neighbours, 101 samples would need to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "be forward propagated through the network for each training sample. However, we find that is is not", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 600, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 611 + ], + "score": 1.0, + "content": "necessary for the RBF centres to be up to date at all times in order for the model to converge. A bank", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 609, + 345, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 345, + 622 + ], + "score": 1.0, + "content": "of the RBF centres is stored and updated at a fixed interval.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 104, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "A further consequence of the RBF centres moving during training is that the nearest neighbours also", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "change. It is intractable to find to correct nearest neighbours each time the weights are updated. This", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "is simply remedied by considering a larger number of nearest neighbours than would be required if", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "all centres and neighbour lists were up-to-date at all times. The embedding space changes slowly", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "enough that it is highly likely many of the previously neighbouring RBF centres will remain relevant.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "score": 1.0, + "content": "Since the Gaussian RBF decays to zero as the distance between the points becomes large, it does not", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 694, + 491, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 491, + 704 + ], + "score": 1.0, + "content": "matter if an RBF centre that is no longer near the sample remains a candidate nearest neighbour.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We call the frequency at which the RBF centres are updated and the nearest neighbours found the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "update interval. During training, at a fixed number of epochs we forward pass the entire training set", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 133, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 141, + 94 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 83, + 190, + 95 + ], + "score": 1.0, + "content": "is the RBF,", + "type": "text" + }, + { + "bbox": [ + 191, + 83, + 217, + 94 + ], + "score": 0.91, + "content": "i \\in Q", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 83, + 323, + 95 + ], + "score": 1.0, + "content": "are the centres with label", + "type": "text" + }, + { + "bbox": [ + 324, + 83, + 333, + 94 + ], + "score": 0.77, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 83, + 336, + 95 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 337, + 84, + 347, + 93 + ], + "score": 0.6, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "is the number of training samples and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 118, + 105 + ], + "score": 0.83, + "content": "w _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 93, + 273, + 106 + ], + "score": 1.0, + "content": "is a learnable weight for RBF centre", + "type": "text" + }, + { + "bbox": [ + 273, + 95, + 278, + 104 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ". Of course, if a sample is in the training set and has a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "corresponding RBF centre, the distance calculation to itself is omitted during the computation of the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 359, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 359, + 127 + ], + "score": 1.0, + "content": "classification distribution, the loss function and the derivatives.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 83, + 506, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 132, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "The loss function used for optimisation is simply the summed negative logarithm of the probabilities", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 465, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 288, + 155 + ], + "score": 1.0, + "content": "of the true class labels. For example, the loss", + "type": "text" + }, + { + "bbox": [ + 288, + 144, + 296, + 153 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 144, + 342, + 155 + ], + "score": 1.0, + "content": "for sample", + "type": "text" + }, + { + "bbox": [ + 342, + 145, + 349, + 153 + ], + "score": 0.51, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 144, + 443, + 155 + ], + "score": 1.0, + "content": "with ground truth label", + "type": "text" + }, + { + "bbox": [ + 444, + 144, + 453, + 153 + ], + "score": 0.83, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 144, + 465, + 155 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 131, + 506, + 155 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 157, + 373, + 171 + ], + "lines": [ + { + "bbox": [ + 237, + 157, + 373, + 171 + ], + "spans": [ + { + "bbox": [ + 237, + 157, + 373, + 171 + ], + "score": 0.93, + "content": "L ( \\mathbf { x } ) = - \\ln \\left( P r ( \\mathbf { x } \\in \\operatorname { c l a s s } R ) \\right) .", + "type": "interline_equation", + "image_path": "9bb32f330f211af7618a01c37cd98921b95b2bf14d564756505ef2e151c55f82.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 237, + 157, + 373, + 171 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 505, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "score": 1.0, + "content": "The same loss function is used regardless of whether the network is being trained for classification,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "as above, or for embedding space building (distance metric learning). This is possible since the RBF", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "classifier is directly computed from distances between features in the embedding space. This means", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "that a network trained for classification will result in features of the same class being located near", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "one another, and similarly a network trained for metric learning will result in an embedding space in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 306, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 306, + 240 + ], + "score": 1.0, + "content": "which features can be well classified using RBFs.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 172, + 506, + 240 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 251, + 287, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 251, + 288, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 288, + 264 + ], + "score": 1.0, + "content": "3.2 NEAREST NEIGHBOUR RBF SOLVER", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 505, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "In Equation 2, the distribution is calculated by summing over all RBF centres. However, since these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 296 + ], + "score": 1.0, + "content": "centres are attached to training samples, of which there could be any large number, computing that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "sum is both intractable and unnecessary. The majority of RBF values for a given feature embedding", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 504, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 504, + 317 + ], + "score": 1.0, + "content": "will be effectively zero, as the sample feature will lie only within a subset of the RBF centres’", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 317, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 328 + ], + "score": 1.0, + "content": "Gaussian windows. As such, only the local neighbourhood around a feature embedding should be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "considered. Operating on the set of the nearest RBF centres to a feature ensures that most of the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 488, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 488, + 351 + ], + "score": 1.0, + "content": "distance values computed are pertinent to the loss calculation. The classifier equation becomes:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 272, + 506, + 351 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 351, + 390, + 382 + ], + "lines": [ + { + "bbox": [ + 218, + 351, + 390, + 382 + ], + "spans": [ + { + "bbox": [ + 218, + 351, + 390, + 382 + ], + "score": 0.95, + "content": "P r ( \\mathbf x \\in \\mathrm { c l a s s } Q ) = \\frac { \\sum _ { i \\in Q \\cap \\mathcal { N } } w _ { i } f ( \\mathbf x , \\mathbf c _ { \\mathbf i } ) } { \\sum _ { j \\in \\mathcal { N } } w _ { j } f ( \\mathbf x , \\mathbf c _ { \\mathbf j } ) } ,", + "type": "interline_equation", + "image_path": "325b18d1e53eff7c6d216b7383e0c9df86986896d30805f7d46d60dbc51063fc.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 351, + 390, + 366.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 218, + 366.5, + 390, + 382.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 133, + 396 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 384, + 144, + 394 + ], + "score": 0.81, + "content": "\\mathcal { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 383, + 370, + 396 + ], + "score": 1.0, + "content": "is the set of approximate nearest neighbours for sample", + "type": "text" + }, + { + "bbox": [ + 371, + 386, + 378, + 394 + ], + "score": 0.73, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 383, + 434, + 396 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 434, + 384, + 480, + 395 + ], + "score": 0.92, + "content": "i \\in Q \\cap \\mathcal N", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 325, + 408 + ], + "score": 1.0, + "content": "set of approximate nearest neighbours that have label", + "type": "text" + }, + { + "bbox": [ + 326, + 395, + 335, + 406 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 394, + 505, + 408 + ], + "score": 1.0, + "content": ". Again, we note that training set samples", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 361, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 361, + 418 + ], + "score": 1.0, + "content": "exclude their own RBF centre from their nearest neighbour list.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 383, + 505, + 418 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "In the interest of providing a scalable solution, we use approximate nearest neighbour search to obtain", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "candidate nearest neighbour lists. This allows for a trade off between precision and computational", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "efficiency. Specifically, we use a Fast Approximate Nearest Neighbour Graph (FANNG) (Harwood &", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "Drummond, 2016), as it provides the most efficiency when needing a high probability of finding the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "true nearest neighbours of a query point. Importantly, FANNG provides scalability in terms of the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 477, + 343, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 343, + 490 + ], + "score": 1.0, + "content": "number of dimensions and the number of training samples.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 423, + 506, + 490 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 502, + 236, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 237, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 237, + 514 + ], + "score": 1.0, + "content": "3.3 END-TO-END LEARNING", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "The network and solver weights are learned end-to-end. As the weights are constantly being updated", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "score": 1.0, + "content": "during training, the locations of the RBF centres are changing. This leads to complications in the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "computation of the derivatives of the loss with respect to the embeddings. This calculation requires", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "dimension by dimension differences between the training embeddings and the RBF centres. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "centres are moving as the network is being updated, but computing the current RBF centre locations", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "online is intractable. For example, if considering 100 nearest neighbours, 101 samples would need to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "be forward propagated through the network for each training sample. However, we find that is is not", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 600, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 611 + ], + "score": 1.0, + "content": "necessary for the RBF centres to be up to date at all times in order for the model to converge. A bank", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 609, + 345, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 345, + 622 + ], + "score": 1.0, + "content": "of the RBF centres is stored and updated at a fixed interval.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 523, + 506, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 104, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "A further consequence of the RBF centres moving during training is that the nearest neighbours also", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "change. It is intractable to find to correct nearest neighbours each time the weights are updated. This", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "is simply remedied by considering a larger number of nearest neighbours than would be required if", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "all centres and neighbour lists were up-to-date at all times. The embedding space changes slowly", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "enough that it is highly likely many of the previously neighbouring RBF centres will remain relevant.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "score": 1.0, + "content": "Since the Gaussian RBF decays to zero as the distance between the points becomes large, it does not", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 694, + 491, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 491, + 704 + ], + "score": 1.0, + "content": "matter if an RBF centre that is no longer near the sample remains a candidate nearest neighbour.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45, + "bbox_fs": [ + 104, + 626, + 506, + 704 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We call the frequency at which the RBF centres are updated and the nearest neighbours found the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "update interval. During training, at a fixed number of epochs we forward pass the entire training set", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "through the network, storing the new RBF centres. The up-to-date nearest neighbours can now be", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "found. If FANNG is used, a rebuild of the graph is required. Note that the stored RBF centres do not", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "have dropout (Srivastava et al., 2014) applied, but the current training embeddings may. The effect of", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 497, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 497, + 127 + ], + "score": 1.0, + "content": "the number of nearest neighbours considered and the update interval are discussed in Section 4.2.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 49.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "through the network, storing the new RBF centres. The up-to-date nearest neighbours can now be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "found. If FANNG is used, a rebuild of the graph is required. Note that the stored RBF centres do not", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "have dropout (Srivastava et al., 2014) applied, but the current training embeddings may. The effect of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 497, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 497, + 127 + ], + "score": 1.0, + "content": "the number of nearest neighbours considered and the update interval are discussed in Section 4.2.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 139, + 505, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 420, + 152 + ], + "score": 1.0, + "content": "Radial Basis Function Parameters A global standard deviation parameter", + "type": "text" + }, + { + "bbox": [ + 421, + 142, + 428, + 149 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "is shared amongst", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 163 + ], + "score": 1.0, + "content": "the RBFs. This ensures that the assumption made about samples only being influenced by their", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 173 + ], + "score": 1.0, + "content": "nearest RBF centres holds. Although the parameter is learnable, we find that fixing the standard", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "deviation value before training is a suitable approach. We treat the standard deviation as an additional", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 197 + ], + "score": 1.0, + "content": "hyperparameter to tune, however it can also be learned independently before full network training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "commences. As seen in Equation 4, each RBF centre has a weight, which is learned end-to-end with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 217 + ], + "score": 1.0, + "content": "the network weights. These weights are initialised at values of one. Note that in our experiments we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 502, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 502, + 228 + ], + "score": 1.0, + "content": "only tune the RBF weights for the classification task; they remain fixed for metric learning problems.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 244, + 200, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 201, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 201, + 259 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 108, + 269, + 495, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 268, + 496, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 496, + 282 + ], + "score": 1.0, + "content": "We detail our experimental results in two tasks; distance metric learning and image classification.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 295, + 261, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 263, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 263, + 307 + ], + "score": 1.0, + "content": "4.1 DISTANCE METRIC LEARNING", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "Experimental Set-up We evaluate our approach on two datasets; Stanford Cars196 (Krause et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "score": 1.0, + "content": "2013) and CUB-200-2011 (Birds200) (Welinder et al., 2010). Cars196 consists of 16,185 images", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "of 196 different car makes and models, while Birds200 consists of 11,788 images of 200 different", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "bird species. In this problem, the network is trained and evaluated on different sets of classes. We", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "follow the experimental set-up used in Song et al. (2016a); Sohn (2016); Song et al. (2016b); Kumar", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "et al. (2017). For the Cars196 dataset, we train the network on the first 98 classes and evaluate on the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 380, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 380, + 506, + 395 + ], + "score": 1.0, + "content": "remaining 98. For the Birds200 dataset we train on the first 100 classes and evaluate on the remaining", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "score": 1.0, + "content": "100. Stochastic gradient descent optimisation is used. All images are first resized to be 256x256 and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "data is augmented by random cropping and horizontal mirroring. Note that we do not crop the images", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 415, + 251, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 251, + 428 + ], + "score": 1.0, + "content": "using the provided bounding boxes.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 431, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "score": 1.0, + "content": "Our method is compared to state-of-the-art approaches on the considered datasets; semi-hard mining", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "for triplet networks (Schroff et al., 2015), lifted structured feature embedding (Song et al., 2016a),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "N-pair loss (Sohn, 2016), clustering (Song et al., 2016b), global loss with triplet networks (Kumar", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "et al., 2016) and smart mining for triplet networks (Kumar et al., 2017). For fair comparison to these", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "score": 1.0, + "content": "methods, we use the same base architecture for our experiments; GoogLeNet (Szegedy et al., 2015).", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "Network weights are initialised from ImageNet (Russakovsky et al., 2015) pre-trained weights. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "use 100 nearest neighbours and an update interval of 10 epochs. RBF weights are fixed at a value of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "score": 1.0, + "content": "one for this task. We train for 50 epochs on Cars196 and 30 epochs on Birds200. A batch size of 20,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 519, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 506, + 531 + ], + "score": 1.0, + "content": "base learning of 0.00001 and weight decay of 0.0002 are used. The RBF standard deviation used", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 530, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 506, + 542 + ], + "score": 1.0, + "content": "depends on size of the embedding dimension. We find values between 10 and 30 work well for this", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 128, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 128, + 553 + ], + "score": 1.0, + "content": "task.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 577 + ], + "score": 1.0, + "content": "Evaluation Metrics Following Song et al. (2016a), we evaluate the embedding space using two", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 443, + 588 + ], + "score": 1.0, + "content": "metrics; Normalised Mutual Information (NMI) (Manning et al., 2008) and Recall", + "type": "text" + }, + { + "bbox": [ + 443, + 576, + 460, + 586 + ], + "score": 0.71, + "content": "@ \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 576, + 506, + 588 + ], + "score": 1.0, + "content": ". The NMI", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "score is the ratio of mutual information and average entropy of a set of clusters and labels. It evaluates", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "only for the number of clusters equal to the number of classes. As discussed in Section 1, a good", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "embedding space does not necessarily have only one cluster per class, but may have multiple well", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "formed clusters in the space. This means that our mutual information may be higher than reported", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 645 + ], + "score": 1.0, + "content": "with this metric. Nevertheless, we present results on the NMI score in the interest of comparing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 349, + 654 + ], + "score": 1.0, + "content": "to existing methods that evaluate on this metric. The Recall", + "type": "text" + }, + { + "bbox": [ + 349, + 642, + 367, + 652 + ], + "score": 0.62, + "content": "@ \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 642, + 399, + 653 + ], + "score": 0.73, + "content": "( \\mathbb { R } ^ { \\ @ \\mathbb { K } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "metric is better suited for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 653, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 665 + ], + "score": 1.0, + "content": "evaluating an embedding space. A true positive is defined as a sample that has at least one of its true", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 664, + 405, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 405, + 676 + ], + "score": 1.0, + "content": "nearest K neighbours in the embedding space with the same class as itself.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Embedding Space Dimension We investigate the importance of the embedding dimension. A", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "similar study in Song et al. (2016a) suggests that the number of dimensions is not important for", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "triplet networks, in fact, increasing the number of dimensions can be detrimental to performance.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "score": 1.0, + "content": "We compare our method with increasing dimension size against triplet loss (Weinberger et al., 2006;", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 83, + 506, + 127 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 139, + 505, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 420, + 152 + ], + "score": 1.0, + "content": "Radial Basis Function Parameters A global standard deviation parameter", + "type": "text" + }, + { + "bbox": [ + 421, + 142, + 428, + 149 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "is shared amongst", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 163 + ], + "score": 1.0, + "content": "the RBFs. This ensures that the assumption made about samples only being influenced by their", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 173 + ], + "score": 1.0, + "content": "nearest RBF centres holds. Although the parameter is learnable, we find that fixing the standard", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "deviation value before training is a suitable approach. We treat the standard deviation as an additional", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 197 + ], + "score": 1.0, + "content": "hyperparameter to tune, however it can also be learned independently before full network training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "commences. As seen in Equation 4, each RBF centre has a weight, which is learned end-to-end with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 217 + ], + "score": 1.0, + "content": "the network weights. These weights are initialised at values of one. Note that in our experiments we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 502, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 502, + 228 + ], + "score": 1.0, + "content": "only tune the RBF weights for the classification task; they remain fixed for metric learning problems.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 138, + 506, + 228 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 244, + 200, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 201, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 201, + 259 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 108, + 269, + 495, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 268, + 496, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 496, + 282 + ], + "score": 1.0, + "content": "We detail our experimental results in two tasks; distance metric learning and image classification.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 268, + 496, + 282 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 295, + 261, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 263, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 263, + 307 + ], + "score": 1.0, + "content": "4.1 DISTANCE METRIC LEARNING", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "Experimental Set-up We evaluate our approach on two datasets; Stanford Cars196 (Krause et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "score": 1.0, + "content": "2013) and CUB-200-2011 (Birds200) (Welinder et al., 2010). Cars196 consists of 16,185 images", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "of 196 different car makes and models, while Birds200 consists of 11,788 images of 200 different", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "bird species. In this problem, the network is trained and evaluated on different sets of classes. We", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "follow the experimental set-up used in Song et al. (2016a); Sohn (2016); Song et al. (2016b); Kumar", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "et al. (2017). For the Cars196 dataset, we train the network on the first 98 classes and evaluate on the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 380, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 380, + 506, + 395 + ], + "score": 1.0, + "content": "remaining 98. For the Birds200 dataset we train on the first 100 classes and evaluate on the remaining", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "score": 1.0, + "content": "100. Stochastic gradient descent optimisation is used. All images are first resized to be 256x256 and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "data is augmented by random cropping and horizontal mirroring. Note that we do not crop the images", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 415, + 251, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 251, + 428 + ], + "score": 1.0, + "content": "using the provided bounding boxes.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 315, + 506, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 431, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "score": 1.0, + "content": "Our method is compared to state-of-the-art approaches on the considered datasets; semi-hard mining", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "for triplet networks (Schroff et al., 2015), lifted structured feature embedding (Song et al., 2016a),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "N-pair loss (Sohn, 2016), clustering (Song et al., 2016b), global loss with triplet networks (Kumar", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "et al., 2016) and smart mining for triplet networks (Kumar et al., 2017). For fair comparison to these", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "score": 1.0, + "content": "methods, we use the same base architecture for our experiments; GoogLeNet (Szegedy et al., 2015).", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "Network weights are initialised from ImageNet (Russakovsky et al., 2015) pre-trained weights. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "use 100 nearest neighbours and an update interval of 10 epochs. RBF weights are fixed at a value of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "score": 1.0, + "content": "one for this task. We train for 50 epochs on Cars196 and 30 epochs on Birds200. A batch size of 20,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 519, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 506, + 531 + ], + "score": 1.0, + "content": "base learning of 0.00001 and weight decay of 0.0002 are used. The RBF standard deviation used", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 530, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 506, + 542 + ], + "score": 1.0, + "content": "depends on size of the embedding dimension. We find values between 10 and 30 work well for this", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 128, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 128, + 553 + ], + "score": 1.0, + "content": "task.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 430, + 506, + 553 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 577 + ], + "score": 1.0, + "content": "Evaluation Metrics Following Song et al. (2016a), we evaluate the embedding space using two", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 443, + 588 + ], + "score": 1.0, + "content": "metrics; Normalised Mutual Information (NMI) (Manning et al., 2008) and Recall", + "type": "text" + }, + { + "bbox": [ + 443, + 576, + 460, + 586 + ], + "score": 0.71, + "content": "@ \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 576, + 506, + 588 + ], + "score": 1.0, + "content": ". The NMI", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "score is the ratio of mutual information and average entropy of a set of clusters and labels. It evaluates", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "only for the number of clusters equal to the number of classes. As discussed in Section 1, a good", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "embedding space does not necessarily have only one cluster per class, but may have multiple well", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "formed clusters in the space. This means that our mutual information may be higher than reported", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 645 + ], + "score": 1.0, + "content": "with this metric. Nevertheless, we present results on the NMI score in the interest of comparing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 349, + 654 + ], + "score": 1.0, + "content": "to existing methods that evaluate on this metric. The Recall", + "type": "text" + }, + { + "bbox": [ + 349, + 642, + 367, + 652 + ], + "score": 0.62, + "content": "@ \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 642, + 399, + 653 + ], + "score": 0.73, + "content": "( \\mathbb { R } ^ { \\ @ \\mathbb { K } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "metric is better suited for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 653, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 665 + ], + "score": 1.0, + "content": "evaluating an embedding space. A true positive is defined as a sample that has at least one of its true", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 664, + 405, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 405, + 676 + ], + "score": 1.0, + "content": "nearest K neighbours in the embedding space with the same class as itself.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 565, + 506, + 676 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Embedding Space Dimension We investigate the importance of the embedding dimension. A", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "similar study in Song et al. (2016a) suggests that the number of dimensions is not important for", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "triplet networks, in fact, increasing the number of dimensions can be detrimental to performance.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "score": 1.0, + "content": "We compare our method with increasing dimension size against triplet loss (Weinberger et al., 2006;", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 688, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 78, + 504, + 230 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 78, + 504, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 78, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 504, + 230 + ], + "score": 0.966, + "type": "image", + "image_path": "c12dcaaee4263341e5a94a15e639afe7c1aabf716e1c6bfa48752a0b28b68e8c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 78, + 504, + 128.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 128.66666666666666, + 504, + 179.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 179.33333333333331, + 504, + 229.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 237, + 505, + 282 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 238, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 505, + 250 + ], + "score": 1.0, + "content": "Figure 2: Effect of embedding size on NMI score on the test set of Cars196 (left) and Birds200", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "(right). The NMI of our RBF approach improves with increasing embedding size, while performance", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "degrades or oscillates for triplet (Weinberger et al., 2006; Schroff et al., 2015) and lifted structured", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 271, + 237, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 237, + 284 + ], + "score": 1.0, + "content": "embedding (Song et al., 2016a).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "image", + "bbox": [ + 106, + 303, + 505, + 455 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 303, + 505, + 455 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 455 + ], + "score": 0.968, + "type": "image", + "image_path": "6ee4762a10f3fa17ca4f97739f1a0d53a7d49d6330e5cecafd50aaa8f77d629c.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 106, + 303, + 505, + 353.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 106, + 353.6666666666667, + 505, + 404.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 106, + 404.33333333333337, + 505, + 455.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 463, + 506, + 486 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "Figure 3: Recall of our RBF solver at 1, 2, 4 and 8 nearest neighbours on the test set of Cars196 (left)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 475, + 465, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 465, + 486 + ], + "score": 1.0, + "content": "and Birds200 (right). Recall performance of our approach increases with embedding size.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + } + ], + "index": 9.25 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 532 + ], + "score": 1.0, + "content": "Schroff et al., 2015) and lifted structured embedding (Song et al., 2016a), both taken from the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "study in Song et al. (2016a). Figure 2 shows the effect of the embedding size on NMI score. It’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "clear that while increasing the number of dimensions does not necessarily improve performance for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "score": 1.0, + "content": "triplet-based networks, the dimensionality is important for our RBF approach. The NMI score for our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 563, + 507, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 507, + 576 + ], + "score": 1.0, + "content": "approach improves with increasing numbers of dimensions. 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Note that the final data point in the plot refers to the entire training set;", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 289, + 403, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 403, + 301 + ], + "score": 1.0, + "content": "while most classes have 24 training samples per class, some have only 23.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + } + ], + "index": 10.0 + }, + { + "type": "table", + "bbox": [ + 205, + 332, + 402, + 408 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 234, + 312, + 376, + 324 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 233, + 311, + 378, + 326 + ], + "spans": [ + { + "bbox": [ + 233, + 311, + 378, + 326 + ], + "score": 1.0, + "content": "Table 2: Birds200 test set accuracy.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 205, + 332, + 402, + 408 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 205, + 332, + 402, + 408 + ], + "spans": [ + { + "bbox": [ + 205, + 332, + 402, + 408 + ], + "score": 0.976, + "html": "
Base NetworkSolver
SoftmaxRBF (Ours)
AlexNet62.4166.95
VGG1675.3778.63
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In Table 2, we evaluate with three network architectures; AlexNet (Krizhevsky et al., 2012),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "VGG16 (Simonyan & Zisserman, 2014) and ResNet50 (He et al., 2016). Our approach outperforms", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "the softmax counterpart for each network. The performance gain over softmax is larger for AlexNet", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "and VGG than for ResNet. 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The importance of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 444, + 700 + ], + "score": 1.0, + "content": "the following components of learning are shown; tuning the RBF standard deviation", + "type": "text" + }, + { + "bbox": [ + 444, + 690, + 451, + 698 + ], + "score": 0.67, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 687, + 505, + 700 + ], + "score": 1.0, + "content": ", learning the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "RBF weights and fine-tuning the network weights. 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Initial Network WeightsTune σLearn RBF WeightsFine-tune Network WeightsTest Accuracy
RandomYesNoNo1.35
ImageNetYesNoNo47.32
ImageNetYesYesNo49.22
ImageNetYesNoYes77.94
ImageNetYesYesYes78.63
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Initial Network WeightsTune σLearn RBF WeightsFine-tune Network WeightsTest Accuracy
RandomYesNoNo1.35
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ImageNetYesYesNo49.22
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DatasetSoftmaxRBF (Ours)
Oxford 102 Flowers82.7986.26
Stanford Cars19685.6786.52
Leafsnap Field73.8075.96
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The dataset", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 366, + 468 + ], + "score": 1.0, + "content": "contains 185 classes of leaf species and we split the data into", + "type": "text" + }, + { + "bbox": [ + 367, + 454, + 387, + 465 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 453, + 391, + 468 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 392, + 454, + 412, + 465 + ], + "score": 0.85, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 453, + 432, + 468 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 432, + 454, + 452, + 465 + ], + "score": 0.86, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 453, + 506, + 468 + ], + "score": 1.0, + "content": "for training,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 466, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 477 + ], + "score": 1.0, + "content": "validation and testing, respectively. 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DatasetSoftmaxRBF (Ours)
Oxford 102 Flowers82.7986.26
Stanford Cars19685.6786.52
Leafsnap Field73.8075.96
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This", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "appears to hold independently of the base network architecture (Table 2) and the improvement is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 439, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 439, + 596 + ], + "score": 1.0, + "content": "particularly strong when limited quantities of training data are available (Figure 5).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 528, + 506, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 507, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 507, + 613 + ], + "score": 1.0, + "content": "Metric learning inherently pulls samples together into high density regions of the embedding space,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "whereas softmax is content to allow samples to fill a very large region of space, provided that the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "logit dimension corresponding to the correct class is larger than the others. 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2Harvard University, 3Toyota Research Institute {kaidicao,cynnjjs,tengyuma}@stanford.edu + +# ABSTRACT + +Real-world large-scale datasets are heteroskedastic and imbalanced – labels have varying levels of uncertainty and label distributions are long-tailed. Heteroskedasticity and imbalance challenge deep learning algorithms due to the difficulty of distinguishing among mislabeled, ambiguous, and rare examples. Addressing heteroskedasticity and imbalance simultaneously is under-explored. We propose a data-dependent regularization technique for heteroskedastic datasets that regularizes different regions of the input space differently. Inspired by the theoretical derivation of the optimal regularization strength in a one-dimensional nonparametric classification setting, our approach adaptively regularizes the data points in higher-uncertainty, lower-density regions more heavily. We test our method on several benchmark tasks, including a real-world heteroskedastic and imbalanced dataset, WebVision. Our experiments corroborate our theory and demonstrate a significant improvement over other methods in noise-robust deep learning. + +# 1 INTRODUCTION + +In real-world machine learning applications, even well-curated training datasets have various types of heterogeneity. Two main types of heterogeneity are: (1) data imbalance: the input or label distribution often has a long-tailed density, and (2) heteroskedasticity: the labels given inputs have varying levels of uncertainties across subsets of data stemming from various sources such as the intrinsic ambiguity of the data or annotation errors. Many deep learning algorithms have been proposed for imbalanced datasets (e.g., see (Wang et al., 2017; Cao et al., 2019; Cui et al., 2019; Liu et al., 2019) and the reference therein). However, heteroskedasticity, a classical notion studied extensively in the statistical community (Pintore et al., 2006; Wang et al., 2013; Tibshirani et al., 2014), has so far been under-explored in deep learning. This paper focuses on addressing heteroskedasticity and its interaction with data imbalance in deep learning. + +Heteroskedasticity is often studied in regression analysis and refers to the property that the distribution of the error varies across inputs. In this work, we mostly focus on classification, though the developed technique also applies to regression. Here, heteroskedasticity reflects how the uncertainty in the conditional distribution $y \mid x$ , or the entropy of $y \mid x$ , varies as a function of $x$ . Real-world datasets are often heteroskedastic. For example, Li et al. (2017) shows that the WebVision dataset has a varying number of ambiguous or true noisy examples across classes.2 + +Conversely, we consider a dataset to be homoscedastic if every example is mislabeled with a fixed probably $\epsilon$ , as assumed by many prior theoretical and empirical works on label corruption (Ghosh et al., 2017; Han et al., 2018; Jiang et al., 2018; Mirzasoleiman et al., 2020). We note that varying uncertainty in $y \mid x$ can come from at least two sources: the intrinsic semantic ambiguity of the input, and the (data-dependent) mislabeling introduced by the annotation process. Our approach can handle both types of noisy examples in a unified way, but for the sake of comparisons with past methods, we call them “ambiguous examples” and “mislabeled examples” respectively, and refer to both of them as “noisy examples”. + +![](images/b0aa3f523002d1c7bc925b084339bea809fac9e5918ef2d25e1e376f5cd78fab.jpg) +Figure 1: Histogram of the distributions of losses on an imbalanced and noisy CIFAR-10 dataset. Clean but rare examples tend to have larger losses, similar to the noisy examples in frequent classes. + +![](images/9499897cf2f9e6c9f97933d2b6e3c3eb440e25eb54680fc3ee92050128d75b8b.jpg) +Figure 2: Real-world datasets have various sources of heterogeneity and it could be hard to distinguish one from another. They require mutuallyexclusive reweighting strategy, but they all benefit from stronger regularization. + +Overparameterized deep learning models tend to overfit more to the noisy examples (Arpit et al., 2017; Frénay & Verleysen, 2013; Zhang et al., 2016). To address this issue, a common approach is to detect noisy examples by selecting those with large training losses, and then remove them from the (re-)training process. However, an input’s training loss can also be big because it is rare or ambiguous (Hacohen & Weinshall, 2019; Wang et al., 2019), as shown in Figure 1. Noise-cleaning methods could fail to distinguish mislabeled from rare/ambiguous examples (see Section 3.1 for empirical proofs). Though dropping the former is desirable, dropping the latter loses important information. Another popular approach is reweighting methods that reduce the contribution of noisy examples in optimization. However, failing to distinguish between mislabeled and rare/ambiguous examples makes the decision of the weights tricky – mislabeled examples require small weights, whereas rare / ambiguous examples benefit from larger weights (Cao et al., 2019; Shu et al., 2019). + +We propose a regularization method that deals with noisy and rare examples in a unified way. We observe that mislabeled, ambiguous, and rare examples all benefit from stronger regularization (Hu et al., 2020; Cao et al., 2019). We apply a Lipschitz regularizer (Wei & Ma, 2019a;b) with varying regularization strength depending on the particular data point. Through theoretical analysis in the one-dimensional setting, we derive the optimal regularization strength for each training example. The optimal strength is larger for rarer and noisier examples. Our proposed algorithm, heteroskedastic adaptive regularization (HAR), first estimates the noise level and density of each example, and then optimizes a Lipschitz-regularized objective with input-dependent regularization with strength provided by the theoretical formula. + +In summary, our main contributions are: (i) we propose to learn heteroskedastic imbalanced datasets under a unified framework, and theoretically study the optimal regularization strength on onedimensional data. (ii) we propose an algorithm, heteroskedastic adaptive regularization (HAR), which applies stronger regularization to data points with high uncertainty and low density. (iii) we experimentally show that HAR achieves significant improvements over other noise-robust deep learning methods on simulated vision and language datasets with controllable degrees of data noise and data imbalance, as well as a real-world heteroskedastic and imbalanced dataset, WebVision. + +# 2 ADAPTIVE REGULARIZATION FOR HETEROSKEDASTIC DATASETS + +# 2.1 BACKGROUNDS + +We first introduce general nonparametric tools that we use in our analysis, and review the dependency of optimal regularization strength on the sample size and noise level. + +Over-parameterized neural networks as nonparametric methods. We use nonparametric method as a surrogate for neural networks because they have been shown to be closely related. Recent work (Savarese et al., 2019) shows that the minimum norm two-layer ReLU network that fits + +![](images/9f6b9f919811d48d690fed50dff8093090a7d6a6528fc0487af6d4f5ee64ac11.jpg) +Figure 3: A one-dimensional example with a three-layer neural network in heteroskedastic and imbalanced regression setting. The curve in blue is the underlying ground truth and the dots are observations with heteroskedastic noise. This example shows that uniformly weak regularization overfits on noisy and rare data (on the right half), whereas uniformly strong regularization causes underfitting on the frequent and oscillating data (on the left half). The adaptive regularization does not underfit the oscillating data but still denoise the noisy data. We note that standard nonparametric methods such as cubic spline do not work here because they also use uniform regularization. + +the training data is in fact a linear spline interpolation. Parhi & Nowak (2019) extend this result to a broader family of neural networks with a broader family of activations. + +Given a training dataset $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ , nonparametric method with penalty works as follows. Let $\mathcal { F } : \mathbb { R } \mathbb { R }$ be a twice-differentiable model family. We aim to fit the data with smoothness penalty + +$$ +\operatorname* { m i n } _ { f } { \ } { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \ell ( f ( x _ { i } ) , y _ { i } ) + \lambda \int ( f ^ { \prime } ( x ) ) ^ { 2 } d x +$$ + +Lipschitz regularization for neural networks. Lipschitz regularization has been shown to be effective for deep neural networks as well. Wei & Ma (2019a) proves a generalization bound of neural networks dependent on the Lipschitzness of each layer with respect to all intermediate layers on the training data, and show that, empirically, regularizing the Lipschitzness improve the generalization. Sokolic et al. ´ (2017) shows similar results in data-limited settings. In Section 2.3, we extend the Lipschitz regularization technique to heteroskedastic setting. + +Regularization strength as a function of noise level and sample size. Finally, we briefly review existing theoretical insights on the optimal choice of regularization strength. Generally, the optimal regularization strength for a given model family increases with the label noise level and decreases in the sample size. As a simple example, consider linear ridge regression $\begin{array} { r } { \operatorname* { m i n } _ { \theta } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } { ( x _ { i } ^ { \top } \theta - y _ { i } ) ^ { 2 } } + } \end{array}$ $\lambda \| \theta \| ^ { 2 }$ , where $x _ { i }$ , $\theta \in \mathbb { R } ^ { d }$ and $y _ { i } \in \mathbb { R }$ . We assume $y _ { i } = x _ { i } ^ { \top } \theta ^ { * } + \xi$ for some ground truth parameter $\theta ^ { * }$ , and $\Dot { \xi } \sim \mathcal { N } ( 0 , \sigma ^ { 2 } )$ . Then the optimal regularization strength $\lambda _ { o p t } = d \sigma ^ { 2 } / \bar { n } \| \theta ^ { * } \| _ { 2 } ^ { 2 }$ . Results of similar nature can also be found in nonparametric statistics (Wang et al., 2013; Tibshirani et al., 2014). + +# 2.2 HETEROSKEDASTIC NONPARAMETRIC CLASSIFICATION ON ONE-DIMENSIONAL DATA + +We consider a one-dimensional binary classification problem where $\mathcal { X } = [ 0 , 1 ] \subset \mathbb { R }$ and $\mathcal { V } = \{ - 1 , 1 \}$ We assume $Y$ given $X$ follows a logistic model with ground-truth function $f ^ { \star }$ , i.e. + +$$ +\operatorname* { P r } \left[ Y = y | X = x \right] = { \frac { 1 } { 1 + \exp ( - y f ^ { \star } ( x ) ) } } . +$$ + +The training objective is cross-entropy loss plus Lipschitz regularization, i.e. + +$$ +\widehat { f } = \mathrm { a r g m i n } _ { f } \widehat { L } ( f ) \triangleq \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( f ( x _ { i } ) , y _ { i } ) + \lambda \int _ { 0 } ^ { 1 } \rho ( x ) ( f ^ { \prime } ( x ) ) ^ { 2 } d x , +$$ + +where $\ell ( a , y ) = - \log ( 1 + \exp ( - y a ) )$ , and $\rho ( x )$ is a smoothing parameter as a function of the noise level and density of $x$ . Let $I ( x )$ be the fisher information matrix conditioned on the input, i.e. $I ( x ) \triangleq \mathbb { E } [ \nabla _ { a } ^ { 2 } \ell ( a , Y ) | _ { a = f ^ { \star } ( X ) } | X = x ]$ . When $( X , Y )$ follows the logistic model in equation 2, + +$$ +I ( x ) = { \frac { 1 } { ( 1 + \exp ( f ^ { \star } ( x ) ) ( 1 + \exp ( - f ^ { \star } ( x ) ) } } = \mathrm { V a r } ( Y | X = x ) . +$$ + +Therefore, $I ( x )$ captures the aleatoric uncertainty of $x$ . For example, when $Y$ is deterministic conditioned on $X = x$ , we have $I ( x ) = 0$ , indicating perfect certainty. + +Define the test metric as the mean-squared-error on the test set $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ , i.e.,1 + +$$ +\operatorname { M S E } ( \hat { f } ) \triangleq \operatorname * { \mathbb { E } } _ { \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n } } \int _ { 0 } ^ { 1 } ( \hat { f } ( t ) - f ^ { \star } ( t ) ) ^ { 2 } d t +$$ + +Our main goal is to derive the optimal choice of $\rho ( x )$ that minimizes the MSE. We start with an analytical characterization of the test error. Let $W _ { 2 } ^ { 2 } = \{ f ^ { \prime }$ is absolute continuous and $f ^ { \prime \prime } \in L ^ { 2 } [ 0 , 1 ] \}$ . We denote the density of $X$ as $q ( x )$ . The following theorem analytically computes the MSE under the regularization strength $\rho ( \cdot )$ , building upon (Wang et al., 2013) for regression problems. The proof of the Theorem is deferred to Appendix A. + +Theorem 1. Assume that $f ^ { \star } , q , I \in W _ { 2 } ^ { 2 }$ . Let $r ( t ) = - 1 / ( q ( t ) I ( t ) )$ and $\begin{array} { r } { L _ { 0 } = \int _ { - \infty } ^ { \infty } \frac { 1 } { 4 } \exp ( - 2 | t | ) d t } \end{array}$ If we choose $\lambda = C _ { 0 } n ^ { - 2 / 5 }$ for some constant $C _ { 0 } > 0$ , the asymptotic mean squared error is + +$$ +\operatorname* { l i m } _ { n \to \infty } M S E ( \hat { f } ) = C _ { n } \int _ { 0 } ^ { 1 } \lambda ^ { 2 } r ^ { 2 } ( t ) \left[ \frac { d } { d t } ( \rho ( t ) ( f ^ { * } ) ^ { \prime } ( t ) ) \right] ^ { 2 } + L _ { 0 } r ( t ) ^ { 1 / 2 } \rho ( t ) ^ { - 1 / 2 } d t +$$ + +in probability, where $C _ { n }$ is a scalar that only depends on $n$ + +Using the analytical formula of the test error above, we want to derive an approximately optimal choice of $\rho ( x )$ . A precise computation is infeasible, so we restrict ourselves to consider only $\rho ( x )$ that is constant within groups of examples. We introduce an additional structure – we assume the data can be divided into $k$ groups $[ a _ { 0 } , a _ { 1 } ) , \bar { [ } a _ { 1 } , a _ { 2 } ) , \cdot \cdot \cdot , [ a _ { k - 1 } , a _ { k } )$ . Each group $[ a _ { j } , a _ { j + 1 } )$ consists of an interval of data with approximately the same aleatoric uncertainty. We approximate $\rho ( t )$ is constant on each of the group $[ a _ { i } , a _ { i + 1 } )$ with value $\rho _ { i }$ . Plugging this piece-wise constant $\rho$ into the asymptotic MSE in Theorem 1, we obtain + +$$ +\underset { n \infty } { \operatorname* { l i m } } \mathrm { M S E } ( \hat { f } ) = \sum _ { j } [ \rho _ { j } ^ { 2 } \int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 2 } ( t ) [ \frac { d ^ { 2 } } { d t ^ { 2 } } f ^ { \star } ( t ) ] ^ { 2 } d t + \rho _ { j } ^ { - 1 / 2 } L _ { 0 } \int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 1 / 2 } ( t ) d t ] . +$$ + +Minimizing the above formula over $\rho _ { 1 } , \ldots , \rho _ { k }$ separately, we derive the optimal weights, $\rho _ { j } =$ $\left[ \frac { L _ { 0 } \int _ { a _ { j } } ^ { a _ { j + 1 } } r ( t ) ^ { 1 / 2 } d t } { 4 \int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 2 } ( t ) \left[ \frac { d ^ { 2 } } { d t ^ { 2 } } f ^ { \star } ( t ) \right] ^ { 2 } d t } \right] ^ { 2 / }$ 5 . In practice, we do not know $f ^ { \star }$ and $q ( x )$ , so we make the following simplifications. We assume that $q ( t )$ and $I ( t )$ are constant on each interval $[ a _ { j } , a _ { j + 1 } ]$ . In other words, we assume that $q ( t ) = q _ { j }$ and $I ( t ) = I _ { j }$ for all $t \in [ a _ { j } , a _ { j + 1 } ]$ . We further assume that $\textstyle { \frac { d ^ { 2 } } { d t ^ { 2 } } } f ^ { \star } ( t )$ is close to a constant on the entire space, because estimating the curvature in high dimension is difficult. This simplification yields $\rho _ { j } \propto \left[ \frac { q _ { j } ^ { - 1 / 2 } I _ { j } ^ { - 1 / 2 } } { q _ { j } ^ { - 2 } I _ { j } ^ { - 2 } } \right] ^ { 2 / 5 } = q _ { j } ^ { 3 / 5 } I _ { j } ^ { 3 / 5 }$ q3/5j I3/5j . We find the simplification works well in practice. + +Adaptive regularization with importance sampling. It is practically infeasible to implement the integration in equation 3 for high-dimensional data. We use importance sampling to approximate the integral: + +$$ +{ \mathrm { m i n i m i z e } } _ { f } L ( f ) \triangleq { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \ell ( f ( x _ { i } ) , y _ { i } ) + \lambda \sum _ { i = 1 } ^ { n } \tau _ { i } f ^ { \prime } ( x _ { i } ) ^ { 2 } +$$ + +Suppose $x _ { i } \in [ a _ { j } , a _ { j + 1 } )$ , we have that $\tau _ { i }$ should satisfy that $\tau _ { i } q _ { j } = \rho _ { j }$ so that the expectation of the regularization term in equation 5 is equal to that in equation 3. Hence, + +$$ +\tau _ { i } = I _ { j } ^ { 3 / 5 } q _ { j } ^ { - 2 / 5 } = I ( x _ { i } ) ^ { 3 / 5 } q ( x _ { i } ) ^ { - 2 / 5 } . +$$ + +Adaptive regularization for multi-class classification and regression. In fact, the proof of Theorem 1 is proved for general loss $\ell ( a , y )$ . Therefore, we can directly generalize it to multiclass classification and regression problems. For a regression problem, $\ell ( a , y )$ is the square loss: $\ell ( y , a ) = 0 . 5 ( y - a ) ^ { 2 }$ , the Fisher information $I ( x ) = 1$ . Therefore, for a regression problem, we can choose regularization weight $\tau _ { i } = q ( x _ { i } ) ^ { - 2 / 5 }$ . + +We heuristically extend the Lipschitz regularization technique discussed in Section 2.2 from nonparametric models to over-parameterized deep neural networks. Let $( x , y )$ be an example and $f _ { \theta }$ be an $r$ -layer neural network. We denote by $h ^ { ( j ) }$ the $j$ -th hidden layer of the network, by $\begin{array} { r } { \dot { J } ^ { ( j ) } ( x ) \triangleq \frac { \hat { \partial } } { \partial h ^ { ( j ) } } \mathcal { L } ( f ( x ) , y ) } \end{array}$ , i.e., the Jacobian of the loss w.r.t $h ^ { ( j ) }$ . We replace the regularization term $f ^ { \prime } ( x ) ^ { 2 }$ in equation 5 by $\begin{array} { r } { R ( x ) = { \left( \sum _ { j = 1 } ^ { r } | | J ^ { ( j ) } ( x ) | | _ { F } ^ { 2 } \right) } ^ { 1 / 2 } } \end{array}$ , which was proposed by (Wei & Ma, 2019a). As a proof of concept, we visualize the behavior of our algorithm in Figure 3, where we observe that the rare and noisy examples have significantly improved error due to stronger regularization. In contrast, a uniform regularization either overfits or underfits different subsets. + +Note that the differences from the 1-D case include the following three aspects. 1. The derivative is taken w.r.t to all the hidden layers for deep models, which has been shown to have superior generalization guarantees for neural networks by (Wei & Ma, 2019a;b). 2. An additional square root is taken in computing $R ( x )$ . This modified version may have milder curvature and be easier to tune. 3. We take the derivative of the loss instead of the derivative of the model, which outputs $k$ numbers for multi-class classification. This is because the derivative of the model requires $k$ times more time to compute. The regularized training objective is consequently + +$$ +\mathrm { m i n i m i z e } _ { f } L ( f ) \triangleq \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( \ell ( f ( x _ { i } ) , y _ { i } ) + \lambda \tau _ { i } R ( x _ { i } ) \right) , +$$ + +where $\tau _ { i }$ is chosen to be $\tau _ { i } = I ( x _ { i } ) ^ { 3 / 5 } / q ( x _ { i } ) ^ { 2 / 5 }$ following the formula equation 5 in Section 2.2 and $\lambda$ is a hyperparameter to control the overall scale of the regularization strength. We note that we do not expect this choice of $\tau _ { i }$ to be optimal for the high-dimensional case with all the modifications above – the optimal choice does depend on the nuances. However, we also observe that the empirical performance is not sensitive to the form of $\tau$ as long as it’s increasing in $I ( x )$ and decreasing in $q ( x )$ . That is, the more uncertain or rare an example is, the stronger regularization should be applied. + +In order to estimate the relative regularization strength $\tau _ { i }$ , the key difficulty lies in the estimation of uncertainty $I ( x )$ . As in the 1-D setting, we divide the examples into $k$ groups $G _ { 1 } , \ldots , G _ { k }$ (e.g., each group can correspond to a class), and estimate the uncertainty on each group. In the binary setting, $I ( x ) = \mathrm { \mathrm { V a r } } ( { \bar { Y } } | X = x ) = \operatorname* { P r } [ Y = 1 \mid X ] \cdot \operatorname* { P r } [ Y = 0 \ { \bar { | } } \ X ]$ can be approximated by $\begin{array} { r } { \tilde { I } ( x ) = 1 - \operatorname* { m a x } _ { k \in \{ 0 , 1 \} } \operatorname* { P r } [ Y = k \mid X = x ] } \end{array}$ up to a factor of at most 2. We use the same formula for multi-class setting as the approximation of the uncertainty. (As a sanity check, when $Y$ is concentrated on a single outcome, the uncertainty is 0.) Note that $\tilde { I } ( \stackrel { \cdot } { x } )$ is essentially the minimum possible error of any deterministic prediction on the data point $x$ . Assume that we have a sufficiently accurate pre-trained model, we can use its validation error to estimate $\tilde { I } ( x )$ : + +Then for all $x \in G _ { j }$ , we estimate $q ( x )$ and $I ( x )$ by $\forall x \in G _ { j } , q ( x ) \propto | G _ { j } | , I$ $I ( x ) \propto$ average validation error of a pre-trained model $f _ { \tilde { \theta } }$ on $G _ { j }$ + +The whole training pipeline is summarized in Algorithm 1. + +# Algorithm 1 Heteroskedastic Adaptive Regularization (HAR) + +Require: Dataset $\mathcal { D } = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ . A parameterized model $f _ { \theta }$ +1: Split training set $\mathcal { D }$ into $\mathcal { D } _ { \mathrm { t r a i n } }$ and $\mathcal { D } _ { \mathrm { v a l } }$ +2: $f _ { \tilde { \theta } } \gets$ Standard SGD Training on $\mathcal { D } _ { \mathrm { t r a i n } }$ +3: Estimate $I ( x ) , q ( x )$ with equation 7 using $f _ { \tilde { \theta } }$ on $\mathcal { D } _ { \mathrm { v a l } }$ , and compute $\tau _ { i } = I ( x _ { i } ) ^ { 3 / 5 } / q ( x _ { i } ) ^ { 2 / 5 }$ +4: +5: Initialize the model parameters $\theta$ randomly +6: $f _ { \boldsymbol { \theta } } \gets \mathsf { S G D }$ with the regularized objective as in equation 6 on the full dataset $\mathcal { D }$ + +# 3 EXPERIMENTS + +We experimentally show that our proposed algorithm HAR(Algorithm 1) improves the test performance of the noisier and rarer groups of examples (by stronger regularization) without negatively affecting the training and test performance of the other groups. We evaluate our algorithms on three vision datasets and one NLP dataset: CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009), IMDB-review (Maas et al., 2011) (see Appendix C.1), and WebVision (Li et al., 2017), a real-world heteroskedastic and imbalanced dataset. Please refer to Appendix B for low-level implementation details. + +Baselines. We compare our proposed HAR with the following baselines. The simplest one is (1) Empirical risk minimization (ERM): the vanilla cross-entropy loss with all examples having the same weights of losses. We select two representatives from the noise-cleaning line of approach. (2) Co-teaching (Han et al., 2018): two deep networks are trained simultaneously. Each network aims to identify clean data points that have small losses and use them to guide the training of the other network. (3) INCV (Chen et al., 2019): it extends Co-teacing to an interative version to estimate the noise ratio and select data. We consider three representatives from the reweighting-based methods, including two that learn the weighting using meta-learning. (4) MentorNet (Jiang et al., 2018): it pretrains a teacher network that outputs weights for examples that are used to train the student network with reweighting. (5) L2RW (Ren et al., 2018): it directly optimizes weights of each example in the training set by minimizing its corresponding loss on a small meta validation set. (6) MW-Net (Shu et al., 2019): it extends L2RW by explicitly defining a weighting function which depends only on the loss of the example. We also compare against two representatives from the robust loss function. (7) GCE (Zhang & Sabuncu, 2018): it generalizes mean average error and cross-entropy loss to obtain a new loss function. (8) DMI ( $\mathrm { X u }$ et al., 2019): it designs a new loss function based on generalized mutual information. In addition, as an essential ablation study, we consider vanilla uniform regularization. (9) Unif-reg: we apply the Jacobian regularizer on all examples with equal strength, and tune the strength to get the best possible validation accuracy. + +# 3.1 SIMULATING HETEROSKEDASTIC AND IMBALANCED DATASETS ON CIFAR + +Setting. Unlike previous works that test on uniform random or asymmetric noise, which is often not the case in reality, in this paper we test our method on more realistic noisy settings, as suggested by Patrini et al. (2017); Zhang & Sabuncu (2018). In order to simulate heteroskedasticity, we only corrupt semantically-similar classes. For CIFAR-10, we exchange $40 \%$ of the labels between classes ‘cat’ and ‘dog’, and between ‘truck’ and ‘automobile’. CIFAR-100 has 100 classes grouped into 20 super classes. For each class of the 5 classes under the super class ‘vehicles $_ { 1 } ,$ and ‘vehicles $_ { 2 } \cdot$ , we corrupt the labels with $40 \%$ probability uniformly randomly to the rest of four classes under the same super class. As a result, the 10 classes under super class ‘vehicle $_ { 1 } ,$ and ‘vehicle $_ { 2 } \cdot$ have high label noise level and the corruption are only within the same super class. Heteroskedasticity of the labels and imbalance of the inputs commonly coexist in the real world settings. HAR can take both of them into account. To understand the challenge imposed by the entanglements of heteroskedasticity and imbalance, and compare HAR with the aforementioned baselines, we inject data imbalance concurrently with the heteroskedastic noise. We remove samples from the corrupted classes to simulate the most difficult scenario — the rare and noisy groups overfit significantly. (A more benign interaction between the noises and imbalance is that the rare classes have lower noise level, we defer it to Appendix C.3.) We use the imbalance ratio to denote the frequency ratio between the frequent (and clean) classes to the rare (and corrupted) classes. We consider imbalance ratio to be 10 and 100. + +Result. Table 1 summarizes the results. Since examples from rare classes tend to have larger training and validation loss regardless of whether the labels are correct or not, noise-cleaning based methods might drop excessive examples with correct labels. We examined the noise ratio of dropped samples for INCV under the setting of imbalance ratio equals 10. Among all dropped examples, there is only $1 9 . 2 \%$ of true noise examples. In addition, the rare class examples selected still have $2 9 . 8 \%$ of label noise. This explains that the significant decrease of accuracies of Co-teaching and INCV on corrupted and rare classes. Reweighting-based methods tend to suffer from the loss of accuracy in other more frequent classes, which is aligned with the findings in Cao et al. (2019). While the aforementioned baselines struggle to deal with heteroskedasticity and imbalance together, HAR is able to put them under the same regularization framework and achieve significant improvements. Notably, HAR also shows improvement over uniform regularization with optimally tuned strength. This clearly demonstrates the importance of introducing adaptive regularization among all examples for a better trade-off. A more detailed ablation study on the trade-off between training accuracy and validation accuracy can be found in Section 3.3. + +Table 1: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic and imbalanced CIFAR-10. HAR significantly improves noisy and rare classes, while keeping the accuracy on other classes almost unaffected. + +
Imbalance ratio Method10100
Noisy&Rare Cls.Clean Cls.Noisy&Rare Cls.Clean Cls.
ERM52.9 ±1.294.4 ± 0.118.9 ± 1.094.2 ± 0.1
Co-teaching30.2 ± 2.388.9± 0.315.4± 2.886.4± 0.7
INCV48.9 ± 1.794.0± 0.225.8 ± 1.893.8 ± 0.2
MentorNet54.1 ± 1.090.3 ± 0.528.3 ± 1.590.2 ±0.4
L2RW44.3 ± 2.090.1± 0.531.2 ± 1.989.7±0.7
MW-Net55.4 ± 1.191.7 ± 0.535.6 ± 1.692.3 ± 0.5
GCE48.2±0.691.6 ± 0.314.1 ± 2.091.7 ± 0.4
DMI44.7 ± 2.390.7 ±0.814.0 ± 2.191.8 ± 0.6
Unif-reg (optimal)53.9 ± 0.992.1± 0.236.7 ± 1.092.4± 0.3
Ours (HAR)63.5 ± 0.894.3 ± 0.242.4±0.794.0±0.2
+ +# 3.2 ABLATION STUDY ON CIFAR + +We disentangle the problem setting to show the effectiveness of our unified framework. + +Simulating heteroskedastic noise on CIFAR. We study the uncertainty part of HAR by testing under the setting with only heteroskedastic noise. The type of noise injection is the same as Section 3.1. + +We report the top-1 validation accuracy of various methods in Table 2. Aligned with our analysis in Section 4, we observe that both noise-cleaning and reweighting based methods don’t get a comparable accuracy on noisy classes with applying strong regularization $\lambda = 0 . 1$ ) under this heteroskedastic setting. We observe the behavior that too strong regularization impede the model from fitting informative samples, thus it could lead to a decrease on clean classes’ accuracy. On the contrary, too weak regularization leads to overfitting the noisy examples thus the accuracy on noisy classes do not reach the optimal. + +Interestingly, we find that even the well-studied CIFAR-100 dataset has intrinsic heteroskedasticity and HAR can improve over uniform regularization to some extent. Please refer to Appendix C.2 for the results on CIFAR-100 and Appendix C.1 for results on IMDB-review. + +Table 2: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic CIFAR-10 and CIFAR-100 for the noisy classes and the clean classes. + +
Dataset MethodCIFAR-10CIFAR-100
Avg. Noisy Cls.Avg. Clean Cls.Avg. Noisy Cls.Avg. Clean Cls.
ERM68.6±0.293.6±0.265.3 ± 0.367.8 ± 0.2
Co-teaching64.7 ± 0.489.1± 0.359.8 ± 0.465.3± 0.3
INCV76.7 ± 0.693.0±0.266.2±0.368.6 ±0.2
MentorNet71.1 ± 0.493.7± 0.265.9 ± 0.367.5 ± 0.3
L2RW70.1± 0.392.5 ± 0.365.1± 0.567.0± 0.3
MW-Net75.0± 0.394.4± 0.265.7 ± 0.369.1± 0.2
GCE62.6 ±1.190.2±0.261.2 ±0.666.9 ±0.2
DMI73.2 ± 0.790.8 ± 0.264.8 ± 0.567.1 ± 0.2
Unif-reg (入= 0.1)77.5 ± 0.692.3± 0.269.3 ± 0.566.6 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1±0.268.5 ± 0.368.6 ±0.2
Ours (HAR)80.7 ± 0.394.5 ± 0.274.2 ± 0.369.3±0.2
+ +Simulating data imbalance on CIFAR. We study the density part of HAR by testing under the setting with only data imbalance. We follow the same setting as Cao et al. (2019) to create imbalanced CIFAR. Long-tailed imbalance follows an exponential decay in sample sizes across different classes. For step imbalance setting, all rare classes have the same sample size, as do all frequent classes. Our approach achieves better results than LDAM-DRW and is comparable to recent state-of-the-art methods under the imbalanced setting. + +Table 3: Top-1 validation errors of ResNet-32 on imbalanced CIFAR-10 and CIFAR-100. + +
DatasetImbalanced CIFAR-10Imbalanced CIFAR-100
Imbalance Typelong-tailedsteplong-tailedstep
Imbalance Ratio10010100101001010010
ERM29.6413.6136.7017.5061.6844.3061.4545.37
Focal29.6213.3436.0916.3661.5944.2261.4346.54
CB Focal25.4312.9039.7316.5463.9842.0180.2449.98
LDAM-DRW22.9711.8423.0812.1957.9641.2954.6440.54
BBN (Zhou et al., 2020)20.1811.6821.6411.9957.4440.8857.4440.36
HAR-DRW20.4610.6220.2711.5855.3538.9851.7337.54
+ +Table 4: Validation accuracy of ResNet-50 when tuning the regularization strength on mini WebVision. HAR stands out of the trade-off constraint of fitting and generalization. + +
Train AccVal Acc
Reg StrengthTop1Top5Top1Top 5
069.0188.6459.4080.84
Unif-reg (入= 0.01)68.9688.5464.3286.11
Unif-reg (λ= 0.02)67.0287.5164.4085.92
Unif-reg (入= 0.05)65.1186.3365.8086.84
Unif-reg (λ = 0.1)63.3584.9865.0486.56
Adaptive (HAR)69.1288.4169.2088.96
+ +# 3.3 EVALUATION ON WEBVISION WITH REAL-WORLD HETEROGENEITY + +WebVision (Li et al., 2017) contains 2.4 million images crawled from Google and Flickr using 1,000 labels shared with the ImageNet dataset. Its training set is both heteroskedastic and imbalanced (detailed statistics can be found in (Li et al., 2017)), and it is considered as a popular benchmark for noise robust learning. As the full dataset is very large, we follow (Jiang et al., 2018) to use a mini version, which contains the first 50 classes of the Google subset of the data. Following the standard protocol (Jiang et al., 2018), we test the trained model on the WebVision validation set and the ImageNet validation set. We use ResNet-50 for ablation study and InceptionResNet-v2 for a fair comparison with the baselines. We report results comparing against other state-of-the-art approaches in Table 5. Strikingly, HAR achieves significant improvement. + +Ablation study. We demonstrate the trade-off between training accuracy and validation accuracy on mini WebVision with various uniform regularization strength and HAR in Table 4. It’s evident that when we gradually increase the overall uniform regularization strength, the training accuracy continues to decrease, and the validation accuracy reaches its peak at 5e-2. While a strong regularization could improve generalization, it reduces deep networks’ capacity to fit the training data. However, with our proposed HAR, we only enforce strong regularization on a subset so that we improve the generalization on noisier groups while maintaining the overall training accuracy not affected. + +# 4 RELATED WORK + +Our work is closely related to the following methods and directions. + +Noise-cleaning. The key idea of noise-cleaning is to identify and remove (or re-label) examples with wrong annotations. The general procedure for identifying mislabeled instances has a long history (Brodley & Friedl, 1999; Wilson & Martinez, 1997; Zhao & Nishida, 1995). Some recent works tailored this idea for deep neural networks. Veit et al. (2017) trains a label cleaning network on a small set of data with clean labels, and uses this model to identify noises in large datasets. To circumvent the requirement of a clean subset, Malach & Shalev-Shwartz (2017) train two networks simultaneously and perform update steps only in case of disagreement. Similarly, in co-teaching (Han et al., 2018), each network selects a certain number of small-loss samples and feeds them to its peer network. Chen et al. (2019) further extends the co-training strategy and comes up with an iterative version that performs even better empirically. Recently Song et al. (2020) discovers that it is not necessary to maintain two networks. Removing examples whose training loss exceeds a certain threshold before learning rate decay can also get robust performance. + +Table 5: Validation accuracy of InceptionResNet-v2 on WebVision and ImageNet validation sets. HAR demonstrates significant improvements over the previous state-of-the-arts. + +
Train Test Methodmini WebVisionfull WebVision
WebVisionImageNetWebVisionImageNet
Top 1Top5Top 1Top 5Top 1Top 5Top 1Top 5
ERM62.580.858.581.869.787.062.983.6
Co-teaching63.685.261.584.71-11
INCV65.285.361.685.0--1-
MentorNet63.081.457.879.970.888.062.583.0
Ours (HAR)75.590.770.390.075.090.667.186.7
+ +Reweighting. Reweighting training data has shown its effectiveness on noisy data (Liu & Tao, 2015). Its challenge lies in the difficulty of weights estimation. Ren et al. (2018) proposes a meta-learning algorithm to assign weights to training examples based on its gradient direction with the one on a clean validation set. Recently, Shu et al. (2019) proposes to learn an explicit loss-weight function to mitigate the optimizing issue of (Ren et al., 2018). Another line of work resorts to curriculum learning by either designing an easy-to-hard strategy of training (Guo et al., 2018) or introducing an extra network (Jiang et al., 2018) to assign weights. + +Noise-cleaning and reweighting methods usually rely on the empirical loss to determine if a sample is noisy. However, when the dataset is heteroskedastic, each example’s training/validation loss no longer correlates well with its noise level. In such cases, we argue that changing the strength of regularization is a more conservative adaption and suffers less from uncertain estimation, compared to changing the weights of losses (Please refer to Section C.4 for empirical justifications). + +Robust loss function. Another line of works has attempted to design robust loss functions (Ghosh et al., 2017; Xu et al., 2019; Zhang & Sabuncu, 2018; Patrini et al., 2017; Cheng et al., 2017; Menon et al., 2016). They usually rely on prior assumption about latent transition matrix that might not hold in practice. On the contrary, we focus on more realistic settings. + +Regularization. Regularization based techniques have also been explored to combat label noise. Li et al. (2019) proves that SGD with early stopping is robust to label noise. Hu et al. (2020) provides theoretical analysis of two additional regularization methods. While these methods consider a uniform regularization on all training examples, our work emphasizes on adjusting the weights of regularizers in search of a better generalization than uniform assignment. + +# 5 CONCLUSION + +We propose a unified framework (HAR) for training on heteroskedastic and imbalanced datasets. Our method achieves significant improvements over the previous state-of-the-arts on a variety of benchmark vision and language tasks. We provide theoretical results as well as empirical justifications by showing that ambiguous, mislabeled, and rare examples all benefit from stronger regularization. We further provide the formula for optimal weighting of regularization. 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Journal of Artificial Intelligence Research, 3:119–145, 1995. +Boyan Zhou, Quan Cui, Xiu-Shen Wei, and Zhao-Min Chen. Bbn: Bilateral-branch network with cumulative learning for long-tailed visual recognition. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9719–9728, 2020. + +# A PROOFS OF THEOREM 1 + +We prove a general theorem here. In particular, we have the general theorem below. + +Theorem 2. Assume that $f ^ { \star } , q , I \in W _ { 2 } ^ { 2 }$ . Suppose (1) $\ell ( a , y )$ is convex and three times continously differentiable with respect to $a$ , (2) there exist constants $0 < c < C < \infty$ such that $c \leq I ( X ) \leq \bar { C }$ almost surely, and $\epsilon \triangleq \nabla _ { a } \ell ( f ^ { \star } ( X ) , Y )$ satisfies $\mathbb { E } [ \epsilon | X ] = 0 \mathrm { \it \Omega }$ , $\mathbb { E } [ \epsilon ^ { 2 } | X ] = I ( X )$ and $\mathbb { E } [ \epsilon ^ { 4 } | X ] < \infty$ almost surely. Let $r ( t ) = - 1 / ( q ( t ) I ( t ) )$ and $\begin{array} { r } { L _ { 0 } = \int _ { - \infty } ^ { \infty } \frac { 1 } { 4 } \exp ( - 2 | t | ) d t . } \end{array}$ . If we choose $\lambda = C _ { 0 } n ^ { - 2 / 5 }$ for some constant $C _ { 0 } > 0$ , the asymptotic mean squared error of $\hat { f }$ by equation 4 is + +$$ +\operatorname* { l i m } _ { n \to \infty } M S E ( \hat { f } ) = C _ { n } \int _ { 0 } ^ { 1 } \lambda ^ { 2 } r ^ { 2 } ( t ) \left[ \frac { d } { d t } ( \rho ( t ) ( f ^ { * } ) ^ { \prime } ( t ) ) \right] ^ { 2 } + L _ { 0 } r ( t ) ^ { 1 / 2 } \rho ( t ) ^ { - 1 / 2 } d t +$$ + +in probability, where $C _ { n }$ is a scalar that only depends on $n$ . + +It is easy to check that the logistic loss satisfies the condition of the theorem. + +The proof strategy of Theorem 2 is adopted from the proof of Theorem 2 of (Wang et al., 2013) by generalizing it from the least square loss to logistic loss. + +The high level idea is to reformulate $\hat { f }$ as solutions to ordinary differential equations. Let $( \gamma _ { v } , h _ { v } )$ be the (normalized) solution of the following equation + +$$ +\begin{array} { r l } & { - \rho ( t ) h _ { v } ^ { \prime \prime } ( t ) = \gamma _ { v } I ( t ) q ( t ) h _ { v } ( t ) , } \\ & { h _ { v } ^ { ' } ( 0 ) = h _ { v } ^ { ' } ( 1 ) = 0 , \mathrm { ~ a n d ~ } h _ { v } ^ { ' \prime } ( 0 ) = h _ { v } ^ { ' \prime } ( 1 ) = 0 . } \end{array} +$$ + +We define the the leading term of $\hat { f } - f ^ { \star }$ as $S _ { n , \lambda } ( f ^ { \star } )$ as + +$$ +S _ { n , \lambda } ( f ^ { \star } ) = { \frac { 1 } { n } } \epsilon _ { i } K _ { X _ { i } } - W _ { \lambda } f ^ { \star } , +$$ + +$$ +K _ { t } ( \cdot ) = \sum _ { v } \frac { h _ { v } ( t ) } { 1 + \lambda \gamma _ { v } } h _ { v } ( \cdot ) \mathrm { a n d } W _ { \lambda } h _ { v } ( \cdot ) = \frac { \lambda \gamma _ { v } } { 1 + \lambda \gamma _ { v } } h _ { v } ( \cdot ) . +$$ + +By Proposition 2.1 and Theorem 3.4 of Shang et al. (2013), we have + +$$ +\operatorname* { s u p } _ { x } | \hat { f } ( x ) - f ^ { \star } ( x ) - S _ { n , \lambda } ( f ^ { \star } ) ( x ) | = o _ { P } ( n ^ { - 1 / 3 } ) . +$$ + +Following the same proof of Theorem 2 of (Wang et al., 2013), we can simplify the definition of $K _ { t }$ and $W ^ { \lambda }$ as + +$$ +K _ { t } ( x ) = \frac { I ( t ) } { q ( t ) } J ( t , x ) \mathrm { a n d } W _ { \lambda } f ^ { \star } ( t ) = \lambda r ( t ) \left[ \frac { d } { d t } ( \rho ( t ) ( f ^ { \ast } ) ^ { \prime } ( t ) ) \right] , +$$ + +where $J ( t , s ) ~ = ~ \beta \rho ( s ) Q _ { \beta } ^ { \prime } ( s ) L _ { 0 } ( \beta | Q _ { \beta } ( t ) - Q _ { \beta } ( s ) | )$ and $\begin{array} { r } { Q _ { \beta } ( t , s ) = \int _ { 0 } ^ { t } ( r ( s ) \rho ( s ) ) ^ { - 1 / 2 } ( 1 + } \end{array}$ $O ( \beta ^ { - 1 } ) ) d s$ and $\beta = 1 / \sqrt { \lambda }$ . Plugging equation 11 into equation 10, we then have + +$$ +\operatorname* { l i m } _ { n \to \infty } \mathrm { M S E } ( \hat { f } ) = C _ { n } \int _ { 0 } ^ { 1 } \lambda ^ { 2 } r ^ { 2 } ( t ) \left[ \frac { d } { d t } ( \rho ( t ) ( f ^ { \ast } ) ^ { \prime } ( t ) ) \right] ^ { 2 } + L _ { 0 } r ( t ) ^ { 1 / 2 } \rho ( t ) ^ { - 1 / 2 } d t +$$ + +# B IMPLEMENTATION DETAILS + +We develop our core algorithm in PyTorch (Paszke et al., 2017). + +Implementation details for CIFAR. We follow the simple data augmentation used in (He et al., 2016) with only random crop and horizontal flip. We use ResNet-32 as our base network and repeat all experiments for 3 runs. We use standard SGD with momentum of 0.9, weight decay of $1 \times \bar { 1 0 } ^ { - 4 }$ for training. The model is trained with a batch size of 128 for 120 epochs. We anneal the learning rate by a factor of 10 at 80 and 100 epochs. We group the data by class labels, and by default we split $\mathcal { D }$ equally and randomly into $\mathcal { D } _ { \mathrm { t r a i n } }$ and $\mathcal { D } _ { \mathrm { v a l } }$ . As for the Jacobian regularizer, we sum over the frobenius norm of the gradients of all normalization layers’ (BN by default) activations with respect to the classification loss. For experiments of HAR, we tune $\lambda$ so that the largest enforced regularization strength $( \lambda \tau _ { i } )$ is 0.1. We train each model with 1 NVIDIA GeForce RTX 2080 Ti. + +Implementation details for IMDB-review. We train a two-layer bidirectional LSTM (Huang et al., 2015) with 256 units followed with 0.5 dropout before the linear classifier. The network is trained for 20 epochs with Adam optimizer (Kingma & Ba, 2014). For HAR, we tune $\lambda$ so that the largest enforced regularization strength $( \lambda \tau _ { i } )$ is 0.1. We train each model with 1 NVIDIA GeForce RTX 2080 Ti. + +Implementation details for WebVision. We use the standard data augmentation same as (He et al., 2016) including random crop and horizontal flip. For mini WebVision, We train the network for 90 epochs using standard SGD with a batch size of 128. The initial learning rate is 0.1 and is annealed by a factor of 10 at epoch 60 and 90. For full WebVision, We train the network for 50 epochs using standard SGD with a batch size of 256. The initial learning rate is 0.1 and is annealed by a factor of 10 at epoch 30 and 40. For experiments of HAR, we tune $\lambda$ so that the largest enforced regularization strength $( \lambda \tau _ { i } )$ is 0.1. We train each model with 8 NVIDIA Tesla V100 GPUs. + +Runtime analysis. Because the pre-trained model only trains on half of the training data and is only done once, the run-time of HAR is at most twice of the time for ERM. Many baselines in our paper use sophisticated pipelines and are slower than HAR. For example, INCV trains 2 models simultaneously for 4 times from random initialization to get a clean training set. MW-Net has a very slow convergence rate, which is a common issue for meta-learning. + +# C ADDITIONAL RESULTS + +# C.1 SIMULATING HETEROSKEDASTIC NOISE ON IMDB-REVIEW. + +IMDB-review dataset has a total of 50,000 (25,000 positive and 25,000 negative reviews) movie reviews for binary sentiment classification (Maas et al., 2011). To simulate heteroskedastic noise for this binary classification problem, we project $5 \%$ of the labels of negative reviews to positive, and $40 \%$ in the reverse direction. Table 6 summarizes the results. The proposed HAR outperforms the ERM baseline with various strength of uniform regularization. + +Table 6: Top-1 validation accuracy (averaged over 3 runs) on heteroskedastic IMDB-review dataset. + +
Reg StrengthAcc. of neg. reviewsAcc. of pos. reviewsMean Acc
091.9 ± 2.050.9 ±1.871.4± 0.5
Unif-reg (入= 0.01)94.3 ±1.851.9 ± 2.073.1± 0.3
Unif-reg (入= 0.1)91.5 ±1.964.3 ±1.677.9 ± 0.4
Ours (HAR)93.1 ± 1.572.8±1.783.0 ± 0.3
+ +C.2 EVALUATION ON CIFAR-100 WITH REAL-WORLD HETEROSKEDASTICITY + +It is acknowledged that CIFAR-100 training set contains noisy examples. For instance, some “tiger” examples are labeled as “leopard” (“tiger” is a defined class as well). There are also noisy examples that contain multiple objects, or are more ambiguous in terms of indentity (Song et al., 2020). We find that HAR can improve over uniform regularization on the well-studied CIFAR-100 due to its heteroskedasticity and the results are reported in Table 7. + +Table 7: Top-1 validation accuracy (average over 3 runs) of ResNet-32 on the original CIFAR-100. + +
Reg StrengthTrain AccVal Acc
096.0±0.169.8± 0.2
Unif-reg (入= 0.001)96.4± 0.170.0± 0.2
Unif-reg (入= 0.01)95.7 ± 0.170.6 ±0.1
Unif-reg (入= 0.1)88.8±0.170.5 ± 0.1
Adaptive (HAR)96.2 ± 0.171.4±0.1
+ +As mentioned in Section 3.1, we consider another variant of heteroskedastic and imbalanced dataset such that the rare classes have low noise level. To simulate this setting, we make the clean classes have fewer labels than the corrupted classes on the heteroskedastic CIFAR-10 we created in Section ??. + +Table 8 summarizes the results. For the setting of imbalance ratio equals 10, INCV automatically drops $34 . 1 \%$ of examples from the clean and rare classes, which results in a decrease of mean accuracy on the rare and clean classes. HAR is able to achieve improvements on both noisy classes and rare classes by enforcing the optimal regularization. + +Table 8: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic and imbalanced CIFAR-10. + +
Imbalance ratio Method10100
Noisy Cls.Rare Cls.Noisy Cls.Rare Cls.
ERM59.9 ± 1.165.2 ± 0.760.4± 0.97.9 ± 1.3
Co-teaching63.1 ± 2.359.4 ± 1.453.8 ± 1.94.4±1.8
INCV74.5 ± 1.263.7 ± 0.868.3 ± 1.82.1 ±1.3
MentorNet67.3 ± 1.665.5 ± 1.263.3 ± 1.510.8 ±1.9
L2RW65.8 ±1.466.3 ± 1.262.4 ± 2.111.3 ± 2.9
MW-Net71.4± 0.667.7 ± 0.665.0± 1.613.5 ± 2.4
GCE64.6 ± 1.160.2 ± 1.371.2 ± 1.92.6±1.4
DMI72.3 ± 1.563.3 ±1.270.8 ±1.76.2 ±1.9
Ours (HAR)76.1 ± 0.872.1 ± 1.073.0 ± 1.626.1±0.8
+ +# C.4 COMPARING THE EFFECT OF WEIGHTS ON LOSSES AND REGULARIZERS + +As discussed in Section 4, we train several classifiers with alternative weights selection scheme which are not optimal. We consider the following two alternatives. (1) random: we draw the weights from a uniform distribution with the same mean as the weights of MW-Net and HAR. (2) inverse: we take the inverse of the weights learned by MW-Net and HAR and then normalize them to ensure the average reweighting/regularization strength remains the same. + +We conducted experiments on the heteroskedastic CIFAR-10 introduced in Section ?? and the results are summarized in Table 9. We could conclude that changing the weights of the regularizer is a more conservative adaption and less susceptible to uncertain estimation than reweighting. + +Table 9: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic CIFAR-10 by changing the weighting scheme. + +
MethodAvg. Noisy Cls.Avg. . Clean Cls.
ERM68.6±0.293.6 ± 0.2
Reweight (MW-Net)75.0 ± 0.394.4± 0.2
Reweight (random)62.1±0.592.9 ± 0.5
Reweight (inverse)13.1 ± 0.990.9 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1± 0.2
Adap-reg (HAR)80.7±0.394.5 ±0.2
Adap-reg (random)74.8 ± 0.494.2±0.2
Adap-reg (inverse)73.2 ± 0.594.0± 0.2
+ +# C.5 VISUALIZATION + +In order to better understand how the proposed HAR works on real-world heteroskedastic datasets, we plot the per-class key statistics used by HAR and validation errors in Figure 4. We observe that HAR outperforms the tuned uniform regularization baseline on the majority of the classes. + +![](images/c076753ebcd5b8f3da8f361db87212d171eb3c37ef46afdae950c98d61c3d805.jpg) +Figure 4: Visualizations of per-class top-1 error and regularization strength of HAR on mini WebVision dataset. The classes are sorted in the ascending order of applied regularization strength. \ No newline at end of file diff --git a/parse/train/mEdwVCRJuX4/mEdwVCRJuX4_content_list.json b/parse/train/mEdwVCRJuX4/mEdwVCRJuX4_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a635a32732af04c37084d356d1c77f41082e781f --- /dev/null +++ b/parse/train/mEdwVCRJuX4/mEdwVCRJuX4_content_list.json @@ -0,0 +1,1966 @@ +[ + { + "type": "text", + "text": "HETEROSKEDASTIC AND IMBALANCED DEEP LEARNING WITH ADAPTIVE REGULARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 99, + 826, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Kaidi $\\mathbf { C a o ^ { 1 } }$ , Yining Chen1, Junwei $\\mathbf { L } \\mathbf { u } ^ { 2 }$ , Nikos Arechiga3, Adrien Gaidon3, Tengyu Ma1 1Stanford University, 2Harvard University, 3Toyota Research Institute {kaidicao,cynnjjs,tengyuma}@stanford.edu ", + "bbox": [ + 186, + 167, + 789, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 250, + 544, + 265 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Real-world large-scale datasets are heteroskedastic and imbalanced – labels have varying levels of uncertainty and label distributions are long-tailed. Heteroskedasticity and imbalance challenge deep learning algorithms due to the difficulty of distinguishing among mislabeled, ambiguous, and rare examples. Addressing heteroskedasticity and imbalance simultaneously is under-explored. We propose a data-dependent regularization technique for heteroskedastic datasets that regularizes different regions of the input space differently. Inspired by the theoretical derivation of the optimal regularization strength in a one-dimensional nonparametric classification setting, our approach adaptively regularizes the data points in higher-uncertainty, lower-density regions more heavily. We test our method on several benchmark tasks, including a real-world heteroskedastic and imbalanced dataset, WebVision. Our experiments corroborate our theory and demonstrate a significant improvement over other methods in noise-robust deep learning. ", + "bbox": [ + 233, + 282, + 766, + 463 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 489, + 336, + 505 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In real-world machine learning applications, even well-curated training datasets have various types of heterogeneity. Two main types of heterogeneity are: (1) data imbalance: the input or label distribution often has a long-tailed density, and (2) heteroskedasticity: the labels given inputs have varying levels of uncertainties across subsets of data stemming from various sources such as the intrinsic ambiguity of the data or annotation errors. Many deep learning algorithms have been proposed for imbalanced datasets (e.g., see (Wang et al., 2017; Cao et al., 2019; Cui et al., 2019; Liu et al., 2019) and the reference therein). However, heteroskedasticity, a classical notion studied extensively in the statistical community (Pintore et al., 2006; Wang et al., 2013; Tibshirani et al., 2014), has so far been under-explored in deep learning. This paper focuses on addressing heteroskedasticity and its interaction with data imbalance in deep learning. ", + "bbox": [ + 174, + 521, + 825, + 660 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Heteroskedasticity is often studied in regression analysis and refers to the property that the distribution of the error varies across inputs. In this work, we mostly focus on classification, though the developed technique also applies to regression. Here, heteroskedasticity reflects how the uncertainty in the conditional distribution $y \\mid x$ , or the entropy of $y \\mid x$ , varies as a function of $x$ . Real-world datasets are often heteroskedastic. For example, Li et al. (2017) shows that the WebVision dataset has a varying number of ambiguous or true noisy examples across classes.2 ", + "bbox": [ + 174, + 667, + 825, + 751 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Conversely, we consider a dataset to be homoscedastic if every example is mislabeled with a fixed probably $\\epsilon$ , as assumed by many prior theoretical and empirical works on label corruption (Ghosh et al., 2017; Han et al., 2018; Jiang et al., 2018; Mirzasoleiman et al., 2020). We note that varying uncertainty in $y \\mid x$ can come from at least two sources: the intrinsic semantic ambiguity of the input, and the (data-dependent) mislabeling introduced by the annotation process. Our approach can handle both types of noisy examples in a unified way, but for the sake of comparisons with past methods, we call them “ambiguous examples” and “mislabeled examples” respectively, and refer to both of them as “noisy examples”. ", + "bbox": [ + 174, + 758, + 825, + 869 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/b0aa3f523002d1c7bc925b084339bea809fac9e5918ef2d25e1e376f5cd78fab.jpg", + "image_caption": [ + "Figure 1: Histogram of the distributions of losses on an imbalanced and noisy CIFAR-10 dataset. Clean but rare examples tend to have larger losses, similar to the noisy examples in frequent classes. " + ], + "image_footnote": [], + "bbox": [ + 176, + 104, + 464, + 268 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/9499897cf2f9e6c9f97933d2b6e3c3eb440e25eb54680fc3ee92050128d75b8b.jpg", + "image_caption": [ + "Figure 2: Real-world datasets have various sources of heterogeneity and it could be hard to distinguish one from another. They require mutuallyexclusive reweighting strategy, but they all benefit from stronger regularization. " + ], + "image_footnote": [], + "bbox": [ + 493, + 106, + 815, + 268 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Overparameterized deep learning models tend to overfit more to the noisy examples (Arpit et al., 2017; Frénay & Verleysen, 2013; Zhang et al., 2016). To address this issue, a common approach is to detect noisy examples by selecting those with large training losses, and then remove them from the (re-)training process. However, an input’s training loss can also be big because it is rare or ambiguous (Hacohen & Weinshall, 2019; Wang et al., 2019), as shown in Figure 1. Noise-cleaning methods could fail to distinguish mislabeled from rare/ambiguous examples (see Section 3.1 for empirical proofs). Though dropping the former is desirable, dropping the latter loses important information. Another popular approach is reweighting methods that reduce the contribution of noisy examples in optimization. However, failing to distinguish between mislabeled and rare/ambiguous examples makes the decision of the weights tricky – mislabeled examples require small weights, whereas rare / ambiguous examples benefit from larger weights (Cao et al., 2019; Shu et al., 2019). ", + "bbox": [ + 174, + 380, + 825, + 532 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We propose a regularization method that deals with noisy and rare examples in a unified way. We observe that mislabeled, ambiguous, and rare examples all benefit from stronger regularization (Hu et al., 2020; Cao et al., 2019). We apply a Lipschitz regularizer (Wei & Ma, 2019a;b) with varying regularization strength depending on the particular data point. Through theoretical analysis in the one-dimensional setting, we derive the optimal regularization strength for each training example. The optimal strength is larger for rarer and noisier examples. Our proposed algorithm, heteroskedastic adaptive regularization (HAR), first estimates the noise level and density of each example, and then optimizes a Lipschitz-regularized objective with input-dependent regularization with strength provided by the theoretical formula. ", + "bbox": [ + 174, + 539, + 825, + 664 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In summary, our main contributions are: (i) we propose to learn heteroskedastic imbalanced datasets under a unified framework, and theoretically study the optimal regularization strength on onedimensional data. (ii) we propose an algorithm, heteroskedastic adaptive regularization (HAR), which applies stronger regularization to data points with high uncertainty and low density. (iii) we experimentally show that HAR achieves significant improvements over other noise-robust deep learning methods on simulated vision and language datasets with controllable degrees of data noise and data imbalance, as well as a real-world heteroskedastic and imbalanced dataset, WebVision. ", + "bbox": [ + 174, + 671, + 825, + 768 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 ADAPTIVE REGULARIZATION FOR HETEROSKEDASTIC DATASETS ", + "text_level": 1, + "bbox": [ + 178, + 789, + 746, + 805 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 BACKGROUNDS ", + "text_level": 1, + "bbox": [ + 174, + 820, + 323, + 834 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We first introduce general nonparametric tools that we use in our analysis, and review the dependency of optimal regularization strength on the sample size and noise level. ", + "bbox": [ + 174, + 847, + 823, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Over-parameterized neural networks as nonparametric methods. We use nonparametric method as a surrogate for neural networks because they have been shown to be closely related. Recent work (Savarese et al., 2019) shows that the minimum norm two-layer ReLU network that fits ", + "bbox": [ + 176, + 882, + 825, + 922 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/9f6b9f919811d48d690fed50dff8093090a7d6a6528fc0487af6d4f5ee64ac11.jpg", + "image_caption": [ + "Figure 3: A one-dimensional example with a three-layer neural network in heteroskedastic and imbalanced regression setting. The curve in blue is the underlying ground truth and the dots are observations with heteroskedastic noise. This example shows that uniformly weak regularization overfits on noisy and rare data (on the right half), whereas uniformly strong regularization causes underfitting on the frequent and oscillating data (on the left half). The adaptive regularization does not underfit the oscillating data but still denoise the noisy data. We note that standard nonparametric methods such as cubic spline do not work here because they also use uniform regularization. " + ], + "image_footnote": [], + "bbox": [ + 184, + 111, + 482, + 299 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "the training data is in fact a linear spline interpolation. Parhi & Nowak (2019) extend this result to a broader family of neural networks with a broader family of activations. ", + "bbox": [ + 173, + 334, + 823, + 363 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given a training dataset $\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ , nonparametric method with penalty works as follows. Let $\\mathcal { F } : \\mathbb { R } \\mathbb { R }$ be a twice-differentiable model family. We aim to fit the data with smoothness penalty ", + "bbox": [ + 173, + 369, + 823, + 398 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1751915e4717c596fc2cbe14189e3caed5fc245ba2a2e06d1bdf3e706fe7a006.jpg", + "text": "$$\n\\operatorname* { m i n } _ { f } { \\ } { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } \\ell ( f ( x _ { i } ) , y _ { i } ) + \\lambda \\int ( f ^ { \\prime } ( x ) ) ^ { 2 } d x\n$$", + "text_format": "latex", + "bbox": [ + 354, + 405, + 643, + 446 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Lipschitz regularization for neural networks. Lipschitz regularization has been shown to be effective for deep neural networks as well. Wei & Ma (2019a) proves a generalization bound of neural networks dependent on the Lipschitzness of each layer with respect to all intermediate layers on the training data, and show that, empirically, regularizing the Lipschitzness improve the generalization. Sokolic et al. ´ (2017) shows similar results in data-limited settings. In Section 2.3, we extend the Lipschitz regularization technique to heteroskedastic setting. ", + "bbox": [ + 173, + 459, + 825, + 545 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Regularization strength as a function of noise level and sample size. Finally, we briefly review existing theoretical insights on the optimal choice of regularization strength. Generally, the optimal regularization strength for a given model family increases with the label noise level and decreases in the sample size. As a simple example, consider linear ridge regression $\\begin{array} { r } { \\operatorname* { m i n } _ { \\theta } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } { ( x _ { i } ^ { \\top } \\theta - y _ { i } ) ^ { 2 } } + } \\end{array}$ $\\lambda \\| \\theta \\| ^ { 2 }$ , where $x _ { i }$ , $\\theta \\in \\mathbb { R } ^ { d }$ and $y _ { i } \\in \\mathbb { R }$ . We assume $y _ { i } = x _ { i } ^ { \\top } \\theta ^ { * } + \\xi$ for some ground truth parameter $\\theta ^ { * }$ , and $\\Dot { \\xi } \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } )$ . Then the optimal regularization strength $\\lambda _ { o p t } = d \\sigma ^ { 2 } / \\bar { n } \\| \\theta ^ { * } \\| _ { 2 } ^ { 2 }$ . Results of similar nature can also be found in nonparametric statistics (Wang et al., 2013; Tibshirani et al., 2014). ", + "bbox": [ + 173, + 550, + 826, + 651 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 HETEROSKEDASTIC NONPARAMETRIC CLASSIFICATION ON ONE-DIMENSIONAL DATA", + "text_level": 1, + "bbox": [ + 173, + 666, + 810, + 683 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We consider a one-dimensional binary classification problem where $\\mathcal { X } = [ 0 , 1 ] \\subset \\mathbb { R }$ and $\\mathcal { V } = \\{ - 1 , 1 \\}$ We assume $Y$ given $X$ follows a logistic model with ground-truth function $f ^ { \\star }$ , i.e. ", + "bbox": [ + 173, + 693, + 825, + 723 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e0c56091aa38ee67473106ccb5ad7073c2ca99f8218de48fbdd7db070a37abdd.jpg", + "text": "$$\n\\operatorname* { P r } \\left[ Y = y | X = x \\right] = { \\frac { 1 } { 1 + \\exp ( - y f ^ { \\star } ( x ) ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 728, + 640, + 762 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The training objective is cross-entropy loss plus Lipschitz regularization, i.e. ", + "bbox": [ + 173, + 767, + 673, + 782 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9885407285e56151abf867dd47a02aeb31f1d5f4aac06c2b5936c3636135b79f.jpg", + "text": "$$\n\\widehat { f } = \\mathrm { a r g m i n } _ { f } \\widehat { L } ( f ) \\triangleq \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell ( f ( x _ { i } ) , y _ { i } ) + \\lambda \\int _ { 0 } ^ { 1 } \\rho ( x ) ( f ^ { \\prime } ( x ) ) ^ { 2 } d x ,\n$$", + "text_format": "latex", + "bbox": [ + 276, + 787, + 720, + 830 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\ell ( a , y ) = - \\log ( 1 + \\exp ( - y a ) )$ , and $\\rho ( x )$ is a smoothing parameter as a function of the noise level and density of $x$ . Let $I ( x )$ be the fisher information matrix conditioned on the input, i.e. $I ( x ) \\triangleq \\mathbb { E } [ \\nabla _ { a } ^ { 2 } \\ell ( a , Y ) | _ { a = f ^ { \\star } ( X ) } | X = x ]$ . When $( X , Y )$ follows the logistic model in equation 2, ", + "bbox": [ + 174, + 835, + 825, + 882 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d3dbb5e0813dd9fdb189f51f81497cdf7b8cd7d59f804cf99f227c66c1e23380.jpg", + "text": "$$\nI ( x ) = { \\frac { 1 } { ( 1 + \\exp ( f ^ { \\star } ( x ) ) ( 1 + \\exp ( - f ^ { \\star } ( x ) ) } } = \\mathrm { V a r } ( Y | X = x ) .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 888, + 709, + 922 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Therefore, $I ( x )$ captures the aleatoric uncertainty of $x$ . For example, when $Y$ is deterministic conditioned on $X = x$ , we have $I ( x ) = 0$ , indicating perfect certainty. ", + "bbox": [ + 173, + 103, + 823, + 133 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Define the test metric as the mean-squared-error on the test set $\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ , i.e.,1 ", + "bbox": [ + 176, + 138, + 712, + 155 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/dc65b3d0381b17b7227b7e0887d508a8f14cd239093f208efd3c6d380930865b.jpg", + "text": "$$\n\\operatorname { M S E } ( \\hat { f } ) \\triangleq \\operatorname * { \\mathbb { E } } _ { \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n } } \\int _ { 0 } ^ { 1 } ( \\hat { f } ( t ) - f ^ { \\star } ( t ) ) ^ { 2 } d t\n$$", + "text_format": "latex", + "bbox": [ + 349, + 156, + 647, + 194 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our main goal is to derive the optimal choice of $\\rho ( x )$ that minimizes the MSE. We start with an analytical characterization of the test error. Let $W _ { 2 } ^ { 2 } = \\{ f ^ { \\prime }$ is absolute continuous and $f ^ { \\prime \\prime } \\in L ^ { 2 } [ 0 , 1 ] \\}$ . We denote the density of $X$ as $q ( x )$ . The following theorem analytically computes the MSE under the regularization strength $\\rho ( \\cdot )$ , building upon (Wang et al., 2013) for regression problems. The proof of the Theorem is deferred to Appendix A. ", + "bbox": [ + 173, + 203, + 826, + 275 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1. Assume that $f ^ { \\star } , q , I \\in W _ { 2 } ^ { 2 }$ . Let $r ( t ) = - 1 / ( q ( t ) I ( t ) )$ and $\\begin{array} { r } { L _ { 0 } = \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { 4 } \\exp ( - 2 | t | ) d t } \\end{array}$ If we choose $\\lambda = C _ { 0 } n ^ { - 2 / 5 }$ for some constant $C _ { 0 } > 0$ , the asymptotic mean squared error is ", + "bbox": [ + 169, + 276, + 823, + 310 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/98777cadf03fe227c6b48fbefae5d79c1864cbea2f9861fd994c07066c9f0efb.jpg", + "text": "$$\n\\operatorname* { l i m } _ { n \\to \\infty } M S E ( \\hat { f } ) = C _ { n } \\int _ { 0 } ^ { 1 } \\lambda ^ { 2 } r ^ { 2 } ( t ) \\left[ \\frac { d } { d t } ( \\rho ( t ) ( f ^ { * } ) ^ { \\prime } ( t ) ) \\right] ^ { 2 } + L _ { 0 } r ( t ) ^ { 1 / 2 } \\rho ( t ) ^ { - 1 / 2 } d t\n$$", + "text_format": "latex", + "bbox": [ + 241, + 313, + 756, + 349 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "in probability, where $C _ { n }$ is a scalar that only depends on $n$ ", + "bbox": [ + 174, + 351, + 563, + 366 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Using the analytical formula of the test error above, we want to derive an approximately optimal choice of $\\rho ( x )$ . A precise computation is infeasible, so we restrict ourselves to consider only $\\rho ( x )$ that is constant within groups of examples. We introduce an additional structure – we assume the data can be divided into $k$ groups $[ a _ { 0 } , a _ { 1 } ) , \\bar { [ } a _ { 1 } , a _ { 2 } ) , \\cdot \\cdot \\cdot , [ a _ { k - 1 } , a _ { k } )$ . Each group $[ a _ { j } , a _ { j + 1 } )$ consists of an interval of data with approximately the same aleatoric uncertainty. We approximate $\\rho ( t )$ is constant on each of the group $[ a _ { i } , a _ { i + 1 } )$ with value $\\rho _ { i }$ . Plugging this piece-wise constant $\\rho$ into the asymptotic MSE in Theorem 1, we obtain ", + "bbox": [ + 173, + 375, + 826, + 473 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5587924fdba5e89247fe3f42a6f90416d13bf1d281e1d1ed0d6f3c0c3f4118d0.jpg", + "text": "$$\n\\underset { n \\infty } { \\operatorname* { l i m } } \\mathrm { M S E } ( \\hat { f } ) = \\sum _ { j } [ \\rho _ { j } ^ { 2 } \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 2 } ( t ) [ \\frac { d ^ { 2 } } { d t ^ { 2 } } f ^ { \\star } ( t ) ] ^ { 2 } d t + \\rho _ { j } ^ { - 1 / 2 } L _ { 0 } \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 1 / 2 } ( t ) d t ] .\n$$", + "text_format": "latex", + "bbox": [ + 215, + 474, + 781, + 518 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Minimizing the above formula over $\\rho _ { 1 } , \\ldots , \\rho _ { k }$ separately, we derive the optimal weights, $\\rho _ { j } =$ $\\left[ \\frac { L _ { 0 } \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ( t ) ^ { 1 / 2 } d t } { 4 \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 2 } ( t ) \\left[ \\frac { d ^ { 2 } } { d t ^ { 2 } } f ^ { \\star } ( t ) \\right] ^ { 2 } d t } \\right] ^ { 2 / }$ 5 . In practice, we do not know $f ^ { \\star }$ and $q ( x )$ , so we make the following simplifications. We assume that $q ( t )$ and $I ( t )$ are constant on each interval $[ a _ { j } , a _ { j + 1 } ]$ . In other words, we assume that $q ( t ) = q _ { j }$ and $I ( t ) = I _ { j }$ for all $t \\in [ a _ { j } , a _ { j + 1 } ]$ . We further assume that $\\textstyle { \\frac { d ^ { 2 } } { d t ^ { 2 } } } f ^ { \\star } ( t )$ is close to a constant on the entire space, because estimating the curvature in high dimension is difficult. This simplification yields $\\rho _ { j } \\propto \\left[ \\frac { q _ { j } ^ { - 1 / 2 } I _ { j } ^ { - 1 / 2 } } { q _ { j } ^ { - 2 } I _ { j } ^ { - 2 } } \\right] ^ { 2 / 5 } = q _ { j } ^ { 3 / 5 } I _ { j } ^ { 3 / 5 }$ q3/5j I3/5j . We find the simplification works well in practice. ", + "bbox": [ + 173, + 520, + 826, + 671 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Adaptive regularization with importance sampling. It is practically infeasible to implement the integration in equation 3 for high-dimensional data. We use importance sampling to approximate the integral: ", + "bbox": [ + 173, + 678, + 823, + 720 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2e1a6787dd6550c48149ec5dcb7d9acd394ea64aa6edf08473a3b48b21a5a890.jpg", + "text": "$$\n{ \\mathrm { m i n i m i z e } } _ { f } L ( f ) \\triangleq { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } \\ell ( f ( x _ { i } ) , y _ { i } ) + \\lambda \\sum _ { i = 1 } ^ { n } \\tau _ { i } f ^ { \\prime } ( x _ { i } ) ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 305, + 722, + 691, + 763 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Suppose $x _ { i } \\in [ a _ { j } , a _ { j + 1 } )$ , we have that $\\tau _ { i }$ should satisfy that $\\tau _ { i } q _ { j } = \\rho _ { j }$ so that the expectation of the regularization term in equation 5 is equal to that in equation 3. Hence, ", + "bbox": [ + 174, + 766, + 821, + 795 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e07633c49d25d6bffcf960cf99ca3e0f38f2d0e6eed54789f39998d3cc6781ba.jpg", + "text": "$$\n\\tau _ { i } = I _ { j } ^ { 3 / 5 } q _ { j } ^ { - 2 / 5 } = I ( x _ { i } ) ^ { 3 / 5 } q ( x _ { i } ) ^ { - 2 / 5 } .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 796, + 625, + 820 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Adaptive regularization for multi-class classification and regression. In fact, the proof of Theorem 1 is proved for general loss $\\ell ( a , y )$ . Therefore, we can directly generalize it to multiclass classification and regression problems. For a regression problem, $\\ell ( a , y )$ is the square loss: $\\ell ( y , a ) = 0 . 5 ( y - a ) ^ { 2 }$ , the Fisher information $I ( x ) = 1$ . Therefore, for a regression problem, we can choose regularization weight $\\tau _ { i } = q ( x _ { i } ) ^ { - 2 / 5 }$ . ", + "bbox": [ + 173, + 829, + 825, + 902 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We heuristically extend the Lipschitz regularization technique discussed in Section 2.2 from nonparametric models to over-parameterized deep neural networks. Let $( x , y )$ be an example and $f _ { \\theta }$ be an $r$ -layer neural network. We denote by $h ^ { ( j ) }$ the $j$ -th hidden layer of the network, by $\\begin{array} { r } { \\dot { J } ^ { ( j ) } ( x ) \\triangleq \\frac { \\hat { \\partial } } { \\partial h ^ { ( j ) } } \\mathcal { L } ( f ( x ) , y ) } \\end{array}$ , i.e., the Jacobian of the loss w.r.t $h ^ { ( j ) }$ . We replace the regularization term $f ^ { \\prime } ( x ) ^ { 2 }$ in equation 5 by $\\begin{array} { r } { R ( x ) = { \\left( \\sum _ { j = 1 } ^ { r } | | J ^ { ( j ) } ( x ) | | _ { F } ^ { 2 } \\right) } ^ { 1 / 2 } } \\end{array}$ , which was proposed by (Wei & Ma, 2019a). As a proof of concept, we visualize the behavior of our algorithm in Figure 3, where we observe that the rare and noisy examples have significantly improved error due to stronger regularization. In contrast, a uniform regularization either overfits or underfits different subsets. ", + "bbox": [ + 173, + 128, + 826, + 258 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that the differences from the 1-D case include the following three aspects. 1. The derivative is taken w.r.t to all the hidden layers for deep models, which has been shown to have superior generalization guarantees for neural networks by (Wei & Ma, 2019a;b). 2. An additional square root is taken in computing $R ( x )$ . This modified version may have milder curvature and be easier to tune. 3. We take the derivative of the loss instead of the derivative of the model, which outputs $k$ numbers for multi-class classification. This is because the derivative of the model requires $k$ times more time to compute. The regularized training objective is consequently ", + "bbox": [ + 173, + 265, + 825, + 363 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/9c0f87a88a7df27d97e5a419a545e647849d7ceef67136637f7d116cf57bf9ae.jpg", + "text": "$$\n\\mathrm { m i n i m i z e } _ { f } L ( f ) \\triangleq \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\left( \\ell ( f ( x _ { i } ) , y _ { i } ) + \\lambda \\tau _ { i } R ( x _ { i } ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 366, + 678, + 406 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\tau _ { i }$ is chosen to be $\\tau _ { i } = I ( x _ { i } ) ^ { 3 / 5 } / q ( x _ { i } ) ^ { 2 / 5 }$ following the formula equation 5 in Section 2.2 and $\\lambda$ is a hyperparameter to control the overall scale of the regularization strength. We note that we do not expect this choice of $\\tau _ { i }$ to be optimal for the high-dimensional case with all the modifications above – the optimal choice does depend on the nuances. However, we also observe that the empirical performance is not sensitive to the form of $\\tau$ as long as it’s increasing in $I ( x )$ and decreasing in $q ( x )$ . That is, the more uncertain or rare an example is, the stronger regularization should be applied. ", + "bbox": [ + 173, + 410, + 825, + 496 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In order to estimate the relative regularization strength $\\tau _ { i }$ , the key difficulty lies in the estimation of uncertainty $I ( x )$ . As in the 1-D setting, we divide the examples into $k$ groups $G _ { 1 } , \\ldots , G _ { k }$ (e.g., each group can correspond to a class), and estimate the uncertainty on each group. In the binary setting, $I ( x ) = \\mathrm { \\mathrm { V a r } } ( { \\bar { Y } } | X = x ) = \\operatorname* { P r } [ Y = 1 \\mid X ] \\cdot \\operatorname* { P r } [ Y = 0 \\ { \\bar { | } } \\ X ]$ can be approximated by $\\begin{array} { r } { \\tilde { I } ( x ) = 1 - \\operatorname* { m a x } _ { k \\in \\{ 0 , 1 \\} } \\operatorname* { P r } [ Y = k \\mid X = x ] } \\end{array}$ up to a factor of at most 2. We use the same formula for multi-class setting as the approximation of the uncertainty. (As a sanity check, when $Y$ is concentrated on a single outcome, the uncertainty is 0.) Note that $\\tilde { I } ( \\stackrel { \\cdot } { x } )$ is essentially the minimum possible error of any deterministic prediction on the data point $x$ . Assume that we have a sufficiently accurate pre-trained model, we can use its validation error to estimate $\\tilde { I } ( x )$ : ", + "bbox": [ + 173, + 501, + 825, + 633 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Then for all $x \\in G _ { j }$ , we estimate $q ( x )$ and $I ( x )$ by $\\forall x \\in G _ { j } , q ( x ) \\propto | G _ { j } | , I$ $I ( x ) \\propto$ average validation error of a pre-trained model $f _ { \\tilde { \\theta } }$ on $G _ { j }$ ", + "bbox": [ + 173, + 640, + 508, + 655 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 194, + 659, + 774, + 674 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The whole training pipeline is summarized in Algorithm 1. ", + "bbox": [ + 173, + 684, + 560, + 700 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 Heteroskedastic Adaptive Regularization (HAR) ", + "text_level": 1, + "bbox": [ + 173, + 713, + 581, + 728 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Require: Dataset $\\mathcal { D } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ . A parameterized model $f _ { \\theta }$ \n1: Split training set $\\mathcal { D }$ into $\\mathcal { D } _ { \\mathrm { t r a i n } }$ and $\\mathcal { D } _ { \\mathrm { v a l } }$ \n2: $f _ { \\tilde { \\theta } } \\gets$ Standard SGD Training on $\\mathcal { D } _ { \\mathrm { t r a i n } }$ \n3: Estimate $I ( x ) , q ( x )$ with equation 7 using $f _ { \\tilde { \\theta } }$ on $\\mathcal { D } _ { \\mathrm { v a l } }$ , and compute $\\tau _ { i } = I ( x _ { i } ) ^ { 3 / 5 } / q ( x _ { i } ) ^ { 2 / 5 }$ \n4: \n5: Initialize the model parameters $\\theta$ randomly \n6: $f _ { \\boldsymbol { \\theta } } \\gets \\mathsf { S G D }$ with the regularized objective as in equation 6 on the full dataset $\\mathcal { D }$ ", + "bbox": [ + 176, + 733, + 802, + 837 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 864, + 328, + 881 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We experimentally show that our proposed algorithm HAR(Algorithm 1) improves the test performance of the noisier and rarer groups of examples (by stronger regularization) without negatively affecting the training and test performance of the other groups. We evaluate our algorithms on three vision datasets and one NLP dataset: CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009), IMDB-review (Maas et al., 2011) (see Appendix C.1), and WebVision (Li et al., 2017), a real-world heteroskedastic and imbalanced dataset. Please refer to Appendix B for low-level implementation details. ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines. We compare our proposed HAR with the following baselines. The simplest one is (1) Empirical risk minimization (ERM): the vanilla cross-entropy loss with all examples having the same weights of losses. We select two representatives from the noise-cleaning line of approach. (2) Co-teaching (Han et al., 2018): two deep networks are trained simultaneously. Each network aims to identify clean data points that have small losses and use them to guide the training of the other network. (3) INCV (Chen et al., 2019): it extends Co-teacing to an interative version to estimate the noise ratio and select data. We consider three representatives from the reweighting-based methods, including two that learn the weighting using meta-learning. (4) MentorNet (Jiang et al., 2018): it pretrains a teacher network that outputs weights for examples that are used to train the student network with reweighting. (5) L2RW (Ren et al., 2018): it directly optimizes weights of each example in the training set by minimizing its corresponding loss on a small meta validation set. (6) MW-Net (Shu et al., 2019): it extends L2RW by explicitly defining a weighting function which depends only on the loss of the example. We also compare against two representatives from the robust loss function. (7) GCE (Zhang & Sabuncu, 2018): it generalizes mean average error and cross-entropy loss to obtain a new loss function. (8) DMI ( $\\mathrm { X u }$ et al., 2019): it designs a new loss function based on generalized mutual information. In addition, as an essential ablation study, we consider vanilla uniform regularization. (9) Unif-reg: we apply the Jacobian regularizer on all examples with equal strength, and tune the strength to get the best possible validation accuracy. ", + "bbox": [ + 174, + 181, + 825, + 429 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 SIMULATING HETEROSKEDASTIC AND IMBALANCED DATASETS ON CIFAR ", + "text_level": 1, + "bbox": [ + 178, + 458, + 725, + 470 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Setting. Unlike previous works that test on uniform random or asymmetric noise, which is often not the case in reality, in this paper we test our method on more realistic noisy settings, as suggested by Patrini et al. (2017); Zhang & Sabuncu (2018). In order to simulate heteroskedasticity, we only corrupt semantically-similar classes. For CIFAR-10, we exchange $40 \\%$ of the labels between classes ‘cat’ and ‘dog’, and between ‘truck’ and ‘automobile’. CIFAR-100 has 100 classes grouped into 20 super classes. For each class of the 5 classes under the super class ‘vehicles $_ { 1 } ,$ and ‘vehicles $_ { 2 } \\cdot$ , we corrupt the labels with $40 \\%$ probability uniformly randomly to the rest of four classes under the same super class. As a result, the 10 classes under super class ‘vehicle $_ { 1 } ,$ and ‘vehicle $_ { 2 } \\cdot$ have high label noise level and the corruption are only within the same super class. Heteroskedasticity of the labels and imbalance of the inputs commonly coexist in the real world settings. HAR can take both of them into account. To understand the challenge imposed by the entanglements of heteroskedasticity and imbalance, and compare HAR with the aforementioned baselines, we inject data imbalance concurrently with the heteroskedastic noise. We remove samples from the corrupted classes to simulate the most difficult scenario — the rare and noisy groups overfit significantly. (A more benign interaction between the noises and imbalance is that the rare classes have lower noise level, we defer it to Appendix C.3.) We use the imbalance ratio to denote the frequency ratio between the frequent (and clean) classes to the rare (and corrupted) classes. We consider imbalance ratio to be 10 and 100. ", + "bbox": [ + 174, + 487, + 825, + 722 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Result. Table 1 summarizes the results. Since examples from rare classes tend to have larger training and validation loss regardless of whether the labels are correct or not, noise-cleaning based methods might drop excessive examples with correct labels. We examined the noise ratio of dropped samples for INCV under the setting of imbalance ratio equals 10. Among all dropped examples, there is only $1 9 . 2 \\%$ of true noise examples. In addition, the rare class examples selected still have $2 9 . 8 \\%$ of label noise. This explains that the significant decrease of accuracies of Co-teaching and INCV on corrupted and rare classes. Reweighting-based methods tend to suffer from the loss of accuracy in other more frequent classes, which is aligned with the findings in Cao et al. (2019). While the aforementioned baselines struggle to deal with heteroskedasticity and imbalance together, HAR is able to put them under the same regularization framework and achieve significant improvements. Notably, HAR also shows improvement over uniform regularization with optimally tuned strength. This clearly demonstrates the importance of introducing adaptive regularization among all examples for a better trade-off. A more detailed ablation study on the trade-off between training accuracy and validation accuracy can be found in Section 3.3. ", + "bbox": [ + 174, + 729, + 825, + 922 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/0925ad5ac39325ab86f25dafa4c34eda39aa731e7e0496f7aa29d570620eacad.jpg", + "table_caption": [ + "Table 1: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic and imbalanced CIFAR-10. HAR significantly improves noisy and rare classes, while keeping the accuracy on other classes almost unaffected. " + ], + "table_footnote": [], + "table_body": "
Imbalance ratio Method10100
Noisy&Rare Cls.Clean Cls.Noisy&Rare Cls.Clean Cls.
