diff --git a/parse/train/Bke6vTVYwH/Bke6vTVYwH.md b/parse/train/Bke6vTVYwH/Bke6vTVYwH.md new file mode 100644 index 0000000000000000000000000000000000000000..c352a67376cf735b255c562e7c94a3a27f4953ea --- /dev/null +++ b/parse/train/Bke6vTVYwH/Bke6vTVYwH.md @@ -0,0 +1,291 @@ +# GRAPH CONVOLUTIONAL NETWORKS FOR LEARNING WITH FEW CLEAN AND MANY NOISY LABELS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +In this work we consider the problem of learning a classifier from noisy labels when a few clean labeled examples are given. The structure of clean and noisy data is modeled by a graph per class and Graph Convolutional Networks (GCN) are used to predict class relevance of noisy examples. For each class, the GCN is treated as a binary classifier learning to discriminate clean from noisy examples using a weighted binary cross-entropy loss function, and then the GCN-inferred “clean” probability is exploited as a relevance measure. Each noisy example is weighted by its relevance when learning a classifier for the end task. We evaluate our method on an extended version of a few-shot learning problem, where the few clean examples of novel classes are supplemented with additional noisy data. Experimental results show that our GCN-based cleaning process significantly improves the classification accuracy over not cleaning the noisy data and standard few-shot classification where only few clean examples are used. The proposed GCN-based method outperforms the transductive approach (Douze et al., 2018) that is using the same additional data without labels. + +# 1 INTRODUCTION + +State-of-the-art deep learning methods require a large amount of manually labeled data. The need for supervision may be reduced by decoupling representation learning from the end task and/or using additional training data that are unlabeled, weakly labeled (with noisy labels), or belong to different domains or classes. Example approaches are transfer learning (Wang & Gupta, 2015), unsupervised representation learning (Wang & Gupta, 2015), semi-supervised learning (Weston et al., 2008), learning from noisy labels (Joulin et al., 2016) and few-shot learning (Snell et al., 2017). + +Learning from noisy labels allows using large-scale data and labels from the web without human annotation effort. Most work focuses on learning the representation jointly with the end task, assuming there is still a considerable amount of clean labeled data (Patrini et al., 2017; Lee et al., 2018; Li et al., 2017). However, for a number of classes only very few or even no clean labeled examples might be available at the representation learning stage. Few-shot learning limits the labeled data to very few on the end task, while the representation is learned on a large training set of different classes (Hariharan & Girshick, 2017; Snell et al., 2017; Vinyals et al., 2016). Nevertheless, in many situations, more data with noisy labels are available or can be acquired for the end task. + +One interesting mix of few-shot learning with additional large-scale data is the work of Douze et al. (2018), where labels are propagated from few clean labeled examples to a large-scale collection. This collection is unlabeled and actually contains data of many more classes than the end task. Their method overall improves the classification accuracy, but at an additional computational cost; it is a transductive method, i.e., instead of learning a parametric classifier, the large-scale collection is still necessary at inference. + +In this work, we learn a classifier from few clean labeled examples and additional weakly labeled data, while the representation is learned on different classes, as in few-shot learning. We assume the class names are known, and we use them to search an existing large collection of images with textual description. The result is a set of images with potentially relevant, but noisy labels. As shown in Figure 1, we clean this data using a graph convolutional network (GCN) (Kipf & Welling, 2017), which learns to predict a class relevance score per image based on the source (clean vs. noisy) of its connections in the graph. Both the clean and the noisy images are then used to learn a classifier, where the noisy examples are weighted by relevance. Unlike most existing work, our method operates independently per class and applies when clean labeled examples are few or even one per class. + +![](images/7eed9d45df54f8a4f9857ca30afe31d8710a4404ec6cdacce1d0af912c397be2.jpg) +Figure 1: Overview of our cleaning approach for 1-shot learning with noisy examples. We