ERM52.9 ±1.294.4 ± 0.118.9 ± 1.094.2 ± 0.1
Co-teaching30.2 ± 2.388.9± 0.315.4± 2.886.4± 0.7
INCV48.9 ± 1.794.0± 0.225.8 ± 1.893.8 ± 0.2
MentorNet54.1 ± 1.090.3 ± 0.528.3 ± 1.590.2 ±0.4
L2RW44.3 ± 2.090.1± 0.531.2 ± 1.989.7±0.7
MW-Net55.4 ± 1.191.7 ± 0.535.6 ± 1.692.3 ± 0.5
GCE48.2±0.691.6 ± 0.314.1 ± 2.091.7 ± 0.4
DMI44.7 ± 2.390.7 ±0.814.0 ± 2.191.8 ± 0.6
Unif-reg (optimal)53.9 ± 0.992.1± 0.236.7 ± 1.092.4± 0.3
Ours (HAR)63.5 ± 0.894.3 ± 0.242.4±0.794.0±0.2
", + "bbox": [ + 222, + 155, + 776, + 329 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 ABLATION STUDY ON CIFAR ", + "text_level": 1, + "bbox": [ + 174, + 354, + 419, + 369 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We disentangle the problem setting to show the effectiveness of our unified framework. ", + "bbox": [ + 174, + 381, + 743, + 396 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Simulating heteroskedastic noise on CIFAR. We study the uncertainty part of HAR by testing under the setting with only heteroskedastic noise. The type of noise injection is the same as Section 3.1. ", + "bbox": [ + 173, + 402, + 823, + 431 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We report the top-1 validation accuracy of various methods in Table 2. Aligned with our analysis in Section 4, we observe that both noise-cleaning and reweighting based methods don’t get a comparable accuracy on noisy classes with applying strong regularization $\\lambda = 0 . 1$ ) under this heteroskedastic setting. We observe the behavior that too strong regularization impede the model from fitting informative samples, thus it could lead to a decrease on clean classes’ accuracy. On the contrary, too weak regularization leads to overfitting the noisy examples thus the accuracy on noisy classes do not reach the optimal. ", + "bbox": [ + 174, + 438, + 825, + 535 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Interestingly, we find that even the well-studied CIFAR-100 dataset has intrinsic heteroskedasticity and HAR can improve over uniform regularization to some extent. Please refer to Appendix C.2 for the results on CIFAR-100 and Appendix C.1 for results on IMDB-review. ", + "bbox": [ + 176, + 541, + 821, + 584 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/74ceeaf106f364eaae7ce788d873447dcbe93d01bc6ee4833908bc8686f25d51.jpg", + "table_caption": [ + "Table 2: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic CIFAR-10 and CIFAR-100 for the noisy classes and the clean classes. " + ], + "table_footnote": [], + "table_body": "
Dataset MethodCIFAR-10CIFAR-100
Avg. Noisy Cls.Avg. Clean Cls.Avg. Noisy Cls.Avg. Clean Cls.
ERM68.6±0.293.6±0.265.3 ± 0.367.8 ± 0.2
Co-teaching64.7 ± 0.489.1± 0.359.8 ± 0.465.3± 0.3
INCV76.7 ± 0.693.0±0.266.2±0.368.6 ±0.2
MentorNet71.1 ± 0.493.7± 0.265.9 ± 0.367.5 ± 0.3
L2RW70.1± 0.392.5 ± 0.365.1± 0.567.0± 0.3
MW-Net75.0± 0.394.4± 0.265.7 ± 0.369.1± 0.2
GCE62.6 ±1.190.2±0.261.2 ±0.666.9 ±0.2
DMI73.2 ± 0.790.8 ± 0.264.8 ± 0.567.1 ± 0.2
Unif-reg (入= 0.1)77.5 ± 0.692.3± 0.269.3 ± 0.566.6 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1±0.268.5 ± 0.368.6 ±0.2
Ours (HAR)80.7 ± 0.394.5 ± 0.274.2 ± 0.369.3±0.2
", + "bbox": [ + 204, + 638, + 792, + 824 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Simulating data imbalance on CIFAR. We study the density part of HAR by testing under the setting with only data imbalance. We follow the same setting as Cao et al. (2019) to create imbalanced CIFAR. Long-tailed imbalance follows an exponential decay in sample sizes across different classes. For step imbalance setting, all rare classes have the same sample size, as do all frequent classes. Our approach achieves better results than LDAM-DRW and is comparable to recent state-of-the-art methods under the imbalanced setting. ", + "bbox": [ + 173, + 839, + 826, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f742ad4dae7372693001d94c3f17026d72ec63c55bd3190cf9813b46aba88127.jpg", + "table_caption": [ + "Table 3: Top-1 validation errors of ResNet-32 on imbalanced CIFAR-10 and CIFAR-100. " + ], + "table_footnote": [], + "table_body": "
DatasetImbalanced CIFAR-10Imbalanced CIFAR-100
Imbalance Typelong-tailedsteplong-tailedstep
Imbalance Ratio10010100101001010010
ERM29.6413.6136.7017.5061.6844.3061.4545.37
Focal29.6213.3436.0916.3661.5944.2261.4346.54
CB Focal25.4312.9039.7316.5463.9842.0180.2449.98
LDAM-DRW22.9711.8423.0812.1957.9641.2954.6440.54
BBN (Zhou et al., 2020)20.1811.6821.6411.9957.4440.8857.4440.36
HAR-DRW20.4610.6220.2711.5855.3538.9851.7337.54
", + "bbox": [ + 204, + 127, + 794, + 270 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/ae1989ab59fa908f6a93e08e6192c658ef0ac8e459c9608d4c224e971f100d7a.jpg", + "table_caption": [ + "Table 4: Validation accuracy of ResNet-50 when tuning the regularization strength on mini WebVision. HAR stands out of the trade-off constraint of fitting and generalization. " + ], + "table_footnote": [], + "table_body": "
Train AccVal Acc
Reg StrengthTop1Top5Top1Top 5
069.0188.6459.4080.84
Unif-reg (入= 0.01)68.9688.5464.3286.11
Unif-reg (λ= 0.02)67.0287.5164.4085.92
Unif-reg (入= 0.05)65.1186.3365.8086.84
Unif-reg (λ = 0.1)63.3584.9865.0486.56
Adaptive (HAR)69.1288.4169.2088.96
", + "bbox": [ + 323, + 323, + 674, + 440 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.3 EVALUATION ON WEBVISION WITH REAL-WORLD HETEROGENEITY ", + "text_level": 1, + "bbox": [ + 174, + 467, + 681, + 482 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "WebVision (Li et al., 2017) contains 2.4 million images crawled from Google and Flickr using 1,000 labels shared with the ImageNet dataset. Its training set is both heteroskedastic and imbalanced (detailed statistics can be found in (Li et al., 2017)), and it is considered as a popular benchmark for noise robust learning. As the full dataset is very large, we follow (Jiang et al., 2018) to use a mini version, which contains the first 50 classes of the Google subset of the data. Following the standard protocol (Jiang et al., 2018), we test the trained model on the WebVision validation set and the ImageNet validation set. We use ResNet-50 for ablation study and InceptionResNet-v2 for a fair comparison with the baselines. We report results comparing against other state-of-the-art approaches in Table 5. Strikingly, HAR achieves significant improvement. ", + "bbox": [ + 173, + 492, + 825, + 619 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Ablation study. We demonstrate the trade-off between training accuracy and validation accuracy on mini WebVision with various uniform regularization strength and HAR in Table 4. It’s evident that when we gradually increase the overall uniform regularization strength, the training accuracy continues to decrease, and the validation accuracy reaches its peak at 5e-2. While a strong regularization could improve generalization, it reduces deep networks’ capacity to fit the training data. However, with our proposed HAR, we only enforce strong regularization on a subset so that we improve the generalization on noisier groups while maintaining the overall training accuracy not affected. ", + "bbox": [ + 174, + 626, + 825, + 723 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 744, + 344, + 761 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Our work is closely related to the following methods and directions. ", + "bbox": [ + 174, + 776, + 619, + 791 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Noise-cleaning. The key idea of noise-cleaning is to identify and remove (or re-label) examples with wrong annotations. The general procedure for identifying mislabeled instances has a long history (Brodley & Friedl, 1999; Wilson & Martinez, 1997; Zhao & Nishida, 1995). Some recent works tailored this idea for deep neural networks. Veit et al. (2017) trains a label cleaning network on a small set of data with clean labels, and uses this model to identify noises in large datasets. To circumvent the requirement of a clean subset, Malach & Shalev-Shwartz (2017) train two networks simultaneously and perform update steps only in case of disagreement. Similarly, in co-teaching (Han et al., 2018), each network selects a certain number of small-loss samples and feeds them to its peer network. Chen et al. (2019) further extends the co-training strategy and comes up with an iterative version that performs even better empirically. Recently Song et al. (2020) discovers that it is not necessary to maintain two networks. Removing examples whose training loss exceeds a certain threshold before learning rate decay can also get robust performance. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/98445b556671984cf4d9574355d3cd581207786a7f407bf0421a0b7507660b20.jpg", + "table_caption": [ + "Table 5: Validation accuracy of InceptionResNet-v2 on WebVision and ImageNet validation sets. HAR demonstrates significant improvements over the previous state-of-the-arts. " + ], + "table_footnote": [], + "table_body": "
Train Test Methodmini WebVisionfull WebVision
WebVisionImageNetWebVisionImageNet
Top 1Top5Top 1Top 5Top 1Top 5Top 1Top 5
ERM62.580.858.581.869.787.062.983.6
Co-teaching63.685.261.584.71-11
INCV65.285.361.685.0--1-
MentorNet63.081.457.879.970.888.062.583.0
Ours (HAR)75.590.770.390.075.090.667.186.7
", + "bbox": [ + 235, + 141, + 763, + 258 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 287, + 823, + 330 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Reweighting. Reweighting training data has shown its effectiveness on noisy data (Liu & Tao, 2015). Its challenge lies in the difficulty of weights estimation. Ren et al. (2018) proposes a meta-learning algorithm to assign weights to training examples based on its gradient direction with the one on a clean validation set. Recently, Shu et al. (2019) proposes to learn an explicit loss-weight function to mitigate the optimizing issue of (Ren et al., 2018). Another line of work resorts to curriculum learning by either designing an easy-to-hard strategy of training (Guo et al., 2018) or introducing an extra network (Jiang et al., 2018) to assign weights. ", + "bbox": [ + 174, + 337, + 825, + 434 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Noise-cleaning and reweighting methods usually rely on the empirical loss to determine if a sample is noisy. However, when the dataset is heteroskedastic, each example’s training/validation loss no longer correlates well with its noise level. In such cases, we argue that changing the strength of regularization is a more conservative adaption and suffers less from uncertain estimation, compared to changing the weights of losses (Please refer to Section C.4 for empirical justifications). ", + "bbox": [ + 174, + 441, + 825, + 511 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Robust loss function. Another line of works has attempted to design robust loss functions (Ghosh et al., 2017; Xu et al., 2019; Zhang & Sabuncu, 2018; Patrini et al., 2017; Cheng et al., 2017; Menon et al., 2016). They usually rely on prior assumption about latent transition matrix that might not hold in practice. On the contrary, we focus on more realistic settings. ", + "bbox": [ + 174, + 518, + 825, + 574 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Regularization. Regularization based techniques have also been explored to combat label noise. Li et al. (2019) proves that SGD with early stopping is robust to label noise. Hu et al. (2020) provides theoretical analysis of two additional regularization methods. While these methods consider a uniform regularization on all training examples, our work emphasizes on adjusting the weights of regularizers in search of a better generalization than uniform assignment. ", + "bbox": [ + 174, + 582, + 825, + 651 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 675, + 318, + 691 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We propose a unified framework (HAR) for training on heteroskedastic and imbalanced datasets. Our method achieves significant improvements over the previous state-of-the-arts on a variety of benchmark vision and language tasks. We provide theoretical results as well as empirical justifications by showing that ambiguous, mislabeled, and rare examples all benefit from stronger regularization. We further provide the formula for optimal weighting of regularization. Heteroskedasticity of datasets is a fascinating direction worth exploring, and it is an important step towards a better understanding of real-world scenarios in the wild. ", + "bbox": [ + 174, + 710, + 825, + 808 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 834, + 367, + 849 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Toyota Research Institute (\"TRI\") provided funds and computational resources to assist the authors with their research but this article solely reflects the opinions and conclusions of its authors and not TRI or any other Toyota entity. YC is supported by Stanford Graduate Fellowship. TM acknowledges support of Google Faculty Award. The work is also partially supported by SDSI and SAIL at Stanford. 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", + "bbox": [ + 171, + 92, + 828, + 559 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A PROOFS OF THEOREM 1 ", + "text_level": 1, + "bbox": [ + 176, + 102, + 406, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We prove a general theorem here. In particular, we have the general theorem below. ", + "bbox": [ + 171, + 132, + 722, + 147 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Theorem 2. Assume that $f ^ { \\star } , q , I \\in W _ { 2 } ^ { 2 }$ . Suppose (1) $\\ell ( a , y )$ is convex and three times continously differentiable with respect to $a$ , (2) there exist constants $0 < c < C < \\infty$ such that $c \\leq I ( X ) \\leq \\bar { C }$ almost surely, and $\\epsilon \\triangleq \\nabla _ { a } \\ell ( f ^ { \\star } ( X ) , Y )$ satisfies $\\mathbb { E } [ \\epsilon | X ] = 0 \\mathrm { \\it \\Omega }$ , $\\mathbb { E } [ \\epsilon ^ { 2 } | X ] = I ( X )$ and $\\mathbb { E } [ \\epsilon ^ { 4 } | X ] < \\infty$ almost surely. Let $r ( t ) = - 1 / ( q ( t ) I ( t ) )$ and $\\begin{array} { r } { L _ { 0 } = \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { 4 } \\exp ( - 2 | t | ) d t . } \\end{array}$ . If we choose $\\lambda = C _ { 0 } n ^ { - 2 / 5 }$ for some constant $C _ { 0 } > 0$ , the asymptotic mean squared error of $\\hat { f }$ by equation 4 is ", + "bbox": [ + 173, + 150, + 825, + 231 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c6a2f35daa2ba3524636b23ce7bd61b4395e2910afa2abf98b06bfe63130914d.jpg", + "text": "$$\n\\operatorname* { l i m } _ { n \\to \\infty } M S E ( \\hat { f } ) = C _ { n } \\int _ { 0 } ^ { 1 } \\lambda ^ { 2 } r ^ { 2 } ( t ) \\left[ \\frac { d } { d t } ( \\rho ( t ) ( f ^ { * } ) ^ { \\prime } ( t ) ) \\right] ^ { 2 } + L _ { 0 } r ( t ) ^ { 1 / 2 } \\rho ( t ) ^ { - 1 / 2 } d t\n$$", + "text_format": "latex", + "bbox": [ + 241, + 232, + 756, + 270 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "in probability, where $C _ { n }$ is a scalar that only depends on $n$ . ", + "bbox": [ + 174, + 272, + 565, + 286 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "It is easy to check that the logistic loss satisfies the condition of the theorem. ", + "bbox": [ + 174, + 296, + 676, + 311 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The proof strategy of Theorem 2 is adopted from the proof of Theorem 2 of (Wang et al., 2013) by generalizing it from the least square loss to logistic loss. ", + "bbox": [ + 171, + 318, + 823, + 347 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The high level idea is to reformulate $\\hat { f }$ as solutions to ordinary differential equations. Let $( \\gamma _ { v } , h _ { v } )$ be the (normalized) solution of the following equation ", + "bbox": [ + 171, + 354, + 823, + 385 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/b8a4db4af9d6efa262b60416d90b62e4a53dee6d07c55dc04bfc06a007322697.jpg", + "text": "$$\n\\begin{array} { r l } & { - \\rho ( t ) h _ { v } ^ { \\prime \\prime } ( t ) = \\gamma _ { v } I ( t ) q ( t ) h _ { v } ( t ) , } \\\\ & { h _ { v } ^ { ' } ( 0 ) = h _ { v } ^ { ' } ( 1 ) = 0 , \\mathrm { ~ a n d ~ } h _ { v } ^ { ' \\prime } ( 0 ) = h _ { v } ^ { ' \\prime } ( 1 ) = 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 346, + 386, + 650, + 431 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We define the the leading term of $\\hat { f } - f ^ { \\star }$ as $S _ { n , \\lambda } ( f ^ { \\star } )$ as ", + "bbox": [ + 174, + 441, + 545, + 459 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/bd041acf74e6c9a42afed2581f43e5f369264f8f49037ab84b79a452afbac0d5.jpg", + "text": "$$\nS _ { n , \\lambda } ( f ^ { \\star } ) = { \\frac { 1 } { n } } \\epsilon _ { i } K _ { X _ { i } } - W _ { \\lambda } f ^ { \\star } ,\n$$", + "text_format": "latex", + "bbox": [ + 372, + 462, + 578, + 492 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/d12e0263c5f39d8811659ca77526544c8626609d53d42680e958dbc977ec72c4.jpg", + "text": "$$\nK _ { t } ( \\cdot ) = \\sum _ { v } \\frac { h _ { v } ( t ) } { 1 + \\lambda \\gamma _ { v } } h _ { v } ( \\cdot ) \\mathrm { a n d } W _ { \\lambda } h _ { v } ( \\cdot ) = \\frac { \\lambda \\gamma _ { v } } { 1 + \\lambda \\gamma _ { v } } h _ { v } ( \\cdot ) .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 494, + 694, + 532 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "By Proposition 2.1 and Theorem 3.4 of Shang et al. (2013), we have ", + "bbox": [ + 174, + 539, + 622, + 554 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/62438ae453c0a80b1ab26fe50451e68c366fc458f539c05e18075d90c2c08787.jpg", + "text": "$$\n\\operatorname* { s u p } _ { x } | \\hat { f } ( x ) - f ^ { \\star } ( x ) - S _ { n , \\lambda } ( f ^ { \\star } ) ( x ) | = o _ { P } ( n ^ { - 1 / 3 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 333, + 558, + 666, + 584 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Following the same proof of Theorem 2 of (Wang et al., 2013), we can simplify the definition of $K _ { t }$ and $W ^ { \\lambda }$ as ", + "bbox": [ + 173, + 587, + 823, + 613 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/104042b649f6b38c434cebe849f9a404ec81eb4ebd35bb20c7ea2c3f45facc52.jpg", + "text": "$$\nK _ { t } ( x ) = \\frac { I ( t ) } { q ( t ) } J ( t , x ) \\mathrm { a n d } W _ { \\lambda } f ^ { \\star } ( t ) = \\lambda r ( t ) \\left[ \\frac { d } { d t } ( \\rho ( t ) ( f ^ { \\ast } ) ^ { \\prime } ( t ) ) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 282, + 611, + 714, + 645 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $J ( t , s ) ~ = ~ \\beta \\rho ( s ) Q _ { \\beta } ^ { \\prime } ( s ) L _ { 0 } ( \\beta | Q _ { \\beta } ( t ) - Q _ { \\beta } ( s ) | )$ and $\\begin{array} { r } { Q _ { \\beta } ( t , s ) = \\int _ { 0 } ^ { t } ( r ( s ) \\rho ( s ) ) ^ { - 1 / 2 } ( 1 + } \\end{array}$ $O ( \\beta ^ { - 1 } ) ) d s$ and $\\beta = 1 / \\sqrt { \\lambda }$ . Plugging equation 11 into equation 10, we then have ", + "bbox": [ + 171, + 648, + 830, + 683 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/78e2e5122e513f455d07608408c2c552e1dcef3c0da025e83074f411c9401196.jpg", + "text": "$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\mathrm { M S E } ( \\hat { f } ) = C _ { n } \\int _ { 0 } ^ { 1 } \\lambda ^ { 2 } r ^ { 2 } ( t ) \\left[ \\frac { d } { d t } ( \\rho ( t ) ( f ^ { \\ast } ) ^ { \\prime } ( t ) ) \\right] ^ { 2 } + L _ { 0 } r ( t ) ^ { 1 / 2 } \\rho ( t ) ^ { - 1 / 2 } d t\n$$", + "text_format": "latex", + "bbox": [ + 238, + 685, + 758, + 723 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 738, + 436, + 755 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We develop our core algorithm in PyTorch (Paszke et al., 2017). ", + "bbox": [ + 174, + 768, + 593, + 784 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Implementation details for CIFAR. We follow the simple data augmentation used in (He et al., 2016) with only random crop and horizontal flip. We use ResNet-32 as our base network and repeat all experiments for 3 runs. We use standard SGD with momentum of 0.9, weight decay of $1 \\times \\bar { 1 0 } ^ { - 4 }$ for training. The model is trained with a batch size of 128 for 120 epochs. We anneal the learning rate by a factor of 10 at 80 and 100 epochs. We group the data by class labels, and by default we split $\\mathcal { D }$ equally and randomly into $\\mathcal { D } _ { \\mathrm { t r a i n } }$ and $\\mathcal { D } _ { \\mathrm { v a l } }$ . As for the Jacobian regularizer, we sum over the frobenius norm of the gradients of all normalization layers’ (BN by default) activations with respect to the classification loss. For experiments of HAR, we tune $\\lambda$ so that the largest enforced regularization strength $( \\lambda \\tau _ { i } )$ is 0.1. We train each model with 1 NVIDIA GeForce RTX 2080 Ti. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Implementation details for IMDB-review. We train a two-layer bidirectional LSTM (Huang et al., 2015) with 256 units followed with 0.5 dropout before the linear classifier. The network is trained for 20 epochs with Adam optimizer (Kingma & Ba, 2014). For HAR, we tune $\\lambda$ so that the largest enforced regularization strength $( \\lambda \\tau _ { i } )$ is 0.1. We train each model with 1 NVIDIA GeForce RTX 2080 Ti. ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Implementation details for WebVision. We use the standard data augmentation same as (He et al., 2016) including random crop and horizontal flip. For mini WebVision, We train the network for 90 epochs using standard SGD with a batch size of 128. The initial learning rate is 0.1 and is annealed by a factor of 10 at epoch 60 and 90. For full WebVision, We train the network for 50 epochs using standard SGD with a batch size of 256. The initial learning rate is 0.1 and is annealed by a factor of 10 at epoch 30 and 40. For experiments of HAR, we tune $\\lambda$ so that the largest enforced regularization strength $( \\lambda \\tau _ { i } )$ is 0.1. We train each model with 8 NVIDIA Tesla V100 GPUs. ", + "bbox": [ + 174, + 191, + 825, + 289 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Runtime analysis. Because the pre-trained model only trains on half of the training data and is only done once, the run-time of HAR is at most twice of the time for ERM. Many baselines in our paper use sophisticated pipelines and are slower than HAR. For example, INCV trains 2 models simultaneously for 4 times from random initialization to get a clean training set. MW-Net has a very slow convergence rate, which is a common issue for meta-learning. ", + "bbox": [ + 174, + 306, + 825, + 377 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 400, + 398, + 416 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.1 SIMULATING HETEROSKEDASTIC NOISE ON IMDB-REVIEW. ", + "text_level": 1, + "bbox": [ + 176, + 433, + 630, + 446 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "IMDB-review dataset has a total of 50,000 (25,000 positive and 25,000 negative reviews) movie reviews for binary sentiment classification (Maas et al., 2011). To simulate heteroskedastic noise for this binary classification problem, we project $5 \\%$ of the labels of negative reviews to positive, and $40 \\%$ in the reverse direction. Table 6 summarizes the results. The proposed HAR outperforms the ERM baseline with various strength of uniform regularization. ", + "bbox": [ + 174, + 460, + 825, + 530 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/5e8422ac1ef554c31c234b247a9e25c04caa2cfebce4a0741772ae6e1ab7c16c.jpg", + "table_caption": [ + "Table 6: Top-1 validation accuracy (averaged over 3 runs) on heteroskedastic IMDB-review dataset. " + ], + "table_footnote": [], + "table_body": "
Reg StrengthAcc. of neg. reviewsAcc. of pos. reviewsMean Acc
091.9 ± 2.050.9 ±1.871.4± 0.5
Unif-reg (入= 0.01)94.3 ±1.851.9 ± 2.073.1± 0.3
Unif-reg (入= 0.1)91.5 ±1.964.3 ±1.677.9 ± 0.4
Ours (HAR)93.1 ± 1.572.8±1.783.0 ± 0.3
", + "bbox": [ + 246, + 571, + 750, + 651 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.2 EVALUATION ON CIFAR-100 WITH REAL-WORLD HETEROSKEDASTICITY ", + "bbox": [ + 176, + 681, + 723, + 695 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "It is acknowledged that CIFAR-100 training set contains noisy examples. For instance, some “tiger” examples are labeled as “leopard” (“tiger” is a defined class as well). There are also noisy examples that contain multiple objects, or are more ambiguous in terms of indentity (Song et al., 2020). We find that HAR can improve over uniform regularization on the well-studied CIFAR-100 due to its heteroskedasticity and the results are reported in Table 7. ", + "bbox": [ + 174, + 707, + 826, + 777 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/e0db088405f29470c79b8a73aed317d7d6c82174968cd58b8d000bd25d084a07.jpg", + "table_caption": [ + "Table 7: Top-1 validation accuracy (average over 3 runs) of ResNet-32 on the original CIFAR-100. " + ], + "table_footnote": [], + "table_body": "
Reg StrengthTrain AccVal Acc
096.0±0.169.8± 0.2
Unif-reg (入= 0.001)96.4± 0.170.0± 0.2
Unif-reg (入= 0.01)95.7 ± 0.170.6 ±0.1
Unif-reg (入= 0.1)88.8±0.170.5 ± 0.1
Adaptive (HAR)96.2 ± 0.171.4±0.1
", + "bbox": [ + 339, + 819, + 658, + 911 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "As mentioned in Section 3.1, we consider another variant of heteroskedastic and imbalanced dataset such that the rare classes have low noise level. To simulate this setting, we make the clean classes have fewer labels than the corrupted classes on the heteroskedastic CIFAR-10 we created in Section ??. ", + "bbox": [ + 176, + 128, + 823, + 171 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Table 8 summarizes the results. For the setting of imbalance ratio equals 10, INCV automatically drops $34 . 1 \\%$ of examples from the clean and rare classes, which results in a decrease of mean accuracy on the rare and clean classes. HAR is able to achieve improvements on both noisy classes and rare classes by enforcing the optimal regularization. ", + "bbox": [ + 174, + 178, + 825, + 234 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/594c8ed960a196e922f1e8eecdcf757f4e39bc0335b2df40a0aa0695929b6376.jpg", + "table_caption": [ + "Table 8: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic and imbalanced CIFAR-10. " + ], + "table_footnote": [], + "table_body": "
Imbalance ratio Method10100
Noisy Cls.Rare Cls.Noisy Cls.Rare Cls.
ERM59.9 ± 1.165.2 ± 0.760.4± 0.97.9 ± 1.3
Co-teaching63.1 ± 2.359.4 ± 1.453.8 ± 1.94.4±1.8
INCV74.5 ± 1.263.7 ± 0.868.3 ± 1.82.1 ±1.3
MentorNet67.3 ± 1.665.5 ± 1.263.3 ± 1.510.8 ±1.9
L2RW65.8 ±1.466.3 ± 1.262.4 ± 2.111.3 ± 2.9
MW-Net71.4± 0.667.7 ± 0.665.0± 1.613.5 ± 2.4
GCE64.6 ± 1.160.2 ± 1.371.2 ± 1.92.6±1.4
DMI72.3 ± 1.563.3 ±1.270.8 ±1.76.2 ±1.9
Ours (HAR)76.1 ± 0.872.1 ± 1.073.0 ± 1.626.1±0.8
", + "bbox": [ + 259, + 287, + 736, + 449 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C.4 COMPARING THE EFFECT OF WEIGHTS ON LOSSES AND REGULARIZERS ", + "text_level": 1, + "bbox": [ + 173, + 473, + 707, + 488 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "As discussed in Section 4, we train several classifiers with alternative weights selection scheme which are not optimal. We consider the following two alternatives. (1) random: we draw the weights from a uniform distribution with the same mean as the weights of MW-Net and HAR. (2) inverse: we take the inverse of the weights learned by MW-Net and HAR and then normalize them to ensure the average reweighting/regularization strength remains the same. ", + "bbox": [ + 174, + 498, + 825, + 569 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We conducted experiments on the heteroskedastic CIFAR-10 introduced in Section ?? and the results are summarized in Table 9. We could conclude that changing the weights of the regularizer is a more conservative adaption and less susceptible to uncertain estimation than reweighting. ", + "bbox": [ + 174, + 575, + 825, + 618 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/233559ee95c0dfdc49b8e15f237a00e907ef5f5126428a1e8de37e94b59eb072.jpg", + "table_caption": [ + "Table 9: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic CIFAR-10 by changing the weighting scheme. " + ], + "table_footnote": [], + "table_body": "
MethodAvg. Noisy Cls.Avg. . Clean Cls.
ERM68.6±0.293.6 ± 0.2
Reweight (MW-Net)75.0 ± 0.394.4± 0.2
Reweight (random)62.1±0.592.9 ± 0.5
Reweight (inverse)13.1 ± 0.990.9 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1± 0.2
Adap-reg (HAR)80.7±0.394.5 ±0.2
Adap-reg (random)74.8 ± 0.494.2±0.2
Adap-reg (inverse)73.2 ± 0.594.0± 0.2
", + "bbox": [ + 310, + 670, + 681, + 808 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C.5 VISUALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 824, + 330, + 838 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In order to better understand how the proposed HAR works on real-world heteroskedastic datasets, we plot the per-class key statistics used by HAR and validation errors in Figure 4. We observe that HAR outperforms the tuned uniform regularization baseline on the majority of the classes. ", + "bbox": [ + 174, + 851, + 826, + 893 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/c076753ebcd5b8f3da8f361db87212d171eb3c37ef46afdae950c98d61c3d805.jpg", + "image_caption": [ + "Figure 4: Visualizations of per-class top-1 error and regularization strength of HAR on mini WebVision dataset. The classes are sorted in the ascending order of applied regularization strength. " + ], + "image_footnote": [], + "bbox": [ + 179, + 294, + 820, + 688 + ], + "page_idx": 15 + } +] \ No newline at end of file diff --git a/parse/train/mEdwVCRJuX4/mEdwVCRJuX4_middle.json b/parse/train/mEdwVCRJuX4/mEdwVCRJuX4_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..d12fa574e2ad779c62cb7e214dae915caba9b96d --- /dev/null +++ b/parse/train/mEdwVCRJuX4/mEdwVCRJuX4_middle.json @@ -0,0 +1,42157 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 79, + 506, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 507, + 97 + ], + "score": 1.0, + "content": "HETEROSKEDASTIC AND IMBALANCED DEEP LEARN-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 400, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 400, + 118 + ], + "score": 1.0, + "content": "ING WITH ADAPTIVE REGULARIZATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 114, + 133, + 483, + 169 + ], + "lines": [ + { + "bbox": [ + 114, + 134, + 484, + 148 + ], + "spans": [ + { + "bbox": [ + 114, + 134, + 141, + 148 + ], + "score": 1.0, + "content": "Kaidi", + "type": "text" + }, + { + "bbox": [ + 141, + 134, + 163, + 146 + ], + "score": 0.71, + "content": "\\mathbf { C a o ^ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 134, + 262, + 148 + ], + "score": 1.0, + "content": ", Yining Chen1, Junwei", + "type": "text" + }, + { + "bbox": [ + 262, + 134, + 280, + 146 + ], + "score": 0.83, + "content": "\\mathbf { L } \\mathbf { u } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 134, + 484, + 148 + ], + "score": 1.0, + "content": ", Nikos Arechiga3, Adrien Gaidon3, Tengyu Ma1", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 113, + 145, + 392, + 159 + ], + "spans": [ + { + "bbox": [ + 113, + 145, + 392, + 159 + ], + "score": 1.0, + "content": "1Stanford University, 2Harvard University, 3Toyota Research Institute", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 113, + 158, + 354, + 170 + ], + "spans": [ + { + "bbox": [ + 113, + 158, + 354, + 170 + ], + "score": 1.0, + "content": "{kaidicao,cynnjjs,tengyuma}@stanford.edu", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 278, + 198, + 333, + 210 + ], + "lines": [ + { + "bbox": [ + 276, + 197, + 335, + 211 + ], + "spans": [ + { + "bbox": [ + 276, + 197, + 335, + 211 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 143, + 224, + 469, + 367 + ], + "lines": [ + { + "bbox": [ + 141, + 225, + 469, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 225, + 469, + 236 + ], + "score": 1.0, + "content": "Real-world large-scale datasets are heteroskedastic and imbalanced – labels have", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 235, + 470, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 235, + 470, + 247 + ], + "score": 1.0, + "content": "varying levels of uncertainty and label distributions are long-tailed. 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We propose", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 470, + 292 + ], + "score": 1.0, + "content": "a data-dependent regularization technique for heteroskedastic datasets that regu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "score": 1.0, + "content": "larizes different regions of the input space differently. 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We test our method on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 469, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 469, + 346 + ], + "score": 1.0, + "content": "several benchmark tasks, including a real-world heteroskedastic and imbalanced", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "score": 1.0, + "content": "dataset, WebVision. Our experiments corroborate our theory and demonstrate a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 441, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 441, + 369 + ], + "score": 1.0, + "content": "significant improvement over other methods in noise-robust deep learning.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 109, + 388, + 206, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 208, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 208, + 402 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "In real-world machine learning applications, even well-curated training datasets have various types of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "score": 1.0, + "content": "heterogeneity. 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Many deep learning algorithms have been proposed for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "imbalanced datasets (e.g., see (Wang et al., 2017; Cao et al., 2019; Cui et al., 2019; Liu et al., 2019)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 504, + 491 + ], + "score": 1.0, + "content": "and the reference therein). However, heteroskedasticity, a classical notion studied extensively in the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "statistical community (Pintore et al., 2006; Wang et al., 2013; Tibshirani et al., 2014), has so far", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "been under-explored in deep learning. This paper focuses on addressing heteroskedasticity and its", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 304, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 304, + 525 + ], + "score": 1.0, + "content": "interaction with data imbalance in deep learning.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Heteroskedasticity is often studied in regression analysis and refers to the property that the distribution", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "of the error varies across inputs. In this work, we mostly focus on classification, though the developed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "technique also applies to regression. Here, heteroskedasticity reflects how the uncertainty in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 203, + 574 + ], + "score": 1.0, + "content": "conditional distribution", + "type": "text" + }, + { + "bbox": [ + 203, + 562, + 223, + 574 + ], + "score": 0.9, + "content": "y \\mid x", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 561, + 297, + 574 + ], + "score": 1.0, + "content": ", or the entropy of", + "type": "text" + }, + { + "bbox": [ + 297, + 562, + 317, + 574 + ], + "score": 0.88, + "content": "y \\mid x", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 561, + 412, + 574 + ], + "score": 1.0, + "content": ", varies as a function of", + "type": "text" + }, + { + "bbox": [ + 412, + 564, + 419, + 572 + ], + "score": 0.67, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 561, + 505, + 574 + ], + "score": 1.0, + "content": ". 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(2017) shows that the WebVision dataset has a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 583, + 388, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 388, + 597 + ], + "score": 1.0, + "content": "varying number of ambiguous or true noisy examples across classes.2", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "Conversely, we consider a dataset to be homoscedastic if every example is mislabeled with a fixed", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 145, + 623 + ], + "score": 1.0, + "content": "probably", + "type": "text" + }, + { + "bbox": [ + 145, + 614, + 150, + 622 + ], + "score": 0.37, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 612, + 505, + 623 + ], + "score": 1.0, + "content": ", as assumed by many prior theoretical and empirical works on label corruption (Ghosh", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "score": 1.0, + "content": "et al., 2017; Han et al., 2018; Jiang et al., 2018; Mirzasoleiman et al., 2020). We note that varying", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 162, + 646 + ], + "score": 1.0, + "content": "uncertainty in", + "type": "text" + }, + { + "bbox": [ + 163, + 634, + 184, + 646 + ], + "score": 0.77, + "content": "y \\mid x", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "can come from at least two sources: the intrinsic semantic ambiguity of the input,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "and the (data-dependent) mislabeling introduced by the annotation process. Our approach can handle", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "both types of noisy examples in a unified way, but for the sake of comparisons with past methods, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "call them “ambiguous examples” and “mislabeled examples” respectively, and refer to both of them", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 192, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 192, + 690 + ], + "score": 1.0, + "content": "as “noisy examples”.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 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Heteroskedas-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 247, + 470, + 258 + ], + "spans": [ + { + "bbox": [ + 142, + 247, + 470, + 258 + ], + "score": 1.0, + "content": "ticity and imbalance challenge deep learning algorithms due to the difficulty of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 469, + 271 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 469, + 271 + ], + "score": 1.0, + "content": "distinguishing among mislabeled, ambiguous, and rare examples. Addressing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 268, + 469, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 469, + 281 + ], + "score": 1.0, + "content": "heteroskedasticity and imbalance simultaneously is under-explored. We propose", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 470, + 292 + ], + "score": 1.0, + "content": "a data-dependent regularization technique for heteroskedastic datasets that regu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "score": 1.0, + "content": "larizes different regions of the input space differently. Inspired by the theoretical", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 470, + 313 + ], + "score": 1.0, + "content": "derivation of the optimal regularization strength in a one-dimensional nonpara-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 313, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 470, + 325 + ], + "score": 1.0, + "content": "metric classification setting, our approach adaptively regularizes the data points", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 324, + 469, + 335 + ], + "spans": [ + { + "bbox": [ + 142, + 324, + 469, + 335 + ], + "score": 1.0, + "content": "in higher-uncertainty, lower-density regions more heavily. We test our method on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 469, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 469, + 346 + ], + "score": 1.0, + "content": "several benchmark tasks, including a real-world heteroskedastic and imbalanced", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "score": 1.0, + "content": "dataset, WebVision. Our experiments corroborate our theory and demonstrate a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 441, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 441, + 369 + ], + "score": 1.0, + "content": "significant improvement over other methods in noise-robust deep learning.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 225, + 470, + 369 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 388, + 206, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 208, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 208, + 402 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "In real-world machine learning applications, even well-curated training datasets have various types of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "score": 1.0, + "content": "heterogeneity. Two main types of heterogeneity are: (1) data imbalance: the input or label distribution", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 433, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 449 + ], + "score": 1.0, + "content": "often has a long-tailed density, and (2) heteroskedasticity: the labels given inputs have varying", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "levels of uncertainties across subsets of data stemming from various sources such as the intrinsic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "ambiguity of the data or annotation errors. Many deep learning algorithms have been proposed for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "imbalanced datasets (e.g., see (Wang et al., 2017; Cao et al., 2019; Cui et al., 2019; Liu et al., 2019)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 504, + 491 + ], + "score": 1.0, + "content": "and the reference therein). However, heteroskedasticity, a classical notion studied extensively in the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "statistical community (Pintore et al., 2006; Wang et al., 2013; Tibshirani et al., 2014), has so far", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "been under-explored in deep learning. This paper focuses on addressing heteroskedasticity and its", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 304, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 304, + 525 + ], + "score": 1.0, + "content": "interaction with data imbalance in deep learning.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 414, + 506, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Heteroskedasticity is often studied in regression analysis and refers to the property that the distribution", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "of the error varies across inputs. In this work, we mostly focus on classification, though the developed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "technique also applies to regression. Here, heteroskedasticity reflects how the uncertainty in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 203, + 574 + ], + "score": 1.0, + "content": "conditional distribution", + "type": "text" + }, + { + "bbox": [ + 203, + 562, + 223, + 574 + ], + "score": 0.9, + "content": "y \\mid x", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 561, + 297, + 574 + ], + "score": 1.0, + "content": ", or the entropy of", + "type": "text" + }, + { + "bbox": [ + 297, + 562, + 317, + 574 + ], + "score": 0.88, + "content": "y \\mid x", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 561, + 412, + 574 + ], + "score": 1.0, + "content": ", varies as a function of", + "type": "text" + }, + { + "bbox": [ + 412, + 564, + 419, + 572 + ], + "score": 0.67, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 561, + 505, + 574 + ], + "score": 1.0, + "content": ". Real-world datasets", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "are often heteroskedastic. For example, Li et al. (2017) shows that the WebVision dataset has a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 583, + 388, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 388, + 597 + ], + "score": 1.0, + "content": "varying number of ambiguous or true noisy examples across classes.2", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 529, + 506, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "Conversely, we consider a dataset to be homoscedastic if every example is mislabeled with a fixed", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 145, + 623 + ], + "score": 1.0, + "content": "probably", + "type": "text" + }, + { + "bbox": [ + 145, + 614, + 150, + 622 + ], + "score": 0.37, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 612, + 505, + 623 + ], + "score": 1.0, + "content": ", as assumed by many prior theoretical and empirical works on label corruption (Ghosh", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "score": 1.0, + "content": "et al., 2017; Han et al., 2018; Jiang et al., 2018; Mirzasoleiman et al., 2020). We note that varying", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 162, + 646 + ], + "score": 1.0, + "content": "uncertainty in", + "type": "text" + }, + { + "bbox": [ + 163, + 634, + 184, + 646 + ], + "score": 0.77, + "content": "y \\mid x", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "can come from at least two sources: the intrinsic semantic ambiguity of the input,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "and the (data-dependent) mislabeling introduced by the annotation process. Our approach can handle", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "both types of noisy examples in a unified way, but for the sake of comparisons with past methods, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "call them “ambiguous examples” and “mislabeled examples” respectively, and refer to both of them", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 192, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 192, + 690 + ], + "score": 1.0, + "content": "as “noisy examples”.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 600, + 506, + 690 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 83, + 284, + 213 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 83, + 284, + 213 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 83, + 284, + 213 + ], + "spans": [ + { + "bbox": [ + 108, + 83, + 284, + 213 + ], + "score": 0.962, + "type": "image", + "image_path": "b0aa3f523002d1c7bc925b084339bea809fac9e5918ef2d25e1e376f5cd78fab.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 108, + 83, + 284, + 96.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 96.0, + 284, + 109.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 109.0, + 284, + 122.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 108, + 122.0, + 284, + 135.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 135.0, + 284, + 148.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 108, + 148.0, + 284, + 161.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 108, + 161.0, + 284, + 174.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 108, + 174.0, + 284, + 187.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 108, + 187.0, + 284, + 200.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 108, + 200.0, + 284, + 213.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 224, + 287, + 279 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 224, + 288, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 288, + 236 + ], + "score": 1.0, + "content": "Figure 1: Histogram of the distributions of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 236, + 287, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 287, + 247 + ], + "score": 1.0, + "content": "losses on an imbalanced and noisy CIFAR-10", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 246, + 287, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 287, + 258 + ], + "score": 1.0, + "content": "dataset. 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They require mutually-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 296, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 296, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "exclusive reweighting strategy, but they all benefit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 297, + 269, + 414, + 280 + ], + "spans": [ + { + "bbox": [ + 297, + 269, + 414, + 280 + ], + "score": 1.0, + "content": "from stronger regularization.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + } + ], + "index": 20.0 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 422 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "Overparameterized deep learning models tend to overfit more to the noisy examples (Arpit et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "2017; Frénay & Verleysen, 2013; Zhang et al., 2016). To address this issue, a common approach", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "is to detect noisy examples by selecting those with large training losses, and then remove them", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "from the (re-)training process. However, an input’s training loss can also be big because it is rare or", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 343, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 358 + ], + "score": 1.0, + "content": "ambiguous (Hacohen & Weinshall, 2019; Wang et al., 2019), as shown in Figure 1. Noise-cleaning", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "methods could fail to distinguish mislabeled from rare/ambiguous examples (see Section 3.1 for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "score": 1.0, + "content": "empirical proofs). Though dropping the former is desirable, dropping the latter loses important", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "information. Another popular approach is reweighting methods that reduce the contribution of noisy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "examples in optimization. However, failing to distinguish between mislabeled and rare/ambiguous", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "examples makes the decision of the weights tricky – mislabeled examples require small weights,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 410, + 502, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 502, + 423 + ], + "score": 1.0, + "content": "whereas rare / ambiguous examples benefit from larger weights (Cao et al., 2019; Shu et al., 2019).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "We propose a regularization method that deals with noisy and rare examples in a unified way. We", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "observe that mislabeled, ambiguous, and rare examples all benefit from stronger regularization (Hu", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 446, + 507, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 507, + 464 + ], + "score": 1.0, + "content": "et al., 2020; Cao et al., 2019). We apply a Lipschitz regularizer (Wei & Ma, 2019a;b) with varying", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "regularization strength depending on the particular data point. Through theoretical analysis in the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "one-dimensional setting, we derive the optimal regularization strength for each training example. The", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "optimal strength is larger for rarer and noisier examples. Our proposed algorithm, heteroskedastic", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "adaptive regularization (HAR), first estimates the noise level and density of each example, and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "then optimizes a Lipschitz-regularized objective with input-dependent regularization with strength", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 515, + 252, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 252, + 528 + ], + "score": 1.0, + "content": "provided by the theoretical formula.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "In summary, our main contributions are: (i) we propose to learn heteroskedastic imbalanced datasets", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 543, + 507, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 507, + 556 + ], + "score": 1.0, + "content": "under a unified framework, and theoretically study the optimal regularization strength on one-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "dimensional data. (ii) we propose an algorithm, heteroskedastic adaptive regularization (HAR),", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "which applies stronger regularization to data points with high uncertainty and low density. (iii)", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 576, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 589 + ], + "score": 1.0, + "content": "we experimentally show that HAR achieves significant improvements over other noise-robust deep", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "learning methods on simulated vision and language datasets with controllable degrees of data noise", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 598, + 491, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 491, + 610 + ], + "score": 1.0, + "content": "and data imbalance, as well as a real-world heteroskedastic and imbalanced dataset, WebVision.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 52 + }, + { + "type": "title", + "bbox": [ + 109, + 625, + 457, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 459, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 459, + 640 + ], + "score": 1.0, + "content": "2 ADAPTIVE REGULARIZATION FOR HETEROSKEDASTIC DATASETS", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 56 + }, + { + "type": "title", + "bbox": [ + 107, + 650, + 198, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 199, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 199, + 663 + ], + "score": 1.0, + "content": "2.1 BACKGROUNDS", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 57 + }, + { + "type": "text", + "bbox": [ + 107, + 671, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 685 + ], + "score": 1.0, + "content": "We first introduce general nonparametric tools that we use in our analysis, and review the dependency", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 682, + 383, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 383, + 693 + ], + "score": 1.0, + "content": "of optimal regularization strength on the sample size and noise level.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 58.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Over-parameterized neural networks as nonparametric methods. 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They require mutually-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 296, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 296, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "exclusive reweighting strategy, but they all benefit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 297, + 269, + 414, + 280 + ], + "spans": [ + { + "bbox": [ + 297, + 269, + 414, + 280 + ], + "score": 1.0, + "content": "from stronger regularization.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + } + ], + "index": 20.0 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 422 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "Overparameterized deep learning models tend to overfit more to the noisy examples (Arpit et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "2017; Frénay & Verleysen, 2013; Zhang et al., 2016). To address this issue, a common approach", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "is to detect noisy examples by selecting those with large training losses, and then remove them", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "from the (re-)training process. However, an input’s training loss can also be big because it is rare or", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 343, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 358 + ], + "score": 1.0, + "content": "ambiguous (Hacohen & Weinshall, 2019; Wang et al., 2019), as shown in Figure 1. 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Another popular approach is reweighting methods that reduce the contribution of noisy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "examples in optimization. However, failing to distinguish between mislabeled and rare/ambiguous", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "examples makes the decision of the weights tricky – mislabeled examples require small weights,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 410, + 502, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 502, + 423 + ], + "score": 1.0, + "content": "whereas rare / ambiguous examples benefit from larger weights (Cao et al., 2019; Shu et al., 2019).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 301, + 506, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "We propose a regularization method that deals with noisy and rare examples in a unified way. We", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "observe that mislabeled, ambiguous, and rare examples all benefit from stronger regularization (Hu", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 446, + 507, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 507, + 464 + ], + "score": 1.0, + "content": "et al., 2020; Cao et al., 2019). We apply a Lipschitz regularizer (Wei & Ma, 2019a;b) with varying", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "regularization strength depending on the particular data point. Through theoretical analysis in the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "one-dimensional setting, we derive the optimal regularization strength for each training example. The", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "optimal strength is larger for rarer and noisier examples. Our proposed algorithm, heteroskedastic", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "adaptive regularization (HAR), first estimates the noise level and density of each example, and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "then optimizes a Lipschitz-regularized objective with input-dependent regularization with strength", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 515, + 252, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 252, + 528 + ], + "score": 1.0, + "content": "provided by the theoretical formula.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44, + "bbox_fs": [ + 104, + 427, + 507, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "In summary, our main contributions are: (i) we propose to learn heteroskedastic imbalanced datasets", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 543, + 507, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 507, + 556 + ], + "score": 1.0, + "content": "under a unified framework, and theoretically study the optimal regularization strength on one-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "dimensional data. (ii) we propose an algorithm, heteroskedastic adaptive regularization (HAR),", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "which applies stronger regularization to data points with high uncertainty and low density. (iii)", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 576, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 589 + ], + "score": 1.0, + "content": "we experimentally show that HAR achieves significant improvements over other noise-robust deep", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "learning methods on simulated vision and language datasets with controllable degrees of data noise", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 598, + 491, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 491, + 610 + ], + "score": 1.0, + "content": "and data imbalance, as well as a real-world heteroskedastic and imbalanced dataset, WebVision.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 532, + 507, + 610 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 625, + 457, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 459, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 459, + 640 + ], + "score": 1.0, + "content": "2 ADAPTIVE REGULARIZATION FOR HETEROSKEDASTIC DATASETS", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 56 + }, + { + "type": "title", + "bbox": [ + 107, + 650, + 198, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 199, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 199, + 663 + ], + "score": 1.0, + "content": "2.1 BACKGROUNDS", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 57 + }, + { + "type": "text", + "bbox": [ + 107, + 671, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 685 + ], + "score": 1.0, + "content": "We first introduce general nonparametric tools that we use in our analysis, and review the dependency", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 682, + 383, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 383, + 693 + ], + "score": 1.0, + "content": "of optimal regularization strength on the sample size and noise level.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 58.5, + "bbox_fs": [ + 105, + 668, + 505, + 693 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Over-parameterized neural networks as nonparametric methods. We use nonparametric", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "method as a surrogate for neural networks because they have been shown to be closely related.", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "Recent work (Savarese et al., 2019) shows that the minimum norm two-layer ReLU network that fits", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 61, + "bbox_fs": [ + 106, + 699, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 88, + 295, + 237 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 88, + 295, + 237 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 88, + 295, + 237 + ], + "spans": [ + { + "bbox": [ + 113, + 88, + 295, + 237 + ], + "score": 0.966, + "type": "image", + "image_path": "9f6b9f919811d48d690fed50dff8093090a7d6a6528fc0487af6d4f5ee64ac11.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 113, + 88, + 295, + 101.54545454545455 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 101.54545454545455, + 295, + 115.0909090909091 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 115.0909090909091, + 295, + 128.63636363636363 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 113, + 128.63636363636363, + 295, + 142.18181818181816 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 113, + 142.18181818181816, + 295, + 155.7272727272727 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 113, + 155.7272727272727, + 295, + 169.27272727272722 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 113, + 169.27272727272722, + 295, + 182.81818181818176 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 113, + 182.81818181818176, + 295, + 196.3636363636363 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 113, + 196.3636363636363, + 295, + 209.90909090909082 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 113, + 209.90909090909082, + 295, + 223.45454545454535 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 113, + 223.45454545454535, + 295, + 236.9999999999999 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 303, + 80, + 500, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 302, + 80, + 500, + 92 + ], + "spans": [ + { + "bbox": [ + 302, + 80, + 500, + 92 + ], + "score": 1.0, + "content": "Figure 3: A one-dimensional example with a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 303, + 91, + 499, + 102 + ], + "spans": [ + { + "bbox": [ + 303, + 91, + 499, + 102 + ], + "score": 1.0, + "content": "three-layer neural network in heteroskedastic and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 303, + 101, + 499, + 114 + ], + "spans": [ + { + "bbox": [ + 303, + 101, + 499, + 114 + ], + "score": 1.0, + "content": "imbalanced regression setting. The curve in blue", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 302, + 113, + 500, + 125 + ], + "spans": [ + { + "bbox": [ + 302, + 113, + 500, + 125 + ], + "score": 1.0, + "content": "is the underlying ground truth and the dots are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 302, + 124, + 501, + 136 + ], + "spans": [ + { + "bbox": [ + 302, + 124, + 501, + 136 + ], + "score": 1.0, + "content": "observations with heteroskedastic noise. This ex-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 303, + 135, + 500, + 147 + ], + "spans": [ + { + "bbox": [ + 303, + 135, + 500, + 147 + ], + "score": 1.0, + "content": "ample shows that uniformly weak regularization", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 303, + 146, + 500, + 158 + ], + "spans": [ + { + "bbox": [ + 303, + 146, + 500, + 158 + ], + "score": 1.0, + "content": "overfits on noisy and rare data (on the right half),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 303, + 157, + 500, + 169 + ], + "spans": [ + { + "bbox": [ + 303, + 157, + 500, + 169 + ], + "score": 1.0, + "content": "whereas uniformly strong regularization causes", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 302, + 168, + 500, + 180 + ], + "spans": [ + { + "bbox": [ + 302, + 168, + 500, + 180 + ], + "score": 1.0, + "content": "underfitting on the frequent and oscillating data", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 302, + 179, + 500, + 191 + ], + "spans": [ + { + "bbox": [ + 302, + 179, + 500, + 191 + ], + "score": 1.0, + "content": "(on the left half). The adaptive regularization does", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 302, + 190, + 500, + 201 + ], + "spans": [ + { + "bbox": [ + 302, + 190, + 500, + 201 + ], + "score": 1.0, + "content": "not underfit the oscillating data but still denoise", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 302, + 201, + 501, + 213 + ], + "spans": [ + { + "bbox": [ + 302, + 201, + 501, + 213 + ], + "score": 1.0, + "content": "the noisy data. We note that standard nonparamet-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 302, + 212, + 500, + 224 + ], + "spans": [ + { + "bbox": [ + 302, + 212, + 500, + 224 + ], + "score": 1.0, + "content": "ric methods such as cubic spline do not work here", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 302, + 222, + 485, + 236 + ], + "spans": [ + { + "bbox": [ + 302, + 222, + 485, + 236 + ], + "score": 1.0, + "content": "because they also use uniform regularization.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17.5 + } + ], + "index": 11.25 + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 504, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 265, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 506, + 278 + ], + "score": 1.0, + "content": "the training data is in fact a linear spline interpolation. Parhi & Nowak (2019) extend this result to a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 276, + 392, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 392, + 290 + ], + "score": 1.0, + "content": "broader family of neural networks with a broader family of activations.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 207, + 308 + ], + "score": 1.0, + "content": "Given a training dataset", + "type": "text" + }, + { + "bbox": [ + 207, + 293, + 260, + 306 + ], + "score": 0.93, + "content": "\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 292, + 506, + 308 + ], + "score": 1.0, + "content": ", nonparametric method with penalty works as follows. Let", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 303, + 501, + 318 + ], + "spans": [ + { + "bbox": [ + 107, + 305, + 154, + 315 + ], + "score": 0.91, + "content": "\\mathcal { F } : \\mathbb { R } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 303, + 501, + 318 + ], + "score": 1.0, + "content": "be a twice-differentiable model family. 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Lipschitz regularization has been shown to be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "effective for deep neural networks as well. Wei & Ma (2019a) proves a generalization bound of neural", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "networks dependent on the Lipschitzness of each layer with respect to all intermediate layers on the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "training data, and show that, empirically, regularizing the Lipschitzness improve the generalization.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "score": 1.0, + "content": "Sokolic et al. ´ (2017) shows similar results in data-limited settings. 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Generally, the optimal", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "regularization strength for a given model family increases with the label noise level and decreases in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 102, + 464, + 507, + 487 + ], + "spans": [ + { + "bbox": [ + 102, + 464, + 390, + 487 + ], + "score": 1.0, + "content": "the sample size. As a simple example, consider linear ridge regression", + "type": "text" + }, + { + "bbox": [ + 390, + 469, + 507, + 483 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { \\theta } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } { ( x _ { i } ^ { \\top } \\theta - y _ { i } ) ^ { 2 } } + } \\end{array}", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 480, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 107, + 482, + 132, + 495 + ], + "score": 0.91, + "content": "\\lambda \\| \\theta \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 480, + 162, + 496 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 162, + 483, + 173, + 493 + ], + "score": 0.6, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 480, + 176, + 496 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 176, + 481, + 206, + 493 + ], + "score": 0.84, + "content": "\\theta \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 480, + 223, + 496 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 223, + 482, + 252, + 494 + ], + "score": 0.91, + "content": "y _ { i } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 480, + 302, + 496 + ], + "score": 1.0, + "content": ". We assume", + "type": "text" + }, + { + "bbox": [ + 303, + 481, + 362, + 495 + ], + "score": 0.93, + "content": "y _ { i } = x _ { i } ^ { \\top } \\theta ^ { * } + \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 480, + 491, + 496 + ], + "score": 1.0, + "content": "for some ground truth parameter", + "type": "text" + }, + { + "bbox": [ + 492, + 483, + 502, + 493 + ], + "score": 0.83, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 480, + 507, + 496 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 123, + 506 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 493, + 180, + 505 + ], + "score": 0.95, + "content": "\\Dot { \\xi } \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 492, + 346, + 506 + ], + "score": 1.0, + "content": ". Then the optimal regularization strength", + "type": "text" + }, + { + "bbox": [ + 347, + 493, + 428, + 505 + ], + "score": 0.92, + "content": "\\lambda _ { o p t } = d \\sigma ^ { 2 } / \\bar { n } \\| \\theta ^ { * } \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 492, + 506, + 506 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 303, + 501, + 318 + ], + "spans": [ + { + "bbox": [ + 107, + 305, + 154, + 315 + ], + "score": 0.91, + "content": "\\mathcal { F } : \\mathbb { R } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 303, + 501, + 318 + ], + "score": 1.0, + "content": "be a twice-differentiable model family. 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The following theorem analytically computes the MSE under", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 211, + 208 + ], + "score": 1.0, + "content": "the regularization strength", + "type": "text" + }, + { + "bbox": [ + 211, + 196, + 228, + 208 + ], + "score": 0.9, + "content": "\\rho ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 195, + 506, + 208 + ], + "score": 1.0, + "content": ", building upon (Wang et al., 2013) for regression problems. The proof", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 280, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 280, + 219 + ], + "score": 1.0, + "content": "of the Theorem is deferred to Appendix A.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 104, + 219, + 504, + 246 + ], + "lines": [ + { + "bbox": [ + 104, + 216, + 503, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 216, + 210, + 236 + ], + "score": 1.0, + "content": "Theorem 1. Assume that", + "type": "text" + }, + { + "bbox": [ + 210, + 219, + 268, + 232 + ], + "score": 0.93, + "content": "f ^ { \\star } , q , I \\in W _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 216, + 287, + 236 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 288, + 219, + 375, + 232 + ], + "score": 0.92, + "content": "r ( t ) = - 1 / ( q ( t ) I ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 216, + 394, + 236 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 395, + 219, + 503, + 234 + ], + "score": 0.91, + "content": "\\begin{array} { r } { L _ { 0 } = \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { 4 } \\exp ( - 2 | t | ) d t } \\end{array}", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 231, + 477, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 158, + 248 + ], + "score": 1.0, + "content": "If we choose", + "type": "text" + }, + { + "bbox": [ + 159, + 233, + 214, + 245 + ], + "score": 0.93, + "content": "\\lambda = C _ { 0 } n ^ { - 2 / 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 231, + 290, + 248 + ], + "score": 1.0, + "content": "for some constant", + "type": "text" + }, + { + "bbox": [ + 290, + 234, + 321, + 245 + ], + "score": 0.91, + "content": "C _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 231, + 477, + 248 + ], + "score": 1.0, + "content": ", the asymptotic mean squared error is", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 248, + 463, + 277 + ], + "lines": [ + { + "bbox": [ + 148, + 248, + 463, + 277 + ], + "spans": [ + { + "bbox": [ + 148, + 248, + 463, + 277 + ], + "score": 0.93, + "content": "\\operatorname* { l i m } _ { n \\to \\infty } M S E ( \\hat { f } ) = C _ { n } \\int _ { 0 } ^ { 1 } \\lambda ^ { 2 } r ^ { 2 } ( t ) \\left[ \\frac { d } { d t } ( \\rho ( t ) ( f ^ { * } ) ^ { \\prime } ( t ) ) \\right] ^ { 2 } + L _ { 0 } r ( t ) ^ { 1 / 2 } \\rho ( t ) ^ { - 1 / 2 } d t", + "type": "interline_equation", + "image_path": "98777cadf03fe227c6b48fbefae5d79c1864cbea2f9861fd994c07066c9f0efb.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 148, + 248, + 463, + 257.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 148, + 257.6666666666667, + 463, + 267.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 148, + 267.33333333333337, + 463, + 277.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 345, + 290 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 343, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 191, + 292 + ], + "score": 1.0, + "content": "in probability, where", + "type": "text" + }, + { + "bbox": [ + 192, + 279, + 205, + 289 + ], + "score": 0.88, + "content": "C _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 277, + 335, + 292 + ], + "score": 1.0, + "content": "is a scalar that only depends on", + "type": "text" + }, + { + "bbox": [ + 336, + 281, + 343, + 288 + ], + "score": 0.55, + "content": "n", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 506, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "Using the analytical formula of the test error above, we want to derive an approximately optimal", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 146, + 321 + ], + "score": 1.0, + "content": "choice of", + "type": "text" + }, + { + "bbox": [ + 147, + 309, + 166, + 321 + ], + "score": 0.92, + "content": "\\rho ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 309, + 485, + 321 + ], + "score": 1.0, + "content": ". A precise computation is infeasible, so we restrict ourselves to consider only", + "type": "text" + }, + { + "bbox": [ + 486, + 310, + 505, + 321 + ], + "score": 0.9, + "content": "\\rho ( x )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 321, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 332 + ], + "score": 1.0, + "content": "that is constant within groups of examples. We introduce an additional structure – we assume the data", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 184, + 345 + ], + "score": 1.0, + "content": "can be divided into", + "type": "text" + }, + { + "bbox": [ + 184, + 331, + 191, + 341 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 330, + 221, + 345 + ], + "score": 1.0, + "content": "groups", + "type": "text" + }, + { + "bbox": [ + 222, + 331, + 354, + 343 + ], + "score": 0.85, + "content": "[ a _ { 0 } , a _ { 1 } ) , \\bar { [ } a _ { 1 } , a _ { 2 } ) , \\cdot \\cdot \\cdot , [ a _ { k - 1 } , a _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 330, + 406, + 345 + ], + "score": 1.0, + "content": ". Each group", + "type": "text" + }, + { + "bbox": [ + 407, + 331, + 447, + 343 + ], + "score": 0.92, + "content": "[ a _ { j } , a _ { j + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 330, + 506, + 345 + ], + "score": 1.0, + "content": "consists of an", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 441, + 354 + ], + "score": 1.0, + "content": "interval of data with approximately the same aleatoric uncertainty. We approximate", + "type": "text" + }, + { + "bbox": [ + 442, + 342, + 459, + 354 + ], + "score": 0.91, + "content": "\\rho ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "is constant", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 189, + 366 + ], + "score": 1.0, + "content": "on each of the group", + "type": "text" + }, + { + "bbox": [ + 190, + 353, + 228, + 365 + ], + "score": 0.91, + "content": "[ a _ { i } , a _ { i + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 353, + 272, + 366 + ], + "score": 1.0, + "content": "with value", + "type": "text" + }, + { + "bbox": [ + 273, + 354, + 282, + 365 + ], + "score": 0.84, + "content": "\\rho _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 353, + 420, + 366 + ], + "score": 1.0, + "content": ". Plugging this piece-wise constant", + "type": "text" + }, + { + "bbox": [ + 420, + 355, + 427, + 365 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "into the asymptotic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 230, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 230, + 375 + ], + "score": 1.0, + "content": "MSE in Theorem 1, we obtain", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 376, + 478, + 411 + ], + "lines": [ + { + "bbox": [ + 132, + 376, + 478, + 411 + ], + "spans": [ + { + "bbox": [ + 132, + 376, + 478, + 411 + ], + "score": 0.91, + "content": "\\underset { n \\infty } { \\operatorname* { l i m } } \\mathrm { M S E } ( \\hat { f } ) = \\sum _ { j } [ \\rho _ { j } ^ { 2 } \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 2 } ( t ) [ \\frac { d ^ { 2 } } { d t ^ { 2 } } f ^ { \\star } ( t ) ] ^ { 2 } d t + \\rho _ { j } ^ { - 1 / 2 } L _ { 0 } \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 1 / 2 } ( t ) d t ] .", + "type": "interline_equation", + "image_path": "5587924fdba5e89247fe3f42a6f90416d13bf1d281e1d1ed0d6f3c0c3f4118d0.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 132, + 376, + 478, + 387.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 132, + 387.6666666666667, + 478, + 399.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 132, + 399.33333333333337, + 478, + 411.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 506, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 258, + 427 + ], + "score": 1.0, + "content": "Minimizing the above formula over", + "type": "text" + }, + { + "bbox": [ + 258, + 415, + 300, + 425 + ], + "score": 0.91, + "content": "\\rho _ { 1 } , \\ldots , \\rho _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 410, + 481, + 427 + ], + "score": 1.0, + "content": "separately, we derive the optimal weights,", + "type": "text" + }, + { + "bbox": [ + 481, + 414, + 505, + 425 + ], + "score": 0.85, + "content": "\\rho _ { j } =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 108, + 423, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 108, + 424, + 225, + 459 + ], + "score": 0.78, + "content": "\\left[ \\frac { L _ { 0 } \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ( t ) ^ { 1 / 2 } d t } { 4 \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 2 } ( t ) \\left[ \\frac { d ^ { 2 } } { d t ^ { 2 } } f ^ { \\star } ( t ) \\right] ^ { 2 } d t } \\right] ^ { 2 / }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 423, + 232, + 435 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 230, + 436, + 348, + 450 + ], + "score": 1.0, + "content": ". In practice, we do not know", + "type": "text" + }, + { + "bbox": [ + 348, + 437, + 360, + 449 + ], + "score": 0.89, + "content": "f ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 436, + 378, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 378, + 437, + 397, + 449 + ], + "score": 0.91, + "content": "q ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 436, + 505, + 450 + ], + "score": 1.0, + "content": ", so we make the following", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 232, + 471 + ], + "score": 1.0, + "content": "simplifications. We assume that", + "type": "text" + }, + { + "bbox": [ + 233, + 458, + 250, + 470 + ], + "score": 0.91, + "content": "q ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 457, + 267, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 268, + 458, + 285, + 470 + ], + "score": 0.91, + "content": "I ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 457, + 400, + 471 + ], + "score": 1.0, + "content": "are constant on each interval", + "type": "text" + }, + { + "bbox": [ + 401, + 458, + 440, + 470 + ], + "score": 0.92, + "content": "[ a _ { j } , a _ { j + 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 457, + 506, + 471 + ], + "score": 1.0, + "content": ". In other words,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 101, + 468, + 509, + 489 + ], + "spans": [ + { + "bbox": [ + 101, + 468, + 171, + 489 + ], + "score": 1.0, + "content": "we assume that", + "type": "text" + }, + { + "bbox": [ + 172, + 472, + 211, + 484 + ], + "score": 0.93, + "content": "q ( t ) = q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 468, + 230, + 489 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 471, + 270, + 484 + ], + "score": 0.92, + "content": "I ( t ) = I _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 468, + 299, + 489 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 299, + 471, + 356, + 484 + ], + "score": 0.93, + "content": "t \\in [ a _ { j } , a _ { j + 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 468, + 458, + 489 + ], + "score": 1.0, + "content": ". We further assume that", + "type": "text" + }, + { + "bbox": [ + 459, + 469, + 494, + 485 + ], + "score": 0.93, + "content": "\\textstyle { \\frac { d ^ { 2 } } { d t ^ { 2 } } } f ^ { \\star } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 468, + 509, + 489 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "close to a constant on the entire space, because estimating the curvature in high dimension is difficult.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 493, + 507, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 210, + 517 + ], + "score": 1.0, + "content": "This simplification yields", + "type": "text" + }, + { + "bbox": [ + 210, + 493, + 349, + 523 + ], + "score": 0.94, + "content": "\\rho _ { j } \\propto \\left[ \\frac { q _ { j } ^ { - 1 / 2 } I _ { j } ^ { - 1 / 2 } } { q _ { j } ^ { - 2 } I _ { j } ^ { - 2 } } \\right] ^ { 2 / 5 } = q _ { j } ^ { 3 / 5 } I _ { j } ^ { 3 / 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 498, + 507, + 518 + ], + "score": 1.0, + "content": "q3/5j I3/5j . We find the simplification works well", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 155, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 155, + 533 + ], + "score": 1.0, + "content": "in practice.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 504, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Adaptive regularization with importance sampling. It is practically infeasible to implement the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "integration in equation 3 for high-dimensional data. We use importance sampling to approximate the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 559, + 142, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 142, + 572 + ], + "score": 1.0, + "content": "integral:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 572, + 423, + 605 + ], + "lines": [ + { + "bbox": [ + 187, + 572, + 423, + 605 + ], + "spans": [ + { + "bbox": [ + 187, + 572, + 423, + 605 + ], + "score": 0.94, + "content": "{ \\mathrm { m i n i m i z e } } _ { f } L ( f ) \\triangleq { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } \\ell ( f ( x _ { i } ) , y _ { i } ) + \\lambda \\sum _ { i = 1 } ^ { n } \\tau _ { i } f ^ { \\prime } ( x _ { i } ) ^ { 2 }", + "type": "interline_equation", + "image_path": "2e1a6787dd6550c48149ec5dcb7d9acd394ea64aa6edf08473a3b48b21a5a890.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 572, + 423, + 588.5 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 187, + 588.5, + 423, + 605.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 503, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 142, + 621 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 143, + 607, + 205, + 620 + ], + "score": 0.92, + "content": "x _ { i } \\in [ a _ { j } , a _ { j + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 606, + 261, + 621 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 261, + 609, + 270, + 618 + ], + "score": 0.86, + "content": "\\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 606, + 346, + 621 + ], + "score": 1.0, + "content": "should satisfy that", + "type": "text" + }, + { + "bbox": [ + 347, + 609, + 387, + 619 + ], + "score": 0.91, + "content": "\\tau _ { i } q _ { j } = \\rho _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 606, + 506, + 621 + ], + "score": 1.0, + "content": "so that the expectation of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 618, + 389, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 389, + 632 + ], + "score": 1.0, + "content": "regularization term in equation 5 is equal to that in equation 3. 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In fact, the proof of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 264, + 681 + ], + "score": 1.0, + "content": "Theorem 1 is proved for general loss", + "type": "text" + }, + { + "bbox": [ + 264, + 668, + 292, + 680 + ], + "score": 0.92, + "content": "\\ell ( a , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 667, + 506, + 681 + ], + "score": 1.0, + "content": ". Therefore, we can directly generalize it to multi-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 401, + 693 + ], + "score": 1.0, + "content": "class classification and regression problems. 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Let", + "type": "text" + }, + { + "bbox": [ + 290, + 174, + 333, + 186 + ], + "score": 0.92, + "content": "W _ { 2 } ^ { 2 } = \\{ f ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 171, + 443, + 188 + ], + "score": 1.0, + "content": "is absolute continuous and", + "type": "text" + }, + { + "bbox": [ + 443, + 173, + 503, + 186 + ], + "score": 0.93, + "content": "f ^ { \\prime \\prime } \\in L ^ { 2 } [ 0 , 1 ] \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 171, + 507, + 188 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 184, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 208, + 196 + ], + "score": 1.0, + "content": "We denote the density of", + "type": "text" + }, + { + "bbox": [ + 209, + 185, + 219, + 195 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 184, + 231, + 196 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 231, + 185, + 250, + 196 + ], + "score": 0.91, + "content": "q ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 184, + 506, + 196 + ], + "score": 1.0, + "content": ". The following theorem analytically computes the MSE under", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 211, + 208 + ], + "score": 1.0, + "content": "the regularization strength", + "type": "text" + }, + { + "bbox": [ + 211, + 196, + 228, + 208 + ], + "score": 0.9, + "content": "\\rho ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 195, + 506, + 208 + ], + "score": 1.0, + "content": ", building upon (Wang et al., 2013) for regression problems. The proof", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 280, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 280, + 219 + ], + "score": 1.0, + "content": "of the Theorem is deferred to Appendix A.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 162, + 507, + 219 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 219, + 504, + 246 + ], + "lines": [ + { + "bbox": [ + 104, + 216, + 503, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 216, + 210, + 236 + ], + "score": 1.0, + "content": "Theorem 1. Assume that", + "type": "text" + }, + { + "bbox": [ + 210, + 219, + 268, + 232 + ], + "score": 0.93, + "content": "f ^ { \\star } , q , I \\in W _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 216, + 287, + 236 + ], + "score": 1.0, + "content": ". 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A precise computation is infeasible, so we restrict ourselves to consider only", + "type": "text" + }, + { + "bbox": [ + 486, + 310, + 505, + 321 + ], + "score": 0.9, + "content": "\\rho ( x )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 321, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 332 + ], + "score": 1.0, + "content": "that is constant within groups of examples. 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Each group", + "type": "text" + }, + { + "bbox": [ + 407, + 331, + 447, + 343 + ], + "score": 0.92, + "content": "[ a _ { j } , a _ { j + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 330, + 506, + 345 + ], + "score": 1.0, + "content": "consists of an", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 441, + 354 + ], + "score": 1.0, + "content": "interval of data with approximately the same aleatoric uncertainty. We approximate", + "type": "text" + }, + { + "bbox": [ + 442, + 342, + 459, + 354 + ], + "score": 0.91, + "content": "\\rho ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "is constant", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 189, + 366 + ], + "score": 1.0, + "content": "on each of the group", + "type": "text" + }, + { + "bbox": [ + 190, + 353, + 228, + 365 + ], + "score": 0.91, + "content": "[ a _ { i } , a _ { i + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 353, + 272, + 366 + ], + "score": 1.0, + "content": "with value", + "type": "text" + }, + { + "bbox": [ + 273, + 354, + 282, + 365 + ], + "score": 0.84, + "content": "\\rho _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 353, + 420, + 366 + ], + "score": 1.0, + "content": ". Plugging this piece-wise constant", + "type": "text" + }, + { + "bbox": [ + 420, + 355, + 427, + 365 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "into the asymptotic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 230, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 230, + 375 + ], + "score": 1.0, + "content": "MSE in Theorem 1, we obtain", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 298, + 506, + 375 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 376, + 478, + 411 + ], + "lines": [ + { + "bbox": [ + 132, + 376, + 478, + 411 + ], + "spans": [ + { + "bbox": [ + 132, + 376, + 478, + 411 + ], + "score": 0.91, + "content": "\\underset { n \\infty } { \\operatorname* { l i m } } \\mathrm { M S E } ( \\hat { f } ) = \\sum _ { j } [ \\rho _ { j } ^ { 2 } \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 2 } ( t ) [ \\frac { d ^ { 2 } } { d t ^ { 2 } } f ^ { \\star } ( t ) ] ^ { 2 } d t + \\rho _ { j } ^ { - 1 / 2 } L _ { 0 } \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 1 / 2 } ( t ) d t ] .", + "type": "interline_equation", + "image_path": "5587924fdba5e89247fe3f42a6f90416d13bf1d281e1d1ed0d6f3c0c3f4118d0.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 132, + 376, + 478, + 387.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 132, + 387.6666666666667, + 478, + 399.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 132, + 399.33333333333337, + 478, + 411.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 506, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 258, + 427 + ], + "score": 1.0, + "content": "Minimizing the above formula over", + "type": "text" + }, + { + "bbox": [ + 258, + 415, + 300, + 425 + ], + "score": 0.91, + "content": "\\rho _ { 1 } , \\ldots , \\rho _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 410, + 481, + 427 + ], + "score": 1.0, + "content": "separately, we derive the optimal weights,", + "type": "text" + }, + { + "bbox": [ + 481, + 414, + 505, + 425 + ], + "score": 0.85, + "content": "\\rho _ { j } =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 108, + 423, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 108, + 424, + 225, + 459 + ], + "score": 0.78, + "content": "\\left[ \\frac { L _ { 0 } \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ( t ) ^ { 1 / 2 } d t } { 4 \\int _ { a _ { j } } ^ { a _ { j + 1 } } r ^ { 2 } ( t ) \\left[ \\frac { d ^ { 2 } } { d t ^ { 2 } } f ^ { \\star } ( t ) \\right] ^ { 2 } d t } \\right] ^ { 2 / }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 423, + 232, + 435 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 230, + 436, + 348, + 450 + ], + "score": 1.0, + "content": ". In practice, we do not know", + "type": "text" + }, + { + "bbox": [ + 348, + 437, + 360, + 449 + ], + "score": 0.89, + "content": "f ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 436, + 378, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 378, + 437, + 397, + 449 + ], + "score": 0.91, + "content": "q ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 436, + 505, + 450 + ], + "score": 1.0, + "content": ", so we make the following", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 232, + 471 + ], + "score": 1.0, + "content": "simplifications. We assume that", + "type": "text" + }, + { + "bbox": [ + 233, + 458, + 250, + 470 + ], + "score": 0.91, + "content": "q ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 457, + 267, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 268, + 458, + 285, + 470 + ], + "score": 0.91, + "content": "I ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 457, + 400, + 471 + ], + "score": 1.0, + "content": "are constant on each interval", + "type": "text" + }, + { + "bbox": [ + 401, + 458, + 440, + 470 + ], + "score": 0.92, + "content": "[ a _ { j } , a _ { j + 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 457, + 506, + 471 + ], + "score": 1.0, + "content": ". In other words,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 101, + 468, + 509, + 489 + ], + "spans": [ + { + "bbox": [ + 101, + 468, + 171, + 489 + ], + "score": 1.0, + "content": "we assume that", + "type": "text" + }, + { + "bbox": [ + 172, + 472, + 211, + 484 + ], + "score": 0.93, + "content": "q ( t ) = q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 468, + 230, + 489 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 471, + 270, + 484 + ], + "score": 0.92, + "content": "I ( t ) = I _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 468, + 299, + 489 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 299, + 471, + 356, + 484 + ], + "score": 0.93, + "content": "t \\in [ a _ { j } , a _ { j + 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 468, + 458, + 489 + ], + "score": 1.0, + "content": ". We further assume that", + "type": "text" + }, + { + "bbox": [ + 459, + 469, + 494, + 485 + ], + "score": 0.93, + "content": "\\textstyle { \\frac { d ^ { 2 } } { d t ^ { 2 } } } f ^ { \\star } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 468, + 509, + 489 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "close to a constant on the entire space, because estimating the curvature in high dimension is difficult.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 493, + 507, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 210, + 517 + ], + "score": 1.0, + "content": "This simplification yields", + "type": "text" + }, + { + "bbox": [ + 210, + 493, + 349, + 523 + ], + "score": 0.94, + "content": "\\rho _ { j } \\propto \\left[ \\frac { q _ { j } ^ { - 1 / 2 } I _ { j } ^ { - 1 / 2 } } { q _ { j } ^ { - 2 } I _ { j } ^ { - 2 } } \\right] ^ { 2 / 5 } = q _ { j } ^ { 3 / 5 } I _ { j } ^ { 3 / 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 498, + 507, + 518 + ], + "score": 1.0, + "content": "q3/5j I3/5j . We find the simplification works well", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 155, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 155, + 533 + ], + "score": 1.0, + "content": "in practice.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 101, + 410, + 509, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 504, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Adaptive regularization with importance sampling. It is practically infeasible to implement the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "integration in equation 3 for high-dimensional data. We use importance sampling to approximate the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 559, + 142, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 142, + 572 + ], + "score": 1.0, + "content": "integral:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 537, + 506, + 572 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 572, + 423, + 605 + ], + "lines": [ + { + "bbox": [ + 187, + 572, + 423, + 605 + ], + "spans": [ + { + "bbox": [ + 187, + 572, + 423, + 605 + ], + "score": 0.94, + "content": "{ \\mathrm { m i n i m i z e } } _ { f } L ( f ) \\triangleq { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } \\ell ( f ( x _ { i } ) , y _ { i } ) + \\lambda \\sum _ { i = 1 } ^ { n } \\tau _ { i } f ^ { \\prime } ( x _ { i } ) ^ { 2 }", + "type": "interline_equation", + "image_path": "2e1a6787dd6550c48149ec5dcb7d9acd394ea64aa6edf08473a3b48b21a5a890.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 572, + 423, + 588.5 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 187, + 588.5, + 423, + 605.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 503, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 142, + 621 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 143, + 607, + 205, + 620 + ], + "score": 0.92, + "content": "x _ { i } \\in [ a _ { j } , a _ { j + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 606, + 261, + 621 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 261, + 609, + 270, + 618 + ], + "score": 0.86, + "content": "\\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 606, + 346, + 621 + ], + "score": 1.0, + "content": "should satisfy that", + "type": "text" + }, + { + "bbox": [ + 347, + 609, + 387, + 619 + ], + "score": 0.91, + "content": "\\tau _ { i } q _ { j } = \\rho _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 606, + 506, + 621 + ], + "score": 1.0, + "content": "so that the expectation of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 618, + 389, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 389, + 632 + ], + "score": 1.0, + "content": "regularization term in equation 5 is equal to that in equation 3. Hence,", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 606, + 506, + 632 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 631, + 383, + 650 + ], + "lines": [ + { + "bbox": [ + 227, + 631, + 383, + 650 + ], + "spans": [ + { + "bbox": [ + 227, + 631, + 383, + 650 + ], + "score": 0.92, + "content": "\\tau _ { i } = I _ { j } ^ { 3 / 5 } q _ { j } ^ { - 2 / 5 } = I ( x _ { i } ) ^ { 3 / 5 } q ( x _ { i } ) ^ { - 2 / 5 } .", + "type": "interline_equation", + "image_path": "e07633c49d25d6bffcf960cf99ca3e0f38f2d0e6eed54789f39998d3cc6781ba.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 227, + 631, + 383, + 650 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "Adaptive regularization for multi-class classification and regression. In fact, the proof of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 264, + 681 + ], + "score": 1.0, + "content": "Theorem 1 is proved for general loss", + "type": "text" + }, + { + "bbox": [ + 264, + 668, + 292, + 680 + ], + "score": 0.92, + "content": "\\ell ( a , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 667, + 506, + 681 + ], + "score": 1.0, + "content": ". Therefore, we can directly generalize it to multi-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 401, + 693 + ], + "score": 1.0, + "content": "class classification and regression problems. For a regression problem,", + "type": "text" + }, + { + "bbox": [ + 401, + 680, + 429, + 691 + ], + "score": 0.93, + "content": "\\ell ( a , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "is the square loss:", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 195, + 702 + ], + "score": 0.93, + "content": "\\ell ( y , a ) = 0 . 5 ( y - a ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 689, + 289, + 704 + ], + "score": 1.0, + "content": ", the Fisher information", + "type": "text" + }, + { + "bbox": [ + 289, + 690, + 327, + 702 + ], + "score": 0.92, + "content": "I ( x ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 689, + 506, + 704 + ], + "score": 1.0, + "content": ". Therefore, for a regression problem, we can", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 701, + 291, + 716 + ], + "spans": [ + { + "bbox": [ + 104, + 701, + 224, + 716 + ], + "score": 1.0, + "content": "choose regularization weight", + "type": "text" + }, + { + "bbox": [ + 224, + 701, + 286, + 714 + ], + "score": 0.92, + "content": "\\tau _ { i } = q ( x _ { i } ) ^ { - 2 / 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 701, + 291, + 716 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 657, + 506, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 102, + 506, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 507, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 507, + 116 + ], + "score": 1.0, + "content": "We heuristically extend the Lipschitz regularization technique discussed in Section 2.2 from non-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 397, + 127 + ], + "score": 1.0, + "content": "parametric models to over-parameterized deep neural networks. Let", + "type": "text" + }, + { + "bbox": [ + 397, + 114, + 420, + 126 + ], + "score": 0.92, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "be an example and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 124, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 107, + 127, + 117, + 138 + ], + "score": 0.84, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 124, + 146, + 141 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 146, + 128, + 152, + 136 + ], + "score": 0.75, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 124, + 312, + 141 + ], + "score": 1.0, + "content": "-layer neural network. We denote by", + "type": "text" + }, + { + "bbox": [ + 313, + 125, + 330, + 137 + ], + "score": 0.89, + "content": "h ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 124, + 348, + 141 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 349, + 127, + 355, + 138 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 124, + 506, + 141 + ], + "score": 1.0, + "content": "-th hidden layer of the network, by", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 135, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 218, + 153 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\dot { J } ^ { ( j ) } ( x ) \\triangleq \\frac { \\hat { \\partial } } { \\partial h ^ { ( j ) } } \\mathcal { L } ( f ( x ) , y ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 135, + 359, + 153 + ], + "score": 1.0, + "content": ", i.e., the Jacobian of the loss w.r.t", + "type": "text" + }, + { + "bbox": [ + 359, + 137, + 376, + 149 + ], + "score": 0.89, + "content": "h ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 135, + 506, + 153 + ], + "score": 1.0, + "content": ". We replace the regularization", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 102, + 150, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 102, + 150, + 127, + 174 + ], + "score": 1.0, + "content": "term", + "type": "text" + }, + { + "bbox": [ + 127, + 157, + 155, + 170 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( x ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 150, + 223, + 174 + ], + "score": 1.0, + "content": "in equation 5 by", + "type": "text" + }, + { + "bbox": [ + 224, + 151, + 356, + 173 + ], + "score": 0.93, + "content": "\\begin{array} { r } { R ( x ) = { \\left( \\sum _ { j = 1 } ^ { r } | | J ^ { ( j ) } ( x ) | | _ { F } ^ { 2 } \\right) } ^ { 1 / 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 150, + 506, + 174 + ], + "score": 1.0, + "content": ", which was proposed by (Wei & Ma,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 171, + 507, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 507, + 185 + ], + "score": 1.0, + "content": "2019a). As a proof of concept, we visualize the behavior of our algorithm in Figure 3, where we ob-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "serve that the rare and noisy examples have significantly improved error due to stronger regularization.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 194, + 429, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 429, + 206 + ], + "score": 1.0, + "content": "In contrast, a uniform regularization either overfits or underfits different subsets.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 211, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 222 + ], + "score": 1.0, + "content": "Note that the differences from the 1-D case include the following three aspects. 1. The derivative", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "is taken w.r.t to all the hidden layers for deep models, which has been shown to have superior", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "generalization guarantees for neural networks by (Wei & Ma, 2019a;b). 2. An additional square root", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 195, + 256 + ], + "score": 1.0, + "content": "is taken in computing", + "type": "text" + }, + { + "bbox": [ + 195, + 244, + 217, + 255 + ], + "score": 0.92, + "content": "R ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 245, + 506, + 256 + ], + "score": 1.0, + "content": ". This modified version may have milder curvature and be easier to tune.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 460, + 266 + ], + "score": 1.0, + "content": "3. We take the derivative of the loss instead of the derivative of the model, which outputs", + "type": "text" + }, + { + "bbox": [ + 461, + 255, + 467, + 264 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "numbers", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 430, + 278 + ], + "score": 1.0, + "content": "for multi-class classification. This is because the derivative of the model requires", + "type": "text" + }, + { + "bbox": [ + 430, + 266, + 437, + 276 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "times more time", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 277, + 357, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 357, + 289 + ], + "score": 1.0, + "content": "to compute. The regularized training objective is consequently", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 290, + 415, + 322 + ], + "lines": [ + { + "bbox": [ + 194, + 290, + 415, + 322 + ], + "spans": [ + { + "bbox": [ + 194, + 290, + 415, + 322 + ], + "score": 0.93, + "content": "\\mathrm { m i n i m i z e } _ { f } L ( f ) \\triangleq \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\left( \\ell ( f ( x _ { i } ) , y _ { i } ) + \\lambda \\tau _ { i } R ( x _ { i } ) \\right) ,", + "type": "interline_equation", + "image_path": "9c0f87a88a7df27d97e5a419a545e647849d7ceef67136637f7d116cf57bf9ae.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 290, + 415, + 306.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 194, + 306.0, + 415, + 322.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 104, + 324, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 132, + 340 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 329, + 141, + 338 + ], + "score": 0.84, + "content": "\\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 324, + 203, + 340 + ], + "score": 1.0, + "content": "is chosen to be", + "type": "text" + }, + { + "bbox": [ + 204, + 325, + 298, + 339 + ], + "score": 0.93, + "content": "\\tau _ { i } = I ( x _ { i } ) ^ { 3 / 5 } / q ( x _ { i } ) ^ { 2 / 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 324, + 506, + 340 + ], + "score": 1.0, + "content": "following the formula equation 5 in Section 2.2 and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 107, + 338, + 114, + 347 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "is a hyperparameter to control the overall scale of the regularization strength. We note that we do", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 208, + 360 + ], + "score": 1.0, + "content": "not expect this choice of", + "type": "text" + }, + { + "bbox": [ + 209, + 350, + 217, + 360 + ], + "score": 0.85, + "content": "\\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "to be optimal for the high-dimensional case with all the modifications", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "above – the optimal choice does depend on the nuances. However, we also observe that the empirical", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 275, + 383 + ], + "score": 1.0, + "content": "performance is not sensitive to the form of", + "type": "text" + }, + { + "bbox": [ + 275, + 372, + 282, + 380 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 370, + 391, + 383 + ], + "score": 1.0, + "content": "as long as it’s increasing in", + "type": "text" + }, + { + "bbox": [ + 392, + 370, + 411, + 382 + ], + "score": 0.92, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 370, + 484, + 383 + ], + "score": 1.0, + "content": "and decreasing in", + "type": "text" + }, + { + "bbox": [ + 484, + 372, + 503, + 382 + ], + "score": 0.92, + "content": "q ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 370, + 506, + 383 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 380, + 486, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 486, + 394 + ], + "score": 1.0, + "content": "That is, the more uncertain or rare an example is, the stronger regularization should be applied.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 331, + 411 + ], + "score": 1.0, + "content": "In order to estimate the relative regularization strength", + "type": "text" + }, + { + "bbox": [ + 332, + 400, + 341, + 409 + ], + "score": 0.84, + "content": "\\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 397, + 505, + 411 + ], + "score": 1.0, + "content": ", the key difficulty lies in the estimation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 165, + 423 + ], + "score": 1.0, + "content": "of uncertainty", + "type": "text" + }, + { + "bbox": [ + 166, + 409, + 185, + 421 + ], + "score": 0.91, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 408, + 395, + 423 + ], + "score": 1.0, + "content": ". As in the 1-D setting, we divide the examples into", + "type": "text" + }, + { + "bbox": [ + 395, + 410, + 402, + 419 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 408, + 433, + 423 + ], + "score": 1.0, + "content": "groups", + "type": "text" + }, + { + "bbox": [ + 434, + 409, + 482, + 420 + ], + "score": 0.92, + "content": "G _ { 1 } , \\ldots , G _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 408, + 506, + 423 + ], + "score": 1.0, + "content": "(e.g.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "score": 1.0, + "content": "each group can correspond to a class), and estimate the uncertainty on each group. In the binary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 430, + 140, + 444 + ], + "score": 1.0, + "content": "setting,", + "type": "text" + }, + { + "bbox": [ + 140, + 431, + 399, + 443 + ], + "score": 0.89, + "content": "I ( x ) = \\mathrm { \\mathrm { V a r } } ( { \\bar { Y } } | X = x ) = \\operatorname* { P r } [ Y = 1 \\mid X ] \\cdot \\operatorname* { P r } [ Y = 0 \\ { \\bar { | } } \\ X ]", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "can be approximated by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 441, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 285, + 456 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\tilde { I } ( x ) = 1 - \\operatorname* { m a x } _ { k \\in \\{ 0 , 1 \\} } \\operatorname* { P r } [ Y = k \\mid X = x ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 441, + 507, + 459 + ], + "score": 1.0, + "content": "up to a factor of at most 2. We use the same formula for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 434, + 467 + ], + "score": 1.0, + "content": "multi-class setting as the approximation of the uncertainty. (As a sanity check, when", + "type": "text" + }, + { + "bbox": [ + 434, + 455, + 443, + 465 + ], + "score": 0.59, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "is concentrated", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 465, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 316, + 479 + ], + "score": 1.0, + "content": "on a single outcome, the uncertainty is 0.) Note that", + "type": "text" + }, + { + "bbox": [ + 317, + 465, + 336, + 479 + ], + "score": 0.92, + "content": "\\tilde { I } ( \\stackrel { \\cdot } { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "is essentially the minimum possible error", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 308, + 491 + ], + "score": 1.0, + "content": "of any deterministic prediction on the data point", + "type": "text" + }, + { + "bbox": [ + 309, + 480, + 315, + 488 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 478, + 505, + 491 + ], + "score": 1.0, + "content": ". Assume that we have a sufficiently accurate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 489, + 375, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 350, + 503 + ], + "score": 1.0, + "content": "pre-trained model, we can use its validation error to estimate", + "type": "text" + }, + { + "bbox": [ + 351, + 489, + 370, + 502 + ], + "score": 0.92, + "content": "\\tilde { I } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 489, + 375, + 503 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 311, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 311, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 155, + 522 + ], + "score": 1.0, + "content": "Then for all", + "type": "text" + }, + { + "bbox": [ + 156, + 507, + 186, + 520 + ], + "score": 0.92, + "content": "x \\in G _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 504, + 240, + 522 + ], + "score": 1.0, + "content": ", we estimate", + "type": "text" + }, + { + "bbox": [ + 240, + 507, + 260, + 519 + ], + "score": 0.92, + "content": "q ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 504, + 278, + 522 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 278, + 507, + 298, + 519 + ], + "score": 0.91, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 504, + 311, + 522 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 119, + 522, + 474, + 534 + ], + "lines": [ + { + "bbox": [ + 120, + 520, + 475, + 536 + ], + "spans": [ + { + "bbox": [ + 120, + 521, + 222, + 535 + ], + "score": 0.53, + "content": "\\forall x \\in G _ { j } , q ( x ) \\propto | G _ { j } | , I", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 522, + 248, + 534 + ], + "score": 0.77, + "content": "I ( x ) \\propto", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 520, + 437, + 536 + ], + "score": 1.0, + "content": "average validation error of a pre-trained model", + "type": "text" + }, + { + "bbox": [ + 437, + 522, + 448, + 534 + ], + "score": 0.85, + "content": "f _ { \\tilde { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 520, + 461, + 536 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 462, + 522, + 475, + 534 + ], + "score": 0.84, + "content": "G _ { j }", + "type": "inline_equation" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 343, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 344, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 344, + 556 + ], + "score": 1.0, + "content": "The whole training pipeline is summarized in Algorithm 1.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 106, + 565, + 356, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 357, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 357, + 580 + ], + "score": 1.0, + "content": "Algorithm 1 Heteroskedastic Adaptive Regularization (HAR)", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 108, + 581, + 491, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 367, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 181, + 595 + ], + "score": 1.0, + "content": "Require: Dataset", + "type": "text" + }, + { + "bbox": [ + 182, + 581, + 257, + 593 + ], + "score": 0.9, + "content": "\\mathcal { D } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 579, + 357, + 595 + ], + "score": 1.0, + "content": ". 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l } }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 109, + 602, + 280, + 618 + ], + "spans": [ + { + "bbox": [ + 109, + 602, + 123, + 618 + ], + "score": 1.0, + "content": "2:", + "type": "text" + }, + { + "bbox": [ + 123, + 604, + 147, + 616 + ], + "score": 0.83, + "content": "f _ { \\tilde { \\theta } } \\gets", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 602, + 257, + 618 + ], + "score": 1.0, + "content": "Standard SGD Training on", + "type": "text" + }, + { + "bbox": [ + 258, + 604, + 280, + 615 + ], + "score": 0.88, + "content": "\\mathcal { D } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 108, + 613, + 488, + 630 + ], + "spans": [ + { + "bbox": [ + 108, + 613, + 160, + 630 + ], + "score": 1.0, + "content": "3: Estimate", + "type": "text" + }, + { + "bbox": [ + 161, + 616, + 203, + 628 + ], + "score": 0.92, + "content": "I ( x ) , q ( x )", + "type": "inline_equation" + }, + { + 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rarer groups of examples (by stronger regularization) without negatively", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 82, + 504, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "2.3 PRACTICAL IMPLEMENTATION ON NEURAL NETWORKS WITH HIGH-DIMENSIONAL DATA", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 102, + 506, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 507, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 507, + 116 + ], + "score": 1.0, + "content": "We heuristically extend the Lipschitz regularization technique discussed in Section 2.2 from non-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 397, + 127 + ], + "score": 1.0, + "content": "parametric models to over-parameterized deep neural networks. Let", + "type": "text" + }, + { + "bbox": [ + 397, + 114, + 420, + 126 + ], + "score": 0.92, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "be an example and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 124, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 107, + 127, + 117, + 138 + ], + "score": 0.84, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 124, + 146, + 141 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 146, + 128, + 152, + 136 + ], + "score": 0.75, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 124, + 312, + 141 + ], + "score": 1.0, + "content": "-layer neural network. We denote by", + "type": "text" + }, + { + "bbox": [ + 313, + 125, + 330, + 137 + ], + "score": 0.89, + "content": "h ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 124, + 348, + 141 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 349, + 127, + 355, + 138 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 124, + 506, + 141 + ], + "score": 1.0, + "content": "-th hidden layer of the network, by", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 135, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 218, + 153 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\dot { J } ^ { ( j ) } ( x ) \\triangleq \\frac { \\hat { \\partial } } { \\partial h ^ { ( j ) } } \\mathcal { L } ( f ( x ) , y ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 135, + 359, + 153 + ], + "score": 1.0, + "content": ", i.e., the Jacobian of the loss w.r.t", + "type": "text" + }, + { + "bbox": [ + 359, + 137, + 376, + 149 + ], + "score": 0.89, + "content": "h ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 135, + 506, + 153 + ], + "score": 1.0, + "content": ". We replace the regularization", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 102, + 150, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 102, + 150, + 127, + 174 + ], + "score": 1.0, + "content": "term", + "type": "text" + }, + { + "bbox": [ + 127, + 157, + 155, + 170 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( x ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 150, + 223, + 174 + ], + "score": 1.0, + "content": "in equation 5 by", + "type": "text" + }, + { + "bbox": [ + 224, + 151, + 356, + 173 + ], + "score": 0.93, + "content": "\\begin{array} { r } { R ( x ) = { \\left( \\sum _ { j = 1 } ^ { r } | | J ^ { ( j ) } ( x ) | | _ { F } ^ { 2 } \\right) } ^ { 1 / 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 150, + 506, + 174 + ], + "score": 1.0, + "content": ", which was proposed by (Wei & Ma,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 171, + 507, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 507, + 185 + ], + "score": 1.0, + "content": "2019a). As a proof of concept, we visualize the behavior of our algorithm in Figure 3, where we ob-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "serve that the rare and noisy examples have significantly improved error due to stronger regularization.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 194, + 429, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 429, + 206 + ], + "score": 1.0, + "content": "In contrast, a uniform regularization either overfits or underfits different subsets.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 102, + 102, + 507, + 206 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 211, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 222 + ], + "score": 1.0, + "content": "Note that the differences from the 1-D case include the following three aspects. 1. The derivative", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "is taken w.r.t to all the hidden layers for deep models, which has been shown to have superior", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "generalization guarantees for neural networks by (Wei & Ma, 2019a;b). 2. An additional square root", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 195, + 256 + ], + "score": 1.0, + "content": "is taken in computing", + "type": "text" + }, + { + "bbox": [ + 195, + 244, + 217, + 255 + ], + "score": 0.92, + "content": "R ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 245, + 506, + 256 + ], + "score": 1.0, + "content": ". This modified version may have milder curvature and be easier to tune.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 460, + 266 + ], + "score": 1.0, + "content": "3. We take the derivative of the loss instead of the derivative of the model, which outputs", + "type": "text" + }, + { + "bbox": [ + 461, + 255, + 467, + 264 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "numbers", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 430, + 278 + ], + "score": 1.0, + "content": "for multi-class classification. This is because the derivative of the model requires", + "type": "text" + }, + { + "bbox": [ + 430, + 266, + 437, + 276 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "times more time", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 277, + 357, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 357, + 289 + ], + "score": 1.0, + "content": "to compute. The regularized training objective is consequently", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 211, + 506, + 289 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 290, + 415, + 322 + ], + "lines": [ + { + "bbox": [ + 194, + 290, + 415, + 322 + ], + "spans": [ + { + "bbox": [ + 194, + 290, + 415, + 322 + ], + "score": 0.93, + "content": "\\mathrm { m i n i m i z e } _ { f } L ( f ) \\triangleq \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\left( \\ell ( f ( x _ { i } ) , y _ { i } ) + \\lambda \\tau _ { i } R ( x _ { i } ) \\right) ,", + "type": "interline_equation", + "image_path": "9c0f87a88a7df27d97e5a419a545e647849d7ceef67136637f7d116cf57bf9ae.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 290, + 415, + 306.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 194, + 306.0, + 415, + 322.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 104, + 324, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 132, + 340 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 329, + 141, + 338 + ], + "score": 0.84, + "content": "\\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 324, + 203, + 340 + ], + "score": 1.0, + "content": "is chosen to be", + "type": "text" + }, + { + "bbox": [ + 204, + 325, + 298, + 339 + ], + "score": 0.93, + "content": "\\tau _ { i } = I ( x _ { i } ) ^ { 3 / 5 } / q ( x _ { i } ) ^ { 2 / 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 324, + 506, + 340 + ], + "score": 1.0, + "content": "following the formula equation 5 in Section 2.2 and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 107, + 338, + 114, + 347 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "is a hyperparameter to control the overall scale of the regularization strength. We note that we do", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 208, + 360 + ], + "score": 1.0, + "content": "not expect this choice of", + "type": "text" + }, + { + "bbox": [ + 209, + 350, + 217, + 360 + ], + "score": 0.85, + "content": "\\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "to be optimal for the high-dimensional case with all the modifications", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "above – the optimal choice does depend on the nuances. However, we also observe that the empirical", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 275, + 383 + ], + "score": 1.0, + "content": "performance is not sensitive to the form of", + "type": "text" + }, + { + "bbox": [ + 275, + 372, + 282, + 380 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 370, + 391, + 383 + ], + "score": 1.0, + "content": "as long as it’s increasing in", + "type": "text" + }, + { + "bbox": [ + 392, + 370, + 411, + 382 + ], + "score": 0.92, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 370, + 484, + 383 + ], + "score": 1.0, + "content": "and decreasing in", + "type": "text" + }, + { + "bbox": [ + 484, + 372, + 503, + 382 + ], + "score": 0.92, + "content": "q ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 370, + 506, + 383 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 380, + 486, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 486, + 394 + ], + "score": 1.0, + "content": "That is, the more uncertain or rare an example is, the stronger regularization should be applied.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 324, + 506, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 331, + 411 + ], + "score": 1.0, + "content": "In order to estimate the relative regularization strength", + "type": "text" + }, + { + "bbox": [ + 332, + 400, + 341, + 409 + ], + "score": 0.84, + "content": "\\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 397, + 505, + 411 + ], + "score": 1.0, + "content": ", the key difficulty lies in the estimation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 165, + 423 + ], + "score": 1.0, + "content": "of uncertainty", + "type": "text" + }, + { + "bbox": [ + 166, + 409, + 185, + 421 + ], + "score": 0.91, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 408, + 395, + 423 + ], + "score": 1.0, + "content": ". As in the 1-D setting, we divide the examples into", + "type": "text" + }, + { + "bbox": [ + 395, + 410, + 402, + 419 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 408, + 433, + 423 + ], + "score": 1.0, + "content": "groups", + "type": "text" + }, + { + "bbox": [ + 434, + 409, + 482, + 420 + ], + "score": 0.92, + "content": "G _ { 1 } , \\ldots , G _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 408, + 506, + 423 + ], + "score": 1.0, + "content": "(e.g.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "score": 1.0, + "content": "each group can correspond to a class), and estimate the uncertainty on each group. In the binary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 430, + 140, + 444 + ], + "score": 1.0, + "content": "setting,", + "type": "text" + }, + { + "bbox": [ + 140, + 431, + 399, + 443 + ], + "score": 0.89, + "content": "I ( x ) = \\mathrm { \\mathrm { V a r } } ( { \\bar { Y } } | X = x ) = \\operatorname* { P r } [ Y = 1 \\mid X ] \\cdot \\operatorname* { P r } [ Y = 0 \\ { \\bar { | } } \\ X ]", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "can be approximated by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 441, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 285, + 456 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\tilde { I } ( x ) = 1 - \\operatorname* { m a x } _ { k \\in \\{ 0 , 1 \\} } \\operatorname* { P r } [ Y = k \\mid X = x ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 441, + 507, + 459 + ], + "score": 1.0, + "content": "up to a factor of at most 2. 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We evaluate our algorithms on", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "three vision datasets and one NLP dataset: CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009),", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "IMDB-review (Maas et al., 2011) (see Appendix C.1), and WebVision (Li et al., 2017), a real-world", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "heteroskedastic and imbalanced dataset. Please refer to Appendix B for low-level implementation", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 138, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 138, + 138 + ], + "score": 1.0, + "content": "details.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "affecting the training and test performance of the other groups. We evaluate our algorithms on", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "three vision datasets and one NLP dataset: CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009),", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "IMDB-review (Maas et al., 2011) (see Appendix C.1), and WebVision (Li et al., 2017), a real-world", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "heteroskedastic and imbalanced dataset. 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The simplest one is (1)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 506, + 166 + ], + "score": 1.0, + "content": "Empirical risk minimization (ERM): the vanilla cross-entropy loss with all examples having the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "same weights of losses. We select two representatives from the noise-cleaning line of approach. (2)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "Co-teaching (Han et al., 2018): two deep networks are trained simultaneously. Each network aims", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "to identify clean data points that have small losses and use them to guide the training of the other", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "network. (3) INCV (Chen et al., 2019): it extends Co-teacing to an interative version to estimate the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "noise ratio and select data. We consider three representatives from the reweighting-based methods,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "including two that learn the weighting using meta-learning. (4) MentorNet (Jiang et al., 2018): it", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "pretrains a teacher network that outputs weights for examples that are used to train the student network", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 506, + 254 + ], + "score": 1.0, + "content": "with reweighting. (5) L2RW (Ren et al., 2018): it directly optimizes weights of each example in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "training set by minimizing its corresponding loss on a small meta validation set. (6) MW-Net (Shu", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "et al., 2019): it extends L2RW by explicitly defining a weighting function which depends only on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "score": 1.0, + "content": "the loss of the example. We also compare against two representatives from the robust loss function.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "(7) GCE (Zhang & Sabuncu, 2018): it generalizes mean average error and cross-entropy loss to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 266, + 309 + ], + "score": 1.0, + "content": "obtain a new loss function. (8) DMI (", + "type": "text" + }, + { + "bbox": [ + 266, + 297, + 280, + 307 + ], + "score": 0.28, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "et al., 2019): it designs a new loss function based on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "generalized mutual information. In addition, as an essential ablation study, we consider vanilla", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "uniform regularization. (9) Unif-reg: we apply the Jacobian regularizer on all examples with equal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 405, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 405, + 343 + ], + "score": 1.0, + "content": "strength, and tune the strength to get the best possible validation accuracy.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 109, + 363, + 444, + 373 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 447, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 447, + 374 + ], + "score": 1.0, + "content": "3.1 SIMULATING HETEROSKEDASTIC AND IMBALANCED DATASETS ON CIFAR", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "Setting. Unlike previous works that test on uniform random or asymmetric noise, which is often not", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "the case in reality, in this paper we test our method on more realistic noisy settings, as suggested", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "by Patrini et al. (2017); Zhang & Sabuncu (2018). In order to simulate heteroskedasticity, we only", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 368, + 432 + ], + "score": 1.0, + "content": "corrupt semantically-similar classes. For CIFAR-10, we exchange", + "type": "text" + }, + { + "bbox": [ + 369, + 419, + 388, + 429 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "of the labels between classes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "‘cat’ and ‘dog’, and between ‘truck’ and ‘automobile’. CIFAR-100 has 100 classes grouped into", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 426, + 453 + ], + "score": 1.0, + "content": "20 super classes. For each class of the 5 classes under the super class ‘vehicles", + "type": "text" + }, + { + "bbox": [ + 426, + 441, + 436, + 451 + ], + "score": 0.26, + "content": "_ { 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 441, + 492, + 453 + ], + "score": 1.0, + "content": "and ‘vehicles", + "type": "text" + }, + { + "bbox": [ + 493, + 441, + 502, + 451 + ], + "score": 0.57, + "content": "_ { 2 } \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 441, + 506, + 453 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 212, + 464 + ], + "score": 1.0, + "content": "we corrupt the labels with", + "type": "text" + }, + { + "bbox": [ + 212, + 452, + 232, + 462 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "probability uniformly randomly to the rest of four classes under the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 461, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 390, + 476 + ], + "score": 1.0, + "content": "same super class. As a result, the 10 classes under super class ‘vehicle", + "type": "text" + }, + { + "bbox": [ + 390, + 463, + 400, + 473 + ], + "score": 0.39, + "content": "_ { 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 461, + 452, + 476 + ], + "score": 1.0, + "content": "and ‘vehicle", + "type": "text" + }, + { + "bbox": [ + 452, + 463, + 462, + 473 + ], + "score": 0.53, + "content": "_ { 2 } \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 461, + 505, + 476 + ], + "score": 1.0, + "content": "have high", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "label noise level and the corruption are only within the same super class. Heteroskedasticity of the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "labels and imbalance of the inputs commonly coexist in the real world settings. HAR can take both of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "them into account. To understand the challenge imposed by the entanglements of heteroskedasticity", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "and imbalance, and compare HAR with the aforementioned baselines, we inject data imbalance", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "concurrently with the heteroskedastic noise. We remove samples from the corrupted classes to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "score": 1.0, + "content": "simulate the most difficult scenario — the rare and noisy groups overfit significantly. (A more benign", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "interaction between the noises and imbalance is that the rare classes have lower noise level, we defer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "it to Appendix C.3.) We use the imbalance ratio to denote the frequency ratio between the frequent", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "score": 1.0, + "content": "(and clean) classes to the rare (and corrupted) classes. We consider imbalance ratio to be 10 and 100.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "Result. Table 1 summarizes the results. Since examples from rare classes tend to have larger training", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 590, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 504, + 601 + ], + "score": 1.0, + "content": "and validation loss regardless of whether the labels are correct or not, noise-cleaning based methods", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "might drop excessive examples with correct labels. We examined the noise ratio of dropped samples", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "for INCV under the setting of imbalance ratio equals 10. Among all dropped examples, there is", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 127, + 634 + ], + "score": 1.0, + "content": "only", + "type": "text" + }, + { + "bbox": [ + 128, + 622, + 155, + 632 + ], + "score": 0.86, + "content": "1 9 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 622, + 477, + 634 + ], + "score": 1.0, + "content": "of true noise examples. In addition, the rare class examples selected still have", + "type": "text" + }, + { + "bbox": [ + 477, + 622, + 505, + 632 + ], + "score": 0.86, + "content": "2 9 . 8 \\%", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "of label noise. This explains that the significant decrease of accuracies of Co-teaching and INCV", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 643, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 643, + 506, + 658 + ], + "score": 1.0, + "content": "on corrupted and rare classes. Reweighting-based methods tend to suffer from the loss of accuracy", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "in other more frequent classes, which is aligned with the findings in Cao et al. (2019). While the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "aforementioned baselines struggle to deal with heteroskedasticity and imbalance together, HAR is", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "able to put them under the same regularization framework and achieve significant improvements.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Notably, HAR also shows improvement over uniform regularization with optimally tuned strength.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "This clearly demonstrates the importance of introducing adaptive regularization among all examples", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "for a better trade-off. A more detailed ablation study on the trade-off between training accuracy and", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 721, + 299, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 299, + 732 + ], + "score": 1.0, + "content": "validation accuracy can be found in Section 3.3.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 47.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 505, + 138 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "Baselines. We compare our proposed HAR with the following baselines. The simplest one is (1)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 506, + 166 + ], + "score": 1.0, + "content": "Empirical risk minimization (ERM): the vanilla cross-entropy loss with all examples having the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "same weights of losses. We select two representatives from the noise-cleaning line of approach. (2)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "Co-teaching (Han et al., 2018): two deep networks are trained simultaneously. Each network aims", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "to identify clean data points that have small losses and use them to guide the training of the other", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "network. (3) INCV (Chen et al., 2019): it extends Co-teacing to an interative version to estimate the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "noise ratio and select data. We consider three representatives from the reweighting-based methods,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "including two that learn the weighting using meta-learning. (4) MentorNet (Jiang et al., 2018): it", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "pretrains a teacher network that outputs weights for examples that are used to train the student network", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 506, + 254 + ], + "score": 1.0, + "content": "with reweighting. (5) L2RW (Ren et al., 2018): it directly optimizes weights of each example in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "training set by minimizing its corresponding loss on a small meta validation set. (6) MW-Net (Shu", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "et al., 2019): it extends L2RW by explicitly defining a weighting function which depends only on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "score": 1.0, + "content": "the loss of the example. We also compare against two representatives from the robust loss function.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "(7) GCE (Zhang & Sabuncu, 2018): it generalizes mean average error and cross-entropy loss to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 266, + 309 + ], + "score": 1.0, + "content": "obtain a new loss function. (8) DMI (", + "type": "text" + }, + { + "bbox": [ + 266, + 297, + 280, + 307 + ], + "score": 0.28, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "et al., 2019): it designs a new loss function based on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "generalized mutual information. In addition, as an essential ablation study, we consider vanilla", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "uniform regularization. (9) Unif-reg: we apply the Jacobian regularizer on all examples with equal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 405, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 405, + 343 + ], + "score": 1.0, + "content": "strength, and tune the strength to get the best possible validation accuracy.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 142, + 506, + 343 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 363, + 444, + 373 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 447, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 447, + 374 + ], + "score": 1.0, + "content": "3.1 SIMULATING HETEROSKEDASTIC AND IMBALANCED DATASETS ON CIFAR", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "Setting. Unlike previous works that test on uniform random or asymmetric noise, which is often not", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "the case in reality, in this paper we test our method on more realistic noisy settings, as suggested", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "by Patrini et al. (2017); Zhang & Sabuncu (2018). In order to simulate heteroskedasticity, we only", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 368, + 432 + ], + "score": 1.0, + "content": "corrupt semantically-similar classes. For CIFAR-10, we exchange", + "type": "text" + }, + { + "bbox": [ + 369, + 419, + 388, + 429 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "of the labels between classes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "‘cat’ and ‘dog’, and between ‘truck’ and ‘automobile’. CIFAR-100 has 100 classes grouped into", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 426, + 453 + ], + "score": 1.0, + "content": "20 super classes. For each class of the 5 classes under the super class ‘vehicles", + "type": "text" + }, + { + "bbox": [ + 426, + 441, + 436, + 451 + ], + "score": 0.26, + "content": "_ { 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 441, + 492, + 453 + ], + "score": 1.0, + "content": "and ‘vehicles", + "type": "text" + }, + { + "bbox": [ + 493, + 441, + 502, + 451 + ], + "score": 0.57, + "content": "_ { 2 } \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 441, + 506, + 453 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 212, + 464 + ], + "score": 1.0, + "content": "we corrupt the labels with", + "type": "text" + }, + { + "bbox": [ + 212, + 452, + 232, + 462 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "probability uniformly randomly to the rest of four classes under the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 461, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 390, + 476 + ], + "score": 1.0, + "content": "same super class. As a result, the 10 classes under super class ‘vehicle", + "type": "text" + }, + { + "bbox": [ + 390, + 463, + 400, + 473 + ], + "score": 0.39, + "content": "_ { 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 461, + 452, + 476 + ], + "score": 1.0, + "content": "and ‘vehicle", + "type": "text" + }, + { + "bbox": [ + 452, + 463, + 462, + 473 + ], + "score": 0.53, + "content": "_ { 2 } \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 461, + 505, + 476 + ], + "score": 1.0, + "content": "have high", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "label noise level and the corruption are only within the same super class. Heteroskedasticity of the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "labels and imbalance of the inputs commonly coexist in the real world settings. HAR can take both of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "them into account. To understand the challenge imposed by the entanglements of heteroskedasticity", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "and imbalance, and compare HAR with the aforementioned baselines, we inject data imbalance", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "concurrently with the heteroskedastic noise. We remove samples from the corrupted classes to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "score": 1.0, + "content": "simulate the most difficult scenario — the rare and noisy groups overfit significantly. (A more benign", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "interaction between the noises and imbalance is that the rare classes have lower noise level, we defer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "it to Appendix C.3.) We use the imbalance ratio to denote the frequency ratio between the frequent", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "score": 1.0, + "content": "(and clean) classes to the rare (and corrupted) classes. We consider imbalance ratio to be 10 and 100.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 386, + 506, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "Result. Table 1 summarizes the results. Since examples from rare classes tend to have larger training", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 590, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 504, + 601 + ], + "score": 1.0, + "content": "and validation loss regardless of whether the labels are correct or not, noise-cleaning based methods", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "might drop excessive examples with correct labels. We examined the noise ratio of dropped samples", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "for INCV under the setting of imbalance ratio equals 10. Among all dropped examples, there is", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 127, + 634 + ], + "score": 1.0, + "content": "only", + "type": "text" + }, + { + "bbox": [ + 128, + 622, + 155, + 632 + ], + "score": 0.86, + "content": "1 9 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 622, + 477, + 634 + ], + "score": 1.0, + "content": "of true noise examples. In addition, the rare class examples selected still have", + "type": "text" + }, + { + "bbox": [ + 477, + 622, + 505, + 632 + ], + "score": 0.86, + "content": "2 9 . 8 \\%", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "of label noise. This explains that the significant decrease of accuracies of Co-teaching and INCV", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 643, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 643, + 506, + 658 + ], + "score": 1.0, + "content": "on corrupted and rare classes. Reweighting-based methods tend to suffer from the loss of accuracy", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "in other more frequent classes, which is aligned with the findings in Cao et al. (2019). While the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "aforementioned baselines struggle to deal with heteroskedasticity and imbalance together, HAR is", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "able to put them under the same regularization framework and achieve significant improvements.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Notably, HAR also shows improvement over uniform regularization with optimally tuned strength.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "This clearly demonstrates the importance of introducing adaptive regularization among all examples", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "for a better trade-off. A more detailed ablation study on the trade-off between training accuracy and", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 721, + 299, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 299, + 732 + ], + "score": 1.0, + "content": "validation accuracy can be found in Section 3.3.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 47.5, + "bbox_fs": [ + 104, + 576, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 136, + 123, + 475, + 261 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 505, + 92 + ], + "score": 1.0, + "content": "Table 1: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 505, + 103 + ], + "score": 1.0, + "content": "imbalanced CIFAR-10. HAR significantly improves noisy and rare classes, while keeping the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 285, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 285, + 114 + ], + "score": 1.0, + "content": "accuracy on other classes almost unaffected.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 136, + 123, + 475, + 261 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 136, + 123, + 475, + 261 + ], + "spans": [ + { + "bbox": [ + 136, + 123, + 475, + 261 + ], + "score": 0.985, + "html": "
Imbalance ratio Method10100
Noisy&Rare Cls.Clean Cls.Noisy&Rare Cls.Clean Cls.
ERM52.9 ±1.294.4 ± 0.118.9 ± 1.094.2 ± 0.1
Co-teaching30.2 ± 2.388.9± 0.315.4± 2.886.4± 0.7
INCV48.9 ± 1.794.0± 0.225.8 ± 1.893.8 ± 0.2
MentorNet54.1 ± 1.090.3 ± 0.528.3 ± 1.590.2 ±0.4
L2RW44.3 ± 2.090.1± 0.531.2 ± 1.989.7±0.7
MW-Net55.4 ± 1.191.7 ± 0.535.6 ± 1.692.3 ± 0.5
GCE48.2±0.691.6 ± 0.314.1 ± 2.091.7 ± 0.4
DMI44.7 ± 2.390.7 ±0.814.0 ± 2.191.8 ± 0.6
Unif-reg (optimal)53.9 ± 0.992.1± 0.236.7 ± 1.092.4± 0.3
Ours (HAR)63.5 ± 0.894.3 ± 0.242.4±0.794.0±0.2
", + "type": "table", + "image_path": "0925ad5ac39325ab86f25dafa4c34eda39aa731e7e0496f7aa29d570620eacad.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 136, + 123, + 475, + 169.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 136, + 169.0, + 475, + 215.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 136, + 215.0, + 475, + 261.0 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 107, + 281, + 257, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 258, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 258, + 295 + ], + "score": 1.0, + "content": "3.2 ABLATION STUDY ON CIFAR", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 455, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 456, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 456, + 315 + ], + "score": 1.0, + "content": "We disentangle the problem setting to show the effectiveness of our unified framework.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 504, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 504, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 504, + 330 + ], + "score": 1.0, + "content": "Simulating heteroskedastic noise on CIFAR. We study the uncertainty part of HAR by testing under", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 330, + 498, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 498, + 343 + ], + "score": 1.0, + "content": "the setting with only heteroskedastic noise. The type of noise injection is the same as Section 3.1.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "We report the top-1 validation accuracy of various methods in Table 2. Aligned with our analysis in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "Section 4, we observe that both noise-cleaning and reweighting based methods don’t get a comparable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 359, + 382 + ], + "score": 1.0, + "content": "accuracy on noisy classes with applying strong regularization", + "type": "text" + }, + { + "bbox": [ + 359, + 370, + 393, + 381 + ], + "score": 0.84, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 370, + 505, + 382 + ], + "score": 1.0, + "content": ") under this heteroskedastic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 379, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 394 + ], + "score": 1.0, + "content": "setting. We observe the behavior that too strong regularization impede the model from fitting", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "informative samples, thus it could lead to a decrease on clean classes’ accuracy. On the contrary, too", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "weak regularization leads to overfitting the noisy examples thus the accuracy on noisy classes do not", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 413, + 181, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 181, + 425 + ], + "score": 1.0, + "content": "reach the optimal.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 429, + 503, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Interestingly, we find that even the well-studied CIFAR-100 dataset has intrinsic heteroskedasticity", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "and HAR can improve over uniform regularization to some extent. Please refer to Appendix C.2 for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 452, + 402, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 402, + 464 + ], + "score": 1.0, + "content": "the results on CIFAR-100 and Appendix C.1 for results on IMDB-review.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "table", + "bbox": [ + 125, + 506, + 485, + 653 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 474, + 504, + 497 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "Table 2: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic CIFAR-10", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 486, + 343, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 343, + 497 + ], + "score": 1.0, + "content": "and CIFAR-100 for the noisy classes and the clean classes.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "table_body", + "bbox": [ + 125, + 506, + 485, + 653 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 125, + 506, + 485, + 653 + ], + "spans": [ + { + "bbox": [ + 125, + 506, + 485, + 653 + ], + "score": 0.985, + "html": "
Dataset MethodCIFAR-10CIFAR-100
Avg. Noisy Cls.Avg. Clean Cls.Avg. Noisy Cls.Avg. Clean Cls.
ERM68.6±0.293.6±0.265.3 ± 0.367.8 ± 0.2
Co-teaching64.7 ± 0.489.1± 0.359.8 ± 0.465.3± 0.3
INCV76.7 ± 0.693.0±0.266.2±0.368.6 ±0.2
MentorNet71.1 ± 0.493.7± 0.265.9 ± 0.367.5 ± 0.3
L2RW70.1± 0.392.5 ± 0.365.1± 0.567.0± 0.3
MW-Net75.0± 0.394.4± 0.265.7 ± 0.369.1± 0.2
GCE62.6 ±1.190.2±0.261.2 ±0.666.9 ±0.2
DMI73.2 ± 0.790.8 ± 0.264.8 ± 0.567.1 ± 0.2
Unif-reg (入= 0.1)77.5 ± 0.692.3± 0.269.3 ± 0.566.6 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1±0.268.5 ± 0.368.6 ±0.2
Ours (HAR)80.7 ± 0.394.5 ± 0.274.2 ± 0.369.3±0.2
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Imbalance ratio Method10100
Noisy&Rare Cls.Clean Cls.Noisy&Rare Cls.Clean Cls.
ERM52.9 ±1.294.4 ± 0.118.9 ± 1.094.2 ± 0.1
Co-teaching30.2 ± 2.388.9± 0.315.4± 2.886.4± 0.7
INCV48.9 ± 1.794.0± 0.225.8 ± 1.893.8 ± 0.2
MentorNet54.1 ± 1.090.3 ± 0.528.3 ± 1.590.2 ±0.4
L2RW44.3 ± 2.090.1± 0.531.2 ± 1.989.7±0.7
MW-Net55.4 ± 1.191.7 ± 0.535.6 ± 1.692.3 ± 0.5
GCE48.2±0.691.6 ± 0.314.1 ± 2.091.7 ± 0.4
DMI44.7 ± 2.390.7 ±0.814.0 ± 2.191.8 ± 0.6
Unif-reg (optimal)53.9 ± 0.992.1± 0.236.7 ± 1.092.4± 0.3
Ours (HAR)63.5 ± 0.894.3 ± 0.242.4±0.794.0±0.2
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We study the uncertainty part of HAR by testing under", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 330, + 498, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 498, + 343 + ], + "score": 1.0, + "content": "the setting with only heteroskedastic noise. The type of noise injection is the same as Section 3.1.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 106, + 319, + 504, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "We report the top-1 validation accuracy of various methods in Table 2. Aligned with our analysis in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "Section 4, we observe that both noise-cleaning and reweighting based methods don’t get a comparable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 359, + 382 + ], + "score": 1.0, + "content": "accuracy on noisy classes with applying strong regularization", + "type": "text" + }, + { + "bbox": [ + 359, + 370, + 393, + 381 + ], + "score": 0.84, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 370, + 505, + 382 + ], + "score": 1.0, + "content": ") under this heteroskedastic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 379, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 394 + ], + "score": 1.0, + "content": "setting. We observe the behavior that too strong regularization impede the model from fitting", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "informative samples, thus it could lead to a decrease on clean classes’ accuracy. On the contrary, too", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "weak regularization leads to overfitting the noisy examples thus the accuracy on noisy classes do not", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 413, + 181, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 181, + 425 + ], + "score": 1.0, + "content": "reach the optimal.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 348, + 506, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 429, + 503, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Interestingly, we find that even the well-studied CIFAR-100 dataset has intrinsic heteroskedasticity", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "and HAR can improve over uniform regularization to some extent. Please refer to Appendix C.2 for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 452, + 402, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 402, + 464 + ], + "score": 1.0, + "content": "the results on CIFAR-100 and Appendix C.1 for results on IMDB-review.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 429, + 505, + 464 + ] + }, + { + "type": "table", + "bbox": [ + 125, + 506, + 485, + 653 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 474, + 504, + 497 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "Table 2: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic CIFAR-10", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 486, + 343, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 343, + 497 + ], + "score": 1.0, + "content": "and CIFAR-100 for the noisy classes and the clean classes.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "table_body", + "bbox": [ + 125, + 506, + 485, + 653 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 125, + 506, + 485, + 653 + ], + "spans": [ + { + "bbox": [ + 125, + 506, + 485, + 653 + ], + "score": 0.985, + "html": "
Dataset MethodCIFAR-10CIFAR-100
Avg. Noisy Cls.Avg. Clean Cls.Avg. Noisy Cls.Avg. Clean Cls.
ERM68.6±0.293.6±0.265.3 ± 0.367.8 ± 0.2
Co-teaching64.7 ± 0.489.1± 0.359.8 ± 0.465.3± 0.3
INCV76.7 ± 0.693.0±0.266.2±0.368.6 ±0.2
MentorNet71.1 ± 0.493.7± 0.265.9 ± 0.367.5 ± 0.3
L2RW70.1± 0.392.5 ± 0.365.1± 0.567.0± 0.3
MW-Net75.0± 0.394.4± 0.265.7 ± 0.369.1± 0.2
GCE62.6 ±1.190.2±0.261.2 ±0.666.9 ±0.2
DMI73.2 ± 0.790.8 ± 0.264.8 ± 0.567.1 ± 0.2
Unif-reg (入= 0.1)77.5 ± 0.692.3± 0.269.3 ± 0.566.6 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1±0.268.5 ± 0.368.6 ±0.2
Ours (HAR)80.7 ± 0.394.5 ± 0.274.2 ± 0.369.3±0.2
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DatasetImbalanced CIFAR-10Imbalanced CIFAR-100
Imbalance Typelong-tailedsteplong-tailedstep
Imbalance Ratio10010100101001010010
ERM29.6413.6136.7017.5061.6844.3061.4545.37
Focal29.6213.3436.0916.3661.5944.2261.4346.54
CB Focal25.4312.9039.7316.5463.9842.0180.2449.98
LDAM-DRW22.9711.8423.0812.1957.9641.2954.6440.54
BBN (Zhou et al., 2020)20.1811.6821.6411.9957.4440.8857.4440.36
HAR-DRW20.4610.6220.2711.5855.3538.9851.7337.54
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Train AccVal Acc
Reg StrengthTop1Top5Top1Top 5
069.0188.6459.4080.84
Unif-reg (入= 0.01)68.9688.5464.3286.11
Unif-reg (λ= 0.02)67.0287.5164.4085.92
Unif-reg (入= 0.05)65.1186.3365.8086.84
Unif-reg (λ = 0.1)63.3584.9865.0486.56
Adaptive (HAR)69.1288.4169.2088.96
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Its training set is both heteroskedastic and imbalanced", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "(detailed statistics can be found in (Li et al., 2017)), and it is considered as a popular benchmark", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "for noise robust learning. As the full dataset is very large, we follow (Jiang et al., 2018) to use a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "mini version, which contains the first 50 classes of the Google subset of the data. 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DatasetImbalanced CIFAR-10Imbalanced CIFAR-100
Imbalance Typelong-tailedsteplong-tailedstep
Imbalance Ratio10010100101001010010
ERM29.6413.6136.7017.5061.6844.3061.4545.37
Focal29.6213.3436.0916.3661.5944.2261.4346.54
CB Focal25.4312.9039.7316.5463.9842.0180.2449.98
LDAM-DRW22.9711.8423.0812.1957.9641.2954.6440.54
BBN (Zhou et al., 2020)20.1811.6821.6411.9957.4440.8857.4440.36
HAR-DRW20.4610.6220.2711.5855.3538.9851.7337.54
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Train AccVal Acc
Reg StrengthTop1Top5Top1Top 5
069.0188.6459.4080.84
Unif-reg (入= 0.01)68.9688.5464.3286.11
Unif-reg (λ= 0.02)67.0287.5164.4085.92
Unif-reg (入= 0.05)65.1186.3365.8086.84
Unif-reg (λ = 0.1)63.3584.9865.0486.56
Adaptive (HAR)69.1288.4169.2088.96
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(2017) trains a label cleaning network", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "on a small set of data with clean labels, and uses this model to identify noises in large datasets. To", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "circumvent the requirement of a clean subset, Malach & Shalev-Shwartz (2017) train two networks", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 699, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "simultaneously and perform update steps only in case of disagreement. 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Train Test Methodmini WebVisionfull WebVision
WebVisionImageNetWebVisionImageNet
Top 1Top5Top 1Top 5Top 1Top 5Top 1Top 5
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INCV65.285.361.685.0--1-
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Heteroskedasticity of datasets", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "is a fascinating direction worth exploring, and it is an important step towards a better understanding", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 629, + 248, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 248, + 640 + ], + "score": 1.0, + "content": "of real-world scenarios in the wild.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 661, + 225, + 673 + ], + "lines": [ + { + "bbox": [ + 107, + 659, + 226, + 676 + ], + "spans": [ + { + "bbox": [ + 107, + 659, + 226, + 676 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Toyota Research Institute (\"TRI\") provided funds and computational resources to assist the authors", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "with their research but this article solely reflects the opinions and conclusions of its authors and not", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "TRI or any other Toyota entity. YC is supported by Stanford Graduate Fellowship. TM acknowledges", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 721, + 507, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 507, + 732 + ], + "score": 1.0, + "content": "support of Google Faculty Award. The work is also partially supported by SDSI and SAIL at Stanford.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 144, + 112, + 467, + 205 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 92 + ], + "score": 1.0, + "content": "Table 5: Validation accuracy of InceptionResNet-v2 on WebVision and ImageNet validation sets.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 428, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 428, + 103 + ], + "score": 1.0, + "content": "HAR demonstrates significant improvements over the previous state-of-the-arts.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 144, + 112, + 467, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 112, + 467, + 205 + ], + "spans": [ + { + "bbox": [ + 144, + 112, + 467, + 205 + ], + "score": 0.982, + "html": "
Train Test Methodmini WebVisionfull WebVision
WebVisionImageNetWebVisionImageNet
Top 1Top5Top 1Top 5Top 1Top 5Top 1Top 5
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INCV65.285.361.685.0--1-
MentorNet63.081.457.879.970.888.062.583.0
Ours (HAR)75.590.770.390.075.090.667.186.7
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(2019) proposes to learn an explicit loss-weight function", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "to mitigate the optimizing issue of (Ren et al., 2018). Another line of work resorts to curriculum", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "learning by either designing an easy-to-hard strategy of training (Guo et al., 2018) or introducing an", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 333, + 315, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 315, + 345 + ], + "score": 1.0, + "content": "extra network (Jiang et al., 2018) to assign weights.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 267, + 506, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "Noise-cleaning and reweighting methods usually rely on the empirical loss to determine if a sample", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "is noisy. 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In such cases, we argue that changing the strength of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "regularization is a more conservative adaption and suffers less from uncertain estimation, compared", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 394, + 465, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 465, + 406 + ], + "score": 1.0, + "content": "to changing the weights of losses (Please refer to Section C.4 for empirical justifications).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 349, + 506, + 406 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "Robust loss function. Another line of works has attempted to design robust loss functions (Ghosh", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "et al., 2017; Xu et al., 2019; Zhang & Sabuncu, 2018; Patrini et al., 2017; Cheng et al., 2017; Menon", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "et al., 2016). They usually rely on prior assumption about latent transition matrix that might not hold", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 443, + 365, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 365, + 456 + ], + "score": 1.0, + "content": "in practice. 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(2020) provides", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "theoretical analysis of two additional regularization methods. 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Heteroskedasticity of datasets", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "is a fascinating direction worth exploring, and it is an important step towards a better understanding", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 629, + 248, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 248, + 640 + ], + "score": 1.0, + "content": "of real-world scenarios in the wild.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 563, + 506, + 640 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 661, + 225, + 673 + ], + "lines": [ + { + "bbox": [ + 107, + 659, + 226, + 676 + ], + "spans": [ + { + "bbox": [ + 107, + 659, + 226, + 676 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Toyota Research Institute (\"TRI\") provided funds and computational resources to assist the authors", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "with their research but this article solely reflects the opinions and conclusions of its authors and not", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "TRI or any other Toyota entity. YC is supported by Stanford Graduate Fellowship. TM acknowledges", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 721, + 507, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 507, + 732 + ], + "score": 1.0, + "content": "support of Google Faculty Award. 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Let", + "type": "text" + }, + { + "bbox": [ + 459, + 282, + 492, + 294 + ], + "score": 0.94, + "content": "( \\gamma _ { v } , h _ { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 279, + 505, + 295 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 293, + 312, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 312, + 304 + ], + "score": 1.0, + "content": "the (normalized) solution of the following equation", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 306, + 398, + 342 + ], + "lines": [ + { + "bbox": [ + 212, + 306, + 398, + 342 + ], + "spans": [ + { + "bbox": [ + 212, + 306, + 398, + 342 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { - \\rho ( t ) h _ { v } ^ { \\prime \\prime } ( t ) = \\gamma _ { v } I ( t ) q ( t ) h _ { v } ( t ) , } \\\\ & { h _ { v } ^ { ' } ( 0 ) = h _ { v } ^ { ' } ( 1 ) = 0 , \\mathrm { ~ a n d ~ } h _ { v } ^ { ' \\prime } ( 0 ) = h _ { v } ^ { ' \\prime } ( 1 ) = 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "b8a4db4af9d6efa262b60416d90b62e4a53dee6d07c55dc04bfc06a007322697.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 306, + 398, + 324.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 212, + 324.0, + 398, + 342.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 334, + 364 + ], + "lines": [ + { + "bbox": [ + 104, + 347, + 334, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 241, + 367 + ], + "score": 1.0, + "content": "We define the the leading term of", + "type": "text" + }, + { + "bbox": [ + 241, + 349, + 271, + 363 + ], + "score": 0.93, + "content": "\\hat { f } - f ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 347, + 283, + 367 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 283, + 351, + 321, + 364 + ], + "score": 0.93, + "content": "S _ { n , \\lambda } ( f ^ { \\star } )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 347, + 334, + 367 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 366, + 354, + 390 + ], + "lines": [ + { + "bbox": [ + 228, + 366, + 354, + 390 + ], + "spans": [ + { + "bbox": [ + 228, + 366, + 354, + 390 + ], + "score": 0.32, + "content": "S _ { n , \\lambda } ( f ^ { \\star } ) = { \\frac { 1 } { n } } \\epsilon _ { i } K _ { X _ { i } } - W _ { \\lambda } f ^ { \\star } ,", + "type": "interline_equation", + "image_path": "bd041acf74e6c9a42afed2581f43e5f369264f8f49037ab84b79a452afbac0d5.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 228, + 366, + 354, + 390 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 392, + 425, + 422 + ], + "lines": [ + { + "bbox": [ + 186, + 392, + 425, + 422 + ], + "spans": [ + { + "bbox": [ + 186, + 392, + 425, + 422 + ], + "score": 0.66, + "content": "K _ { t } ( \\cdot ) = \\sum _ { v } \\frac { h _ { v } ( t ) } { 1 + \\lambda \\gamma _ { v } } h _ { v } ( \\cdot ) \\mathrm { a n d } W _ { \\lambda } h _ { v } ( \\cdot ) = \\frac { \\lambda \\gamma _ { v } } { 1 + \\lambda \\gamma _ { v } } h _ { v } ( \\cdot ) .", + "type": "interline_equation", + "image_path": "d12e0263c5f39d8811659ca77526544c8626609d53d42680e958dbc977ec72c4.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 392, + 425, + 407.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 186, + 407.0, + 425, + 422.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 381, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 381, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 381, + 441 + ], + "score": 1.0, + "content": "By Proposition 2.1 and Theorem 3.4 of Shang et al. (2013), we have", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 442, + 408, + 463 + ], + "lines": [ + { + "bbox": [ + 204, + 442, + 408, + 463 + ], + "spans": [ + { + "bbox": [ + 204, + 442, + 408, + 463 + ], + "score": 0.91, + "content": "\\operatorname* { s u p } _ { x } | \\hat { f } ( x ) - f ^ { \\star } ( x ) - S _ { n , \\lambda } ( f ^ { \\star } ) ( x ) | = o _ { P } ( n ^ { - 1 / 3 } ) .", + "type": "interline_equation", + "image_path": "62438ae453c0a80b1ab26fe50451e68c366fc458f539c05e18075d90c2c08787.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 204, + 442, + 408, + 463 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 504, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 504, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 491, + 478 + ], + "score": 1.0, + "content": "Following the same proof of Theorem 2 of (Wang et al., 2013), we can simplify the definition of", + "type": "text" + }, + { + "bbox": [ + 491, + 466, + 504, + 477 + ], + "score": 0.87, + "content": "K _ { t }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 475, + 154, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 123, + 489 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 476, + 140, + 487 + ], + "score": 0.89, + "content": "W ^ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 475, + 154, + 489 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 484, + 437, + 511 + ], + "lines": [ + { + "bbox": [ + 173, + 484, + 437, + 511 + ], + "spans": [ + { + "bbox": [ + 173, + 484, + 437, + 511 + ], + "score": 0.91, + "content": "K _ { t } ( x ) = \\frac { I ( t ) } { q ( t ) } J ( t , x ) \\mathrm { a n d } W _ { \\lambda } f ^ { \\star } ( t ) = \\lambda r ( t ) \\left[ \\frac { d } { d t } ( \\rho ( t ) ( f ^ { \\ast } ) ^ { \\prime } ( t ) ) \\right] ,", + "type": "interline_equation", + "image_path": "104042b649f6b38c434cebe849f9a404ec81eb4ebd35bb20c7ea2c3f45facc52.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 173, + 484, + 437, + 511 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 514, + 508, + 541 + ], + "lines": [ + { + "bbox": [ + 104, + 512, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 512, + 136, + 529 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 136, + 514, + 333, + 528 + ], + "score": 0.9, + "content": "J ( t , s ) ~ = ~ \\beta \\rho ( s ) Q _ { \\beta } ^ { \\prime } ( s ) L _ { 0 } ( \\beta | Q _ { \\beta } ( t ) - Q _ { \\beta } ( s ) | )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 512, + 356, + 529 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 356, + 513, + 506, + 528 + ], + "score": 0.9, + "content": "\\begin{array} { r } { Q _ { \\beta } ( t , s ) = \\int _ { 0 } ^ { t } ( r ( s ) \\rho ( s ) ) ^ { - 1 / 2 } ( 1 + } \\end{array}", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 527, + 434, + 543 + ], + "spans": [ + { + "bbox": [ + 107, + 528, + 154, + 541 + ], + "score": 0.92, + "content": "O ( \\beta ^ { - 1 } ) ) d s", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 527, + 172, + 543 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 528, + 217, + 541 + ], + "score": 0.93, + "content": "\\beta = 1 / \\sqrt { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 527, + 434, + 543 + ], + "score": 1.0, + "content": ". 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We follow the simple data augmentation used in (He et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "2016) with only random crop and horizontal flip. We use ResNet-32 as our base network and repeat", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 654, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 465, + 667 + ], + "score": 1.0, + "content": "all experiments for 3 runs. We use standard SGD with momentum of 0.9, weight decay of", + "type": "text" + }, + { + "bbox": [ + 465, + 654, + 504, + 666 + ], + "score": 0.91, + "content": "1 \\times \\bar { 1 0 } ^ { - 4 }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "for training. The model is trained with a batch size of 128 for 120 epochs. We anneal the learning rate", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 494, + 690 + ], + "score": 1.0, + "content": "by a factor of 10 at 80 and 100 epochs. We group the data by class labels, and by default we split", + "type": "text" + }, + { + "bbox": [ + 495, + 677, + 504, + 687 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 212, + 701 + ], + "score": 1.0, + "content": "equally and randomly into", + "type": "text" + }, + { + "bbox": [ + 212, + 688, + 234, + 699 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 687, + 252, + 701 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 688, + 270, + 699 + ], + "score": 0.9, + "content": "\\mathcal { D } _ { \\mathrm { v a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 687, + 505, + 701 + ], + "score": 1.0, + "content": ". As for the Jacobian regularizer, we sum over the frobenius", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "norm of the gradients of all normalization layers’ (BN by default) activations with respect to the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 325, + 722 + ], + "score": 1.0, + "content": "classification loss. For experiments of HAR, we tune", + "type": "text" + }, + { + "bbox": [ + 325, + 710, + 333, + 720 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "so that the largest enforced regularization", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 435, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 141, + 732 + ], + "score": 1.0, + "content": "strength", + "type": "text" + }, + { + "bbox": [ + 141, + 721, + 162, + 732 + ], + "score": 0.87, + "content": "( \\lambda \\tau _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 720, + 435, + 732 + ], + "score": 1.0, + "content": "is 0.1. We train each model with 1 NVIDIA GeForce RTX 2080 Ti.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 249, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 251, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 251, + 95 + ], + "score": 1.0, + "content": "A PROOFS OF THEOREM 1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 105, + 442, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 442, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 442, + 119 + ], + "score": 1.0, + "content": "We prove a general theorem here. In particular, we have the general theorem below.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 104, + 442, + 119 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 506, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 211, + 132 + ], + "score": 1.0, + "content": "Theorem 2. Assume that", + "type": "text" + }, + { + "bbox": [ + 212, + 119, + 269, + 132 + ], + "score": 0.92, + "content": "f ^ { \\star } , q , I \\in W _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 119, + 324, + 132 + ], + "score": 1.0, + "content": ". Suppose (1)", + "type": "text" + }, + { + "bbox": [ + 324, + 119, + 352, + 131 + ], + "score": 0.66, + "content": "\\ell ( a , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 119, + 506, + 132 + ], + "score": 1.0, + "content": "is convex and three times continously", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 130, + 504, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 224, + 143 + ], + "score": 1.0, + "content": "differentiable with respect to", + "type": "text" + }, + { + "bbox": [ + 225, + 133, + 231, + 140 + ], + "score": 0.61, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 130, + 334, + 143 + ], + "score": 1.0, + "content": ", (2) there exist constants", + "type": "text" + }, + { + "bbox": [ + 334, + 131, + 402, + 141 + ], + "score": 0.89, + "content": "0 < c < C < \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 130, + 442, + 143 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 442, + 130, + 504, + 142 + ], + "score": 0.9, + "content": "c \\leq I ( X ) \\leq \\bar { C }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 504, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 183, + 156 + ], + "score": 1.0, + "content": "almost surely, and", + "type": "text" + }, + { + "bbox": [ + 183, + 142, + 267, + 155 + ], + "score": 0.84, + "content": "\\epsilon \\triangleq \\nabla _ { a } \\ell ( f ^ { \\star } ( X ) , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 142, + 304, + 156 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 304, + 142, + 353, + 155 + ], + "score": 0.89, + "content": "\\mathbb { E } [ \\epsilon | X ] = 0 \\mathrm { \\it \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 142, + 357, + 156 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 357, + 142, + 426, + 155 + ], + "score": 0.91, + "content": "\\mathbb { E } [ \\epsilon ^ { 2 } | X ] = I ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 142, + 445, + 156 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 446, + 143, + 504, + 155 + ], + "score": 0.9, + "content": "\\mathbb { E } [ \\epsilon ^ { 4 } | X ] < \\infty", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 103, + 153, + 504, + 171 + ], + "spans": [ + { + "bbox": [ + 103, + 153, + 178, + 171 + ], + "score": 1.0, + "content": "almost surely. Let", + "type": "text" + }, + { + "bbox": [ + 178, + 155, + 266, + 168 + ], + "score": 0.91, + "content": "r ( t ) = - 1 / ( q ( t ) I ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 153, + 284, + 171 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 155, + 393, + 170 + ], + "score": 0.9, + "content": "\\begin{array} { r } { L _ { 0 } = \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { 4 } \\exp ( - 2 | t | ) d t . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 153, + 447, + 171 + ], + "score": 1.0, + "content": ". If we choose", + "type": "text" + }, + { + "bbox": [ + 447, + 155, + 504, + 167 + ], + "score": 0.89, + "content": "\\lambda = C _ { 0 } n ^ { - 2 / 5 }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 169, + 442, + 183 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 180, + 183 + ], + "score": 1.0, + "content": "for some constant", + "type": "text" + }, + { + "bbox": [ + 180, + 171, + 211, + 182 + ], + "score": 0.91, + "content": "C _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 169, + 367, + 183 + ], + "score": 1.0, + "content": ", the asymptotic mean squared error of", + "type": "text" + }, + { + "bbox": [ + 367, + 169, + 374, + 182 + ], + "score": 0.82, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 169, + 442, + 183 + ], + "score": 1.0, + "content": "by equation 4 is", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 103, + 119, + 506, + 183 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 184, + 463, + 214 + ], + "lines": [ + { + "bbox": [ + 148, + 184, + 463, + 214 + ], + "spans": [ + { + "bbox": [ + 148, + 184, + 463, + 214 + ], + "score": 0.92, + "content": "\\operatorname* { l i m } _ { n \\to \\infty } M S E ( \\hat { f } ) = C _ { n } \\int _ { 0 } ^ { 1 } \\lambda ^ { 2 } r ^ { 2 } ( t ) \\left[ \\frac { d } { d t } ( \\rho ( t ) ( f ^ { * } ) ^ { \\prime } ( t ) ) \\right] ^ { 2 } + L _ { 0 } r ( t ) ^ { 1 / 2 } \\rho ( t ) ^ { - 1 / 2 } d t", + "type": "interline_equation", + "image_path": "c6a2f35daa2ba3524636b23ce7bd61b4395e2910afa2abf98b06bfe63130914d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 148, + 184, + 463, + 194.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 148, + 194.0, + 463, + 204.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 148, + 204.0, + 463, + 214.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 346, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 347, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 191, + 229 + ], + "score": 1.0, + "content": "in probability, where", + "type": "text" + }, + { + "bbox": [ + 192, + 216, + 205, + 227 + ], + "score": 0.89, + "content": "C _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 214, + 336, + 229 + ], + "score": 1.0, + "content": "is a scalar that only depends on", + "type": "text" + }, + { + "bbox": [ + 336, + 218, + 343, + 226 + ], + "score": 0.47, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 214, + 347, + 229 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 214, + 347, + 229 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 414, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 414, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 414, + 249 + ], + "score": 1.0, + "content": "It is easy to check that the logistic loss satisfies the condition of the theorem.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 235, + 414, + 249 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 252, + 504, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 266 + ], + "score": 1.0, + "content": "The proof strategy of Theorem 2 is adopted from the proof of Theorem 2 of (Wang et al., 2013) by", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 333, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 333, + 276 + ], + "score": 1.0, + "content": "generalizing it from the least square loss to logistic loss.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 250, + 505, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 281, + 504, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 251, + 295 + ], + "score": 1.0, + "content": "The high level idea is to reformulate", + "type": "text" + }, + { + "bbox": [ + 251, + 280, + 258, + 294 + ], + "score": 0.86, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 279, + 459, + 295 + ], + "score": 1.0, + "content": "as solutions to ordinary differential equations. Let", + "type": "text" + }, + { + "bbox": [ + 459, + 282, + 492, + 294 + ], + "score": 0.94, + "content": "( \\gamma _ { v } , h _ { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 279, + 505, + 295 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 293, + 312, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 312, + 304 + ], + "score": 1.0, + "content": "the (normalized) solution of the following equation", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 279, + 505, + 304 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 306, + 398, + 342 + ], + "lines": [ + { + "bbox": [ + 212, + 306, + 398, + 342 + ], + "spans": [ + { + "bbox": [ + 212, + 306, + 398, + 342 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { - \\rho ( t ) h _ { v } ^ { \\prime \\prime } ( t ) = \\gamma _ { v } I ( t ) q ( t ) h _ { v } ( t ) , } \\\\ & { h _ { v } ^ { ' } ( 0 ) = h _ { v } ^ { ' } ( 1 ) = 0 , \\mathrm { ~ a n d ~ } h _ { v } ^ { ' \\prime } ( 0 ) = h _ { v } ^ { ' \\prime } ( 1 ) = 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "b8a4db4af9d6efa262b60416d90b62e4a53dee6d07c55dc04bfc06a007322697.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 306, + 398, + 324.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 212, + 324.0, + 398, + 342.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 334, + 364 + ], + "lines": [ + { + "bbox": [ + 104, + 347, + 334, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 241, + 367 + ], + "score": 1.0, + "content": "We define the the leading term of", + "type": "text" + }, + { + "bbox": [ + 241, + 349, + 271, + 363 + ], + "score": 0.93, + "content": "\\hat { f } - f ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 347, + 283, + 367 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 283, + 351, + 321, + 364 + ], + "score": 0.93, + "content": "S _ { n , \\lambda } ( f ^ { \\star } )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 347, + 334, + 367 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 347, + 334, + 367 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 366, + 354, + 390 + ], + "lines": [ + { + "bbox": [ + 228, + 366, + 354, + 390 + ], + "spans": [ + { + "bbox": [ + 228, + 366, + 354, + 390 + ], + "score": 0.32, + "content": "S _ { n , \\lambda } ( f ^ { \\star } ) = { \\frac { 1 } { n } } \\epsilon _ { i } K _ { X _ { i } } - W _ { \\lambda } f ^ { \\star } ,", + "type": "interline_equation", + "image_path": "bd041acf74e6c9a42afed2581f43e5f369264f8f49037ab84b79a452afbac0d5.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 228, + 366, + 354, + 390 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 392, + 425, + 422 + ], + "lines": [ + { + "bbox": [ + 186, + 392, + 425, + 422 + ], + "spans": [ + { + "bbox": [ + 186, + 392, + 425, + 422 + ], + "score": 0.66, + "content": "K _ { t } ( \\cdot ) = \\sum _ { v } \\frac { h _ { v } ( t ) } { 1 + \\lambda \\gamma _ { v } } h _ { v } ( \\cdot ) \\mathrm { a n d } W _ { \\lambda } h _ { v } ( \\cdot ) = \\frac { \\lambda \\gamma _ { v } } { 1 + \\lambda \\gamma _ { v } } h _ { v } ( \\cdot ) .", + "type": "interline_equation", + "image_path": "d12e0263c5f39d8811659ca77526544c8626609d53d42680e958dbc977ec72c4.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 392, + 425, + 407.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 186, + 407.0, + 425, + 422.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 381, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 381, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 381, + 441 + ], + "score": 1.0, + "content": "By Proposition 2.1 and Theorem 3.4 of Shang et al. (2013), we have", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 426, + 381, + 441 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 442, + 408, + 463 + ], + "lines": [ + { + "bbox": [ + 204, + 442, + 408, + 463 + ], + "spans": [ + { + "bbox": [ + 204, + 442, + 408, + 463 + ], + "score": 0.91, + "content": "\\operatorname* { s u p } _ { x } | \\hat { f } ( x ) - f ^ { \\star } ( x ) - S _ { n , \\lambda } ( f ^ { \\star } ) ( x ) | = o _ { P } ( n ^ { - 1 / 3 } ) .", + "type": "interline_equation", + "image_path": "62438ae453c0a80b1ab26fe50451e68c366fc458f539c05e18075d90c2c08787.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 204, + 442, + 408, + 463 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 504, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 504, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 491, + 478 + ], + "score": 1.0, + "content": "Following the same proof of Theorem 2 of (Wang et al., 2013), we can simplify the definition of", + "type": "text" + }, + { + "bbox": [ + 491, + 466, + 504, + 477 + ], + "score": 0.87, + "content": "K _ { t }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 475, + 154, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 123, + 489 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 476, + 140, + 487 + ], + "score": 0.89, + "content": "W ^ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 475, + 154, + 489 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 464, + 504, + 489 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 484, + 437, + 511 + ], + "lines": [ + { + "bbox": [ + 173, + 484, + 437, + 511 + ], + "spans": [ + { + "bbox": [ + 173, + 484, + 437, + 511 + ], + "score": 0.91, + "content": "K _ { t } ( x ) = \\frac { I ( t ) } { q ( t ) } J ( t , x ) \\mathrm { a n d } W _ { \\lambda } f ^ { \\star } ( t ) = \\lambda r ( t ) \\left[ \\frac { d } { d t } ( \\rho ( t ) ( f ^ { \\ast } ) ^ { \\prime } ( t ) ) \\right] ,", + "type": "interline_equation", + "image_path": "104042b649f6b38c434cebe849f9a404ec81eb4ebd35bb20c7ea2c3f45facc52.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 173, + 484, + 437, + 511 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 514, + 508, + 541 + ], + "lines": [ + { + "bbox": [ + 104, + 512, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 512, + 136, + 529 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 136, + 514, + 333, + 528 + ], + "score": 0.9, + "content": "J ( t , s ) ~ = ~ \\beta \\rho ( s ) Q _ { \\beta } ^ { \\prime } ( s ) L _ { 0 } ( \\beta | Q _ { \\beta } ( t ) - Q _ { \\beta } ( s ) | )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 512, + 356, + 529 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 356, + 513, + 506, + 528 + ], + "score": 0.9, + "content": "\\begin{array} { r } { Q _ { \\beta } ( t , s ) = \\int _ { 0 } ^ { t } ( r ( s ) \\rho ( s ) ) ^ { - 1 / 2 } ( 1 + } \\end{array}", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 527, + 434, + 543 + ], + "spans": [ + { + "bbox": [ + 107, + 528, + 154, + 541 + ], + "score": 0.92, + "content": "O ( \\beta ^ { - 1 } ) ) d s", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 527, + 172, + 543 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 528, + 217, + 541 + ], + "score": 0.93, + "content": "\\beta = 1 / \\sqrt { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 527, + 434, + 543 + ], + "score": 1.0, + "content": ". 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For experiments of HAR, we tune", + "type": "text" + }, + { + "bbox": [ + 325, + 710, + 333, + 720 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "so that the largest enforced regularization", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 435, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 141, + 732 + ], + "score": 1.0, + "content": "strength", + "type": "text" + }, + { + "bbox": [ + 141, + 721, + 162, + 732 + ], + "score": 0.87, + "content": "( \\lambda \\tau _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 720, + 435, + 732 + ], + "score": 1.0, + "content": "is 0.1. We train each model with 1 NVIDIA GeForce RTX 2080 Ti.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 632, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 95 + ], + "score": 1.0, + "content": "Implementation details for IMDB-review. We train a two-layer bidirectional LSTM (Huang et al.,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "2015) with 256 units followed with 0.5 dropout before the linear classifier. 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For mini WebVision, We train the network for 90", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "epochs using standard SGD with a batch size of 128. The initial learning rate is 0.1 and is annealed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 184, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 506, + 199 + ], + "score": 1.0, + "content": "by a factor of 10 at epoch 60 and 90. For full WebVision, We train the network for 50 epochs using", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 196, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 207 + ], + "score": 1.0, + "content": "standard SGD with a batch size of 256. 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For experiments of HAR, we tune", + "type": "text" + }, + { + "bbox": [ + 333, + 208, + 340, + 217 + ], + "score": 0.64, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "so that the largest enforced regularization", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 418, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 141, + 230 + ], + "score": 1.0, + "content": "strength", + "type": "text" + }, + { + "bbox": [ + 141, + 218, + 162, + 229 + ], + "score": 0.85, + "content": "( \\lambda \\tau _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 218, + 418, + 230 + ], + "score": 1.0, + "content": "is 0.1. We train each model with 8 NVIDIA Tesla V100 GPUs.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "Runtime analysis. Because the pre-trained model only trains on half of the training data and is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "only done once, the run-time of HAR is at most twice of the time for ERM. Many baselines in our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "paper use sophisticated pipelines and are slower than HAR. For example, INCV trains 2 models", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 275, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 291 + ], + "score": 1.0, + "content": "simultaneously for 4 times from random initialization to get a clean training set. MW-Net has a very", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 376, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 376, + 301 + ], + "score": 1.0, + "content": "slow convergence rate, which is a common issue for meta-learning.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 317, + 244, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 246, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 246, + 332 + ], + "score": 1.0, + "content": "C ADDITIONAL RESULTS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 343, + 386, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 388, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 388, + 356 + ], + "score": 1.0, + "content": "C.1 SIMULATING HETEROSKEDASTIC NOISE ON IMDB-REVIEW.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "IMDB-review dataset has a total of 50,000 (25,000 positive and 25,000 negative reviews) movie", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "reviews for binary sentiment classification (Maas et al., 2011). To simulate heteroskedastic noise for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 291, + 400 + ], + "score": 1.0, + "content": "this binary classification problem, we project", + "type": "text" + }, + { + "bbox": [ + 291, + 387, + 307, + 398 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "of the labels of negative reviews to positive, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 398, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 126, + 408 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 398, + 504, + 410 + ], + "score": 1.0, + "content": "in the reverse direction. Table 6 summarizes the results. The proposed HAR outperforms the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 408, + 357, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 357, + 421 + ], + "score": 1.0, + "content": "ERM baseline with various strength of uniform regularization.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "table", + "bbox": [ + 151, + 453, + 459, + 516 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 432, + 503, + 444 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "Table 6: Top-1 validation accuracy (averaged over 3 runs) on heteroskedastic IMDB-review dataset.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 151, + 453, + 459, + 516 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 151, + 453, + 459, + 516 + ], + "spans": [ + { + "bbox": [ + 151, + 453, + 459, + 516 + ], + "score": 0.979, + "html": "
Reg StrengthAcc. of neg. reviewsAcc. of pos. reviewsMean Acc
091.9 ± 2.050.9 ±1.871.4± 0.5
Unif-reg (入= 0.01)94.3 ±1.851.9 ± 2.073.1± 0.3
Unif-reg (入= 0.1)91.5 ±1.964.3 ±1.677.9 ± 0.4
Ours (HAR)93.1 ± 1.572.8±1.783.0 ± 0.3
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Reg StrengthTrain AccVal Acc
096.0±0.169.8± 0.2
Unif-reg (入= 0.001)96.4± 0.170.0± 0.2
Unif-reg (入= 0.01)95.7 ± 0.170.6 ±0.1
Unif-reg (入= 0.1)88.8±0.170.5 ± 0.1
Adaptive (HAR)96.2 ± 0.171.4±0.1
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We train a two-layer bidirectional LSTM (Huang et al.,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "2015) with 256 units followed with 0.5 dropout before the linear classifier. The network is trained", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 423, + 117 + ], + "score": 1.0, + "content": "for 20 epochs with Adam optimizer (Kingma & Ba, 2014). 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We train each model with 8 NVIDIA Tesla V100 GPUs.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 152, + 506, + 230 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "Runtime analysis. Because the pre-trained model only trains on half of the training data and is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "only done once, the run-time of HAR is at most twice of the time for ERM. Many baselines in our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "paper use sophisticated pipelines and are slower than HAR. For example, INCV trains 2 models", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 275, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 291 + ], + "score": 1.0, + "content": "simultaneously for 4 times from random initialization to get a clean training set. MW-Net has a very", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 376, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 376, + 301 + ], + "score": 1.0, + "content": "slow convergence rate, which is a common issue for meta-learning.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 244, + 506, + 301 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 317, + 244, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 246, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 246, + 332 + ], + "score": 1.0, + "content": "C ADDITIONAL RESULTS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 343, + 386, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 388, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 388, + 356 + ], + "score": 1.0, + "content": "C.1 SIMULATING HETEROSKEDASTIC NOISE ON IMDB-REVIEW.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "IMDB-review dataset has a total of 50,000 (25,000 positive and 25,000 negative reviews) movie", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "reviews for binary sentiment classification (Maas et al., 2011). To simulate heteroskedastic noise for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 291, + 400 + ], + "score": 1.0, + "content": "this binary classification problem, we project", + "type": "text" + }, + { + "bbox": [ + 291, + 387, + 307, + 398 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "of the labels of negative reviews to positive, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 398, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 126, + 408 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 398, + 504, + 410 + ], + "score": 1.0, + "content": "in the reverse direction. Table 6 summarizes the results. The proposed HAR outperforms the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 408, + 357, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 357, + 421 + ], + "score": 1.0, + "content": "ERM baseline with various strength of uniform regularization.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 365, + 505, + 421 + ] + }, + { + "type": "table", + "bbox": [ + 151, + 453, + 459, + 516 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 432, + 503, + 444 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "Table 6: Top-1 validation accuracy (averaged over 3 runs) on heteroskedastic IMDB-review dataset.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 151, + 453, + 459, + 516 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 151, + 453, + 459, + 516 + ], + "spans": [ + { + "bbox": [ + 151, + 453, + 459, + 516 + ], + "score": 0.979, + "html": "
Reg StrengthAcc. of neg. reviewsAcc. of pos. reviewsMean Acc
091.9 ± 2.050.9 ±1.871.4± 0.5
Unif-reg (入= 0.01)94.3 ±1.851.9 ± 2.073.1± 0.3
Unif-reg (入= 0.1)91.5 ±1.964.3 ±1.677.9 ± 0.4
Ours (HAR)93.1 ± 1.572.8±1.783.0 ± 0.3
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Reg StrengthTrain AccVal Acc
096.0±0.169.8± 0.2
Unif-reg (入= 0.001)96.4± 0.170.0± 0.2
Unif-reg (入= 0.01)95.7 ± 0.170.6 ±0.1
Unif-reg (入= 0.1)88.8±0.170.5 ± 0.1
Adaptive (HAR)96.2 ± 0.171.4±0.1
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Imbalance ratio Method10100
Noisy Cls.Rare Cls.Noisy Cls.Rare Cls.
ERM59.9 ± 1.165.2 ± 0.760.4± 0.97.9 ± 1.3
Co-teaching63.1 ± 2.359.4 ± 1.453.8 ± 1.94.4±1.8
INCV74.5 ± 1.263.7 ± 0.868.3 ± 1.82.1 ±1.3
MentorNet67.3 ± 1.665.5 ± 1.263.3 ± 1.510.8 ±1.9
L2RW65.8 ±1.466.3 ± 1.262.4 ± 2.111.3 ± 2.9
MW-Net71.4± 0.667.7 ± 0.665.0± 1.613.5 ± 2.4
GCE64.6 ± 1.160.2 ± 1.371.2 ± 1.92.6±1.4
DMI72.3 ± 1.563.3 ±1.270.8 ±1.76.2 ±1.9
Ours (HAR)76.1 ± 0.872.1 ± 1.073.0 ± 1.626.1±0.8
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MethodAvg. Noisy Cls.Avg. . Clean Cls.
ERM68.6±0.293.6 ± 0.2
Reweight (MW-Net)75.0 ± 0.394.4± 0.2
Reweight (random)62.1±0.592.9 ± 0.5
Reweight (inverse)13.1 ± 0.990.9 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1± 0.2
Adap-reg (HAR)80.7±0.394.5 ±0.2
Adap-reg (random)74.8 ± 0.494.2±0.2
Adap-reg (inverse)73.2 ± 0.594.0± 0.2
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We observe that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 695, + 469, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 469, + 709 + ], + "score": 1.0, + "content": "HAR outperforms the tuned uniform regularization baseline on the majority of the classes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 111, + 83, + 444, + 93 + ], + "lines": [ + { + "bbox": [ + 109, + 82, + 447, + 95 + ], + "spans": [ + { + "bbox": [ + 109, + 82, + 447, + 95 + ], + "score": 1.0, + "content": "C.3 SIMULATING HETEROSKEDASTIC AND IMBALANCED DATASETS ON CIFAR", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 102, + 504, + 136 + ], + "lines": [ + { + "bbox": [ + 107, + 103, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 107, + 103, + 505, + 114 + ], + "score": 1.0, + "content": "As mentioned in Section 3.1, we consider another variant of heteroskedastic and imbalanced dataset", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 115, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 107, + 115, + 505, + 126 + ], + "score": 1.0, + "content": "such that the rare classes have low noise level. 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For the setting of imbalance ratio equals 10, INCV automatically", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 151, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 130, + 166 + ], + "score": 1.0, + "content": "drops", + "type": "text" + }, + { + "bbox": [ + 131, + 153, + 157, + 163 + ], + "score": 0.86, + "content": "34 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 151, + 505, + 166 + ], + "score": 1.0, + "content": "of examples from the clean and rare classes, which results in a decrease of mean accuracy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 164, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 176 + ], + "score": 1.0, + "content": "on the rare and clean classes. HAR is able to achieve improvements on both noisy classes and rare", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 175, + 297, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 297, + 187 + ], + "score": 1.0, + "content": "classes by enforcing the optimal regularization.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 142, + 505, + 187 + ] + }, + { + "type": "table", + "bbox": [ + 159, + 228, + 451, + 356 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 196, + 504, + 219 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 195, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 210 + ], + "score": 1.0, + "content": "Table 8: Top-1 validation accuracy (averaged over 3 runs) of ResNet-32 on heteroskedastic and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 207, + 202, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 202, + 219 + ], + "score": 1.0, + "content": "imbalanced CIFAR-10.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "table_body", + "bbox": [ + 159, + 228, + 451, + 356 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 159, + 228, + 451, + 356 + ], + "spans": [ + { + "bbox": [ + 159, + 228, + 451, + 356 + ], + "score": 0.985, + "html": "
Imbalance ratio Method10100
Noisy Cls.Rare Cls.Noisy Cls.Rare Cls.
ERM59.9 ± 1.165.2 ± 0.760.4± 0.97.9 ± 1.3
Co-teaching63.1 ± 2.359.4 ± 1.453.8 ± 1.94.4±1.8
INCV74.5 ± 1.263.7 ± 0.868.3 ± 1.82.1 ±1.3
MentorNet67.3 ± 1.665.5 ± 1.263.3 ± 1.510.8 ±1.9
L2RW65.8 ±1.466.3 ± 1.262.4 ± 2.111.3 ± 2.9
MW-Net71.4± 0.667.7 ± 0.665.0± 1.613.5 ± 2.4
GCE64.6 ± 1.160.2 ± 1.371.2 ± 1.92.6±1.4
DMI72.3 ± 1.563.3 ±1.270.8 ±1.76.2 ±1.9
Ours (HAR)76.1 ± 0.872.1 ± 1.073.0 ± 1.626.1±0.8
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MethodAvg. Noisy Cls.Avg. . Clean Cls.
ERM68.6±0.293.6 ± 0.2
Reweight (MW-Net)75.0 ± 0.394.4± 0.2
Reweight (random)62.1±0.592.9 ± 0.5
Reweight (inverse)13.1 ± 0.990.9 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1± 0.2
Adap-reg (HAR)80.7±0.394.5 ±0.2
Adap-reg (random)74.8 ± 0.494.2±0.2
Adap-reg (inverse)73.2 ± 0.594.0± 0.2
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We observe that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 695, + 469, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 469, + 709 + ], + "score": 1.0, + "content": "HAR outperforms the tuned uniform regularization baseline on the majority of the classes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 672, + 506, + 709 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 233, + 502, + 545 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 233, + 502, + 545 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 233, + 502, + 545 + ], + "spans": [ + { + "bbox": [ + 110, + 233, + 502, + 545 + ], + "score": 0.972, + "type": "image", + "image_path": "c076753ebcd5b8f3da8f361db87212d171eb3c37ef46afdae950c98d61c3d805.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 233, + 502, + 337.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 337.0, + 502, + 441.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 441.0, + 502, + 545.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 555, + 506, + 579 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "Figure 4: Visualizations of per-class top-1 error and regularization strength of HAR on mini WebVi-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 566, + 475, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 475, + 579 + ], + "score": 1.0, + "content": "sion dataset. 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Require: Dataset D = {(xi, yi)}i=1: A parameterized model fe
1: Split training set D into Dtrain and Dval
2:ff ← Standard SGD Training on Dtrain
3: Estimate I(x),q(x) with equation 7 using f on Dval, and compute Ti = I(xi)3/5/q(𝑥)2/5
4:
5: Initialize the model parameters θ randomly
6:fe ← SGD with the regularized objective as in equation 6 on the full dataset D
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Imbalance ratio Method10100
Noisy&Rare Cls.Clean Cls.Noisy&Rare Cls.Clean Cls.
ERM52.9 ±1.294.4 ± 0.118.9 ± 1.094.2 ± 0.1
Co-teaching30.2 ± 2.388.9± 0.315.4± 2.886.4± 0.7
INCV48.9 ± 1.794.0± 0.225.8 ± 1.893.8 ± 0.2
MentorNet54.1 ± 1.090.3 ± 0.528.3 ± 1.590.2 ±0.4
L2RW44.3 ± 2.090.1± 0.531.2 ± 1.989.7±0.7
MW-Net55.4 ± 1.191.7 ± 0.535.6 ± 1.692.3 ± 0.5
GCE48.2±0.691.6 ± 0.314.1 ± 2.091.7 ± 0.4
DMI44.7 ± 2.390.7 ±0.814.0 ± 2.191.8 ± 0.6
Unif-reg (optimal)53.9 ± 0.992.1± 0.236.7 ± 1.092.4± 0.3
Ours (HAR)63.5 ± 0.894.3 ± 0.242.4±0.794.0±0.2
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Dataset MethodCIFAR-10CIFAR-100
Avg. Noisy Cls.Avg. Clean Cls.Avg. Noisy Cls.Avg. Clean Cls.
ERM68.6±0.293.6±0.265.3 ± 0.367.8 ± 0.2
Co-teaching64.7 ± 0.489.1± 0.359.8 ± 0.465.3± 0.3
INCV76.7 ± 0.693.0±0.266.2±0.368.6 ±0.2
MentorNet71.1 ± 0.493.7± 0.265.9 ± 0.367.5 ± 0.3
L2RW70.1± 0.392.5 ± 0.365.1± 0.567.0± 0.3
MW-Net75.0± 0.394.4± 0.265.7 ± 0.369.1± 0.2
GCE62.6 ±1.190.2±0.261.2 ±0.666.9 ±0.2
DMI73.2 ± 0.790.8 ± 0.264.8 ± 0.567.1 ± 0.2
Unif-reg (入= 0.1)77.5 ± 0.692.3± 0.269.3 ± 0.566.6 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1±0.268.5 ± 0.368.6 ±0.2
Ours (HAR)80.7 ± 0.394.5 ± 0.274.2 ± 0.369.3±0.2
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DatasetImbalanced CIFAR-10Imbalanced CIFAR-100
Imbalance Typelong-tailedsteplong-tailedstep
Imbalance Ratio10010100101001010010
ERM29.6413.6136.7017.5061.6844.3061.4545.37
Focal29.6213.3436.0916.3661.5944.2261.4346.54
CB Focal25.4312.9039.7316.5463.9842.0180.2449.98
LDAM-DRW22.9711.8423.0812.1957.9641.2954.6440.54
BBN (Zhou et al., 2020)20.1811.6821.6411.9957.4440.8857.4440.36
HAR-DRW20.4610.6220.2711.5855.3538.9851.7337.54
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Train AccVal Acc
Reg StrengthTop1Top5Top1Top 5
069.0188.6459.4080.84
Unif-reg (入= 0.01)68.9688.5464.3286.11
Unif-reg (λ= 0.02)67.0287.5164.4085.92
Unif-reg (入= 0.05)65.1186.3365.8086.84
Unif-reg (λ = 0.1)63.3584.9865.0486.56
Adaptive (HAR)69.1288.4169.2088.96
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Train Test Methodmini WebVisionfull WebVision
WebVisionImageNetWebVisionImageNet
Top 1Top5Top 1Top 5Top 1Top 5Top 1Top 5
ERM62.580.858.581.869.787.062.983.6
Co-teaching63.685.261.584.71-11
INCV65.285.361.685.0--1-
MentorNet63.081.457.879.970.888.062.583.0
Ours (HAR)75.590.770.390.075.090.667.186.7
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Reg StrengthTrain AccVal Acc
096.0±0.169.8± 0.2
Unif-reg (入= 0.001)96.4± 0.170.0± 0.2
Unif-reg (入= 0.01)95.7 ± 0.170.6 ±0.1
Unif-reg (入= 0.1)88.8±0.170.5 ± 0.1
Adaptive (HAR)96.2 ± 0.171.4±0.1
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Reg StrengthAcc. of neg. reviewsAcc. of pos. reviewsMean Acc
091.9 ± 2.050.9 ±1.871.4± 0.5
Unif-reg (入= 0.01)94.3 ±1.851.9 ± 2.073.1± 0.3
Unif-reg (入= 0.1)91.5 ±1.964.3 ±1.677.9 ± 0.4
Ours (HAR)93.1 ± 1.572.8±1.783.0 ± 0.3
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Imbalance ratio Method10100
Noisy Cls.Rare Cls.Noisy Cls.Rare Cls.
ERM59.9 ± 1.165.2 ± 0.760.4± 0.97.9 ± 1.3
Co-teaching63.1 ± 2.359.4 ± 1.453.8 ± 1.94.4±1.8
INCV74.5 ± 1.263.7 ± 0.868.3 ± 1.82.1 ±1.3
MentorNet67.3 ± 1.665.5 ± 1.263.3 ± 1.510.8 ±1.9
L2RW65.8 ±1.466.3 ± 1.262.4 ± 2.111.3 ± 2.9
MW-Net71.4± 0.667.7 ± 0.665.0± 1.613.5 ± 2.4
GCE64.6 ± 1.160.2 ± 1.371.2 ± 1.92.6±1.4
DMI72.3 ± 1.563.3 ±1.270.8 ±1.76.2 ±1.9
Ours (HAR)76.1 ± 0.872.1 ± 1.073.0 ± 1.626.1±0.8
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MethodAvg. Noisy Cls.Avg. . Clean Cls.