use the class name admiral to crawl noisy images from web and create an adjacency graph based on visual similarity. We then assign a relevance score to each noisy example with a graph convolutional network (GCN). Relevance scores are displayed next to the images. + +We make the following contributions: + +• We learn a classifier on a large-scale weakly-labeled collection jointly with only few clean labeled examples. +• To our knowledge, we are the first to use a GCN to clean noisy data: we cast a GCN as a binary classifier learning to discriminate clean from noisy data, and we use its inferred probabilities for the “clean” class as a relevance score per example. +We apply our method to a few-shot learning benchmark and show significant improvement in accuracy, while outperforming the method by Douze et al. (2018) using the same large-scale collection of data. + +# 2 RELATED WORK + +Learning with noisy labels is often concerned with estimating or learning a transition matrix (Natarajan et al., 2013; Patrini et al., 2017; Sukhbaatar et al., 2014) or knowledge graph (Li et al., 2017) between labels and correcting the loss function, which does not apply in our case since the classes in the noisy data are unknown. Most recent work on learning from large-scale weakly-labeled data focuses on learning the representation e.g. by metric learning (Lee et al., 2018; Wang et al., 2018a), bootstrapping (Reed et al., 2015), or distillation (Li et al., 2017). In our case however, since the clean labeled examples are few, we need to keep the representation mostly fixed. + +Dealing with the noise, e.g. by thresholding (Lee et al., 2018), outlier detection (Wang et al., 2018a) or reweighting (Liu & Tao, 2015), is applicable while the representation is learned, based e.g. on the gradient of the loss (Ren et al., 2018b). In contrast, the relatively-shallow GCN that we propose effectively decouples reweighting from both representation learning and classifier learning. Learning to clean the noisy labels (Veit et al., 2017) typically assumes adequate human verified labels for training, which again is not the case in this work. + +Few-shot learning. Meta-learning (Vilalta & Drissi, 2002) refers to learning at two levels, where generic knowledge is acquired before adapting to more specific tasks. In few-shot learning, this translates to learning on a set of base classes how to learn from few examples on a distinct set of novel classes without overfitting. For instance, optimization meta-learning (Finn et al., 2017; 2018; Ravi & Larochelle, 2017) amounts to learning a model that is easy to fine-tune in few steps. In our work, we study an extension of few-shot learning where more data are available on novel classes, reducing the risk of overfitting when fine-tuning the model. Metric learning approaches learn how to compare queries for instance to few examples (Vinyals et al., 2016) or to the corresponding class prototypes (Snell et al., 2017). Hariharan & Girshick (2017) and Wang et al. (2018b) learn how to generate novel-class examples, which is not needed when more data are actually available. + +Gidaris & Komodakis (2018) learn on base classes a simpler cosine similarity-based parametric classifier, or simply cosine classifier, without meta-learning. The same classifier has been introduced independently by Qi et al. (2018), who further fine-tune the network, assuming access to the base class training set. A recent survey (Chen et al., 2019) confirms the superiority of the cosine classifier to previous work including meta-learning (Finn et al., 2017). We use the cosine classifier in this work, both for base and novel classes. All of the above use only the few labeled examples of the novel classes. + +Making use of unlabeled data has been little explored in few-shot learning until recently. Ren et al. (2018a) introduce a semi-supervised few-shot classification task, where some labels are unknown. Liu et al. (2019) follow the same semi-supervised setup, but use graph-based label propagation (LP) (Zhou et al., 2003a) for classification and consider jointly all test images. These methods assume a meta-learning scenario, where only small-scale data is available at each training episode; arguably, such a small amount of data limits the representation adaptation and generalization to unseen data. Similarly, Rohrbach et al. (2013) use label propagation in a transductive setting, but at a larger scale assuming that all examples come from a set of known classes. Douze et al. (2018) extend to even larger scale, leveraging 100M unlabeled