ERM68.6±0.293.6 ± 0.2
Reweight (MW-Net)75.0 ± 0.394.4± 0.2
Reweight (random)62.1±0.592.9 ± 0.5
Reweight (inverse)13.1 ± 0.990.9 ± 0.3
Unif-reg (optimal)75.3 ± 0.394.1± 0.2
Adap-reg (HAR)80.7±0.394.5 ±0.2
Adap-reg (random)74.8 ± 0.494.2±0.2
Adap-reg (inverse)73.2 ± 0.594.0± 0.2
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0000000000000000000000000000000000000000..a96f6fbbf3c6c8bc33ac6486791eb3b923c726be --- /dev/null +++ b/parse/train/rJVorjCcKQ/rJVorjCcKQ.md @@ -0,0 +1,391 @@ +# SLALOM: FAST, VERIFIABLE AND PRIVATE EXECUTION OF NEURAL NETWORKS IN TRUSTED HARDWARE + +Florian Tramèr +Stanford University +tramer@cs.stanford.edu + +Dan Boneh Stanford University dabo@cs.stanford.edu + +# ABSTRACT + +As Machine Learning (ML) gets applied to security-critical or sensitive domains, there is a growing need for integrity and privacy for outsourced ML computations. A pragmatic solution comes from Trusted Execution Environments (TEEs), which use hardware and software protections to isolate sensitive computations from the untrusted software stack. However, these isolation guarantees come at a price in performance, compared to untrusted alternatives. This paper initiates the study of high performance execution of Deep Neural Networks (DNNs) in TEEs by efficiently partitioning DNN computations between trusted and untrusted devices. Building upon an efficient outsourcing scheme for matrix multiplication, we propose Slalom, a framework that securely delegates execution of all linear layers in a DNN from a TEE (e.g., Intel SGX or Sanctum) to a faster, yet untrusted, co-located processor. We evaluate Slalom by running DNNs in an Intel SGX enclave, which selectively delegates work to an untrusted GPU. For canonical DNNs (VGG16, MobileNet and ResNet variants) we obtain $6 \times$ to $2 0 \times$ increases in throughput for verifiable inference, and $4 \times$ to $1 1 \times$ for verifiable and private inference. + +# 1 INTRODUCTION + +Machine learning is increasingly used in sensitive decision making and security-critical settings. At the same time, the growth in both cloud offerings and software stack complexity widens the attack surface for ML applications. This raises the question of integrity and privacy guarantees for ML computations in untrusted environments, in particular for ML tasks outsourced by a client to a remote server. Prominent examples include cloud-based ML APIs (e.g., a speech-to-text application that consumes user-provided data) or general ML-as-a-Service platforms. + +Trusted Execution Environments (TEEs), e.g, Intel SGX (McKeen et al., 2013), ARM TrustZone (Alves & Felton, 2004) or Sanctum (Costan et al., 2016) offer a pragmatic solution to this problem. TEEs use hardware and software protections to isolate sensitive code from other applications, while attesting to its correct execution. Running outsourced ML computations in TEEs provides remote clients with strong privacy and integrity guarantees. + +For outsourced ML computations, TEEs outperform pure cryptographic approaches (e.g, (Gilad-Bachrach et al., 2016; Mohassel & Zhang, 2017; Ghodsi et al., 2017; Juvekar et al., 2018)) by multiple orders of magnitude. At the same time, the isolation guarantees of TEEs still come at a steep price in performance, compared to untrusted alternatives (i.e., running ML models on contemporary hardware with no security guarantees). For instance, Intel SGX (Intel Corp., 2015) incurs significant overhead for memory intensive tasks (Orenbach et al., 2017; Harnik & Tsfadia, 2017), has difficulties exploiting multi-threading, and is currently limited to desktop CPUs that are outmatched by untrusted alternatives (e.g., GPUs or server CPUs). Thus, our thesis is that for modern ML workloads, TEEs will be at least an order of magnitude less efficient than the best available untrusted hardware. + +This leads us to the main question of this paper: How can we most efficiently leverage TEEs for secure machine learning? This was posed by Stoica et al. (2017) as one of nine open research problems for system challenges in AI. A specific challenge they raised is that of appropriately splitting ML computations between trusted and untrusted components, to increase efficiency as well as security by minimizing the Trusted Computing Base. + +This paper explores a novel approach to this challenge, wherein a Deep Neural Network (DNN) execution is partially outsourced from a TEE to a co-located, untrusted but faster device. Our approach, inspired by the verifiable ASICs of Wahby et al. (2016), differs from cryptographic ML outsourcing. In our case, work is delegated between two co-located parties, thus allowing for highly interactive—yet conceptually simpler— outsourcing protocols with orders-of-magnitude better efficiency. Our work also departs from prior systems that execute DNNs fully in a TEE (Ohrimenko et al., 2016; Hunt et al., 2018; Cheng et al., 2018; Hanzlik et al., 2018). + +The main observation that guides our approach is that matrix multiplication—the main bottleneck in DNNs— admits a concretely efficient verifiable outsourcing scheme known as Freivalds’ algorithm (Freivalds, 1977), which can also be turned private in our setting. Our TEE selectively outsources these CPU intensive steps to a fast untrusted co-processor (and runs the remaining steps itself) therefore achieving much better performance than running the entire computation in the enclave, without compromising security. + +Contributions. We propose Slalom, a framework for efficient DNN inference in any trusted execution environment (e.g., SGX or Sanctum). To evaluate Slalom, we build a lightweight DNN library for Intel SGX, which may be of independent interest. Our library allows for outsourcing all linear layers to an untrusted GPU without compromising integrity or privacy. Our code is available at https://github.com/ftramer/slalom. + +We formally prove Slalom’s security, and evaluate it on multiple canonical DNNs with a variety of computational costs—VGG16 (Simonyan & Zisserman, 2014), MobileNet (Howard et al., 2017), and ResNets (He et al., 2016). Compared to running all computations in SGX, outsourcing linear layers to an untrusted GPU increases throughput (as well as energy efficiency) by $6 \times$ to $2 0 \times$ for verifiable inference, and by $4 \times$ to $1 1 \times$ for verifiable and private inference. Finally, we discuss open challenges towards efficient verifiable training of DNNs in TEEs. + +# 2 BACKGROUND + +# 2.1 PROBLEM SETTING + +We consider an outsourcing scheme between a client $\mathcal { C }$ and a server $s$ , where $s$ executes a DNN $F ( x ) : \mathcal { X } \mathcal { Y }$ on data provided by $\mathcal { C }$ . The DNN can either belong to the user (e.g., as in some ML-as-a-service platforms), or to the server (e.g., as in a cloud-based ML API). Depending on the application, this scheme should satisfy one or more of the following security properties (see Appendix $\mathbf { B }$ for formal definitions): + +• t-Integrity: For any $s$ and input $x$ , the probability that a user interacting with $s$ does not abort (i.e., output ⊥) and outputs an incorrect value $\tilde { y } \ne F ( x )$ is less than $t$ . +• Privacy: The server $s$ learns no information about the user’s input $x$ . +• Model privacy: If the model $F$ is provided by the user, $s$ learns no information about $F$ (beyond e.g., its approximate size). If $F$ belongs to the server, $\mathcal { C }$ learns no more about $F$ than what is revealed by $y = F ( x )$ . 1 + +2.2 TRUSTED EXECUTION ENVIRONMENTS (TEES), INTEL SGX, AND A STRONG BASELINE + +Trusted Execution Environments (TEE) such as Intel SGX, ARM TrustZone or Sanctum (Costan et al., 2016) enable execution of programs in secure enclaves. Hardware protections isolate computations in enclaves from all programs on the same host, including the operating system. Enclaves can produce remote attestations—digital signatures over an enclave’s code—that a remote party can verify using the manufacturer’s public key. Our experiments with Slalom use hardware enclaves provided by Intel SGX (see Appendix A for details).2 + +Table 1: Security guarantees and performance (relative to baseline) of different ML outsourcing schemes. + +
ModelPrivacy
ApproachTEEIntegrityPrivacyw.r.t. Serverw.r.t. ClientThroughput (relative)
SafetyNets (Ghodsi et al.,2017)1.OOO≤1/200 ×
Gazelle (Juvekar et al.,2018)1O. *O1≤1/1000 ×
Secure baseline (run DNN in TEE)√.•1×
Insecure baseline (run DNN on GPU)O0≥ 50x
Slalom (Ours)√..O4× - 20×
+ +\* With an offline preprocessing phase. + +TEEs offer an efficient solution for ML outsourcing: The server runs an enclave that initiates a secure communication with $\mathcal { C }$ and evaluates a model $F$ on $\mathcal { C }$ ’s input data. This simple scheme (which we implemented in SGX, see Section 4) outperforms cryptographic ML outsourcing protocols by 2-3 orders of magnitude (albeit under a different trust model). See Table 1 and Appendix C for a comparison to two representative works. + +Yet, SGX’s security comes at a performance cost, and there remains a large gap between TEEs and untrusted devices. For example, current SGX CPUs are limited to $1 2 8 \mathrm { M B }$ of Processor Reserved Memory (PRM) (Costan & Devadas, 2016) and incur severe paging overheads when exceeding this allowance (Orenbach et al., 2017). We also failed to achieve noticeable speed ups for multi-threaded DNN evaluations in SGX enclaves (see Appendix H). For DNN computations, current SGX enclaves thus cannot compete—in terms of performance or energy efficiency (see Appendix C)—with contemporary untrusted hardware, such as a GPU or server CPU. + +In this work, we treat the above simple (yet powerful) TEE scheme as a baseline, and identify settings where we can still improve upon it. We will show that our system, Slalom, substantially outperforms this baseline when the server has access to the model $F$ (e.g., $F$ belongs to $s$ as in cloud ML APIs, or $F$ is public). Slalom performs best for verifiable inference (the setting considered in SafetyNets (Ghodsi et al., 2017)). If the TEE can run some offline data-independent preprocessing (e.g., as in Gazelle (Juvekar et al., 2018)), Slalom also outperforms the baseline for private (and verifiable) outsourced computations in a later online phase. Such a two-stage approach is viable if user data is sent at irregular intervals yet has to be processed with high throughput when available. + +2.3 OUTSOURCING OUTSOURCED DNNS AND FREIVALDS’ ALGORITHM + +Our idea for speeding up DNN inference in TEEs is to further outsource work from the TEE to a co-located faster untrusted processor. Improving upon the above baseline thus requires that the combined cost of doing work on the untrusted device and verifying it in the TEE be cheaper than evaluating the full DNN in the TEE. + +Wahby et al. (2016; 2017) aim at this goal for arbitrary computations outsourced between co-located ASICs. The generic non-interactive proofs they use for integrity are similar to those used in SafetyNets (Ghodsi et al., 2017), which incur overheads that are too large to warrant outsourcing in our setting (e.g., Wahby et al. (2016) find that the technology gap between trusted and untrusted devices needs to be of over two decades for their scheme to break even). Similarly for privacy, standard cryptographic outsourcing protocols (e.g., (Juvekar et al., 2018)) are unusable in our setting as simply running the computation in the TEE is much more efficient (see Table 1). + +To overcome this barrier, we design outsourcing protocols tailored to DNNs, leveraging two insights: + +1. In our setting, the TEE is co-located with the server’s faster untrusted processors, thus widening the design space to interactive outsourcing protocols with high communication but better efficiency. 2. The TEE always has knowledge of the model and can selectively outsource part of the DNN evaluation and compute others—for which outsourcing is harder—itself. + +DNNs are a class of functions that are particularly well suited for selective outsourcing. Indeed, non-linearities— which are hard to securely outsource (with integrity or privacy)—represent a small fraction of the computation in a DNN so we can evaluate these in the TEE (e.g., for VGG16 inference on a single CPU thread, about $1 . 5 \%$ of the computation is spent on non-linearities). In contrast, linear operators—the main computational bottleneck in DNNs—admit for a conceptually simple yet concretely efficient secure delegation scheme, described below. + +Integrity. We verify integrity of outsourced linear layers using variants of an algorithm by Freivalds (1977). + +Lemma 2.1 (Freivalds). Let $A , B$ and $C$ be $n \times n$ matrices over a field $\mathbb { F }$ and let s be a uniformly random vector in $\mathbb { S } ^ { n }$ , for $\mathbb { S } \subseteq \mathbb { F }$ . Then, $\operatorname* { P r } [ C s = A ( B s ) \mid C \neq A B ] = \operatorname* { P r } [ ( C - A B ) s = \mathbf { 0 } \mid ( C - A B ) \neq \mathbf { 0 } ] \leq ^ { 1 } / | \mathbb { S } |$ . + +The randomized check requires $3 n ^ { 2 }$ multiplications, a significant reduction (both in concrete terms and asymptotically) over evaluating the product directly. The algorithm has no false negatives and trivially extends to rectangular matrices. Independently repeating the check $k$ times yields soundness error $1 / | \mathbb { S } | ^ { k }$ . + +Privacy. Input privacy for outsourced linear operators could be achieved with linearly homomorphic encryption, but the overhead (see the micro-benchmarks in (Juvekar et al., 2018)) is too high to compete with our baseline (i.e., computing the function directly in the TEE would be faster than outsourcing it over encrypted data). + +We instead propose a very efficient two-stage approach based on symmetric cryptography, i.e., an additive stream cipher. Let $f : \mathbb { F } ^ { m } \mathbb { F } ^ { n }$ be a linear function over a field $\mathbb { F }$ . In an offline phase, the TEE generates a stream of one-time-use pseudorandom elements $r \in \mathbb { F } ^ { m }$ , and pre-computes $u = f ( r )$ . Then, in the online phase when the remote client sends an input $x$ , the TEE computes $\mathtt { E n c } ( x ) = x + r$ over $\mathbb { F } ^ { m }$ (i.e., a secure encryption of $x$ with a stream cipher), and outsources the computation of $f ( \mathsf { E n c } ( x ) )$ to the faster processor. Given the result $f ( \mathsf { E n c } ( x ) ) = f ( x + r ) = f ( x ) + f ( r ) = f ( x ) \overset { \cdot } { + } u$ , the TEE recovers $f ( x )$ using the pre-computed $u$ . + +Communication. Using Freivalds’ algorithm and symmetric encryption for each linear layer in a DNN incurs high interaction and communication between the TEE and untrusted co-processor (e.g., over 50MB per inference for VGG16, see Table 3). This would be prohibitive if they were not co-located. There are protocols with lower communication than repeatedly using Freivalds’ ((Fiore & Gennaro, 2012; Thaler, 2013; Ghodsi et al., 2017)). Yet, these incur a high overhead on the prover in practice and are thus not suitable in our setting. + +# 3 SLALOM + +We introduce Slalom, a three-step approach for outsourcing DNNs from a TEE to an untrusted but faster device: (1) Inputs and weights are quantized and embedded in a field $\mathbb { F }$ ; (2) Linear layers are outsourced and verified using Freivalds’ algorithm; (3) Inputs of linear layers are encrypted with a pre-computed pseudorandom stream to guarantee privacy. Figure 1 shows two Slalom variants, one to achieve integrity, and one to also achieve privacy. + +We focus on feed-forward networks with fully connected layers, convolutions, separable convolutions, pooling layers and activations. Slalom can be extended to other architectures (e.g., residual networks, see Section 4.3). + +
Slalomwith integrity TEE(F,x1) S(F)
for i∈ [1,n] do
yi=xiWi
xi+1=σ(yi)
1...yn
fori∈[1,n] do assert Freivalds(yi,x,W)
xi+1=σ(yi)
return yn
+ +
Slalom with integrity& privacy TEE(F,x1) S(F)
Preproc: for i∈[1,n] dori ← Fmi,ui =riWi
for i∈[1,n] do 主 xi=xi+ri
起 yi=xW
yi=yi-ui
assert Freivalds(yi,xi,Wi)
xi+1=σ(yi) return yn
+ +# 3.1 QUANTIZATION + +The techniques we use for integrity and privacy (Freivalds’ algorithm and stream ciphers) work over a field $\mathbb { F }$ . We thus quantize all inputs and weights of a DNN to integers, and embed these integers in the field $\mathbb { Z } _ { p }$ of integers modulo a prime $p$ (where $p$ is larger than all values computed in a DNN evaluation, so as to avoid wrap-around). + +As in (Gupta et al., 2015), we convert floating point numbers $x$ to a fixed-point representation as $\tilde { x } = \mathtt { F P } ( x ; l ) : =$ $\mathtt { r o u n d } ( 2 ^ { l } \cdot x ) .$ . For a linear layer with kernel $W$ and bias $b$ , we define integer parameters $\tilde { W } = \mathrm { F P } ( W , l ) , \tilde { b } =$ $\mathrm { E P } ( b , 2 l )$ . After applying the layer to a quantized input $\tilde { x }$ , we scale the output by $2 ^ { - l }$ and re-round to an integer. + +For efficiency reasons, we perform integer arithmetic using floats (so-called fake quantization), and choose $p < 2 ^ { 2 4 }$ to avoid loss of precision (we use $p = 2 ^ { 2 4 } - 3 )$ . For the models we evaluate, setting $l = 8$ for all weights and inputs ensures that all DNN values are bounded by $2 ^ { 2 4 }$ , with less than a $0 . 5 \%$ drop in accuracy (see Table 3). When performing arithmetic modulo $p$ (e.g., for Freivalds’ algorithm or when computing on encrypted data), we use double-precision floats, to reduce the number of modular reductions required (details are in Appendix F). + +# 3.2 VERIFYING COMMON LINEAR OPERATORS + +We now describe Slalom’s approach to verifying the integrity of outsourced linear layers. We describe these layers in detail in Appendix D and summarize this section’s results in Table 2. + +Freivalds’ Algorithm for Batches. The most direct way of applying Freivalds’ algorithm to arbitrary linear layers of a DNN is by exploiting batching. Any linear layer $f ( x )$ from inputs of size $m$ to outputs of size $n$ can be represented (with appropriate reshaping) as $f ( x ) = x ^ { \top } W$ for a (often sparse and implicit) $m \times n$ matrix $W$ + +For a batch $X$ of size $B$ , we can outsource $f ( X )$ and check that the output $Y$ satisfies $f ( s ^ { \top } X ) = s ^ { \top } Y$ , for a random vector $s$ (we are implicitly applying Freivalds to the matrix product $X W = Y$ ). As the batch size $B$ grows, the cost of evaluating $f$ is amortized and the total verification cost is $| X | + | Y | + \mathsf { c o s t } _ { f }$ multiplications (i.e., we approach one operation per input and output). Yet, as we show in Section 4.3, while batched verification is worthwhile for processors with larger memory, it is prohibitive in SGX enclaves due to the limited PRM. + +For full convolutions (and pointwise convolutions), a direct application of Freivalds’ check is worthwhile even for single-element batches. For $f ( x ) = \mathbf { C o n v } ( x , W )$ and purported output $y$ , we can sample a random vector $s$ of + +Table 2: Complexity (number of multiplications) for evaluating and verifying linear functions. The layers are “Fully Connected”, ”Convolution”, ”Depthwise Convolution” and ”Pointwise Convolution”, defined in Appendix D. Each layer $f$ has an input $x$ , output $y$ and kernel $W$ . We assume a batch size of $B \geq 1$ . + +
Layer|x|,|yl|W|costf (B=1)Batched verificationWith preproc.
FChin,houthin·hout|a|·lyB·(lx|+lyl)+cost fB·(lx|+lyl)
Convh·w·Cin,h·w·Coutk² · Cin ·Coutal·k2 · CoutB·(lx|+lyl)+Cin·Cout+ |x| · k²B·(|xl+lyl)
Depth.Convh·w·Cin,h·w·Cink2.Cinlac|.k²B·(lx|+lyl)+cost fB·(|x|+ lyl)
Point. Convh·w ·Cin,h · w· CoutCin·Cout|c| · CoutB·(x|+lyl)+Cin·CoutB·(lx|+|yl)
+ +dimension $c _ { \mathrm { o u t } }$ (the number of output channels), and check that $\operatorname { C o n v } ( x , W s ) = y s$ (with appropriate reshaping). +For a batch of inputs $X$ , we can also apply Freivalds’ algorithm twice to reduce both $W$ and $X$ . + +Preprocessing. We now show how to obtain an outsourcing scheme for linear layers that has optimal verification complexity (i.e., $| x | + | y |$ operations) for single-element batches and arbitrary linear operators, while at the same time compressing the DNN’s weights (a welcome property in our memory-limited TEE model). + +We leverage two facts: (1) DNN weights are fixed at inference time, so part of Freivalds’ check can be precomputed; (2) the TEE can keep secrets from the host $s$ , so the random values $s$ can be re-used across layers or inputs (if we run Freivalds’ check $n$ times with the same secret randomness, the soundness errors grows at most by a factor $n$ ). Our verification scheme with preprocessing follows from a reformulation of Lemma (2.1): + +Lemma 3.1. Let $f : \mathbb { F } ^ { m } \mathbb { F } ^ { n }$ be a linear operator, $f ( x ) : = x ^ { \top } W$ . Let s be uniformly random in $\mathbb { S } ^ { n }$ , for $\mathbb { S } \subseteq \mathbb { F }$ , and let $\tilde { s } : = \nabla F _ { x } ( s ) = W s$ . For any $\boldsymbol { x } \in \mathbb { F } ^ { m }$ , $y \in \mathbb { F } ^ { n }$ , we have $\operatorname* { P r } \left[ y ^ { \top } s = x ^ { \top } { \tilde { s } } \mid y \neq f ( x ) \right] \leq { \boldsymbol { \mathrm { 1 } } } / { \boldsymbol { \mathrm { | \mathbb { S } } } } { \boldsymbol { \mathrm { | \nabla } } }$ . + +The check requires $| x | + | y |$ multiplications, and storage for $s$ and ${ \tilde { s } } : = W s$ (of size $| x |$ and $| y | ,$ ). To save space, we can reuse the same random $s$ for every layer. The memory footprint of a model is then equal to the size of the inputs of all its linear layers (e.g., for VGG16 the footprint is reduced from 550MB to 36MB, see Table 3). + +# 3.3 INPUT PRIVACY + +To guarantee privacy of the client’s inputs, we use precomputed blinding factors for each outsourced computation, as described in Section 2.3. The TEE uses a cryptographic Pseudo Random Number Generator (PRNG) to generate blinding factors. The precomputed “unblinding factors” are encrypted and stored in untrusted memory or disk. In the online phase, the TEE regenerates the blinding factors using the same PRNG seed, and uses the precomputed unblinding factors to decrypt the output of the outsourced linear layer. + +This blinding process incurs several overheads: (1) the computations on the untrusted device have to be performed over $\mathbb { Z } _ { p }$ so we use double-precision arithmetic. (2) The trusted and untrusted processors exchange data in-between each layer, rather than at the end of a full inference pass. (3) The TEE has to efficiently load precomputed unblinding factors, which requires either a large amount of RAM, or a fast access to disk (e.g., a PCIe SSD). + +Slalom’s security is given by the following results. Formal definitions and proofs are in Appendix B. Let negl be a negligible function (for any integer $c > 0$ there exists an integer $N _ { c }$ such that for all $x > N _ { c }$ , $| \mathrm { n e g l } ( x ) | < \bar { 1 } / { x ^ { c } } )$ + +Theorem 3.2. Let Slalom be the protocol from Figure 1 (right), where $F$ is an $n$ -layer DNN, and Freivalds’ algorithm is repeated k times per layer with random vectors drawn from $\mathbb { S } \subseteq \mathbb { F }$ . Assume all random values are generated using a secure PRNG with security parameter $\lambda$ . Then, Slalom is a secure outsourcing scheme for $F$ between a TEE and an untrusted co-processor $s$ with privacy and $t$ -integrity for $t = n / \vert \mathbb { S } \vert ^ { k } - \mathrm { n e g l } ( \lambda )$ . + +Corollary 3.3. Assuming the TEE is secure (i.e., it acts as a trusted third party hosted by $s$ ), Slalom is a secure outsourcing scheme between a remote client $\mathcal { C }$ and server $\boldsymbol { S }$ with privacy and $t$ -integrity for $t = { \boldsymbol { \mathscr { n } } } / { \boldsymbol { | \mathbb { S } } } { \boldsymbol { \vert \sp k - \mathrm { n e g l } } } ( \lambda )$ . If the model $F$ is the property of $\boldsymbol { S }$ , the scheme further satisfies model privacy. + +# 4 EMPIRICAL EVALUATION + +We evaluate Slalom on real Intel SGX hardware, on micro-benchmarks and a sample application (ImageNet inference with VGG16, MobileNet and ResNet models). Our aim is to show that, compared to a baseline that runs inference fully in the TEE, outsourcing linear layers increases performance without sacrificing security. + +# 4.1 IMPLEMENTATION + +As enclaves cannot access most OS features (e.g., multi-threading, disk and driver IO), porting a large framework such as TensorFlow or Intel’s MKL-DNN to SGX is hard. Instead, we designed a lightweight $\mathrm { C } { + + }$ library for feed-forward networks based on Eigen, a linear-algebra library which TensorFlow uses as a CPU backend. Our library implements the forward pass of DNNs, with support for dense layers, standard and separable convolutions, pooling, and activations. When run on a native CPU (without SGX), its performance is comparable to TensorFlow on CPU (compiled with AVX). Our code is available at https://github.com/ftramer/slalom. + +Slalom performs arithmetic over $\mathbb { Z } _ { p }$ , for $p = 2 ^ { 2 4 } - 3$ . For integrity, we apply Freivalds’ check twice to each layer $k = 2 ,$ ), with random values from $\mathbb { S } = [ - 2 ^ { 1 9 } , 2 ^ { 1 9 } ]$ , to achieve 40 bits of statistical soundness per layer (see Appendix F for details on the selection of these parameters). For a 50-layer DNN, $s$ has a chance of less than 1 in 22 billion of fooling the TEE on any incorrect DNN evaluation (a slightly better guarantee than in SafetyNets). For privacy, we use AES-CTR and AES-GCM to generate, encrypt and authenticate blinding factors. + +# 4.2 SETUP + +We use an Intel Core i7-6700 Skylake 3.40GHz processor with 8GB of RAM, a desktop processor with SGX support. The outsourced computations are performed on a co-located Nvidia TITAN XP GPU. Due to a lack of native internal multi-threading in SGX, we run our TEE in a single CPU thread. We discuss challenges for efficient parallelization in Appendix H. We evaluate Slalom on the following workloads: + +• Synthetic benchmarks for matrix products, convolutions and separable convolutions, where we compare the enclave’s running time for computing a linear operation to that of solely verifying the result. • ImageNet (Deng et al., 2009) classification with VGG16 (Simonyan & Zisserman, 2014), MobileNet (Howard et al., 2017), and ResNet He et al. (2016) models (with fused Batch Normalization layers when applicable). + +MobileNet, a model tailored for low compute devices, serves as a worst-case benchmark for Slalom, as the model’s design aggressively minimizes the amount of computation performed per layer. We also consider a “fused” variant of MobileNet with no activation between depthwise and pointwise convolutions. Removing these activations improves convergence and accuracy (Chollet, 2017; Sheng et al., 2018), while also making the network more outsourcing-friendly (i.e., it is possible to verify a separable convolution in a single step). + +Our evaluation focuses on throughput (number of forward passes per second). We also discuss energy efficiency in Appendix C to account for hardware differences between our baseline (TEE only) and Slalom $\mathrm { ( T E E + G P U ) }$ ). + +# 4.3 RESULTS + +Micro-Benchmarks. Our micro-benchmark suite consists of square matrix products of increasing dimensions, convolutional operations performed by VGG16, and separable convolutions performed by MobileNet. In all cases, the data is pre-loaded inside an enclave, so we only measure the in-enclave execution time. Figure 2 plots the relative speedups of various verification strategies over the cost of computing the linear operation directly. In all cases, the baseline computation is performed in single-precision floating point, and the verification algorithms repeat Freivalds’ check so as to attain at least 40 bits of statistical soundness. + +![](images/e479efd0e691ddcb877c1c08f199812ca305e3227b9c63d4af690b286c7b0522.jpg) +Figure 2: Micro benchmarks on Intel SGX. We plot the relative speedup of verifying the result of a linear operator compared to computing it entirely in the enclave. The dotted line shows the throughput obtained for a direct computation. “Fused” separable convolutions contain no intermediate activation. + +For square matrices of dimensions up to 2048, verifying an outsourced result is $4 \times$ to $8 \times$ faster than computing it. For larger matrices, we exceed the limit of SGX’s DRAM, so the enclave resorts to expensive paging which drastically reduces performance both for computation and verification. + +For convolutions (standard or separable), we achieve large savings with outsourcing if Freivalds’ algorithm is applied with preprocessing. The savings get higher as the number of channels increases. Without preprocessing, Freivalds’ algorithm results in savings when $c _ { \mathrm { o u t } }$ is large. Due to SGX’s small PRM, batched verification is only effective for operators with small memory footprints. As expected, “truly” separable convolutions (with no intermediate non-linearity) are much faster to verify, as they can be viewed as a single linear operator. + +Verifiable Inference. Figure 3 shows the throughout of end-to-end forward passes in two neural networks, VGG16 and MobileNet. For integrity, we compare the secure baseline (executing the DNN fully in the enclave) to two variants of the Slalom algorithm in Figure 1. The first (in red) applies Freivalds’ algorithm “on-the-fly”, while the second more efficient variant (in orange) pre-computes part of Freivalds’ check as described in Section 3.2. + +The VGG16 network is much larger (500MB) than SGX’s PRM. As a result, there is a large overhead on the forward pass and verification without preprocessing. If the enclave securely stores preprocessed products $W r$ for all network weights, we drastically reduce the memory footprint and achieve up to a $2 0 . 3 \times$ increase in throughput. We also ran the lower-half of the VGG16 network (without the fully connected layers), a common approach for extracting features for transfer learning or object recognition (Liu et al., 2016). This part fits in the PRM, and we thus achieve higher throughput for in-enclave forward passes and on-the-fly verification. + +For MobileNet, we achieve between $3 . 6 \times$ and $6 . 4 \times$ speedups when using Slalom for verifiable inference (for the standard or “fused” model, respectively). The speedups are smaller than for VGG16, as MobileNet performs much fewer operations per layer (verifying a linear layer requires computing at least two multiplications for each input and output. The closer the forward pass gets to that lower-bound, the less we can save by outsourcing). + +Private Inference. We further benchmark the cost of private DNN inference, where inputs of outsourced linear layers are additionally blinded. Blinding and unblinding each layer’s inputs and outputs is costly, especially in SGX due to the extra in-enclave memory reads and writes. Nevertheless, for VGG16 and the fused MobileNet variant without intermediate activations, we achieve respective speedups of $1 3 . 0 \times$ and $5 . 0 \times$ for private outsourcing (in black in Figure 3), and speedups of $1 0 . 7 \times$ and $4 . 1 \times$ when also ensuring integrity (in purple). For this benchmark, the precomputed unblinding factor are stored in untrusted memory. + +We performed the same experiments on a standard CPU (i.e., without SGX) and find that Slalom’s improvements are even higher in non-resource-constrained or multi-threaded environments (see Appendix G-H). Slalom’s improvements over the baseline also hold when accounting for energy efficiency (see Section C). + +![](images/5df56a4274ed6e233912c5524664ae350b78563790ca22e417ac562aad5df2f1.jpg) +Figure 3: Verifiable and private inference with Intel SGX. We show results for VGG16, VGG16 without the fully connected layers, MobileNet, and a fused MobileNet variant with no intermediate activation for separable convolutions. We compare the baseline of fully executing the DNN in the enclave (blue) to different secure outsourcing schemes: integrity with Freivalds (red); integrity with Freivalds and precomputed secrets (yellow); privacy only (black); privacy and integrity (purple). + +![](images/9ceb8a1cf9a5c8503bd599b1f66c4bf349f3d490915536e0c23e1b7269842915.jpg) +Figure 4: Secure outsourcing of ResNet models with Intel SGX. We compare the baseline of fully executing the DNN in the enclave (blue) to secure outsourcing with integrity (yellow) and privacy and integrity (purple). + +Extending Slalom to Deep Residual Networks. The Slalom algorithm in Figure 1 and our evaluations above focus on feed-forward architectures. Extending Slalom to more complex DNNs is quite simple. To illustrate, we consider the family of ResNet models (He et al., 2016), which use residual blocks $f ( x ) = \sigma ( f _ { 1 } ( x ) + f _ { 2 } ( x ) )$ that merge two feed-forward “paths” $f _ { 1 }$ and $f _ { 2 }$ into a final activation $\sigma$ . To verify integrity of $f ( x )$ , the TEE simply verifies all linear layers in $f _ { 1 }$ and $f _ { 2 }$ and computes $\sigma$ directly. For privacy, the TEE applies the interactive Slalom protocol in Figure 1 (right) in turn to $f _ { 1 }$ and $f _ { 2 }$ , and then computes $\sigma$ . The results for the privacy-preserving Slalom variant in Figure 4 use a preliminary implementation that performs all required operations—and thus provides meaningful performance numbers—but without properly constructed unblinding factors. + +We use the ResNet implementation from Keras Chollet et al. (2015), which contains a pre-trained 50-layer variant. For this model, we find that our quantization scheme results in less than a $0 . 5 \%$ decrease in accuracy (see Table 3). For other variants (i.e., with 18, 34, 101 and 152 layers) we compute throughput on untrained models. Figure 4 shows benchmarks for different ResNet variants when executed fully in the enclave (our baseline) as well as secure outsourcing with integrity or privacy and integrity. For all models, we achieve $6 . 6 \times$ to $1 4 . 4 \times$ speedups for verifiable inference and $4 . 4 \times$ to $9 . 0 \times$ speedups when adding privacy. + +Comparing results for different models is illustrative of how Slalom’s savings scale with model size and architectural design choices. The 18 and 34-layer ResNets use convolutions with $3 \times 3$ kernels, whereas the larger models mainly use pointwise convolutions. As shown in Table 2 verifying a convolution is about a factor $k ^ { 2 } \cdot c _ { \mathrm { o u t } }$ than computing it, which explains the higher savings for models that use convolutions with large kernel windows. When adding more layers to a model, we expect Slalom’s speedup over the baseline to remain constant (e.g., if we duplicate each layer, the baseline computation and the verification should both take twice as long). Yet we find that Slalom’s speedups usually increase as layers get added to the ResNet architecture. This is because the deeper ResNet variants are obtained by duplicating layers towards the end of the pipeline, which have the largest number of channels and for which Slalom achieves the highest savings. + +# 5 CHALLENGES FOR VERIFIABLE AND PRIVATE TRAINING + +Our techniques for secure outsourcing of DNN inference might also apply to DNN training. Indeed, a backward pass consists of similar linear operators as a forward pass, and can thus be verified with Freivalds’ algorithm. Yet, applying Slalom to DNN training is challenging, as described below, and we leave this problem open. + +• Quantizing DNNs for training is harder than for inference, due to large changes in weight magnitudes (Micikevicius et al., 2018). Thus, a more flexible quantization scheme than the one we used would be necessary. • Because the DNN’s weights change during training, the same preprocessed random vectors for Freivalds’ check cannot be re-used indefinitely. The most efficient approach would presumably be to train with very large batches than can then be verified simultaneously. • Finally, the pre-computation techniques we employ for protecting input privacy do not apply for training, as the weights change after every processed batch. Moreover, Slalom does not try to hide the model weights from the untrusted processor, which might be a requirement for private training. + +# 6 CONCLUSION + +This paper has studied the efficiency of evaluating a DNN in a Trusted Execution Environment (TEE) to provide strong integrity and privacy guarantees. We explored new approaches for segmenting a DNN evaluation to securely outsource work from a trusted environment to a faster co-located but untrusted processor. + +We designed Slalom, a framework for efficient DNN evaluation that outsources all linear layers from a TEE to a GPU. Slalom leverage Freivalds’ algorithm for verifying correctness of linear operators, and additionally encrypts inputs with precomputed blinding factors to preserve privacy. Slalom can work with any TEE and we evaluated its performance using Intel SGX on various workloads. 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ACM, 2014. + +# A DETAILS ON INTEL SGX SECURITY + +SGX enclaves isolate execution of a program from all other processes on a same host, including a potentially malicious OS. In particular, enclave memory is fully encrypted and authenticated. When a word is read from memory into a CPU register, a Memory Management Engine handles the decryption (Costan & Devadas, 2016). + +While SGX covers many software and hardware attack vectors, there is a large and prominent class of sidechannel attacks that it explicitly does not address (Costan & Devadas, 2016; Tramèr et al., 2017). In the past years, many attacks have been proposed, with the goal of undermining privacy of enclave computations $\mathrm { { X u } }$ et al., 2015; Brasser et al., 2017; Moghimi et al., 2017; Götzfried et al., 2017; Van Bulck et al., 2017; Lee et al., 2017). Most of these attacks rely on data dependent code behavior in an enclave (e.g., branching or memory access) that can be partially observed by other processes running on the same host. These side-channels are a minor concern for the DNN computations considered in this paper, as the standard computations in a DNN are data-oblivious (i.e., the same operations are applied regardless of the input data) (Ohrimenko et al., 2016). + +The recent Spectre attacks on speculative execution (Kocher et al., 2018) also prove damaging to SGX (as well as to most other processors), as recently shown (Chen et al., 2018; Dall et al., 2018; Van Bulck et al., 2018). Mitigations for these side-channel attacks are being developed (Shinde et al., 2016; Shih et al., 2017; Chen et al., 2017; Intel Corp., 2018) but a truly secure solution might require some architectural changes, e.g., as in the proposed Sanctum processor (Costan et al., 2016). + +We refrain from formally modeling SGX’s (or other TEE’s) security in this paper, as Slalom is mostly concerned with outsourcing protocols wherein the TEE acts as a client. We refer the interested reader to (Pass et al., 2017; Fisch et al., 2017; Subramanyan et al., 2017) for different attempts at such formalisms. + +# B FORMAL SECURITY DEFINITIONS AND PROOFS + +We define a secure outsourcing scheme, between a client $\mathcal { C }$ and a server $s$ , for a DNN $F ( x ) : \mathcal { X } \mathcal { Y }$ from some family $\mathcal { F }$ (e.g., all DNNs of a given size). We first assume that the model $F$ is known to both $\mathcal { C }$ and $s$ : + +Definition B.1 (Secure Outsourcing Schemes). A secure outsourcing scheme consists of an offline preprocessing algorithm Preproc, as well as an interactive online protocol Outsource $\langle \mathcal { C } , \mathcal { S } \rangle$ , defined as follows: + +• $\mathsf { s t } \gets \mathsf { P r e p r o c } ( F , 1 ^ { \lambda } )$ : The preprocessing algorithm is run by $\mathcal { C }$ and generates some data-independent state $\mathtt { s t }$ (e.g., cryptographic keys or precomputed values to accelerate the online outsourcing protocol.) • $\mathcal { V } \cup \{ \bot \} \mathsf { O u t s o u r c e } \langle \mathcal { C } ( F , x , \mathrm { s t } ) , \mathcal { S } ( F ) \rangle \mathrm { : }$ : The online outsourcing protocol is initiated by $\mathcal { C }$ with inputs $( F , x , s \ t )$ . At the end of the protocol, $\mathcal { C }$ either outputs a value $y \in \mathcal { V }$ or aborts (i.e., $\mathcal { C }$ outputs $\perp$ ). + +The properties that we may require from a secure outsourcing scheme are: + +• Correctness: For any $\begin{array} { r l r l } { F } & { { } \in } & { { \mathcal { F } } } \end{array}$ and $\begin{array} { r l r l } { x } & { { } \in } & { { \mathcal { X } } } \end{array}$ , running $\begin{array} { r l r } { \mathsf { s t } } & { { } \gets } & { \mathsf { P r e p r o c } ( F , 1 ^ { \lambda } ) } \end{array}$ and $y $ Outsource $\langle \mathcal { C } ( F , x , \mathrm { s t } ) , \mathcal { S } ( F ) \rangle$ yields $y = F ( x )$ . + +• t-Integrity: For any $F \in { \mathcal { F } }$ , input $x \in \mathcal { X }$ and probabilistic polynomial-time adversary $S ^ { * }$ , the probability that $\tilde { y } = \mathsf { O u t s o u r c e } \langle \mathcal { C } ( F , x , \mathsf { s t } ) , \mathcal { S } ^ { \ast } ( F ) \rangle$ and $\tilde { y } \notin \{ F ( x ) , \bot \}$ is less than $t$ . + +• Input privacy: For any $F \in { \mathcal { F } }$ , inputs $x , x ^ { \prime } \in { \mathcal { X } }$ and probabilistic poly-time adversary ${ \boldsymbol { S } } ^ { * }$ , the views of $S ^ { * }$ in Outsource $\langle \mathcal { C } ( F , x , \mathsf { s t } ) , \mathcal { S } ^ { * } ( F ) \rangle$ and Outsource $\langle \mathcal { C } ( F , \overline { { x ^ { \prime } } } , \mathrm { { s t } } ) , S ^ { \ast } ( \mathrm { { \bar { F } } } ) \rangle$ are computationally indistinguishable. + +• Efficiency: The online computation of $\mathcal { C }$ in Outsource should be less than the cost for $\mathcal { C }$ to evaluate $F \in { \mathcal { F } }$ . + +Model Privacy. In some applications a secure outsourcing scheme may also require to hide the model $F$ from either $s$ or $\mathcal { C }$ (in which case that party would obviously not take $F$ as input in the above scheme). + +Privacy with respect to an adversarial server ${ \boldsymbol { S } } ^ { * }$ (which Slalom does not provide), is defined as the indistinguishability of $S ^ { * }$ ’s views in Outsource $\langle \mathcal { C } ( F , x , \mathsf { s t } ) , \mathcal { S } ^ { * } \rangle$ and Outsource $\langle \mathcal { C } ( F ^ { \prime } , x , \mathrm { s t } ) , S ^ { * } \rangle$ for any $F , F ^ { \prime } \in \mathcal { F }$ . + +As noted in Section 2.1, a meaningful model-privacy guarantee with respect to $\mathcal { C }$ requires that $s$ first commit to a specific DNN $F$ , and then convinces $\mathcal { C }$ that her outputs were produced with the same model as all other clients’. We refer the reader to Canetti et al. (2002) for formal definitions for such commit-and-prove schemes, and to Tramèr et al. (2017) who show how to trivially instantiate them using a TEE. + +Proof of Theorem 3.2. Let $\mathrm { s t ~ ~ }$ Preproc and Outsource $\langle \mathrm { T E E } ( F , x , \mathrm { s t } ) , S \rangle$ be the outsourcing scheme defined in Figure 1 (right). We assume that all random values sampled by the TEE are produced by a secure cryptographically secure pseudorandom number generator (PRNG) (with elements in $\mathbb { S } \subseteq \mathbb { F }$ for the integritycheck vectors $s$ used in Freivalds’ algorithm, and in $\mathbb { F }$ for the blinding vectors $r _ { i }$ ). + +We first consider integrity. Assume that the scheme is run with input $x _ { 1 }$ and that the TEE outputs $y _ { n }$ . We will bound $\operatorname* { P r } [ y _ { n } \neq F ( x _ { 1 } ) \ | \ { \dot { y } } _ { n } \neq \bot ]$ . By the security of the PRNG, we can replace the vectors $s$ used in Freivalds’ algorithm by truly uniformly random values in $\mathbb { S } \subseteq \mathbb { F }$ , via a simple hybrid argument. For the $i$ -th linear layer, with operator $W _ { i }$ , input $x _ { i }$ and purported output $y _ { i }$ , we then have that $y _ { i } \neq x _ { i } W _ { i }$ with probability at most $1 / | \mathbb { S } | ^ { k }$ . By a simple union bound, we thus have that $\operatorname* { P r } [ y _ { n } \neq F ( x _ { 1 } ) ] \leq { n } / { | \mathbb { S } | ^ { k } } - \operatorname { n e g l } ( \lambda ) .$ . Note that this bound holds even if the same (secret) random values $s$ are re-used across layers. + +For privacy, consider the views of an adversary $S ^ { * }$ when Slalom is run with inputs $x _ { 1 }$ and $x _ { 1 } ^ { \prime }$ . Again, by the security of the PRNG, we consider a hybrid protocol where we replace the pre-computed blinding vectors $r _ { i }$ by truly uniformly random values in $\mathbb { F }$ . In this hybrid protocol, $\tilde { x } _ { i } = x _ { i } + r _ { i }$ is simply a “one-time-pad” encryption of $x _ { i }$ over the field $\mathbb { F }$ , so $S ^ { * }$ ’s views in both executions of the hybrid protocol are equal (information theoretically). Thus, $S ^ { * }$ ’s views in both executions of the original protocol are computationally indistinguishable. □ + +Proof of Corollary 3.3. The outsourcing protocol between the remote client $\mathcal { C }$ and server $s$ hosting the TEE is simply defined as follows (we assume the model belongs to $s$ ): + +• $\mathsf { s t } \gets \mathsf { P r e p r o c ( ) }$ : $\mathcal { C }$ and the TEE setup a secure authenticated communication channel, using the TEE’s remote attestation property. The TEE receives the model $F$ from $s$ and initializes the Slalom protocol. + +• Outsource $\langle { \mathcal { C } } ( x , { \mathrm { s t } } ) , S ( F ) \rangle$ : – $\mathcal { C }$ sends $x$ to the TEE over the secure channel. – The TEE securely computes $y = F ( x )$ using Slalom. – The TEE sends $y$ (and a publicly verifiable commitment to $F$ ) to $\mathcal { C }$ over the secure channel. + +If the TEE is secure (i.e., it acts as a trusted third party hosted by $s$ ), then the result follows. + +# C PERFORMANCE COMPARISON OF DNN OUTSOURCING SCHEMES + +We provide a brief overview of the outsourcing approaches compared in Table 1. Our baseline runs a DNN in a TEE (a single-threaded Intel SGX enclave) and can provide all the security guarantees of an ML outsourcing scheme. On a high-end GPU (an Nvidia TITAN XP), we achieve over $5 0 \times$ higher throughput but no security. For example, for MobileNet, the enclave evaluates 16 images/sec and the GPU 900 images/sec $5 6 \times$ higher). + +SafetyNets (Ghodsi et al., 2017) and Gazelle (Juvekar et al., 2018) are two representative works that achieve respectively integrity and privacy using purely cryptographic approaches (without a TEE). SafetyNets does not hide the model from either party, while Gazelle leaks some architectural details to the client. The cryptographic techniques used by these systems incur large computation and communication overheads in practice. The largest model evaluated by SafetyNets is a 4-layer TIMIT model with quadratic activations which runs at about 13 images/sec (on a notebook CPU). In our baseline enclave, the same model runs at over 3,500 images/sec. The largest model evaluated by Gazelle is an 8-layer CIFAR10 model. In the enclave, we can evaluate 450 images/sec whereas Gazelle evaluates a single image in 3.5 sec with $3 0 0 \mathrm { M B }$ of communication between client and server. + +A Note on Energy Efficiency. When comparing approaches with different hardware (e.g., our single-core CPU baseline versus Slalom which also uses a GPU), throughput alone is not the fairest metric. E.g., the baseline’s throughput could also be increased by adding more SGX CPUs. A more accurate comparison considers the energy efficiency of a particular approach, a more direct measure of the recurrent costs to the server $s$ . + +For example, when evaluating MobileNet or VGG16, our GPU draws 85W of power, whereas our baseline SGX CPU draws 30W. As noted above, the GPU also achieves more than $5 0 \times$ higher throughput, and thus is at least $1 8 \times$ more energy efficient (e.g., measured in Joules per image) than the enclave. + +For Slalom, we must consider the cost of running both the enclave and GPU. In our evaluations, the outsourced computations on the GPU account for at most $1 0 \%$ of the total running time of Slalom (i.e., the integrity checks and data encryption/decryption in the enclave are the main bottleneck). Thus, the power consumption attributed to Slalom is roughly $1 0 \% \cdot 8 5 \mathrm { W } + 9 0 \% \cdot 3 0 \mathrm { W } = 3 5 . 5 \mathrm { W } .$ Note that when not being in use by Slalom, the trusted CPU or untrusted GPU can be used by other tasks running on the server. As Slalom achieves $4 \times - 2 0 \times$ higher throughput than our baseline for the tasks we evaluate, it is also about $3 . 4 \times - 1 7 . 1 \times$ more energy efficient. + +# D NOTATION FOR STANDARD LINEAR OPERATORS + +Below we describe some common linear operators used in deep neural networks. For simplicity, we omit additive bias terms, and assume that convolutional operators preserve the spatial height and width of their inputs. Our techniques easily extend to convolutions with arbitrary strides, paddings, and window sizes. + +For a fully-connected layer $f _ { \mathrm { F C } }$ , the kernel $W$ has dimension $\left( h _ { \mathrm { i n } } \times h _ { \mathrm { o u t } } \right)$ . For an input $x$ of dimension $h _ { \mathrm { i n } }$ , we have $f _ { \mathrm { F C } } ( \dot { \boldsymbol { x } } ) = \boldsymbol { x } ^ { \top } \boldsymbol { W }$ . The cost of the layer is $h _ { \mathrm { i n } } \cdot h _ { \mathrm { o u t } }$ multiplications. + +A convolutional layer has kernel $W$ of size $( k \times k \times c _ { \mathrm { i n } } \times c _ { \mathrm { o u t } } )$ . On input $x$ of size $( h \times w \times c _ { \mathrm { i n } } )$ , $f _ { \mathrm { c o n v } } ( x ) =$ $\mathsf { C o n v } ( x ; W )$ produces an output of size $( h \times w \times c _ { \mathrm { o u t } } )$ . A convolution can be seen as the combination of two linear operators: a “patch-extraction” process that transforms the input $x$ into an intermediate input $x ^ { \prime }$ of dimension $( \bar { h } \cdot w , k ^ { 2 } \cdot c _ { \mathrm { i n } } ) $ by extracting $k \times k$ patches, followed by a matrix multiplication with $W$ . The cost of this layer is thus $\boldsymbol { k } ^ { 2 } \cdot \boldsymbol { h } \cdot \boldsymbol { w } \cdot \boldsymbol { c } _ { \mathrm { i n } } \cdot \boldsymbol { c } _ { \mathrm { o u t } }$ multiplications. + +A separable convolution has two kernels, $W _ { 1 }$ of size $( k \times k \times c _ { \mathrm { i n } } )$ and $W _ { 2 }$ of size $\left( { c _ { \mathrm { i n } } } \times { c _ { \mathrm { o u t } } } \right)$ . On input $x$ of size $( h \times w \times c _ { \mathrm { i n } } )$ , $f _ { \mathrm { s e p - c o n v } } ( x )$ produces an output of size $( h \times w \times c _ { \mathrm { o u t } } )$ , by applying a depthwise convolution $f _ { \mathrm { d p - c o n v } } ( x )$ with kernel $W _ { 1 }$ followed by a pointwise convolution $f _ { \mathrm { p t - c o n v } } ( x )$ with kernel $W _ { 2 }$ . The depthwise convolution consists of $c _ { \mathrm { i n } }$ independent convolutions with filters of size $k \times k \times 1 \times 1$ , applied to a single input channel, which requires $k ^ { 2 } \cdot h \cdot w \cdot c _ { \mathrm { i n } }$ multiplications. A pointwise convolution is simply a matrix product with an input of size $( h \cdot w ) \times c _ { \mathrm { i n } }$ , and thus requires $h \cdot w \cdot c _ { \mathrm { i n } } \cdot c _ { \mathrm { o u t } }$ multiplications. + +# E NEURAL NETWORK DETAILS + +Table 3 provides details about the two DNNs we use in our evaluation (all pre-trained models are taken from Keras Chollet et al. (2015)). We report top 1 and top 5 accuracy on ImageNet with and without the simple quantization scheme described in Section 3.1. Quantization results in at most a $0 . 5 \%$ drop in top 1 and top 5 accuracy. More elaborate quantization schemes exist (e.g., Micikevicius et al. (2018)) that we have not experimented with in this work. + +We report the number of model parameters, which is relevant to the memory constraints of TEEs such as Intel SGX. We also list the total size of the inputs and outputs of all the model’s linear layers, which impact the amount of communication between trusted and untrusted co-processors in Slalom, as well as the amount of data stored in the TEE when using Freivalds’ algorithm with preprocessing. + +Table 3: Details of models used in our evaluation. Accuracies are computed on the ImageNet validation set. Pre-trained models are from Keras Chollet et al. (2015). + +
AccuracyQuantized
ModelTop 1Top5Top1Top 5LayersParameters (M)Size of layer inputs/outputs (M)
VGG1671.090.070.689.516138.49.1 / 13.6
VGG16 (no top)=-1314.79.1/13.5
MobileNet70.789.670.589.5284.25.5/ 5.0
MobileNet (fused)=1154.23.6/3.1
ResNet 5076.992.476.492.25025.510.0 /10.4
+ +# F MODULAR ARITHMETIC WITH FLOATING POINT OPERATIONS + +In this section, we briefly describe how Slalom performs modular arithmetic over a field $\mathbb { Z } _ { p }$ in the TEE, while leveraging standard floating point operations to maximize computational efficiency. The main computations in the TEE are inner products over $\mathbb { Z } _ { p }$ for Freivalds’ check (a matrix product is itself a set of inner products). + +Our quantization scheme (see Section 3.1) ensures that all DNN values can be represented in $\mathbb { Z } _ { p }$ , for $p \lessapprox 2 ^ { 2 4 }$ , which fits in a standard float. To compute inner products, we first cast elements to doubles (as a single multiplication in $\mathbb { Z } _ { p }$ would exceed the range of integers exactly representable as floats). Single or double precision floats are preferable to integer types on Intel architectures due to the availability of much more efficient SIMD instructions, at a minor reduction in the range of exactly representable integers. + +In our evaluation, we target a soundness error of $2 ^ { - 4 0 }$ for each layer. This leads to a tradeoff between the number of repetitions $k$ of Freivalds’ check, and the size of the set $\mathbb { S }$ from which we draw random values. One check with $| \mathbb { S } | = 2 ^ { 4 0 }$ is problematic, as multiplying elements in $\mathbb { Z } _ { p }$ and $\mathbb { S }$ can exceed the range of integers exactly representable as doubles $( 2 ^ { 5 3 } )$ . With $k = 2$ repetitions, we can set $\mathbb { S } = [ - 2 ^ { 1 9 } , 2 ^ { 1 9 } ]$ . Multiplications are then bounded by $2 ^ { 2 4 + 1 9 } = 2 ^ { 4 3 }$ , and we can accumulate $2 ^ { 1 0 }$ terms in the inner-product before needing a modular reduction. In practice, we find that increasing $k$ further (and thus reducing $| \mathbb { S } | )$ is not worthwhile, as the cost of performing more inner products trumps the savings from reducing the number of modulos. + +# G RESULTS ON A STANDARD CPU + +For completeness, and to asses how our outsourcing scheme fairs in an environment devoid of Intel SGX’s performance quirks, we rerun the evaluations in Section 4 on the same CPU but outside of SGX’s enclave mode. + +![](images/9e3a7cd6a779288993a211826b1f16979c45c88d6f37c278fede67cb508715d1.jpg) +Figure 5: Micro benchmarks on an untrusted CPU. For three different linear operators, we plot the relative speedup of verifying a result compared to computing it. The dotted line in each plot shows the throughput obtained for computing the operation. + +Figure 5 show the results of the micro-benchmarks for matrix multiplication, convolution and separable convolutions. In all cases, verifying a computation becomes 1-2 orders of magnitude faster than computing it as the outer dimension grows. Compared to the SGX benchmarks, we also see a much better viability of batched verification (we haven’t optimized batched verifications much, as they are inherently slow on SGX. It is likely that these numbers could be improved significantly, to approach those of verification with preprocessing). + +Figure 6 shows benchmarks for VGG16 and MobileNet on a single core with either direct computation or various secure outsourcing strategies. For integrity alone, we achieve savings up to $8 . 9 \times$ and $1 9 . 5 \times$ for MobileNet and VGG16 respectively. Even without storing any secrets in the enclave, we obtain good speedups using batched verification. As noted above, it is likely that the batched results could be further improved. With additional blinding to preserve privacy, we achieve speedups of $3 . 9 \times$ and $8 . 1 \times$ for MobileNet and VGG16 respectively. + +![](images/c8154e92d0e65f4edf1599b73db1e4f3db8f57d0f186fd57a3cf3863327d0a07.jpg) +Figure 6: Inference with integrity and/or privacy on an untrusted CPU. We compare the baseline inference throughput (blue) to that obtained with “on-the-fly” integrity checks (red); batched integrity checks (green); integrity checks with precomputed secrets (yellow); privacy only (black); and privacy and integrity (purple). The fused MobileNet model has no intermediate activation for separable convolutions. + +# H PARALLELIZATION + +Our experiments on SGX in Section 4 where performed using a single execution thread, as SGX enclaves do not have the ability to create threads. We have also experimented with techniques for achieving parallelism in SGX, both for standard computations and outsourced ones, but with little success. + +To optimize for throughput, a simple approach is to run multiple forward passes simultaneously. On a standard CPU, this form of “outer-parallelism” achieves close to linear scaling as we increase the number of threads from 1 to 4 on our quad-core machine. With SGX however, we did not manage to achieve any parallel speedup for VGG16—whether for direct computation or verifying outsourced results—presumably because each independent thread requires extra memory that quickly exceeds the PRM limit. For the smaller MobileNet model, we get less than a $1 . 5 \times$ speedup using up to 4 threads, for direct computation or outsourced verification alike. + +DNNs typically also make use of intra-operation parallelism, i.e., computing the output of a given layer using multiple threads. Our DNN library currently does not support intra-operation parallelism, but implementing a dedicated thread pool for SGX could be an interesting extension for future work. Instead, we evaluate the potential benefits of intra-op parallelism on a standard untrusted CPU, for our matrix-product and convolution benchmarks. We make use of Eigen’s internal multi-threading support to speed up these operations, and custom OpenMP code to parallelize dot products, as Eigen does not do this on its own. + +![](images/827a1dd46e9ddda18510395cc72fb7d148bced674096122d688675387980e22d.jpg) +Figure 7: Multi-threaded micro benchmarks on an untrusted CPU. Reiterates benchmarks for matrix products and convolutions using 4 threads. + +Figure 7 shows the results using 4 threads. For convolutions, we have currently only implemented multi-threading for the verification with preprocessing (which requires only standard dot products). Surprisingly maybe, we find that multi-threading increases the gap between direct and verified computations of matrix products, probably because dot products are extremely easy to parallelize efficiently (compared to full convolutions). We also obtain close to linear speedups for verifiable separable convolutions, but omit the results as we currently do not have an implementation of multi-threaded direct computation for depthwise convolutions, which renders the comparison unfair. Due to the various memory-access overheads in SGX, it is unclear whether similar speedups could be obtained by using intra-op parallelism in an enclave, but this is an avenue worth exploring. \ No newline at end of file diff --git a/parse/train/rJVorjCcKQ/rJVorjCcKQ_content_list.json b/parse/train/rJVorjCcKQ/rJVorjCcKQ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d6eb52999283290e7391e3dde87fa52421653f12 --- /dev/null +++ b/parse/train/rJVorjCcKQ/rJVorjCcKQ_content_list.json @@ -0,0 +1,1987 @@ +[ + { + "type": "text", + "text": "SLALOM: FAST, VERIFIABLE AND PRIVATE EXECUTION OF NEURAL NETWORKS IN TRUSTED HARDWARE ", + "text_level": 1, + "bbox": [ + 138, + 171, + 861, + 219 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Florian Tramèr \nStanford University \ntramer@cs.stanford.edu ", + "bbox": [ + 145, + 242, + 364, + 285 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Dan Boneh Stanford University dabo@cs.stanford.edu ", + "bbox": [ + 516, + 243, + 715, + 285 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 321, + 544, + 337 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As Machine Learning (ML) gets applied to security-critical or sensitive domains, there is a growing need for integrity and privacy for outsourced ML computations. A pragmatic solution comes from Trusted Execution Environments (TEEs), which use hardware and software protections to isolate sensitive computations from the untrusted software stack. However, these isolation guarantees come at a price in performance, compared to untrusted alternatives. This paper initiates the study of high performance execution of Deep Neural Networks (DNNs) in TEEs by efficiently partitioning DNN computations between trusted and untrusted devices. Building upon an efficient outsourcing scheme for matrix multiplication, we propose Slalom, a framework that securely delegates execution of all linear layers in a DNN from a TEE (e.g., Intel SGX or Sanctum) to a faster, yet untrusted, co-located processor. We evaluate Slalom by running DNNs in an Intel SGX enclave, which selectively delegates work to an untrusted GPU. For canonical DNNs (VGG16, MobileNet and ResNet variants) we obtain $6 \\times$ to $2 0 \\times$ increases in throughput for verifiable inference, and $4 \\times$ to $1 1 \\times$ for verifiable and private inference. ", + "bbox": [ + 196, + 357, + 803, + 537 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 140, + 571, + 299, + 588 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Machine learning is increasingly used in sensitive decision making and security-critical settings. At the same time, the growth in both cloud offerings and software stack complexity widens the attack surface for ML applications. This raises the question of integrity and privacy guarantees for ML computations in untrusted environments, in particular for ML tasks outsourced by a client to a remote server. Prominent examples include cloud-based ML APIs (e.g., a speech-to-text application that consumes user-provided data) or general ML-as-a-Service platforms. ", + "bbox": [ + 138, + 606, + 862, + 676 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Trusted Execution Environments (TEEs), e.g, Intel SGX (McKeen et al., 2013), ARM TrustZone (Alves & Felton, 2004) or Sanctum (Costan et al., 2016) offer a pragmatic solution to this problem. TEEs use hardware and software protections to isolate sensitive code from other applications, while attesting to its correct execution. Running outsourced ML computations in TEEs provides remote clients with strong privacy and integrity guarantees. ", + "bbox": [ + 138, + 681, + 862, + 738 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "For outsourced ML computations, TEEs outperform pure cryptographic approaches (e.g, (Gilad-Bachrach et al., 2016; Mohassel & Zhang, 2017; Ghodsi et al., 2017; Juvekar et al., 2018)) by multiple orders of magnitude. At the same time, the isolation guarantees of TEEs still come at a steep price in performance, compared to untrusted alternatives (i.e., running ML models on contemporary hardware with no security guarantees). For instance, Intel SGX (Intel Corp., 2015) incurs significant overhead for memory intensive tasks (Orenbach et al., 2017; Harnik & Tsfadia, 2017), has difficulties exploiting multi-threading, and is currently limited to desktop CPUs that are outmatched by untrusted alternatives (e.g., GPUs or server CPUs). Thus, our thesis is that for modern ML workloads, TEEs will be at least an order of magnitude less efficient than the best available untrusted hardware. ", + "bbox": [ + 137, + 746, + 862, + 857 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This leads us to the main question of this paper: How can we most efficiently leverage TEEs for secure machine learning? This was posed by Stoica et al. (2017) as one of nine open research problems for system challenges in AI. A specific challenge they raised is that of appropriately splitting ML computations between trusted and untrusted components, to increase efficiency as well as security by minimizing the Trusted Computing Base. ", + "bbox": [ + 137, + 176, + 861, + 233 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This paper explores a novel approach to this challenge, wherein a Deep Neural Network (DNN) execution is partially outsourced from a TEE to a co-located, untrusted but faster device. Our approach, inspired by the verifiable ASICs of Wahby et al. (2016), differs from cryptographic ML outsourcing. In our case, work is delegated between two co-located parties, thus allowing for highly interactive—yet conceptually simpler— outsourcing protocols with orders-of-magnitude better efficiency. Our work also departs from prior systems that execute DNNs fully in a TEE (Ohrimenko et al., 2016; Hunt et al., 2018; Cheng et al., 2018; Hanzlik et al., 2018). ", + "bbox": [ + 137, + 239, + 862, + 324 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main observation that guides our approach is that matrix multiplication—the main bottleneck in DNNs— admits a concretely efficient verifiable outsourcing scheme known as Freivalds’ algorithm (Freivalds, 1977), which can also be turned private in our setting. Our TEE selectively outsources these CPU intensive steps to a fast untrusted co-processor (and runs the remaining steps itself) therefore achieving much better performance than running the entire computation in the enclave, without compromising security. ", + "bbox": [ + 137, + 330, + 864, + 400 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions. We propose Slalom, a framework for efficient DNN inference in any trusted execution environment (e.g., SGX or Sanctum). To evaluate Slalom, we build a lightweight DNN library for Intel SGX, which may be of independent interest. Our library allows for outsourcing all linear layers to an untrusted GPU without compromising integrity or privacy. Our code is available at https://github.com/ftramer/slalom. ", + "bbox": [ + 137, + 409, + 862, + 465 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We formally prove Slalom’s security, and evaluate it on multiple canonical DNNs with a variety of computational costs—VGG16 (Simonyan & Zisserman, 2014), MobileNet (Howard et al., 2017), and ResNets (He et al., 2016). Compared to running all computations in SGX, outsourcing linear layers to an untrusted GPU increases throughput (as well as energy efficiency) by $6 \\times$ to $2 0 \\times$ for verifiable inference, and by $4 \\times$ to $1 1 \\times$ for verifiable and private inference. Finally, we discuss open challenges towards efficient verifiable training of DNNs in TEEs. ", + "bbox": [ + 137, + 472, + 862, + 541 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 137, + 555, + 289, + 571 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 PROBLEM SETTING ", + "text_level": 1, + "bbox": [ + 137, + 585, + 312, + 601 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We consider an outsourcing scheme between a client $\\mathcal { C }$ and a server $s$ , where $s$ executes a DNN $F ( x ) : \\mathcal { X } \\mathcal { Y }$ on data provided by $\\mathcal { C }$ . The DNN can either belong to the user (e.g., as in some ML-as-a-service platforms), or to the server (e.g., as in a cloud-based ML API). Depending on the application, this scheme should satisfy one or more of the following security properties (see Appendix $\\mathbf { B }$ for formal definitions): ", + "bbox": [ + 137, + 612, + 862, + 667 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• t-Integrity: For any $s$ and input $x$ , the probability that a user interacting with $s$ does not abort (i.e., output ⊥) and outputs an incorrect value $\\tilde { y } \\ne F ( x )$ is less than $t$ . \n• Privacy: The server $s$ learns no information about the user’s input $x$ . \n• Model privacy: If the model $F$ is provided by the user, $s$ learns no information about $F$ (beyond e.g., its approximate size). If $F$ belongs to the server, $\\mathcal { C }$ learns no more about $F$ than what is revealed by $y = F ( x )$ . 1 ", + "bbox": [ + 143, + 672, + 862, + 751 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 TRUSTED EXECUTION ENVIRONMENTS (TEES), INTEL SGX, AND A STRONG BASELINE ", + "bbox": [ + 137, + 767, + 789, + 782 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Trusted Execution Environments (TEE) such as Intel SGX, ARM TrustZone or Sanctum (Costan et al., 2016) enable execution of programs in secure enclaves. Hardware protections isolate computations in enclaves from all programs on the same host, including the operating system. Enclaves can produce remote attestations—digital signatures over an enclave’s code—that a remote party can verify using the manufacturer’s public key. Our experiments with Slalom use hardware enclaves provided by Intel SGX (see Appendix A for details).2 ", + "bbox": [ + 133, + 792, + 861, + 821 + ], + "page_idx": 1 + }, + { + "type": "table", + "img_path": "images/d0f6cc00c0b1482c5f0835cda6bb82c1a97e49dbc4922eca8057b926edb979e8.jpg", + "table_caption": [ + "Table 1: Security guarantees and performance (relative to baseline) of different ML outsourcing schemes. " + ], + "table_footnote": [ + "\\* With an offline preprocessing phase. " + ], + "table_body": "
ModelPrivacy
ApproachTEEIntegrityPrivacyw.r.t. Serverw.r.t. ClientThroughput (relative)
SafetyNets (Ghodsi et al.,2017)1.OOO≤1/200 ×
Gazelle (Juvekar et al.,2018)1O. *O1≤1/1000 ×
Secure baseline (run DNN in TEE)√.•1×
Insecure baseline (run DNN on GPU)O0≥ 50x
Slalom (Ours)√..O4× - 20×
", + "bbox": [ + 150, + 194, + 846, + 292 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 315, + 862, + 358 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "TEEs offer an efficient solution for ML outsourcing: The server runs an enclave that initiates a secure communication with $\\mathcal { C }$ and evaluates a model $F$ on $\\mathcal { C }$ ’s input data. This simple scheme (which we implemented in SGX, see Section 4) outperforms cryptographic ML outsourcing protocols by 2-3 orders of magnitude (albeit under a different trust model). See Table 1 and Appendix C for a comparison to two representative works. ", + "bbox": [ + 137, + 364, + 862, + 420 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Yet, SGX’s security comes at a performance cost, and there remains a large gap between TEEs and untrusted devices. For example, current SGX CPUs are limited to $1 2 8 \\mathrm { M B }$ of Processor Reserved Memory (PRM) (Costan & Devadas, 2016) and incur severe paging overheads when exceeding this allowance (Orenbach et al., 2017). We also failed to achieve noticeable speed ups for multi-threaded DNN evaluations in SGX enclaves (see Appendix H). For DNN computations, current SGX enclaves thus cannot compete—in terms of performance or energy efficiency (see Appendix C)—with contemporary untrusted hardware, such as a GPU or server CPU. ", + "bbox": [ + 135, + 428, + 862, + 511 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work, we treat the above simple (yet powerful) TEE scheme as a baseline, and identify settings where we can still improve upon it. We will show that our system, Slalom, substantially outperforms this baseline when the server has access to the model $F$ (e.g., $F$ belongs to $s$ as in cloud ML APIs, or $F$ is public). Slalom performs best for verifiable inference (the setting considered in SafetyNets (Ghodsi et al., 2017)). If the TEE can run some offline data-independent preprocessing (e.g., as in Gazelle (Juvekar et al., 2018)), Slalom also outperforms the baseline for private (and verifiable) outsourced computations in a later online phase. Such a two-stage approach is viable if user data is sent at irregular intervals yet has to be processed with high throughput when available. ", + "bbox": [ + 137, + 517, + 862, + 616 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.3 OUTSOURCING OUTSOURCED DNNS AND FREIVALDS’ ALGORITHM ", + "bbox": [ + 138, + 633, + 648, + 647 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our idea for speeding up DNN inference in TEEs is to further outsource work from the TEE to a co-located faster untrusted processor. Improving upon the above baseline thus requires that the combined cost of doing work on the untrusted device and verifying it in the TEE be cheaper than evaluating the full DNN in the TEE. ", + "bbox": [ + 135, + 660, + 862, + 702 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Wahby et al. (2016; 2017) aim at this goal for arbitrary computations outsourced between co-located ASICs. The generic non-interactive proofs they use for integrity are similar to those used in SafetyNets (Ghodsi et al., 2017), which incur overheads that are too large to warrant outsourcing in our setting (e.g., Wahby et al. (2016) find that the technology gap between trusted and untrusted devices needs to be of over two decades for their scheme to break even). Similarly for privacy, standard cryptographic outsourcing protocols (e.g., (Juvekar et al., 2018)) are unusable in our setting as simply running the computation in the TEE is much more efficient (see Table 1). ", + "bbox": [ + 137, + 709, + 862, + 792 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To overcome this barrier, we design outsourcing protocols tailored to DNNs, leveraging two insights: ", + "bbox": [ + 135, + 176, + 795, + 191 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. In our setting, the TEE is co-located with the server’s faster untrusted processors, thus widening the design space to interactive outsourcing protocols with high communication but better efficiency. 2. The TEE always has knowledge of the model and can selectively outsource part of the DNN evaluation and compute others—for which outsourcing is harder—itself. ", + "bbox": [ + 137, + 195, + 862, + 260 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "DNNs are a class of functions that are particularly well suited for selective outsourcing. Indeed, non-linearities— which are hard to securely outsource (with integrity or privacy)—represent a small fraction of the computation in a DNN so we can evaluate these in the TEE (e.g., for VGG16 inference on a single CPU thread, about $1 . 5 \\%$ of the computation is spent on non-linearities). In contrast, linear operators—the main computational bottleneck in DNNs—admit for a conceptually simple yet concretely efficient secure delegation scheme, described below. ", + "bbox": [ + 137, + 263, + 862, + 334 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Integrity. We verify integrity of outsourced linear layers using variants of an algorithm by Freivalds (1977). ", + "bbox": [ + 132, + 352, + 852, + 368 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 2.1 (Freivalds). Let $A , B$ and $C$ be $n \\times n$ matrices over a field $\\mathbb { F }$ and let s be a uniformly random vector in $\\mathbb { S } ^ { n }$ , for $\\mathbb { S } \\subseteq \\mathbb { F }$ . Then, $\\operatorname* { P r } [ C s = A ( B s ) \\mid C \\neq A B ] = \\operatorname* { P r } [ ( C - A B ) s = \\mathbf { 0 } \\mid ( C - A B ) \\neq \\mathbf { 0 } ] \\leq ^ { 1 } / | \\mathbb { S } |$ . ", + "bbox": [ + 130, + 376, + 857, + 406 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The randomized check requires $3 n ^ { 2 }$ multiplications, a significant reduction (both in concrete terms and asymptotically) over evaluating the product directly. The algorithm has no false negatives and trivially extends to rectangular matrices. Independently repeating the check $k$ times yields soundness error $1 / | \\mathbb { S } | ^ { k }$ . ", + "bbox": [ + 137, + 415, + 862, + 458 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Privacy. Input privacy for outsourced linear operators could be achieved with linearly homomorphic encryption, but the overhead (see the micro-benchmarks in (Juvekar et al., 2018)) is too high to compete with our baseline (i.e., computing the function directly in the TEE would be faster than outsourcing it over encrypted data). ", + "bbox": [ + 135, + 476, + 864, + 518 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We instead propose a very efficient two-stage approach based on symmetric cryptography, i.e., an additive stream cipher. Let $f : \\mathbb { F } ^ { m } \\mathbb { F } ^ { n }$ be a linear function over a field $\\mathbb { F }$ . In an offline phase, the TEE generates a stream of one-time-use pseudorandom elements $r \\in \\mathbb { F } ^ { m }$ , and pre-computes $u = f ( r )$ . Then, in the online phase when the remote client sends an input $x$ , the TEE computes $\\mathtt { E n c } ( x ) = x + r$ over $\\mathbb { F } ^ { m }$ (i.e., a secure encryption of $x$ with a stream cipher), and outsources the computation of $f ( \\mathsf { E n c } ( x ) )$ to the faster processor. Given the result $f ( \\mathsf { E n c } ( x ) ) = f ( x + r ) = f ( x ) + f ( r ) = f ( x ) \\overset { \\cdot } { + } u$ , the TEE recovers $f ( x )$ using the pre-computed $u$ . ", + "bbox": [ + 137, + 525, + 862, + 611 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Communication. Using Freivalds’ algorithm and symmetric encryption for each linear layer in a DNN incurs high interaction and communication between the TEE and untrusted co-processor (e.g., over 50MB per inference for VGG16, see Table 3). This would be prohibitive if they were not co-located. There are protocols with lower communication than repeatedly using Freivalds’ ((Fiore & Gennaro, 2012; Thaler, 2013; Ghodsi et al., 2017)). Yet, these incur a high overhead on the prover in practice and are thus not suitable in our setting. ", + "bbox": [ + 137, + 627, + 862, + 699 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 SLALOM ", + "text_level": 1, + "bbox": [ + 137, + 728, + 240, + 744 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We introduce Slalom, a three-step approach for outsourcing DNNs from a TEE to an untrusted but faster device: (1) Inputs and weights are quantized and embedded in a field $\\mathbb { F }$ ; (2) Linear layers are outsourced and verified using Freivalds’ algorithm; (3) Inputs of linear layers are encrypted with a pre-computed pseudorandom stream to guarantee privacy. Figure 1 shows two Slalom variants, one to achieve integrity, and one to also achieve privacy. ", + "bbox": [ + 135, + 765, + 864, + 821 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We focus on feed-forward networks with fully connected layers, convolutions, separable convolutions, pooling layers and activations. Slalom can be extended to other architectures (e.g., residual networks, see Section 4.3). ", + "bbox": [ + 133, + 829, + 861, + 858 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/68e3dc2240594a3fa26a194d36ba32950ab09ff2dc6712e779ab23c590da496f.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Slalomwith integrity TEE(F,x1) S(F)
for i∈ [1,n] do
yi=xiWi
xi+1=σ(yi)
1...yn
fori∈[1,n] do assert Freivalds(yi,x,W)
xi+1=σ(yi)
return yn
", + "bbox": [ + 137, + 174, + 477, + 325 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/e4e070963d356a31639df7aca4a78879b76895ffe4fc201b38e685066e487465.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Slalom with integrity& privacy TEE(F,x1) S(F)
Preproc: for i∈[1,n] dori ← Fmi,ui =riWi
for i∈[1,n] do 主 xi=xi+ri
起 yi=xW
yi=yi-ui
assert Freivalds(yi,xi,Wi)
xi+1=σ(yi) return yn
", + "bbox": [ + 495, + 174, + 854, + 327 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 QUANTIZATION ", + "text_level": 1, + "bbox": [ + 137, + 420, + 285, + 434 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The techniques we use for integrity and privacy (Freivalds’ algorithm and stream ciphers) work over a field $\\mathbb { F }$ . We thus quantize all inputs and weights of a DNN to integers, and embed these integers in the field $\\mathbb { Z } _ { p }$ of integers modulo a prime $p$ (where $p$ is larger than all values computed in a DNN evaluation, so as to avoid wrap-around). ", + "bbox": [ + 137, + 446, + 864, + 489 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As in (Gupta et al., 2015), we convert floating point numbers $x$ to a fixed-point representation as $\\tilde { x } = \\mathtt { F P } ( x ; l ) : =$ $\\mathtt { r o u n d } ( 2 ^ { l } \\cdot x ) .$ . For a linear layer with kernel $W$ and bias $b$ , we define integer parameters $\\tilde { W } = \\mathrm { F P } ( W , l ) , \\tilde { b } =$ $\\mathrm { E P } ( b , 2 l )$ . After applying the layer to a quantized input $\\tilde { x }$ , we scale the output by $2 ^ { - l }$ and re-round to an integer. ", + "bbox": [ + 137, + 496, + 864, + 540 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For efficiency reasons, we perform integer arithmetic using floats (so-called fake quantization), and choose $p < 2 ^ { 2 4 }$ to avoid loss of precision (we use $p = 2 ^ { 2 4 } - 3 )$ . For the models we evaluate, setting $l = 8$ for all weights and inputs ensures that all DNN values are bounded by $2 ^ { 2 4 }$ , with less than a $0 . 5 \\%$ drop in accuracy (see Table 3). When performing arithmetic modulo $p$ (e.g., for Freivalds’ algorithm or when computing on encrypted data), we use double-precision floats, to reduce the number of modular reductions required (details are in Appendix F). ", + "bbox": [ + 135, + 546, + 862, + 617 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 VERIFYING COMMON LINEAR OPERATORS ", + "text_level": 1, + "bbox": [ + 137, + 636, + 475, + 650 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now describe Slalom’s approach to verifying the integrity of outsourced linear layers. We describe these layers in detail in Appendix D and summarize this section’s results in Table 2. ", + "bbox": [ + 133, + 662, + 862, + 691 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Freivalds’ Algorithm for Batches. The most direct way of applying Freivalds’ algorithm to arbitrary linear layers of a DNN is by exploiting batching. Any linear layer $f ( x )$ from inputs of size $m$ to outputs of size $n$ can be represented (with appropriate reshaping) as $f ( x ) = x ^ { \\top } W$ for a (often sparse and implicit) $m \\times n$ matrix $W$ ", + "bbox": [ + 133, + 703, + 862, + 746 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For a batch $X$ of size $B$ , we can outsource $f ( X )$ and check that the output $Y$ satisfies $f ( s ^ { \\top } X ) = s ^ { \\top } Y$ , for a random vector $s$ (we are implicitly applying Freivalds to the matrix product $X W = Y$ ). As the batch size $B$ grows, the cost of evaluating $f$ is amortized and the total verification cost is $| X | + | Y | + \\mathsf { c o s t } _ { f }$ multiplications (i.e., we approach one operation per input and output). Yet, as we show in Section 4.3, while batched verification is worthwhile for processors with larger memory, it is prohibitive in SGX enclaves due to the limited PRM. ", + "bbox": [ + 137, + 751, + 862, + 821 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For full convolutions (and pointwise convolutions), a direct application of Freivalds’ check is worthwhile even for single-element batches. For $f ( x ) = \\mathbf { C o n v } ( x , W )$ and purported output $y$ , we can sample a random vector $s$ of ", + "bbox": [ + 133, + 828, + 862, + 858 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/8093e25c85b772cb9b41d04b60e16481ee83a0c5698d0450951e485d347eae61.jpg", + "table_caption": [ + "Table 2: Complexity (number of multiplications) for evaluating and verifying linear functions. The layers are “Fully Connected”, ”Convolution”, ”Depthwise Convolution” and ”Pointwise Convolution”, defined in Appendix D. Each layer $f$ has an input $x$ , output $y$ and kernel $W$ . We assume a batch size of $B \\geq 1$ . " + ], + "table_footnote": [], + "table_body": "
Layer|x|,|yl|W|costf (B=1)Batched verificationWith preproc.
FChin,houthin·hout|a|·lyB·(lx|+lyl)+cost fB·(lx|+lyl)
Convh·w·Cin,h·w·Coutk² · Cin ·Coutal·k2 · CoutB·(lx|+lyl)+Cin·Cout+ |x| · k²B·(|xl+lyl)
Depth.Convh·w·Cin,h·w·Cink2.Cinlac|.k²B·(lx|+lyl)+cost fB·(|x|+ lyl)
Point. Convh·w ·Cin,h · w· CoutCin·Cout|c| · CoutB·(x|+lyl)+Cin·CoutB·(lx|+|yl)
", + "bbox": [ + 140, + 220, + 854, + 290 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "dimension $c _ { \\mathrm { o u t } }$ (the number of output channels), and check that $\\operatorname { C o n v } ( x , W s ) = y s$ (with appropriate reshaping). \nFor a batch of inputs $X$ , we can also apply Freivalds’ algorithm twice to reduce both $W$ and $X$ . ", + "bbox": [ + 135, + 309, + 861, + 338 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Preprocessing. We now show how to obtain an outsourcing scheme for linear layers that has optimal verification complexity (i.e., $| x | + | y |$ operations) for single-element batches and arbitrary linear operators, while at the same time compressing the DNN’s weights (a welcome property in our memory-limited TEE model). ", + "bbox": [ + 135, + 347, + 862, + 390 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We leverage two facts: (1) DNN weights are fixed at inference time, so part of Freivalds’ check can be precomputed; (2) the TEE can keep secrets from the host $s$ , so the random values $s$ can be re-used across layers or inputs (if we run Freivalds’ check $n$ times with the same secret randomness, the soundness errors grows at most by a factor $n$ ). Our verification scheme with preprocessing follows from a reformulation of Lemma (2.1): ", + "bbox": [ + 135, + 396, + 862, + 453 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Lemma 3.1. Let $f : \\mathbb { F } ^ { m } \\mathbb { F } ^ { n }$ be a linear operator, $f ( x ) : = x ^ { \\top } W$ . Let s be uniformly random in $\\mathbb { S } ^ { n }$ , for $\\mathbb { S } \\subseteq \\mathbb { F }$ , and let $\\tilde { s } : = \\nabla F _ { x } ( s ) = W s$ . For any $\\boldsymbol { x } \\in \\mathbb { F } ^ { m }$ , $y \\in \\mathbb { F } ^ { n }$ , we have $\\operatorname* { P r } \\left[ y ^ { \\top } s = x ^ { \\top } { \\tilde { s } } \\mid y \\neq f ( x ) \\right] \\leq { \\boldsymbol { \\mathrm { 1 } } } / { \\boldsymbol { \\mathrm { | \\mathbb { S } } } } { \\boldsymbol { \\mathrm { | \\nabla } } }$ . ", + "bbox": [ + 133, + 455, + 857, + 488 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The check requires $| x | + | y |$ multiplications, and storage for $s$ and ${ \\tilde { s } } : = W s$ (of size $| x |$ and $| y | ,$ ). To save space, we can reuse the same random $s$ for every layer. The memory footprint of a model is then equal to the size of the inputs of all its linear layers (e.g., for VGG16 the footprint is reduced from 550MB to 36MB, see Table 3). ", + "bbox": [ + 135, + 500, + 864, + 542 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 INPUT PRIVACY ", + "text_level": 1, + "bbox": [ + 137, + 553, + 287, + 568 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To guarantee privacy of the client’s inputs, we use precomputed blinding factors for each outsourced computation, as described in Section 2.3. The TEE uses a cryptographic Pseudo Random Number Generator (PRNG) to generate blinding factors. The precomputed “unblinding factors” are encrypted and stored in untrusted memory or disk. In the online phase, the TEE regenerates the blinding factors using the same PRNG seed, and uses the precomputed unblinding factors to decrypt the output of the outsourced linear layer. ", + "bbox": [ + 137, + 579, + 864, + 650 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This blinding process incurs several overheads: (1) the computations on the untrusted device have to be performed over $\\mathbb { Z } _ { p }$ so we use double-precision arithmetic. (2) The trusted and untrusted processors exchange data in-between each layer, rather than at the end of a full inference pass. (3) The TEE has to efficiently load precomputed unblinding factors, which requires either a large amount of RAM, or a fast access to disk (e.g., a PCIe SSD). ", + "bbox": [ + 137, + 655, + 862, + 713 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Slalom’s security is given by the following results. Formal definitions and proofs are in Appendix B. Let negl be a negligible function (for any integer $c > 0$ there exists an integer $N _ { c }$ such that for all $x > N _ { c }$ , $| \\mathrm { n e g l } ( x ) | < \\bar { 1 } / { x ^ { c } } )$ ", + "bbox": [ + 135, + 718, + 861, + 747 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.2. Let Slalom be the protocol from Figure 1 (right), where $F$ is an $n$ -layer DNN, and Freivalds’ algorithm is repeated k times per layer with random vectors drawn from $\\mathbb { S } \\subseteq \\mathbb { F }$ . Assume all random values are generated using a secure PRNG with security parameter $\\lambda$ . Then, Slalom is a secure outsourcing scheme for $F$ between a TEE and an untrusted co-processor $s$ with privacy and $t$ -integrity for $t = n / \\vert \\mathbb { S } \\vert ^ { k } - \\mathrm { n e g l } ( \\lambda )$ . ", + "bbox": [ + 137, + 751, + 864, + 808 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Corollary 3.3. Assuming the TEE is secure (i.e., it acts as a trusted third party hosted by $s$ ), Slalom is a secure outsourcing scheme between a remote client $\\mathcal { C }$ and server $\\boldsymbol { S }$ with privacy and $t$ -integrity for $t = { \\boldsymbol { \\mathscr { n } } } / { \\boldsymbol { | \\mathbb { S } } } { \\boldsymbol { \\vert \\sp k - \\mathrm { n e g l } } } ( \\lambda )$ . If the model $F$ is the property of $\\boldsymbol { S }$ , the scheme further satisfies model privacy. ", + "bbox": [ + 137, + 813, + 862, + 854 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EMPIRICAL EVALUATION ", + "text_level": 1, + "bbox": [ + 138, + 175, + 377, + 191 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate Slalom on real Intel SGX hardware, on micro-benchmarks and a sample application (ImageNet inference with VGG16, MobileNet and ResNet models). Our aim is to show that, compared to a baseline that runs inference fully in the TEE, outsourcing linear layers increases performance without sacrificing security. ", + "bbox": [ + 137, + 207, + 862, + 250 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 IMPLEMENTATION ", + "text_level": 1, + "bbox": [ + 137, + 268, + 303, + 282 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As enclaves cannot access most OS features (e.g., multi-threading, disk and driver IO), porting a large framework such as TensorFlow or Intel’s MKL-DNN to SGX is hard. Instead, we designed a lightweight $\\mathrm { C } { + + }$ library for feed-forward networks based on Eigen, a linear-algebra library which TensorFlow uses as a CPU backend. Our library implements the forward pass of DNNs, with support for dense layers, standard and separable convolutions, pooling, and activations. When run on a native CPU (without SGX), its performance is comparable to TensorFlow on CPU (compiled with AVX). Our code is available at https://github.com/ftramer/slalom. ", + "bbox": [ + 137, + 295, + 862, + 378 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Slalom performs arithmetic over $\\mathbb { Z } _ { p }$ , for $p = 2 ^ { 2 4 } - 3$ . For integrity, we apply Freivalds’ check twice to each layer $k = 2 ,$ ), with random values from $\\mathbb { S } = [ - 2 ^ { 1 9 } , 2 ^ { 1 9 } ]$ , to achieve 40 bits of statistical soundness per layer (see Appendix F for details on the selection of these parameters). For a 50-layer DNN, $s$ has a chance of less than 1 in 22 billion of fooling the TEE on any incorrect DNN evaluation (a slightly better guarantee than in SafetyNets). For privacy, we use AES-CTR and AES-GCM to generate, encrypt and authenticate blinding factors. ", + "bbox": [ + 137, + 385, + 862, + 457 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 SETUP ", + "text_level": 1, + "bbox": [ + 137, + 474, + 223, + 488 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use an Intel Core i7-6700 Skylake 3.40GHz processor with 8GB of RAM, a desktop processor with SGX support. The outsourced computations are performed on a co-located Nvidia TITAN XP GPU. Due to a lack of native internal multi-threading in SGX, we run our TEE in a single CPU thread. We discuss challenges for efficient parallelization in Appendix H. We evaluate Slalom on the following workloads: ", + "bbox": [ + 137, + 501, + 862, + 558 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• Synthetic benchmarks for matrix products, convolutions and separable convolutions, where we compare the enclave’s running time for computing a linear operation to that of solely verifying the result. • ImageNet (Deng et al., 2009) classification with VGG16 (Simonyan & Zisserman, 2014), MobileNet (Howard et al., 2017), and ResNet He et al. (2016) models (with fused Batch Normalization layers when applicable). ", + "bbox": [ + 140, + 563, + 862, + 623 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "MobileNet, a model tailored for low compute devices, serves as a worst-case benchmark for Slalom, as the model’s design aggressively minimizes the amount of computation performed per layer. We also consider a “fused” variant of MobileNet with no activation between depthwise and pointwise convolutions. Removing these activations improves convergence and accuracy (Chollet, 2017; Sheng et al., 2018), while also making the network more outsourcing-friendly (i.e., it is possible to verify a separable convolution in a single step). ", + "bbox": [ + 135, + 630, + 862, + 700 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our evaluation focuses on throughput (number of forward passes per second). We also discuss energy efficiency in Appendix C to account for hardware differences between our baseline (TEE only) and Slalom $\\mathrm { ( T E E + G P U ) }$ ). ", + "bbox": [ + 135, + 705, + 861, + 734 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 RESULTS ", + "text_level": 1, + "bbox": [ + 137, + 753, + 240, + 767 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Micro-Benchmarks. Our micro-benchmark suite consists of square matrix products of increasing dimensions, convolutional operations performed by VGG16, and separable convolutions performed by MobileNet. In all cases, the data is pre-loaded inside an enclave, so we only measure the in-enclave execution time. Figure 2 plots the relative speedups of various verification strategies over the cost of computing the linear operation directly. In all cases, the baseline computation is performed in single-precision floating point, and the verification algorithms repeat Freivalds’ check so as to attain at least 40 bits of statistical soundness. ", + "bbox": [ + 137, + 773, + 862, + 857 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e479efd0e691ddcb877c1c08f199812ca305e3227b9c63d4af690b286c7b0522.jpg", + "image_caption": [ + "Figure 2: Micro benchmarks on Intel SGX. We plot the relative speedup of verifying the result of a linear operator compared to computing it entirely in the enclave. The dotted line shows the throughput obtained for a direct computation. “Fused” separable convolutions contain no intermediate activation. " + ], + "image_footnote": [], + "bbox": [ + 138, + 174, + 856, + 314 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For square matrices of dimensions up to 2048, verifying an outsourced result is $4 \\times$ to $8 \\times$ faster than computing it. For larger matrices, we exceed the limit of SGX’s DRAM, so the enclave resorts to expensive paging which drastically reduces performance both for computation and verification. ", + "bbox": [ + 137, + 372, + 862, + 415 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For convolutions (standard or separable), we achieve large savings with outsourcing if Freivalds’ algorithm is applied with preprocessing. The savings get higher as the number of channels increases. Without preprocessing, Freivalds’ algorithm results in savings when $c _ { \\mathrm { o u t } }$ is large. Due to SGX’s small PRM, batched verification is only effective for operators with small memory footprints. As expected, “truly” separable convolutions (with no intermediate non-linearity) are much faster to verify, as they can be viewed as a single linear operator. ", + "bbox": [ + 137, + 421, + 862, + 492 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Verifiable Inference. Figure 3 shows the throughout of end-to-end forward passes in two neural networks, VGG16 and MobileNet. For integrity, we compare the secure baseline (executing the DNN fully in the enclave) to two variants of the Slalom algorithm in Figure 1. The first (in red) applies Freivalds’ algorithm “on-the-fly”, while the second more efficient variant (in orange) pre-computes part of Freivalds’ check as described in Section 3.2. ", + "bbox": [ + 137, + 502, + 862, + 559 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The VGG16 network is much larger (500MB) than SGX’s PRM. As a result, there is a large overhead on the forward pass and verification without preprocessing. If the enclave securely stores preprocessed products $W r$ for all network weights, we drastically reduce the memory footprint and achieve up to a $2 0 . 3 \\times$ increase in throughput. We also ran the lower-half of the VGG16 network (without the fully connected layers), a common approach for extracting features for transfer learning or object recognition (Liu et al., 2016). This part fits in the PRM, and we thus achieve higher throughput for in-enclave forward passes and on-the-fly verification. ", + "bbox": [ + 137, + 565, + 862, + 650 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For MobileNet, we achieve between $3 . 6 \\times$ and $6 . 4 \\times$ speedups when using Slalom for verifiable inference (for the standard or “fused” model, respectively). The speedups are smaller than for VGG16, as MobileNet performs much fewer operations per layer (verifying a linear layer requires computing at least two multiplications for each input and output. The closer the forward pass gets to that lower-bound, the less we can save by outsourcing). ", + "bbox": [ + 137, + 656, + 862, + 713 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Private Inference. We further benchmark the cost of private DNN inference, where inputs of outsourced linear layers are additionally blinded. Blinding and unblinding each layer’s inputs and outputs is costly, especially in SGX due to the extra in-enclave memory reads and writes. Nevertheless, for VGG16 and the fused MobileNet variant without intermediate activations, we achieve respective speedups of $1 3 . 0 \\times$ and $5 . 0 \\times$ for private outsourcing (in black in Figure 3), and speedups of $1 0 . 7 \\times$ and $4 . 1 \\times$ when also ensuring integrity (in purple). For this benchmark, the precomputed unblinding factor are stored in untrusted memory. ", + "bbox": [ + 137, + 724, + 862, + 809 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We performed the same experiments on a standard CPU (i.e., without SGX) and find that Slalom’s improvements are even higher in non-resource-constrained or multi-threaded environments (see Appendix G-H). Slalom’s improvements over the baseline also hold when accounting for energy efficiency (see Section C). ", + "bbox": [ + 138, + 815, + 861, + 857 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/5df56a4274ed6e233912c5524664ae350b78563790ca22e417ac562aad5df2f1.jpg", + "image_caption": [ + "Figure 3: Verifiable and private inference with Intel SGX. We show results for VGG16, VGG16 without the fully connected layers, MobileNet, and a fused MobileNet variant with no intermediate activation for separable convolutions. We compare the baseline of fully executing the DNN in the enclave (blue) to different secure outsourcing schemes: integrity with Freivalds (red); integrity with Freivalds and precomputed secrets (yellow); privacy only (black); privacy and integrity (purple). " + ], + "image_footnote": [], + "bbox": [ + 138, + 174, + 861, + 272 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/9ceb8a1cf9a5c8503bd599b1f66c4bf349f3d490915536e0c23e1b7269842915.jpg", + "image_caption": [ + "Figure 4: Secure outsourcing of ResNet models with Intel SGX. We compare the baseline of fully executing the DNN in the enclave (blue) to secure outsourcing with integrity (yellow) and privacy and integrity (purple). " + ], + "image_footnote": [], + "bbox": [ + 137, + 371, + 862, + 457 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Extending Slalom to Deep Residual Networks. The Slalom algorithm in Figure 1 and our evaluations above focus on feed-forward architectures. Extending Slalom to more complex DNNs is quite simple. To illustrate, we consider the family of ResNet models (He et al., 2016), which use residual blocks $f ( x ) = \\sigma ( f _ { 1 } ( x ) + f _ { 2 } ( x ) )$ that merge two feed-forward “paths” $f _ { 1 }$ and $f _ { 2 }$ into a final activation $\\sigma$ . To verify integrity of $f ( x )$ , the TEE simply verifies all linear layers in $f _ { 1 }$ and $f _ { 2 }$ and computes $\\sigma$ directly. For privacy, the TEE applies the interactive Slalom protocol in Figure 1 (right) in turn to $f _ { 1 }$ and $f _ { 2 }$ , and then computes $\\sigma$ . The results for the privacy-preserving Slalom variant in Figure 4 use a preliminary implementation that performs all required operations—and thus provides meaningful performance numbers—but without properly constructed unblinding factors. ", + "bbox": [ + 137, + 522, + 862, + 635 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We use the ResNet implementation from Keras Chollet et al. (2015), which contains a pre-trained 50-layer variant. For this model, we find that our quantization scheme results in less than a $0 . 5 \\%$ decrease in accuracy (see Table 3). For other variants (i.e., with 18, 34, 101 and 152 layers) we compute throughput on untrained models. Figure 4 shows benchmarks for different ResNet variants when executed fully in the enclave (our baseline) as well as secure outsourcing with integrity or privacy and integrity. For all models, we achieve $6 . 6 \\times$ to $1 4 . 4 \\times$ speedups for verifiable inference and $4 . 4 \\times$ to $9 . 0 \\times$ speedups when adding privacy. ", + "bbox": [ + 135, + 641, + 862, + 726 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Comparing results for different models is illustrative of how Slalom’s savings scale with model size and architectural design choices. The 18 and 34-layer ResNets use convolutions with $3 \\times 3$ kernels, whereas the larger models mainly use pointwise convolutions. As shown in Table 2 verifying a convolution is about a factor $k ^ { 2 } \\cdot c _ { \\mathrm { o u t } }$ than computing it, which explains the higher savings for models that use convolutions with large kernel windows. When adding more layers to a model, we expect Slalom’s speedup over the baseline to remain constant (e.g., if we duplicate each layer, the baseline computation and the verification should both take twice as long). Yet we find that Slalom’s speedups usually increase as layers get added to the ResNet architecture. This is because the deeper ResNet variants are obtained by duplicating layers towards the end of the pipeline, which have the largest number of channels and for which Slalom achieves the highest savings. ", + "bbox": [ + 135, + 731, + 862, + 858 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CHALLENGES FOR VERIFIABLE AND PRIVATE TRAINING ", + "text_level": 1, + "bbox": [ + 137, + 174, + 637, + 191 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Our techniques for secure outsourcing of DNN inference might also apply to DNN training. Indeed, a backward pass consists of similar linear operators as a forward pass, and can thus be verified with Freivalds’ algorithm. Yet, applying Slalom to DNN training is challenging, as described below, and we leave this problem open. ", + "bbox": [ + 137, + 207, + 864, + 250 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "• Quantizing DNNs for training is harder than for inference, due to large changes in weight magnitudes (Micikevicius et al., 2018). Thus, a more flexible quantization scheme than the one we used would be necessary. • Because the DNN’s weights change during training, the same preprocessed random vectors for Freivalds’ check cannot be re-used indefinitely. The most efficient approach would presumably be to train with very large batches than can then be verified simultaneously. • Finally, the pre-computation techniques we employ for protecting input privacy do not apply for training, as the weights change after every processed batch. Moreover, Slalom does not try to hide the model weights from the untrusted processor, which might be a requirement for private training. ", + "bbox": [ + 143, + 253, + 862, + 375 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 137, + 393, + 282, + 410 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This paper has studied the efficiency of evaluating a DNN in a Trusted Execution Environment (TEE) to provide strong integrity and privacy guarantees. We explored new approaches for segmenting a DNN evaluation to securely outsource work from a trusted environment to a faster co-located but untrusted processor. ", + "bbox": [ + 137, + 425, + 862, + 468 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We designed Slalom, a framework for efficient DNN evaluation that outsources all linear layers from a TEE to a GPU. Slalom leverage Freivalds’ algorithm for verifying correctness of linear operators, and additionally encrypts inputs with precomputed blinding factors to preserve privacy. Slalom can work with any TEE and we evaluated its performance using Intel SGX on various workloads. For canonical DNNs (VGG16, MobileNet and ResNet variants), we have shown that Slalom boosts inference throughput without compromising security. ", + "bbox": [ + 137, + 474, + 861, + 545 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Securely outsourcing matrix products from a TEE has applications in ML beyond DNNs (e.g., non negative matrix factorization, dimensionality reduction, etc.) We have also explored avenues and challenges towards applying similar techniques to DNN training, an interesting direction for future work. Finally, our general approach of outsourcing work from a TEE to a faster co-processor could be applied to other problems which have fast verification algorithms, e.g., those considered in (McConnell et al., 2011; Zhang et al., 2014). 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", + "bbox": [ + 132, + 154, + 866, + 861 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 132, + 178, + 866, + 566 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A DETAILS ON INTEL SGX SECURITY ", + "text_level": 1, + "bbox": [ + 137, + 588, + 470, + 604 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "SGX enclaves isolate execution of a program from all other processes on a same host, including a potentially malicious OS. In particular, enclave memory is fully encrypted and authenticated. When a word is read from memory into a CPU register, a Memory Management Engine handles the decryption (Costan & Devadas, 2016). ", + "bbox": [ + 137, + 619, + 862, + 662 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "While SGX covers many software and hardware attack vectors, there is a large and prominent class of sidechannel attacks that it explicitly does not address (Costan & Devadas, 2016; Tramèr et al., 2017). In the past years, many attacks have been proposed, with the goal of undermining privacy of enclave computations $\\mathrm { { X u } }$ et al., 2015; Brasser et al., 2017; Moghimi et al., 2017; Götzfried et al., 2017; Van Bulck et al., 2017; Lee et al., 2017). Most of these attacks rely on data dependent code behavior in an enclave (e.g., branching or memory access) that can be partially observed by other processes running on the same host. These side-channels are a minor concern for the DNN computations considered in this paper, as the standard computations in a DNN are data-oblivious (i.e., the same operations are applied regardless of the input data) (Ohrimenko et al., 2016). ", + "bbox": [ + 137, + 669, + 862, + 781 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The recent Spectre attacks on speculative execution (Kocher et al., 2018) also prove damaging to SGX (as well as to most other processors), as recently shown (Chen et al., 2018; Dall et al., 2018; Van Bulck et al., 2018). Mitigations for these side-channel attacks are being developed (Shinde et al., 2016; Shih et al., 2017; Chen et al., 2017; Intel Corp., 2018) but a truly secure solution might require some architectural changes, e.g., as in the proposed Sanctum processor (Costan et al., 2016). ", + "bbox": [ + 137, + 786, + 862, + 857 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We refrain from formally modeling SGX’s (or other TEE’s) security in this paper, as Slalom is mostly concerned with outsourcing protocols wherein the TEE acts as a client. We refer the interested reader to (Pass et al., 2017; Fisch et al., 2017; Subramanyan et al., 2017) for different attempts at such formalisms. ", + "bbox": [ + 135, + 176, + 864, + 219 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B FORMAL SECURITY DEFINITIONS AND PROOFS ", + "text_level": 1, + "bbox": [ + 137, + 241, + 563, + 257 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We define a secure outsourcing scheme, between a client $\\mathcal { C }$ and a server $s$ , for a DNN $F ( x ) : \\mathcal { X } \\mathcal { Y }$ from some family $\\mathcal { F }$ (e.g., all DNNs of a given size). We first assume that the model $F$ is known to both $\\mathcal { C }$ and $s$ : ", + "bbox": [ + 135, + 273, + 862, + 303 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Definition B.1 (Secure Outsourcing Schemes). A secure outsourcing scheme consists of an offline preprocessing algorithm Preproc, as well as an interactive online protocol Outsource $\\langle \\mathcal { C } , \\mathcal { S } \\rangle$ , defined as follows: ", + "bbox": [ + 135, + 306, + 859, + 335 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• $\\mathsf { s t } \\gets \\mathsf { P r e p r o c } ( F , 1 ^ { \\lambda } )$ : The preprocessing algorithm is run by $\\mathcal { C }$ and generates some data-independent state $\\mathtt { s t }$ (e.g., cryptographic keys or precomputed values to accelerate the online outsourcing protocol.) • $\\mathcal { V } \\cup \\{ \\bot \\} \\mathsf { O u t s o u r c e } \\langle \\mathcal { C } ( F , x , \\mathrm { s t } ) , \\mathcal { S } ( F ) \\rangle \\mathrm { : }$ : The online outsourcing protocol is initiated by $\\mathcal { C }$ with inputs $( F , x , s \\ t )$ . At the end of the protocol, $\\mathcal { C }$ either outputs a value $y \\in \\mathcal { V }$ or aborts (i.e., $\\mathcal { C }$ outputs $\\perp$ ). ", + "bbox": [ + 138, + 340, + 862, + 407 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The properties that we may require from a secure outsourcing scheme are: ", + "bbox": [ + 140, + 412, + 622, + 428 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Correctness: For any $\\begin{array} { r l r l } { F } & { { } \\in } & { { \\mathcal { F } } } \\end{array}$ and $\\begin{array} { r l r l } { x } & { { } \\in } & { { \\mathcal { X } } } \\end{array}$ , running $\\begin{array} { r l r } { \\mathsf { s t } } & { { } \\gets } & { \\mathsf { P r e p r o c } ( F , 1 ^ { \\lambda } ) } \\end{array}$ and $y $ Outsource $\\langle \\mathcal { C } ( F , x , \\mathrm { s t } ) , \\mathcal { S } ( F ) \\rangle$ yields $y = F ( x )$ . ", + "bbox": [ + 143, + 431, + 856, + 463 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• t-Integrity: For any $F \\in { \\mathcal { F } }$ , input $x \\in \\mathcal { X }$ and probabilistic polynomial-time adversary $S ^ { * }$ , the probability that $\\tilde { y } = \\mathsf { O u t s o u r c e } \\langle \\mathcal { C } ( F , x , \\mathsf { s t } ) , \\mathcal { S } ^ { \\ast } ( F ) \\rangle$ and $\\tilde { y } \\notin \\{ F ( x ) , \\bot \\}$ is less than $t$ . ", + "bbox": [ + 145, + 468, + 857, + 500 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Input privacy: For any $F \\in { \\mathcal { F } }$ , inputs $x , x ^ { \\prime } \\in { \\mathcal { X } }$ and probabilistic poly-time adversary ${ \\boldsymbol { S } } ^ { * }$ , the views of $S ^ { * }$ in Outsource $\\langle \\mathcal { C } ( F , x , \\mathsf { s t } ) , \\mathcal { S } ^ { * } ( F ) \\rangle$ and Outsource $\\langle \\mathcal { C } ( F , \\overline { { x ^ { \\prime } } } , \\mathrm { { s t } } ) , S ^ { \\ast } ( \\mathrm { { \\bar { F } } } ) \\rangle$ are computationally indistinguishable. ", + "bbox": [ + 147, + 506, + 857, + 535 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Efficiency: The online computation of $\\mathcal { C }$ in Outsource should be less than the cost for $\\mathcal { C }$ to evaluate $F \\in { \\mathcal { F } }$ . ", + "bbox": [ + 142, + 541, + 862, + 558 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Model Privacy. In some applications a secure outsourcing scheme may also require to hide the model $F$ from either $s$ or $\\mathcal { C }$ (in which case that party would obviously not take $F$ as input in the above scheme). ", + "bbox": [ + 137, + 573, + 862, + 602 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Privacy with respect to an adversarial server ${ \\boldsymbol { S } } ^ { * }$ (which Slalom does not provide), is defined as the indistinguishability of $S ^ { * }$ ’s views in Outsource $\\langle \\mathcal { C } ( F , x , \\mathsf { s t } ) , \\mathcal { S } ^ { * } \\rangle$ and Outsource $\\langle \\mathcal { C } ( F ^ { \\prime } , x , \\mathrm { s t } ) , S ^ { * } \\rangle$ for any $F , F ^ { \\prime } \\in \\mathcal { F }$ . ", + "bbox": [ + 135, + 608, + 862, + 637 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "As noted in Section 2.1, a meaningful model-privacy guarantee with respect to $\\mathcal { C }$ requires that $s$ first commit to a specific DNN $F$ , and then convinces $\\mathcal { C }$ that her outputs were produced with the same model as all other clients’. We refer the reader to Canetti et al. (2002) for formal definitions for such commit-and-prove schemes, and to Tramèr et al. (2017) who show how to trivially instantiate them using a TEE. ", + "bbox": [ + 137, + 643, + 862, + 700 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof of Theorem 3.2. Let $\\mathrm { s t ~ ~ }$ Preproc and Outsource $\\langle \\mathrm { T E E } ( F , x , \\mathrm { s t } ) , S \\rangle$ be the outsourcing scheme defined in Figure 1 (right). We assume that all random values sampled by the TEE are produced by a secure cryptographically secure pseudorandom number generator (PRNG) (with elements in $\\mathbb { S } \\subseteq \\mathbb { F }$ for the integritycheck vectors $s$ used in Freivalds’ algorithm, and in $\\mathbb { F }$ for the blinding vectors $r _ { i }$ ). ", + "bbox": [ + 137, + 709, + 862, + 767 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We first consider integrity. Assume that the scheme is run with input $x _ { 1 }$ and that the TEE outputs $y _ { n }$ . We will bound $\\operatorname* { P r } [ y _ { n } \\neq F ( x _ { 1 } ) \\ | \\ { \\dot { y } } _ { n } \\neq \\bot ]$ . By the security of the PRNG, we can replace the vectors $s$ used in Freivalds’ algorithm by truly uniformly random values in $\\mathbb { S } \\subseteq \\mathbb { F }$ , via a simple hybrid argument. For the $i$ -th linear layer, with operator $W _ { i }$ , input $x _ { i }$ and purported output $y _ { i }$ , we then have that $y _ { i } \\neq x _ { i } W _ { i }$ with probability at most $1 / | \\mathbb { S } | ^ { k }$ . By a simple union bound, we thus have that $\\operatorname* { P r } [ y _ { n } \\neq F ( x _ { 1 } ) ] \\leq { n } / { | \\mathbb { S } | ^ { k } } - \\operatorname { n e g l } ( \\lambda ) .$ . Note that this bound holds even if the same (secret) random values $s$ are re-used across layers. ", + "bbox": [ + 137, + 773, + 862, + 858 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For privacy, consider the views of an adversary $S ^ { * }$ when Slalom is run with inputs $x _ { 1 }$ and $x _ { 1 } ^ { \\prime }$ . Again, by the security of the PRNG, we consider a hybrid protocol where we replace the pre-computed blinding vectors $r _ { i }$ by truly uniformly random values in $\\mathbb { F }$ . In this hybrid protocol, $\\tilde { x } _ { i } = x _ { i } + r _ { i }$ is simply a “one-time-pad” encryption of $x _ { i }$ over the field $\\mathbb { F }$ , so $S ^ { * }$ ’s views in both executions of the hybrid protocol are equal (information theoretically). Thus, $S ^ { * }$ ’s views in both executions of the original protocol are computationally indistinguishable. □ ", + "bbox": [ + 137, + 176, + 862, + 247 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof of Corollary 3.3. The outsourcing protocol between the remote client $\\mathcal { C }$ and server $s$ hosting the TEE is simply defined as follows (we assume the model belongs to $s$ ): ", + "bbox": [ + 137, + 261, + 859, + 289 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• $\\mathsf { s t } \\gets \\mathsf { P r e p r o c ( ) }$ : $\\mathcal { C }$ and the TEE setup a secure authenticated communication channel, using the TEE’s remote attestation property. The TEE receives the model $F$ from $s$ and initializes the Slalom protocol. ", + "bbox": [ + 140, + 295, + 859, + 324 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• Outsource $\\langle { \\mathcal { C } } ( x , { \\mathrm { s t } } ) , S ( F ) \\rangle$ : – $\\mathcal { C }$ sends $x$ to the TEE over the secure channel. – The TEE securely computes $y = F ( x )$ using Slalom. – The TEE sends $y$ (and a publicly verifiable commitment to $F$ ) to $\\mathcal { C }$ over the secure channel. ", + "bbox": [ + 145, + 329, + 785, + 393 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "If the TEE is secure (i.e., it acts as a trusted third party hosted by $s$ ), then the result follows. ", + "bbox": [ + 135, + 400, + 740, + 415 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C PERFORMANCE COMPARISON OF DNN OUTSOURCING SCHEMES ", + "text_level": 1, + "bbox": [ + 140, + 439, + 710, + 455 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We provide a brief overview of the outsourcing approaches compared in Table 1. Our baseline runs a DNN in a TEE (a single-threaded Intel SGX enclave) and can provide all the security guarantees of an ML outsourcing scheme. On a high-end GPU (an Nvidia TITAN XP), we achieve over $5 0 \\times$ higher throughput but no security. For example, for MobileNet, the enclave evaluates 16 images/sec and the GPU 900 images/sec $5 6 \\times$ higher). ", + "bbox": [ + 137, + 473, + 862, + 529 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "SafetyNets (Ghodsi et al., 2017) and Gazelle (Juvekar et al., 2018) are two representative works that achieve respectively integrity and privacy using purely cryptographic approaches (without a TEE). SafetyNets does not hide the model from either party, while Gazelle leaks some architectural details to the client. The cryptographic techniques used by these systems incur large computation and communication overheads in practice. The largest model evaluated by SafetyNets is a 4-layer TIMIT model with quadratic activations which runs at about 13 images/sec (on a notebook CPU). In our baseline enclave, the same model runs at over 3,500 images/sec. The largest model evaluated by Gazelle is an 8-layer CIFAR10 model. In the enclave, we can evaluate 450 images/sec whereas Gazelle evaluates a single image in 3.5 sec with $3 0 0 \\mathrm { M B }$ of communication between client and server. ", + "bbox": [ + 137, + 536, + 861, + 648 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A Note on Energy Efficiency. When comparing approaches with different hardware (e.g., our single-core CPU baseline versus Slalom which also uses a GPU), throughput alone is not the fairest metric. E.g., the baseline’s throughput could also be increased by adding more SGX CPUs. A more accurate comparison considers the energy efficiency of a particular approach, a more direct measure of the recurrent costs to the server $s$ . ", + "bbox": [ + 137, + 661, + 862, + 718 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For example, when evaluating MobileNet or VGG16, our GPU draws 85W of power, whereas our baseline SGX CPU draws 30W. As noted above, the GPU also achieves more than $5 0 \\times$ higher throughput, and thus is at least $1 8 \\times$ more energy efficient (e.g., measured in Joules per image) than the enclave. ", + "bbox": [ + 137, + 723, + 862, + 767 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For Slalom, we must consider the cost of running both the enclave and GPU. In our evaluations, the outsourced computations on the GPU account for at most $1 0 \\%$ of the total running time of Slalom (i.e., the integrity checks and data encryption/decryption in the enclave are the main bottleneck). Thus, the power consumption attributed to Slalom is roughly $1 0 \\% \\cdot 8 5 \\mathrm { W } + 9 0 \\% \\cdot 3 0 \\mathrm { W } = 3 5 . 5 \\mathrm { W } .$ Note that when not being in use by Slalom, the trusted CPU or untrusted GPU can be used by other tasks running on the server. As Slalom achieves $4 \\times - 2 0 \\times$ higher throughput than our baseline for the tasks we evaluate, it is also about $3 . 4 \\times - 1 7 . 1 \\times$ more energy efficient. ", + "bbox": [ + 137, + 773, + 861, + 858 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D NOTATION FOR STANDARD LINEAR OPERATORS ", + "text_level": 1, + "bbox": [ + 135, + 174, + 573, + 191 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Below we describe some common linear operators used in deep neural networks. For simplicity, we omit additive bias terms, and assume that convolutional operators preserve the spatial height and width of their inputs. Our techniques easily extend to convolutions with arbitrary strides, paddings, and window sizes. ", + "bbox": [ + 135, + 217, + 862, + 260 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For a fully-connected layer $f _ { \\mathrm { F C } }$ , the kernel $W$ has dimension $\\left( h _ { \\mathrm { i n } } \\times h _ { \\mathrm { o u t } } \\right)$ . For an input $x$ of dimension $h _ { \\mathrm { i n } }$ , we have $f _ { \\mathrm { F C } } ( \\dot { \\boldsymbol { x } } ) = \\boldsymbol { x } ^ { \\top } \\boldsymbol { W }$ . The cost of the layer is $h _ { \\mathrm { i n } } \\cdot h _ { \\mathrm { o u t } }$ multiplications. ", + "bbox": [ + 135, + 266, + 861, + 295 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A convolutional layer has kernel $W$ of size $( k \\times k \\times c _ { \\mathrm { i n } } \\times c _ { \\mathrm { o u t } } )$ . On input $x$ of size $( h \\times w \\times c _ { \\mathrm { i n } } )$ , $f _ { \\mathrm { c o n v } } ( x ) =$ $\\mathsf { C o n v } ( x ; W )$ produces an output of size $( h \\times w \\times c _ { \\mathrm { o u t } } )$ . A convolution can be seen as the combination of two linear operators: a “patch-extraction” process that transforms the input $x$ into an intermediate input $x ^ { \\prime }$ of dimension $( \\bar { h } \\cdot w , k ^ { 2 } \\cdot c _ { \\mathrm { i n } } ) $ by extracting $k \\times k$ patches, followed by a matrix multiplication with $W$ . The cost of this layer is thus $\\boldsymbol { k } ^ { 2 } \\cdot \\boldsymbol { h } \\cdot \\boldsymbol { w } \\cdot \\boldsymbol { c } _ { \\mathrm { i n } } \\cdot \\boldsymbol { c } _ { \\mathrm { o u t } }$ multiplications. ", + "bbox": [ + 137, + 301, + 862, + 372 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A separable convolution has two kernels, $W _ { 1 }$ of size $( k \\times k \\times c _ { \\mathrm { i n } } )$ and $W _ { 2 }$ of size $\\left( { c _ { \\mathrm { i n } } } \\times { c _ { \\mathrm { o u t } } } \\right)$ . On input $x$ of size $( h \\times w \\times c _ { \\mathrm { i n } } )$ , $f _ { \\mathrm { s e p - c o n v } } ( x )$ produces an output of size $( h \\times w \\times c _ { \\mathrm { o u t } } )$ , by applying a depthwise convolution $f _ { \\mathrm { d p - c o n v } } ( x )$ with kernel $W _ { 1 }$ followed by a pointwise convolution $f _ { \\mathrm { p t - c o n v } } ( x )$ with kernel $W _ { 2 }$ . The depthwise convolution consists of $c _ { \\mathrm { i n } }$ independent convolutions with filters of size $k \\times k \\times 1 \\times 1$ , applied to a single input channel, which requires $k ^ { 2 } \\cdot h \\cdot w \\cdot c _ { \\mathrm { i n } }$ multiplications. A pointwise convolution is simply a matrix product with an input of size $( h \\cdot w ) \\times c _ { \\mathrm { i n } }$ , and thus requires $h \\cdot w \\cdot c _ { \\mathrm { i n } } \\cdot c _ { \\mathrm { o u t } }$ multiplications. ", + "bbox": [ + 135, + 377, + 862, + 463 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E NEURAL NETWORK DETAILS ", + "text_level": 1, + "bbox": [ + 137, + 501, + 413, + 517 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Table 3 provides details about the two DNNs we use in our evaluation (all pre-trained models are taken from Keras Chollet et al. (2015)). We report top 1 and top 5 accuracy on ImageNet with and without the simple quantization scheme described in Section 3.1. Quantization results in at most a $0 . 5 \\%$ drop in top 1 and top 5 accuracy. More elaborate quantization schemes exist (e.g., Micikevicius et al. (2018)) that we have not experimented with in this work. ", + "bbox": [ + 137, + 542, + 862, + 613 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We report the number of model parameters, which is relevant to the memory constraints of TEEs such as Intel SGX. We also list the total size of the inputs and outputs of all the model’s linear layers, which impact the amount of communication between trusted and untrusted co-processors in Slalom, as well as the amount of data stored in the TEE when using Freivalds’ algorithm with preprocessing. ", + "bbox": [ + 137, + 619, + 862, + 676 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/0f91fa022782d0ac5f385e372dd812fdd263b672ff72a2a75427796980f9477e.jpg", + "table_caption": [ + "Table 3: Details of models used in our evaluation. Accuracies are computed on the ImageNet validation set. Pre-trained models are from Keras Chollet et al. (2015). " + ], + "table_footnote": [], + "table_body": "
AccuracyQuantized
ModelTop 1Top5Top1Top 5LayersParameters (M)Size of layer inputs/outputs (M)
VGG1671.090.070.689.516138.49.1 / 13.6
VGG16 (no top)=-1314.79.1/13.5
MobileNet70.789.670.589.5284.25.5/ 5.0
MobileNet (fused)=1154.23.6/3.1
ResNet 5076.992.476.492.25025.510.0 /10.4
", + "bbox": [ + 150, + 742, + 843, + 844 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "F MODULAR ARITHMETIC WITH FLOATING POINT OPERATIONS", + "text_level": 1, + "bbox": [ + 135, + 174, + 683, + 191 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In this section, we briefly describe how Slalom performs modular arithmetic over a field $\\mathbb { Z } _ { p }$ in the TEE, while leveraging standard floating point operations to maximize computational efficiency. The main computations in the TEE are inner products over $\\mathbb { Z } _ { p }$ for Freivalds’ check (a matrix product is itself a set of inner products). ", + "bbox": [ + 137, + 208, + 861, + 251 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Our quantization scheme (see Section 3.1) ensures that all DNN values can be represented in $\\mathbb { Z } _ { p }$ , for $p \\lessapprox 2 ^ { 2 4 }$ , which fits in a standard float. To compute inner products, we first cast elements to doubles (as a single multiplication in $\\mathbb { Z } _ { p }$ would exceed the range of integers exactly representable as floats). Single or double precision floats are preferable to integer types on Intel architectures due to the availability of much more efficient SIMD instructions, at a minor reduction in the range of exactly representable integers. ", + "bbox": [ + 137, + 258, + 862, + 329 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In our evaluation, we target a soundness error of $2 ^ { - 4 0 }$ for each layer. This leads to a tradeoff between the number of repetitions $k$ of Freivalds’ check, and the size of the set $\\mathbb { S }$ from which we draw random values. One check with $| \\mathbb { S } | = 2 ^ { 4 0 }$ is problematic, as multiplying elements in $\\mathbb { Z } _ { p }$ and $\\mathbb { S }$ can exceed the range of integers exactly representable as doubles $( 2 ^ { 5 3 } )$ . With $k = 2$ repetitions, we can set $\\mathbb { S } = [ - 2 ^ { 1 9 } , 2 ^ { 1 9 } ]$ . Multiplications are then bounded by $2 ^ { 2 4 + 1 9 } = 2 ^ { 4 3 }$ , and we can accumulate $2 ^ { 1 0 }$ terms in the inner-product before needing a modular reduction. In practice, we find that increasing $k$ further (and thus reducing $| \\mathbb { S } | )$ is not worthwhile, as the cost of performing more inner products trumps the savings from reducing the number of modulos. ", + "bbox": [ + 137, + 335, + 862, + 435 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "G RESULTS ON A STANDARD CPU ", + "text_level": 1, + "bbox": [ + 138, + 458, + 437, + 474 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For completeness, and to asses how our outsourcing scheme fairs in an environment devoid of Intel SGX’s performance quirks, we rerun the evaluations in Section 4 on the same CPU but outside of SGX’s enclave mode. ", + "bbox": [ + 133, + 492, + 862, + 521 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/9e3a7cd6a779288993a211826b1f16979c45c88d6f37c278fede67cb508715d1.jpg", + "image_caption": [ + "Figure 5: Micro benchmarks on an untrusted CPU. For three different linear operators, we plot the relative speedup of verifying a result compared to computing it. The dotted line in each plot shows the throughput obtained for computing the operation. " + ], + "image_footnote": [], + "bbox": [ + 137, + 536, + 859, + 678 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Figure 5 show the results of the micro-benchmarks for matrix multiplication, convolution and separable convolutions. In all cases, verifying a computation becomes 1-2 orders of magnitude faster than computing it as the outer dimension grows. Compared to the SGX benchmarks, we also see a much better viability of batched verification (we haven’t optimized batched verifications much, as they are inherently slow on SGX. It is likely that these numbers could be improved significantly, to approach those of verification with preprocessing). ", + "bbox": [ + 135, + 738, + 862, + 809 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Figure 6 shows benchmarks for VGG16 and MobileNet on a single core with either direct computation or various secure outsourcing strategies. For integrity alone, we achieve savings up to $8 . 9 \\times$ and $1 9 . 5 \\times$ for MobileNet and VGG16 respectively. Even without storing any secrets in the enclave, we obtain good speedups using batched verification. As noted above, it is likely that the batched results could be further improved. With additional blinding to preserve privacy, we achieve speedups of $3 . 9 \\times$ and $8 . 1 \\times$ for MobileNet and VGG16 respectively. ", + "bbox": [ + 137, + 814, + 861, + 858 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/c8154e92d0e65f4edf1599b73db1e4f3db8f57d0f186fd57a3cf3863327d0a07.jpg", + "image_caption": [ + "Figure 6: Inference with integrity and/or privacy on an untrusted CPU. We compare the baseline inference throughput (blue) to that obtained with “on-the-fly” integrity checks (red); batched integrity checks (green); integrity checks with precomputed secrets (yellow); privacy only (black); and privacy and integrity (purple). The fused MobileNet model has no intermediate activation for separable convolutions. " + ], + "image_footnote": [], + "bbox": [ + 147, + 174, + 861, + 270 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 133, + 348, + 862, + 377 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "H PARALLELIZATION ", + "text_level": 1, + "bbox": [ + 137, + 397, + 330, + 414 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Our experiments on SGX in Section 4 where performed using a single execution thread, as SGX enclaves do not have the ability to create threads. We have also experimented with techniques for achieving parallelism in SGX, both for standard computations and outsourced ones, but with little success. ", + "bbox": [ + 133, + 429, + 864, + 472 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "To optimize for throughput, a simple approach is to run multiple forward passes simultaneously. On a standard CPU, this form of “outer-parallelism” achieves close to linear scaling as we increase the number of threads from 1 to 4 on our quad-core machine. With SGX however, we did not manage to achieve any parallel speedup for VGG16—whether for direct computation or verifying outsourced results—presumably because each independent thread requires extra memory that quickly exceeds the PRM limit. For the smaller MobileNet model, we get less than a $1 . 5 \\times$ speedup using up to 4 threads, for direct computation or outsourced verification alike. ", + "bbox": [ + 135, + 478, + 862, + 563 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "DNNs typically also make use of intra-operation parallelism, i.e., computing the output of a given layer using multiple threads. Our DNN library currently does not support intra-operation parallelism, but implementing a dedicated thread pool for SGX could be an interesting extension for future work. Instead, we evaluate the potential benefits of intra-op parallelism on a standard untrusted CPU, for our matrix-product and convolution benchmarks. We make use of Eigen’s internal multi-threading support to speed up these operations, and custom OpenMP code to parallelize dot products, as Eigen does not do this on its own. ", + "bbox": [ + 135, + 568, + 862, + 652 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/827a1dd46e9ddda18510395cc72fb7d148bced674096122d688675387980e22d.jpg", + "image_caption": [ + "Figure 7: Multi-threaded micro benchmarks on an untrusted CPU. Reiterates benchmarks for matrix products and convolutions using 4 threads. " + ], + "image_footnote": [], + "bbox": [ + 232, + 665, + 763, + 819 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Figure 7 shows the results using 4 threads. For convolutions, we have currently only implemented multi-threading for the verification with preprocessing (which requires only standard dot products). Surprisingly maybe, we find that multi-threading increases the gap between direct and verified computations of matrix products, probably because dot products are extremely easy to parallelize efficiently (compared to full convolutions). We also obtain close to linear speedups for verifiable separable convolutions, but omit the results as we currently do not have an implementation of multi-threaded direct computation for depthwise convolutions, which renders the comparison unfair. Due to the various memory-access overheads in SGX, it is unclear whether similar speedups could be obtained by using intra-op parallelism in an enclave, but this is an avenue worth exploring. 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A pragmatic solution", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 118, + 305, + 492, + 317 + ], + "spans": [ + { + "bbox": [ + 118, + 305, + 492, + 317 + ], + "score": 1.0, + "content": "comes from Trusted Execution Environments (TEEs), which use hardware and software", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 118, + 316, + 492, + 328 + ], + "spans": [ + { + "bbox": [ + 118, + 316, + 492, + 328 + ], + "score": 1.0, + "content": "protections to isolate sensitive computations from the untrusted software stack. However, these", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 118, + 327, + 492, + 339 + ], + "spans": [ + { + "bbox": [ + 118, + 327, + 492, + 339 + ], + "score": 1.0, + "content": "isolation guarantees come at a price in performance, compared to untrusted alternatives. This", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 118, + 338, + 493, + 351 + ], + "spans": [ + { + "bbox": [ + 118, + 338, + 493, + 351 + ], + "score": 1.0, + "content": "paper initiates the study of high performance execution of Deep Neural Networks (DNNs) in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 118, + 348, + 493, + 362 + ], + "spans": [ + { + "bbox": [ + 118, + 348, + 493, + 362 + ], + "score": 1.0, + "content": "TEEs by efficiently partitioning DNN computations between trusted and untrusted devices.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 118, + 358, + 494, + 373 + ], + "spans": [ + { + "bbox": [ + 118, + 358, + 494, + 373 + ], + "score": 1.0, + "content": "Building upon an efficient outsourcing scheme for matrix multiplication, we propose Slalom, a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 118, + 368, + 494, + 385 + ], + "spans": [ + { + "bbox": [ + 118, + 368, + 494, + 385 + ], + "score": 1.0, + "content": "framework that securely delegates execution of all linear layers in a DNN from a TEE (e.g.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 118, + 380, + 493, + 395 + ], + "spans": [ + { + "bbox": [ + 118, + 380, + 493, + 395 + ], + "score": 1.0, + "content": "Intel SGX or Sanctum) to a faster, yet untrusted, co-located processor. We evaluate Slalom by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 119, + 393, + 493, + 405 + ], + "spans": [ + { + "bbox": [ + 119, + 393, + 493, + 405 + ], + "score": 1.0, + "content": "running DNNs in an Intel SGX enclave, which selectively delegates work to an untrusted GPU.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 118, + 404, + 492, + 415 + ], + "spans": [ + { + "bbox": [ + 118, + 404, + 408, + 415 + ], + "score": 1.0, + "content": "For canonical DNNs (VGG16, MobileNet and ResNet variants) we obtain", + "type": "text" + }, + { + "bbox": [ + 409, + 404, + 423, + 414 + ], + "score": 0.87, + "content": "6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 404, + 433, + 415 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 434, + 404, + 453, + 414 + ], + "score": 0.87, + "content": "2 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 404, + 492, + 415 + ], + "score": 1.0, + "content": "increases", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 118, + 415, + 477, + 427 + ], + "spans": [ + { + "bbox": [ + 118, + 415, + 288, + 427 + ], + "score": 1.0, + "content": "in throughput for verifiable inference, and", + "type": "text" + }, + { + "bbox": [ + 289, + 415, + 303, + 425 + ], + "score": 0.87, + "content": "4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 415, + 314, + 427 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 314, + 415, + 333, + 425 + ], + "score": 0.89, + "content": "1 1 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 415, + 477, + 427 + ], + "score": 1.0, + "content": "for verifiable and private inference.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 86, + 453, + 183, + 466 + ], + "lines": [ + { + "bbox": [ + 82, + 452, + 185, + 469 + ], + "spans": [ + { + "bbox": [ + 82, + 452, + 185, + 469 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 85, + 480, + 528, + 536 + ], + "lines": [ + { + "bbox": [ + 82, + 479, + 530, + 493 + ], + "spans": [ + { + "bbox": [ + 82, + 479, + 530, + 493 + ], + "score": 1.0, + "content": "Machine learning is increasingly used in sensitive decision making and security-critical settings. At the same time,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 83, + 491, + 529, + 503 + ], + "spans": [ + { + "bbox": [ + 83, + 491, + 529, + 503 + ], + "score": 1.0, + "content": "the growth in both cloud offerings and software stack complexity widens the attack surface for ML applications.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 501, + 529, + 515 + ], + "spans": [ + { + "bbox": [ + 82, + 501, + 529, + 515 + ], + "score": 1.0, + "content": "This raises the question of integrity and privacy guarantees for ML computations in untrusted environments, in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 513, + 529, + 526 + ], + "spans": [ + { + "bbox": [ + 82, + 513, + 529, + 526 + ], + "score": 1.0, + "content": "particular for ML tasks outsourced by a client to a remote server. Prominent examples include cloud-based ML", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 524, + 530, + 537 + ], + "spans": [ + { + "bbox": [ + 82, + 524, + 530, + 537 + ], + "score": 1.0, + "content": "APIs (e.g., a speech-to-text application that consumes user-provided data) or general ML-as-a-Service platforms.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 85, + 540, + 528, + 585 + ], + "lines": [ + { + "bbox": [ + 83, + 541, + 529, + 554 + ], + "spans": [ + { + "bbox": [ + 83, + 541, + 529, + 554 + ], + "score": 1.0, + "content": "Trusted Execution Environments (TEEs), e.g, Intel SGX (McKeen et al., 2013), ARM TrustZone (Alves & Felton,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 552, + 528, + 565 + ], + "spans": [ + { + "bbox": [ + 83, + 552, + 528, + 565 + ], + "score": 1.0, + "content": "2004) or Sanctum (Costan et al., 2016) offer a pragmatic solution to this problem. TEEs use hardware and software", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 82, + 561, + 528, + 577 + ], + "spans": [ + { + "bbox": [ + 82, + 561, + 528, + 577 + ], + "score": 1.0, + "content": "protections to isolate sensitive code from other applications, while attesting to its correct execution. Running", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 81, + 573, + 514, + 588 + ], + "spans": [ + { + "bbox": [ + 81, + 573, + 514, + 588 + ], + "score": 1.0, + "content": "outsourced ML computations in TEEs provides remote clients with strong privacy and integrity guarantees.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 84, + 591, + 528, + 679 + ], + "lines": [ + { + "bbox": [ + 82, + 591, + 530, + 604 + ], + "spans": [ + { + "bbox": [ + 82, + 591, + 530, + 604 + ], + "score": 1.0, + "content": "For outsourced ML computations, TEEs outperform pure cryptographic approaches (e.g, (Gilad-Bachrach et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 82, + 601, + 529, + 615 + ], + "spans": [ + { + "bbox": [ + 82, + 601, + 529, + 615 + ], + "score": 1.0, + "content": "2016; Mohassel & Zhang, 2017; Ghodsi et al., 2017; Juvekar et al., 2018)) by multiple orders of magnitude. At", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 83, + 614, + 529, + 626 + ], + "spans": [ + { + "bbox": [ + 83, + 614, + 529, + 626 + ], + "score": 1.0, + "content": "the same time, the isolation guarantees of TEEs still come at a steep price in performance, compared to untrusted", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 624, + 530, + 637 + ], + "spans": [ + { + "bbox": [ + 82, + 624, + 530, + 637 + ], + "score": 1.0, + "content": "alternatives (i.e., running ML models on contemporary hardware with no security guarantees). For instance,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 83, + 635, + 529, + 647 + ], + "spans": [ + { + "bbox": [ + 83, + 635, + 529, + 647 + ], + "score": 1.0, + "content": "Intel SGX (Intel Corp., 2015) incurs significant overhead for memory intensive tasks (Orenbach et al., 2017;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 83, + 646, + 528, + 658 + ], + "spans": [ + { + "bbox": [ + 83, + 646, + 528, + 658 + ], + "score": 1.0, + "content": "Harnik & Tsfadia, 2017), has difficulties exploiting multi-threading, and is currently limited to desktop CPUs", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 83, + 657, + 528, + 669 + ], + "spans": [ + { + "bbox": [ + 83, + 657, + 528, + 669 + ], + "score": 1.0, + "content": "that are outmatched by untrusted alternatives (e.g., GPUs or server CPUs). 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A pragmatic solution", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 118, + 305, + 492, + 317 + ], + "spans": [ + { + "bbox": [ + 118, + 305, + 492, + 317 + ], + "score": 1.0, + "content": "comes from Trusted Execution Environments (TEEs), which use hardware and software", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 118, + 316, + 492, + 328 + ], + "spans": [ + { + "bbox": [ + 118, + 316, + 492, + 328 + ], + "score": 1.0, + "content": "protections to isolate sensitive computations from the untrusted software stack. However, these", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 118, + 327, + 492, + 339 + ], + "spans": [ + { + "bbox": [ + 118, + 327, + 492, + 339 + ], + "score": 1.0, + "content": "isolation guarantees come at a price in performance, compared to untrusted alternatives. This", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 118, + 338, + 493, + 351 + ], + "spans": [ + { + "bbox": [ + 118, + 338, + 493, + 351 + ], + "score": 1.0, + "content": "paper initiates the study of high performance execution of Deep Neural Networks (DNNs) in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 118, + 348, + 493, + 362 + ], + "spans": [ + { + "bbox": [ + 118, + 348, + 493, + 362 + ], + "score": 1.0, + "content": "TEEs by efficiently partitioning DNN computations between trusted and untrusted devices.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 118, + 358, + 494, + 373 + ], + "spans": [ + { + "bbox": [ + 118, + 358, + 494, + 373 + ], + "score": 1.0, + "content": "Building upon an efficient outsourcing scheme for matrix multiplication, we propose Slalom, a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 118, + 368, + 494, + 385 + ], + "spans": [ + { + "bbox": [ + 118, + 368, + 494, + 385 + ], + "score": 1.0, + "content": "framework that securely delegates execution of all linear layers in a DNN from a TEE (e.g.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 118, + 380, + 493, + 395 + ], + "spans": [ + { + "bbox": [ + 118, + 380, + 493, + 395 + ], + "score": 1.0, + "content": "Intel SGX or Sanctum) to a faster, yet untrusted, co-located processor. We evaluate Slalom by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 119, + 393, + 493, + 405 + ], + "spans": [ + { + "bbox": [ + 119, + 393, + 493, + 405 + ], + "score": 1.0, + "content": "running DNNs in an Intel SGX enclave, which selectively delegates work to an untrusted GPU.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 118, + 404, + 492, + 415 + ], + "spans": [ + { + "bbox": [ + 118, + 404, + 408, + 415 + ], + "score": 1.0, + "content": "For canonical DNNs (VGG16, MobileNet and ResNet variants) we obtain", + "type": "text" + }, + { + "bbox": [ + 409, + 404, + 423, + 414 + ], + "score": 0.87, + "content": "6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 404, + 433, + 415 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 434, + 404, + 453, + 414 + ], + "score": 0.87, + "content": "2 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 404, + 492, + 415 + ], + "score": 1.0, + "content": "increases", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 118, + 415, + 477, + 427 + ], + "spans": [ + { + "bbox": [ + 118, + 415, + 288, + 427 + ], + "score": 1.0, + "content": "in throughput for verifiable inference, and", + "type": "text" + }, + { + "bbox": [ + 289, + 415, + 303, + 425 + ], + "score": 0.87, + "content": "4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 415, + 314, + 427 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 314, + 415, + 333, + 425 + ], + "score": 0.89, + "content": "1 1 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 415, + 477, + 427 + ], + "score": 1.0, + "content": "for verifiable and private inference.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15, + "bbox_fs": [ + 118, + 283, + 494, + 427 + ] + }, + { + "type": "title", + "bbox": [ + 86, + 453, + 183, + 466 + ], + "lines": [ + { + "bbox": [ + 82, + 452, + 185, + 469 + ], + "spans": [ + { + "bbox": [ + 82, + 452, + 185, + 469 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 85, + 480, + 528, + 536 + ], + "lines": [ + { + "bbox": [ + 82, + 479, + 530, + 493 + ], + "spans": [ + { + "bbox": [ + 82, + 479, + 530, + 493 + ], + "score": 1.0, + "content": "Machine learning is increasingly used in sensitive decision making and security-critical settings. At the same time,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 83, + 491, + 529, + 503 + ], + "spans": [ + { + "bbox": [ + 83, + 491, + 529, + 503 + ], + "score": 1.0, + "content": "the growth in both cloud offerings and software stack complexity widens the attack surface for ML applications.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 501, + 529, + 515 + ], + "spans": [ + { + "bbox": [ + 82, + 501, + 529, + 515 + ], + "score": 1.0, + "content": "This raises the question of integrity and privacy guarantees for ML computations in untrusted environments, in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 513, + 529, + 526 + ], + "spans": [ + { + "bbox": [ + 82, + 513, + 529, + 526 + ], + "score": 1.0, + "content": "particular for ML tasks outsourced by a client to a remote server. Prominent examples include cloud-based ML", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 524, + 530, + 537 + ], + "spans": [ + { + "bbox": [ + 82, + 524, + 530, + 537 + ], + "score": 1.0, + "content": "APIs (e.g., a speech-to-text application that consumes user-provided data) or general ML-as-a-Service platforms.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 82, + 479, + 530, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 540, + 528, + 585 + ], + "lines": [ + { + "bbox": [ + 83, + 541, + 529, + 554 + ], + "spans": [ + { + "bbox": [ + 83, + 541, + 529, + 554 + ], + "score": 1.0, + "content": "Trusted Execution Environments (TEEs), e.g, Intel SGX (McKeen et al., 2013), ARM TrustZone (Alves & Felton,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 552, + 528, + 565 + ], + "spans": [ + { + "bbox": [ + 83, + 552, + 528, + 565 + ], + "score": 1.0, + "content": "2004) or Sanctum (Costan et al., 2016) offer a pragmatic solution to this problem. TEEs use hardware and software", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 82, + 561, + 528, + 577 + ], + "spans": [ + { + "bbox": [ + 82, + 561, + 528, + 577 + ], + "score": 1.0, + "content": "protections to isolate sensitive code from other applications, while attesting to its correct execution. Running", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 81, + 573, + 514, + 588 + ], + "spans": [ + { + "bbox": [ + 81, + 573, + 514, + 588 + ], + "score": 1.0, + "content": "outsourced ML computations in TEEs provides remote clients with strong privacy and integrity guarantees.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 81, + 541, + 529, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 591, + 528, + 679 + ], + "lines": [ + { + "bbox": [ + 82, + 591, + 530, + 604 + ], + "spans": [ + { + "bbox": [ + 82, + 591, + 530, + 604 + ], + "score": 1.0, + "content": "For outsourced ML computations, TEEs outperform pure cryptographic approaches (e.g, (Gilad-Bachrach et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 82, + 601, + 529, + 615 + ], + "spans": [ + { + "bbox": [ + 82, + 601, + 529, + 615 + ], + "score": 1.0, + "content": "2016; Mohassel & Zhang, 2017; Ghodsi et al., 2017; Juvekar et al., 2018)) by multiple orders of magnitude. At", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 83, + 614, + 529, + 626 + ], + "spans": [ + { + "bbox": [ + 83, + 614, + 529, + 626 + ], + "score": 1.0, + "content": "the same time, the isolation guarantees of TEEs still come at a steep price in performance, compared to untrusted", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 624, + 530, + 637 + ], + "spans": [ + { + "bbox": [ + 82, + 624, + 530, + 637 + ], + "score": 1.0, + "content": "alternatives (i.e., running ML models on contemporary hardware with no security guarantees). For instance,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 83, + 635, + 529, + 647 + ], + "spans": [ + { + "bbox": [ + 83, + 635, + 529, + 647 + ], + "score": 1.0, + "content": "Intel SGX (Intel Corp., 2015) incurs significant overhead for memory intensive tasks (Orenbach et al., 2017;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 83, + 646, + 528, + 658 + ], + "spans": [ + { + "bbox": [ + 83, + 646, + 528, + 658 + ], + "score": 1.0, + "content": "Harnik & Tsfadia, 2017), has difficulties exploiting multi-threading, and is currently limited to desktop CPUs", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 83, + 657, + 528, + 669 + ], + "spans": [ + { + "bbox": [ + 83, + 657, + 528, + 669 + ], + "score": 1.0, + "content": "that are outmatched by untrusted alternatives (e.g., GPUs or server CPUs). Thus, our thesis is that for modern ML", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 82, + 667, + 531, + 681 + ], + "spans": [ + { + "bbox": [ + 82, + 667, + 531, + 681 + ], + "score": 1.0, + "content": "workloads, TEEs will be at least an order of magnitude less efficient than the best available untrusted hardware.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 82, + 591, + 531, + 681 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 84, + 140, + 527, + 185 + ], + "lines": [ + { + "bbox": [ + 82, + 139, + 529, + 154 + ], + "spans": [ + { + "bbox": [ + 82, + 139, + 529, + 154 + ], + "score": 1.0, + "content": "This leads us to the main question of this paper: How can we most efficiently leverage TEEs for secure machine", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 82, + 150, + 529, + 165 + ], + "spans": [ + { + "bbox": [ + 82, + 150, + 529, + 165 + ], + "score": 1.0, + "content": "learning? This was posed by Stoica et al. (2017) as one of nine open research problems for system challenges", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 83, + 162, + 528, + 175 + ], + "spans": [ + { + "bbox": [ + 83, + 162, + 528, + 175 + ], + "score": 1.0, + "content": "in AI. A specific challenge they raised is that of appropriately splitting ML computations between trusted and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 82, + 173, + 519, + 187 + ], + "spans": [ + { + "bbox": [ + 82, + 173, + 519, + 187 + ], + "score": 1.0, + "content": "untrusted components, to increase efficiency as well as security by minimizing the Trusted Computing Base.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 84, + 190, + 528, + 257 + ], + "lines": [ + { + "bbox": [ + 83, + 190, + 529, + 204 + ], + "spans": [ + { + "bbox": [ + 83, + 190, + 529, + 204 + ], + "score": 1.0, + "content": "This paper explores a novel approach to this challenge, wherein a Deep Neural Network (DNN) execution", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 82, + 201, + 528, + 215 + ], + "spans": [ + { + "bbox": [ + 82, + 201, + 528, + 215 + ], + "score": 1.0, + "content": "is partially outsourced from a TEE to a co-located, untrusted but faster device. Our approach, inspired by", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 82, + 212, + 529, + 225 + ], + "spans": [ + { + "bbox": [ + 82, + 212, + 529, + 225 + ], + "score": 1.0, + "content": "the verifiable ASICs of Wahby et al. (2016), differs from cryptographic ML outsourcing. In our case, work", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 223, + 527, + 236 + ], + "spans": [ + { + "bbox": [ + 82, + 223, + 527, + 236 + ], + "score": 1.0, + "content": "is delegated between two co-located parties, thus allowing for highly interactive—yet conceptually simpler—", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 235, + 528, + 246 + ], + "spans": [ + { + "bbox": [ + 83, + 235, + 528, + 246 + ], + "score": 1.0, + "content": "outsourcing protocols with orders-of-magnitude better efficiency. Our work also departs from prior systems that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 82, + 245, + 530, + 258 + ], + "spans": [ + { + "bbox": [ + 82, + 245, + 530, + 258 + ], + "score": 1.0, + "content": "execute DNNs fully in a TEE (Ohrimenko et al., 2016; Hunt et al., 2018; Cheng et al., 2018; Hanzlik et al., 2018).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 84, + 262, + 529, + 317 + ], + "lines": [ + { + "bbox": [ + 83, + 262, + 529, + 274 + ], + "spans": [ + { + "bbox": [ + 83, + 262, + 529, + 274 + ], + "score": 1.0, + "content": "The main observation that guides our approach is that matrix multiplication—the main bottleneck in DNNs—", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 83, + 273, + 529, + 285 + ], + "spans": [ + { + "bbox": [ + 83, + 273, + 529, + 285 + ], + "score": 1.0, + "content": "admits a concretely efficient verifiable outsourcing scheme known as Freivalds’ algorithm (Freivalds, 1977),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 82, + 283, + 530, + 297 + ], + "spans": [ + { + "bbox": [ + 82, + 283, + 530, + 297 + ], + "score": 1.0, + "content": "which can also be turned private in our setting. Our TEE selectively outsources these CPU intensive steps to a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 83, + 295, + 528, + 307 + ], + "spans": [ + { + "bbox": [ + 83, + 295, + 528, + 307 + ], + "score": 1.0, + "content": "fast untrusted co-processor (and runs the remaining steps itself) therefore achieving much better performance", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 305, + 417, + 320 + ], + "spans": [ + { + "bbox": [ + 82, + 305, + 417, + 320 + ], + "score": 1.0, + "content": "than running the entire computation in the enclave, without compromising security.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 84, + 324, + 528, + 369 + ], + "lines": [ + { + "bbox": [ + 83, + 324, + 529, + 337 + ], + "spans": [ + { + "bbox": [ + 83, + 324, + 529, + 337 + ], + "score": 1.0, + "content": "Contributions. We propose Slalom, a framework for efficient DNN inference in any trusted execution environ-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 82, + 335, + 528, + 347 + ], + "spans": [ + { + "bbox": [ + 82, + 335, + 528, + 347 + ], + "score": 1.0, + "content": "ment (e.g., SGX or Sanctum). To evaluate Slalom, we build a lightweight DNN library for Intel SGX, which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 346, + 529, + 359 + ], + "spans": [ + { + "bbox": [ + 82, + 346, + 529, + 359 + ], + "score": 1.0, + "content": "may be of independent interest. Our library allows for outsourcing all linear layers to an untrusted GPU without", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 82, + 357, + 524, + 370 + ], + "spans": [ + { + "bbox": [ + 82, + 357, + 524, + 370 + ], + "score": 1.0, + "content": "compromising integrity or privacy. Our code is available at https://github.com/ftramer/slalom.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 84, + 374, + 528, + 429 + ], + "lines": [ + { + "bbox": [ + 83, + 374, + 527, + 386 + ], + "spans": [ + { + "bbox": [ + 83, + 374, + 527, + 386 + ], + "score": 1.0, + "content": "We formally prove Slalom’s security, and evaluate it on multiple canonical DNNs with a variety of computational", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 384, + 530, + 398 + ], + "spans": [ + { + "bbox": [ + 82, + 384, + 530, + 398 + ], + "score": 1.0, + "content": "costs—VGG16 (Simonyan & Zisserman, 2014), MobileNet (Howard et al., 2017), and ResNets (He et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 396, + 529, + 409 + ], + "spans": [ + { + "bbox": [ + 82, + 396, + 529, + 409 + ], + "score": 1.0, + "content": "2016). Compared to running all computations in SGX, outsourcing linear layers to an untrusted GPU increases", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 83, + 407, + 528, + 419 + ], + "spans": [ + { + "bbox": [ + 83, + 407, + 258, + 419 + ], + "score": 1.0, + "content": "throughput (as well as energy efficiency) by", + "type": "text" + }, + { + "bbox": [ + 259, + 407, + 273, + 417 + ], + "score": 0.87, + "content": "6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 407, + 284, + 419 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 285, + 407, + 303, + 417 + ], + "score": 0.89, + "content": "2 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 407, + 428, + 419 + ], + "score": 1.0, + "content": "for verifiable inference, and by", + "type": "text" + }, + { + "bbox": [ + 429, + 407, + 443, + 417 + ], + "score": 0.86, + "content": "4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 407, + 454, + 419 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 454, + 407, + 473, + 417 + ], + "score": 0.88, + "content": "1 1 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 407, + 528, + 419 + ], + "score": 1.0, + "content": "for verifiable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 83, + 418, + 530, + 431 + ], + "spans": [ + { + "bbox": [ + 83, + 418, + 530, + 431 + ], + "score": 1.0, + "content": "and private inference. Finally, we discuss open challenges towards efficient verifiable training of DNNs in TEEs.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 84, + 440, + 177, + 453 + ], + "lines": [ + { + "bbox": [ + 81, + 438, + 178, + 456 + ], + "spans": [ + { + "bbox": [ + 81, + 438, + 178, + 456 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 84, + 464, + 191, + 476 + ], + "lines": [ + { + "bbox": [ + 83, + 465, + 192, + 477 + ], + "spans": [ + { + "bbox": [ + 83, + 465, + 192, + 477 + ], + "score": 1.0, + "content": "2.1 PROBLEM SETTING", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 84, + 485, + 528, + 529 + ], + "lines": [ + { + "bbox": [ + 82, + 485, + 527, + 499 + ], + "spans": [ + { + "bbox": [ + 82, + 485, + 292, + 499 + ], + "score": 1.0, + "content": "We consider an outsourcing scheme between a client", + "type": "text" + }, + { + "bbox": [ + 293, + 486, + 300, + 495 + ], + "score": 0.78, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 485, + 350, + 499 + ], + "score": 1.0, + "content": "and a server", + "type": "text" + }, + { + "bbox": [ + 351, + 486, + 358, + 495 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 485, + 388, + 499 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 389, + 486, + 397, + 495 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 485, + 464, + 499 + ], + "score": 1.0, + "content": "executes a DNN", + "type": "text" + }, + { + "bbox": [ + 465, + 485, + 527, + 497 + ], + "score": 0.9, + "content": "F ( x ) : \\mathcal { X } \\mathcal { Y }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 495, + 529, + 509 + ], + "spans": [ + { + "bbox": [ + 82, + 495, + 163, + 509 + ], + "score": 1.0, + "content": "on data provided by", + "type": "text" + }, + { + "bbox": [ + 164, + 497, + 171, + 506 + ], + "score": 0.76, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 495, + 529, + 509 + ], + "score": 1.0, + "content": ". The DNN can either belong to the user (e.g., as in some ML-as-a-service platforms), or to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 82, + 506, + 529, + 521 + ], + "spans": [ + { + "bbox": [ + 82, + 506, + 529, + 521 + ], + "score": 1.0, + "content": "the server (e.g., as in a cloud-based ML API). 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A specific challenge they raised is that of appropriately splitting ML computations between trusted and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 82, + 173, + 519, + 187 + ], + "spans": [ + { + "bbox": [ + 82, + 173, + 519, + 187 + ], + "score": 1.0, + "content": "untrusted components, to increase efficiency as well as security by minimizing the Trusted Computing Base.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 82, + 139, + 529, + 187 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 190, + 528, + 257 + ], + "lines": [ + { + "bbox": [ + 83, + 190, + 529, + 204 + ], + "spans": [ + { + "bbox": [ + 83, + 190, + 529, + 204 + ], + "score": 1.0, + "content": "This paper explores a novel approach to this challenge, wherein a Deep Neural Network (DNN) execution", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 82, + 201, + 528, + 215 + ], + "spans": [ + { + "bbox": [ + 82, + 201, + 528, + 215 + ], + "score": 1.0, + "content": "is partially outsourced from a TEE to a co-located, untrusted but faster device. Our approach, inspired by", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 82, + 212, + 529, + 225 + ], + "spans": [ + { + "bbox": [ + 82, + 212, + 529, + 225 + ], + "score": 1.0, + "content": "the verifiable ASICs of Wahby et al. (2016), differs from cryptographic ML outsourcing. In our case, work", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 223, + 527, + 236 + ], + "spans": [ + { + "bbox": [ + 82, + 223, + 527, + 236 + ], + "score": 1.0, + "content": "is delegated between two co-located parties, thus allowing for highly interactive—yet conceptually simpler—", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 235, + 528, + 246 + ], + "spans": [ + { + "bbox": [ + 83, + 235, + 528, + 246 + ], + "score": 1.0, + "content": "outsourcing protocols with orders-of-magnitude better efficiency. Our work also departs from prior systems that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 82, + 245, + 530, + 258 + ], + "spans": [ + { + "bbox": [ + 82, + 245, + 530, + 258 + ], + "score": 1.0, + "content": "execute DNNs fully in a TEE (Ohrimenko et al., 2016; Hunt et al., 2018; Cheng et al., 2018; Hanzlik et al., 2018).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5, + "bbox_fs": [ + 82, + 190, + 530, + 258 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 262, + 529, + 317 + ], + "lines": [ + { + "bbox": [ + 83, + 262, + 529, + 274 + ], + "spans": [ + { + "bbox": [ + 83, + 262, + 529, + 274 + ], + "score": 1.0, + "content": "The main observation that guides our approach is that matrix multiplication—the main bottleneck in DNNs—", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 83, + 273, + 529, + 285 + ], + "spans": [ + { + "bbox": [ + 83, + 273, + 529, + 285 + ], + "score": 1.0, + "content": "admits a concretely efficient verifiable outsourcing scheme known as Freivalds’ algorithm (Freivalds, 1977),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 82, + 283, + 530, + 297 + ], + "spans": [ + { + "bbox": [ + 82, + 283, + 530, + 297 + ], + "score": 1.0, + "content": "which can also be turned private in our setting. Our TEE selectively outsources these CPU intensive steps to a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 83, + 295, + 528, + 307 + ], + "spans": [ + { + "bbox": [ + 83, + 295, + 528, + 307 + ], + "score": 1.0, + "content": "fast untrusted co-processor (and runs the remaining steps itself) therefore achieving much better performance", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 305, + 417, + 320 + ], + "spans": [ + { + "bbox": [ + 82, + 305, + 417, + 320 + ], + "score": 1.0, + "content": "than running the entire computation in the enclave, without compromising security.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 82, + 262, + 530, + 320 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 324, + 528, + 369 + ], + "lines": [ + { + "bbox": [ + 83, + 324, + 529, + 337 + ], + "spans": [ + { + "bbox": [ + 83, + 324, + 529, + 337 + ], + "score": 1.0, + "content": "Contributions. We propose Slalom, a framework for efficient DNN inference in any trusted execution environ-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 82, + 335, + 528, + 347 + ], + "spans": [ + { + "bbox": [ + 82, + 335, + 528, + 347 + ], + "score": 1.0, + "content": "ment (e.g., SGX or Sanctum). To evaluate Slalom, we build a lightweight DNN library for Intel SGX, which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 346, + 529, + 359 + ], + "spans": [ + { + "bbox": [ + 82, + 346, + 529, + 359 + ], + "score": 1.0, + "content": "may be of independent interest. Our library allows for outsourcing all linear layers to an untrusted GPU without", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 82, + 357, + 524, + 370 + ], + "spans": [ + { + "bbox": [ + 82, + 357, + 524, + 370 + ], + "score": 1.0, + "content": "compromising integrity or privacy. Our code is available at https://github.com/ftramer/slalom.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 82, + 324, + 529, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 374, + 528, + 429 + ], + "lines": [ + { + "bbox": [ + 83, + 374, + 527, + 386 + ], + "spans": [ + { + "bbox": [ + 83, + 374, + 527, + 386 + ], + "score": 1.0, + "content": "We formally prove Slalom’s security, and evaluate it on multiple canonical DNNs with a variety of computational", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 384, + 530, + 398 + ], + "spans": [ + { + "bbox": [ + 82, + 384, + 530, + 398 + ], + "score": 1.0, + "content": "costs—VGG16 (Simonyan & Zisserman, 2014), MobileNet (Howard et al., 2017), and ResNets (He et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 396, + 529, + 409 + ], + "spans": [ + { + "bbox": [ + 82, + 396, + 529, + 409 + ], + "score": 1.0, + "content": "2016). Compared to running all computations in SGX, outsourcing linear layers to an untrusted GPU increases", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 83, + 407, + 528, + 419 + ], + "spans": [ + { + "bbox": [ + 83, + 407, + 258, + 419 + ], + "score": 1.0, + "content": "throughput (as well as energy efficiency) by", + "type": "text" + }, + { + "bbox": [ + 259, + 407, + 273, + 417 + ], + "score": 0.87, + "content": "6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 407, + 284, + 419 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 285, + 407, + 303, + 417 + ], + "score": 0.89, + "content": "2 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 407, + 428, + 419 + ], + "score": 1.0, + "content": "for verifiable inference, and by", + "type": "text" + }, + { + "bbox": [ + 429, + 407, + 443, + 417 + ], + "score": 0.86, + "content": "4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 407, + 454, + 419 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 454, + 407, + 473, + 417 + ], + "score": 0.88, + "content": "1 1 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 407, + 528, + 419 + ], + "score": 1.0, + "content": "for verifiable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 83, + 418, + 530, + 431 + ], + "spans": [ + { + "bbox": [ + 83, + 418, + 530, + 431 + ], + "score": 1.0, + "content": "and private inference. Finally, we discuss open challenges towards efficient verifiable training of DNNs in TEEs.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 82, + 374, + 530, + 431 + ] + }, + { + "type": "title", + "bbox": [ + 84, + 440, + 177, + 453 + ], + "lines": [ + { + "bbox": [ + 81, + 438, + 178, + 456 + ], + "spans": [ + { + "bbox": [ + 81, + 438, + 178, + 456 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 84, + 464, + 191, + 476 + ], + "lines": [ + { + "bbox": [ + 83, + 465, + 192, + 477 + ], + "spans": [ + { + "bbox": [ + 83, + 465, + 192, + 477 + ], + "score": 1.0, + "content": "2.1 PROBLEM SETTING", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 84, + 485, + 528, + 529 + ], + "lines": [ + { + "bbox": [ + 82, + 485, + 527, + 499 + ], + "spans": [ + { + "bbox": [ + 82, + 485, + 292, + 499 + ], + "score": 1.0, + "content": "We consider an outsourcing scheme between a client", + "type": "text" + }, + { + "bbox": [ + 293, + 486, + 300, + 495 + ], + "score": 0.78, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 485, + 350, + 499 + ], + "score": 1.0, + "content": "and a server", + "type": "text" + }, + { + "bbox": [ + 351, + 486, + 358, + 495 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 485, + 388, + 499 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 389, + 486, + 397, + 495 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 485, + 464, + 499 + ], + "score": 1.0, + "content": "executes a DNN", + "type": "text" + }, + { + "bbox": [ + 465, + 485, + 527, + 497 + ], + "score": 0.9, + "content": "F ( x ) : \\mathcal { X } \\mathcal { Y }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 495, + 529, + 509 + ], + "spans": [ + { + "bbox": [ + 82, + 495, + 163, + 509 + ], + "score": 1.0, + "content": "on data provided by", + "type": "text" + }, + { + "bbox": [ + 164, + 497, + 171, + 506 + ], + "score": 0.76, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 495, + 529, + 509 + ], + "score": 1.0, + "content": ". The DNN can either belong to the user (e.g., as in some ML-as-a-service platforms), or to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 82, + 506, + 529, + 521 + ], + "spans": [ + { + "bbox": [ + 82, + 506, + 529, + 521 + ], + "score": 1.0, + "content": "the server (e.g., as in a cloud-based ML API). Depending on the application, this scheme should satisfy one or", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 517, + 413, + 531 + ], + "spans": [ + { + "bbox": [ + 82, + 517, + 309, + 531 + ], + "score": 1.0, + "content": "more of the following security properties (see Appendix", + "type": "text" + }, + { + "bbox": [ + 309, + 519, + 317, + 528 + ], + "score": 0.25, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 517, + 413, + 531 + ], + "score": 1.0, + "content": "for formal definitions):", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 82, + 485, + 529, + 531 + ] + }, + { + "type": "list", + "bbox": [ + 88, + 533, + 528, + 595 + ], + "lines": [ + { + "bbox": [ + 88, + 534, + 528, + 546 + ], + "spans": [ + { + "bbox": [ + 88, + 534, + 181, + 546 + ], + "score": 1.0, + "content": "• t-Integrity: For any", + "type": "text" + }, + { + "bbox": [ + 181, + 534, + 189, + 544 + ], + "score": 0.8, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 534, + 230, + 546 + ], + "score": 1.0, + "content": "and input", + "type": "text" + }, + { + "bbox": [ + 231, + 536, + 237, + 544 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 534, + 411, + 546 + ], + "score": 1.0, + "content": ", the probability that a user interacting with", + "type": "text" + }, + { + "bbox": [ + 411, + 534, + 419, + 544 + ], + "score": 0.8, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 534, + 528, + 546 + ], + "score": 1.0, + "content": "does not abort (i.e., output", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 96, + 545, + 331, + 558 + ], + "spans": [ + { + "bbox": [ + 96, + 545, + 234, + 558 + ], + "score": 1.0, + "content": "⊥) and outputs an incorrect value", + "type": "text" + }, + { + "bbox": [ + 234, + 546, + 274, + 557 + ], + "score": 0.93, + "content": "\\tilde { y } \\ne F ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 545, + 322, + 558 + ], + "score": 1.0, + "content": "is less than", + "type": "text" + }, + { + "bbox": [ + 322, + 546, + 327, + 555 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 545, + 331, + 558 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + }, + { + "bbox": [ + 88, + 557, + 378, + 572 + ], + "spans": [ + { + "bbox": [ + 88, + 557, + 181, + 572 + ], + "score": 1.0, + "content": "• Privacy: The server", + "type": "text" + }, + { + "bbox": [ + 181, + 559, + 189, + 569 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 557, + 367, + 572 + ], + "score": 1.0, + "content": "learns no information about the user’s input", + "type": "text" + }, + { + "bbox": [ + 367, + 561, + 374, + 569 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 557, + 378, + 572 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 88, + 571, + 528, + 585 + ], + "spans": [ + { + "bbox": [ + 88, + 571, + 218, + 585 + ], + "score": 1.0, + "content": "• Model privacy: If the model", + "type": "text" + }, + { + "bbox": [ + 218, + 573, + 227, + 582 + ], + "score": 0.81, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 571, + 325, + 585 + ], + "score": 1.0, + "content": "is provided by the user,", + "type": "text" + }, + { + "bbox": [ + 326, + 573, + 334, + 582 + ], + "score": 0.8, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 571, + 450, + 585 + ], + "score": 1.0, + "content": "learns no information about", + "type": "text" + }, + { + "bbox": [ + 450, + 573, + 459, + 582 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 571, + 528, + 585 + ], + "score": 1.0, + "content": "(beyond e.g., its", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 96, + 582, + 528, + 596 + ], + "spans": [ + { + "bbox": [ + 96, + 582, + 182, + 596 + ], + "score": 1.0, + "content": "approximate size). 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ModelPrivacy
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ModelPrivacy
ApproachTEEIntegrityPrivacyw.r.t. Serverw.r.t. ClientThroughput (relative)
SafetyNets (Ghodsi et al.,2017)1.OOO≤1/200 ×
Gazelle (Juvekar et al.,2018)1O. *O1≤1/1000 ×
Secure baseline (run DNN in TEE)√.•1×
Insecure baseline (run DNN on GPU)O0≥ 50x
Slalom (Ours)√..O4× - 20×
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Improving upon the above baseline thus requires that the combined cost of doing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 83, + 545, + 523, + 557 + ], + "spans": [ + { + "bbox": [ + 83, + 545, + 523, + 557 + ], + "score": 1.0, + "content": "work on the untrusted device and verifying it in the TEE be cheaper than evaluating the full DNN in the TEE.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 82, + 523, + 528, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 562, + 528, + 628 + ], + "lines": [ + { + "bbox": [ + 83, + 561, + 528, + 575 + ], + "spans": [ + { + "bbox": [ + 83, + 561, + 528, + 575 + ], + "score": 1.0, + "content": "Wahby et al. (2016; 2017) aim at this goal for arbitrary computations outsourced between co-located ASICs. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 82, + 573, + 529, + 586 + ], + "spans": [ + { + "bbox": [ + 82, + 573, + 529, + 586 + ], + "score": 1.0, + "content": "generic non-interactive proofs they use for integrity are similar to those used in SafetyNets (Ghodsi et al., 2017),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 583, + 529, + 598 + ], + "spans": [ + { + "bbox": [ + 82, + 583, + 529, + 598 + ], + "score": 1.0, + "content": "which incur overheads that are too large to warrant outsourcing in our setting (e.g., Wahby et al. (2016) find that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 83, + 595, + 528, + 608 + ], + "spans": [ + { + "bbox": [ + 83, + 595, + 528, + 608 + ], + "score": 1.0, + "content": "the technology gap between trusted and untrusted devices needs to be of over two decades for their scheme to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 606, + 528, + 618 + ], + "spans": [ + { + "bbox": [ + 83, + 606, + 528, + 618 + ], + "score": 1.0, + "content": "break even). Similarly for privacy, standard cryptographic outsourcing protocols (e.g., (Juvekar et al., 2018)) are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 82, + 616, + 511, + 630 + ], + "spans": [ + { + "bbox": [ + 82, + 616, + 511, + 630 + ], + "score": 1.0, + "content": "unusable in our setting as simply running the computation in the TEE is much more efficient (see Table 1).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 82, + 561, + 529, + 630 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 83, + 140, + 487, + 152 + ], + "lines": [ + { + "bbox": [ + 82, + 138, + 488, + 155 + ], + "spans": [ + { + "bbox": [ + 82, + 138, + 488, + 155 + ], + "score": 1.0, + "content": "To overcome this barrier, we design outsourcing protocols tailored to DNNs, leveraging two insights:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 155, + 528, + 206 + ], + "lines": [ + { + "bbox": [ + 84, + 154, + 528, + 169 + ], + "spans": [ + { + "bbox": [ + 84, + 154, + 528, + 169 + ], + "score": 1.0, + "content": "1. In our setting, the TEE is co-located with the server’s faster untrusted processors, thus widening the design", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 96, + 165, + 455, + 181 + ], + "spans": [ + { + "bbox": [ + 96, + 165, + 455, + 181 + ], + "score": 1.0, + "content": "space to interactive outsourcing protocols with high communication but better efficiency.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 183, + 528, + 197 + ], + "spans": [ + { + "bbox": [ + 83, + 183, + 528, + 197 + ], + "score": 1.0, + "content": "2. The TEE always has knowledge of the model and can selectively outsource part of the DNN evaluation and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 96, + 195, + 328, + 207 + ], + "spans": [ + { + "bbox": [ + 96, + 195, + 328, + 207 + ], + "score": 1.0, + "content": "compute others—for which outsourcing is harder—itself.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 84, + 209, + 528, + 265 + ], + "lines": [ + { + "bbox": [ + 83, + 210, + 529, + 222 + ], + "spans": [ + { + "bbox": [ + 83, + 210, + 529, + 222 + ], + "score": 1.0, + "content": "DNNs are a class of functions that are particularly well suited for selective outsourcing. Indeed, non-linearities—", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 83, + 221, + 529, + 233 + ], + "spans": [ + { + "bbox": [ + 83, + 221, + 529, + 233 + ], + "score": 1.0, + "content": "which are hard to securely outsource (with integrity or privacy)—represent a small fraction of the computation in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 232, + 528, + 244 + ], + "spans": [ + { + "bbox": [ + 82, + 232, + 493, + 244 + ], + "score": 1.0, + "content": "a DNN so we can evaluate these in the TEE (e.g., for VGG16 inference on a single CPU thread, about", + "type": "text" + }, + { + "bbox": [ + 494, + 232, + 516, + 243 + ], + "score": 0.88, + "content": "1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 516, + 232, + 528, + 244 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 243, + 529, + 255 + ], + "spans": [ + { + "bbox": [ + 83, + 243, + 529, + 255 + ], + "score": 1.0, + "content": "the computation is spent on non-linearities). In contrast, linear operators—the main computational bottleneck in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 82, + 253, + 517, + 267 + ], + "spans": [ + { + "bbox": [ + 82, + 253, + 517, + 267 + ], + "score": 1.0, + "content": "DNNs—admit for a conceptually simple yet concretely efficient secure delegation scheme, described below.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 81, + 279, + 522, + 292 + ], + "lines": [ + { + "bbox": [ + 83, + 279, + 523, + 293 + ], + "spans": [ + { + "bbox": [ + 83, + 279, + 523, + 293 + ], + "score": 1.0, + "content": "Integrity. We verify integrity of outsourced linear layers using variants of an algorithm by Freivalds (1977).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 80, + 298, + 525, + 322 + ], + "lines": [ + { + "bbox": [ + 82, + 298, + 527, + 312 + ], + "spans": [ + { + "bbox": [ + 82, + 298, + 198, + 312 + ], + "score": 1.0, + "content": "Lemma 2.1 (Freivalds). Let", + "type": "text" + }, + { + "bbox": [ + 198, + 299, + 218, + 310 + ], + "score": 0.66, + "content": "A , B", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 298, + 237, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 299, + 246, + 309 + ], + "score": 0.77, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 298, + 258, + 312 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 258, + 300, + 282, + 309 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 298, + 365, + 312 + ], + "score": 1.0, + "content": "matrices over a field", + "type": "text" + }, + { + "bbox": [ + 365, + 299, + 373, + 309 + ], + "score": 0.66, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 298, + 527, + 312 + ], + "score": 1.0, + "content": "and let s be a uniformly random vector", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 81, + 308, + 498, + 324 + ], + "spans": [ + { + "bbox": [ + 81, + 308, + 93, + 324 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 93, + 310, + 105, + 320 + ], + "score": 0.82, + "content": "\\mathbb { S } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 106, + 308, + 123, + 324 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 124, + 310, + 150, + 321 + ], + "score": 0.88, + "content": "\\mathbb { S } \\subseteq \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 308, + 178, + 324 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 179, + 310, + 493, + 322 + ], + "score": 0.85, + "content": "\\operatorname* { P r } [ C s = A ( B s ) \\mid C \\neq A B ] = \\operatorname* { P r } [ ( C - A B ) s = \\mathbf { 0 } \\mid ( C - A B ) \\neq \\mathbf { 0 } ] \\leq ^ { 1 } / | \\mathbb { S } |", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 308, + 498, + 324 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 84, + 329, + 528, + 363 + ], + "lines": [ + { + "bbox": [ + 82, + 327, + 529, + 342 + ], + "spans": [ + { + "bbox": [ + 82, + 327, + 211, + 342 + ], + "score": 1.0, + "content": "The randomized check requires", + "type": "text" + }, + { + "bbox": [ + 212, + 328, + 228, + 339 + ], + "score": 0.88, + "content": "3 n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 327, + 529, + 342 + ], + "score": 1.0, + "content": "multiplications, a significant reduction (both in concrete terms and asymp-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 83, + 341, + 528, + 353 + ], + "spans": [ + { + "bbox": [ + 83, + 341, + 528, + 353 + ], + "score": 1.0, + "content": "totically) over evaluating the product directly. The algorithm has no false negatives and trivially extends to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 351, + 459, + 364 + ], + "spans": [ + { + "bbox": [ + 82, + 351, + 309, + 364 + ], + "score": 1.0, + "content": "rectangular matrices. Independently repeating the check", + "type": "text" + }, + { + "bbox": [ + 309, + 352, + 317, + 361 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 351, + 433, + 364 + ], + "score": 1.0, + "content": "times yields soundness error", + "type": "text" + }, + { + "bbox": [ + 433, + 351, + 454, + 363 + ], + "score": 0.92, + "content": "1 / | \\mathbb { S } | ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 351, + 459, + 364 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 83, + 377, + 529, + 411 + ], + "lines": [ + { + "bbox": [ + 82, + 377, + 529, + 391 + ], + "spans": [ + { + "bbox": [ + 82, + 377, + 529, + 391 + ], + "score": 1.0, + "content": "Privacy. Input privacy for outsourced linear operators could be achieved with linearly homomorphic encryption,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 83, + 389, + 528, + 401 + ], + "spans": [ + { + "bbox": [ + 83, + 389, + 528, + 401 + ], + "score": 1.0, + "content": "but the overhead (see the micro-benchmarks in (Juvekar et al., 2018)) is too high to compete with our baseline", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 83, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 83, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "(i.e., computing the function directly in the TEE would be faster than outsourcing it over encrypted data).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 84, + 416, + 528, + 484 + ], + "lines": [ + { + "bbox": [ + 82, + 415, + 529, + 429 + ], + "spans": [ + { + "bbox": [ + 82, + 415, + 529, + 429 + ], + "score": 1.0, + "content": "We instead propose a very efficient two-stage approach based on symmetric cryptography, i.e., an additive stream", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 425, + 529, + 441 + ], + "spans": [ + { + "bbox": [ + 82, + 425, + 131, + 441 + ], + "score": 1.0, + "content": "cipher. Let", + "type": "text" + }, + { + "bbox": [ + 131, + 428, + 189, + 439 + ], + "score": 0.92, + "content": "f : \\mathbb { F } ^ { m } \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 425, + 320, + 441 + ], + "score": 1.0, + "content": "be a linear function over a field", + "type": "text" + }, + { + "bbox": [ + 320, + 428, + 328, + 437 + ], + "score": 0.54, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 425, + 529, + 441 + ], + "score": 1.0, + "content": ". In an offline phase, the TEE generates a stream", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 437, + 529, + 452 + ], + "spans": [ + { + "bbox": [ + 82, + 437, + 248, + 452 + ], + "score": 1.0, + "content": "of one-time-use pseudorandom elements", + "type": "text" + }, + { + "bbox": [ + 248, + 439, + 279, + 449 + ], + "score": 0.9, + "content": "r \\in \\mathbb { F } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 437, + 357, + 452 + ], + "score": 1.0, + "content": ", and pre-computes", + "type": "text" + }, + { + "bbox": [ + 358, + 439, + 396, + 450 + ], + "score": 0.93, + "content": "u = f ( r )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 437, + 529, + 452 + ], + "score": 1.0, + "content": ". Then, in the online phase when", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 448, + 527, + 463 + ], + "spans": [ + { + "bbox": [ + 82, + 448, + 214, + 463 + ], + "score": 1.0, + "content": "the remote client sends an input", + "type": "text" + }, + { + "bbox": [ + 215, + 451, + 222, + 459 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 448, + 303, + 463 + ], + "score": 1.0, + "content": ", the TEE computes", + "type": "text" + }, + { + "bbox": [ + 303, + 450, + 369, + 461 + ], + "score": 0.9, + "content": "\\mathtt { E n c } ( x ) = x + r", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 448, + 390, + 463 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 390, + 450, + 405, + 460 + ], + "score": 0.87, + "content": "\\mathbb { F } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 448, + 520, + 463 + ], + "score": 1.0, + "content": "(i.e., a secure encryption of", + "type": "text" + }, + { + "bbox": [ + 520, + 451, + 527, + 459 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 460, + 529, + 474 + ], + "spans": [ + { + "bbox": [ + 82, + 460, + 319, + 474 + ], + "score": 1.0, + "content": "with a stream cipher), and outsources the computation of", + "type": "text" + }, + { + "bbox": [ + 320, + 461, + 362, + 472 + ], + "score": 0.9, + "content": "f ( \\mathsf { E n c } ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 460, + 529, + 474 + ], + "score": 1.0, + "content": "to the faster processor. Given the result", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 83, + 471, + 496, + 484 + ], + "spans": [ + { + "bbox": [ + 83, + 471, + 291, + 484 + ], + "score": 0.84, + "content": "f ( \\mathsf { E n c } ( x ) ) = f ( x + r ) = f ( x ) + f ( r ) = f ( x ) \\overset { \\cdot } { + } u", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 471, + 366, + 484 + ], + "score": 1.0, + "content": ", the TEE recovers", + "type": "text" + }, + { + "bbox": [ + 366, + 471, + 387, + 483 + ], + "score": 0.92, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 471, + 484, + 484 + ], + "score": 1.0, + "content": "using the pre-computed", + "type": "text" + }, + { + "bbox": [ + 485, + 473, + 491, + 481 + ], + "score": 0.7, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 471, + 496, + 484 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 84, + 497, + 528, + 554 + ], + "lines": [ + { + "bbox": [ + 83, + 498, + 528, + 510 + ], + "spans": [ + { + "bbox": [ + 83, + 498, + 528, + 510 + ], + "score": 1.0, + "content": "Communication. Using Freivalds’ algorithm and symmetric encryption for each linear layer in a DNN incurs", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 508, + 528, + 522 + ], + "spans": [ + { + "bbox": [ + 83, + 508, + 528, + 522 + ], + "score": 1.0, + "content": "high interaction and communication between the TEE and untrusted co-processor (e.g., over 50MB per inference", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 519, + 529, + 532 + ], + "spans": [ + { + "bbox": [ + 82, + 519, + 529, + 532 + ], + "score": 1.0, + "content": "for VGG16, see Table 3). This would be prohibitive if they were not co-located. There are protocols with lower", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 81, + 530, + 530, + 544 + ], + "spans": [ + { + "bbox": [ + 81, + 530, + 530, + 544 + ], + "score": 1.0, + "content": "communication than repeatedly using Freivalds’ ((Fiore & Gennaro, 2012; Thaler, 2013; Ghodsi et al., 2017)).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 541, + 469, + 555 + ], + "spans": [ + { + "bbox": [ + 82, + 541, + 469, + 555 + ], + "score": 1.0, + "content": "Yet, these incur a high overhead on the prover in practice and are thus not suitable in our setting.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 84, + 577, + 147, + 590 + ], + "lines": [ + { + "bbox": [ + 81, + 574, + 150, + 592 + ], + "spans": [ + { + "bbox": [ + 81, + 574, + 150, + 592 + ], + "score": 1.0, + "content": "3 SLALOM", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 83, + 606, + 529, + 651 + ], + "lines": [ + { + "bbox": [ + 82, + 606, + 529, + 620 + ], + "spans": [ + { + "bbox": [ + 82, + 606, + 529, + 620 + ], + "score": 1.0, + "content": "We introduce Slalom, a three-step approach for outsourcing DNNs from a TEE to an untrusted but faster device:", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 82, + 617, + 529, + 630 + ], + "spans": [ + { + "bbox": [ + 82, + 617, + 333, + 630 + ], + "score": 1.0, + "content": "(1) Inputs and weights are quantized and embedded in a field", + "type": "text" + }, + { + "bbox": [ + 334, + 618, + 341, + 628 + ], + "score": 0.27, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 617, + 529, + 630 + ], + "score": 1.0, + "content": "; (2) Linear layers are outsourced and verified", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 82, + 628, + 529, + 642 + ], + "spans": [ + { + "bbox": [ + 82, + 628, + 529, + 642 + ], + "score": 1.0, + "content": "using Freivalds’ algorithm; (3) Inputs of linear layers are encrypted with a pre-computed pseudorandom stream to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 81, + 639, + 531, + 654 + ], + "spans": [ + { + "bbox": [ + 81, + 639, + 531, + 654 + ], + "score": 1.0, + "content": "guarantee privacy. Figure 1 shows two Slalom variants, one to achieve integrity, and one to also achieve privacy.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 82, + 657, + 527, + 680 + ], + "lines": [ + { + "bbox": [ + 82, + 655, + 528, + 669 + ], + "spans": [ + { + "bbox": [ + 82, + 655, + 528, + 669 + ], + "score": 1.0, + "content": "We focus on feed-forward networks with fully connected layers, convolutions, separable convolutions, pooling", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 82, + 668, + 526, + 681 + ], + "spans": [ + { + "bbox": [ + 82, + 668, + 526, + 681 + ], + "score": 1.0, + "content": "layers and activations. Slalom can be extended to other architectures (e.g., residual networks, see Section 4.3).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 84, + 84, + 270, + 95 + ], + "lines": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 279, + 699, + 285, + 707 + ], + "lines": [ + { + "bbox": [ + 278, + 698, + 286, + 710 + ], + "spans": [ + { + "bbox": [ + 278, + 698, + 286, + 710 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 83, + 140, + 487, + 152 + ], + "lines": [ + { + "bbox": [ + 82, + 138, + 488, + 155 + ], + "spans": [ + { + "bbox": [ + 82, + 138, + 488, + 155 + ], + "score": 1.0, + "content": "To overcome this barrier, we design outsourcing protocols tailored to DNNs, leveraging two insights:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 82, + 138, + 488, + 155 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 155, + 528, + 206 + ], + "lines": [ + { + "bbox": [ + 84, + 154, + 528, + 169 + ], + "spans": [ + { + "bbox": [ + 84, + 154, + 528, + 169 + ], + "score": 1.0, + "content": "1. In our setting, the TEE is co-located with the server’s faster untrusted processors, thus widening the design", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 96, + 165, + 455, + 181 + ], + "spans": [ + { + "bbox": [ + 96, + 165, + 455, + 181 + ], + "score": 1.0, + "content": "space to interactive outsourcing protocols with high communication but better efficiency.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 183, + 528, + 197 + ], + "spans": [ + { + "bbox": [ + 83, + 183, + 528, + 197 + ], + "score": 1.0, + "content": "2. The TEE always has knowledge of the model and can selectively outsource part of the DNN evaluation and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 96, + 195, + 328, + 207 + ], + "spans": [ + { + "bbox": [ + 96, + 195, + 328, + 207 + ], + "score": 1.0, + "content": "compute others—for which outsourcing is harder—itself.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 83, + 154, + 528, + 207 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 209, + 528, + 265 + ], + "lines": [ + { + "bbox": [ + 83, + 210, + 529, + 222 + ], + "spans": [ + { + "bbox": [ + 83, + 210, + 529, + 222 + ], + "score": 1.0, + "content": "DNNs are a class of functions that are particularly well suited for selective outsourcing. Indeed, non-linearities—", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 83, + 221, + 529, + 233 + ], + "spans": [ + { + "bbox": [ + 83, + 221, + 529, + 233 + ], + "score": 1.0, + "content": "which are hard to securely outsource (with integrity or privacy)—represent a small fraction of the computation in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 232, + 528, + 244 + ], + "spans": [ + { + "bbox": [ + 82, + 232, + 493, + 244 + ], + "score": 1.0, + "content": "a DNN so we can evaluate these in the TEE (e.g., for VGG16 inference on a single CPU thread, about", + "type": "text" + }, + { + "bbox": [ + 494, + 232, + 516, + 243 + ], + "score": 0.88, + "content": "1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 516, + 232, + 528, + 244 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 243, + 529, + 255 + ], + "spans": [ + { + "bbox": [ + 83, + 243, + 529, + 255 + ], + "score": 1.0, + "content": "the computation is spent on non-linearities). In contrast, linear operators—the main computational bottleneck in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 82, + 253, + 517, + 267 + ], + "spans": [ + { + "bbox": [ + 82, + 253, + 517, + 267 + ], + "score": 1.0, + "content": "DNNs—admit for a conceptually simple yet concretely efficient secure delegation scheme, described below.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 82, + 210, + 529, + 267 + ] + }, + { + "type": "text", + "bbox": [ + 81, + 279, + 522, + 292 + ], + "lines": [ + { + "bbox": [ + 83, + 279, + 523, + 293 + ], + "spans": [ + { + "bbox": [ + 83, + 279, + 523, + 293 + ], + "score": 1.0, + "content": "Integrity. We verify integrity of outsourced linear layers using variants of an algorithm by Freivalds (1977).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 83, + 279, + 523, + 293 + ] + }, + { + "type": "text", + "bbox": [ + 80, + 298, + 525, + 322 + ], + "lines": [ + { + "bbox": [ + 82, + 298, + 527, + 312 + ], + "spans": [ + { + "bbox": [ + 82, + 298, + 198, + 312 + ], + "score": 1.0, + "content": "Lemma 2.1 (Freivalds). Let", + "type": "text" + }, + { + "bbox": [ + 198, + 299, + 218, + 310 + ], + "score": 0.66, + "content": "A , B", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 298, + 237, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 299, + 246, + 309 + ], + "score": 0.77, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 298, + 258, + 312 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 258, + 300, + 282, + 309 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 298, + 365, + 312 + ], + "score": 1.0, + "content": "matrices over a field", + "type": "text" + }, + { + "bbox": [ + 365, + 299, + 373, + 309 + ], + "score": 0.66, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 298, + 527, + 312 + ], + "score": 1.0, + "content": "and let s be a uniformly random vector", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 81, + 308, + 498, + 324 + ], + "spans": [ + { + "bbox": [ + 81, + 308, + 93, + 324 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 93, + 310, + 105, + 320 + ], + "score": 0.82, + "content": "\\mathbb { S } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 106, + 308, + 123, + 324 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 124, + 310, + 150, + 321 + ], + "score": 0.88, + "content": "\\mathbb { S } \\subseteq \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 308, + 178, + 324 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 179, + 310, + 493, + 322 + ], + "score": 0.85, + "content": "\\operatorname* { P r } [ C s = A ( B s ) \\mid C \\neq A B ] = \\operatorname* { P r } [ ( C - A B ) s = \\mathbf { 0 } \\mid ( C - A B ) \\neq \\mathbf { 0 } ] \\leq ^ { 1 } / | \\mathbb { S } |", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 308, + 498, + 324 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 81, + 298, + 527, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 329, + 528, + 363 + ], + "lines": [ + { + "bbox": [ + 82, + 327, + 529, + 342 + ], + "spans": [ + { + "bbox": [ + 82, + 327, + 211, + 342 + ], + "score": 1.0, + "content": "The randomized check requires", + "type": "text" + }, + { + "bbox": [ + 212, + 328, + 228, + 339 + ], + "score": 0.88, + "content": "3 n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 327, + 529, + 342 + ], + "score": 1.0, + "content": "multiplications, a significant reduction (both in concrete terms and asymp-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 83, + 341, + 528, + 353 + ], + "spans": [ + { + "bbox": [ + 83, + 341, + 528, + 353 + ], + "score": 1.0, + "content": "totically) over evaluating the product directly. The algorithm has no false negatives and trivially extends to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 351, + 459, + 364 + ], + "spans": [ + { + "bbox": [ + 82, + 351, + 309, + 364 + ], + "score": 1.0, + "content": "rectangular matrices. Independently repeating the check", + "type": "text" + }, + { + "bbox": [ + 309, + 352, + 317, + 361 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 351, + 433, + 364 + ], + "score": 1.0, + "content": "times yields soundness error", + "type": "text" + }, + { + "bbox": [ + 433, + 351, + 454, + 363 + ], + "score": 0.92, + "content": "1 / | \\mathbb { S } | ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 351, + 459, + 364 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 82, + 327, + 529, + 364 + ] + }, + { + "type": "text", + "bbox": [ + 83, + 377, + 529, + 411 + ], + "lines": [ + { + "bbox": [ + 82, + 377, + 529, + 391 + ], + "spans": [ + { + "bbox": [ + 82, + 377, + 529, + 391 + ], + "score": 1.0, + "content": "Privacy. Input privacy for outsourced linear operators could be achieved with linearly homomorphic encryption,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 83, + 389, + 528, + 401 + ], + "spans": [ + { + "bbox": [ + 83, + 389, + 528, + 401 + ], + "score": 1.0, + "content": "but the overhead (see the micro-benchmarks in (Juvekar et al., 2018)) is too high to compete with our baseline", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 83, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 83, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "(i.e., computing the function directly in the TEE would be faster than outsourcing it over encrypted data).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 82, + 377, + 529, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 416, + 528, + 484 + ], + "lines": [ + { + "bbox": [ + 82, + 415, + 529, + 429 + ], + "spans": [ + { + "bbox": [ + 82, + 415, + 529, + 429 + ], + "score": 1.0, + "content": "We instead propose a very efficient two-stage approach based on symmetric cryptography, i.e., an additive stream", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 425, + 529, + 441 + ], + "spans": [ + { + "bbox": [ + 82, + 425, + 131, + 441 + ], + "score": 1.0, + "content": "cipher. Let", + "type": "text" + }, + { + "bbox": [ + 131, + 428, + 189, + 439 + ], + "score": 0.92, + "content": "f : \\mathbb { F } ^ { m } \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 425, + 320, + 441 + ], + "score": 1.0, + "content": "be a linear function over a field", + "type": "text" + }, + { + "bbox": [ + 320, + 428, + 328, + 437 + ], + "score": 0.54, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 425, + 529, + 441 + ], + "score": 1.0, + "content": ". In an offline phase, the TEE generates a stream", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 437, + 529, + 452 + ], + "spans": [ + { + "bbox": [ + 82, + 437, + 248, + 452 + ], + "score": 1.0, + "content": "of one-time-use pseudorandom elements", + "type": "text" + }, + { + "bbox": [ + 248, + 439, + 279, + 449 + ], + "score": 0.9, + "content": "r \\in \\mathbb { F } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 437, + 357, + 452 + ], + "score": 1.0, + "content": ", and pre-computes", + "type": "text" + }, + { + "bbox": [ + 358, + 439, + 396, + 450 + ], + "score": 0.93, + "content": "u = f ( r )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 437, + 529, + 452 + ], + "score": 1.0, + "content": ". Then, in the online phase when", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 448, + 527, + 463 + ], + "spans": [ + { + "bbox": [ + 82, + 448, + 214, + 463 + ], + "score": 1.0, + "content": "the remote client sends an input", + "type": "text" + }, + { + "bbox": [ + 215, + 451, + 222, + 459 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 448, + 303, + 463 + ], + "score": 1.0, + "content": ", the TEE computes", + "type": "text" + }, + { + "bbox": [ + 303, + 450, + 369, + 461 + ], + "score": 0.9, + "content": "\\mathtt { E n c } ( x ) = x + r", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 448, + 390, + 463 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 390, + 450, + 405, + 460 + ], + "score": 0.87, + "content": "\\mathbb { F } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 448, + 520, + 463 + ], + "score": 1.0, + "content": "(i.e., a secure encryption of", + "type": "text" + }, + { + "bbox": [ + 520, + 451, + 527, + 459 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 460, + 529, + 474 + ], + "spans": [ + { + "bbox": [ + 82, + 460, + 319, + 474 + ], + "score": 1.0, + "content": "with a stream cipher), and outsources the computation of", + "type": "text" + }, + { + "bbox": [ + 320, + 461, + 362, + 472 + ], + "score": 0.9, + "content": "f ( \\mathsf { E n c } ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 460, + 529, + 474 + ], + "score": 1.0, + "content": "to the faster processor. Given the result", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 83, + 471, + 496, + 484 + ], + "spans": [ + { + "bbox": [ + 83, + 471, + 291, + 484 + ], + "score": 0.84, + "content": "f ( \\mathsf { E n c } ( x ) ) = f ( x + r ) = f ( x ) + f ( r ) = f ( x ) \\overset { \\cdot } { + } u", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 471, + 366, + 484 + ], + "score": 1.0, + "content": ", the TEE recovers", + "type": "text" + }, + { + "bbox": [ + 366, + 471, + 387, + 483 + ], + "score": 0.92, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 471, + 484, + 484 + ], + "score": 1.0, + "content": "using the pre-computed", + "type": "text" + }, + { + "bbox": [ + 485, + 473, + 491, + 481 + ], + "score": 0.7, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 471, + 496, + 484 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 82, + 415, + 529, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 497, + 528, + 554 + ], + "lines": [ + { + "bbox": [ + 83, + 498, + 528, + 510 + ], + "spans": [ + { + "bbox": [ + 83, + 498, + 528, + 510 + ], + "score": 1.0, + "content": "Communication. Using Freivalds’ algorithm and symmetric encryption for each linear layer in a DNN incurs", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 508, + 528, + 522 + ], + "spans": [ + { + "bbox": [ + 83, + 508, + 528, + 522 + ], + "score": 1.0, + "content": "high interaction and communication between the TEE and untrusted co-processor (e.g., over 50MB per inference", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 519, + 529, + 532 + ], + "spans": [ + { + "bbox": [ + 82, + 519, + 529, + 532 + ], + "score": 1.0, + "content": "for VGG16, see Table 3). This would be prohibitive if they were not co-located. There are protocols with lower", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 81, + 530, + 530, + 544 + ], + "spans": [ + { + "bbox": [ + 81, + 530, + 530, + 544 + ], + "score": 1.0, + "content": "communication than repeatedly using Freivalds’ ((Fiore & Gennaro, 2012; Thaler, 2013; Ghodsi et al., 2017)).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 541, + 469, + 555 + ], + "spans": [ + { + "bbox": [ + 82, + 541, + 469, + 555 + ], + "score": 1.0, + "content": "Yet, these incur a high overhead on the prover in practice and are thus not suitable in our setting.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 81, + 498, + 530, + 555 + ] + }, + { + "type": "title", + "bbox": [ + 84, + 577, + 147, + 590 + ], + "lines": [ + { + "bbox": [ + 81, + 574, + 150, + 592 + ], + "spans": [ + { + "bbox": [ + 81, + 574, + 150, + 592 + ], + "score": 1.0, + "content": "3 SLALOM", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 83, + 606, + 529, + 651 + ], + "lines": [ + { + "bbox": [ + 82, + 606, + 529, + 620 + ], + "spans": [ + { + "bbox": [ + 82, + 606, + 529, + 620 + ], + "score": 1.0, + "content": "We introduce Slalom, a three-step approach for outsourcing DNNs from a TEE to an untrusted but faster device:", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 82, + 617, + 529, + 630 + ], + "spans": [ + { + "bbox": [ + 82, + 617, + 333, + 630 + ], + "score": 1.0, + "content": "(1) Inputs and weights are quantized and embedded in a field", + "type": "text" + }, + { + "bbox": [ + 334, + 618, + 341, + 628 + ], + "score": 0.27, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 617, + 529, + 630 + ], + "score": 1.0, + "content": "; (2) Linear layers are outsourced and verified", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 82, + 628, + 529, + 642 + ], + "spans": [ + { + "bbox": [ + 82, + 628, + 529, + 642 + ], + "score": 1.0, + "content": "using Freivalds’ algorithm; (3) Inputs of linear layers are encrypted with a pre-computed pseudorandom stream to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 81, + 639, + 531, + 654 + ], + "spans": [ + { + "bbox": [ + 81, + 639, + 531, + 654 + ], + "score": 1.0, + "content": "guarantee privacy. Figure 1 shows two Slalom variants, one to achieve integrity, and one to also achieve privacy.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 81, + 606, + 531, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 82, + 657, + 527, + 680 + ], + "lines": [ + { + "bbox": [ + 82, + 655, + 528, + 669 + ], + "spans": [ + { + "bbox": [ + 82, + 655, + 528, + 669 + ], + "score": 1.0, + "content": "We focus on feed-forward networks with fully connected layers, convolutions, separable convolutions, pooling", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 82, + 668, + 526, + 681 + ], + "spans": [ + { + "bbox": [ + 82, + 668, + 526, + 681 + ], + "score": 1.0, + "content": "layers and activations. Slalom can be extended to other architectures (e.g., residual networks, see Section 4.3).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 82, + 655, + 528, + 681 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 84, + 138, + 292, + 258 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 84, + 138, + 292, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 84, + 138, + 292, + 258 + ], + "spans": [ + { + "bbox": [ + 84, + 138, + 292, + 258 + ], + "score": 0.422, + "html": "
Slalomwith integrity TEE(F,x1) S(F)
for i∈ [1,n] do
yi=xiWi
xi+1=σ(yi)
1...yn
fori∈[1,n] do assert Freivalds(yi,x,W)
xi+1=σ(yi)
return yn
", + "type": "table", + "image_path": "68e3dc2240594a3fa26a194d36ba32950ab09ff2dc6712e779ab23c590da496f.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 84, + 138, + 292, + 153.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 84, + 153.0, + 292, + 168.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 84, + 168.0, + 292, + 183.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 84, + 183.0, + 292, + 198.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 84, + 198.0, + 292, + 213.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 84, + 213.0, + 292, + 228.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 84, + 228.0, + 292, + 243.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 84, + 243.0, + 292, + 258.0 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 3.5 + }, + { + "type": "table", + "bbox": [ + 303, + 138, + 523, + 259 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 303, + 138, + 523, + 259 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 303, + 138, + 523, + 259 + ], + "spans": [ + { + "bbox": [ + 303, + 138, + 523, + 259 + ], + "score": 0.481, + "html": "
Slalom with integrity& privacy TEE(F,x1) S(F)
Preproc: for i∈[1,n] dori ← Fmi,ui =riWi
for i∈[1,n] do 主 xi=xi+ri
起 yi=xW
yi=yi-ui
assert Freivalds(yi,xi,Wi)
xi+1=σ(yi) return yn
", + "type": "table", + "image_path": "e4e070963d356a31639df7aca4a78879b76895ffe4fc201b38e685066e487465.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 303, + 138, + 523, + 153.125 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 303, + 153.125, + 523, + 168.25 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 303, + 168.25, + 523, + 183.375 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 303, + 183.375, + 523, + 198.5 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 303, + 198.5, + 523, + 213.625 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 303, + 213.625, + 523, + 228.75 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 303, + 228.75, + 523, + 243.875 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 303, + 243.875, + 523, + 259.0 + ], + "spans": [], + "index": 15 + } + ] + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 84, + 333, + 175, + 344 + ], + "lines": [ + { + "bbox": [ + 82, + 331, + 177, + 347 + ], + "spans": [ + { + "bbox": [ + 82, + 331, + 177, + 347 + ], + "score": 1.0, + "content": "3.1 QUANTIZATION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 84, + 354, + 529, + 388 + ], + "lines": [ + { + "bbox": [ + 83, + 353, + 529, + 367 + ], + "spans": [ + { + "bbox": [ + 83, + 353, + 518, + 367 + ], + "score": 1.0, + "content": "The techniques we use for integrity and privacy (Freivalds’ algorithm and stream ciphers) work over a field", + "type": "text" + }, + { + "bbox": [ + 519, + 354, + 526, + 364 + ], + "score": 0.59, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 526, + 353, + 529, + 367 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 82, + 363, + 529, + 379 + ], + "spans": [ + { + "bbox": [ + 82, + 363, + 470, + 379 + ], + "score": 1.0, + "content": "We thus quantize all inputs and weights of a DNN to integers, and embed these integers in the field", + "type": "text" + }, + { + "bbox": [ + 471, + 366, + 483, + 378 + ], + "score": 0.89, + "content": "\\mathbb { Z } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 363, + 529, + 379 + ], + "score": 1.0, + "content": "of integers", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 83, + 376, + 530, + 389 + ], + "spans": [ + { + "bbox": [ + 83, + 376, + 148, + 389 + ], + "score": 1.0, + "content": "modulo a prime", + "type": "text" + }, + { + "bbox": [ + 149, + 378, + 155, + 388 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 376, + 185, + 389 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 186, + 378, + 192, + 388 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 376, + 530, + 389 + ], + "score": 1.0, + "content": "is larger than all values computed in a DNN evaluation, so as to avoid wrap-around).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 84, + 393, + 529, + 428 + ], + "lines": [ + { + "bbox": [ + 83, + 393, + 528, + 406 + ], + "spans": [ + { + "bbox": [ + 83, + 393, + 322, + 406 + ], + "score": 1.0, + "content": "As in (Gupta et al., 2015), we convert floating point numbers", + "type": "text" + }, + { + "bbox": [ + 323, + 396, + 330, + 403 + ], + "score": 0.79, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 393, + 461, + 406 + ], + "score": 1.0, + "content": "to a fixed-point representation as", + "type": "text" + }, + { + "bbox": [ + 461, + 393, + 528, + 405 + ], + "score": 0.87, + "content": "\\tilde { x } = \\mathtt { F P } ( x ; l ) : =", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 84, + 404, + 529, + 419 + ], + "spans": [ + { + "bbox": [ + 84, + 405, + 144, + 417 + ], + "score": 0.54, + "content": "\\mathtt { r o u n d } ( 2 ^ { l } \\cdot x ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 404, + 268, + 419 + ], + "score": 1.0, + "content": ". For a linear layer with kernel", + "type": "text" + }, + { + "bbox": [ + 268, + 406, + 280, + 416 + ], + "score": 0.64, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 404, + 317, + 419 + ], + "score": 1.0, + "content": "and bias", + "type": "text" + }, + { + "bbox": [ + 317, + 406, + 323, + 416 + ], + "score": 0.75, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 404, + 446, + 419 + ], + "score": 1.0, + "content": ", we define integer parameters", + "type": "text" + }, + { + "bbox": [ + 446, + 405, + 529, + 418 + ], + "score": 0.89, + "content": "\\tilde { W } = \\mathrm { F P } ( W , l ) , \\tilde { b } =", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 83, + 415, + 530, + 430 + ], + "spans": [ + { + "bbox": [ + 83, + 417, + 121, + 429 + ], + "score": 0.91, + "content": "\\mathrm { E P } ( b , 2 l )", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 415, + 304, + 430 + ], + "score": 1.0, + "content": ". After applying the layer to a quantized input", + "type": "text" + }, + { + "bbox": [ + 304, + 418, + 311, + 427 + ], + "score": 0.81, + "content": "\\tilde { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 415, + 406, + 430 + ], + "score": 1.0, + "content": ", we scale the output by", + "type": "text" + }, + { + "bbox": [ + 406, + 416, + 421, + 427 + ], + "score": 0.88, + "content": "2 ^ { - l }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 415, + 530, + 430 + ], + "score": 1.0, + "content": "and re-round to an integer.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 83, + 433, + 528, + 489 + ], + "lines": [ + { + "bbox": [ + 82, + 433, + 529, + 447 + ], + "spans": [ + { + "bbox": [ + 82, + 433, + 529, + 447 + ], + "score": 1.0, + "content": "For efficiency reasons, we perform integer arithmetic using floats (so-called fake quantization), and choose", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 83, + 442, + 529, + 459 + ], + "spans": [ + { + "bbox": [ + 83, + 444, + 116, + 456 + ], + "score": 0.91, + "content": "p < 2 ^ { 2 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 442, + 249, + 459 + ], + "score": 1.0, + "content": "to avoid loss of precision (we use", + "type": "text" + }, + { + "bbox": [ + 250, + 444, + 301, + 456 + ], + "score": 0.9, + "content": "p = 2 ^ { 2 4 } - 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 442, + 445, + 459 + ], + "score": 1.0, + "content": ". For the models we evaluate, setting", + "type": "text" + }, + { + "bbox": [ + 446, + 445, + 469, + 455 + ], + "score": 0.89, + "content": "l = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 442, + 529, + 459 + ], + "score": 1.0, + "content": "for all weights", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 81, + 453, + 530, + 469 + ], + "spans": [ + { + "bbox": [ + 81, + 453, + 302, + 469 + ], + "score": 1.0, + "content": "and inputs ensures that all DNN values are bounded by", + "type": "text" + }, + { + "bbox": [ + 302, + 455, + 316, + 466 + ], + "score": 0.84, + "content": "2 ^ { 2 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 453, + 383, + 469 + ], + "score": 1.0, + "content": ", with less than a", + "type": "text" + }, + { + "bbox": [ + 383, + 456, + 406, + 467 + ], + "score": 0.87, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 453, + 530, + 469 + ], + "score": 1.0, + "content": "drop in accuracy (see Table 3).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 465, + 529, + 480 + ], + "spans": [ + { + "bbox": [ + 82, + 465, + 230, + 480 + ], + "score": 1.0, + "content": "When performing arithmetic modulo", + "type": "text" + }, + { + "bbox": [ + 230, + 468, + 237, + 478 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 465, + 529, + 480 + ], + "score": 1.0, + "content": "(e.g., for Freivalds’ algorithm or when computing on encrypted data), we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 83, + 478, + 520, + 489 + ], + "spans": [ + { + "bbox": [ + 83, + 478, + 520, + 489 + ], + "score": 1.0, + "content": "use double-precision floats, to reduce the number of modular reductions required (details are in Appendix F).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 84, + 504, + 291, + 515 + ], + "lines": [ + { + "bbox": [ + 82, + 504, + 292, + 517 + ], + "spans": [ + { + "bbox": [ + 82, + 504, + 292, + 517 + ], + "score": 1.0, + "content": "3.2 VERIFYING COMMON LINEAR OPERATORS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 82, + 525, + 528, + 548 + ], + "lines": [ + { + "bbox": [ + 82, + 524, + 529, + 539 + ], + "spans": [ + { + "bbox": [ + 82, + 524, + 529, + 539 + ], + "score": 1.0, + "content": "We now describe Slalom’s approach to verifying the integrity of outsourced linear layers. We describe these", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 82, + 537, + 397, + 549 + ], + "spans": [ + { + "bbox": [ + 82, + 537, + 397, + 549 + ], + "score": 1.0, + "content": "layers in detail in Appendix D and summarize this section’s results in Table 2.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 82, + 557, + 528, + 591 + ], + "lines": [ + { + "bbox": [ + 82, + 556, + 528, + 570 + ], + "spans": [ + { + "bbox": [ + 82, + 556, + 528, + 570 + ], + "score": 1.0, + "content": "Freivalds’ Algorithm for Batches. The most direct way of applying Freivalds’ algorithm to arbitrary linear", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 82, + 567, + 528, + 582 + ], + "spans": [ + { + "bbox": [ + 82, + 567, + 322, + 582 + ], + "score": 1.0, + "content": "layers of a DNN is by exploiting batching. 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Slalomwith integrity TEE(F,x1) S(F)
for i∈ [1,n] do
yi=xiWi
xi+1=σ(yi)
1...yn
fori∈[1,n] do assert Freivalds(yi,x,W)
xi+1=σ(yi)
return yn
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Slalom with integrity& privacy TEE(F,x1) S(F)
Preproc: for i∈[1,n] dori ← Fmi,ui =riWi
for i∈[1,n] do 主 xi=xi+ri
起 yi=xW
yi=yi-ui
assert Freivalds(yi,xi,Wi)
xi+1=σ(yi) return yn
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Layer|x|,|yl|W|costf (B=1)Batched verificationWith preproc.
FChin,houthin·hout|a|·lyB·(lx|+lyl)+cost fB·(lx|+lyl)
Convh·w·Cin,h·w·Coutk² · Cin ·Coutal·k2 · CoutB·(lx|+lyl)+Cin·Cout+ |x| · k²B·(|xl+lyl)
Depth.Convh·w·Cin,h·w·Cink2.Cinlac|.k²B·(lx|+lyl)+cost fB·(|x|+ lyl)
Point. Convh·w ·Cin,h · w· CoutCin·Cout|c| · CoutB·(x|+lyl)+Cin·CoutB·(lx|+|yl)
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We now show how to obtain an outsourcing scheme for linear layers that has optimal verification", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 83, + 286, + 529, + 300 + ], + "spans": [ + { + "bbox": [ + 83, + 286, + 150, + 300 + ], + "score": 1.0, + "content": "complexity (i.e.,", + "type": "text" + }, + { + "bbox": [ + 150, + 286, + 185, + 299 + ], + "score": 0.92, + "content": "| x | + | y |", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 286, + 529, + 300 + ], + "score": 1.0, + "content": "operations) for single-element batches and arbitrary linear operators, while at the same", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 83, + 297, + 467, + 311 + ], + "spans": [ + { + "bbox": [ + 83, + 297, + 467, + 311 + ], + "score": 1.0, + "content": "time compressing the DNN’s weights (a welcome property in our memory-limited TEE model).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 83, + 314, + 528, + 359 + ], + "lines": [ + { + "bbox": [ + 83, + 314, + 529, + 327 + ], + "spans": [ + { + "bbox": [ + 83, + 314, + 529, + 327 + ], + "score": 1.0, + "content": "We leverage two facts: (1) DNN weights are fixed at inference time, so part of Freivalds’ check can be pre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 82, + 325, + 528, + 337 + ], + "spans": [ + { + "bbox": [ + 82, + 325, + 299, + 337 + ], + "score": 1.0, + "content": "computed; (2) the TEE can keep secrets from the host", + "type": "text" + }, + { + "bbox": [ + 299, + 326, + 307, + 335 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 325, + 397, + 337 + ], + "score": 1.0, + "content": ", so the random values", + "type": "text" + }, + { + "bbox": [ + 397, + 328, + 403, + 335 + ], + "score": 0.68, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 325, + 528, + 337 + ], + "score": 1.0, + "content": "can be re-used across layers or", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 84, + 336, + 528, + 347 + ], + "spans": [ + { + "bbox": [ + 84, + 336, + 219, + 347 + ], + "score": 1.0, + "content": "inputs (if we run Freivalds’ check", + "type": "text" + }, + { + "bbox": [ + 219, + 338, + 227, + 346 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 336, + 528, + 347 + ], + "score": 1.0, + "content": "times with the same secret randomness, the soundness errors grows at most", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 83, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 83, + 347, + 128, + 360 + ], + "score": 1.0, + "content": "by a factor", + "type": "text" + }, + { + "bbox": [ + 129, + 350, + 136, + 358 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "). Our verification scheme with preprocessing follows from a reformulation of Lemma (2.1):", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 82, + 361, + 525, + 387 + ], + "lines": [ + { + "bbox": [ + 82, + 360, + 528, + 375 + ], + "spans": [ + { + "bbox": [ + 82, + 360, + 156, + 375 + ], + "score": 1.0, + "content": "Lemma 3.1. Let", + "type": "text" + }, + { + "bbox": [ + 156, + 362, + 215, + 374 + ], + "score": 0.91, + "content": "f : \\mathbb { F } ^ { m } \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 360, + 305, + 375 + ], + "score": 1.0, + "content": "be a linear operator,", + "type": "text" + }, + { + "bbox": [ + 306, + 361, + 366, + 374 + ], + "score": 0.92, + "content": "f ( x ) : = x ^ { \\top } W", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 360, + 496, + 375 + ], + "score": 1.0, + "content": ". Let s be uniformly random in", + "type": "text" + }, + { + "bbox": [ + 497, + 362, + 509, + 372 + ], + "score": 0.82, + "content": "\\mathbb { S } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 509, + 360, + 528, + 375 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 83, + 371, + 518, + 388 + ], + "spans": [ + { + "bbox": [ + 83, + 374, + 110, + 386 + ], + "score": 0.86, + "content": "\\mathbb { S } \\subseteq \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 110, + 371, + 143, + 388 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 144, + 374, + 226, + 387 + ], + "score": 0.92, + "content": "\\tilde { s } : = \\nabla F _ { x } ( s ) = W s", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 371, + 263, + 388 + ], + "score": 1.0, + "content": ". 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To save space,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 83, + 407, + 528, + 419 + ], + "spans": [ + { + "bbox": [ + 83, + 407, + 205, + 419 + ], + "score": 1.0, + "content": "we can reuse the same random", + "type": "text" + }, + { + "bbox": [ + 206, + 410, + 212, + 417 + ], + "score": 0.69, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 407, + 528, + 419 + ], + "score": 1.0, + "content": "for every layer. The memory footprint of a model is then equal to the size of the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 83, + 418, + 511, + 432 + ], + "spans": [ + { + "bbox": [ + 83, + 418, + 511, + 432 + ], + "score": 1.0, + "content": "inputs of all its linear layers (e.g., for VGG16 the footprint is reduced from 550MB to 36MB, see Table 3).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 84, + 438, + 176, + 450 + ], + "lines": [ + { + "bbox": [ + 82, + 437, + 178, + 451 + ], + "spans": [ + { + "bbox": [ + 82, + 437, + 178, + 451 + ], + "score": 1.0, + "content": "3.3 INPUT PRIVACY", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 84, + 459, + 529, + 515 + ], + "lines": [ + { + "bbox": [ + 82, + 459, + 530, + 473 + ], + "spans": [ + { + "bbox": [ + 82, + 459, + 530, + 473 + ], + "score": 1.0, + "content": "To guarantee privacy of the client’s inputs, we use precomputed blinding factors for each outsourced computation,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 470, + 529, + 483 + ], + "spans": [ + { + "bbox": [ + 82, + 470, + 529, + 483 + ], + "score": 1.0, + "content": "as described in Section 2.3. The TEE uses a cryptographic Pseudo Random Number Generator (PRNG) to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 81, + 480, + 529, + 495 + ], + "spans": [ + { + "bbox": [ + 81, + 480, + 529, + 495 + ], + "score": 1.0, + "content": "generate blinding factors. The precomputed “unblinding factors” are encrypted and stored in untrusted memory", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 82, + 492, + 528, + 505 + ], + "spans": [ + { + "bbox": [ + 82, + 492, + 528, + 505 + ], + "score": 1.0, + "content": "or disk. In the online phase, the TEE regenerates the blinding factors using the same PRNG seed, and uses the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 503, + 420, + 516 + ], + "spans": [ + { + "bbox": [ + 82, + 503, + 420, + 516 + ], + "score": 1.0, + "content": "precomputed unblinding factors to decrypt the output of the outsourced linear layer.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 84, + 519, + 528, + 565 + ], + "lines": [ + { + "bbox": [ + 82, + 519, + 528, + 533 + ], + "spans": [ + { + "bbox": [ + 82, + 519, + 528, + 533 + ], + "score": 1.0, + "content": "This blinding process incurs several overheads: (1) the computations on the untrusted device have to be performed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 83, + 531, + 528, + 543 + ], + "spans": [ + { + "bbox": [ + 83, + 531, + 102, + 543 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 103, + 532, + 115, + 543 + ], + "score": 0.89, + "content": "\\mathbb { Z } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 531, + 528, + 543 + ], + "score": 1.0, + "content": "so we use double-precision arithmetic. (2) The trusted and untrusted processors exchange data in-between", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 83, + 542, + 528, + 554 + ], + "spans": [ + { + "bbox": [ + 83, + 542, + 528, + 554 + ], + "score": 1.0, + "content": "each layer, rather than at the end of a full inference pass. (3) The TEE has to efficiently load precomputed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 553, + 518, + 565 + ], + "spans": [ + { + "bbox": [ + 83, + 553, + 518, + 565 + ], + "score": 1.0, + "content": "unblinding factors, which requires either a large amount of RAM, or a fast access to disk (e.g., a PCIe SSD).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 83, + 569, + 527, + 592 + ], + "lines": [ + { + "bbox": [ + 83, + 570, + 527, + 582 + ], + "spans": [ + { + "bbox": [ + 83, + 570, + 527, + 582 + ], + "score": 1.0, + "content": "Slalom’s security is given by the following results. Formal definitions and proofs are in Appendix B. Let negl be", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 580, + 525, + 593 + ], + "spans": [ + { + "bbox": [ + 82, + 580, + 229, + 593 + ], + "score": 1.0, + "content": "a negligible function (for any integer", + "type": "text" + }, + { + "bbox": [ + 230, + 581, + 253, + 591 + ], + "score": 0.89, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 580, + 342, + 593 + ], + "score": 1.0, + "content": "there exists an integer", + "type": "text" + }, + { + "bbox": [ + 342, + 581, + 356, + 592 + ], + "score": 0.89, + "content": "N _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 580, + 420, + 593 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 420, + 581, + 452, + 592 + ], + "score": 0.89, + "content": "x > N _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 580, + 456, + 593 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 456, + 581, + 525, + 593 + ], + "score": 0.93, + "content": "| \\mathrm { n e g l } ( x ) | < \\bar { 1 } / { x ^ { c } } )", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 84, + 595, + 529, + 640 + ], + "lines": [ + { + "bbox": [ + 82, + 595, + 529, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 595, + 375, + 609 + ], + "score": 1.0, + "content": "Theorem 3.2. Let Slalom be the protocol from Figure 1 (right), where", + "type": "text" + }, + { + "bbox": [ + 375, + 596, + 384, + 606 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 595, + 408, + 609 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 408, + 598, + 415, + 606 + ], + "score": 0.63, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 595, + 529, + 609 + ], + "score": 1.0, + "content": "-layer DNN, and Freivalds’", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 607, + 528, + 619 + ], + "spans": [ + { + "bbox": [ + 83, + 607, + 374, + 619 + ], + "score": 1.0, + "content": "algorithm is repeated k times per layer with random vectors drawn from", + "type": "text" + }, + { + "bbox": [ + 375, + 607, + 401, + 618 + ], + "score": 0.87, + "content": "\\mathbb { S } \\subseteq \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 607, + 528, + 619 + ], + "score": 1.0, + "content": ". Assume all random values are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 81, + 617, + 527, + 631 + ], + "spans": [ + { + "bbox": [ + 81, + 617, + 311, + 631 + ], + "score": 1.0, + "content": "generated using a secure PRNG with security parameter", + "type": "text" + }, + { + "bbox": [ + 311, + 619, + 318, + 627 + ], + "score": 0.72, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 617, + 518, + 631 + ], + "score": 1.0, + "content": ". 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Layer|x|,|yl|W|costf (B=1)Batched verificationWith preproc.
FChin,houthin·hout|a|·lyB·(lx|+lyl)+cost fB·(lx|+lyl)
Convh·w·Cin,h·w·Coutk² · Cin ·Coutal·k2 · CoutB·(lx|+lyl)+Cin·Cout+ |x| · k²B·(|xl+lyl)
Depth.Convh·w·Cin,h·w·Cink2.Cinlac|.k²B·(lx|+lyl)+cost fB·(|x|+ lyl)
Point. Convh·w ·Cin,h · w· CoutCin·Cout|c| · CoutB·(x|+lyl)+Cin·CoutB·(lx|+|yl)
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The TEE uses a cryptographic Pseudo Random Number Generator (PRNG) to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 81, + 480, + 529, + 495 + ], + "spans": [ + { + "bbox": [ + 81, + 480, + 529, + 495 + ], + "score": 1.0, + "content": "generate blinding factors. The precomputed “unblinding factors” are encrypted and stored in untrusted memory", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 82, + 492, + 528, + 505 + ], + "spans": [ + { + "bbox": [ + 82, + 492, + 528, + 505 + ], + "score": 1.0, + "content": "or disk. In the online phase, the TEE regenerates the blinding factors using the same PRNG seed, and uses the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 503, + 420, + 516 + ], + "spans": [ + { + "bbox": [ + 82, + 503, + 420, + 516 + ], + "score": 1.0, + "content": "precomputed unblinding factors to decrypt the output of the outsourced linear layer.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 81, + 459, + 530, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 519, + 528, + 565 + ], + "lines": [ + { + "bbox": [ + 82, + 519, + 528, + 533 + ], + "spans": [ + { + "bbox": [ + 82, + 519, + 528, + 533 + ], + "score": 1.0, + "content": "This blinding process incurs several overheads: (1) the computations on the untrusted device have to be performed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 83, + 531, + 528, + 543 + ], + "spans": [ + { + "bbox": [ + 83, + 531, + 102, + 543 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 103, + 532, + 115, + 543 + ], + "score": 0.89, + "content": "\\mathbb { Z } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 531, + 528, + 543 + ], + "score": 1.0, + "content": "so we use double-precision arithmetic. (2) The trusted and untrusted processors exchange data in-between", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 83, + 542, + 528, + 554 + ], + "spans": [ + { + "bbox": [ + 83, + 542, + 528, + 554 + ], + "score": 1.0, + "content": "each layer, rather than at the end of a full inference pass. (3) The TEE has to efficiently load precomputed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 553, + 518, + 565 + ], + "spans": [ + { + "bbox": [ + 83, + 553, + 518, + 565 + ], + "score": 1.0, + "content": "unblinding factors, which requires either a large amount of RAM, or a fast access to disk (e.g., a PCIe SSD).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 82, + 519, + 528, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 83, + 569, + 527, + 592 + ], + "lines": [ + { + "bbox": [ + 83, + 570, + 527, + 582 + ], + "spans": [ + { + "bbox": [ + 83, + 570, + 527, + 582 + ], + "score": 1.0, + "content": "Slalom’s security is given by the following results. Formal definitions and proofs are in Appendix B. Let negl be", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 580, + 525, + 593 + ], + "spans": [ + { + "bbox": [ + 82, + 580, + 229, + 593 + ], + "score": 1.0, + "content": "a negligible function (for any integer", + "type": "text" + }, + { + "bbox": [ + 230, + 581, + 253, + 591 + ], + "score": 0.89, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 580, + 342, + 593 + ], + "score": 1.0, + "content": "there exists an integer", + "type": "text" + }, + { + "bbox": [ + 342, + 581, + 356, + 592 + ], + "score": 0.89, + "content": "N _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 580, + 420, + 593 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 420, + 581, + 452, + 592 + ], + "score": 0.89, + "content": "x > N _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 580, + 456, + 593 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 456, + 581, + 525, + 593 + ], + "score": 0.93, + "content": "| \\mathrm { n e g l } ( x ) | < \\bar { 1 } / { x ^ { c } } )", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 82, + 570, + 527, + 593 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 595, + 529, + 640 + ], + "lines": [ + { + "bbox": [ + 82, + 595, + 529, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 595, + 375, + 609 + ], + "score": 1.0, + "content": "Theorem 3.2. Let Slalom be the protocol from Figure 1 (right), where", + "type": "text" + }, + { + "bbox": [ + 375, + 596, + 384, + 606 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 595, + 408, + 609 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 408, + 598, + 415, + 606 + ], + "score": 0.63, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 595, + 529, + 609 + ], + "score": 1.0, + "content": "-layer DNN, and Freivalds’", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 607, + 528, + 619 + ], + "spans": [ + { + "bbox": [ + 83, + 607, + 374, + 619 + ], + "score": 1.0, + "content": "algorithm is repeated k times per layer with random vectors drawn from", + "type": "text" + }, + { + "bbox": [ + 375, + 607, + 401, + 618 + ], + "score": 0.87, + "content": "\\mathbb { S } \\subseteq \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 607, + 528, + 619 + ], + "score": 1.0, + "content": ". Assume all random values are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 81, + 617, + 527, + 631 + ], + "spans": [ + { + "bbox": [ + 81, + 617, + 311, + 631 + ], + "score": 1.0, + "content": "generated using a secure PRNG with security parameter", + "type": "text" + }, + { + "bbox": [ + 311, + 619, + 318, + 627 + ], + "score": 0.72, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 617, + 518, + 631 + ], + "score": 1.0, + "content": ". Then, Slalom is a secure outsourcing scheme for", + "type": "text" + }, + { + "bbox": [ + 518, + 618, + 527, + 628 + ], + "score": 0.79, + "content": "F", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 627, + 492, + 642 + ], + "spans": [ + { + "bbox": [ + 82, + 627, + 271, + 642 + ], + "score": 1.0, + "content": "between a TEE and an untrusted co-processor", + "type": "text" + }, + { + "bbox": [ + 271, + 630, + 278, + 639 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 627, + 349, + 642 + ], + "score": 1.0, + "content": "with privacy and", + "type": "text" + }, + { + "bbox": [ + 350, + 630, + 354, + 639 + ], + "score": 0.66, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 627, + 406, + 642 + ], + "score": 1.0, + "content": "-integrity for", + "type": "text" + }, + { + "bbox": [ + 406, + 629, + 488, + 641 + ], + "score": 0.92, + "content": "t = n / \\vert \\mathbb { S } \\vert ^ { k } - \\mathrm { n e g l } ( \\lambda )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 627, + 492, + 642 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 81, + 595, + 529, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 644, + 528, + 677 + ], + "lines": [ + { + "bbox": [ + 83, + 643, + 528, + 657 + ], + "spans": [ + { + "bbox": [ + 83, + 643, + 438, + 657 + ], + "score": 1.0, + "content": "Corollary 3.3. Assuming the TEE is secure (i.e., it acts as a trusted third party hosted by", + "type": "text" + }, + { + "bbox": [ + 439, + 644, + 447, + 654 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 643, + 528, + 657 + ], + "score": 1.0, + "content": "), Slalom is a secure", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 82, + 654, + 529, + 668 + ], + "spans": [ + { + "bbox": [ + 82, + 654, + 259, + 668 + ], + "score": 1.0, + "content": "outsourcing scheme between a remote client", + "type": "text" + }, + { + "bbox": [ + 259, + 657, + 265, + 664 + ], + "score": 0.8, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 654, + 311, + 668 + ], + "score": 1.0, + "content": "and server", + "type": "text" + }, + { + "bbox": [ + 311, + 657, + 318, + 665 + ], + "score": 0.82, + "content": "\\boldsymbol { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 654, + 388, + 668 + ], + "score": 1.0, + "content": "with privacy and", + "type": "text" + }, + { + "bbox": [ + 388, + 656, + 393, + 665 + ], + "score": 0.59, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 654, + 443, + 668 + ], + "score": 1.0, + "content": "-integrity for", + "type": "text" + }, + { + "bbox": [ + 443, + 655, + 525, + 667 + ], + "score": 0.92, + "content": "t = { \\boldsymbol { \\mathscr { n } } } / { \\boldsymbol { | \\mathbb { S } } } { \\boldsymbol { \\vert \\sp k - \\mathrm { n e g l } } } ( \\lambda )", + "type": "inline_equation" + }, + { + "bbox": [ + 526, + 654, + 529, + 668 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 82, + 665, + 397, + 680 + ], + "spans": [ + { + "bbox": [ + 82, + 665, + 133, + 680 + ], + "score": 1.0, + "content": "If the model", + "type": "text" + }, + { + "bbox": [ + 134, + 667, + 142, + 676 + ], + "score": 0.62, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 665, + 214, + 680 + ], + "score": 1.0, + "content": "is the property of", + "type": "text" + }, + { + "bbox": [ + 214, + 668, + 222, + 676 + ], + "score": 0.73, + "content": "\\boldsymbol { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 665, + 397, + 680 + ], + "score": 1.0, + "content": ", the scheme further satisfies model privacy.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 82, + 643, + 529, + 680 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 85, + 139, + 231, + 152 + ], + "lines": [ + { + "bbox": [ + 82, + 137, + 233, + 154 + ], + "spans": [ + { + "bbox": [ + 82, + 137, + 233, + 154 + ], + "score": 1.0, + "content": "4 EMPIRICAL EVALUATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 164, + 528, + 198 + ], + "lines": [ + { + "bbox": [ + 83, + 165, + 528, + 177 + ], + "spans": [ + { + "bbox": [ + 83, + 165, + 528, + 177 + ], + "score": 1.0, + "content": "We evaluate Slalom on real Intel SGX hardware, on micro-benchmarks and a sample application (ImageNet", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 83, + 176, + 528, + 188 + ], + "spans": [ + { + "bbox": [ + 83, + 176, + 528, + 188 + ], + "score": 1.0, + "content": "inference with VGG16, MobileNet and ResNet models). Our aim is to show that, compared to a baseline that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 82, + 186, + 518, + 200 + ], + "spans": [ + { + "bbox": [ + 82, + 186, + 518, + 200 + ], + "score": 1.0, + "content": "runs inference fully in the TEE, outsourcing linear layers increases performance without sacrificing security.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 84, + 213, + 186, + 224 + ], + "lines": [ + { + "bbox": [ + 83, + 213, + 187, + 225 + ], + "spans": [ + { + "bbox": [ + 83, + 213, + 187, + 225 + ], + "score": 1.0, + "content": "4.1 IMPLEMENTATION", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 84, + 234, + 528, + 300 + ], + "lines": [ + { + "bbox": [ + 82, + 234, + 528, + 247 + ], + "spans": [ + { + "bbox": [ + 82, + 234, + 528, + 247 + ], + "score": 1.0, + "content": "As enclaves cannot access most OS features (e.g., multi-threading, disk and driver IO), porting a large framework", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 82, + 244, + 528, + 258 + ], + "spans": [ + { + "bbox": [ + 82, + 244, + 463, + 258 + ], + "score": 1.0, + "content": "such as TensorFlow or Intel’s MKL-DNN to SGX is hard. Instead, we designed a lightweight", + "type": "text" + }, + { + "bbox": [ + 464, + 245, + 484, + 255 + ], + "score": 0.84, + "content": "\\mathrm { C } { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 244, + 528, + 258 + ], + "score": 1.0, + "content": "library for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 255, + 528, + 268 + ], + "spans": [ + { + "bbox": [ + 82, + 255, + 528, + 268 + ], + "score": 1.0, + "content": "feed-forward networks based on Eigen, a linear-algebra library which TensorFlow uses as a CPU backend. Our", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 82, + 265, + 530, + 281 + ], + "spans": [ + { + "bbox": [ + 82, + 265, + 530, + 281 + ], + "score": 1.0, + "content": "library implements the forward pass of DNNs, with support for dense layers, standard and separable convolutions,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 82, + 277, + 529, + 290 + ], + "spans": [ + { + "bbox": [ + 82, + 277, + 529, + 290 + ], + "score": 1.0, + "content": "pooling, and activations. When run on a native CPU (without SGX), its performance is comparable to TensorFlow", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 83, + 289, + 507, + 301 + ], + "spans": [ + { + "bbox": [ + 83, + 289, + 507, + 301 + ], + "score": 1.0, + "content": "on CPU (compiled with AVX). Our code is available at https://github.com/ftramer/slalom.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 84, + 305, + 528, + 362 + ], + "lines": [ + { + "bbox": [ + 82, + 304, + 529, + 319 + ], + "spans": [ + { + "bbox": [ + 82, + 304, + 218, + 319 + ], + "score": 1.0, + "content": "Slalom performs arithmetic over", + "type": "text" + }, + { + "bbox": [ + 219, + 306, + 231, + 318 + ], + "score": 0.9, + "content": "\\mathbb { Z } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 304, + 250, + 319 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 250, + 305, + 302, + 317 + ], + "score": 0.9, + "content": "p = 2 ^ { 2 4 } - 3", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 304, + 529, + 319 + ], + "score": 1.0, + "content": ". For integrity, we apply Freivalds’ check twice to each", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 81, + 314, + 529, + 330 + ], + "spans": [ + { + "bbox": [ + 81, + 314, + 108, + 330 + ], + "score": 1.0, + "content": "layer", + "type": "text" + }, + { + "bbox": [ + 108, + 318, + 135, + 328 + ], + "score": 0.86, + "content": "k = 2 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 314, + 240, + 330 + ], + "score": 1.0, + "content": "), with random values from", + "type": "text" + }, + { + "bbox": [ + 241, + 317, + 306, + 330 + ], + "score": 0.89, + "content": "\\mathbb { S } = [ - 2 ^ { 1 9 } , 2 ^ { 1 9 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 314, + 529, + 330 + ], + "score": 1.0, + "content": ", to achieve 40 bits of statistical soundness per layer (see", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 83, + 329, + 529, + 341 + ], + "spans": [ + { + "bbox": [ + 83, + 329, + 411, + 341 + ], + "score": 1.0, + "content": "Appendix F for details on the selection of these parameters). For a 50-layer DNN,", + "type": "text" + }, + { + "bbox": [ + 412, + 329, + 420, + 339 + ], + "score": 0.74, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 329, + 529, + 341 + ], + "score": 1.0, + "content": "has a chance of less than 1", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 339, + 530, + 352 + ], + "spans": [ + { + "bbox": [ + 82, + 339, + 530, + 352 + ], + "score": 1.0, + "content": "in 22 billion of fooling the TEE on any incorrect DNN evaluation (a slightly better guarantee than in SafetyNets).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 83, + 351, + 488, + 362 + ], + "spans": [ + { + "bbox": [ + 83, + 351, + 488, + 362 + ], + "score": 1.0, + "content": "For privacy, we use AES-CTR and AES-GCM to generate, encrypt and authenticate blinding factors.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 84, + 376, + 137, + 387 + ], + "lines": [ + { + "bbox": [ + 82, + 375, + 138, + 389 + ], + "spans": [ + { + "bbox": [ + 82, + 375, + 138, + 389 + ], + "score": 1.0, + "content": "4.2 SETUP", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 84, + 397, + 528, + 442 + ], + "lines": [ + { + "bbox": [ + 82, + 397, + 529, + 411 + ], + "spans": [ + { + "bbox": [ + 82, + 397, + 529, + 411 + ], + "score": 1.0, + "content": "We use an Intel Core i7-6700 Skylake 3.40GHz processor with 8GB of RAM, a desktop processor with SGX", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 82, + 407, + 529, + 421 + ], + "spans": [ + { + "bbox": [ + 82, + 407, + 529, + 421 + ], + "score": 1.0, + "content": "support. The outsourced computations are performed on a co-located Nvidia TITAN XP GPU. Due to a lack", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 82, + 418, + 529, + 433 + ], + "spans": [ + { + "bbox": [ + 82, + 418, + 529, + 433 + ], + "score": 1.0, + "content": "of native internal multi-threading in SGX, we run our TEE in a single CPU thread. We discuss challenges for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 83, + 430, + 438, + 443 + ], + "spans": [ + { + "bbox": [ + 83, + 430, + 438, + 443 + ], + "score": 1.0, + "content": "efficient parallelization in Appendix H. 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(2016) models (with fused Batch Normalization layers when applicable).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 83, + 499, + 528, + 555 + ], + "lines": [ + { + "bbox": [ + 83, + 499, + 528, + 511 + ], + "spans": [ + { + "bbox": [ + 83, + 499, + 528, + 511 + ], + "score": 1.0, + "content": "MobileNet, a model tailored for low compute devices, serves as a worst-case benchmark for Slalom, as the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 510, + 529, + 523 + ], + "spans": [ + { + "bbox": [ + 82, + 510, + 529, + 523 + ], + "score": 1.0, + "content": "model’s design aggressively minimizes the amount of computation performed per layer. 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In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 82, + 656, + 528, + 670 + ], + "spans": [ + { + "bbox": [ + 82, + 656, + 528, + 670 + ], + "score": 1.0, + "content": "all cases, the baseline computation is performed in single-precision floating point, and the verification algorithms", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 83, + 667, + 392, + 681 + ], + "spans": [ + { + "bbox": [ + 83, + 667, + 392, + 681 + ], + "score": 1.0, + "content": "repeat Freivalds’ check so as to attain at least 40 bits of statistical soundness.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 84, + 84, + 270, + 95 + ], + "lines": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 279, + 699, + 285, + 707 + ], + "lines": [ + { + "bbox": [ + 279, + 698, + 286, + 710 + ], + "spans": [ + { + "bbox": [ + 279, + 698, + 286, + 710 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 85, + 139, + 231, + 152 + ], + "lines": [ + { + "bbox": [ + 82, + 137, + 233, + 154 + ], + "spans": [ + { + "bbox": [ + 82, + 137, + 233, + 154 + ], + "score": 1.0, + "content": "4 EMPIRICAL EVALUATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 164, + 528, + 198 + ], + "lines": [ + { + "bbox": [ + 83, + 165, + 528, + 177 + ], + "spans": [ + { + "bbox": [ + 83, + 165, + 528, + 177 + ], + "score": 1.0, + "content": "We evaluate Slalom on real Intel SGX hardware, on micro-benchmarks and a sample application (ImageNet", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 83, + 176, + 528, + 188 + ], + "spans": [ + { + "bbox": [ + 83, + 176, + 528, + 188 + ], + "score": 1.0, + "content": "inference with VGG16, MobileNet and ResNet models). 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Instead, we designed a lightweight", + "type": "text" + }, + { + "bbox": [ + 464, + 245, + 484, + 255 + ], + "score": 0.84, + "content": "\\mathrm { C } { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 244, + 528, + 258 + ], + "score": 1.0, + "content": "library for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 255, + 528, + 268 + ], + "spans": [ + { + "bbox": [ + 82, + 255, + 528, + 268 + ], + "score": 1.0, + "content": "feed-forward networks based on Eigen, a linear-algebra library which TensorFlow uses as a CPU backend. 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For integrity, we apply Freivalds’ check twice to each", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 81, + 314, + 529, + 330 + ], + "spans": [ + { + "bbox": [ + 81, + 314, + 108, + 330 + ], + "score": 1.0, + "content": "layer", + "type": "text" + }, + { + "bbox": [ + 108, + 318, + 135, + 328 + ], + "score": 0.86, + "content": "k = 2 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 314, + 240, + 330 + ], + "score": 1.0, + "content": "), with random values from", + "type": "text" + }, + { + "bbox": [ + 241, + 317, + 306, + 330 + ], + "score": 0.89, + "content": "\\mathbb { S } = [ - 2 ^ { 1 9 } , 2 ^ { 1 9 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 314, + 529, + 330 + ], + "score": 1.0, + "content": ", to achieve 40 bits of statistical soundness per layer (see", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 83, + 329, + 529, + 341 + ], + "spans": [ + { + "bbox": [ + 83, + 329, + 411, + 341 + ], + "score": 1.0, + "content": "Appendix F for details on the selection of these parameters). For a 50-layer DNN,", + "type": "text" + }, + { + "bbox": [ + 412, + 329, + 420, + 339 + ], + "score": 0.74, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 329, + 529, + 341 + ], + "score": 1.0, + "content": "has a chance of less than 1", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 339, + 530, + 352 + ], + "spans": [ + { + "bbox": [ + 82, + 339, + 530, + 352 + ], + "score": 1.0, + "content": "in 22 billion of fooling the TEE on any incorrect DNN evaluation (a slightly better guarantee than in SafetyNets).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 83, + 351, + 488, + 362 + ], + "spans": [ + { + "bbox": [ + 83, + 351, + 488, + 362 + ], + "score": 1.0, + "content": "For privacy, we use AES-CTR and AES-GCM to generate, encrypt and authenticate blinding factors.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 81, + 304, + 530, + 362 + ] + }, + { + "type": "title", + "bbox": [ + 84, + 376, + 137, + 387 + ], + "lines": [ + { + "bbox": [ + 82, + 375, + 138, + 389 + ], + "spans": [ + { + "bbox": [ + 82, + 375, + 138, + 389 + ], + "score": 1.0, + "content": "4.2 SETUP", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 84, + 397, + 528, + 442 + ], + "lines": [ + { + "bbox": [ + 82, + 397, + 529, + 411 + ], + "spans": [ + { + "bbox": [ + 82, + 397, + 529, + 411 + ], + "score": 1.0, + "content": "We use an Intel Core i7-6700 Skylake 3.40GHz processor with 8GB of RAM, a desktop processor with SGX", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 82, + 407, + 529, + 421 + ], + "spans": [ + { + "bbox": [ + 82, + 407, + 529, + 421 + ], + "score": 1.0, + "content": "support. The outsourced computations are performed on a co-located Nvidia TITAN XP GPU. Due to a lack", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 82, + 418, + 529, + 433 + ], + "spans": [ + { + "bbox": [ + 82, + 418, + 529, + 433 + ], + "score": 1.0, + "content": "of native internal multi-threading in SGX, we run our TEE in a single CPU thread. We discuss challenges for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 83, + 430, + 438, + 443 + ], + "spans": [ + { + "bbox": [ + 83, + 430, + 438, + 443 + ], + "score": 1.0, + "content": "efficient parallelization in Appendix H. 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We also consider a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 520, + 528, + 534 + ], + "spans": [ + { + "bbox": [ + 82, + 520, + 528, + 534 + ], + "score": 1.0, + "content": "“fused” variant of MobileNet with no activation between depthwise and pointwise convolutions. Removing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 82, + 532, + 529, + 545 + ], + "spans": [ + { + "bbox": [ + 82, + 532, + 529, + 545 + ], + "score": 1.0, + "content": "these activations improves convergence and accuracy (Chollet, 2017; Sheng et al., 2018), while also making the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 543, + 499, + 555 + ], + "spans": [ + { + "bbox": [ + 82, + 543, + 499, + 555 + ], + "score": 1.0, + "content": "network more outsourcing-friendly (i.e., it is possible to verify a separable convolution in a single step).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 82, + 499, + 529, + 555 + ] + }, + { + "type": "text", + "bbox": [ + 83, + 559, + 527, + 582 + ], + "lines": [ + { + "bbox": [ + 83, + 558, + 528, + 573 + ], + "spans": [ + { + "bbox": [ + 83, + 558, + 528, + 573 + ], + "score": 1.0, + "content": "Our evaluation focuses on throughput (number of forward passes per second). We also discuss energy efficiency", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 571, + 529, + 584 + ], + "spans": [ + { + "bbox": [ + 82, + 571, + 470, + 584 + ], + "score": 1.0, + "content": "in Appendix C to account for hardware differences between our baseline (TEE only) and Slalom", + "type": "text" + }, + { + "bbox": [ + 471, + 571, + 524, + 582 + ], + "score": 0.52, + "content": "\\mathrm { ( T E E + G P U ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 524, + 571, + 529, + 584 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 82, + 558, + 529, + 584 + ] + }, + { + "type": "title", + "bbox": [ + 84, + 597, + 147, + 608 + ], + "lines": [ + { + "bbox": [ + 82, + 595, + 149, + 610 + ], + "spans": [ + { + "bbox": [ + 82, + 595, + 149, + 610 + ], + "score": 1.0, + "content": "4.3 RESULTS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 84, + 613, + 528, + 679 + ], + "lines": [ + { + "bbox": [ + 82, + 612, + 530, + 626 + ], + "spans": [ + { + "bbox": [ + 82, + 612, + 530, + 626 + ], + "score": 1.0, + "content": "Micro-Benchmarks. Our micro-benchmark suite consists of square matrix products of increasing dimensions,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 82, + 624, + 529, + 636 + ], + "spans": [ + { + "bbox": [ + 82, + 624, + 529, + 636 + ], + "score": 1.0, + "content": "convolutional operations performed by VGG16, and separable convolutions performed by MobileNet. In all", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 634, + 529, + 648 + ], + "spans": [ + { + "bbox": [ + 82, + 634, + 529, + 648 + ], + "score": 1.0, + "content": "cases, the data is pre-loaded inside an enclave, so we only measure the in-enclave execution time. Figure 2 plots", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 83, + 646, + 528, + 659 + ], + "spans": [ + { + "bbox": [ + 83, + 646, + 528, + 659 + ], + "score": 1.0, + "content": "the relative speedups of various verification strategies over the cost of computing the linear operation directly. In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 82, + 656, + 528, + 670 + ], + "spans": [ + { + "bbox": [ + 82, + 656, + 528, + 670 + ], + "score": 1.0, + "content": "all cases, the baseline computation is performed in single-precision floating point, and the verification algorithms", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 83, + 667, + 392, + 681 + ], + "spans": [ + { + "bbox": [ + 83, + 667, + 392, + 681 + ], + "score": 1.0, + "content": "repeat Freivalds’ check so as to attain at least 40 bits of statistical soundness.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 82, + 612, + 530, + 681 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 85, + 138, + 524, + 249 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 85, + 138, + 524, + 249 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 85, + 138, + 524, + 249 + ], + "spans": [ + { + "bbox": [ + 85, + 138, + 524, + 249 + ], + "score": 0.968, + "type": "image", + "image_path": "e479efd0e691ddcb877c1c08f199812ca305e3227b9c63d4af690b286c7b0522.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 85, + 138, + 524, + 175.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 85, + 175.0, + 524, + 212.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 85, + 212.0, + 524, + 249.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 84, + 253, + 528, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 82, + 252, + 528, + 266 + ], + "spans": [ + { + "bbox": [ + 82, + 252, + 528, + 266 + ], + "score": 1.0, + "content": "Figure 2: Micro benchmarks on Intel SGX. We plot the relative speedup of verifying the result of a linear", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 82, + 263, + 529, + 277 + ], + "spans": [ + { + "bbox": [ + 82, + 263, + 529, + 277 + ], + "score": 1.0, + "content": "operator compared to computing it entirely in the enclave. The dotted line shows the throughput obtained for a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 275, + 433, + 288 + ], + "spans": [ + { + "bbox": [ + 83, + 275, + 433, + 288 + ], + "score": 1.0, + "content": "direct computation. “Fused” separable convolutions contain no intermediate activation.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 84, + 295, + 528, + 329 + ], + "lines": [ + { + "bbox": [ + 82, + 293, + 528, + 310 + ], + "spans": [ + { + "bbox": [ + 82, + 293, + 398, + 310 + ], + "score": 1.0, + "content": "For square matrices of dimensions up to 2048, verifying an outsourced result is", + "type": "text" + }, + { + "bbox": [ + 399, + 296, + 413, + 306 + ], + "score": 0.87, + "content": "4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 293, + 424, + 310 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 424, + 296, + 438, + 306 + ], + "score": 0.86, + "content": "8 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 293, + 528, + 310 + ], + "score": 1.0, + "content": "faster than computing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 305, + 528, + 320 + ], + "spans": [ + { + "bbox": [ + 82, + 305, + 528, + 320 + ], + "score": 1.0, + "content": "it. For larger matrices, we exceed the limit of SGX’s DRAM, so the enclave resorts to expensive paging which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 82, + 317, + 367, + 331 + ], + "spans": [ + { + "bbox": [ + 82, + 317, + 367, + 331 + ], + "score": 1.0, + "content": "drastically reduces performance both for computation and verification.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 84, + 334, + 528, + 390 + ], + "lines": [ + { + "bbox": [ + 82, + 334, + 529, + 347 + ], + "spans": [ + { + "bbox": [ + 82, + 334, + 529, + 347 + ], + "score": 1.0, + "content": "For convolutions (standard or separable), we achieve large savings with outsourcing if Freivalds’ algorithm is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 82, + 344, + 529, + 359 + ], + "spans": [ + { + "bbox": [ + 82, + 344, + 529, + 359 + ], + "score": 1.0, + "content": "applied with preprocessing. The savings get higher as the number of channels increases. Without preprocessing,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 82, + 356, + 529, + 369 + ], + "spans": [ + { + "bbox": [ + 82, + 356, + 268, + 369 + ], + "score": 1.0, + "content": "Freivalds’ algorithm results in savings when", + "type": "text" + }, + { + "bbox": [ + 268, + 358, + 284, + 367 + ], + "score": 0.86, + "content": "c _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 356, + 529, + 369 + ], + "score": 1.0, + "content": "is large. Due to SGX’s small PRM, batched verification is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 83, + 368, + 528, + 380 + ], + "spans": [ + { + "bbox": [ + 83, + 368, + 528, + 380 + ], + "score": 1.0, + "content": "only effective for operators with small memory footprints. As expected, “truly” separable convolutions (with no", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 83, + 379, + 491, + 390 + ], + "spans": [ + { + "bbox": [ + 83, + 379, + 491, + 390 + ], + "score": 1.0, + "content": "intermediate non-linearity) are much faster to verify, as they can be viewed as a single linear operator.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 84, + 398, + 528, + 443 + ], + "lines": [ + { + "bbox": [ + 82, + 398, + 529, + 412 + ], + "spans": [ + { + "bbox": [ + 82, + 398, + 529, + 412 + ], + "score": 1.0, + "content": "Verifiable Inference. Figure 3 shows the throughout of end-to-end forward passes in two neural networks,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 408, + 529, + 424 + ], + "spans": [ + { + "bbox": [ + 82, + 408, + 529, + 424 + ], + "score": 1.0, + "content": "VGG16 and MobileNet. For integrity, we compare the secure baseline (executing the DNN fully in the enclave) to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 83, + 420, + 528, + 433 + ], + "spans": [ + { + "bbox": [ + 83, + 420, + 528, + 433 + ], + "score": 1.0, + "content": "two variants of the Slalom algorithm in Figure 1. The first (in red) applies Freivalds’ algorithm “on-the-fly”, while", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 432, + 528, + 446 + ], + "spans": [ + { + "bbox": [ + 82, + 432, + 528, + 446 + ], + "score": 1.0, + "content": "the second more efficient variant (in orange) pre-computes part of Freivalds’ check as described in Section 3.2.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 84, + 448, + 528, + 515 + ], + "lines": [ + { + "bbox": [ + 82, + 448, + 529, + 462 + ], + "spans": [ + { + "bbox": [ + 82, + 448, + 529, + 462 + ], + "score": 1.0, + "content": "The VGG16 network is much larger (500MB) than SGX’s PRM. As a result, there is a large overhead on the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 82, + 460, + 527, + 473 + ], + "spans": [ + { + "bbox": [ + 82, + 460, + 510, + 473 + ], + "score": 1.0, + "content": "forward pass and verification without preprocessing. If the enclave securely stores preprocessed products", + "type": "text" + }, + { + "bbox": [ + 510, + 460, + 527, + 470 + ], + "score": 0.79, + "content": "W r", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 470, + 529, + 484 + ], + "spans": [ + { + "bbox": [ + 82, + 470, + 452, + 484 + ], + "score": 1.0, + "content": "for all network weights, we drastically reduce the memory footprint and achieve up to a", + "type": "text" + }, + { + "bbox": [ + 452, + 471, + 479, + 481 + ], + "score": 0.87, + "content": "2 0 . 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 470, + 529, + 484 + ], + "score": 1.0, + "content": "increase in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 481, + 529, + 495 + ], + "spans": [ + { + "bbox": [ + 82, + 481, + 529, + 495 + ], + "score": 1.0, + "content": "throughput. We also ran the lower-half of the VGG16 network (without the fully connected layers), a common", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 492, + 529, + 505 + ], + "spans": [ + { + "bbox": [ + 82, + 492, + 529, + 505 + ], + "score": 1.0, + "content": "approach for extracting features for transfer learning or object recognition (Liu et al., 2016). This part fits in the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 502, + 495, + 516 + ], + "spans": [ + { + "bbox": [ + 82, + 502, + 495, + 516 + ], + "score": 1.0, + "content": "PRM, and we thus achieve higher throughput for in-enclave forward passes and on-the-fly verification.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 84, + 520, + 528, + 565 + ], + "lines": [ + { + "bbox": [ + 82, + 520, + 528, + 533 + ], + "spans": [ + { + "bbox": [ + 82, + 520, + 227, + 533 + ], + "score": 1.0, + "content": "For MobileNet, we achieve between", + "type": "text" + }, + { + "bbox": [ + 227, + 521, + 249, + 531 + ], + "score": 0.88, + "content": "3 . 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 520, + 266, + 533 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 267, + 521, + 289, + 531 + ], + "score": 0.87, + "content": "6 . 4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 520, + 528, + 533 + ], + "score": 1.0, + "content": "speedups when using Slalom for verifiable inference (for the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 531, + 528, + 543 + ], + "spans": [ + { + "bbox": [ + 82, + 531, + 528, + 543 + ], + "score": 1.0, + "content": "standard or “fused” model, respectively). The speedups are smaller than for VGG16, as MobileNet performs", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 542, + 528, + 555 + ], + "spans": [ + { + "bbox": [ + 83, + 542, + 528, + 555 + ], + "score": 1.0, + "content": "much fewer operations per layer (verifying a linear layer requires computing at least two multiplications for each", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 552, + 519, + 567 + ], + "spans": [ + { + "bbox": [ + 82, + 552, + 519, + 567 + ], + "score": 1.0, + "content": "input and output. The closer the forward pass gets to that lower-bound, the less we can save by outsourcing).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 84, + 574, + 528, + 641 + ], + "lines": [ + { + "bbox": [ + 83, + 573, + 528, + 587 + ], + "spans": [ + { + "bbox": [ + 83, + 573, + 528, + 587 + ], + "score": 1.0, + "content": "Private Inference. We further benchmark the cost of private DNN inference, where inputs of outsourced", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 583, + 529, + 599 + ], + "spans": [ + { + "bbox": [ + 82, + 583, + 529, + 599 + ], + "score": 1.0, + "content": "linear layers are additionally blinded. Blinding and unblinding each layer’s inputs and outputs is costly,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 82, + 595, + 528, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 595, + 528, + 609 + ], + "score": 1.0, + "content": "especially in SGX due to the extra in-enclave memory reads and writes. Nevertheless, for VGG16 and the fused", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 83, + 607, + 529, + 620 + ], + "spans": [ + { + "bbox": [ + 83, + 607, + 444, + 620 + ], + "score": 1.0, + "content": "MobileNet variant without intermediate activations, we achieve respective speedups of", + "type": "text" + }, + { + "bbox": [ + 444, + 607, + 471, + 618 + ], + "score": 0.87, + "content": "1 3 . 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 607, + 490, + 620 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 607, + 512, + 618 + ], + "score": 0.87, + "content": "5 . 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 513, + 607, + 529, + 620 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 82, + 617, + 529, + 631 + ], + "spans": [ + { + "bbox": [ + 82, + 617, + 326, + 631 + ], + "score": 1.0, + "content": "private outsourcing (in black in Figure 3), and speedups of", + "type": "text" + }, + { + "bbox": [ + 327, + 618, + 353, + 629 + ], + "score": 0.88, + "content": "1 0 . 7 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 617, + 372, + 631 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 372, + 618, + 394, + 628 + ], + "score": 0.88, + "content": "4 . 1 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 617, + 529, + 631 + ], + "score": 1.0, + "content": "when also ensuring integrity (in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 82, + 628, + 470, + 643 + ], + "spans": [ + { + "bbox": [ + 82, + 628, + 470, + 643 + ], + "score": 1.0, + "content": "purple). For this benchmark, the precomputed unblinding factor are stored in untrusted memory.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 85, + 646, + 527, + 679 + ], + "lines": [ + { + "bbox": [ + 82, + 644, + 529, + 660 + ], + "spans": [ + { + "bbox": [ + 82, + 644, + 529, + 660 + ], + "score": 1.0, + "content": "We performed the same experiments on a standard CPU (i.e., without SGX) and find that Slalom’s improvements", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 656, + 529, + 669 + ], + "spans": [ + { + "bbox": [ + 82, + 656, + 529, + 669 + ], + "score": 1.0, + "content": "are even higher in non-resource-constrained or multi-threaded environments (see Appendix G-H). Slalom’s", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 83, + 668, + 471, + 680 + ], + "spans": [ + { + "bbox": [ + 83, + 668, + 471, + 680 + ], + "score": 1.0, + "content": "improvements over the baseline also hold when accounting for energy efficiency (see Section C).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 84, + 85, + 270, + 95 + ], + "lines": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 279, + 699, + 285, + 707 + ], + "lines": [ + { + "bbox": [ + 279, + 698, + 286, + 709 + ], + "spans": [ + { + "bbox": [ + 279, + 698, + 286, + 709 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 85, + 138, + 524, + 249 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 85, + 138, + 524, + 249 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 85, + 138, + 524, + 249 + ], + "spans": [ + { + "bbox": [ + 85, + 138, + 524, + 249 + ], + "score": 0.968, + "type": "image", + "image_path": "e479efd0e691ddcb877c1c08f199812ca305e3227b9c63d4af690b286c7b0522.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 85, + 138, + 524, + 175.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 85, + 175.0, + 524, + 212.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 85, + 212.0, + 524, + 249.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 84, + 253, + 528, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 82, + 252, + 528, + 266 + ], + "spans": [ + { + "bbox": [ + 82, + 252, + 528, + 266 + ], + "score": 1.0, + "content": "Figure 2: Micro benchmarks on Intel SGX. We plot the relative speedup of verifying the result of a linear", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 82, + 263, + 529, + 277 + ], + "spans": [ + { + "bbox": [ + 82, + 263, + 529, + 277 + ], + "score": 1.0, + "content": "operator compared to computing it entirely in the enclave. The dotted line shows the throughput obtained for a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 275, + 433, + 288 + ], + "spans": [ + { + "bbox": [ + 83, + 275, + 433, + 288 + ], + "score": 1.0, + "content": "direct computation. “Fused” separable convolutions contain no intermediate activation.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 84, + 295, + 528, + 329 + ], + "lines": [ + { + "bbox": [ + 82, + 293, + 528, + 310 + ], + "spans": [ + { + "bbox": [ + 82, + 293, + 398, + 310 + ], + "score": 1.0, + "content": "For square matrices of dimensions up to 2048, verifying an outsourced result is", + "type": "text" + }, + { + "bbox": [ + 399, + 296, + 413, + 306 + ], + "score": 0.87, + "content": "4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 293, + 424, + 310 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 424, + 296, + 438, + 306 + ], + "score": 0.86, + "content": "8 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 293, + 528, + 310 + ], + "score": 1.0, + "content": "faster than computing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 305, + 528, + 320 + ], + "spans": [ + { + "bbox": [ + 82, + 305, + 528, + 320 + ], + "score": 1.0, + "content": "it. For larger matrices, we exceed the limit of SGX’s DRAM, so the enclave resorts to expensive paging which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 82, + 317, + 367, + 331 + ], + "spans": [ + { + "bbox": [ + 82, + 317, + 367, + 331 + ], + "score": 1.0, + "content": "drastically reduces performance both for computation and verification.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 82, + 293, + 528, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 334, + 528, + 390 + ], + "lines": [ + { + "bbox": [ + 82, + 334, + 529, + 347 + ], + "spans": [ + { + "bbox": [ + 82, + 334, + 529, + 347 + ], + "score": 1.0, + "content": "For convolutions (standard or separable), we achieve large savings with outsourcing if Freivalds’ algorithm is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 82, + 344, + 529, + 359 + ], + "spans": [ + { + "bbox": [ + 82, + 344, + 529, + 359 + ], + "score": 1.0, + "content": "applied with preprocessing. The savings get higher as the number of channels increases. Without preprocessing,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 82, + 356, + 529, + 369 + ], + "spans": [ + { + "bbox": [ + 82, + 356, + 268, + 369 + ], + "score": 1.0, + "content": "Freivalds’ algorithm results in savings when", + "type": "text" + }, + { + "bbox": [ + 268, + 358, + 284, + 367 + ], + "score": 0.86, + "content": "c _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 356, + 529, + 369 + ], + "score": 1.0, + "content": "is large. Due to SGX’s small PRM, batched verification is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 83, + 368, + 528, + 380 + ], + "spans": [ + { + "bbox": [ + 83, + 368, + 528, + 380 + ], + "score": 1.0, + "content": "only effective for operators with small memory footprints. As expected, “truly” separable convolutions (with no", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 83, + 379, + 491, + 390 + ], + "spans": [ + { + "bbox": [ + 83, + 379, + 491, + 390 + ], + "score": 1.0, + "content": "intermediate non-linearity) are much faster to verify, as they can be viewed as a single linear operator.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 82, + 334, + 529, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 398, + 528, + 443 + ], + "lines": [ + { + "bbox": [ + 82, + 398, + 529, + 412 + ], + "spans": [ + { + "bbox": [ + 82, + 398, + 529, + 412 + ], + "score": 1.0, + "content": "Verifiable Inference. Figure 3 shows the throughout of end-to-end forward passes in two neural networks,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 408, + 529, + 424 + ], + "spans": [ + { + "bbox": [ + 82, + 408, + 529, + 424 + ], + "score": 1.0, + "content": "VGG16 and MobileNet. For integrity, we compare the secure baseline (executing the DNN fully in the enclave) to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 83, + 420, + 528, + 433 + ], + "spans": [ + { + "bbox": [ + 83, + 420, + 528, + 433 + ], + "score": 1.0, + "content": "two variants of the Slalom algorithm in Figure 1. The first (in red) applies Freivalds’ algorithm “on-the-fly”, while", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 432, + 528, + 446 + ], + "spans": [ + { + "bbox": [ + 82, + 432, + 528, + 446 + ], + "score": 1.0, + "content": "the second more efficient variant (in orange) pre-computes part of Freivalds’ check as described in Section 3.2.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 82, + 398, + 529, + 446 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 448, + 528, + 515 + ], + "lines": [ + { + "bbox": [ + 82, + 448, + 529, + 462 + ], + "spans": [ + { + "bbox": [ + 82, + 448, + 529, + 462 + ], + "score": 1.0, + "content": "The VGG16 network is much larger (500MB) than SGX’s PRM. As a result, there is a large overhead on the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 82, + 460, + 527, + 473 + ], + "spans": [ + { + "bbox": [ + 82, + 460, + 510, + 473 + ], + "score": 1.0, + "content": "forward pass and verification without preprocessing. If the enclave securely stores preprocessed products", + "type": "text" + }, + { + "bbox": [ + 510, + 460, + 527, + 470 + ], + "score": 0.79, + "content": "W r", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 470, + 529, + 484 + ], + "spans": [ + { + "bbox": [ + 82, + 470, + 452, + 484 + ], + "score": 1.0, + "content": "for all network weights, we drastically reduce the memory footprint and achieve up to a", + "type": "text" + }, + { + "bbox": [ + 452, + 471, + 479, + 481 + ], + "score": 0.87, + "content": "2 0 . 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 470, + 529, + 484 + ], + "score": 1.0, + "content": "increase in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 481, + 529, + 495 + ], + "spans": [ + { + "bbox": [ + 82, + 481, + 529, + 495 + ], + "score": 1.0, + "content": "throughput. We also ran the lower-half of the VGG16 network (without the fully connected layers), a common", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 492, + 529, + 505 + ], + "spans": [ + { + "bbox": [ + 82, + 492, + 529, + 505 + ], + "score": 1.0, + "content": "approach for extracting features for transfer learning or object recognition (Liu et al., 2016). This part fits in the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 502, + 495, + 516 + ], + "spans": [ + { + "bbox": [ + 82, + 502, + 495, + 516 + ], + "score": 1.0, + "content": "PRM, and we thus achieve higher throughput for in-enclave forward passes and on-the-fly verification.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 82, + 448, + 529, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 520, + 528, + 565 + ], + "lines": [ + { + "bbox": [ + 82, + 520, + 528, + 533 + ], + "spans": [ + { + "bbox": [ + 82, + 520, + 227, + 533 + ], + "score": 1.0, + "content": "For MobileNet, we achieve between", + "type": "text" + }, + { + "bbox": [ + 227, + 521, + 249, + 531 + ], + "score": 0.88, + "content": "3 . 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 520, + 266, + 533 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 267, + 521, + 289, + 531 + ], + "score": 0.87, + "content": "6 . 4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 520, + 528, + 533 + ], + "score": 1.0, + "content": "speedups when using Slalom for verifiable inference (for the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 531, + 528, + 543 + ], + "spans": [ + { + "bbox": [ + 82, + 531, + 528, + 543 + ], + "score": 1.0, + "content": "standard or “fused” model, respectively). The speedups are smaller than for VGG16, as MobileNet performs", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 542, + 528, + 555 + ], + "spans": [ + { + "bbox": [ + 83, + 542, + 528, + 555 + ], + "score": 1.0, + "content": "much fewer operations per layer (verifying a linear layer requires computing at least two multiplications for each", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 552, + 519, + 567 + ], + "spans": [ + { + "bbox": [ + 82, + 552, + 519, + 567 + ], + "score": 1.0, + "content": "input and output. The closer the forward pass gets to that lower-bound, the less we can save by outsourcing).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 82, + 520, + 528, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 574, + 528, + 641 + ], + "lines": [ + { + "bbox": [ + 83, + 573, + 528, + 587 + ], + "spans": [ + { + "bbox": [ + 83, + 573, + 528, + 587 + ], + "score": 1.0, + "content": "Private Inference. We further benchmark the cost of private DNN inference, where inputs of outsourced", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 583, + 529, + 599 + ], + "spans": [ + { + "bbox": [ + 82, + 583, + 529, + 599 + ], + "score": 1.0, + "content": "linear layers are additionally blinded. Blinding and unblinding each layer’s inputs and outputs is costly,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 82, + 595, + 528, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 595, + 528, + 609 + ], + "score": 1.0, + "content": "especially in SGX due to the extra in-enclave memory reads and writes. Nevertheless, for VGG16 and the fused", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 83, + 607, + 529, + 620 + ], + "spans": [ + { + "bbox": [ + 83, + 607, + 444, + 620 + ], + "score": 1.0, + "content": "MobileNet variant without intermediate activations, we achieve respective speedups of", + "type": "text" + }, + { + "bbox": [ + 444, + 607, + 471, + 618 + ], + "score": 0.87, + "content": "1 3 . 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 607, + 490, + 620 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 607, + 512, + 618 + ], + "score": 0.87, + "content": "5 . 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 513, + 607, + 529, + 620 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 82, + 617, + 529, + 631 + ], + "spans": [ + { + "bbox": [ + 82, + 617, + 326, + 631 + ], + "score": 1.0, + "content": "private outsourcing (in black in Figure 3), and speedups of", + "type": "text" + }, + { + "bbox": [ + 327, + 618, + 353, + 629 + ], + "score": 0.88, + "content": "1 0 . 7 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 617, + 372, + 631 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 372, + 618, + 394, + 628 + ], + "score": 0.88, + "content": "4 . 1 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 617, + 529, + 631 + ], + "score": 1.0, + "content": "when also ensuring integrity (in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 82, + 628, + 470, + 643 + ], + "spans": [ + { + "bbox": [ + 82, + 628, + 470, + 643 + ], + "score": 1.0, + "content": "purple). For this benchmark, the precomputed unblinding factor are stored in untrusted memory.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 82, + 573, + 529, + 643 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 646, + 527, + 679 + ], + "lines": [ + { + "bbox": [ + 82, + 644, + 529, + 660 + ], + "spans": [ + { + "bbox": [ + 82, + 644, + 529, + 660 + ], + "score": 1.0, + "content": "We performed the same experiments on a standard CPU (i.e., without SGX) and find that Slalom’s improvements", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 656, + 529, + 669 + ], + "spans": [ + { + "bbox": [ + 82, + 656, + 529, + 669 + ], + "score": 1.0, + "content": "are even higher in non-resource-constrained or multi-threaded environments (see Appendix G-H). Slalom’s", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 83, + 668, + 471, + 680 + ], + "spans": [ + { + "bbox": [ + 83, + 668, + 471, + 680 + ], + "score": 1.0, + "content": "improvements over the baseline also hold when accounting for energy efficiency (see Section C).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 82, + 644, + 529, + 680 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 85, + 138, + 527, + 216 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 85, + 138, + 527, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 85, + 138, + 527, + 216 + ], + "spans": [ + { + "bbox": [ + 85, + 138, + 527, + 216 + ], + "score": 0.964, + "type": "image", + "image_path": "5df56a4274ed6e233912c5524664ae350b78563790ca22e417ac562aad5df2f1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 85, + 138, + 527, + 164.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 85, + 164.0, + 527, + 190.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 85, + 190.0, + 527, + 216.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 83, + 223, + 528, + 280 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 83, + 224, + 528, + 236 + ], + "spans": [ + { + "bbox": [ + 83, + 224, + 528, + 236 + ], + "score": 1.0, + "content": "Figure 3: Verifiable and private inference with Intel SGX. We show results for VGG16, VGG16 without the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 82, + 235, + 529, + 248 + ], + "spans": [ + { + "bbox": [ + 82, + 235, + 529, + 248 + ], + "score": 1.0, + "content": "fully connected layers, MobileNet, and a fused MobileNet variant with no intermediate activation for separable", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 246, + 528, + 258 + ], + "spans": [ + { + "bbox": [ + 83, + 246, + 528, + 258 + ], + "score": 1.0, + "content": "convolutions. We compare the baseline of fully executing the DNN in the enclave (blue) to different secure", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 82, + 257, + 529, + 270 + ], + "spans": [ + { + "bbox": [ + 82, + 257, + 529, + 270 + ], + "score": 1.0, + "content": "outsourcing schemes: integrity with Freivalds (red); integrity with Freivalds and precomputed secrets (yellow);", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 82, + 268, + 291, + 281 + ], + "spans": [ + { + "bbox": [ + 82, + 268, + 291, + 281 + ], + "score": 1.0, + "content": "privacy only (black); privacy and integrity (purple).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "image", + "bbox": [ + 84, + 294, + 528, + 362 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 84, + 294, + 528, + 362 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 84, + 294, + 528, + 362 + ], + "spans": [ + { + "bbox": [ + 84, + 294, + 528, + 362 + ], + "score": 0.967, + "type": "image", + "image_path": "9ceb8a1cf9a5c8503bd599b1f66c4bf349f3d490915536e0c23e1b7269842915.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 84, + 294, + 528, + 316.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 84, + 316.6666666666667, + 528, + 339.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 84, + 339.33333333333337, + 528, + 362.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 82, + 367, + 528, + 390 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 82, + 366, + 528, + 380 + ], + "spans": [ + { + "bbox": [ + 82, + 366, + 528, + 380 + ], + "score": 1.0, + "content": "Figure 4: Secure outsourcing of ResNet models with Intel SGX. We compare the baseline of fully executing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 82, + 378, + 525, + 392 + ], + "spans": [ + { + "bbox": [ + 82, + 378, + 525, + 392 + ], + "score": 1.0, + "content": "the DNN in the enclave (blue) to secure outsourcing with integrity (yellow) and privacy and integrity (purple).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + } + ], + "index": 10.25 + }, + { + "type": "text", + "bbox": [ + 84, + 414, + 528, + 503 + ], + "lines": [ + { + "bbox": [ + 83, + 415, + 528, + 426 + ], + "spans": [ + { + "bbox": [ + 83, + 415, + 528, + 426 + ], + "score": 1.0, + "content": "Extending Slalom to Deep Residual Networks. The Slalom algorithm in Figure 1 and our evaluations above", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 425, + 529, + 439 + ], + "spans": [ + { + "bbox": [ + 82, + 425, + 529, + 439 + ], + "score": 1.0, + "content": "focus on feed-forward architectures. Extending Slalom to more complex DNNs is quite simple. To illustrate, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 437, + 529, + 450 + ], + "spans": [ + { + "bbox": [ + 82, + 437, + 405, + 450 + ], + "score": 1.0, + "content": "consider the family of ResNet models (He et al., 2016), which use residual blocks", + "type": "text" + }, + { + "bbox": [ + 405, + 437, + 509, + 448 + ], + "score": 0.91, + "content": "f ( x ) = \\sigma ( f _ { 1 } ( x ) + f _ { 2 } ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 510, + 437, + 529, + 450 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 82, + 447, + 528, + 461 + ], + "spans": [ + { + "bbox": [ + 82, + 447, + 214, + 461 + ], + "score": 1.0, + "content": "merge two feed-forward “paths”", + "type": "text" + }, + { + "bbox": [ + 215, + 448, + 225, + 459 + ], + "score": 0.87, + "content": "f _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 447, + 243, + 461 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 448, + 254, + 459 + ], + "score": 0.88, + "content": "f _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 447, + 341, + 461 + ], + "score": 1.0, + "content": "into a final activation", + "type": "text" + }, + { + "bbox": [ + 342, + 450, + 349, + 457 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 447, + 439, + 461 + ], + "score": 1.0, + "content": ". To verify integrity of", + "type": "text" + }, + { + "bbox": [ + 439, + 448, + 459, + 459 + ], + "score": 0.91, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 447, + 528, + 461 + ], + "score": 1.0, + "content": ", the TEE simply", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 83, + 459, + 528, + 471 + ], + "spans": [ + { + "bbox": [ + 83, + 459, + 187, + 471 + ], + "score": 1.0, + "content": "verifies all linear layers in", + "type": "text" + }, + { + "bbox": [ + 187, + 459, + 198, + 470 + ], + "score": 0.88, + "content": "f _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 459, + 215, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 460, + 226, + 470 + ], + "score": 0.87, + "content": "f _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 459, + 283, + 471 + ], + "score": 1.0, + "content": "and computes", + "type": "text" + }, + { + "bbox": [ + 283, + 461, + 291, + 469 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 459, + 528, + 471 + ], + "score": 1.0, + "content": "directly. For privacy, the TEE applies the interactive Slalom", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 81, + 468, + 529, + 484 + ], + "spans": [ + { + "bbox": [ + 81, + 468, + 236, + 484 + ], + "score": 1.0, + "content": "protocol in Figure 1 (right) in turn to", + "type": "text" + }, + { + "bbox": [ + 236, + 470, + 247, + 481 + ], + "score": 0.88, + "content": "f _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 468, + 265, + 484 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 266, + 470, + 276, + 481 + ], + "score": 0.89, + "content": "f _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 468, + 359, + 484 + ], + "score": 1.0, + "content": ", and then computes", + "type": "text" + }, + { + "bbox": [ + 359, + 471, + 366, + 479 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 468, + 529, + 484 + ], + "score": 1.0, + "content": ". The results for the privacy-preserving", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 82, + 480, + 529, + 494 + ], + "spans": [ + { + "bbox": [ + 82, + 480, + 529, + 494 + ], + "score": 1.0, + "content": "Slalom variant in Figure 4 use a preliminary implementation that performs all required operations—and thus", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 491, + 475, + 505 + ], + "spans": [ + { + "bbox": [ + 82, + 491, + 475, + 505 + ], + "score": 1.0, + "content": "provides meaningful performance numbers—but without properly constructed unblinding factors.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 83, + 508, + 528, + 575 + ], + "lines": [ + { + "bbox": [ + 82, + 507, + 529, + 521 + ], + "spans": [ + { + "bbox": [ + 82, + 507, + 529, + 521 + ], + "score": 1.0, + "content": "We use the ResNet implementation from Keras Chollet et al. (2015), which contains a pre-trained 50-layer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 518, + 529, + 532 + ], + "spans": [ + { + "bbox": [ + 82, + 518, + 403, + 532 + ], + "score": 1.0, + "content": "variant. For this model, we find that our quantization scheme results in less than a", + "type": "text" + }, + { + "bbox": [ + 403, + 519, + 426, + 530 + ], + "score": 0.88, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 518, + 529, + 532 + ], + "score": 1.0, + "content": "decrease in accuracy (see", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 529, + 530, + 543 + ], + "spans": [ + { + "bbox": [ + 82, + 529, + 530, + 543 + ], + "score": 1.0, + "content": "Table 3). For other variants (i.e., with 18, 34, 101 and 152 layers) we compute throughput on untrained models.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 82, + 540, + 529, + 554 + ], + "spans": [ + { + "bbox": [ + 82, + 540, + 529, + 554 + ], + "score": 1.0, + "content": "Figure 4 shows benchmarks for different ResNet variants when executed fully in the enclave (our baseline) as", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 551, + 527, + 565 + ], + "spans": [ + { + "bbox": [ + 82, + 551, + 466, + 565 + ], + "score": 1.0, + "content": "well as secure outsourcing with integrity or privacy and integrity. 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The 18 and 34-layer ResNets use convolutions with", + "type": "text" + }, + { + "bbox": [ + 417, + 591, + 441, + 602 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 591, + 529, + 604 + ], + "score": 1.0, + "content": "kernels, whereas the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 602, + 528, + 614 + ], + "spans": [ + { + "bbox": [ + 83, + 602, + 528, + 614 + ], + "score": 1.0, + "content": "larger models mainly use pointwise convolutions. As shown in Table 2 verifying a convolution is about a factor", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 83, + 612, + 529, + 627 + ], + "spans": [ + { + "bbox": [ + 83, + 613, + 116, + 624 + ], + "score": 0.9, + "content": "k ^ { 2 } \\cdot c _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 612, + 529, + 627 + ], + "score": 1.0, + "content": "than computing it, which explains the higher savings for models that use convolutions with large kernel", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 624, + 529, + 637 + ], + "spans": [ + { + "bbox": [ + 82, + 624, + 529, + 637 + ], + "score": 1.0, + "content": "windows. When adding more layers to a model, we expect Slalom’s speedup over the baseline to remain constant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 81, + 634, + 529, + 648 + ], + "spans": [ + { + "bbox": [ + 81, + 634, + 529, + 648 + ], + "score": 1.0, + "content": "(e.g., if we duplicate each layer, the baseline computation and the verification should both take twice as long). Yet", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 81, + 645, + 529, + 660 + ], + "spans": [ + { + "bbox": [ + 81, + 645, + 529, + 660 + ], + "score": 1.0, + "content": "we find that Slalom’s speedups usually increase as layers get added to the ResNet architecture. 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We show results for VGG16, VGG16 without the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 82, + 235, + 529, + 248 + ], + "spans": [ + { + "bbox": [ + 82, + 235, + 529, + 248 + ], + "score": 1.0, + "content": "fully connected layers, MobileNet, and a fused MobileNet variant with no intermediate activation for separable", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 246, + 528, + 258 + ], + "spans": [ + { + "bbox": [ + 83, + 246, + 528, + 258 + ], + "score": 1.0, + "content": "convolutions. 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Extending Slalom to more complex DNNs is quite simple. To illustrate, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 437, + 529, + 450 + ], + "spans": [ + { + "bbox": [ + 82, + 437, + 405, + 450 + ], + "score": 1.0, + "content": "consider the family of ResNet models (He et al., 2016), which use residual blocks", + "type": "text" + }, + { + "bbox": [ + 405, + 437, + 509, + 448 + ], + "score": 0.91, + "content": "f ( x ) = \\sigma ( f _ { 1 } ( x ) + f _ { 2 } ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 510, + 437, + 529, + 450 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 82, + 447, + 528, + 461 + ], + "spans": [ + { + "bbox": [ + 82, + 447, + 214, + 461 + ], + "score": 1.0, + "content": "merge two feed-forward “paths”", + "type": "text" + }, + { + "bbox": [ + 215, + 448, + 225, + 459 + ], + "score": 0.87, + "content": "f _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 447, + 243, + 461 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 448, + 254, + 459 + ], + "score": 0.88, + "content": "f _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 447, + 341, + 461 + ], + "score": 1.0, + "content": "into a final activation", + "type": "text" + }, + { + "bbox": [ + 342, + 450, + 349, + 457 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 447, + 439, + 461 + ], + "score": 1.0, + "content": ". 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(2015), which contains a pre-trained 50-layer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 518, + 529, + 532 + ], + "spans": [ + { + "bbox": [ + 82, + 518, + 403, + 532 + ], + "score": 1.0, + "content": "variant. For this model, we find that our quantization scheme results in less than a", + "type": "text" + }, + { + "bbox": [ + 403, + 519, + 426, + 530 + ], + "score": 0.88, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 518, + 529, + 532 + ], + "score": 1.0, + "content": "decrease in accuracy (see", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 529, + 530, + 543 + ], + "spans": [ + { + "bbox": [ + 82, + 529, + 530, + 543 + ], + "score": 1.0, + "content": "Table 3). For other variants (i.e., with 18, 34, 101 and 152 layers) we compute throughput on untrained models.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 82, + 540, + 529, + 554 + ], + "spans": [ + { + "bbox": [ + 82, + 540, + 529, + 554 + ], + "score": 1.0, + "content": "Figure 4 shows benchmarks for different ResNet variants when executed fully in the enclave (our baseline) as", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 551, + 527, + 565 + ], + "spans": [ + { + "bbox": [ + 82, + 551, + 466, + 565 + ], + "score": 1.0, + "content": "well as secure outsourcing with integrity or privacy and integrity. For all models, we achieve", + "type": "text" + }, + { + "bbox": [ + 466, + 552, + 488, + 563 + ], + "score": 0.87, + "content": "6 . 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 551, + 500, + 565 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 500, + 552, + 527, + 563 + ], + "score": 0.85, + "content": "1 4 . 4 \\times", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 561, + 416, + 577 + ], + "spans": [ + { + "bbox": [ + 82, + 561, + 233, + 577 + ], + "score": 1.0, + "content": "speedups for verifiable inference and", + "type": "text" + }, + { + "bbox": [ + 233, + 564, + 255, + 574 + ], + "score": 0.85, + "content": "4 . 4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 561, + 266, + 577 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 266, + 564, + 288, + 574 + ], + "score": 0.9, + "content": "9 . 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 561, + 416, + 577 + ], + "score": 1.0, + "content": "speedups when adding privacy.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 82, + 507, + 530, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 83, + 579, + 528, + 680 + ], + "lines": [ + { + "bbox": [ + 82, + 579, + 529, + 592 + ], + "spans": [ + { + "bbox": [ + 82, + 579, + 529, + 592 + ], + "score": 1.0, + "content": "Comparing results for different models is illustrative of how Slalom’s savings scale with model size and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 82, + 591, + 529, + 604 + ], + "spans": [ + { + "bbox": [ + 82, + 591, + 417, + 604 + ], + "score": 1.0, + "content": "architectural design choices. The 18 and 34-layer ResNets use convolutions with", + "type": "text" + }, + { + "bbox": [ + 417, + 591, + 441, + 602 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 591, + 529, + 604 + ], + "score": 1.0, + "content": "kernels, whereas the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 602, + 528, + 614 + ], + "spans": [ + { + "bbox": [ + 83, + 602, + 528, + 614 + ], + "score": 1.0, + "content": "larger models mainly use pointwise convolutions. As shown in Table 2 verifying a convolution is about a factor", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 83, + 612, + 529, + 627 + ], + "spans": [ + { + "bbox": [ + 83, + 613, + 116, + 624 + ], + "score": 0.9, + "content": "k ^ { 2 } \\cdot c _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 612, + 529, + 627 + ], + "score": 1.0, + "content": "than computing it, which explains the higher savings for models that use convolutions with large kernel", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 624, + 529, + 637 + ], + "spans": [ + { + "bbox": [ + 82, + 624, + 529, + 637 + ], + "score": 1.0, + "content": "windows. When adding more layers to a model, we expect Slalom’s speedup over the baseline to remain constant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 81, + 634, + 529, + 648 + ], + "spans": [ + { + "bbox": [ + 81, + 634, + 529, + 648 + ], + "score": 1.0, + "content": "(e.g., if we duplicate each layer, the baseline computation and the verification should both take twice as long). Yet", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 81, + 645, + 529, + 660 + ], + "spans": [ + { + "bbox": [ + 81, + 645, + 529, + 660 + ], + "score": 1.0, + "content": "we find that Slalom’s speedups usually increase as layers get added to the ResNet architecture. This is because", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 82, + 657, + 528, + 670 + ], + "spans": [ + { + "bbox": [ + 82, + 657, + 528, + 670 + ], + "score": 1.0, + "content": "the deeper ResNet variants are obtained by duplicating layers towards the end of the pipeline, which have the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 667, + 399, + 681 + ], + "spans": [ + { + "bbox": [ + 82, + 667, + 399, + 681 + ], + "score": 1.0, + "content": "largest number of channels and for which Slalom achieves the highest savings.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31, + "bbox_fs": [ + 81, + 579, + 529, + 681 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 84, + 138, + 390, + 152 + ], + "lines": [ + { + "bbox": [ + 82, + 137, + 392, + 154 + ], + "spans": [ + { + "bbox": [ + 82, + 137, + 392, + 154 + ], + "score": 1.0, + "content": "5 CHALLENGES FOR VERIFIABLE AND PRIVATE TRAINING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 164, + 529, + 198 + ], + "lines": [ + { + "bbox": [ + 83, + 164, + 528, + 176 + ], + "spans": [ + { + "bbox": [ + 83, + 164, + 528, + 176 + ], + "score": 1.0, + "content": "Our techniques for secure outsourcing of DNN inference might also apply to DNN training. Indeed, a backward", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 82, + 174, + 530, + 189 + ], + "spans": [ + { + "bbox": [ + 82, + 174, + 530, + 189 + ], + "score": 1.0, + "content": "pass consists of similar linear operators as a forward pass, and can thus be verified with Freivalds’ algorithm. Yet,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 186, + 491, + 200 + ], + "spans": [ + { + "bbox": [ + 83, + 186, + 491, + 200 + ], + "score": 1.0, + "content": "applying Slalom to DNN training is challenging, as described below, and we leave this problem open.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 88, + 201, + 528, + 297 + ], + "lines": [ + { + "bbox": [ + 88, + 202, + 529, + 215 + ], + "spans": [ + { + "bbox": [ + 88, + 202, + 529, + 215 + ], + "score": 1.0, + "content": "• Quantizing DNNs for training is harder than for inference, due to large changes in weight magnitudes (Mi-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 95, + 211, + 530, + 227 + ], + "spans": [ + { + "bbox": [ + 95, + 211, + 530, + 227 + ], + "score": 1.0, + "content": "cikevicius et al., 2018). Thus, a more flexible quantization scheme than the one we used would be necessary.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 87, + 226, + 529, + 241 + ], + "spans": [ + { + "bbox": [ + 87, + 226, + 529, + 241 + ], + "score": 1.0, + "content": "• Because the DNN’s weights change during training, the same preprocessed random vectors for Freivalds’", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 95, + 236, + 528, + 252 + ], + "spans": [ + { + "bbox": [ + 95, + 236, + 528, + 252 + ], + "score": 1.0, + "content": "check cannot be re-used indefinitely. The most efficient approach would presumably be to train with very", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 95, + 248, + 317, + 263 + ], + "spans": [ + { + "bbox": [ + 95, + 248, + 317, + 263 + ], + "score": 1.0, + "content": "large batches than can then be verified simultaneously.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 91, + 262, + 528, + 277 + ], + "spans": [ + { + "bbox": [ + 91, + 262, + 528, + 277 + ], + "score": 1.0, + "content": "• Finally, the pre-computation techniques we employ for protecting input privacy do not apply for training, as", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 96, + 274, + 528, + 287 + ], + "spans": [ + { + "bbox": [ + 96, + 274, + 528, + 287 + ], + "score": 1.0, + "content": "the weights change after every processed batch. Moreover, Slalom does not try to hide the model weights", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 95, + 283, + 419, + 300 + ], + "spans": [ + { + "bbox": [ + 95, + 283, + 419, + 300 + ], + "score": 1.0, + "content": "from the untrusted processor, which might be a requirement for private training.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 84, + 312, + 173, + 325 + ], + "lines": [ + { + "bbox": [ + 82, + 311, + 174, + 328 + ], + "spans": [ + { + "bbox": [ + 82, + 311, + 174, + 328 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 84, + 337, + 528, + 371 + ], + "lines": [ + { + "bbox": [ + 83, + 337, + 528, + 351 + ], + "spans": [ + { + "bbox": [ + 83, + 337, + 528, + 351 + ], + "score": 1.0, + "content": "This paper has studied the efficiency of evaluating a DNN in a Trusted Execution Environment (TEE) to provide", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 348, + 529, + 362 + ], + "spans": [ + { + "bbox": [ + 82, + 348, + 529, + 362 + ], + "score": 1.0, + "content": "strong integrity and privacy guarantees. We explored new approaches for segmenting a DNN evaluation to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 360, + 477, + 373 + ], + "spans": [ + { + "bbox": [ + 82, + 360, + 477, + 373 + ], + "score": 1.0, + "content": "securely outsource work from a trusted environment to a faster co-located but untrusted processor.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 84, + 376, + 527, + 432 + ], + "lines": [ + { + "bbox": [ + 83, + 377, + 528, + 389 + ], + "spans": [ + { + "bbox": [ + 83, + 377, + 528, + 389 + ], + "score": 1.0, + "content": "We designed Slalom, a framework for efficient DNN evaluation that outsources all linear layers from a TEE", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 387, + 528, + 401 + ], + "spans": [ + { + "bbox": [ + 82, + 387, + 528, + 401 + ], + "score": 1.0, + "content": "to a GPU. Slalom leverage Freivalds’ algorithm for verifying correctness of linear operators, and additionally", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 82, + 398, + 529, + 411 + ], + "spans": [ + { + "bbox": [ + 82, + 398, + 529, + 411 + ], + "score": 1.0, + "content": "encrypts inputs with precomputed blinding factors to preserve privacy. Slalom can work with any TEE and we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 83, + 409, + 529, + 423 + ], + "spans": [ + { + "bbox": [ + 83, + 409, + 529, + 423 + ], + "score": 1.0, + "content": "evaluated its performance using Intel SGX on various workloads. For canonical DNNs (VGG16, MobileNet and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 420, + 509, + 434 + ], + "spans": [ + { + "bbox": [ + 82, + 420, + 509, + 434 + ], + "score": 1.0, + "content": "ResNet variants), we have shown that Slalom boosts inference throughput without compromising security.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 84, + 437, + 528, + 493 + ], + "lines": [ + { + "bbox": [ + 82, + 437, + 529, + 451 + ], + "spans": [ + { + "bbox": [ + 82, + 437, + 529, + 451 + ], + "score": 1.0, + "content": "Securely outsourcing matrix products from a TEE has applications in ML beyond DNNs (e.g., non negative", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 83, + 448, + 528, + 461 + ], + "spans": [ + { + "bbox": [ + 83, + 448, + 528, + 461 + ], + "score": 1.0, + "content": "matrix factorization, dimensionality reduction, etc.) We have also explored avenues and challenges towards", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 459, + 529, + 473 + ], + "spans": [ + { + "bbox": [ + 82, + 459, + 529, + 473 + ], + "score": 1.0, + "content": "applying similar techniques to DNN training, an interesting direction for future work. Finally, our general", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 82, + 470, + 529, + 483 + ], + "spans": [ + { + "bbox": [ + 82, + 470, + 529, + 483 + ], + "score": 1.0, + "content": "approach of outsourcing work from a TEE to a faster co-processor could be applied to other problems which", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 481, + 496, + 495 + ], + "spans": [ + { + "bbox": [ + 82, + 481, + 496, + 495 + ], + "score": 1.0, + "content": "have fast verification algorithms, e.g., those considered in (McConnell et al., 2011; Zhang et al., 2014).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 84, + 509, + 153, + 520 + ], + "lines": [ + { + "bbox": [ + 83, + 509, + 153, + 522 + ], + "spans": [ + { + "bbox": [ + 83, + 509, + 153, + 522 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 82, + 526, + 528, + 679 + ], + "lines": [ + { + "bbox": [ + 82, + 525, + 528, + 539 + ], + "spans": [ + { + "bbox": [ + 82, + 525, + 528, + 539 + ], + "score": 1.0, + "content": "Tiago Alves and Don Felton. 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ACM, 2017.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 84, + 84, + 270, + 95 + ], + "lines": [ + { + "bbox": [ + 83, + 83, + 271, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 83, + 271, + 96 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 277, + 699, + 288, + 708 + ], + "lines": [ + { + "bbox": [ + 276, + 698, + 290, + 711 + ], + "spans": [ + { + "bbox": [ + 276, + 698, + 290, + 711 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 84, + 138, + 390, + 152 + ], + "lines": [ + { + "bbox": [ + 82, + 137, + 392, + 154 + ], + "spans": [ + { + "bbox": [ + 82, + 137, + 392, + 154 + ], + "score": 1.0, + "content": "5 CHALLENGES FOR VERIFIABLE AND PRIVATE TRAINING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 164, + 529, + 198 + ], + "lines": [ + { + "bbox": [ + 83, + 164, + 528, + 176 + ], + "spans": [ + { + "bbox": [ + 83, + 164, + 528, + 176 + ], + "score": 1.0, + "content": "Our techniques for secure outsourcing of DNN inference might also apply to DNN training. Indeed, a backward", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 82, + 174, + 530, + 189 + ], + "spans": [ + { + "bbox": [ + 82, + 174, + 530, + 189 + ], + "score": 1.0, + "content": "pass consists of similar linear operators as a forward pass, and can thus be verified with Freivalds’ algorithm. Yet,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 186, + 491, + 200 + ], + "spans": [ + { + "bbox": [ + 83, + 186, + 491, + 200 + ], + "score": 1.0, + "content": "applying Slalom to DNN training is challenging, as described below, and we leave this problem open.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 82, + 164, + 530, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 88, + 201, + 528, + 297 + ], + "lines": [ + { + "bbox": [ + 88, + 202, + 529, + 215 + ], + "spans": [ + { + "bbox": [ + 88, + 202, + 529, + 215 + ], + "score": 1.0, + "content": "• Quantizing DNNs for training is harder than for inference, due to large changes in weight magnitudes (Mi-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 95, + 211, + 530, + 227 + ], + "spans": [ + { + "bbox": [ + 95, + 211, + 530, + 227 + ], + "score": 1.0, + "content": "cikevicius et al., 2018). Thus, a more flexible quantization scheme than the one we used would be necessary.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 87, + 226, + 529, + 241 + ], + "spans": [ + { + "bbox": [ + 87, + 226, + 529, + 241 + ], + "score": 1.0, + "content": "• Because the DNN’s weights change during training, the same preprocessed random vectors for Freivalds’", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 95, + 236, + 528, + 252 + ], + "spans": [ + { + "bbox": [ + 95, + 236, + 528, + 252 + ], + "score": 1.0, + "content": "check cannot be re-used indefinitely. The most efficient approach would presumably be to train with very", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 95, + 248, + 317, + 263 + ], + "spans": [ + { + "bbox": [ + 95, + 248, + 317, + 263 + ], + "score": 1.0, + "content": "large batches than can then be verified simultaneously.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 91, + 262, + 528, + 277 + ], + "spans": [ + { + "bbox": [ + 91, + 262, + 528, + 277 + ], + "score": 1.0, + "content": "• Finally, the pre-computation techniques we employ for protecting input privacy do not apply for training, as", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 96, + 274, + 528, + 287 + ], + "spans": [ + { + "bbox": [ + 96, + 274, + 528, + 287 + ], + "score": 1.0, + "content": "the weights change after every processed batch. Moreover, Slalom does not try to hide the model weights", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 95, + 283, + 419, + 300 + ], + "spans": [ + { + "bbox": [ + 95, + 283, + 419, + 300 + ], + "score": 1.0, + "content": "from the untrusted processor, which might be a requirement for private training.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 87, + 202, + 530, + 300 + ] + }, + { + "type": "title", + "bbox": [ + 84, + 312, + 173, + 325 + ], + "lines": [ + { + "bbox": [ + 82, + 311, + 174, + 328 + ], + "spans": [ + { + "bbox": [ + 82, + 311, + 174, + 328 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 84, + 337, + 528, + 371 + ], + "lines": [ + { + "bbox": [ + 83, + 337, + 528, + 351 + ], + "spans": [ + { + "bbox": [ + 83, + 337, + 528, + 351 + ], + "score": 1.0, + "content": "This paper has studied the efficiency of evaluating a DNN in a Trusted Execution Environment (TEE) to provide", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 348, + 529, + 362 + ], + "spans": [ + { + "bbox": [ + 82, + 348, + 529, + 362 + ], + "score": 1.0, + "content": "strong integrity and privacy guarantees. We explored new approaches for segmenting a DNN evaluation to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 360, + 477, + 373 + ], + "spans": [ + { + "bbox": [ + 82, + 360, + 477, + 373 + ], + "score": 1.0, + "content": "securely outsource work from a trusted environment to a faster co-located but untrusted processor.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 82, + 337, + 529, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 376, + 527, + 432 + ], + "lines": [ + { + "bbox": [ + 83, + 377, + 528, + 389 + ], + "spans": [ + { + "bbox": [ + 83, + 377, + 528, + 389 + ], + "score": 1.0, + "content": "We designed Slalom, a framework for efficient DNN evaluation that outsources all linear layers from a TEE", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 387, + 528, + 401 + ], + "spans": [ + { + "bbox": [ + 82, + 387, + 528, + 401 + ], + "score": 1.0, + "content": "to a GPU. Slalom leverage Freivalds’ algorithm for verifying correctness of linear operators, and additionally", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 82, + 398, + 529, + 411 + ], + "spans": [ + { + "bbox": [ + 82, + 398, + 529, + 411 + ], + "score": 1.0, + "content": "encrypts inputs with precomputed blinding factors to preserve privacy. Slalom can work with any TEE and we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 83, + 409, + 529, + 423 + ], + "spans": [ + { + "bbox": [ + 83, + 409, + 529, + 423 + ], + "score": 1.0, + "content": "evaluated its performance using Intel SGX on various workloads. For canonical DNNs (VGG16, MobileNet and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 420, + 509, + 434 + ], + "spans": [ + { + "bbox": [ + 82, + 420, + 509, + 434 + ], + "score": 1.0, + "content": "ResNet variants), we have shown that Slalom boosts inference throughput without compromising security.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 82, + 377, + 529, + 434 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 437, + 528, + 493 + ], + "lines": [ + { + "bbox": [ + 82, + 437, + 529, + 451 + ], + "spans": [ + { + "bbox": [ + 82, + 437, + 529, + 451 + ], + "score": 1.0, + "content": "Securely outsourcing matrix products from a TEE has applications in ML beyond DNNs (e.g., non negative", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 83, + 448, + 528, + 461 + ], + "spans": [ + { + "bbox": [ + 83, + 448, + 528, + 461 + ], + "score": 1.0, + "content": "matrix factorization, dimensionality reduction, etc.) We have also explored avenues and challenges towards", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 459, + 529, + 473 + ], + "spans": [ + { + "bbox": [ + 82, + 459, + 529, + 473 + ], + "score": 1.0, + "content": "applying similar techniques to DNN training, an interesting direction for future work. 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In the past", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 81, + 552, + 528, + 565 + ], + "spans": [ + { + "bbox": [ + 81, + 552, + 512, + 565 + ], + "score": 1.0, + "content": "years, many attacks have been proposed, with the goal of undermining privacy of enclave computations", + "type": "text" + }, + { + "bbox": [ + 513, + 552, + 528, + 563 + ], + "score": 0.29, + "content": "\\mathrm { { X u } }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 563, + 529, + 575 + ], + "spans": [ + { + "bbox": [ + 82, + 563, + 529, + 575 + ], + "score": 1.0, + "content": "et al., 2015; Brasser et al., 2017; Moghimi et al., 2017; Götzfried et al., 2017; Van Bulck et al., 2017; Lee et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 82, + 573, + 529, + 588 + ], + "spans": [ + { + "bbox": [ + 82, + 573, + 529, + 588 + ], + "score": 1.0, + "content": "2017). Most of these attacks rely on data dependent code behavior in an enclave (e.g., branching or memory", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 82, + 584, + 529, + 598 + ], + "spans": [ + { + "bbox": [ + 82, + 584, + 529, + 598 + ], + "score": 1.0, + "content": "access) that can be partially observed by other processes running on the same host. These side-channels are a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 82, + 596, + 529, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 596, + 529, + 609 + ], + "score": 1.0, + "content": "minor concern for the DNN computations considered in this paper, as the standard computations in a DNN are", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 83, + 608, + 508, + 620 + ], + "spans": [ + { + "bbox": [ + 83, + 608, + 508, + 620 + ], + "score": 1.0, + "content": "data-oblivious (i.e., the same operations are applied regardless of the input data) (Ohrimenko et al., 2016).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 84, + 623, + 528, + 679 + ], + "lines": [ + { + "bbox": [ + 83, + 624, + 528, + 636 + ], + "spans": [ + { + "bbox": [ + 83, + 624, + 528, + 636 + ], + "score": 1.0, + "content": "The recent Spectre attacks on speculative execution (Kocher et al., 2018) also prove damaging to SGX (as well", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 81, + 634, + 530, + 647 + ], + "spans": [ + { + "bbox": [ + 81, + 634, + 530, + 647 + ], + "score": 1.0, + "content": "as to most other processors), as recently shown (Chen et al., 2018; Dall et al., 2018; Van Bulck et al., 2018).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 82, + 645, + 530, + 658 + ], + "spans": [ + { + "bbox": [ + 82, + 645, + 530, + 658 + ], + "score": 1.0, + "content": "Mitigations for these side-channel attacks are being developed (Shinde et al., 2016; Shih et al., 2017; Chen et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 82, + 655, + 528, + 669 + ], + "spans": [ + { + "bbox": [ + 82, + 655, + 528, + 669 + ], + "score": 1.0, + "content": "2017; Intel Corp., 2018) but a truly secure solution might require some architectural changes, e.g., as in the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 82, + 668, + 286, + 680 + ], + "spans": [ + { + "bbox": [ + 82, + 668, + 286, + 680 + ], + "score": 1.0, + "content": "proposed Sanctum processor (Costan et al., 2016).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 84, + 84, + 270, + 95 + ], + "lines": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 277, + 699, + 287, + 708 + ], + "lines": [ + { + "bbox": [ + 276, + 698, + 289, + 711 + ], + "spans": [ + { + "bbox": [ + 276, + 698, + 289, + 711 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 81, + 141, + 530, + 449 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 81, + 141, + 530, + 448 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 84, + 466, + 288, + 479 + ], + "lines": [ + { + "bbox": [ + 82, + 465, + 289, + 482 + ], + "spans": [ + { + "bbox": [ + 82, + 465, + 289, + 482 + ], + "score": 1.0, + "content": "A DETAILS ON INTEL SGX SECURITY", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 84, + 491, + 528, + 525 + ], + "lines": [ + { + "bbox": [ + 83, + 491, + 528, + 505 + ], + "spans": [ + { + "bbox": [ + 83, + 491, + 528, + 505 + ], + "score": 1.0, + "content": "SGX enclaves isolate execution of a program from all other processes on a same host, including a potentially", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 502, + 528, + 514 + ], + "spans": [ + { + "bbox": [ + 83, + 502, + 528, + 514 + ], + "score": 1.0, + "content": "malicious OS. In particular, enclave memory is fully encrypted and authenticated. When a word is read from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 513, + 530, + 527 + ], + "spans": [ + { + "bbox": [ + 82, + 513, + 530, + 527 + ], + "score": 1.0, + "content": "memory into a CPU register, a Memory Management Engine handles the decryption (Costan & Devadas, 2016).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 82, + 491, + 530, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 530, + 528, + 619 + ], + "lines": [ + { + "bbox": [ + 83, + 531, + 529, + 542 + ], + "spans": [ + { + "bbox": [ + 83, + 531, + 529, + 542 + ], + "score": 1.0, + "content": "While SGX covers many software and hardware attack vectors, there is a large and prominent class of side-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 540, + 529, + 554 + ], + "spans": [ + { + "bbox": [ + 82, + 540, + 529, + 554 + ], + "score": 1.0, + "content": "channel attacks that it explicitly does not address (Costan & Devadas, 2016; Tramèr et al., 2017). In the past", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 81, + 552, + 528, + 565 + ], + "spans": [ + { + "bbox": [ + 81, + 552, + 512, + 565 + ], + "score": 1.0, + "content": "years, many attacks have been proposed, with the goal of undermining privacy of enclave computations", + "type": "text" + }, + { + "bbox": [ + 513, + 552, + 528, + 563 + ], + "score": 0.29, + "content": "\\mathrm { { X u } }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 563, + 529, + 575 + ], + "spans": [ + { + "bbox": [ + 82, + 563, + 529, + 575 + ], + "score": 1.0, + "content": "et al., 2015; Brasser et al., 2017; Moghimi et al., 2017; Götzfried et al., 2017; Van Bulck et al., 2017; Lee et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 82, + 573, + 529, + 588 + ], + "spans": [ + { + "bbox": [ + 82, + 573, + 529, + 588 + ], + "score": 1.0, + "content": "2017). 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These side-channels are a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 82, + 596, + 529, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 596, + 529, + 609 + ], + "score": 1.0, + "content": "minor concern for the DNN computations considered in this paper, as the standard computations in a DNN are", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 83, + 608, + 508, + 620 + ], + "spans": [ + { + "bbox": [ + 83, + 608, + 508, + 620 + ], + "score": 1.0, + "content": "data-oblivious (i.e., the same operations are applied regardless of the input data) (Ohrimenko et al., 2016).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 81, + 531, + 529, + 620 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 623, + 528, + 679 + ], + "lines": [ + { + "bbox": [ + 83, + 624, + 528, + 636 + ], + "spans": [ + { + "bbox": [ + 83, + 624, + 528, + 636 + ], + "score": 1.0, + "content": "The recent Spectre attacks on speculative execution (Kocher et al., 2018) also prove damaging to SGX (as well", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 81, + 634, + 530, + 647 + ], + "spans": [ + { + "bbox": [ + 81, + 634, + 530, + 647 + ], + "score": 1.0, + "content": "as to most other processors), as recently shown (Chen et al., 2018; Dall et al., 2018; Van Bulck et al., 2018).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 82, + 645, + 530, + 658 + ], + "spans": [ + { + "bbox": [ + 82, + 645, + 530, + 658 + ], + "score": 1.0, + "content": "Mitigations for these side-channel attacks are being developed (Shinde et al., 2016; Shih et al., 2017; Chen et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 82, + 655, + 528, + 669 + ], + "spans": [ + { + "bbox": [ + 82, + 655, + 528, + 669 + ], + "score": 1.0, + "content": "2017; Intel Corp., 2018) but a truly secure solution might require some architectural changes, e.g., as in the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 82, + 668, + 286, + 680 + ], + "spans": [ + { + "bbox": [ + 82, + 668, + 286, + 680 + ], + "score": 1.0, + "content": "proposed Sanctum processor (Costan et al., 2016).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38, + "bbox_fs": [ + 81, + 624, + 530, + 680 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 83, + 140, + 529, + 174 + ], + "lines": [ + { + "bbox": [ + 83, + 140, + 527, + 152 + ], + "spans": [ + { + "bbox": [ + 83, + 140, + 527, + 152 + ], + "score": 1.0, + "content": "We refrain from formally modeling SGX’s (or other TEE’s) security in this paper, as Slalom is mostly concerned", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 83, + 152, + 529, + 164 + ], + "spans": [ + { + "bbox": [ + 83, + 152, + 529, + 164 + ], + "score": 1.0, + "content": "with outsourcing protocols wherein the TEE acts as a client. We refer the interested reader to (Pass et al., 2017;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 83, + 163, + 431, + 175 + ], + "spans": [ + { + "bbox": [ + 83, + 163, + 431, + 175 + ], + "score": 1.0, + "content": "Fisch et al., 2017; Subramanyan et al., 2017) for different attempts at such formalisms.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 84, + 191, + 345, + 204 + ], + "lines": [ + { + "bbox": [ + 82, + 190, + 346, + 206 + ], + "spans": [ + { + "bbox": [ + 82, + 190, + 346, + 206 + ], + "score": 1.0, + "content": "B FORMAL SECURITY DEFINITIONS AND PROOFS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 83, + 217, + 528, + 240 + ], + "lines": [ + { + "bbox": [ + 83, + 217, + 528, + 230 + ], + "spans": [ + { + "bbox": [ + 83, + 217, + 307, + 230 + ], + "score": 1.0, + "content": "We define a secure outsourcing scheme, between a client", + "type": "text" + }, + { + "bbox": [ + 307, + 218, + 314, + 227 + ], + "score": 0.78, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 217, + 364, + 230 + ], + "score": 1.0, + "content": "and a server", + "type": "text" + }, + { + "bbox": [ + 365, + 218, + 372, + 227 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 217, + 420, + 230 + ], + "score": 1.0, + "content": ", for a DNN", + "type": "text" + }, + { + "bbox": [ + 420, + 217, + 483, + 229 + ], + "score": 0.91, + "content": "F ( x ) : \\mathcal { X } \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 217, + 528, + 230 + ], + "score": 1.0, + "content": "from some", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 228, + 493, + 240 + ], + "spans": [ + { + "bbox": [ + 83, + 228, + 112, + 240 + ], + "score": 1.0, + "content": "family", + "type": "text" + }, + { + "bbox": [ + 112, + 228, + 122, + 239 + ], + "score": 0.81, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 228, + 376, + 240 + ], + "score": 1.0, + "content": "(e.g., all DNNs of a given size). We first assume that the model", + "type": "text" + }, + { + "bbox": [ + 377, + 229, + 386, + 238 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 228, + 456, + 240 + ], + "score": 1.0, + "content": "is known to both", + "type": "text" + }, + { + "bbox": [ + 456, + 229, + 463, + 238 + ], + "score": 0.82, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 228, + 481, + 240 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 482, + 229, + 489, + 238 + ], + "score": 0.83, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 228, + 493, + 240 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 83, + 243, + 526, + 266 + ], + "lines": [ + { + "bbox": [ + 82, + 242, + 528, + 257 + ], + "spans": [ + { + "bbox": [ + 82, + 242, + 528, + 257 + ], + "score": 1.0, + "content": "Definition B.1 (Secure Outsourcing Schemes). A secure outsourcing scheme consists of an offline preprocessing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 83, + 254, + 472, + 267 + ], + "spans": [ + { + "bbox": [ + 83, + 254, + 365, + 267 + ], + "score": 1.0, + "content": "algorithm Preproc, as well as an interactive online protocol Outsource", + "type": "text" + }, + { + "bbox": [ + 366, + 255, + 390, + 266 + ], + "score": 0.91, + "content": "\\langle \\mathcal { C } , \\mathcal { S } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 254, + 472, + 267 + ], + "score": 1.0, + "content": ", defined as follows:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 85, + 270, + 528, + 323 + ], + "lines": [ + { + "bbox": [ + 87, + 270, + 528, + 284 + ], + "spans": [ + { + "bbox": [ + 87, + 270, + 97, + 284 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 98, + 270, + 188, + 283 + ], + "score": 0.85, + "content": "\\mathsf { s t } \\gets \\mathsf { P r e p r o c } ( F , 1 ^ { \\lambda } )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 270, + 347, + 284 + ], + "score": 1.0, + "content": ": The preprocessing algorithm is run by", + "type": "text" + }, + { + "bbox": [ + 347, + 271, + 354, + 281 + ], + "score": 0.76, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 270, + 528, + 284 + ], + "score": 1.0, + "content": "and generates some data-independent state", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 98, + 282, + 492, + 295 + ], + "spans": [ + { + "bbox": [ + 98, + 283, + 111, + 293 + ], + "score": 0.34, + "content": "\\mathtt { s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 282, + 492, + 295 + ], + "score": 1.0, + "content": "(e.g., cryptographic keys or precomputed values to accelerate the online outsourcing protocol.)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 87, + 298, + 528, + 313 + ], + "spans": [ + { + "bbox": [ + 87, + 298, + 96, + 313 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 96, + 299, + 276, + 312 + ], + "score": 0.8, + "content": "\\mathcal { V } \\cup \\{ \\bot \\} \\mathsf { O u t s o u r c e } \\langle \\mathcal { C } ( F , x , \\mathrm { s t } ) , \\mathcal { S } ( F ) \\rangle \\mathrm { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 298, + 471, + 313 + ], + "score": 1.0, + "content": ": The online outsourcing protocol is initiated by", + "type": "text" + }, + { + "bbox": [ + 471, + 300, + 479, + 310 + ], + "score": 0.78, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 298, + 528, + 313 + ], + "score": 1.0, + "content": "with inputs", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 97, + 311, + 488, + 324 + ], + "spans": [ + { + "bbox": [ + 97, + 311, + 138, + 323 + ], + "score": 0.85, + "content": "( F , x , s \\ t )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 311, + 250, + 324 + ], + "score": 1.0, + "content": ". 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{ \\ast } ( F ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 383, + 281, + 396 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 383, + 343, + 395 + ], + "score": 0.93, + "content": "\\tilde { y } \\notin \\{ F ( x ) , \\bot \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 383, + 390, + 396 + ], + "score": 1.0, + "content": "is less than", + "type": "text" + }, + { + "bbox": [ + 391, + 384, + 396, + 393 + ], + "score": 0.69, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 383, + 400, + 396 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 90, + 401, + 525, + 424 + ], + "lines": [ + { + "bbox": [ + 89, + 400, + 527, + 414 + ], + "spans": [ + { + "bbox": [ + 89, + 400, + 193, + 414 + ], + "score": 1.0, + "content": "• Input privacy: For any", + "type": "text" + }, + { + "bbox": [ + 193, + 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In some applications a secure outsourcing scheme may also require to hide the model", + "type": "text" + }, + { + "bbox": [ + 496, + 455, + 505, + 464 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 505, + 454, + 528, + 467 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 83, + 466, + 473, + 477 + ], + "spans": [ + { + "bbox": [ + 83, + 466, + 108, + 477 + ], + "score": 1.0, + "content": "either", + "type": "text" + }, + { + "bbox": [ + 109, + 466, + 117, + 475 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 466, + 128, + 477 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 129, + 466, + 136, + 475 + ], + "score": 0.79, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 466, + 340, + 477 + ], + "score": 1.0, + "content": "(in which case that party would obviously not take", + "type": "text" + }, + { + "bbox": [ + 341, + 466, + 350, + 475 + ], + "score": 0.81, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 466, + 473, + 477 + ], + "score": 1.0, + "content": "as input in the above scheme).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 83, + 482, + 528, + 505 + ], + "lines": [ + { + "bbox": [ + 83, + 482, + 529, + 495 + ], + "spans": [ + { + "bbox": [ + 83, + 482, + 258, + 495 + ], + "score": 1.0, + "content": "Privacy with respect to an adversarial server", + "type": "text" + }, + { + "bbox": [ + 258, + 483, + 270, + 492 + ], + "score": 0.87, + "content": "{ \\boldsymbol { S } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 482, + 529, + 495 + ], + "score": 1.0, + "content": "(which Slalom does not provide), is defined as the indistinguisha-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 83, + 493, + 504, + 506 + ], + 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"clients’. We refer the reader to Canetti et al. (2002) for formal definitions for such commit-and-prove schemes,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 543, + 419, + 556 + ], + "spans": [ + { + "bbox": [ + 82, + 543, + 419, + 556 + ], + "score": 1.0, + "content": "and to Tramèr et al. (2017) who show how to trivially instantiate them using a TEE.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 84, + 562, + 528, + 608 + ], + "lines": [ + { + "bbox": [ + 81, + 561, + 528, + 578 + ], + "spans": [ + { + "bbox": [ + 81, + 561, + 205, + 578 + ], + "score": 1.0, + "content": "Proof of Theorem 3.2. Let", + "type": "text" + }, + { + "bbox": [ + 206, + 564, + 237, + 574 + ], + "score": 0.32, + "content": "\\mathrm { s t ~ ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 561, + 333, + 578 + ], + "score": 1.0, + "content": "Preproc and Outsource", + "type": "text" + }, + { + "bbox": [ + 334, + 563, + 413, + 575 + ], + "score": 0.81, + "content": "\\langle \\mathrm { T E E } ( F , x , \\mathrm { s t } ) , S \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 561, + 528, + 578 + ], + "score": 1.0, + "content": "be the outsourcing scheme", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 573, + 528, + 587 + ], + "spans": [ + { + "bbox": [ + 82, + 573, + 528, + 587 + ], + "score": 1.0, + "content": "defined in Figure 1 (right). We assume that all random values sampled by the TEE are produced by a secure", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 82, + 585, + 529, + 599 + ], + "spans": [ + { + "bbox": [ + 82, + 585, + 432, + 599 + ], + "score": 1.0, + "content": "cryptographically secure pseudorandom number generator (PRNG) (with elements in", + "type": "text" + }, + { + "bbox": [ + 433, + 585, + 459, + 596 + ], + "score": 0.9, + "content": "\\mathbb { S } \\subseteq \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 585, + 529, + 599 + ], + "score": 1.0, + "content": "for the integrity-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 595, + 412, + 610 + ], + "spans": [ + { + "bbox": [ + 82, + 595, + 140, + 610 + ], + "score": 1.0, + "content": "check vectors", + "type": "text" + }, + { + "bbox": [ + 141, + 599, + 146, + 606 + ], + "score": 0.53, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 595, + 291, + 610 + ], + "score": 1.0, + "content": "used in Freivalds’ algorithm, and in", + "type": "text" + }, + { + "bbox": [ + 291, + 596, + 299, + 606 + ], + "score": 0.77, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 595, + 396, + 610 + ], + "score": 1.0, + "content": "for the blinding vectors", + "type": "text" + }, + { + "bbox": [ + 396, + 598, + 405, + 607 + ], + "score": 0.79, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 595, + 412, + 610 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 84, + 613, + 528, + 680 + ], + "lines": [ + { + "bbox": [ + 82, + 612, + 529, + 627 + ], + "spans": [ + { + "bbox": [ + 82, + 612, + 362, + 627 + ], + "score": 1.0, + "content": "We first consider integrity. Assume that the scheme is run with input", + "type": "text" + }, + { + "bbox": [ + 362, + 615, + 374, + 624 + ], + "score": 0.86, + "content": "x _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 612, + 478, + 627 + ], + "score": 1.0, + "content": "and that the TEE outputs", + "type": "text" + }, + { + "bbox": [ + 478, + 615, + 489, + 624 + ], + "score": 0.84, + "content": "y _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 612, + 529, + 627 + ], + "score": 1.0, + "content": ". We will", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 623, + 529, + 637 + ], + "spans": [ + { + "bbox": [ + 83, + 623, + 111, + 637 + ], + "score": 1.0, + "content": "bound", + "type": "text" + }, + { + "bbox": [ + 111, + 624, + 217, + 636 + ], + "score": 0.91, + "content": "\\operatorname* { P r } [ y _ { n } \\neq F ( x _ { 1 } ) \\ | \\ { \\dot { y } } _ { n } \\neq \\bot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 623, + 449, + 637 + ], + "score": 1.0, + "content": ". By the security of the PRNG, we can replace the vectors", + "type": "text" + }, + { + "bbox": [ + 449, + 626, + 455, + 634 + ], + "score": 0.73, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 623, + 529, + 637 + ], + "score": 1.0, + "content": "used in Freivalds’", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 82, + 634, + 529, + 648 + ], + "spans": [ + { + "bbox": [ + 82, + 634, + 275, + 648 + ], + "score": 1.0, + "content": "algorithm by truly uniformly random values in", + "type": "text" + }, + { + "bbox": [ + 275, + 635, + 302, + 646 + ], + "score": 0.9, + "content": "\\mathbb { S } \\subseteq \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 634, + 461, + 648 + ], + "score": 1.0, + "content": ", via a simple hybrid argument. For the", + "type": "text" + }, + { + "bbox": [ + 461, + 636, + 466, + 645 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 634, + 529, + 648 + ], + "score": 1.0, + "content": "-th linear layer,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 646, + 529, + 659 + ], + "spans": [ + { + "bbox": [ + 82, + 646, + 138, + 659 + ], + "score": 1.0, + "content": "with operator", + "type": "text" + }, + { + "bbox": [ + 139, + 646, + 152, + 657 + ], + "score": 0.88, + "content": "W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 646, + 179, + 659 + ], + "score": 1.0, + "content": ", input", + "type": "text" + }, + { + "bbox": [ + 179, + 648, + 190, + 657 + ], + "score": 0.86, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 646, + 276, + 659 + ], + "score": 1.0, + "content": "and purported output", + "type": "text" + }, + { + "bbox": [ + 277, + 648, + 285, + 657 + ], + "score": 0.81, + "content": "y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 646, + 361, + 659 + ], + "score": 1.0, + "content": ", we then have that", + "type": "text" + }, + { + "bbox": [ + 362, + 646, + 406, + 657 + ], + "score": 0.91, + "content": "y _ { i } \\neq x _ { i } W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 646, + 504, + 659 + ], + "score": 1.0, + "content": "with probability at most", + "type": "text" + }, + { + "bbox": [ + 505, + 646, + 525, + 658 + ], + "score": 0.91, + "content": "1 / | \\mathbb { S } | ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 526, + 646, + 529, + 659 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 83, + 656, + 529, + 670 + ], + "spans": [ + { + "bbox": [ + 83, + 656, + 266, + 670 + ], + "score": 1.0, + "content": "By a simple union bound, we thus have that", + "type": "text" + }, + { + "bbox": [ + 266, + 657, + 413, + 669 + ], + "score": 0.91, + "content": "\\operatorname* { P r } [ y _ { n } \\neq F ( x _ { 1 } ) ] \\leq { n } / { | \\mathbb { S } | ^ { k } } - \\operatorname { n e g l } ( \\lambda ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 656, + 529, + 670 + ], + "score": 1.0, + "content": ". Note that this bound holds", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 82, + 667, + 354, + 681 + ], + "spans": [ + { + "bbox": [ + 82, + 667, + 244, + 681 + ], + "score": 1.0, + "content": "even if the same (secret) random values", + "type": "text" + }, + { + "bbox": [ + 244, + 671, + 249, + 678 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 667, + 354, + 681 + ], + "score": 1.0, + "content": "are re-used across layers.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 84, + 84, + 270, + 95 + ], + "lines": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 277, + 699, + 288, + 708 + ], + "lines": [ + { + "bbox": [ + 276, + 698, + 289, + 711 + ], + "spans": [ + { + "bbox": [ + 276, + 698, + 289, + 711 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 83, + 140, + 529, + 174 + ], + "lines": [ + { + "bbox": [ + 83, + 140, + 527, + 152 + ], + "spans": [ + { + "bbox": [ + 83, + 140, + 527, + 152 + ], + "score": 1.0, + "content": "We refrain from formally modeling SGX’s (or other TEE’s) security in this paper, as Slalom is mostly concerned", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 83, + 152, + 529, + 164 + ], + "spans": [ + { + "bbox": [ + 83, + 152, + 529, + 164 + ], + "score": 1.0, + "content": "with outsourcing protocols wherein the TEE acts as a client. We refer the interested reader to (Pass et al., 2017;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 83, + 163, + 431, + 175 + ], + "spans": [ + { + "bbox": [ + 83, + 163, + 431, + 175 + ], + "score": 1.0, + "content": "Fisch et al., 2017; Subramanyan et al., 2017) for different attempts at such formalisms.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 83, + 140, + 529, + 175 + ] + }, + { + "type": "title", + "bbox": [ + 84, + 191, + 345, + 204 + ], + "lines": [ + { + "bbox": [ + 82, + 190, + 346, + 206 + ], + "spans": [ + { + "bbox": [ + 82, + 190, + 346, + 206 + ], + "score": 1.0, + "content": "B FORMAL SECURITY DEFINITIONS AND PROOFS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 83, + 217, + 528, + 240 + ], + "lines": [ + { + "bbox": [ + 83, + 217, + 528, + 230 + ], + "spans": [ + { + "bbox": [ + 83, + 217, + 307, + 230 + ], + "score": 1.0, + "content": "We define a secure outsourcing scheme, between a client", + "type": "text" + }, + { + "bbox": [ + 307, + 218, + 314, + 227 + ], + "score": 0.78, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 217, + 364, + 230 + ], + "score": 1.0, + "content": "and a server", + "type": "text" + }, + { + "bbox": [ + 365, + 218, + 372, + 227 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 217, + 420, + 230 + ], + "score": 1.0, + "content": ", for a DNN", + "type": "text" + }, + { + "bbox": [ + 420, + 217, + 483, + 229 + ], + "score": 0.91, + "content": "F ( x ) : \\mathcal { X } \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 217, + 528, + 230 + ], + "score": 1.0, + "content": "from some", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 228, + 493, + 240 + ], + "spans": [ + { + "bbox": [ + 83, + 228, + 112, + 240 + ], + "score": 1.0, + "content": "family", + "type": "text" + }, + { + "bbox": [ + 112, + 228, + 122, + 239 + ], + "score": 0.81, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 228, + 376, + 240 + ], + "score": 1.0, + "content": "(e.g., all DNNs of a given size). We first assume that the model", + "type": "text" + }, + { + "bbox": [ + 377, + 229, + 386, + 238 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 228, + 456, + 240 + ], + "score": 1.0, + "content": "is known to both", + "type": "text" + }, + { + "bbox": [ + 456, + 229, + 463, + 238 + ], + "score": 0.82, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 228, + 481, + 240 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 482, + 229, + 489, + 238 + ], + "score": 0.83, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 228, + 493, + 240 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 83, + 217, + 528, + 240 + ] + }, + { + "type": "text", + "bbox": [ + 83, + 243, + 526, + 266 + ], + "lines": [ + { + "bbox": [ + 82, + 242, + 528, + 257 + ], + "spans": [ + { + "bbox": [ + 82, + 242, + 528, + 257 + ], + "score": 1.0, + "content": "Definition B.1 (Secure Outsourcing Schemes). A secure outsourcing scheme consists of an offline preprocessing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 83, + 254, + 472, + 267 + ], + "spans": [ + { + "bbox": [ + 83, + 254, + 365, + 267 + ], + "score": 1.0, + "content": "algorithm Preproc, as well as an interactive online protocol Outsource", + "type": "text" + }, + { + "bbox": [ + 366, + 255, + 390, + 266 + ], + "score": 0.91, + "content": "\\langle \\mathcal { C } , \\mathcal { S } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 254, + 472, + 267 + ], + "score": 1.0, + "content": ", defined as follows:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 82, + 242, + 528, + 267 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 270, + 528, + 323 + ], + "lines": [ + { + "bbox": [ + 87, + 270, + 528, + 284 + ], + "spans": [ + { + "bbox": [ + 87, + 270, + 97, + 284 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 98, + 270, + 188, + 283 + ], + "score": 0.85, + "content": "\\mathsf { s t } \\gets \\mathsf { P r e p r o c } ( F , 1 ^ { \\lambda } )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 270, + 347, + 284 + ], + "score": 1.0, + "content": ": The preprocessing algorithm is run by", + "type": "text" + }, + { + "bbox": [ + 347, + 271, + 354, + 281 + ], + "score": 0.76, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 270, + 528, + 284 + ], + "score": 1.0, + "content": "and generates some data-independent state", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 98, + 282, + 492, + 295 + ], + "spans": [ + { + "bbox": [ + 98, + 283, + 111, + 293 + ], + "score": 0.34, + "content": "\\mathtt { s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 282, + 492, + 295 + ], + "score": 1.0, + "content": "(e.g., cryptographic keys or precomputed values to accelerate the online outsourcing protocol.)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 87, + 298, + 528, + 313 + ], + "spans": [ + { + "bbox": [ + 87, + 298, + 96, + 313 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 96, + 299, + 276, + 312 + ], + "score": 0.8, + "content": "\\mathcal { V } \\cup \\{ \\bot \\} \\mathsf { O u t s o u r c e } \\langle \\mathcal { C } ( F , x , \\mathrm { s t } ) , \\mathcal { S } ( F ) \\rangle \\mathrm { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 298, + 471, + 313 + ], + "score": 1.0, + "content": ": The online outsourcing protocol is initiated by", + "type": "text" + }, + { + "bbox": [ + 471, + 300, + 479, + 310 + ], + "score": 0.78, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 298, + 528, + 313 + ], + "score": 1.0, + "content": "with inputs", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 97, + 311, + 488, + 324 + ], + "spans": [ + { + "bbox": [ + 97, + 311, + 138, + 323 + ], + "score": 0.85, + "content": "( F , x , s \\ t )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 311, + 250, + 324 + ], + "score": 1.0, + "content": ". At the end of the protocol,", + "type": "text" + }, + { + "bbox": [ + 251, + 312, + 258, + 321 + ], + "score": 0.77, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 311, + 347, + 324 + ], + "score": 1.0, + "content": "either outputs a value", + "type": "text" + }, + { + "bbox": [ + 347, + 311, + 373, + 322 + ], + "score": 0.91, + "content": "y \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 311, + 432, + 324 + ], + "score": 1.0, + "content": "or aborts (i.e.,", + "type": "text" + }, + { + "bbox": [ + 432, + 311, + 439, + 321 + ], + "score": 0.76, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 311, + 472, + 324 + ], + "score": 1.0, + "content": "outputs", + "type": "text" + }, + { + "bbox": [ + 473, + 312, + 481, + 321 + ], + "score": 0.64, + "content": "\\perp", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 311, + 488, + 324 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 87, + 270, + 528, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 327, + 381, + 339 + ], + "lines": [ + { + "bbox": [ + 84, + 326, + 382, + 341 + ], + "spans": [ + { + "bbox": [ + 84, + 326, + 382, + 341 + ], + "score": 1.0, + "content": "The properties that we may require from a secure outsourcing scheme are:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 84, + 326, + 382, + 341 + ] + }, + { + "type": "text", + "bbox": [ + 88, + 342, + 524, + 367 + ], + "lines": [ + { + "bbox": [ + 88, + 340, + 528, + 357 + ], + "spans": [ + { + "bbox": [ + 88, + 340, + 205, + 357 + ], + "score": 1.0, + "content": "• Correctness: For any", + "type": "text" + }, + { + "bbox": [ + 205, + 343, + 251, + 353 + ], + "score": 0.84, + "content": "\\begin{array} { r l r l } { F } & { { } \\in } & { { \\mathcal { F } } } \\end{array}", + "type": "inline_equation" + }, + { + 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In some applications a secure outsourcing scheme may also require to hide the model", + "type": "text" + }, + { + "bbox": [ + 496, + 455, + 505, + 464 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 505, + 454, + 528, + 467 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 83, + 466, + 473, + 477 + ], + "spans": [ + { + "bbox": [ + 83, + 466, + 108, + 477 + ], + "score": 1.0, + "content": "either", + "type": "text" + }, + { + "bbox": [ + 109, + 466, + 117, + 475 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 466, + 128, + 477 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 129, + 466, + 136, + 475 + ], + "score": 0.79, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 466, + 340, + 477 + ], + "score": 1.0, + "content": "(in which case that party would obviously not take", + "type": "text" + }, + { + "bbox": [ + 341, + 466, + 350, + 475 + ], + "score": 0.81, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 466, + 473, + 477 + ], + "score": 1.0, + "content": "as input in the above scheme).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 83, + 454, + 528, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 83, + 482, + 528, + 505 + ], + "lines": [ + { + "bbox": [ + 83, + 482, + 529, + 495 + ], + "spans": [ + { + "bbox": [ + 83, + 482, + 258, + 495 + ], + "score": 1.0, + "content": "Privacy with respect to an adversarial server", + "type": "text" + }, + { + "bbox": [ + 258, + 483, + 270, + 492 + ], + "score": 0.87, + "content": "{ \\boldsymbol { S } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 482, + 529, + 495 + ], + "score": 1.0, + "content": "(which Slalom does not provide), is defined as the indistinguisha-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 83, + 493, + 504, + 506 + ], + "spans": [ + { + "bbox": [ + 83, + 493, + 118, + 506 + ], + "score": 1.0, + "content": "bility of", + "type": "text" + }, + { + "bbox": [ + 118, + 494, + 130, + 504 + ], + "score": 0.85, + "content": "S ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 493, + 217, + 506 + ], + "score": 1.0, + "content": "’s views in Outsource", + "type": "text" + }, + { + "bbox": [ + 217, + 493, + 288, + 505 + ], + "score": 0.9, + "content": "\\langle \\mathcal { C } ( F , x , \\mathsf { s t } ) , \\mathcal { S } ^ { * } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 493, + 349, + 506 + ], + "score": 1.0, + "content": "and Outsource", + "type": "text" + }, + { + "bbox": [ + 349, + 493, + 423, + 505 + ], + "score": 0.9, + "content": "\\langle \\mathcal { C } ( F ^ { \\prime } , x , \\mathrm { s t } ) , S ^ { * } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 493, + 456, + 506 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 456, + 493, + 499, + 505 + ], + "score": 0.92, + "content": "F , F ^ { \\prime } \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 493, + 504, + 506 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 83, + 482, + 529, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 510, + 528, + 555 + ], + "lines": [ + { + "bbox": [ + 82, + 509, + 529, + 524 + ], + "spans": [ + { + "bbox": [ + 82, + 509, + 405, + 524 + ], + "score": 1.0, + "content": "As noted in Section 2.1, a meaningful model-privacy guarantee with respect to", + "type": "text" + }, + { + "bbox": [ + 405, + 511, + 412, + 520 + ], + "score": 0.76, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 509, + 466, + 524 + ], + "score": 1.0, + "content": "requires that", + "type": "text" + }, + { + "bbox": [ + 467, + 511, + 475, + 520 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 509, + 529, + 524 + ], + "score": 1.0, + "content": "first commit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 520, + 528, + 533 + ], + "spans": [ + { + "bbox": [ + 82, + 520, + 160, + 533 + ], + "score": 1.0, + "content": "to a specific DNN", + "type": "text" + }, + { + "bbox": [ + 160, + 522, + 169, + 531 + ], + "score": 0.79, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 520, + 254, + 533 + ], + "score": 1.0, + "content": ", and then convinces", + "type": "text" + }, + { + "bbox": [ + 254, + 522, + 262, + 531 + ], + "score": 0.77, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 520, + 528, + 533 + ], + "score": 1.0, + "content": "that her outputs were produced with the same model as all other", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 531, + 530, + 546 + ], + "spans": [ + { + "bbox": [ + 82, + 531, + 530, + 546 + ], + "score": 1.0, + "content": "clients’. We refer the reader to Canetti et al. (2002) for formal definitions for such commit-and-prove schemes,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 543, + 419, + 556 + ], + "spans": [ + { + "bbox": [ + 82, + 543, + 419, + 556 + ], + "score": 1.0, + "content": "and to Tramèr et al. (2017) who show how to trivially instantiate them using a TEE.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 82, + 509, + 530, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 562, + 528, + 608 + ], + "lines": [ + { + "bbox": [ + 81, + 561, + 528, + 578 + ], + "spans": [ + { + "bbox": [ + 81, + 561, + 205, + 578 + ], + "score": 1.0, + "content": "Proof of Theorem 3.2. Let", + "type": "text" + }, + { + "bbox": [ + 206, + 564, + 237, + 574 + ], + "score": 0.32, + "content": "\\mathrm { s t ~ ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 561, + 333, + 578 + ], + "score": 1.0, + "content": "Preproc and Outsource", + "type": "text" + }, + { + "bbox": [ + 334, + 563, + 413, + 575 + ], + "score": 0.81, + "content": "\\langle \\mathrm { T E E } ( F , x , \\mathrm { s t } ) , S \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 561, + 528, + 578 + ], + "score": 1.0, + "content": "be the outsourcing scheme", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 573, + 528, + 587 + ], + "spans": [ + { + "bbox": [ + 82, + 573, + 528, + 587 + ], + "score": 1.0, + "content": "defined in Figure 1 (right). We assume that all random values sampled by the TEE are produced by a secure", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 82, + 585, + 529, + 599 + ], + "spans": [ + { + "bbox": [ + 82, + 585, + 432, + 599 + ], + "score": 1.0, + "content": "cryptographically secure pseudorandom number generator (PRNG) (with elements in", + "type": "text" + }, + { + "bbox": [ + 433, + 585, + 459, + 596 + ], + "score": 0.9, + "content": "\\mathbb { S } \\subseteq \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 585, + 529, + 599 + ], + "score": 1.0, + "content": "for the integrity-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 595, + 412, + 610 + ], + "spans": [ + { + "bbox": [ + 82, + 595, + 140, + 610 + ], + "score": 1.0, + "content": "check vectors", + "type": "text" + }, + { + "bbox": [ + 141, + 599, + 146, + 606 + ], + "score": 0.53, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 595, + 291, + 610 + ], + "score": 1.0, + "content": "used in Freivalds’ algorithm, and in", + "type": "text" + }, + { + "bbox": [ + 291, + 596, + 299, + 606 + ], + "score": 0.77, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 595, + 396, + 610 + ], + "score": 1.0, + "content": "for the blinding vectors", + "type": "text" + }, + { + "bbox": [ + 396, + 598, + 405, + 607 + ], + "score": 0.79, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 595, + 412, + 610 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 81, + 561, + 529, + 610 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 613, + 528, + 680 + ], + "lines": [ + { + "bbox": [ + 82, + 612, + 529, + 627 + ], + "spans": [ + { + "bbox": [ + 82, + 612, + 362, + 627 + ], + "score": 1.0, + "content": "We first consider integrity. Assume that the scheme is run with input", + "type": "text" + }, + { + "bbox": [ + 362, + 615, + 374, + 624 + ], + "score": 0.86, + "content": "x _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 612, + 478, + 627 + ], + "score": 1.0, + "content": "and that the TEE outputs", + "type": "text" + }, + { + "bbox": [ + 478, + 615, + 489, + 624 + ], + "score": 0.84, + "content": "y _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 612, + 529, + 627 + ], + "score": 1.0, + "content": ". We will", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 623, + 529, + 637 + ], + "spans": [ + { + "bbox": [ + 83, + 623, + 111, + 637 + ], + "score": 1.0, + "content": "bound", + "type": "text" + }, + { + "bbox": [ + 111, + 624, + 217, + 636 + ], + "score": 0.91, + "content": "\\operatorname* { P r } [ y _ { n } \\neq F ( x _ { 1 } ) \\ | \\ { \\dot { y } } _ { n } \\neq \\bot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 623, + 449, + 637 + ], + "score": 1.0, + "content": ". By the security of the PRNG, we can replace the vectors", + "type": "text" + }, + { + "bbox": [ + 449, + 626, + 455, + 634 + ], + "score": 0.73, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 623, + 529, + 637 + ], + "score": 1.0, + "content": "used in Freivalds’", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 82, + 634, + 529, + 648 + ], + "spans": [ + { + "bbox": [ + 82, + 634, + 275, + 648 + ], + "score": 1.0, + "content": "algorithm by truly uniformly random values in", + "type": "text" + }, + { + "bbox": [ + 275, + 635, + 302, + 646 + ], + "score": 0.9, + "content": "\\mathbb { S } \\subseteq \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 634, + 461, + 648 + ], + "score": 1.0, + "content": ", via a simple hybrid argument. For the", + "type": "text" + }, + { + "bbox": [ + 461, + 636, + 466, + 645 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 634, + 529, + 648 + ], + "score": 1.0, + "content": "-th linear layer,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 82, + 646, + 529, + 659 + ], + "spans": [ + { + "bbox": [ + 82, + 646, + 138, + 659 + ], + "score": 1.0, + "content": "with operator", + "type": "text" + }, + { + "bbox": [ + 139, + 646, + 152, + 657 + ], + "score": 0.88, + "content": "W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 646, + 179, + 659 + ], + "score": 1.0, + "content": ", input", + "type": "text" + }, + { + "bbox": [ + 179, + 648, + 190, + 657 + ], + "score": 0.86, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 646, + 276, + 659 + ], + "score": 1.0, + "content": "and purported output", + "type": "text" + }, + { + "bbox": [ + 277, + 648, + 285, + 657 + ], + "score": 0.81, + "content": "y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 646, + 361, + 659 + ], + "score": 1.0, + "content": ", we then have that", + "type": "text" + }, + { + "bbox": [ + 362, + 646, + 406, + 657 + ], + "score": 0.91, + "content": "y _ { i } \\neq x _ { i } W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 646, + 504, + 659 + ], + "score": 1.0, + "content": "with probability at most", + "type": "text" + }, + { + "bbox": [ + 505, + 646, + 525, + 658 + ], + "score": 0.91, + "content": "1 / | \\mathbb { S } | ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 526, + 646, + 529, + 659 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 83, + 656, + 529, + 670 + ], + "spans": [ + { + "bbox": [ + 83, + 656, + 266, + 670 + ], + "score": 1.0, + "content": "By a simple union bound, we thus have that", + "type": "text" + }, + { + "bbox": [ + 266, + 657, + 413, + 669 + ], + "score": 0.91, + "content": "\\operatorname* { P r } [ y _ { n } \\neq F ( x _ { 1 } ) ] \\leq { n } / { | \\mathbb { S } | ^ { k } } - \\operatorname { n e g l } ( \\lambda ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 656, + 529, + 670 + ], + "score": 1.0, + "content": ". 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Our baseline runs a DNN in a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 82, + 385, + 528, + 400 + ], + "spans": [ + { + "bbox": [ + 82, + 385, + 528, + 400 + ], + "score": 1.0, + "content": "TEE (a single-threaded Intel SGX enclave) and can provide all the security guarantees of an ML outsourcing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 397, + 529, + 410 + ], + "spans": [ + { + "bbox": [ + 82, + 397, + 369, + 410 + ], + "score": 1.0, + "content": "scheme. On a high-end GPU (an Nvidia TITAN XP), we achieve over", + "type": "text" + }, + { + "bbox": [ + 370, + 398, + 389, + 408 + ], + "score": 0.87, + "content": "5 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 397, + 529, + 410 + ], + "score": 1.0, + "content": "higher throughput but no security.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 81, + 407, + 521, + 422 + ], + "spans": [ + { + "bbox": [ + 81, + 407, + 466, + 422 + ], + "score": 1.0, + "content": "For example, for MobileNet, the enclave evaluates 16 images/sec and the GPU 900 images/sec", + "type": "text" + }, + { + "bbox": [ + 467, + 409, + 486, + 419 + ], + "score": 0.85, + "content": "5 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 407, + 521, + 422 + ], + "score": 1.0, + "content": "higher).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 84, + 425, + 527, + 514 + ], + "lines": [ + { + "bbox": [ + 83, + 425, + 528, + 437 + ], + "spans": [ + { + "bbox": [ + 83, + 425, + 528, + 437 + ], + "score": 1.0, + "content": "SafetyNets (Ghodsi et al., 2017) and Gazelle (Juvekar et al., 2018) are two representative works that achieve", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 436, + 529, + 450 + ], + "spans": [ + { + "bbox": [ + 82, + 436, + 529, + 450 + ], + "score": 1.0, + "content": "respectively integrity and privacy using purely cryptographic approaches (without a TEE). SafetyNets does not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 447, + 528, + 460 + ], + "spans": [ + { + "bbox": [ + 82, + 447, + 528, + 460 + ], + "score": 1.0, + "content": "hide the model from either party, while Gazelle leaks some architectural details to the client. The cryptographic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 459, + 529, + 471 + ], + "spans": [ + { + "bbox": [ + 82, + 459, + 529, + 471 + ], + "score": 1.0, + "content": "techniques used by these systems incur large computation and communication overheads in practice. The largest", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 468, + 529, + 482 + ], + "spans": [ + { + "bbox": [ + 82, + 468, + 529, + 482 + ], + "score": 1.0, + "content": "model evaluated by SafetyNets is a 4-layer TIMIT model with quadratic activations which runs at about 13", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 82, + 480, + 528, + 493 + ], + "spans": [ + { + "bbox": [ + 82, + 480, + 528, + 493 + ], + "score": 1.0, + "content": "images/sec (on a notebook CPU). In our baseline enclave, the same model runs at over 3,500 images/sec. The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 491, + 529, + 504 + ], + "spans": [ + { + "bbox": [ + 82, + 491, + 529, + 504 + ], + "score": 1.0, + "content": "largest model evaluated by Gazelle is an 8-layer CIFAR10 model. In the enclave, we can evaluate 450 images/sec", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 502, + 525, + 515 + ], + "spans": [ + { + "bbox": [ + 83, + 502, + 311, + 515 + ], + "score": 1.0, + "content": "whereas Gazelle evaluates a single image in 3.5 sec with", + "type": "text" + }, + { + "bbox": [ + 311, + 502, + 343, + 513 + ], + "score": 0.29, + "content": "3 0 0 \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 502, + 525, + 515 + ], + "score": 1.0, + "content": "of communication between client and server.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 84, + 524, + 528, + 569 + ], + "lines": [ + { + "bbox": [ + 82, + 523, + 528, + 538 + ], + "spans": [ + { + "bbox": [ + 82, + 523, + 528, + 538 + ], + "score": 1.0, + "content": "A Note on Energy Efficiency. When comparing approaches with different hardware (e.g., our single-core CPU", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 82, + 535, + 529, + 548 + ], + "spans": [ + { + "bbox": [ + 82, + 535, + 529, + 548 + ], + "score": 1.0, + "content": "baseline versus Slalom which also uses a GPU), throughput alone is not the fairest metric. E.g., the baseline’s", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 546, + 529, + 558 + ], + "spans": [ + { + "bbox": [ + 83, + 546, + 529, + 558 + ], + "score": 1.0, + "content": "throughput could also be increased by adding more SGX CPUs. A more accurate comparison considers the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 82, + 558, + 493, + 569 + ], + "spans": [ + { + "bbox": [ + 82, + 558, + 481, + 569 + ], + "score": 1.0, + "content": "energy efficiency of a particular approach, a more direct measure of the recurrent costs to the server", + "type": "text" + }, + { + "bbox": [ + 482, + 558, + 489, + 567 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 558, + 493, + 569 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 84, + 573, + 528, + 608 + ], + "lines": [ + { + "bbox": [ + 82, + 573, + 529, + 587 + ], + "spans": [ + { + "bbox": [ + 82, + 573, + 529, + 587 + ], + "score": 1.0, + "content": "For example, when evaluating MobileNet or VGG16, our GPU draws 85W of power, whereas our baseline SGX", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 82, + 584, + 529, + 598 + ], + "spans": [ + { + "bbox": [ + 82, + 584, + 356, + 598 + ], + "score": 1.0, + "content": "CPU draws 30W. As noted above, the GPU also achieves more than", + "type": "text" + }, + { + "bbox": [ + 356, + 586, + 375, + 596 + ], + "score": 0.87, + "content": "5 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 584, + 529, + 598 + ], + "score": 1.0, + "content": "higher throughput, and thus is at least", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 596, + 406, + 609 + ], + "spans": [ + { + "bbox": [ + 83, + 596, + 102, + 607 + ], + "score": 0.86, + "content": "1 8 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 102, + 596, + 406, + 609 + ], + "score": 1.0, + "content": "more energy efficient (e.g., measured in Joules per image) than the enclave.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 84, + 613, + 527, + 680 + ], + "lines": [ + { + "bbox": [ + 83, + 613, + 528, + 624 + ], + "spans": [ + { + "bbox": [ + 83, + 613, + 528, + 624 + ], + "score": 1.0, + "content": "For Slalom, we must consider the cost of running both the enclave and GPU. 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Thus, the power consumption attributed", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 83, + 645, + 528, + 658 + ], + "spans": [ + { + "bbox": [ + 83, + 646, + 165, + 658 + ], + "score": 1.0, + "content": "to Slalom is roughly", + "type": "text" + }, + { + "bbox": [ + 165, + 645, + 310, + 657 + ], + "score": 0.89, + "content": "1 0 \\% \\cdot 8 5 \\mathrm { W } + 9 0 \\% \\cdot 3 0 \\mathrm { W } = 3 5 . 5 \\mathrm { W } .", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 646, + 528, + 658 + ], + "score": 1.0, + "content": "Note that when not being in use by Slalom, the trusted", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 83, + 656, + 528, + 670 + ], + "spans": [ + { + "bbox": [ + 83, + 656, + 463, + 670 + ], + "score": 1.0, + "content": "CPU or untrusted GPU can be used by other tasks running on the server. 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The outsourcing protocol between the remote client", + "type": "text" + }, + { + "bbox": [ + 393, + 208, + 400, + 217 + ], + "score": 0.78, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 206, + 443, + 219 + ], + "score": 1.0, + "content": "and server", + "type": "text" + }, + { + "bbox": [ + 444, + 207, + 452, + 217 + ], + "score": 0.77, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 206, + 528, + 219 + ], + "score": 1.0, + "content": "hosting the TEE is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 82, + 217, + 339, + 230 + ], + "spans": [ + { + "bbox": [ + 82, + 217, + 322, + 230 + ], + "score": 1.0, + "content": "simply defined as follows (we assume the model belongs to", + "type": "text" + }, + { + "bbox": [ + 323, + 218, + 331, + 228 + ], + "score": 0.74, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 217, + 339, + 230 + ], + "score": 1.0, + "content": "):", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 82, + 206, + 528, + 230 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 234, + 526, + 257 + ], + "lines": [ + { + "bbox": [ + 88, + 234, + 528, + 246 + ], + "spans": [ + { + "bbox": [ + 88, + 235, + 97, + 246 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 97, + 234, + 167, + 246 + ], + "score": 0.56, + "content": "\\mathsf { s t } \\gets \\mathsf { P r e p r o c ( ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 235, + 173, + 246 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 174, + 235, + 181, + 245 + ], + "score": 0.69, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 235, + 528, + 246 + ], + "score": 1.0, + "content": "and the TEE setup a secure authenticated communication channel, using the TEE’s", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 96, + 245, + 509, + 257 + ], + "spans": [ + { + "bbox": [ + 96, + 245, + 325, + 257 + ], + "score": 1.0, + "content": "remote attestation property. 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Our baseline runs a DNN in a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 82, + 385, + 528, + 400 + ], + "spans": [ + { + "bbox": [ + 82, + 385, + 528, + 400 + ], + "score": 1.0, + "content": "TEE (a single-threaded Intel SGX enclave) and can provide all the security guarantees of an ML outsourcing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 397, + 529, + 410 + ], + "spans": [ + { + "bbox": [ + 82, + 397, + 369, + 410 + ], + "score": 1.0, + "content": "scheme. On a high-end GPU (an Nvidia TITAN XP), we achieve over", + "type": "text" + }, + { + "bbox": [ + 370, + 398, + 389, + 408 + ], + "score": 0.87, + "content": "5 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 397, + 529, + 410 + ], + "score": 1.0, + "content": "higher throughput but no security.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 81, + 407, + 521, + 422 + ], + "spans": [ + { + "bbox": [ + 81, + 407, + 466, + 422 + ], + "score": 1.0, + "content": "For example, for MobileNet, the enclave evaluates 16 images/sec and the GPU 900 images/sec", + "type": "text" + }, + { + "bbox": [ + 467, + 409, + 486, + 419 + ], + "score": 0.85, + "content": "5 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 407, + 521, + 422 + ], + "score": 1.0, + "content": "higher).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 81, + 376, + 529, + 422 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 425, + 527, + 514 + ], + "lines": [ + { + "bbox": [ + 83, + 425, + 528, + 437 + ], + "spans": [ + { + "bbox": [ + 83, + 425, + 528, + 437 + ], + "score": 1.0, + "content": "SafetyNets (Ghodsi et al., 2017) and Gazelle (Juvekar et al., 2018) are two representative works that achieve", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 436, + 529, + 450 + ], + "spans": [ + { + "bbox": [ + 82, + 436, + 529, + 450 + ], + "score": 1.0, + "content": "respectively integrity and privacy using purely cryptographic approaches (without a TEE). SafetyNets does not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 447, + 528, + 460 + ], + "spans": [ + { + "bbox": [ + 82, + 447, + 528, + 460 + ], + "score": 1.0, + "content": "hide the model from either party, while Gazelle leaks some architectural details to the client. The cryptographic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 459, + 529, + 471 + ], + "spans": [ + { + "bbox": [ + 82, + 459, + 529, + 471 + ], + "score": 1.0, + "content": "techniques used by these systems incur large computation and communication overheads in practice. The largest", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 468, + 529, + 482 + ], + "spans": [ + { + "bbox": [ + 82, + 468, + 529, + 482 + ], + "score": 1.0, + "content": "model evaluated by SafetyNets is a 4-layer TIMIT model with quadratic activations which runs at about 13", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 82, + 480, + 528, + 493 + ], + "spans": [ + { + "bbox": [ + 82, + 480, + 528, + 493 + ], + "score": 1.0, + "content": "images/sec (on a notebook CPU). In our baseline enclave, the same model runs at over 3,500 images/sec. The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 82, + 491, + 529, + 504 + ], + "spans": [ + { + "bbox": [ + 82, + 491, + 529, + 504 + ], + "score": 1.0, + "content": "largest model evaluated by Gazelle is an 8-layer CIFAR10 model. In the enclave, we can evaluate 450 images/sec", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 502, + 525, + 515 + ], + "spans": [ + { + "bbox": [ + 83, + 502, + 311, + 515 + ], + "score": 1.0, + "content": "whereas Gazelle evaluates a single image in 3.5 sec with", + "type": "text" + }, + { + "bbox": [ + 311, + 502, + 343, + 513 + ], + "score": 0.29, + "content": "3 0 0 \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 502, + 525, + 515 + ], + "score": 1.0, + "content": "of communication between client and server.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 82, + 425, + 529, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 524, + 528, + 569 + ], + "lines": [ + { + "bbox": [ + 82, + 523, + 528, + 538 + ], + "spans": [ + { + "bbox": [ + 82, + 523, + 528, + 538 + ], + "score": 1.0, + "content": "A Note on Energy Efficiency. 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AccuracyQuantized
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VGG1671.090.070.689.516138.49.1 / 13.6
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MobileNet70.789.670.589.5284.25.5/ 5.0
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In all cases, verifying a computation becomes 1-2 orders of magnitude faster than computing it as the outer", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 606, + 528, + 619 + ], + "spans": [ + { + "bbox": [ + 82, + 606, + 528, + 619 + ], + "score": 1.0, + "content": "dimension grows. Compared to the SGX benchmarks, we also see a much better viability of batched verification", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 82, + 617, + 529, + 631 + ], + "spans": [ + { + "bbox": [ + 82, + 617, + 529, + 631 + ], + "score": 1.0, + "content": "(we haven’t optimized batched verifications much, as they are inherently slow on SGX. It is likely that these", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 628, + 466, + 642 + ], + "spans": [ + { + "bbox": [ + 82, + 628, + 466, + 642 + ], + "score": 1.0, + "content": "numbers could be improved significantly, to approach those of verification with preprocessing).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 84, + 645, + 527, + 680 + ], + "lines": [ + { + "bbox": [ + 83, + 644, + 528, + 659 + ], + "spans": [ + { + "bbox": [ + 83, + 644, + 528, + 659 + ], + "score": 1.0, + "content": "Figure 6 shows benchmarks for VGG16 and MobileNet on a single core with either direct computation or various", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 656, + 529, + 670 + ], + "spans": [ + { + "bbox": [ + 82, + 656, + 383, + 670 + ], + "score": 1.0, + "content": "secure outsourcing strategies. 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Even without storing any secrets in the enclave, we obtain good speedups using batched", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 84, + 84, + 270, + 95 + ], + "lines": [ + { + "bbox": [ + 83, + 83, + 271, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 83, + 271, + 96 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 277, + 699, + 287, + 707 + ], + "lines": [ + { + "bbox": [ + 276, + 698, + 289, + 712 + ], + "spans": [ + { + "bbox": [ + 276, + 698, + 289, + 712 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 83, + 138, + 418, + 152 + ], + "lines": [ + { + "bbox": [ + 82, + 138, + 420, + 153 + ], + "spans": [ + { + "bbox": [ + 82, + 138, + 420, + 153 + ], + "score": 1.0, + "content": "F MODULAR ARITHMETIC WITH FLOATING POINT OPERATIONS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 165, + 527, + 199 + ], + "lines": [ + { + "bbox": [ + 83, + 166, + 528, + 178 + ], + "spans": [ + { + "bbox": [ + 83, + 166, + 440, + 178 + ], + "score": 1.0, + "content": "In this section, we briefly describe how Slalom performs modular arithmetic over a field", + "type": "text" + }, + { + "bbox": [ + 441, + 166, + 453, + 178 + ], + "score": 0.89, + "content": "\\mathbb { Z } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 166, + 528, + 178 + ], + "score": 1.0, + "content": "in the TEE, while", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 83, + 176, + 529, + 190 + ], + "spans": [ + { + "bbox": [ + 83, + 176, + 529, + 190 + ], + "score": 1.0, + "content": "leveraging standard floating point operations to maximize computational efficiency. 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The dotted line in each plot shows the throughput", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 82, + 561, + 237, + 576 + ], + "spans": [ + { + "bbox": [ + 82, + 561, + 237, + 576 + ], + "score": 1.0, + "content": "obtained for computing the operation.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 83, + 585, + 528, + 641 + ], + "lines": [ + { + "bbox": [ + 82, + 585, + 529, + 597 + ], + "spans": [ + { + "bbox": [ + 82, + 585, + 529, + 597 + ], + "score": 1.0, + "content": "Figure 5 show the results of the micro-benchmarks for matrix multiplication, convolution and separable convolu-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 595, + 529, + 609 + ], + "spans": [ + { + "bbox": [ + 82, + 595, + 529, + 609 + ], + "score": 1.0, + "content": "tions. In all cases, verifying a computation becomes 1-2 orders of magnitude faster than computing it as the outer", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 82, + 606, + 528, + 619 + ], + "spans": [ + { + "bbox": [ + 82, + 606, + 528, + 619 + ], + "score": 1.0, + "content": "dimension grows. Compared to the SGX benchmarks, we also see a much better viability of batched verification", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 82, + 617, + 529, + 631 + ], + "spans": [ + { + "bbox": [ + 82, + 617, + 529, + 631 + ], + "score": 1.0, + "content": "(we haven’t optimized batched verifications much, as they are inherently slow on SGX. It is likely that these", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 628, + 466, + 642 + ], + "spans": [ + { + "bbox": [ + 82, + 628, + 466, + 642 + ], + "score": 1.0, + "content": "numbers could be improved significantly, to approach those of verification with preprocessing).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 82, + 585, + 529, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 645, + 527, + 680 + ], + "lines": [ + { + "bbox": [ + 83, + 644, + 528, + 659 + ], + "spans": [ + { + "bbox": [ + 83, + 644, + 528, + 659 + ], + "score": 1.0, + "content": "Figure 6 shows benchmarks for VGG16 and MobileNet on a single core with either direct computation or various", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 82, + 656, + 529, + 670 + ], + "spans": [ + { + "bbox": [ + 82, + 656, + 383, + 670 + ], + "score": 1.0, + "content": "secure outsourcing strategies. For integrity alone, we achieve savings up to", + "type": "text" + }, + { + "bbox": [ + 384, + 657, + 406, + 668 + ], + "score": 0.88, + "content": "8 . 9 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 656, + 424, + 670 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 424, + 657, + 450, + 668 + ], + "score": 0.86, + "content": "1 9 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 656, + 529, + 670 + ], + "score": 1.0, + "content": "for MobileNet and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 83, + 667, + 529, + 681 + ], + "spans": [ + { + "bbox": [ + 83, + 667, + 529, + 681 + ], + "score": 1.0, + "content": "VGG16 respectively. Even without storing any secrets in the enclave, we obtain good speedups using batched", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 277, + 528, + 288 + ], + "spans": [ + { + "bbox": [ + 83, + 277, + 528, + 288 + ], + "score": 1.0, + "content": "verification. As noted above, it is likely that the batched results could be further improved. With additional", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 287, + 520, + 300 + ], + "spans": [ + { + "bbox": [ + 83, + 287, + 295, + 300 + ], + "score": 1.0, + "content": "blinding to preserve privacy, we achieve speedups of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 296, + 288, + 317, + 298 + ], + "score": 0.87, + "content": "3 . 9 \\times", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 318, + 287, + 335, + 300 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 335, + 288, + 357, + 299 + ], + "score": 0.86, + "content": "8 . 1 \\times", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 358, + 287, + 520, + 300 + ], + "score": 1.0, + "content": "for MobileNet and VGG16 respectively.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 31, + "bbox_fs": [ + 82, + 644, + 529, + 681 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 90, + 138, + 527, + 214 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 90, + 138, + 527, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 90, + 138, + 527, + 214 + ], + "spans": [ + { + "bbox": [ + 90, + 138, + 527, + 214 + ], + "score": 0.959, + "type": "image", + "image_path": "c8154e92d0e65f4edf1599b73db1e4f3db8f57d0f186fd57a3cf3863327d0a07.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 90, + 138, + 527, + 163.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 90, + 163.33333333333334, + 527, + 188.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 90, + 188.66666666666669, + 527, + 214.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 83, + 219, + 528, + 264 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 82, + 218, + 528, + 232 + ], + "spans": [ + { + "bbox": [ + 82, + 218, + 528, + 232 + ], + "score": 1.0, + "content": "Figure 6: Inference with integrity and/or privacy on an untrusted CPU. We compare the baseline inference", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 82, + 229, + 529, + 244 + ], + "spans": [ + { + "bbox": [ + 82, + 229, + 529, + 244 + ], + "score": 1.0, + "content": "throughput (blue) to that obtained with “on-the-fly” integrity checks (red); batched integrity checks (green);", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 241, + 528, + 254 + ], + "spans": [ + { + "bbox": [ + 83, + 241, + 528, + 254 + ], + "score": 1.0, + "content": "integrity checks with precomputed secrets (yellow); privacy only (black); and privacy and integrity (purple). The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 82, + 252, + 412, + 265 + ], + "spans": [ + { + "bbox": [ + 82, + 252, + 412, + 265 + ], + "score": 1.0, + "content": "fused MobileNet model has no intermediate activation for separable convolutions.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 82, + 276, + 528, + 299 + ], + "lines": [ + { + "bbox": [ + 83, + 277, + 528, + 288 + ], + "spans": [ + { + "bbox": [ + 83, + 277, + 528, + 288 + ], + "score": 1.0, + "content": "verification. As noted above, it is likely that the batched results could be further improved. With additional", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 287, + 520, + 300 + ], + "spans": [ + { + "bbox": [ + 83, + 287, + 295, + 300 + ], + "score": 1.0, + "content": "blinding to preserve privacy, we achieve speedups of", + "type": "text" + }, + { + "bbox": [ + 296, + 288, + 317, + 298 + ], + "score": 0.87, + "content": "3 . 9 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 287, + 335, + 300 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 335, + 288, + 357, + 299 + ], + "score": 0.86, + "content": "8 . 1 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 287, + 520, + 300 + ], + "score": 1.0, + "content": "for MobileNet and VGG16 respectively.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 84, + 315, + 202, + 328 + ], + "lines": [ + { + "bbox": [ + 82, + 315, + 203, + 330 + ], + "spans": [ + { + "bbox": [ + 82, + 315, + 203, + 330 + ], + "score": 1.0, + "content": "H PARALLELIZATION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 82, + 340, + 529, + 374 + ], + "lines": [ + { + "bbox": [ + 82, + 339, + 528, + 353 + ], + "spans": [ + { + "bbox": [ + 82, + 339, + 528, + 353 + ], + "score": 1.0, + "content": "Our experiments on SGX in Section 4 where performed using a single execution thread, as SGX enclaves do not", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 83, + 352, + 529, + 364 + ], + "spans": [ + { + "bbox": [ + 83, + 352, + 529, + 364 + ], + "score": 1.0, + "content": "have the ability to create threads. We have also experimented with techniques for achieving parallelism in SGX,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 82, + 361, + 388, + 375 + ], + "spans": [ + { + "bbox": [ + 82, + 361, + 388, + 375 + ], + "score": 1.0, + "content": "both for standard computations and outsourced ones, but with little success.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 83, + 379, + 528, + 446 + ], + "lines": [ + { + "bbox": [ + 83, + 379, + 528, + 391 + ], + "spans": [ + { + "bbox": [ + 83, + 379, + 528, + 391 + ], + "score": 1.0, + "content": "To optimize for throughput, a simple approach is to run multiple forward passes simultaneously. On a standard", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 389, + 529, + 403 + ], + "spans": [ + { + "bbox": [ + 82, + 389, + 529, + 403 + ], + "score": 1.0, + "content": "CPU, this form of “outer-parallelism” achieves close to linear scaling as we increase the number of threads from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 401, + 529, + 414 + ], + "spans": [ + { + "bbox": [ + 82, + 401, + 529, + 414 + ], + "score": 1.0, + "content": "1 to 4 on our quad-core machine. With SGX however, we did not manage to achieve any parallel speedup for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 82, + 412, + 529, + 425 + ], + "spans": [ + { + "bbox": [ + 82, + 412, + 529, + 425 + ], + "score": 1.0, + "content": "VGG16—whether for direct computation or verifying outsourced results—presumably because each independent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 422, + 528, + 436 + ], + "spans": [ + { + "bbox": [ + 82, + 422, + 528, + 436 + ], + "score": 1.0, + "content": "thread requires extra memory that quickly exceeds the PRM limit. For the smaller MobileNet model, we get less", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 82, + 434, + 477, + 447 + ], + "spans": [ + { + "bbox": [ + 82, + 434, + 110, + 447 + ], + "score": 1.0, + "content": "than a", + "type": "text" + }, + { + "bbox": [ + 110, + 434, + 132, + 445 + ], + "score": 0.87, + "content": "1 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 434, + 477, + 447 + ], + "score": 1.0, + "content": "speedup using up to 4 threads, for direct computation or outsourced verification alike.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 83, + 450, + 528, + 517 + ], + "lines": [ + { + "bbox": [ + 82, + 449, + 529, + 465 + ], + "spans": [ + { + "bbox": [ + 82, + 449, + 529, + 465 + ], + "score": 1.0, + "content": "DNNs typically also make use of intra-operation parallelism, i.e., computing the output of a given layer using", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 460, + 529, + 476 + ], + "spans": [ + { + "bbox": [ + 82, + 460, + 529, + 476 + ], + "score": 1.0, + "content": "multiple threads. Our DNN library currently does not support intra-operation parallelism, but implementing", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 472, + 528, + 485 + ], + "spans": [ + { + "bbox": [ + 82, + 472, + 528, + 485 + ], + "score": 1.0, + "content": "a dedicated thread pool for SGX could be an interesting extension for future work. Instead, we evaluate the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 483, + 529, + 497 + ], + "spans": [ + { + "bbox": [ + 82, + 483, + 529, + 497 + ], + "score": 1.0, + "content": "potential benefits of intra-op parallelism on a standard untrusted CPU, for our matrix-product and convolution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 493, + 528, + 508 + ], + "spans": [ + { + "bbox": [ + 82, + 493, + 528, + 508 + ], + "score": 1.0, + "content": "benchmarks. We make use of Eigen’s internal multi-threading support to speed up these operations, and custom", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 84, + 506, + 400, + 519 + ], + "spans": [ + { + "bbox": [ + 84, + 506, + 400, + 519 + ], + "score": 1.0, + "content": "OpenMP code to parallelize dot products, as Eigen does not do this on its own.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "image", + "bbox": [ + 142, + 527, + 467, + 649 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 142, + 527, + 467, + 649 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 142, + 527, + 467, + 649 + ], + "spans": [ + { + "bbox": [ + 142, + 527, + 467, + 649 + ], + "score": 0.887, + "type": "image", + "image_path": "827a1dd46e9ddda18510395cc72fb7d148bced674096122d688675387980e22d.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 142, + 527, + 467, + 567.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 142, + 567.6666666666666, + 467, + 608.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 142, + 608.3333333333333, + 467, + 648.9999999999999 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 82, + 652, + 528, + 675 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 83, + 652, + 529, + 665 + ], + "spans": [ + { + "bbox": [ + 83, + 652, + 529, + 665 + ], + "score": 1.0, + "content": "Figure 7: Multi-threaded micro benchmarks on an untrusted CPU. Reiterates benchmarks for matrix prod-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 664, + 237, + 675 + ], + "spans": [ + { + "bbox": [ + 82, + 664, + 237, + 675 + ], + "score": 1.0, + "content": "ucts and convolutions using 4 threads.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "index": 27.25 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 84, + 85, + 270, + 95 + ], + "lines": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "spans": [ + { + "bbox": [ + 83, + 84, + 271, + 96 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 277, + 699, + 288, + 708 + ], + "lines": [ + { + "bbox": [ + 276, + 698, + 289, + 711 + ], + "spans": [ + { + "bbox": [ + 276, + 698, + 289, + 711 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 90, + 138, + 527, + 214 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 90, + 138, + 527, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 90, + 138, + 527, + 214 + ], + "spans": [ + { + "bbox": [ + 90, + 138, + 527, + 214 + ], + "score": 0.959, + "type": "image", + "image_path": "c8154e92d0e65f4edf1599b73db1e4f3db8f57d0f186fd57a3cf3863327d0a07.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 90, + 138, + 527, + 163.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 90, + 163.33333333333334, + 527, + 188.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 90, + 188.66666666666669, + 527, + 214.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 83, + 219, + 528, + 264 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 82, + 218, + 528, + 232 + ], + "spans": [ + { + "bbox": [ + 82, + 218, + 528, + 232 + ], + "score": 1.0, + "content": "Figure 6: Inference with integrity and/or privacy on an untrusted CPU. We compare the baseline inference", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 82, + 229, + 529, + 244 + ], + "spans": [ + { + "bbox": [ + 82, + 229, + 529, + 244 + ], + "score": 1.0, + "content": "throughput (blue) to that obtained with “on-the-fly” integrity checks (red); batched integrity checks (green);", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 241, + 528, + 254 + ], + "spans": [ + { + "bbox": [ + 83, + 241, + 528, + 254 + ], + "score": 1.0, + "content": "integrity checks with precomputed secrets (yellow); privacy only (black); and privacy and integrity (purple). The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 82, + 252, + 412, + 265 + ], + "spans": [ + { + "bbox": [ + 82, + 252, + 412, + 265 + ], + "score": 1.0, + "content": "fused MobileNet model has no intermediate activation for separable convolutions.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 82, + 276, + 528, + 299 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 83, + 277, + 528, + 300 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 84, + 315, + 202, + 328 + ], + "lines": [ + { + "bbox": [ + 82, + 315, + 203, + 330 + ], + "spans": [ + { + "bbox": [ + 82, + 315, + 203, + 330 + ], + "score": 1.0, + "content": "H PARALLELIZATION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 82, + 340, + 529, + 374 + ], + "lines": [ + { + "bbox": [ + 82, + 339, + 528, + 353 + ], + "spans": [ + { + "bbox": [ + 82, + 339, + 528, + 353 + ], + "score": 1.0, + "content": "Our experiments on SGX in Section 4 where performed using a single execution thread, as SGX enclaves do not", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 83, + 352, + 529, + 364 + ], + "spans": [ + { + "bbox": [ + 83, + 352, + 529, + 364 + ], + "score": 1.0, + "content": "have the ability to create threads. We have also experimented with techniques for achieving parallelism in SGX,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 82, + 361, + 388, + 375 + ], + "spans": [ + { + "bbox": [ + 82, + 361, + 388, + 375 + ], + "score": 1.0, + "content": "both for standard computations and outsourced ones, but with little success.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 82, + 339, + 529, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 83, + 379, + 528, + 446 + ], + "lines": [ + { + "bbox": [ + 83, + 379, + 528, + 391 + ], + "spans": [ + { + "bbox": [ + 83, + 379, + 528, + 391 + ], + "score": 1.0, + "content": "To optimize for throughput, a simple approach is to run multiple forward passes simultaneously. On a standard", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 82, + 389, + 529, + 403 + ], + "spans": [ + { + "bbox": [ + 82, + 389, + 529, + 403 + ], + "score": 1.0, + "content": "CPU, this form of “outer-parallelism” achieves close to linear scaling as we increase the number of threads from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 82, + 401, + 529, + 414 + ], + "spans": [ + { + "bbox": [ + 82, + 401, + 529, + 414 + ], + "score": 1.0, + "content": "1 to 4 on our quad-core machine. With SGX however, we did not manage to achieve any parallel speedup for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 82, + 412, + 529, + 425 + ], + "spans": [ + { + "bbox": [ + 82, + 412, + 529, + 425 + ], + "score": 1.0, + "content": "VGG16—whether for direct computation or verifying outsourced results—presumably because each independent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 82, + 422, + 528, + 436 + ], + "spans": [ + { + "bbox": [ + 82, + 422, + 528, + 436 + ], + "score": 1.0, + "content": "thread requires extra memory that quickly exceeds the PRM limit. For the smaller MobileNet model, we get less", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 82, + 434, + 477, + 447 + ], + "spans": [ + { + "bbox": [ + 82, + 434, + 110, + 447 + ], + "score": 1.0, + "content": "than a", + "type": "text" + }, + { + "bbox": [ + 110, + 434, + 132, + 445 + ], + "score": 0.87, + "content": "1 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 434, + 477, + 447 + ], + "score": 1.0, + "content": "speedup using up to 4 threads, for direct computation or outsourced verification alike.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 82, + 379, + 529, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 83, + 450, + 528, + 517 + ], + "lines": [ + { + "bbox": [ + 82, + 449, + 529, + 465 + ], + "spans": [ + { + "bbox": [ + 82, + 449, + 529, + 465 + ], + "score": 1.0, + "content": "DNNs typically also make use of intra-operation parallelism, i.e., computing the output of a given layer using", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 82, + 460, + 529, + 476 + ], + "spans": [ + { + "bbox": [ + 82, + 460, + 529, + 476 + ], + "score": 1.0, + "content": "multiple threads. Our DNN library currently does not support intra-operation parallelism, but implementing", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 82, + 472, + 528, + 485 + ], + "spans": [ + { + "bbox": [ + 82, + 472, + 528, + 485 + ], + "score": 1.0, + "content": "a dedicated thread pool for SGX could be an interesting extension for future work. Instead, we evaluate the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 82, + 483, + 529, + 497 + ], + "spans": [ + { + "bbox": [ + 82, + 483, + 529, + 497 + ], + "score": 1.0, + "content": "potential benefits of intra-op parallelism on a standard untrusted CPU, for our matrix-product and convolution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 82, + 493, + 528, + 508 + ], + "spans": [ + { + "bbox": [ + 82, + 493, + 528, + 508 + ], + "score": 1.0, + "content": "benchmarks. We make use of Eigen’s internal multi-threading support to speed up these operations, and custom", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 84, + 506, + 400, + 519 + ], + "spans": [ + { + "bbox": [ + 84, + 506, + 400, + 519 + ], + "score": 1.0, + "content": "OpenMP code to parallelize dot products, as Eigen does not do this on its own.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 82, + 449, + 529, + 519 + ] + }, + { + "type": "image", + "bbox": [ + 142, + 527, + 467, + 649 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 142, + 527, + 467, + 649 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 142, + 527, + 467, + 649 + ], + "spans": [ + { + "bbox": [ + 142, + 527, + 467, + 649 + ], + "score": 0.887, + "type": "image", + "image_path": "827a1dd46e9ddda18510395cc72fb7d148bced674096122d688675387980e22d.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 142, + 527, + 467, + 567.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 142, + 567.6666666666666, + 467, + 608.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 142, + 608.3333333333333, + 467, + 648.9999999999999 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 82, + 652, + 528, + 675 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 83, + 652, + 529, + 665 + ], + "spans": [ + { + "bbox": [ + 83, + 652, + 529, + 665 + ], + "score": 1.0, + "content": "Figure 7: Multi-threaded micro benchmarks on an untrusted CPU. Reiterates benchmarks for matrix prod-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 82, + 664, + 237, + 675 + ], + "spans": [ + { + "bbox": [ + 82, + 664, + 237, + 675 + ], + "score": 1.0, + "content": "ucts and convolutions using 4 threads.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "index": 27.25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 84, + 140, + 528, + 229 + ], + "lines": [ + { + "bbox": [ + 82, + 140, + 528, + 153 + ], + "spans": [ + { + "bbox": [ + 82, + 140, + 528, + 153 + ], + "score": 1.0, + "content": "Figure 7 shows the results using 4 threads. For convolutions, we have currently only implemented multi-threading", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 82, + 151, + 529, + 165 + ], + "spans": [ + { + "bbox": [ + 82, + 151, + 529, + 165 + ], + "score": 1.0, + "content": "for the verification with preprocessing (which requires only standard dot products). Surprisingly maybe, we find", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 81, + 162, + 528, + 176 + ], + "spans": [ + { + "bbox": [ + 81, + 162, + 528, + 176 + ], + "score": 1.0, + "content": "that multi-threading increases the gap between direct and verified computations of matrix products, probably", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 82, + 173, + 529, + 187 + ], + "spans": [ + { + "bbox": [ + 82, + 173, + 529, + 187 + ], + "score": 1.0, + "content": "because dot products are extremely easy to parallelize efficiently (compared to full convolutions). We also obtain", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 82, + 183, + 529, + 198 + ], + "spans": [ + { + "bbox": [ + 82, + 183, + 529, + 198 + ], + "score": 1.0, + "content": "close to linear speedups for verifiable separable convolutions, but omit the results as we currently do not have an", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 82, + 194, + 529, + 208 + ], + "spans": [ + { + "bbox": [ + 82, + 194, + 529, + 208 + ], + "score": 1.0, + "content": "implementation of multi-threaded direct computation for depthwise convolutions, which renders the comparison", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 82, + 205, + 529, + 219 + ], + "spans": [ + { + "bbox": [ + 82, + 205, + 529, + 219 + ], + "score": 1.0, + "content": "unfair. 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For convolutions, we have currently only implemented multi-threading", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 82, + 151, + 529, + 165 + ], + "spans": [ + { + "bbox": [ + 82, + 151, + 529, + 165 + ], + "score": 1.0, + "content": "for the verification with preprocessing (which requires only standard dot products). Surprisingly maybe, we find", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 81, + 162, + 528, + 176 + ], + "spans": [ + { + "bbox": [ + 81, + 162, + 528, + 176 + ], + "score": 1.0, + "content": "that multi-threading increases the gap between direct and verified computations of matrix products, probably", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 82, + 173, + 529, + 187 + ], + "spans": [ + { + "bbox": [ + 82, + 173, + 529, + 187 + ], + "score": 1.0, + "content": "because dot products are extremely easy to parallelize efficiently (compared to full convolutions). 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ModelPrivacy
ApproachTEEIntegrityPrivacyw.r.t. Serverw.r.t. ClientThroughput (relative)
SafetyNets (Ghodsi et al.,2017)1.OOO≤1/200 ×
Gazelle (Juvekar et al.,2018)1O. *O1≤1/1000 ×
Secure baseline (run DNN in TEE)√.•1×
Insecure baseline (run DNN on GPU)O0≥ 50x
Slalom (Ours)√..O4× - 20×
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Slalom with integrity& privacy TEE(F,x1) S(F)
Preproc: for i∈[1,n] dori ← Fmi,ui =riWi
for i∈[1,n] do 主 xi=xi+ri
起 yi=xW
yi=yi-ui
assert Freivalds(yi,xi,Wi)
xi+1=σ(yi) return yn
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Slalomwith integrity TEE(F,x1) S(F)
for i∈ [1,n] do
yi=xiWi
xi+1=σ(yi)
1...yn
fori∈[1,n] do assert Freivalds(yi,x,W)
xi+1=σ(yi)
return yn
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Layer|x|,|yl|W|costf (B=1)Batched verificationWith preproc.
FChin,houthin·hout|a|·lyB·(lx|+lyl)+cost fB·(lx|+lyl)
Convh·w·Cin,h·w·Coutk² · Cin ·Coutal·k2 · CoutB·(lx|+lyl)+Cin·Cout+ |x| · k²B·(|xl+lyl)
Depth.Convh·w·Cin,h·w·Cink2.Cinlac|.k²B·(lx|+lyl)+cost fB·(|x|+ lyl)
Point. Convh·w ·Cin,h · w· CoutCin·Cout|c| · CoutB·(x|+lyl)+Cin·CoutB·(lx|+|yl)
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AccuracyQuantized
ModelTop 1Top5Top1Top 5LayersParameters (M)Size of layer inputs/outputs (M)
VGG1671.090.070.689.516138.49.1 / 13.6
VGG16 (no top)=-1314.79.1/13.5
MobileNet70.789.670.589.5284.25.5/ 5.0
MobileNet (fused)=1154.23.6/3.1
ResNet 5076.992.476.492.25025.510.0 /10.4
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REPRESENTATION LEARNING + +Michael Tschannen∗ Josip Djolonga∗ Paul K. Rubenstein† Sylvain Gelly Mario Lucic Google Research, Brain Team + +# ABSTRACT + +Many recent methods for unsupervised or self-supervised representation learning train feature extractors by maximizing an estimate of the mutual information (MI) between different views of the data. This comes with several immediate problems: For example, MI is notoriously hard to estimate, and using it as an objective for representation learning may lead to highly entangled representations due to its invariance under arbitrary invertible transformations. Nevertheless, these methods have been repeatedly shown to excel in practice. In this paper we argue, and provide empirical evidence, that the success of these methods cannot be attributed to the properties of MI alone, and that they strongly depend on the inductive bias in both the choice of feature extractor architectures and the parametrization of the employed MI estimators. Finally, we establish a connection to deep metric learning and argue that this interpretation may be a plausible explanation for the success of the recently introduced methods. + +# 1 INTRODUCTION + +Unsupervised representation learning is a fundamental problem in machine learning. Intuitively, one aims to learn a function $g$ which maps the data into some, usually lower-dimensional, space where one can solve some (generally a priori unknown) target supervised tasks more efficiently, i.e. with fewer labels. In contrast to supervised and semi-supervised learning, the learner has access only to unlabeled data. Even though the task seems ill-posed as there is no natural objective one should optimize, by leveraging domain knowledge this approach can be successfully applied to a variety of problem areas, including image (Kolesnikov et al., 2019; van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019; Hjelm et al., 2019; Bachman et al., 2019) and video classification (Wang and Gupta, 2015; Sun et al., 2019), and natural language understanding (van den Oord et al., 2018; Peters et al., 2018; Devlin et al., 2019). + +Recently, there has been a revival of approaches inspired by the InfoMax principle (Linsker, 1988): Choose a representation $g ( x )$ maximizing the mutual information (MI) between the input and its representation, possibly subject to some structural constraints. MI measures the amount of information obtained about a random variable $X$ by observing some other random variable $Y ^ { 1 }$ Formally, the MI between $X$ and $Y$ , with joint density $p ( x , y )$ and marginal densities $p ( x )$ and $p ( y )$ , is defined as the Kullback–Leibler (KL) divergence between the joint and the product of the marginals + +$$ +I ( X ; Y ) = D _ { \mathrm { K L } } \left( p ( x , y ) \parallel p ( x ) p ( y ) \right) = \mathbb { E } _ { p ( x , y ) } \left[ \log \frac { p ( x , y ) } { p ( x ) p ( y ) } \right] . +$$ + +The fundamental properties of MI are well understood and have been extensively studied (see e.g. Kraskov et al. (2004)). Firstly, MI is invariant under reparametrization of the variables — namely, if $X ^ { \prime } = f _ { 1 } ( X )$ and $Y ^ { \prime } = f _ { 2 } ( Y )$ are homeomorphisms (i.e. smooth invertible maps), then $I ( X ; Y ) = I ( X ^ { \prime } ; Y ^ { \prime } )$ . Secondly, estimating MI in high-dimensional spaces is a notoriously difficult task, and in practice one often maximizes a tractable lower bound on this quantity (Poole et al., 2019). + +Nonetheless, any distribution-free high-confidence lower bound on entropy requires a sample size exponential in the size of the bound (McAllester and Statos, 2018). + +Despite these fundamental challenges, several recent works have demonstrated promising empirical results in representation learning using MI maximization (van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019; Hjelm et al., 2019; Bachman et al., 2019; Sun et al., 2019). In this work we argue, and provide empirical evidence, that the success of these methods cannot be attributed to the properties of MI alone. In fact, we show that maximizing tighter bounds on MI can result in worse representations. In addition, we establish a connection to deep metric learning and argue that this interpretation may be a plausible explanation of the success of the recently introduced methods.2 + +# 2 BACKGROUND AND RELATED WORK + +Recent progress and the InfoMax principle While promising results in other domains have been presented in the literature, we will focus on unsupervised image representation learning techniques that have achieved state-of-the-art performance on image classification tasks (Hénaff et al., 2019; Tian et al., 2019; Bachman et al., 2019). The usual problem setup dates back at least to Becker and Hinton (1992) and can conceptually be described as follows: For a given image $X$ , let $X ^ { ( 1 ) }$ and $X ^ { ( 2 ) }$ be different, possibly overlapping views of $X$ , for instance the top and bottom halves of the image. These are encoded using encoders $g _ { 1 }$ and $g _ { 2 }$ respectively, and the MI between the two representations $g _ { 1 } ( X ^ { ( 1 ) } )$ and $g _ { 2 } \big ( X ^ { ( 2 ) } \big )$ is maximized, + +$$ +\operatorname* { m a x } _ { g _ { 1 } \in { \mathcal G } _ { 1 } , g _ { 2 } \in { \mathcal G } _ { 2 } } \quad I _ { \mathrm { E S T } } \left( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) \right) , +$$ + +where $I _ { \tt E S T } ( X ; Y )$ is a sample-based estimator of the true MI $I ( X ; Y )$ and the function classes $\mathcal { G } _ { 1 }$ and $\mathcal { G } _ { 2 }$ can be used to specify structural constraints on the encoders. While not explicitly reflected in (2), note that $g _ { 1 }$ and $g _ { 2 }$ can often share parameters. Furthermore, it can be shown that $I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) \le I ( X ; g _ { 1 } ( X ^ { ( 1 ) } ) , g _ { 2 } ( X ^ { ( 2 ) } ) ) ,$ 3 hence the objective in (2) can be seen as a lower bound on the InfoMax objective $\operatorname* { m a x } _ { g \in { \mathcal { G } } } I ( X ; g ( X ) )$ (Linsker, 1988). + +Practical advantages of multi-view formulations There are two main advantages in using (2) rather than the original InfoMax objective. First, the MI has to be estimated only between the learned representations of the two views, which typically lie on a much lower-dimensional space than the one where the original data $X$ lives. Second, it gives us plenty of modeling flexibility, as the two views can be chosen to capture completely different aspects and modalities of the data, for example: + +1. In the basic form of DeepInfoMax (Hjelm et al., 2019) $g _ { 1 }$ extracts global features from the entire image $X ^ { ( 1 ) }$ and $g _ { 2 }$ local features from image patches $X ^ { ( 2 ) }$ , where $g _ { 1 }$ and $g _ { 2 }$ correspond to activations in different layers of the same convolutional network. Bachman et al. (2019) build on this and compute the two views from different augmentations of the same image. +2. Contrastive multiview coding (CMC) (Tian et al., 2019) generalizes the objective in (2) to consider multiple views $X ^ { ( i ) }$ , where each $X ^ { ( i ) }$ corresponds to a different image modality (e.g., different color channels, or the image and its segmentation mask). +3. Contrastive predictive coding (CPC) (van den Oord et al., 2018; Hénaff et al., 2019) incorporates a sequential component of the data. Concretely, one extracts a sequence of patches from an image in some fixed order, maps each patch using an encoder, aggregates the resulting features of the first $t$ patches into a context vector, and maximizes the MI between the context and features extracted from the patch at position $t + k$ . In (2), $X ^ { ( 1 ) }$ would thus correspond to the first $t$ patches and $X ^ { ( 2 ) }$ to the patch at location $t + k$ . Other approaches, such as those presented by Sermanet et al. (2018), Hu et al. (2017), and Ji et al. +(2019), can be similarly subsumed under the same objective. + +Lower bounds on MI As evident from (2), another critical choice is the MI estimator $I _ { \mathrm { E S T } }$ . Given the fundamental limitations of MI estimation (McAllester and Statos, 2018), recent work has focused on deriving lower bounds on MI (Barber and Agakov, 2003; Belghazi et al., 2018; Poole et al., + +2019). Intuitively, these bounds are based on the following idea: If a classifier can accurately distinguish between samples drawn from the joint $p ( x , y )$ and those drawn from the product of marginals $p ( x ) p ( y )$ , then $X$ and $Y$ have a high MI. + +We will focus on two such estimators, which are most commonly used in the representation learning literature. The first of them, termed InfoNCE (van den Oord et al., 2018), is defined as + +$$ +I ( X ; Y ) \geq \mathbb { E } \left[ \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \log \frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \frac { 1 } { K } \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) } } \right] \triangleq I _ { \tt N C E } ( X ; Y ) , +$$ + +where the expectation is over $K$ independent samples $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { K }$ from the joint distribution $p ( x , y )$ (Poole et al., 2019). In practice we estimate (3) using Monte Carlo estimation by averaging over multiple batches of samples. Intuitively, the critic function $f$ tries to predict for each $x _ { i }$ which of the $K$ samples $y _ { 1 } , \ldots , y _ { k }$ it was jointly drawn with, by assigning high values to the jointly drawn pair, and low values to all other pairs. The second estimator is based on the variational form of the KL divergence due to Nguyen, Wainwright, and Jordan (NWJ) (Nguyen et al., 2010) and takes the form + +$$ +I ( X ; Y ) \geq \mathbb { E } _ { p ( x , y ) } [ f ( x , y ) ] - e ^ { - 1 } \mathbb { E } _ { p ( x ) } [ \mathbb { E } _ { p ( y ) } e ^ { f ( x , y ) } ] \triangleq I _ { \mathbb { N } \mathbb { N } } ( X ; Y ) . +$$ + +For detailed derivations we refer the reader to (Ruderman et al., 2012; Poole et al., 2019). Note that these bounds hold for any critic $f$ and when used in (2) one in practice jointly maximizes over $g _ { 1 } , g _ { 2 }$ and $f$ . Furthermore, it can be shown that (3) is maximized by $f ^ { * } ( x , \bar { y } ) = \log p ( y | x )$ and (4) by $f ^ { * } ( x , y ) \mathop { = } 1 + \log p ( y | x )$ (Poole et al., 2019). Common choices for $f$ include bilinear critics $f ( x , y ) = x ^ { \top } W y$ (van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019), separable critics $f ( x , y ) = \phi _ { 1 } ( x ) ^ { \top } \phi _ { 2 } ( y )$ (Bachman et al., 2019), and concatenated critics $f ( x , y ) = \phi ( [ x , y ] )$ (Hjelm et al., 2019) (here $\phi , \phi _ { 1 } , \phi _ { 2 }$ are typically shallow multi-layer perceptrons (MLPs)). When applying these estimators to solve (2), the line between the critic and the encoders $g _ { 1 } , g _ { 2 }$ can be blurry. For example, one can train with an inner product critic $f ( x , y ) = x ^ { \top } y$ , but extract features from an intermediate layer of $g _ { 1 } , g _ { 2 }$ , in which case the top layers of $g _ { 1 } , g _ { 2 }$ form a separable critic. Nevertheless, this boundary is crucial for the interplay between MI estimation and the interpretation of the learned representations. + +# 3 BIASES IN APPROXIMATE INFORMATION MAXIMIZATION + +It is folklore knowledge that maximizing MI does not necessarily lead to useful representations. Already Linsker (1988) talks in his seminal work about constraints, while a manifestation of the problem in clustering approaches using MI criteria has been brought up by Bridle et al. (1992) and subsequently addressed using regularization by Krause et al. (2010). To what can we then attribute the recent success of methods building on the principles of MI maximization? We will argue that their connection to the InfoMax principle might be very loose. Namely, we will show that they behave counter-intuitively if one equates them with MI maximization, and that the performance of these methods depends strongly on the bias that is encoded not only in the encoders, but also on the actual form of the used estimators. + +1. We first consider encoders which are bijective by design. Even though the true MI is maximized for any choice of model parameters, the representation quality (measured by downstream linear classification accuracy) improves during training. Furthermore, there exist invertible encoders for which the representation quality is worse than using raw pixels, despite also maximizing MI. +2. We next consider encoders that can model both invertible and non-invertible functions. When the encoder can be non-invertible, but is initialized to be invertible, $I _ { \mathrm { E S T } }$ still biases the encoders to be very ill-conditioned and hard to invert. +3. For $I _ { \mathrm { N C E } }$ and $I _ { \mathrm { N } \bar { w } \bar { s } }$ , higher-capacity critics admit tighter bounds on MI. We demonstrate that simple critics yielding loose bounds can lead to better representations than high-capacity critics. +4. Finally, we optimize the estimators to the same MI lower-bound value with different encoder architectures and show that the representation quality can be impacted more by the choice of the architecture, than the estimator. + +As a consequence, we argue that the success of these methods and the way they are instantiated in practice is only loosely connected to MI. Then, in Section 4 we provide an alternative explanation for the success of recent methods through a connection to classic triplet losses from metric learning. + +![](images/a84c5f10c5703c8bb062be5139fa94f689d1924b0e49aabbcdefbabec169b769.jpg) +Figure 1: (a, b) Maximizing $I _ { \mathrm { E S T } }$ over a family of invertible models. We can see that during training the downstream classification performance improves (and the testing $I _ { \mathrm { E S T } }$ value increases), even though the true MI remains constant throughout. (c) Downstream classification accuracy of a different invertible encoder (with the same architecture) trained to have poor performance. This demonstrates the existence of encoders that provably maximize MI yet have bad downstream performance. + +Setup Our goal is to provide a minimal set of easily reproducible empirical experiments to understand the role of MI estimators, critic and encoder architectures when learning representations via the objective (2). To this end, we consider a simple setup of learning a representation of the top half of MNIST handwritten digit images (we present results for the experiments from Sections 3.2 and 3.3 on CIFAR10 in Appendix G; the conclusions are analogous). This setup has been used in the context of deep canonical correlation analysis (Andrew et al., 2013), where the target is to maximize the correlation between the representations. Following the widely adopted downstream linear evaluation protocol (Kolesnikov et al., 2019; van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019; Hjelm et al., 2019; Bachman et al., 2019), we train a linear classifier4 for digit classification on the learned representation using all available training labels (other evaluation protocols are discussed in Section 5). To learn the representation we instantiate (2) and split each input MNIST image $x \in [ 0 , 1 ] ^ { 7 8 4 }$ into two parts, the top part of the image $x _ { \mathrm { t o p } } \in [ 0 , 1 ] ^ { 3 9 2 }$ corresponding to $X ^ { ( 1 ) }$ , and the bottom part, $x _ { \mathrm { b o t t o m } } \in [ 0 , 1 ] ^ { 3 9 2 }$ , corresponding to $X ^ { ( 2 ) }$ , respectively. We train $g _ { 1 } , g _ { 2 }$ , and $f$ using the Adam optimizer (Kingma and Ba, 2015), and use $g _ { 1 } \big ( x _ { \mathrm { t o p } } \big )$ as the representation for the linear evaluation. Unless stated otherwise, we use a bilinear critic $f ( x , y ) = x ^ { \top } W y$ (we investigate its effect in a separate ablation study), set the batch size to 128 and the learning rate to $1 0 ^ { - 4 }$ . 5 Throughout, $I _ { \mathrm { E S T } }$ values and downstream classification accuracies are averaged over 20 runs and reported on the testing set (we did not observe large gaps between the training and testing values of $I _ { \mathrm { E S T } } ,$ ). As a common baseline, we rely on a linear classifier in pixel space on $x _ { \mathrm { t o p } }$ , which obtains a testing accuracy of about $8 5 \%$ . For comparison, a simple MLP or ConvNet architecture achieves about $5 4 \%$ (see Section 3.3 for details). + +# 3.1 LARGE MI IS NOT PREDICTIVE OF DOWNSTREAM PERFORMANCE + +We start by investigating the behavior of $I _ { \mathrm { N C E } }$ and $I _ { \mathrm { N W } } { \boldsymbol { \mathbf { \mathit { J } } } }$ when $g _ { 1 }$ and $g _ { 2 }$ are parameterized to be always invertible. Hence, for any choice of the encoder parameters, the MI is constant, i.e. $I ( g _ { 1 } ( X ^ { \acute { ( 1 ) } } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) = I ( X ^ { ( 1 ) } ; X ^ { ( 2 ) } )$ for all $g _ { 1 } , g _ { 2 }$ . This means that if we could exactly compute the MI, any parameter choice would be a global maximizer and thus the gradients vanish everywhere.6 However, as we will empirically show, the estimators we consider are biased and prefer those settings which yield representations useful for the downstream classification task. + +Maximized MI and improved downstream performance We model $g _ { 1 }$ and $g _ { 2 }$ using the invertible RealNVP architecture (Dinh et al., 2016). We use a total of 30 coupling layers, and each of them computes the shift using a separate MLP with two ReLU hidden layers, each with 512 units. + +![](images/47a695e5fcef7059f0d8b7d49d675445acaafdcfd22f6c9b2f1667165ff97877.jpg) +Figure 2: Maximizing $I _ { \mathrm { E S T } }$ using a network architecture that can realize both invertible and noninvertible functions. (a, b) As $I _ { \mathrm { E S T } }$ increases, the linear classification testing performance increases. (c) Meanwhile, the condition number of Jacobian evaluated at inputs randomly sampled from the data distribution deteriorates, i.e. $g _ { 1 }$ becomes increasingly ill-conditioned (lines represent 0th, 20th, . . . , $1 0 0 \mathrm { { t h } }$ percentiles for $I _ { \mathrm { N C E } }$ , the corresponding figure for $I _ { \mathrm { N } \bar { w } \bar { s } }$ can be found in Appendix F; the empirical distribution is obtained by randomly sampling 128 inputs from the data distribution, computing the corresponding condition numbers, and aggregating them across runs). + +Figure 1 shows the testing value of $I _ { \mathrm { E S T } }$ and the testing accuracy on the classification task. Despite the fact that MI is maximized by any instantiation of $g _ { 1 }$ and $g _ { 2 }$ , $I _ { \mathrm { E S T } }$ and downstream accuracy increase during training, implying that the estimators provide gradient feedback leading to a representation useful for linear classification. This confirms our hypothesis that the estimator biases the encoders towards solutions suitable to solve the downstream linear classification task. + +The previous experiment demonstrated that among many invertible encoders, all of which are globally optimal MI maximizers, some give rise to improved linear classification performance over raw pixels, and maximizing $I _ { \mathrm { N C E } }$ and $I _ { \mathrm { N W } } { \boldsymbol { \mathbf { \mathit { J } } } }$ yields such encoders. Next we demonstrate that for the same invertible encoder architecture there are model parameters for which linear classification performance is significantly worse than using raw pixels, despite also being globally optimal MI maximizers. + +Maximized MI and worsened downstream performance The goal is to learn a (bijective) representation maximizing MI such that the optimal linear classifier performs poorly; we achieve this by jointly training a representation and classifier in an adversarial fashion (a separate classifier is trained for the evaluation), without using a MI estimator. Intuitively, we will train the encoder to make the classification task for the linear layer as hard as possible. The experimental details are presented in Appendix B. Figure 1c shows the result of one such training run, displaying the loss of a separately trained classifier on top of the frozen representation. At the beginning of training the network is initialized to be close to the identity mapping, and as such achieves the baseline classification accuracy corresponding to raw pixels. All points beyond this correspond to invertible feature maps with worse classification performance, despite still achieving globally maximal MI. + +Alternatively, the following thought experiment would yield the same conclusion: Using a lossless compression algorithm (e.g. PNG) for $g _ { 1 }$ and $g _ { 2 }$ also satisfies $I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) \ =$ $I ( X ^ { ( 1 ) } ; X ^ { ( 2 ) } )$ . Yet, performing linear classification on the raw compressed bit stream $g _ { 1 } ( X ^ { ( 1 ) } )$ will likely lead to worse performance than the baseline in pixel space. The information content alone is not sufficient to guarantee a useful geometry in the representation space. + +We next investigate the behavior of the model if we use a network architecture that can model both invertible and non-invertible functions. We would like to understand whether $I _ { \mathrm { E S T } }$ prefers the network to remain bijective, thus maximizing the true MI, or to ignore part of the input signal, which can be beneficial for representation learning. + +Bias towards hard-to-invert encoders We use an MLP architecture with 4 hidden layers of the same dimension as the input, and with a skip connection added to each layer (hence by setting all weights to 0 the network becomes the identity function). As quantifying invertibility is hard, we analyze the condition number, i.e. the ratio between the largest and the smallest singular value, of the Jacobian of $g _ { 1 }$ : By the implicit function theorem, the function is invertible if the Jacobian is non-singular.7 However, the data itself might lie on a low-dimensional manifold, so that having a singular Jacobian is not necessarily indicative of losing invertibility on the support of the data distribution. To ensure the support of the data distribution covers the complete input space, we corrupt $X ^ { ( 1 ) }$ and $X ^ { ( 2 ) }$ in a coupled way by adding to each the same 392-dimensional random vector, whose coordinates are sampled (independently of $X ^ { ( 1 ) } , X ^ { ( 2 ) } )$ from a normal with standard deviation 0.05 (the standard deviation of the pixels themselves is 0.3). Hence, non-invertible encoders $g _ { 1 } , g _ { 2 }$ do not maximize $I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) )$ . 8 As a reference point, the linear classification accuracy from pixels drops to about $84 \%$ due to the added noise. + +![](images/8aa32e9c03a77e8657ee9a3fc96b1e03d12ad15997f3d7f40349ab8ce7ec0d00.jpg) +Figure 3: Downstream testing accuracy for $I _ { \mathrm { N C E } }$ and $I _ { \mathrm { N } \bar { w } \bar { s } }$ , and testing $I _ { \mathrm { N } \bar { w } \bar { s } }$ value for MLP encoders $g _ { 1 } , g _ { 1 }$ and different critic architectures (the testing $I _ { \mathrm { N C E } }$ curve can be found in Appendix F). Bilinear and separable critics lead to higher downstream accuracy than MLP critics, while reaching lower $I _ { \mathrm { N } \bar { w } \bar { s } }$ . + +In Figure 2 we can see that the $I _ { \mathrm { E S T } }$ value and the downstream accuracy both increase during training, as before. Moreover, even though $g _ { 1 }$ is initialized very close to the identity function (which maximizes the true MI), the condition number of its Jacobian evaluated at inputs randomly sampled from the data-distribution steadily deteriorates over time, suggesting that in practice (i.e. numerically) inverting the model becomes increasingly hard. It therefore seems that the bounds we consider favor hard-to-invert encoders, which heavily attenuate part of the noise (as the support of the noise is the entire input space), over well conditioned encoders (such as the identity function at initialization), which preserve the noise and hence the entropy of the data well. + +# 3.2 HIGHER CAPACITY CRITICS CAN LEAD TO WORSE DOWNSTREAM PERFORMANCE + +In the previous section we have established that MI and downstream performance are only loosely connected. Clearly, maximizing MI is not sufficient to learn good representations and there is a non-trivial interplay between the architectures of the encoder, critic, and the underlying estimators. In this section, we will focus on how one of these factors, namely the critic architecture, impacts the quality of the learned representation. Recall that it determines how the estimators such as $I _ { \mathrm { N C E } }$ and $I _ { \mathrm { N W } } { \boldsymbol { \mathbf { \mathit { I } } } }$ distinguish between samples from the joint distribution $p ( x , y )$ and the product of the marginals $p ( x ) p ( y )$ , and thereby determines the tightness on the lower bound. A higher capacity critic should allow for a tighter lower-bound on MI (Belghazi et al., 2018). Furthermore, in the context of representation learning where $f$ is instantiated as a neural network, the critic provides gradient feedback to $g _ { 1 }$ and $g _ { 2 }$ and thereby shapes the learned representation. + +Looser bounds with simpler critics can lead to better representations We compare three critic architectures, a bilinear critic, a separable critic $f ( x , y ) \stackrel { \mathrm { ~ } } { = } \phi _ { 1 } ( x ) ^ { \top } \phi _ { 2 } ( y ) ( \phi _ { 1 } , \phi _ { 2 }$ are MLPs with a single hidden layer with 100 units and ReLU activations, followed by a linear layer with 100 units; comprising $4 0 \mathrm { k }$ parameters in total) and an MLP critic with a single hidden layer with 200 units and ReLU activations, applied to the concatenated input $[ x , y ]$ (40k trainable parameters). Further, we use identical MLP architectures for $g _ { 1 }$ and $g _ { 2 }$ with two hidden layers comprising 300 units each, and a third linear layer mapping to a 100-dimensional feature space. + +Figure 3 shows the downstream testing accuracy and the testing $I _ { \mathrm { E S T } }$ value as a function of the iteration (see Appendix G for the corresponding results on CIFAR10). It can be seen that for both lower bounds, representations trained with the MLP critic barely outperform the baseline on pixel space, whereas the same lower bounds with bilinear and separable critics clearly lead to a higher accuracy than the baseline. While the testing $I _ { \mathrm { N C E } }$ value is close to the theoretically achievable maximum value for all critics, the testing $I _ { \mathrm { N } \bar { W } \bar { J } }$ value is higher for the MLP critic than for the separable and bilinear critics, resulting in a tighter bound on the MI. However, despite achieving the smallest + +![](images/bf9d8c2d5e81792c60fb051b5b3799ff17a09863301230629190341d2eb3b09e.jpg) +Figure 4: (a, b) Downstream testing accuracy for different encoder architectures and MI estimators, using a bilinear critic trained to match a given target $I _ { \mathrm { E S T } }$ of $t$ (we minimize $L _ { t } ( g _ { 1 } , g _ { 2 } ) =$ $| I _ { \mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t |$ ; loss curves can be found in Appendix F). For a given estimator and $t$ , ConvNet encoders clearly outperform MLP encoders in terms of downstream testing accuracy. (c) Estimating MI from i.i.d. and non-i.i.d. samples in a synthetic setting (Section 4). If negative samples are not drawn i.i.d., both $I _ { \mathrm { N C E } }$ and $I _ { \mathrm { N } \bar { W } \bar { J } }$ estimators can be greater than the true MI. Despite being commonly justified as a lower bound on MI, $I _ { \mathrm { N C E } }$ is often used in the non-i.i.d. setting in practice. + +$I _ { \mathrm { N } \bar { w } \bar { s } }$ testing value, the simple bilinear critic leads to a better downstream performance than the higher-capacity separable and MLP critics. + +A related phenomenon was observed in the context of variational autoencoders (VAEs) (Kingma and Welling, 2014), where one maximizes a lower bound on the data likelihood: Looser bounds often yield better inference models, i.e. latent representations (Rainforth et al., 2018). + +# 3.3 ENCODER ARCHITECTURE CAN BE MORE IMPORTANT THAN THE SPECIFIC ESTIMATOR + +We will now show that the encoder architecture is a critical design choice and we will investigate its effect on the learned representation. We consider the same MLP architecture (238k parameters) as in Section 3.2, as well as a ConvNet architecture comprising two convolution layers (with a $5 \times 5$ kernel, stride of 2, ReLU activations, and 64 and 128 channels, respectively; $2 2 0 \mathrm { k }$ parameters), followed by spatial average pooling and a fully connected layer. Before the average pooling operation we apply layer normalization (Ba et al., 2016) which greatly reduces the variance of $\bar { I } _ { \mathrm { N W } } ^ { \mathrm { ~ ~ } } \bar { J }$ . 9 To ensure that both network architectures achieve the same lower bound $I _ { \mathrm { E S T } }$ on the MI, we minimize $L _ { t } ( g _ { 1 } , g _ { 2 } ) = | I _ { \mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t |$ instead of solving (2), for two different values $t = 2 , 4$ + +Figure 4 shows the downstream testing accuracy as a function of the training iteration (see Appendix G for the corresponding results on CIFAR10). It can be seen in the testing loss curves in Appendix F that for both architectures and estimators the objective value after 7k iterations matches the target $t$ (i.e., $L _ { t } ( g _ { 1 } , g _ { 2 } ) \approx 0 )$ which implies that they achieve the same lower-bound on the MI. Despite matching lower bounds, ConvNet encoders lead to clearly superior classification accuracy, for both $I _ { \mathrm { N C E } }$ and $I _ { \mathrm { N } \bar { w } \bar { s } }$ . Note that, in contrast, the MLP and ConvNet architectures trained end-to-end in supervised fashion both achieve essentially the same testing accuracy of about $9 4 \%$ . + +In the context of VAEs, Alemi et al. (2018) similarly observed that models achieving the same evidence lower bound value can lead to vastly different representations depending on the employed encoder architecture, and do not necessarily capture useful information about the data (Tschannen et al., 2018; Blau and Michaeli, 2019). + +# 4 CONNECTION TO DEEP METRIC LEARNING AND TRIPLET LOSSES + +In the previous section we empirically demonstrated that there is a disconnect between approximate MI maximization and representation quality. However, many recent works have applied the $I _ { \mathrm { N C E } }$ estimator to obtain state-of-the-art results in practice. We provide some insight on this conundrum by connecting $I _ { \mathrm { N C E } }$ to a popular triplet $k$ -plet) loss known in the deep metric learning community. + +The metric learning view Given sets of triplets, namely an anchor point $x$ , a positive instance $y$ , and a negative instance $z$ , the goal is to learn a representation $g ( x )$ such that the distances (i.e., $\ell _ { 2 }$ ) between $g ( x )$ and $g ( y )$ is smaller than the distance between $g ( x )$ and $g ( z )$ , for each triplet. In the supervised setting, the positive instances are usually sampled from the same class, while the negative instances are sampled from any other class. A major focus in deep metric learning is how to perform (semi-)hard positive mining — we want to present non-trivial triplets to the learning algorithm which become more challenging as $g$ improves. Natural extensions to the unsupervised setting can be obtained by exploiting the structure present in the input data, namely spatial (e.g. patches from the same image should be closer than patches from different images) and temporal information (temporally close video frames should be encoded closer than the ones which are further away in time) (Hoffer and Ailon, 2015). + +Connection to InfoNCE The InfoNCE objective can be rewritten as follows: + +$$ +\tilde { \mathrm { \Sigma } } _ { \mathrm { N C E } } = \mathbb { E } \left[ \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \log \frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \frac { 1 } { K } \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) } } \right] = \log K - \mathbb { E } \left[ \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \log \left( 1 + \sum _ { j \ne i } e ^ { f ( x _ { i } , y _ { j } ) - f ( x _ { i } , y _ { i } ) } \right) \right] . +$$ + +The derivation is presented in Appendix C. In the particular case that $x$ and $y$ take value in the same space and $f$ is constrained to be of the form $f ( x , y ) = \phi ( x ) ^ { \top } \phi ( y )$ , for some function $\phi$ , this coincides (up to constants and change of sign) with the expectation of the multi-class $K$ -pair loss proposed in (Sohn, 2016, Eqn. (7)): + +$$ +L _ { \mathrm { K - p a i r r o c } } \left( \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { K } , \phi \right) = \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \log \left( 1 + \sum _ { j \ne i } e ^ { \phi ( x _ { i } ) ^ { \top } \phi ( y _ { j } ) - \phi ( x _ { i } ) ^ { \top } \phi ( y _ { i } ) } \right) . +$$ + +Representation learning by maximizing $I _ { \mathrm { N C E } }$ using a symmetric separable critic $f ( x , y ) = \phi ( x ) ^ { \top } \phi ( y )$ and an encoder $g = g _ { 1 } = g _ { 2 }$ shared across views is thus equivalent to metric learning based on (5). When using different encoders for different views and asymmetric critics as employed by CPC, DeepInfoMax, and CMC one recovers asymmetric variants of (5), see, e.g. (Yu et al., 2017; Zhang et al., 2019). As a result, one can view (5) as learning encoders with a parameter-less inner product critic, for which the MI lower-bound is very weak in general. + +There are (at least) two immediate benefits of viewing recent representation learning methods based on MI estimators through the lens of metric learning. Firstly, in the MI view, using inner product or bilinear critic functions is sub-optimal since the critic should ideally be as flexible as possible in order to reduce the gap between the lower bound and the true MI. In the metric learning view, the inner product critic corresponds to a simple metric on the embedding space. The metric learning view seems hence in better accordance with the observations from Section 3.2 than the MI view. Secondly, it elucidates the importance of appropriately choosing the negative samples, which is indeed a critical component in deep metric learning based on triplet losses (Norouzi et al., 2012; Schroff et al., 2015). + +InfoNCE and the importance of negative sampling The negative sample mining issue also manifests itself in MI-based contrastive losses. In fact, while InfoNCE is a lower bound on MI if the negative samples are drawn from the true marginal distribution (Poole et al., 2019), i.e. + +$$ +I ( X , Y ) \geq \mathbb { E } _ { \prod _ { k } p ( x _ { k } , y _ { k } ) } { \frac { 1 } { K } } \sum _ { i = 1 } ^ { K } \left[ \log { \frac { e ^ { f ( x _ { i } , y _ { i } ) } } { { \frac { 1 } { K } } \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } } \right] \triangleq I _ { \mathbb { N } \mathbb { C } } , +$$ + +we show that if the negative samples are drawn in a dependent fashion (corresponding to the $( x _ { i } , y _ { i } )$ being drawn identically but not independently), the $I _ { \mathrm { N C E } }$ estimator is in general neither a lower nor an upper bound on the true MI $I ( X , Y )$ . We prove this in Appendix D and present empirical evidence here. Let $( X , Y ) = Z + \epsilon .$ , where $\bar { Z } \sim \mathcal { N } ( 0 , \Sigma _ { Z } )$ and $\epsilon \sim \mathcal { N } ( 0 , \Sigma _ { \epsilon } )$ are two-dimensional Gaussians. We generate batches of data $( X _ { i } , Y _ { i } ) = Z + \epsilon _ { i }$ where each $\epsilon _ { i }$ is sampled independently for each element of the batch, but $Z$ is sampled only once per batch. As such, $( X _ { i } , Y _ { i } )$ has the same marginal distribution for each $i$ , but the elements of the batch are not independent. Although we do not treat it theoretically, we also display results of the same experiment using the $I _ { \mathrm { N } \bar { w } \bar { s } }$ estimator. The experimental details are presented in Appendix E. We observe in Figure $_ \mathrm { 4 c }$ that when using noni.i.d. samples both the $I _ { \mathrm { N C E } }$ and $I _ { \mathrm { N } \bar { w } \bar { s } }$ values are larger than the true MI, and that when i.i.d. samples are used, both are lower bounds on the true MI. Hence, the connection to MI under improper negative sampling is no longer clear and might vanish completely. + +Notwithstanding this fundamental problem, the negative sampling strategy is often treated as a design choice. In Hénaff et al. (2019), CPC is applied to images by partitioning the input image into patches. Then, MI (estimated by InfoNCE) between representations of patches and a context summarizing several patches that are vertically above or below in the same image is minimized. Negative samples are obtained by patches from different images as well as patches from the same image, violating the independence assumption. Similarly, van den Oord et al. (2018) learn representations of speech using samples from a variety of speakers. It was found that using utterances from the same speaker as negative samples is more effective, whereas the “proper” negative samples should be drawn from an appropriate mixture of utterances from all speakers. + +A common observation is that increasing the number of negative examples helps in practice (Hjelm et al., 2019; Tian et al., 2019; Bachman et al., 2019). Indeed, Ma and Collins (2018) show that $I _ { \mathrm { N C E } }$ is consistent for any number of negative samples (under technical conditions), and Poole et al. (2019) show that the signal-to-noise ratio increases with the number of negative samples. On the other hand, (Arora et al., 2019) have demonstrated, both theoretically and empirically, that increasing the number of negative samples does not necessarily help, and can even deteriorate the performance. The intricacies of negative sampling hence remain a key research challenge. + +# 5 CONCLUSION + +Is MI maximization a good objective for learning good representations in an unsupervised fashion? Possibly, but it is clearly not sufficient. In this work we have demonstrated that, under the common linear evaluation protocol, maximizing lower bounds on MI as done in modern incarnations of the InfoMax principle can result in bad representations. We have revealed that the commonly used estimators have strong inductive biases and—perhaps surprisingly—looser bounds can lead to better representations. Furthermore, we have demonstrated that the connection of recent approaches to MI maximization might vanish if negative samples are not drawn independently (as done by some approaches in the literature). As a result, it is unclear whether the connection to MI is a sufficient (or necessary) component for designing powerful unsupervised representation learning algorithms. We propose that the success of these recent methods could be explained through the view of triplet-based metric learning and that leveraging advances in that domain might lead to further improvements. We have several suggestions for future work, which we summarize in the following. + +Alternative measures of information We believe that the question of developing new notions of information suitable for representation learning should receive more attention. While MI has appealing theoretical properties, it is clearly not sufficient for this task—it is hard to estimate, invariant to bijections and can result in suboptimal representations which do not correlate with downstream performance. Therefore, a new notion of information should account for both the amount of information stored in a representation and the geometry of the induced space necessary for good performance on downstream tasks. One possible avenue is to consider extensions to MI which explicitly account for the modeling power and computational constraints of the observer, such as the recently introduced $\mathcal { F }$ -information $\mathrm { X u }$ et al. (2020). Alternatively, one can investigate other statistical divergences to measure the discrepancy between $p ( x , y )$ and $p ( x ) p ( y )$ . For example, using the Wasserstein distance leads to promising results in representation learning as it naturally enforces smoothness in the encoders (Ozair et al., 2019). + +A holistic view We believe that any theory on measuring information for representation learning built on critics should explicitly take into account the function families one uses (e.g. that of the critic and estimator). Most importantly, we would expect some natural trade-offs between the amount of information that can be stored against how hard it is to extract it in the downstream tasks as a function of the architectural choices. While the distribution of downstream tasks is typically assumed unknown in representation learning, it might be possible to rely on weaker assumptions such as a family of invariances relevant for the downstream tasks. Moreover, it seems that in the literature (i) the critics that are used to measure the information, (ii) the encoders, and (iii) the downstream models/evaluation protocol are all mostly chosen independently of each other. Our empirical results show that the downstream performance depends on the intricate balance between these choices and we believe that one should co-design them. This holistic view is currently under-explored and due to the lack of any theory or extensive studies to guide the practitioners. + +Going beyond the widely used linear evaluation protocol While it was shown that learning good representations under the linear evaluation protocol can lead to reduced sample complexity for downstream tasks (Arora et al., 2019), some recent works (Bachman et al., 2019; Tian et al., 2019) report marginal improvements in terms of the downstream performance under a non-linear regime. Related to the previous point, it would hence be interesting to further explore the implications of the evaluation protocol, in particular its importance in the context of other design choices. We stress that a highly-nonlinear evaluation framework may result in better downstream performance, but it defeats the purpose of learning efficiently transferable data representations. + +Systematic investigations into design decisions that matter On the practical side, we believe that the link to metric learning could lead to new methods, that break away from the goal of estimating MI and place more weight on the aspects that have a stronger effect on the performance such as the negative sampling strategy. An example where the metric learning perspective led to similar methods as the MI view is presented by Sermanet et al. 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Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018. + +Xiaolong Wang and Abhinav Gupta. Unsupervised learning of visual representations using videos. In IEEE International Conference on Computer Vision, 2015. + +Yilun Xu, Shengjia Zhao, Jiaming Song, Russell Stewart, and Stefano Ermon. A Theory of Usable Information under Computational Constraints. In International Conference on Learning Representations, 2020. + +Hong-Xing Yu, Ancong Wu, and Wei-Shi Zheng. Cross-view asymmetric metric learning for unsupervised person re-identification. In IEEE International Conference on Computer Vision, 2017. + +Ji Zhang, Yannis Kalantidis, Marcus Rohrbach, Manohar Paluri, Ahmed Elgammal, and Mohamed Elhoseiny. Large-scale visual relationship understanding. In AAAI Conference on Artificial Intelligence, 2019. + +# APPENDIX + +# A RELATION BETWEEN (2) AND THE INFOMAX OBJECTIVE + +Proposition 1. Let $X$ be a random variable and define $X _ { 1 } = g _ { 1 } ( X )$ and $X _ { 2 } = g _ { 2 } ( X )$ be arbitrary functions of $X$ . Then $I ( X _ { 1 } ; X _ { 2 } ) \leq I \left( X ; ( X _ { 1 } , X _ { 2 } ) \right)$ . + +Proof. Follows by two applications of the data processing inequality, which states that for random variables $X , Y$ and $Z$ satisfying the Markov relation $X Y Z$ , the inequality $I ( X ; Z ) \ \leq$ $I ( X ; Y )$ holds. + +The first step is to observe that $X$ , $X _ { 1 }$ and $X _ { 2 }$ satisfy the relation $X _ { 1 } \left. X \right. X _ { 2 }$ , which is Markov equivalent to $X _ { 1 } X X _ { 2 }$ (in particular, $X _ { 1 }$ and $X _ { 2 }$ are conditionally independent given $X$ ). It therefore follows that $I ( X _ { 1 } ; X _ { 2 } ) \leq I ( X ; X _ { 1 } )$ . The second step is to observe that $X ( X _ { 1 } , X _ { 2 } ) X _ { 1 }$ and therefore $I ( X ; X _ { 1 } ) \leq I ( X ; ( X _ { 1 } , X _ { 2 } ) )$ . + +Combining the two inequalities yields $I ( X _ { 1 } ; X _ { 2 } ) \leq I ( X ; ( X _ { 1 } , X _ { 2 } ) )$ , as required. + +# B EXPERIMENT DETAILS: ADVERSARIALLY TRAINED ENCODER(SECTION 3.1) + +In the following, we present the details for training the invertible model from Section 3.1 adversarially. We model $g _ { 1 }$ with the same RealNVP architecture as in the first experiment, and do not model $g _ { 2 }$ . On top of $g _ { 1 } ( X ^ { ( 1 ) } )$ we add a linear layer mapping to 10 outputs (i.e. logits). The parameters of the linear layer trained by minimizing the cross-entropy loss with respect to the true label of $X$ from which $X ^ { ( 1 ) }$ is derived. Conversely, the parameters of the encoder $g _ { 1 }$ are trained to minimize the cross-entropy loss with respect to a uniform probability vector over all 10 classes. We use the Adam optimizer with a learning rate of $1 0 ^ { - 4 }$ for the parameters of the classifier and $1 0 ^ { - 6 }$ for the parameters of the encoder, and perform 10 classifier optimization steps per encoder step. Furthermore, in a warm-up phase we train the classifier for 1k iterations before alternating between classifier and encoder steps. + +# C CONNECTION BETWEEN METRIC LEARNING AND INFONCE + +$I _ { \mathrm { N C E } }$ can be rewritten as follows: + +$$ +\begin{array} { r l } & { I _ { \mathrm { R e x s } } = \mathbb { E } \left[ \displaystyle \frac { 1 } { K } \displaystyle \sum _ { i = 1 } ^ { K } \log \frac { e ^ { f ( x _ { i } , y _ { j } ) } } { \frac { 1 } { K } \displaystyle \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) } } \right] } \\ & { \quad = \mathbb { E } \left[ \displaystyle \frac { 1 } { K } \displaystyle \sum _ { i = 1 } ^ { K } \log \frac { 1 } { \frac { 1 } { K } \displaystyle \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) - f ( x _ { i } , y _ { i } ) } } \right] } \\ & { \quad = \mathbb { E } \left[ - \displaystyle \frac { 1 } { K } \displaystyle \sum _ { i = 1 } ^ { K } \log \frac { 1 } { K } \displaystyle \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) - f ( x _ { i } , y _ { i } ) } \right] } \\ & { \quad = \log K - \mathbb { E } \left[ \displaystyle \frac { 1 } { K } \displaystyle \sum _ { i = 1 } ^ { K } \log \left( 1 + \displaystyle \sum _ { j \neq i } e ^ { f ( x _ { i } , y _ { j } ) - f ( x _ { i } , y _ { i } ) } \right) \right] . } \end{array} +$$ + +# D INFONCE UNDER NON-I.I.D. SAMPLING + +The proof that InfoNCE is a lower bound on MI presented in (Poole et al., 2019) makes crucial use of the assumption that the negative samples are drawn from the true marginal distribution. We briefly review this proof to highlight the importance of the negative sampling distribution. Their proof starts from the NWJ lower bound of the KL divergence, namely that for any function $\tilde { f }$ the following lower bound holds (Nguyen et al., 2010; Nowozin et al., 2016): + +$$ +I ( X ; Y ) = D _ { K L } ( p ( x , y ) | | p ( x ) p ( y ) ) \geq \mathbb { E } _ { p ( x , y ) } [ \tilde { f } ( x , y ) ] - e ^ { - 1 } \mathbb { E } _ { p ( x ) p ( y ) } [ e ^ { \tilde { f } ( x , y ) } ] . +$$ + +Suppose that $( X _ { i } , Y _ { i } ) _ { i = 1 } ^ { K }$ are i.i.d. draws from $p ( x , y )$ and write $X _ { 1 : K } = ( X _ { 1 } , X _ { 2 } , \dots , X _ { K } )$ . Then, for any $i$ we have that $I ( X _ { 1 : K } ; Y _ { i } ) = I ( X _ { i } ; Y _ { i } ) = I ( X ; Y )$ . We thus have + +$$ +\begin{array} { r } { I ( \boldsymbol { X } ; \boldsymbol { Y } ) = I ( X _ { 1 : K } ; Y _ { i } ) \ge \mathbb { E } _ { p ( x _ { i } , y _ { i } ) \prod _ { k \geqslant i } p ( x _ { k } ) } [ \tilde { f } ( x _ { 1 : K } , y _ { i } ) ] - e ^ { - 1 } \mathbb { E } _ { p ( y _ { i } ) \prod _ { k } p ( x _ { k } ) } [ e ^ { \tilde { f } ( x _ { 1 : K } , y _ { i } ) } ] , } \end{array} +$$ + +where the equality follows from the assumption that the $( X _ { i } , Y _ { i } ) _ { i = 1 } ^ { K }$ are i.i.d. and the inequality is (6) applied to $I ( X _ { 1 : K } ; Y _ { i } )$ . In particular, taking $\begin{array} { r } { \tilde { f } ( x _ { 1 : K } , y _ { i } ) = 1 + \log \frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \frac { 1 } { K } \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } } \end{array}$ ef(xj ,yi) yields + +$$ +\textmd { ( X , Y ) \geq 1 + } \mathbb { E } _ { p ( x _ { i } , y _ { i } ) \prod _ { k \geqslant i } p ( x _ { k } ) } \left[ \log \frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \frac { 1 } { K } \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } \right] - \mathbb { E } _ { p ( y _ { i } ) \prod _ { k } p ( x _ { k } ) } \left[ \frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \frac { 1 } { K } \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } \right] . +$$ + +This is then averaged over the $K$ samples $Y _ { i }$ , in which case the third term above cancels with the constant 1 (all occurences of $y _ { i }$ in the last term of (7) can be replaced with $y _ { 1 }$ thanks to $( X _ { i } , Y _ { i } )$ being identically distributed), yielding the familiar $I _ { \mathrm { N C E } }$ lower bound: + +$$ +I ( X , Y ) \geq \mathbb { E } _ { \prod _ { k } p ( x _ { k } , y _ { k } ) } { \frac { 1 } { K } } \sum _ { i = 1 } ^ { K } \left[ \log { \frac { e ^ { f ( x _ { i } , y _ { i } ) } } { { \frac { 1 } { K } } \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } } \right] = I _ { \mathbb { N } \mathbb { C } } . +$$ + +The point in this proof that makes use of the i.i.d. assumption of the negative samples is in the equality $I ( X _ { i } , Y _ { i } ) = I ( X _ { 1 : K } , Y _ { i } )$ , which allowed us to leverage multiple samples when estimating the MI between two variables. If instead the negative samples are drawn in a dependent fashion (corresponding to the $( X _ { i } , Y _ { i } )$ being drawn identically but not independently), we have $I ( X _ { i } , Y _ { i } ) \leq$ $I ( X _ { 1 : K } , Y _ { i } )$ , though the remainder of the proof still holds, resulting in + +$$ +I ( X , Y ) \leq \frac { 1 } { K } \sum _ { i = 1 } ^ { K } I ( X _ { 1 : K } ; Y _ { i } ) \geq \mathbb { E } _ { p ( x _ { 1 : K } , y _ { 1 : K } ) } \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \left[ \log \frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \frac { 1 } { K } \sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } \right] . +$$ + +Therefore the resulting $I _ { \mathrm { N C E } }$ estimator is neither a lower nor an upper bound on the true MI $I ( X , Y )$ . + +E EXPERIMENT DETAILS: NON-I.I.D. SAMPLING (SECTION 4) + +Recall that $( X , Y ) = Z + \epsilon$ . We use $Z \sim { \mathcal { N } } ( 0 , \Sigma _ { Z } )$ and $\epsilon \sim \mathcal { N } ( 0 , \Sigma _ { \epsilon } )$ , where + +$$ +\Sigma _ { Z } = \left( { \begin{array} { c c } { 1 } & { - 0 . 5 } \\ { - 0 . 5 } & { 1 } \end{array} } \right) \qquad { \mathrm { a n d } } \qquad \Sigma _ { \epsilon } = \left( { \begin{array} { c c } { 1 } & { 0 . 9 } \\ { 0 . 9 } & { 1 } \end{array} } \right) . +$$ + +Batches of data are obtained as $( X _ { i } , Y _ { i } ) = Z + \epsilon _ { i }$ where each $\epsilon _ { i }$ is sampled independently for each element of the batch, but $Z$ is sampled only once per batch. The true MI $I ( X , Y )$ can be calculated analytically since $( X , Y )$ is jointly Gaussian with known covariance matrix $\Sigma _ { Z } + \Sigma _ { \epsilon }$ : For two univariate random variables $( X , Y )$ that are jointly Gaussian with covariance $\Sigma$ the MI can be written as + +$$ +I ( X , Y ) = - \frac { 1 } { 2 } \log ( 1 - \frac { \Sigma _ { 1 2 } \Sigma _ { 2 1 } } { \Sigma _ { 1 1 } \Sigma _ { 2 2 } } ) . +$$ + +This can be derived using the decomposition $I ( X , Y ) = H ( X ) + H ( Y ) - H ( X , Y )$ and the analytic expression for the entropy $H$ of a Gaussian. + +We compare the same setting trained using i.i.d. sampled pairs $( X _ { i } , Y _ { i } )$ as a baseline. We parametrize the critic as a MLP with 5 hidden layers, each with 10 units and ReLU activations, followed by a linear layer and maximize $I _ { \mathrm { N C E } }$ using these non-i.i.d. samples with batch size 128. Note that if a batch size of $K$ is used, the bound $I _ { \mathrm { N C E } } \leq \log K$ always holds. We used $K$ sufficiently large so that $I ( X , Y ) \leq \log K$ to avoid $I _ { \mathrm { N C E } }$ trivially lower bounding the true MI. + +# F ADDITIONAL FIGURES + +![](images/e3bc66417ec78d24fd4ad9e7fff6ab986083cec15b4855fa50745c89b7014b91.jpg) +Figure 5: Additional plot Section 3.1: The condition number of the Jacobian evaluated at inputs randomly sampled from the data distribution deteriorates, i.e. $g _ { 1 }$ becomes increasingly ill-conditioned (lines represent 0th, 20th, . . . , 100th percentiles for $I _ { \mathrm { N } \bar { w } \bar { s } }$ ; the empirical distribution is obtained by randomly sampling 128 inputs from the data distribution, computing the corresponding condition numbers, and aggregating them across runs). + +![](images/d77b2c9b62b360687ed9dfc30280d3cb98e2bfeb8717db8ee4b31a6e0e849549.jpg) +Figure 6: Additional plot Section 3.2: Testing $I _ { \mathrm { N C E } }$ value for MLP encoders $g _ { 1 } , g _ { 1 }$ and different critic architectures. + +![](images/bfebaf5b3a9444dbfa6df2d8f53e998a99eed2e5946708bc27f0b283bc9456d2.jpg) +Figure 7: Additional plots Section 3.3: Testing loss for different encoder architectures and MI estimators, using a bilinear critic trained to match a given target $I _ { \mathrm { E S T } }$ of $t$ (we minimize $L _ { t } ( g _ { 1 } , g _ { 2 } ) =$ $| I _ { \mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t | )$ . + +# G RESULTS FOR THE EXPERIMENTS FROM SEC. 3.2 AND 3.3 ON CIFAR10 + +We run the experiments form Sections 3.2 and 3.3 on CIFAR10 with minimal changes. Specifically, we use the same encoder and critic architectures with the only difference that the input layers of the encoders are adapted to process the (flattened) $3 2 \times 1 4 \times 3$ pixel image halves. Furthermore, we reduce the learning rate from $1 0 ^ { - 4 }$ to $1 0 ^ { - 5 }$ and triple the number of training iterations. Linear classification in pixel space from the upper image halves achieves a testing accuracy of about $2 4 \%$ . + +The CIFAR10 results for the experiment investigating the critic architecture (Section 3.2) can be found in Figure 8 and the results for the experiments investigating the encoder architecture (Section 3.3) in Figure 9. The qualitative behavior of the different encoder and critic architectures in terms of downstream testing accuracy and testing $I _ { \mathrm { E S T } }$ is very similar to the one observed for MNIST. The conclusions made for MNIST hence carry over to CIFAR10. + +![](images/56d94b31f65aab6708fdb7dad6d538766c4811675f88ac2a4b08462ae07f14d0.jpg) +Figure 8: Downstream testing accuracy for $I _ { \mathrm { N C E } }$ and $I _ { \mathrm { N } \bar { w } \bar { s } }$ (top row), and corresponding testing $I _ { \mathrm { E S T } }$ value (bottom row) for MLP encoders $g _ { 1 } , g _ { 1 }$ and different critic architectures. Bilinear and separable critics lead to higher downstream accuracy than MLP critics, while reaching lower $I _ { \mathrm { N } \bar { W } \bar { J } }$ . Note that $I _ { \mathrm { N } \bar { w } \bar { s } }$ exhibits high variance (which is a known property of $I _ { \mathrm { N } \bar { w } \bar { s } }$ (Poole et al., 2019)). + +![](images/b2c4285498c9141c7ca5eb06e5fd42aac5a4f766c041749337da5ec7f1b22fcb.jpg) +Figure 9: Downstream testing accuracy (top row) and testing loss value (bottom row) for different encoder architectures and MI estimators, using a bilinear critic trained to match a given target $I _ { \mathrm { E S T } }$ of $t$ (we minimize $L _ { t } ( g _ { 1 } , g _ { 2 } ) = | I _ { \mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t | )$ . For a given estimator and $t$ , ConvNet encoders clearly outperform MLP encoders in terms of downstream testing accuracy. \ No newline at end of file diff --git a/parse/train/rkxoh24FPH/rkxoh24FPH_content_list.json b/parse/train/rkxoh24FPH/rkxoh24FPH_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..1bce61d7b552ee785eb541b6930e0940e904ccf9 --- /dev/null +++ b/parse/train/rkxoh24FPH/rkxoh24FPH_content_list.json @@ -0,0 +1,1891 @@ +[ + { + "type": "text", + "text": "ON MUTUAL INFORMATION MAXIMIZATION FOR REPRESENTATION LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 98, + 826, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Michael Tschannen∗ Josip Djolonga∗ Paul K. Rubenstein† Sylvain Gelly Mario Lucic Google Research, Brain Team ", + "bbox": [ + 183, + 169, + 828, + 199 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 227, + 544, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many recent methods for unsupervised or self-supervised representation learning train feature extractors by maximizing an estimate of the mutual information (MI) between different views of the data. This comes with several immediate problems: For example, MI is notoriously hard to estimate, and using it as an objective for representation learning may lead to highly entangled representations due to its invariance under arbitrary invertible transformations. Nevertheless, these methods have been repeatedly shown to excel in practice. In this paper we argue, and provide empirical evidence, that the success of these methods cannot be attributed to the properties of MI alone, and that they strongly depend on the inductive bias in both the choice of feature extractor architectures and the parametrization of the employed MI estimators. Finally, we establish a connection to deep metric learning and argue that this interpretation may be a plausible explanation for the success of the recently introduced methods. ", + "bbox": [ + 233, + 260, + 766, + 440 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 468, + 334, + 484 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Unsupervised representation learning is a fundamental problem in machine learning. Intuitively, one aims to learn a function $g$ which maps the data into some, usually lower-dimensional, space where one can solve some (generally a priori unknown) target supervised tasks more efficiently, i.e. with fewer labels. In contrast to supervised and semi-supervised learning, the learner has access only to unlabeled data. Even though the task seems ill-posed as there is no natural objective one should optimize, by leveraging domain knowledge this approach can be successfully applied to a variety of problem areas, including image (Kolesnikov et al., 2019; van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019; Hjelm et al., 2019; Bachman et al., 2019) and video classification (Wang and Gupta, 2015; Sun et al., 2019), and natural language understanding (van den Oord et al., 2018; Peters et al., 2018; Devlin et al., 2019). ", + "bbox": [ + 174, + 500, + 825, + 640 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, there has been a revival of approaches inspired by the InfoMax principle (Linsker, 1988): Choose a representation $g ( x )$ maximizing the mutual information (MI) between the input and its representation, possibly subject to some structural constraints. MI measures the amount of information obtained about a random variable $X$ by observing some other random variable $Y ^ { 1 }$ Formally, the MI between $X$ and $Y$ , with joint density $p ( x , y )$ and marginal densities $p ( x )$ and $p ( y )$ , is defined as the Kullback–Leibler (KL) divergence between the joint and the product of the marginals ", + "bbox": [ + 174, + 646, + 825, + 731 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/15728f3d2ad16802b1b65e6e44c1e6759981f83f98327320c511b75cd981c420.jpg", + "text": "$$\nI ( X ; Y ) = D _ { \\mathrm { K L } } \\left( p ( x , y ) \\parallel p ( x ) p ( y ) \\right) = \\mathbb { E } _ { p ( x , y ) } \\left[ \\log \\frac { p ( x , y ) } { p ( x ) p ( y ) } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 737, + 714, + 772 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The fundamental properties of MI are well understood and have been extensively studied (see e.g. Kraskov et al. (2004)). Firstly, MI is invariant under reparametrization of the variables — namely, if $X ^ { \\prime } = f _ { 1 } ( X )$ and $Y ^ { \\prime } = f _ { 2 } ( Y )$ are homeomorphisms (i.e. smooth invertible maps), then $I ( X ; Y ) = I ( X ^ { \\prime } ; Y ^ { \\prime } )$ . Secondly, estimating MI in high-dimensional spaces is a notoriously difficult task, and in practice one often maximizes a tractable lower bound on this quantity (Poole et al., 2019). ", + "bbox": [ + 174, + 779, + 825, + 849 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nonetheless, any distribution-free high-confidence lower bound on entropy requires a sample size exponential in the size of the bound (McAllester and Statos, 2018). ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Despite these fundamental challenges, several recent works have demonstrated promising empirical results in representation learning using MI maximization (van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019; Hjelm et al., 2019; Bachman et al., 2019; Sun et al., 2019). In this work we argue, and provide empirical evidence, that the success of these methods cannot be attributed to the properties of MI alone. In fact, we show that maximizing tighter bounds on MI can result in worse representations. In addition, we establish a connection to deep metric learning and argue that this interpretation may be a plausible explanation of the success of the recently introduced methods.2 ", + "bbox": [ + 174, + 138, + 825, + 236 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND AND RELATED WORK ", + "text_level": 1, + "bbox": [ + 174, + 256, + 513, + 272 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Recent progress and the InfoMax principle While promising results in other domains have been presented in the literature, we will focus on unsupervised image representation learning techniques that have achieved state-of-the-art performance on image classification tasks (Hénaff et al., 2019; Tian et al., 2019; Bachman et al., 2019). The usual problem setup dates back at least to Becker and Hinton (1992) and can conceptually be described as follows: For a given image $X$ , let $X ^ { ( 1 ) }$ and $X ^ { ( 2 ) }$ be different, possibly overlapping views of $X$ , for instance the top and bottom halves of the image. These are encoded using encoders $g _ { 1 }$ and $g _ { 2 }$ respectively, and the MI between the two representations $g _ { 1 } ( X ^ { ( 1 ) } )$ and $g _ { 2 } \\big ( X ^ { ( 2 ) } \\big )$ is maximized, ", + "bbox": [ + 173, + 287, + 825, + 402 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/4bec240137cc50b1fd88007459fc37aa0ebf66a45b8b1fe0f963d2af480aec11.jpg", + "text": "$$\n\\operatorname* { m a x } _ { g _ { 1 } \\in { \\mathcal G } _ { 1 } , g _ { 2 } \\in { \\mathcal G } _ { 2 } } \\quad I _ { \\mathrm { E S T } } \\left( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 356, + 407, + 640, + 438 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $I _ { \\tt E S T } ( X ; Y )$ is a sample-based estimator of the true MI $I ( X ; Y )$ and the function classes $\\mathcal { G } _ { 1 }$ and $\\mathcal { G } _ { 2 }$ can be used to specify structural constraints on the encoders. While not explicitly reflected in (2), note that $g _ { 1 }$ and $g _ { 2 }$ can often share parameters. Furthermore, it can be shown that $I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) \\le I ( X ; g _ { 1 } ( X ^ { ( 1 ) } ) , g _ { 2 } ( X ^ { ( 2 ) } ) ) ,$ 3 hence the objective in (2) can be seen as a lower bound on the InfoMax objective $\\operatorname* { m a x } _ { g \\in { \\mathcal { G } } } I ( X ; g ( X ) )$ (Linsker, 1988). ", + "bbox": [ + 174, + 443, + 825, + 517 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Practical advantages of multi-view formulations There are two main advantages in using (2) rather than the original InfoMax objective. First, the MI has to be estimated only between the learned representations of the two views, which typically lie on a much lower-dimensional space than the one where the original data $X$ lives. Second, it gives us plenty of modeling flexibility, as the two views can be chosen to capture completely different aspects and modalities of the data, for example: ", + "bbox": [ + 174, + 522, + 825, + 592 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. In the basic form of DeepInfoMax (Hjelm et al., 2019) $g _ { 1 }$ extracts global features from the entire image $X ^ { ( 1 ) }$ and $g _ { 2 }$ local features from image patches $X ^ { ( 2 ) }$ , where $g _ { 1 }$ and $g _ { 2 }$ correspond to activations in different layers of the same convolutional network. Bachman et al. (2019) build on this and compute the two views from different augmentations of the same image. \n2. Contrastive multiview coding (CMC) (Tian et al., 2019) generalizes the objective in (2) to consider multiple views $X ^ { ( i ) }$ , where each $X ^ { ( i ) }$ corresponds to a different image modality (e.g., different color channels, or the image and its segmentation mask). \n3. Contrastive predictive coding (CPC) (van den Oord et al., 2018; Hénaff et al., 2019) incorporates a sequential component of the data. Concretely, one extracts a sequence of patches from an image in some fixed order, maps each patch using an encoder, aggregates the resulting features of the first $t$ patches into a context vector, and maximizes the MI between the context and features extracted from the patch at position $t + k$ . In (2), $X ^ { ( 1 ) }$ would thus correspond to the first $t$ patches and $X ^ { ( 2 ) }$ to the patch at location $t + k$ . Other approaches, such as those presented by Sermanet et al. (2018), Hu et al. (2017), and Ji et al. \n(2019), can be similarly subsumed under the same objective. ", + "bbox": [ + 183, + 598, + 825, + 789 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 796, + 823, + 825 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Lower bounds on MI As evident from (2), another critical choice is the MI estimator $I _ { \\mathrm { E S T } }$ . Given the fundamental limitations of MI estimation (McAllester and Statos, 2018), recent work has focused on deriving lower bounds on MI (Barber and Agakov, 2003; Belghazi et al., 2018; Poole et al., ", + "bbox": [ + 174, + 832, + 826, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2019). Intuitively, these bounds are based on the following idea: If a classifier can accurately distinguish between samples drawn from the joint $p ( x , y )$ and those drawn from the product of marginals $p ( x ) p ( y )$ , then $X$ and $Y$ have a high MI. ", + "bbox": [ + 173, + 103, + 823, + 147 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We will focus on two such estimators, which are most commonly used in the representation learning literature. The first of them, termed InfoNCE (van den Oord et al., 2018), is defined as ", + "bbox": [ + 169, + 152, + 823, + 181 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/243e78ea4391040da2daea342569b96683de91841d9344da31ddf2c2eff5e20a.jpg", + "text": "$$\nI ( X ; Y ) \\geq \\mathbb { E } \\left[ \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) } } \\right] \\triangleq I _ { \\tt N C E } ( X ; Y ) ,\n$$", + "text_format": "latex", + "bbox": [ + 294, + 185, + 702, + 229 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the expectation is over $K$ independent samples $\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { K }$ from the joint distribution $p ( x , y )$ (Poole et al., 2019). In practice we estimate (3) using Monte Carlo estimation by averaging over multiple batches of samples. Intuitively, the critic function $f$ tries to predict for each $x _ { i }$ which of the $K$ samples $y _ { 1 } , \\ldots , y _ { k }$ it was jointly drawn with, by assigning high values to the jointly drawn pair, and low values to all other pairs. The second estimator is based on the variational form of the KL divergence due to Nguyen, Wainwright, and Jordan (NWJ) (Nguyen et al., 2010) and takes the form ", + "bbox": [ + 173, + 234, + 825, + 320 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6b3d9b83561e96c5b3c906e840480bf764ed97f34d3354022f149b3be25fbdd1.jpg", + "text": "$$\nI ( X ; Y ) \\geq \\mathbb { E } _ { p ( x , y ) } [ f ( x , y ) ] - e ^ { - 1 } \\mathbb { E } _ { p ( x ) } [ \\mathbb { E } _ { p ( y ) } e ^ { f ( x , y ) } ] \\triangleq I _ { \\mathbb { N } \\mathbb { N } } ( X ; Y ) .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 324, + 727, + 344 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For detailed derivations we refer the reader to (Ruderman et al., 2012; Poole et al., 2019). Note that these bounds hold for any critic $f$ and when used in (2) one in practice jointly maximizes over $g _ { 1 } , g _ { 2 }$ and $f$ . Furthermore, it can be shown that (3) is maximized by $f ^ { * } ( x , \\bar { y } ) = \\log p ( y | x )$ and (4) by $f ^ { * } ( x , y ) \\mathop { = } 1 + \\log p ( y | x )$ (Poole et al., 2019). Common choices for $f$ include bilinear critics $f ( x , y ) = x ^ { \\top } W y$ (van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019), separable critics $f ( x , y ) = \\phi _ { 1 } ( x ) ^ { \\top } \\phi _ { 2 } ( y )$ (Bachman et al., 2019), and concatenated critics $f ( x , y ) = \\phi ( [ x , y ] )$ (Hjelm et al., 2019) (here $\\phi , \\phi _ { 1 } , \\phi _ { 2 }$ are typically shallow multi-layer perceptrons (MLPs)). When applying these estimators to solve (2), the line between the critic and the encoders $g _ { 1 } , g _ { 2 }$ can be blurry. For example, one can train with an inner product critic $f ( x , y ) = x ^ { \\top } y$ , but extract features from an intermediate layer of $g _ { 1 } , g _ { 2 }$ , in which case the top layers of $g _ { 1 } , g _ { 2 }$ form a separable critic. Nevertheless, this boundary is crucial for the interplay between MI estimation and the interpretation of the learned representations. ", + "bbox": [ + 173, + 348, + 825, + 515 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 BIASES IN APPROXIMATE INFORMATION MAXIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 536, + 671, + 551 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "It is folklore knowledge that maximizing MI does not necessarily lead to useful representations. Already Linsker (1988) talks in his seminal work about constraints, while a manifestation of the problem in clustering approaches using MI criteria has been brought up by Bridle et al. (1992) and subsequently addressed using regularization by Krause et al. (2010). To what can we then attribute the recent success of methods building on the principles of MI maximization? We will argue that their connection to the InfoMax principle might be very loose. Namely, we will show that they behave counter-intuitively if one equates them with MI maximization, and that the performance of these methods depends strongly on the bias that is encoded not only in the encoders, but also on the actual form of the used estimators. ", + "bbox": [ + 173, + 565, + 825, + 691 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. We first consider encoders which are bijective by design. Even though the true MI is maximized for any choice of model parameters, the representation quality (measured by downstream linear classification accuracy) improves during training. Furthermore, there exist invertible encoders for which the representation quality is worse than using raw pixels, despite also maximizing MI. \n2. We next consider encoders that can model both invertible and non-invertible functions. When the encoder can be non-invertible, but is initialized to be invertible, $I _ { \\mathrm { E S T } }$ still biases the encoders to be very ill-conditioned and hard to invert. \n3. For $I _ { \\mathrm { N C E } }$ and $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ , higher-capacity critics admit tighter bounds on MI. We demonstrate that simple critics yielding loose bounds can lead to better representations than high-capacity critics. \n4. Finally, we optimize the estimators to the same MI lower-bound value with different encoder architectures and show that the representation quality can be impacted more by the choice of the architecture, than the estimator. ", + "bbox": [ + 186, + 700, + 826, + 873 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As a consequence, we argue that the success of these methods and the way they are instantiated in practice is only loosely connected to MI. Then, in Section 4 we provide an alternative explanation for the success of recent methods through a connection to classic triplet losses from metric learning. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/a84c5f10c5703c8bb062be5139fa94f689d1924b0e49aabbcdefbabec169b769.jpg", + "image_caption": [ + "Figure 1: (a, b) Maximizing $I _ { \\mathrm { E S T } }$ over a family of invertible models. We can see that during training the downstream classification performance improves (and the testing $I _ { \\mathrm { E S T } }$ value increases), even though the true MI remains constant throughout. (c) Downstream classification accuracy of a different invertible encoder (with the same architecture) trained to have poor performance. This demonstrates the existence of encoders that provably maximize MI yet have bad downstream performance. " + ], + "image_footnote": [], + "bbox": [ + 181, + 97, + 816, + 232 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Setup Our goal is to provide a minimal set of easily reproducible empirical experiments to understand the role of MI estimators, critic and encoder architectures when learning representations via the objective (2). To this end, we consider a simple setup of learning a representation of the top half of MNIST handwritten digit images (we present results for the experiments from Sections 3.2 and 3.3 on CIFAR10 in Appendix G; the conclusions are analogous). This setup has been used in the context of deep canonical correlation analysis (Andrew et al., 2013), where the target is to maximize the correlation between the representations. Following the widely adopted downstream linear evaluation protocol (Kolesnikov et al., 2019; van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019; Hjelm et al., 2019; Bachman et al., 2019), we train a linear classifier4 for digit classification on the learned representation using all available training labels (other evaluation protocols are discussed in Section 5). To learn the representation we instantiate (2) and split each input MNIST image $x \\in [ 0 , 1 ] ^ { 7 8 4 }$ into two parts, the top part of the image $x _ { \\mathrm { t o p } } \\in [ 0 , 1 ] ^ { 3 9 2 }$ corresponding to $X ^ { ( 1 ) }$ , and the bottom part, $x _ { \\mathrm { b o t t o m } } \\in [ 0 , 1 ] ^ { 3 9 2 }$ , corresponding to $X ^ { ( 2 ) }$ , respectively. We train $g _ { 1 } , g _ { 2 }$ , and $f$ using the Adam optimizer (Kingma and Ba, 2015), and use $g _ { 1 } \\big ( x _ { \\mathrm { t o p } } \\big )$ as the representation for the linear evaluation. Unless stated otherwise, we use a bilinear critic $f ( x , y ) = x ^ { \\top } W y$ (we investigate its effect in a separate ablation study), set the batch size to 128 and the learning rate to $1 0 ^ { - 4 }$ . 5 Throughout, $I _ { \\mathrm { E S T } }$ values and downstream classification accuracies are averaged over 20 runs and reported on the testing set (we did not observe large gaps between the training and testing values of $I _ { \\mathrm { E S T } } ,$ ). As a common baseline, we rely on a linear classifier in pixel space on $x _ { \\mathrm { t o p } }$ , which obtains a testing accuracy of about $8 5 \\%$ . For comparison, a simple MLP or ConvNet architecture achieves about $5 4 \\%$ (see Section 3.3 for details). ", + "bbox": [ + 173, + 338, + 825, + 635 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 LARGE MI IS NOT PREDICTIVE OF DOWNSTREAM PERFORMANCE ", + "text_level": 1, + "bbox": [ + 176, + 660, + 665, + 672 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We start by investigating the behavior of $I _ { \\mathrm { N C E } }$ and $I _ { \\mathrm { N W } } { \\boldsymbol { \\mathbf { \\mathit { J } } } }$ when $g _ { 1 }$ and $g _ { 2 }$ are parameterized to be always invertible. Hence, for any choice of the encoder parameters, the MI is constant, i.e. $I ( g _ { 1 } ( X ^ { \\acute { ( 1 ) } } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) = I ( X ^ { ( 1 ) } ; X ^ { ( 2 ) } )$ for all $g _ { 1 } , g _ { 2 }$ . This means that if we could exactly compute the MI, any parameter choice would be a global maximizer and thus the gradients vanish everywhere.6 However, as we will empirically show, the estimators we consider are biased and prefer those settings which yield representations useful for the downstream classification task. ", + "bbox": [ + 174, + 688, + 826, + 772 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Maximized MI and improved downstream performance We model $g _ { 1 }$ and $g _ { 2 }$ using the invertible RealNVP architecture (Dinh et al., 2016). We use a total of 30 coupling layers, and each of them computes the shift using a separate MLP with two ReLU hidden layers, each with 512 units. ", + "bbox": [ + 176, + 780, + 823, + 821 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/47a695e5fcef7059f0d8b7d49d675445acaafdcfd22f6c9b2f1667165ff97877.jpg", + "image_caption": [ + "Figure 2: Maximizing $I _ { \\mathrm { E S T } }$ using a network architecture that can realize both invertible and noninvertible functions. (a, b) As $I _ { \\mathrm { E S T } }$ increases, the linear classification testing performance increases. (c) Meanwhile, the condition number of Jacobian evaluated at inputs randomly sampled from the data distribution deteriorates, i.e. $g _ { 1 }$ becomes increasingly ill-conditioned (lines represent 0th, 20th, . . . , $1 0 0 \\mathrm { { t h } }$ percentiles for $I _ { \\mathrm { N C E } }$ , the corresponding figure for $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ can be found in Appendix F; the empirical distribution is obtained by randomly sampling 128 inputs from the data distribution, computing the corresponding condition numbers, and aggregating them across runs). " + ], + "image_footnote": [], + "bbox": [ + 179, + 92, + 813, + 227 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 1 shows the testing value of $I _ { \\mathrm { E S T } }$ and the testing accuracy on the classification task. Despite the fact that MI is maximized by any instantiation of $g _ { 1 }$ and $g _ { 2 }$ , $I _ { \\mathrm { E S T } }$ and downstream accuracy increase during training, implying that the estimators provide gradient feedback leading to a representation useful for linear classification. This confirms our hypothesis that the estimator biases the encoders towards solutions suitable to solve the downstream linear classification task. ", + "bbox": [ + 174, + 348, + 825, + 416 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The previous experiment demonstrated that among many invertible encoders, all of which are globally optimal MI maximizers, some give rise to improved linear classification performance over raw pixels, and maximizing $I _ { \\mathrm { N C E } }$ and $I _ { \\mathrm { N W } } { \\boldsymbol { \\mathbf { \\mathit { J } } } }$ yields such encoders. Next we demonstrate that for the same invertible encoder architecture there are model parameters for which linear classification performance is significantly worse than using raw pixels, despite also being globally optimal MI maximizers. ", + "bbox": [ + 174, + 424, + 825, + 494 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Maximized MI and worsened downstream performance The goal is to learn a (bijective) representation maximizing MI such that the optimal linear classifier performs poorly; we achieve this by jointly training a representation and classifier in an adversarial fashion (a separate classifier is trained for the evaluation), without using a MI estimator. Intuitively, we will train the encoder to make the classification task for the linear layer as hard as possible. The experimental details are presented in Appendix B. Figure 1c shows the result of one such training run, displaying the loss of a separately trained classifier on top of the frozen representation. At the beginning of training the network is initialized to be close to the identity mapping, and as such achieves the baseline classification accuracy corresponding to raw pixels. All points beyond this correspond to invertible feature maps with worse classification performance, despite still achieving globally maximal MI. ", + "bbox": [ + 173, + 501, + 825, + 640 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Alternatively, the following thought experiment would yield the same conclusion: Using a lossless compression algorithm (e.g. PNG) for $g _ { 1 }$ and $g _ { 2 }$ also satisfies $I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) \\ =$ $I ( X ^ { ( 1 ) } ; X ^ { ( 2 ) } )$ . Yet, performing linear classification on the raw compressed bit stream $g _ { 1 } ( X ^ { ( 1 ) } )$ will likely lead to worse performance than the baseline in pixel space. The information content alone is not sufficient to guarantee a useful geometry in the representation space. ", + "bbox": [ + 174, + 647, + 825, + 719 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We next investigate the behavior of the model if we use a network architecture that can model both invertible and non-invertible functions. We would like to understand whether $I _ { \\mathrm { E S T } }$ prefers the network to remain bijective, thus maximizing the true MI, or to ignore part of the input signal, which can be beneficial for representation learning. ", + "bbox": [ + 174, + 727, + 825, + 784 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Bias towards hard-to-invert encoders We use an MLP architecture with 4 hidden layers of the same dimension as the input, and with a skip connection added to each layer (hence by setting all weights to 0 the network becomes the identity function). As quantifying invertibility is hard, we analyze the condition number, i.e. the ratio between the largest and the smallest singular value, of the Jacobian of $g _ { 1 }$ : By the implicit function theorem, the function is invertible if the Jacobian is non-singular.7 However, the data itself might lie on a low-dimensional manifold, so that having a singular Jacobian is not necessarily indicative of losing invertibility on the support of the data distribution. To ensure the support of the data distribution covers the complete input space, we corrupt $X ^ { ( 1 ) }$ and $X ^ { ( 2 ) }$ in a coupled way by adding to each the same 392-dimensional random vector, whose coordinates are sampled (independently of $X ^ { ( 1 ) } , X ^ { ( 2 ) } )$ from a normal with standard deviation 0.05 (the standard deviation of the pixels themselves is 0.3). Hence, non-invertible encoders $g _ { 1 } , g _ { 2 }$ do not maximize $I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) )$ . 8 As a reference point, the linear classification accuracy from pixels drops to about $84 \\%$ due to the added noise. ", + "bbox": [ + 174, + 790, + 825, + 887 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/8aa32e9c03a77e8657ee9a3fc96b1e03d12ad15997f3d7f40349ab8ce7ec0d00.jpg", + "image_caption": [ + "Figure 3: Downstream testing accuracy for $I _ { \\mathrm { N C E } }$ and $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ , and testing $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ value for MLP encoders $g _ { 1 } , g _ { 1 }$ and different critic architectures (the testing $I _ { \\mathrm { N C E } }$ curve can be found in Appendix F). Bilinear and separable critics lead to higher downstream accuracy than MLP critics, while reaching lower $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ . " + ], + "image_footnote": [], + "bbox": [ + 187, + 97, + 810, + 212 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 276, + 825, + 363 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Figure 2 we can see that the $I _ { \\mathrm { E S T } }$ value and the downstream accuracy both increase during training, as before. Moreover, even though $g _ { 1 }$ is initialized very close to the identity function (which maximizes the true MI), the condition number of its Jacobian evaluated at inputs randomly sampled from the data-distribution steadily deteriorates over time, suggesting that in practice (i.e. numerically) inverting the model becomes increasingly hard. It therefore seems that the bounds we consider favor hard-to-invert encoders, which heavily attenuate part of the noise (as the support of the noise is the entire input space), over well conditioned encoders (such as the identity function at initialization), which preserve the noise and hence the entropy of the data well. ", + "bbox": [ + 174, + 371, + 825, + 482 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 HIGHER CAPACITY CRITICS CAN LEAD TO WORSE DOWNSTREAM PERFORMANCE", + "text_level": 1, + "bbox": [ + 176, + 503, + 771, + 517 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the previous section we have established that MI and downstream performance are only loosely connected. Clearly, maximizing MI is not sufficient to learn good representations and there is a non-trivial interplay between the architectures of the encoder, critic, and the underlying estimators. In this section, we will focus on how one of these factors, namely the critic architecture, impacts the quality of the learned representation. Recall that it determines how the estimators such as $I _ { \\mathrm { N C E } }$ and $I _ { \\mathrm { N W } } { \\boldsymbol { \\mathbf { \\mathit { I } } } }$ distinguish between samples from the joint distribution $p ( x , y )$ and the product of the marginals $p ( x ) p ( y )$ , and thereby determines the tightness on the lower bound. A higher capacity critic should allow for a tighter lower-bound on MI (Belghazi et al., 2018). Furthermore, in the context of representation learning where $f$ is instantiated as a neural network, the critic provides gradient feedback to $g _ { 1 }$ and $g _ { 2 }$ and thereby shapes the learned representation. ", + "bbox": [ + 173, + 530, + 825, + 670 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Looser bounds with simpler critics can lead to better representations We compare three critic architectures, a bilinear critic, a separable critic $f ( x , y ) \\stackrel { \\mathrm { ~ } } { = } \\phi _ { 1 } ( x ) ^ { \\top } \\phi _ { 2 } ( y ) ( \\phi _ { 1 } , \\phi _ { 2 }$ are MLPs with a single hidden layer with 100 units and ReLU activations, followed by a linear layer with 100 units; comprising $4 0 \\mathrm { k }$ parameters in total) and an MLP critic with a single hidden layer with 200 units and ReLU activations, applied to the concatenated input $[ x , y ]$ (40k trainable parameters). Further, we use identical MLP architectures for $g _ { 1 }$ and $g _ { 2 }$ with two hidden layers comprising 300 units each, and a third linear layer mapping to a 100-dimensional feature space. ", + "bbox": [ + 173, + 676, + 825, + 773 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 3 shows the downstream testing accuracy and the testing $I _ { \\mathrm { E S T } }$ value as a function of the iteration (see Appendix G for the corresponding results on CIFAR10). It can be seen that for both lower bounds, representations trained with the MLP critic barely outperform the baseline on pixel space, whereas the same lower bounds with bilinear and separable critics clearly lead to a higher accuracy than the baseline. While the testing $I _ { \\mathrm { N C E } }$ value is close to the theoretically achievable maximum value for all critics, the testing $I _ { \\mathrm { N } \\bar { W } \\bar { J } }$ value is higher for the MLP critic than for the separable and bilinear critics, resulting in a tighter bound on the MI. However, despite achieving the smallest ", + "bbox": [ + 173, + 781, + 825, + 878 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/bf9d8c2d5e81792c60fb051b5b3799ff17a09863301230629190341d2eb3b09e.jpg", + "image_caption": [ + "Figure 4: (a, b) Downstream testing accuracy for different encoder architectures and MI estimators, using a bilinear critic trained to match a given target $I _ { \\mathrm { E S T } }$ of $t$ (we minimize $L _ { t } ( g _ { 1 } , g _ { 2 } ) =$ $| I _ { \\mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t |$ ; loss curves can be found in Appendix F). For a given estimator and $t$ , ConvNet encoders clearly outperform MLP encoders in terms of downstream testing accuracy. (c) Estimating MI from i.i.d. and non-i.i.d. samples in a synthetic setting (Section 4). If negative samples are not drawn i.i.d., both $I _ { \\mathrm { N C E } }$ and $I _ { \\mathrm { N } \\bar { W } \\bar { J } }$ estimators can be greater than the true MI. Despite being commonly justified as a lower bound on MI, $I _ { \\mathrm { N C E } }$ is often used in the non-i.i.d. setting in practice. " + ], + "image_footnote": [], + "bbox": [ + 181, + 98, + 815, + 229 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "$I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ testing value, the simple bilinear critic leads to a better downstream performance than the higher-capacity separable and MLP critics. ", + "bbox": [ + 174, + 361, + 821, + 388 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "A related phenomenon was observed in the context of variational autoencoders (VAEs) (Kingma and Welling, 2014), where one maximizes a lower bound on the data likelihood: Looser bounds often yield better inference models, i.e. latent representations (Rainforth et al., 2018). ", + "bbox": [ + 174, + 396, + 825, + 438 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.3 ENCODER ARCHITECTURE CAN BE MORE IMPORTANT THAN THE SPECIFIC ESTIMATOR ", + "text_level": 1, + "bbox": [ + 173, + 460, + 807, + 473 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We will now show that the encoder architecture is a critical design choice and we will investigate its effect on the learned representation. We consider the same MLP architecture (238k parameters) as in Section 3.2, as well as a ConvNet architecture comprising two convolution layers (with a $5 \\times 5$ kernel, stride of 2, ReLU activations, and 64 and 128 channels, respectively; $2 2 0 \\mathrm { k }$ parameters), followed by spatial average pooling and a fully connected layer. Before the average pooling operation we apply layer normalization (Ba et al., 2016) which greatly reduces the variance of $\\bar { I } _ { \\mathrm { N W } } ^ { \\mathrm { ~ ~ } } \\bar { J }$ . 9 To ensure that both network architectures achieve the same lower bound $I _ { \\mathrm { E S T } }$ on the MI, we minimize $L _ { t } ( g _ { 1 } , g _ { 2 } ) = | I _ { \\mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t |$ instead of solving (2), for two different values $t = 2 , 4$ ", + "bbox": [ + 173, + 486, + 825, + 599 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 4 shows the downstream testing accuracy as a function of the training iteration (see Appendix G for the corresponding results on CIFAR10). It can be seen in the testing loss curves in Appendix F that for both architectures and estimators the objective value after 7k iterations matches the target $t$ (i.e., $L _ { t } ( g _ { 1 } , g _ { 2 } ) \\approx 0 )$ which implies that they achieve the same lower-bound on the MI. Despite matching lower bounds, ConvNet encoders lead to clearly superior classification accuracy, for both $I _ { \\mathrm { N C E } }$ and $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ . Note that, in contrast, the MLP and ConvNet architectures trained end-to-end in supervised fashion both achieve essentially the same testing accuracy of about $9 4 \\%$ . ", + "bbox": [ + 173, + 604, + 825, + 703 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the context of VAEs, Alemi et al. (2018) similarly observed that models achieving the same evidence lower bound value can lead to vastly different representations depending on the employed encoder architecture, and do not necessarily capture useful information about the data (Tschannen et al., 2018; Blau and Michaeli, 2019). ", + "bbox": [ + 174, + 709, + 825, + 765 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 CONNECTION TO DEEP METRIC LEARNING AND TRIPLET LOSSES ", + "text_level": 1, + "bbox": [ + 176, + 790, + 738, + 806 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the previous section we empirically demonstrated that there is a disconnect between approximate MI maximization and representation quality. However, many recent works have applied the $I _ { \\mathrm { N C E } }$ estimator to obtain state-of-the-art results in practice. We provide some insight on this conundrum by connecting $I _ { \\mathrm { N C E } }$ to a popular triplet $k$ -plet) loss known in the deep metric learning community. ", + "bbox": [ + 174, + 824, + 825, + 881 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The metric learning view Given sets of triplets, namely an anchor point $x$ , a positive instance $y$ , and a negative instance $z$ , the goal is to learn a representation $g ( x )$ such that the distances (i.e., $\\ell _ { 2 }$ ) between $g ( x )$ and $g ( y )$ is smaller than the distance between $g ( x )$ and $g ( z )$ , for each triplet. In the supervised setting, the positive instances are usually sampled from the same class, while the negative instances are sampled from any other class. A major focus in deep metric learning is how to perform (semi-)hard positive mining — we want to present non-trivial triplets to the learning algorithm which become more challenging as $g$ improves. Natural extensions to the unsupervised setting can be obtained by exploiting the structure present in the input data, namely spatial (e.g. patches from the same image should be closer than patches from different images) and temporal information (temporally close video frames should be encoded closer than the ones which are further away in time) (Hoffer and Ailon, 2015). ", + "bbox": [ + 173, + 103, + 826, + 257 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Connection to InfoNCE The InfoNCE objective can be rewritten as follows: ", + "bbox": [ + 176, + 262, + 692, + 279 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/4865ee3219d7d938dfd53613ca8157fbc691b587965551a9b9a6bd525099ed70.jpg", + "text": "$$\n\\tilde { \\mathrm { \\Sigma } } _ { \\mathrm { N C E } } = \\mathbb { E } \\left[ \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) } } \\right] = \\log K - \\mathbb { E } \\left[ \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\log \\left( 1 + \\sum _ { j \\ne i } e ^ { f ( x _ { i } , y _ { j } ) - f ( x _ { i } , y _ { i } ) } \\right) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 282, + 828, + 333 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The derivation is presented in Appendix C. In the particular case that $x$ and $y$ take value in the same space and $f$ is constrained to be of the form $f ( x , y ) = \\phi ( x ) ^ { \\top } \\phi ( y )$ , for some function $\\phi$ , this coincides (up to constants and change of sign) with the expectation of the multi-class $K$ -pair loss proposed in (Sohn, 2016, Eqn. (7)): ", + "bbox": [ + 174, + 335, + 825, + 392 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/9ce25da6e970470c3dc44ec565657cfb0753a9c0cdc951af563afa335e874c47.jpg", + "text": "$$\nL _ { \\mathrm { K - p a i r r o c } } \\left( \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { K } , \\phi \\right) = \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\log \\left( 1 + \\sum _ { j \\ne i } e ^ { \\phi ( x _ { i } ) ^ { \\top } \\phi ( y _ { j } ) - \\phi ( x _ { i } ) ^ { \\top } \\phi ( y _ { i } ) } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 235, + 397, + 763, + 446 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Representation learning by maximizing $I _ { \\mathrm { N C E } }$ using a symmetric separable critic $f ( x , y ) = \\phi ( x ) ^ { \\top } \\phi ( y )$ and an encoder $g = g _ { 1 } = g _ { 2 }$ shared across views is thus equivalent to metric learning based on (5). When using different encoders for different views and asymmetric critics as employed by CPC, DeepInfoMax, and CMC one recovers asymmetric variants of (5), see, e.g. (Yu et al., 2017; Zhang et al., 2019). As a result, one can view (5) as learning encoders with a parameter-less inner product critic, for which the MI lower-bound is very weak in general. ", + "bbox": [ + 173, + 452, + 825, + 536 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "There are (at least) two immediate benefits of viewing recent representation learning methods based on MI estimators through the lens of metric learning. Firstly, in the MI view, using inner product or bilinear critic functions is sub-optimal since the critic should ideally be as flexible as possible in order to reduce the gap between the lower bound and the true MI. In the metric learning view, the inner product critic corresponds to a simple metric on the embedding space. The metric learning view seems hence in better accordance with the observations from Section 3.2 than the MI view. Secondly, it elucidates the importance of appropriately choosing the negative samples, which is indeed a critical component in deep metric learning based on triplet losses (Norouzi et al., 2012; Schroff et al., 2015). ", + "bbox": [ + 173, + 542, + 825, + 655 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "InfoNCE and the importance of negative sampling The negative sample mining issue also manifests itself in MI-based contrastive losses. In fact, while InfoNCE is a lower bound on MI if the negative samples are drawn from the true marginal distribution (Poole et al., 2019), i.e. ", + "bbox": [ + 174, + 661, + 825, + 704 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/52e91f01da6694fcddd8df01621988aa6ac5cf229695ad2c31dfba41037daae4.jpg", + "text": "$$\nI ( X , Y ) \\geq \\mathbb { E } _ { \\prod _ { k } p ( x _ { k } , y _ { k } ) } { \\frac { 1 } { K } } \\sum _ { i = 1 } ^ { K } \\left[ \\log { \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { { \\frac { 1 } { K } } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } } \\right] \\triangleq I _ { \\mathbb { N } \\mathbb { C } } ,\n$$", + "text_format": "latex", + "bbox": [ + 282, + 708, + 714, + 752 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "we show that if the negative samples are drawn in a dependent fashion (corresponding to the $( x _ { i } , y _ { i } )$ being drawn identically but not independently), the $I _ { \\mathrm { N C E } }$ estimator is in general neither a lower nor an upper bound on the true MI $I ( X , Y )$ . We prove this in Appendix D and present empirical evidence here. Let $( X , Y ) = Z + \\epsilon .$ , where $\\bar { Z } \\sim \\mathcal { N } ( 0 , \\Sigma _ { Z } )$ and $\\epsilon \\sim \\mathcal { N } ( 0 , \\Sigma _ { \\epsilon } )$ are two-dimensional Gaussians. We generate batches of data $( X _ { i } , Y _ { i } ) = Z + \\epsilon _ { i }$ where each $\\epsilon _ { i }$ is sampled independently for each element of the batch, but $Z$ is sampled only once per batch. As such, $( X _ { i } , Y _ { i } )$ has the same marginal distribution for each $i$ , but the elements of the batch are not independent. Although we do not treat it theoretically, we also display results of the same experiment using the $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ estimator. The experimental details are presented in Appendix E. We observe in Figure $_ \\mathrm { 4 c }$ that when using noni.i.d. samples both the $I _ { \\mathrm { N C E } }$ and $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ values are larger than the true MI, and that when i.i.d. samples are used, both are lower bounds on the true MI. Hence, the connection to MI under improper negative sampling is no longer clear and might vanish completely. ", + "bbox": [ + 173, + 756, + 826, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Notwithstanding this fundamental problem, the negative sampling strategy is often treated as a design choice. In Hénaff et al. (2019), CPC is applied to images by partitioning the input image into patches. Then, MI (estimated by InfoNCE) between representations of patches and a context summarizing several patches that are vertically above or below in the same image is minimized. Negative samples are obtained by patches from different images as well as patches from the same image, violating the independence assumption. Similarly, van den Oord et al. (2018) learn representations of speech using samples from a variety of speakers. It was found that using utterances from the same speaker as negative samples is more effective, whereas the “proper” negative samples should be drawn from an appropriate mixture of utterances from all speakers. ", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "A common observation is that increasing the number of negative examples helps in practice (Hjelm et al., 2019; Tian et al., 2019; Bachman et al., 2019). Indeed, Ma and Collins (2018) show that $I _ { \\mathrm { N C E } }$ is consistent for any number of negative samples (under technical conditions), and Poole et al. (2019) show that the signal-to-noise ratio increases with the number of negative samples. On the other hand, (Arora et al., 2019) have demonstrated, both theoretically and empirically, that increasing the number of negative samples does not necessarily help, and can even deteriorate the performance. The intricacies of negative sampling hence remain a key research challenge. ", + "bbox": [ + 174, + 236, + 825, + 333 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 369, + 318, + 385 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Is MI maximization a good objective for learning good representations in an unsupervised fashion? Possibly, but it is clearly not sufficient. In this work we have demonstrated that, under the common linear evaluation protocol, maximizing lower bounds on MI as done in modern incarnations of the InfoMax principle can result in bad representations. We have revealed that the commonly used estimators have strong inductive biases and—perhaps surprisingly—looser bounds can lead to better representations. Furthermore, we have demonstrated that the connection of recent approaches to MI maximization might vanish if negative samples are not drawn independently (as done by some approaches in the literature). As a result, it is unclear whether the connection to MI is a sufficient (or necessary) component for designing powerful unsupervised representation learning algorithms. We propose that the success of these recent methods could be explained through the view of triplet-based metric learning and that leveraging advances in that domain might lead to further improvements. We have several suggestions for future work, which we summarize in the following. ", + "bbox": [ + 174, + 410, + 825, + 577 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Alternative measures of information We believe that the question of developing new notions of information suitable for representation learning should receive more attention. While MI has appealing theoretical properties, it is clearly not sufficient for this task—it is hard to estimate, invariant to bijections and can result in suboptimal representations which do not correlate with downstream performance. Therefore, a new notion of information should account for both the amount of information stored in a representation and the geometry of the induced space necessary for good performance on downstream tasks. One possible avenue is to consider extensions to MI which explicitly account for the modeling power and computational constraints of the observer, such as the recently introduced $\\mathcal { F }$ -information $\\mathrm { X u }$ et al. (2020). Alternatively, one can investigate other statistical divergences to measure the discrepancy between $p ( x , y )$ and $p ( x ) p ( y )$ . For example, using the Wasserstein distance leads to promising results in representation learning as it naturally enforces smoothness in the encoders (Ozair et al., 2019). ", + "bbox": [ + 174, + 583, + 825, + 750 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "A holistic view We believe that any theory on measuring information for representation learning built on critics should explicitly take into account the function families one uses (e.g. that of the critic and estimator). Most importantly, we would expect some natural trade-offs between the amount of information that can be stored against how hard it is to extract it in the downstream tasks as a function of the architectural choices. While the distribution of downstream tasks is typically assumed unknown in representation learning, it might be possible to rely on weaker assumptions such as a family of invariances relevant for the downstream tasks. Moreover, it seems that in the literature (i) the critics that are used to measure the information, (ii) the encoders, and (iii) the downstream models/evaluation protocol are all mostly chosen independently of each other. Our empirical results show that the downstream performance depends on the intricate balance between these choices and we believe that one should co-design them. This holistic view is currently under-explored and due to the lack of any theory or extensive studies to guide the practitioners. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Going beyond the widely used linear evaluation protocol While it was shown that learning good representations under the linear evaluation protocol can lead to reduced sample complexity for downstream tasks (Arora et al., 2019), some recent works (Bachman et al., 2019; Tian et al., 2019) report marginal improvements in terms of the downstream performance under a non-linear regime. Related to the previous point, it would hence be interesting to further explore the implications of the evaluation protocol, in particular its importance in the context of other design choices. We stress that a highly-nonlinear evaluation framework may result in better downstream performance, but it defeats the purpose of learning efficiently transferable data representations. ", + "bbox": [ + 174, + 103, + 825, + 214 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Systematic investigations into design decisions that matter On the practical side, we believe that the link to metric learning could lead to new methods, that break away from the goal of estimating MI and place more weight on the aspects that have a stronger effect on the performance such as the negative sampling strategy. An example where the metric learning perspective led to similar methods as the MI view is presented by Sermanet et al. (2018): They developed a multi-view representation learning approach for video data similar to CMC, but without drawing negative samples independently and seemingly without relying on the MI mental model to motivate their design choices. ", + "bbox": [ + 174, + 222, + 825, + 319 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 343, + 356, + 357 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank Alex Alemi, Ben Poole, Olivier Bachem, and Alexey Dosovitskiy for inspiring discussions and comments on the manuscript. We are grateful for the general support and discussions from other members of Google Brain team in Zurich. ", + "bbox": [ + 176, + 373, + 825, + 415 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 438, + 285, + 454 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Alexander Alemi, Ben Poole, Ian Fischer, Joshua Dillon, Rif A Saurous, and Kevin Murphy. Fixing a Broken ELBO. In International Conference on Machine Learning, 2018. 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Cross-view asymmetric metric learning for unsupervised person re-identification. In IEEE International Conference on Computer Vision, 2017. ", + "bbox": [ + 173, + 780, + 825, + 821 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Ji Zhang, Yannis Kalantidis, Marcus Rohrbach, Manohar Paluri, Ahmed Elgammal, and Mohamed Elhoseiny. Large-scale visual relationship understanding. In AAAI Conference on Artificial Intelligence, 2019. ", + "bbox": [ + 174, + 832, + 825, + 875 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 264, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A RELATION BETWEEN (2) AND THE INFOMAX OBJECTIVE ", + "text_level": 1, + "bbox": [ + 176, + 133, + 678, + 151 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proposition 1. Let $X$ be a random variable and define $X _ { 1 } = g _ { 1 } ( X )$ and $X _ { 2 } = g _ { 2 } ( X )$ be arbitrary functions of $X$ . Then $I ( X _ { 1 } ; X _ { 2 } ) \\leq I \\left( X ; ( X _ { 1 } , X _ { 2 } ) \\right)$ . ", + "bbox": [ + 169, + 165, + 825, + 195 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. Follows by two applications of the data processing inequality, which states that for random variables $X , Y$ and $Z$ satisfying the Markov relation $X Y Z$ , the inequality $I ( X ; Z ) \\ \\leq$ $I ( X ; Y )$ holds. ", + "bbox": [ + 174, + 208, + 825, + 251 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The first step is to observe that $X$ , $X _ { 1 }$ and $X _ { 2 }$ satisfy the relation $X _ { 1 } \\left. X \\right. X _ { 2 }$ , which is Markov equivalent to $X _ { 1 } X X _ { 2 }$ (in particular, $X _ { 1 }$ and $X _ { 2 }$ are conditionally independent given $X$ ). It therefore follows that $I ( X _ { 1 } ; X _ { 2 } ) \\leq I ( X ; X _ { 1 } )$ . The second step is to observe that $X ( X _ { 1 } , X _ { 2 } ) X _ { 1 }$ and therefore $I ( X ; X _ { 1 } ) \\leq I ( X ; ( X _ { 1 } , X _ { 2 } ) )$ . ", + "bbox": [ + 174, + 257, + 825, + 315 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Combining the two inequalities yields $I ( X _ { 1 } ; X _ { 2 } ) \\leq I ( X ; ( X _ { 1 } , X _ { 2 } ) )$ , as required. ", + "bbox": [ + 176, + 319, + 707, + 337 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B EXPERIMENT DETAILS: ADVERSARIALLY TRAINED ENCODER(SECTION 3.1)", + "text_level": 1, + "bbox": [ + 174, + 354, + 718, + 388 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In the following, we present the details for training the invertible model from Section 3.1 adversarially. We model $g _ { 1 }$ with the same RealNVP architecture as in the first experiment, and do not model $g _ { 2 }$ . On top of $g _ { 1 } ( X ^ { ( 1 ) } )$ we add a linear layer mapping to 10 outputs (i.e. logits). The parameters of the linear layer trained by minimizing the cross-entropy loss with respect to the true label of $X$ from which $X ^ { ( 1 ) }$ is derived. Conversely, the parameters of the encoder $g _ { 1 }$ are trained to minimize the cross-entropy loss with respect to a uniform probability vector over all 10 classes. We use the Adam optimizer with a learning rate of $1 0 ^ { - 4 }$ for the parameters of the classifier and $1 0 ^ { - 6 }$ for the parameters of the encoder, and perform 10 classifier optimization steps per encoder step. Furthermore, in a warm-up phase we train the classifier for 1k iterations before alternating between classifier and encoder steps. ", + "bbox": [ + 173, + 401, + 826, + 531 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C CONNECTION BETWEEN METRIC LEARNING AND INFONCE ", + "text_level": 1, + "bbox": [ + 173, + 551, + 702, + 568 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$I _ { \\mathrm { N C E } }$ can be rewritten as follows: ", + "bbox": [ + 174, + 582, + 388, + 595 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/2eb28af347537fd0e83e244a77f15c1471c7792ab1cda7be7526d86672abba99.jpg", + "text": "$$\n\\begin{array} { r l } & { I _ { \\mathrm { R e x s } } = \\mathbb { E } \\left[ \\displaystyle \\frac { 1 } { K } \\displaystyle \\sum _ { i = 1 } ^ { K } \\log \\frac { e ^ { f ( x _ { i } , y _ { j } ) } } { \\frac { 1 } { K } \\displaystyle \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) } } \\right] } \\\\ & { \\quad = \\mathbb { E } \\left[ \\displaystyle \\frac { 1 } { K } \\displaystyle \\sum _ { i = 1 } ^ { K } \\log \\frac { 1 } { \\frac { 1 } { K } \\displaystyle \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) - f ( x _ { i } , y _ { i } ) } } \\right] } \\\\ & { \\quad = \\mathbb { E } \\left[ - \\displaystyle \\frac { 1 } { K } \\displaystyle \\sum _ { i = 1 } ^ { K } \\log \\frac { 1 } { K } \\displaystyle \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) - f ( x _ { i } , y _ { i } ) } \\right] } \\\\ & { \\quad = \\log K - \\mathbb { E } \\left[ \\displaystyle \\frac { 1 } { K } \\displaystyle \\sum _ { i = 1 } ^ { K } \\log \\left( 1 + \\displaystyle \\sum _ { j \\neq i } e ^ { f ( x _ { i } , y _ { j } ) - f ( x _ { i } , y _ { i } ) } \\right) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 597, + 709, + 787 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D INFONCE UNDER NON-I.I.D. SAMPLING ", + "text_level": 1, + "bbox": [ + 173, + 799, + 544, + 816 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The proof that InfoNCE is a lower bound on MI presented in (Poole et al., 2019) makes crucial use of the assumption that the negative samples are drawn from the true marginal distribution. We briefly review this proof to highlight the importance of the negative sampling distribution. Their proof starts from the NWJ lower bound of the KL divergence, namely that for any function $\\tilde { f }$ the following lower bound holds (Nguyen et al., 2010; Nowozin et al., 2016): ", + "bbox": [ + 173, + 830, + 825, + 904 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/25acdbbb3fe5b324ed4038df7893a441aae4431216503c69fa6a1238d55372a6.jpg", + "text": "$$\nI ( X ; Y ) = D _ { K L } ( p ( x , y ) | | p ( x ) p ( y ) ) \\geq \\mathbb { E } _ { p ( x , y ) } [ \\tilde { f } ( x , y ) ] - e ^ { - 1 } \\mathbb { E } _ { p ( x ) p ( y ) } [ e ^ { \\tilde { f } ( x , y ) } ] .\n$$", + "text_format": "latex", + "bbox": [ + 228, + 905, + 767, + 926 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Suppose that $( X _ { i } , Y _ { i } ) _ { i = 1 } ^ { K }$ are i.i.d. draws from $p ( x , y )$ and write $X _ { 1 : K } = ( X _ { 1 } , X _ { 2 } , \\dots , X _ { K } )$ . Then, for any $i$ we have that $I ( X _ { 1 : K } ; Y _ { i } ) = I ( X _ { i } ; Y _ { i } ) = I ( X ; Y )$ . We thus have ", + "bbox": [ + 171, + 102, + 826, + 133 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/6bf011198f44d559a7df9ffdc53a90bda5d0ae203c67a28c57219df12de88740.jpg", + "text": "$$\n\\begin{array} { r } { I ( \\boldsymbol { X } ; \\boldsymbol { Y } ) = I ( X _ { 1 : K } ; Y _ { i } ) \\ge \\mathbb { E } _ { p ( x _ { i } , y _ { i } ) \\prod _ { k \\geqslant i } p ( x _ { k } ) } [ \\tilde { f } ( x _ { 1 : K } , y _ { i } ) ] - e ^ { - 1 } \\mathbb { E } _ { p ( y _ { i } ) \\prod _ { k } p ( x _ { k } ) } [ e ^ { \\tilde { f } ( x _ { 1 : K } , y _ { i } ) } ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 156, + 803, + 180 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where the equality follows from the assumption that the $( X _ { i } , Y _ { i } ) _ { i = 1 } ^ { K }$ are i.i.d. and the inequality is (6) applied to $I ( X _ { 1 : K } ; Y _ { i } )$ . In particular, taking $\\begin{array} { r } { \\tilde { f } ( x _ { 1 : K } , y _ { i } ) = 1 + \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } } \\end{array}$ ef(xj ,yi) yields ", + "bbox": [ + 173, + 185, + 826, + 228 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/827fc1bbe033cd21e55517b744938640b02166889ed26b97ecd5e68b374dac72.jpg", + "text": "$$\n\\textmd { ( X , Y ) \\geq 1 + } \\mathbb { E } _ { p ( x _ { i } , y _ { i } ) \\prod _ { k \\geqslant i } p ( x _ { k } ) } \\left[ \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } \\right] - \\mathbb { E } _ { p ( y _ { i } ) \\prod _ { k } p ( x _ { k } ) } \\left[ \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 243, + 828, + 287 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This is then averaged over the $K$ samples $Y _ { i }$ , in which case the third term above cancels with the constant 1 (all occurences of $y _ { i }$ in the last term of (7) can be replaced with $y _ { 1 }$ thanks to $( X _ { i } , Y _ { i } )$ being identically distributed), yielding the familiar $I _ { \\mathrm { N C E } }$ lower bound: ", + "bbox": [ + 173, + 314, + 826, + 358 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c4b330f710076584231eb17a12344d3100a7adb54423037b1470730523754625.jpg", + "text": "$$\nI ( X , Y ) \\geq \\mathbb { E } _ { \\prod _ { k } p ( x _ { k } , y _ { k } ) } { \\frac { 1 } { K } } \\sum _ { i = 1 } ^ { K } \\left[ \\log { \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { { \\frac { 1 } { K } } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } } \\right] = I _ { \\mathbb { N } \\mathbb { C } } .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 382, + 714, + 426 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The point in this proof that makes use of the i.i.d. assumption of the negative samples is in the equality $I ( X _ { i } , Y _ { i } ) = I ( X _ { 1 : K } , Y _ { i } )$ , which allowed us to leverage multiple samples when estimating the MI between two variables. If instead the negative samples are drawn in a dependent fashion (corresponding to the $( X _ { i } , Y _ { i } )$ being drawn identically but not independently), we have $I ( X _ { i } , Y _ { i } ) \\leq$ $I ( X _ { 1 : K } , Y _ { i } )$ , though the remainder of the proof still holds, resulting in ", + "bbox": [ + 173, + 438, + 826, + 510 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ddf028218118e476dd360f213a65d0eaecbd9f3ba2f90ce6688d40ae085d2206.jpg", + "text": "$$\nI ( X , Y ) \\leq \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } I ( X _ { 1 : K } ; Y _ { i } ) \\geq \\mathbb { E } _ { p ( x _ { 1 : K } , y _ { 1 : K } ) } \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\left[ \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 534, + 764, + 578 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Therefore the resulting $I _ { \\mathrm { N C E } }$ estimator is neither a lower nor an upper bound on the true MI $I ( X , Y )$ . ", + "bbox": [ + 171, + 590, + 826, + 607 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "E EXPERIMENT DETAILS: NON-I.I.D. SAMPLING (SECTION 4) ", + "bbox": [ + 173, + 625, + 702, + 642 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Recall that $( X , Y ) = Z + \\epsilon$ . We use $Z \\sim { \\mathcal { N } } ( 0 , \\Sigma _ { Z } )$ and $\\epsilon \\sim \\mathcal { N } ( 0 , \\Sigma _ { \\epsilon } )$ , where ", + "bbox": [ + 174, + 655, + 689, + 672 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/d77b680826ff2ac675e7533def8b827d7f626eb222c2ee7f04758f02bc9163a2.jpg", + "text": "$$\n\\Sigma _ { Z } = \\left( { \\begin{array} { c c } { 1 } & { - 0 . 5 } \\\\ { - 0 . 5 } & { 1 } \\end{array} } \\right) \\qquad { \\mathrm { a n d } } \\qquad \\Sigma _ { \\epsilon } = \\left( { \\begin{array} { c c } { 1 } & { 0 . 9 } \\\\ { 0 . 9 } & { 1 } \\end{array} } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 678, + 687, + 713 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Batches of data are obtained as $( X _ { i } , Y _ { i } ) = Z + \\epsilon _ { i }$ where each $\\epsilon _ { i }$ is sampled independently for each element of the batch, but $Z$ is sampled only once per batch. The true MI $I ( X , Y )$ can be calculated analytically since $( X , Y )$ is jointly Gaussian with known covariance matrix $\\Sigma _ { Z } + \\Sigma _ { \\epsilon }$ : For two univariate random variables $( X , Y )$ that are jointly Gaussian with covariance $\\Sigma$ the MI can be written as ", + "bbox": [ + 173, + 718, + 826, + 786 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/cd27ab7c0798506aa3961c144bebf1b6ef36dc50baae434841d0ee52b2c08ac8.jpg", + "text": "$$\nI ( X , Y ) = - \\frac { 1 } { 2 } \\log ( 1 - \\frac { \\Sigma _ { 1 2 } \\Sigma _ { 2 1 } } { \\Sigma _ { 1 1 } \\Sigma _ { 2 2 } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 784, + 614, + 815 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This can be derived using the decomposition $I ( X , Y ) = H ( X ) + H ( Y ) - H ( X , Y )$ and the analytic expression for the entropy $H$ of a Gaussian. ", + "bbox": [ + 174, + 818, + 825, + 848 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We compare the same setting trained using i.i.d. sampled pairs $( X _ { i } , Y _ { i } )$ as a baseline. We parametrize the critic as a MLP with 5 hidden layers, each with 10 units and ReLU activations, followed by a linear layer and maximize $I _ { \\mathrm { N C E } }$ using these non-i.i.d. samples with batch size 128. Note that if a batch size of $K$ is used, the bound $I _ { \\mathrm { N C E } } \\leq \\log K$ always holds. We used $K$ sufficiently large so that $I ( X , Y ) \\leq \\log K$ to avoid $I _ { \\mathrm { N C E } }$ trivially lower bounding the true MI. ", + "bbox": [ + 173, + 853, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "F ADDITIONAL FIGURES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 397, + 118 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/e3bc66417ec78d24fd4ad9e7fff6ab986083cec15b4855fa50745c89b7014b91.jpg", + "image_caption": [ + "Figure 5: Additional plot Section 3.1: The condition number of the Jacobian evaluated at inputs randomly sampled from the data distribution deteriorates, i.e. $g _ { 1 }$ becomes increasingly ill-conditioned (lines represent 0th, 20th, . . . , 100th percentiles for $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ ; the empirical distribution is obtained by randomly sampling 128 inputs from the data distribution, computing the corresponding condition numbers, and aggregating them across runs). " + ], + "image_footnote": [], + "bbox": [ + 392, + 136, + 602, + 253 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/d77b2c9b62b360687ed9dfc30280d3cb98e2bfeb8717db8ee4b31a6e0e849549.jpg", + "image_caption": [ + "Figure 6: Additional plot Section 3.2: Testing $I _ { \\mathrm { N C E } }$ value for MLP encoders $g _ { 1 } , g _ { 1 }$ and different critic architectures. " + ], + "image_footnote": [], + "bbox": [ + 383, + 352, + 612, + 478 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/bfebaf5b3a9444dbfa6df2d8f53e998a99eed2e5946708bc27f0b283bc9456d2.jpg", + "image_caption": [ + "Figure 7: Additional plots Section 3.3: Testing loss for different encoder architectures and MI estimators, using a bilinear critic trained to match a given target $I _ { \\mathrm { E S T } }$ of $t$ (we minimize $L _ { t } ( g _ { 1 } , g _ { 2 } ) =$ $| I _ { \\mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t | )$ . " + ], + "image_footnote": [], + "bbox": [ + 253, + 535, + 743, + 661 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "G RESULTS FOR THE EXPERIMENTS FROM SEC. 3.2 AND 3.3 ON CIFAR10 ", + "text_level": 1, + "bbox": [ + 171, + 744, + 805, + 762 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We run the experiments form Sections 3.2 and 3.3 on CIFAR10 with minimal changes. Specifically, we use the same encoder and critic architectures with the only difference that the input layers of the encoders are adapted to process the (flattened) $3 2 \\times 1 4 \\times 3$ pixel image halves. Furthermore, we reduce the learning rate from $1 0 ^ { - 4 }$ to $1 0 ^ { - 5 }$ and triple the number of training iterations. Linear classification in pixel space from the upper image halves achieves a testing accuracy of about $2 4 \\%$ . ", + "bbox": [ + 173, + 776, + 826, + 848 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The CIFAR10 results for the experiment investigating the critic architecture (Section 3.2) can be found in Figure 8 and the results for the experiments investigating the encoder architecture (Section 3.3) in Figure 9. The qualitative behavior of the different encoder and critic architectures in terms of downstream testing accuracy and testing $I _ { \\mathrm { E S T } }$ is very similar to the one observed for MNIST. The conclusions made for MNIST hence carry over to CIFAR10. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/56d94b31f65aab6708fdb7dad6d538766c4811675f88ac2a4b08462ae07f14d0.jpg", + "image_caption": [ + "Figure 8: Downstream testing accuracy for $I _ { \\mathrm { N C E } }$ and $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ (top row), and corresponding testing $I _ { \\mathrm { E S T } }$ value (bottom row) for MLP encoders $g _ { 1 } , g _ { 1 }$ and different critic architectures. Bilinear and separable critics lead to higher downstream accuracy than MLP critics, while reaching lower $I _ { \\mathrm { N } \\bar { W } \\bar { J } }$ . Note that $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ exhibits high variance (which is a known property of $I _ { \\mathrm { N } \\bar { w } \\bar { s } }$ (Poole et al., 2019)). " + ], + "image_footnote": [], + "bbox": [ + 220, + 125, + 779, + 411 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/b2c4285498c9141c7ca5eb06e5fd42aac5a4f766c041749337da5ec7f1b22fcb.jpg", + "image_caption": [ + "Figure 9: Downstream testing accuracy (top row) and testing loss value (bottom row) for different encoder architectures and MI estimators, using a bilinear critic trained to match a given target $I _ { \\mathrm { E S T } }$ of $t$ (we minimize $L _ { t } ( g _ { 1 } , g _ { 2 } ) = | I _ { \\mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t | )$ . For a given estimator and $t$ , ConvNet encoders clearly outperform MLP encoders in terms of downstream testing accuracy. 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