images in a graph without using additional text information. We focus on the latter large-scale scenario using the same 100M dataset. However, we filter by text to obtain noisy labels and follow an inductive approach by training a classifier for novel classes, such that the 100M collection is not needed at inference. + +Graph neural networks are generalizations of convolutional networks to non-Euclidean spaces (Bronstein et al., 2017). Early spectral methods (Bruna et al., 2014; Henaff et al., 2015) have been succeeded by Chebyshev polynomial approximations (Defferrard et al., 2016), which avoid the high computational cost of computing eigenvectors. Graph convolutional networks (GCN) (Kipf & Welling, 2017) provide a further simplification by a first-order approximation of graph filtering and are applied to semi-supervised (Kipf & Welling, 2017) and subsequently to few-shot learning (Garcia & Bruna, 2018). In Kipf & Welling (2017), the loss function is applied to labeled examples to make predictions on unlabeled ones. Similarly in Garcia & Bruna (2018), GCNs make predictions on novel class examples. Gidaris & Komodakis (2019) use Graph Neural Networks as denoising autoencoders to generate class weights for novel classes. In contrast, we cast GCNs as binary classifiers discriminating clean from noisy examples: we apply a loss function to all examples, and then use the inferred probabilities as a class relevance measure, effectively cleaning the data. + +Our counter-intuitive objective of treating all noisy examples as negative can be compared to treating each example as a different class in instance-level discrimination (Wu et al., 2018). In fact, our loss function is similar to noise-contrastive estimation (NCE) (Gutmann & Hyvärinen, 2010) used in that work. According to our experiments, our GCN-based classifier outperforms classical LP (Zhou et al., 2003a) used for a similar purpose by Rohrbach et al. (2013). + +# 3 PROBLEM FORMULATION + +We consider a space $\mathcal { X }$ of examples. We are given a set $X _ { \mathcal { L } } \subset \mathcal { X }$ of examples, each having a clean (manually verified) label in a set $C _ { \mathcal { L } }$ of classes with $| C _ { \mathcal { L } } | = K _ { \mathcal { L } }$ . For any set $X \subset { \mathcal { X } }$ , we denote by $X ^ { c }$ its subset of examples having a label in class $c$ . We assume that the number $| X _ { \mathcal { L } } ^ { c } |$ of examples labeled in each class $c \in C _ { \mathcal { L } }$ is only $k$ , typically in $\{ 1 , 2 , 5 , 1 0 , 2 0 \}$ . We are also given an additional set $X _ { \mathcal { Z } } ^ { c }$ of examples, each with a set of noisy labels in $C _ { \mathcal { L } }$ . The extended set of examples for class $c$ is now $X _ { \mathcal { E } } ^ { c } = X _ { \mathcal { L } } ^ { c } \cup X _ { \mathcal { Z } } ^ { c }$ . Examples or sets of examples having clean (noisy) labels are referred to as clean (noisy) as well. The goal is to train a classifier, using the additional noisy set in order to improve the accuracy compared to only using the small clean set. + +We assume that we are given a feature extractor $g _ { \theta } : \mathcal { X } \mathbb { R } ^ { d }$ , mapping an example to a $d .$ -dimensional vector. For instance, when examples are images, the feature extractor is typically a convolutional neural network (CNN) and $\theta$ are the parameters of all layers. + +In this work, we assume that the noisy set $X z$ is collected via web crawling with examples that are images accompanied with free-form text description and/or user tags originating from community photo collections. To make use of text data, we assume that the names of classes in $C _ { \mathcal { L } }$ are given. An example in $X z$ is given a label in class $c \in C _ { \mathcal { L } }$ if its textual information contains the name of class $c$ ; it may then have none, one or more labels. In this way, we automatically infer labels for $X z$ without human effort, which are however noisy. + +We perform cleaning by predicting a class relevance measure for each noisy example in $X _ { \mathcal { Z } } ^ { c }$ , independently per class $c \in C _ { \mathcal { L } }$ . To simplify notation, we drop superscript $c$ where possible in this subsection and we denote $X _ { \mathcal { E } } ^ { c }$ by $\left\{ x _ { 1 } , \ldots , x _ { k } , x _ { k + 1 } , \ldots , x _ { N } \right\}$ , where $X _ { \mathcal { L } } ^ { c } ~ = ~ \{ \bar { x _ { 1 } } , \ldots , x _ { k } \}$ and $X _ { \mathcal { Z } } ^ { c } = \{ x _ { k + 1 } , \ldots , x _ { N } \}$ . The features of these examples are similarly represented by matrix $V = [ \ b { \mathrm { v } } _ { 1 } , \ b { \mathrm { ~ . ~ . ~ . ~ } } , \ b { \mathrm { v } } _ { k } , \ b { \mathrm { v } } _ { k + 1 } , \ b { \mathrm { ~ . ~ . ~ . ~ } } , \ b { \mathrm { v } } _ { N } ] \in \mathbb { R } ^ { d \times N }$ , where $\mathbf { v } _ { i } = g _ { \theta } ( x _ { i } )$ for $i = 1 , \ldots , N$ . + +We construct an affinity matrix $A \in \mathbb { R } ^ { N \times N }$ with elements $a _ { i j } = [ \mathbf { v } _ { i } ^ { \top } \mathbf { v } _ { j } ] _ { + }$ if examples $\mathbf { v } _ { i }$ and $\mathbf { v } _ { j }$ are reciprocal nearest neighbors in $X _ { \mathcal { E } } ^ { c }$ and 0 otherwise. Matrix $A$ has zero diagonal, but self-connections are added and then $A$ is normalized as $\tilde { A } = D ^ { - 1 } ( A + I )$ with $D = \mathrm { d i a g } ( ( A + I ) \mathbf { 1 } )$ being the degree matrix of $A + I$ and 1 the all-ones vector. + +Graph convolutional networks (GCNs) (Kipf & Welling, 2017) are formed by a sequence of layers. Each layer is a function $f _ { \Theta } : \mathbb { R } ^ { \hat { N } \times N } \times \mathbb { R } ^ { l \times \hat { N } } \mathbb { R } ^ { n \times \hat { N } }$ of the form + +$$ +f _ { \Theta } ( \tilde { A } , Z ) = h ( \Theta ^ { \top } Z \tilde { A } ) , +$$ + +where $Z \in \mathbb { R } ^ { l \times N }$ represents the input features, $\boldsymbol { \Theta } \in \mathbb { R } ^ { l \times n }$ holds the parameters of the layer to be learned, and $h$ is a nonlinear activation function. Function $f _ { \Theta }$ maps $l$ -dimensional input features to $n$ -dimensional output features. + +In this work we consider a two layer GCN with a scalar output per example. This network is a function $F _ { \Theta } : \mathbb { R } ^ { N \times N } \times \mathbb { R } ^ { d \times N } \bar { \mathbb { R } } ^ { N }$ given by + +$$ +F _ { \Theta } ( \tilde { A } , V ) = \sigma ( \Theta _ { 2 } ^ { \top } [ \Theta _ { 1 } ^ { \top } V \tilde { A } ] _ { + } \tilde { A } ) , +$$ + +where $\Theta = \{ \Theta _ { 1 } , \Theta _ { 2 } \}$ , $\Theta _ { 1 } \in \mathbb { R } ^ { d \times m }$ , $\Theta _ { 2 } \in \mathbb { R } ^ { m \times 1 }$ , $[ \cdot ] _ { + }$ is the positive part or ReLU function (Nair & Hinton, 2010) and $\bar { \sigma ( } x ) = ( 1 + e ^ { - x } ) ^ { - 1 }$ for $x \in \mathbb { R }$ is the sigmoid function. Function $F _ { \Theta }$ performs feature propagation through the affinity matrix in an analogy to classical graph-based propagation methods for classification (Zhou et al., 2003a) or search (Zhou et al., 2003b). + +The output $F _ { \Theta } ( \tilde { A } , V )$ is a vector of length $N$ , with element $F _ { \Theta } ( { \tilde { A } } , V ) _ { i }$ in $[ 0 , 1 ]$ representing a relevance value of example $x _ { i }$ for class $c$ . To learn the parameters $\Theta$ , we treat the GCN as a binary classifier where target output 1 corresponds to clean examples and 0 to noisy. In particular, we minimize the loss function + +$$ +L _ { \mathcal { G } } ( V , \tilde { A } ; \Theta ) = - \frac { 1 } { k } \sum _ { i = 1 } ^ { k } \log \left( F _ { \Theta } ( \tilde { A } , V ) _ { i } \right) - \frac { \lambda } { N - k } \sum _ { i = k + 1 } ^ { N } \log \left( 1 - F _ { \Theta } ( \tilde { A } , V ) _ { i } \right) . +$$ + +This is a binary cross-entropy loss function where noisy examples are given an importance weight $\lambda$ . Given the propagation on the nearest neighbor graph, and depending on the relative importance $\lambda$ of the second term, noisy examples that are strongly connected to clean ones are still expected to receive high class relevance, while noisy examples that are not relevant to the current class are expected to get a class relevance near zero. + +The impact of parameter $\lambda$ is validated in Section 6, where we show that the fewer the available clean images are (smaller $k$ ) the smaller the importance weight should be. As is standard practice for GCNs in classification (Kipf & Welling, 2017), training is performed in batches of size $N$ , that is the entire set of features. + +Figure 2 shows examples of clean images, corresponding noisy ones and the predicted relevance. Thanks to the visual similarity to the clean image, we can use relevance to resolve cases of polysemy, e.g. black widow (spider) vs. black widow (superhero), or cases like pineapple vs. pineapple juice. + +Discussion. Loss function (3) is similar to noise-contrastive estimation (NCE) (Gutmann & Hyvärinen, 2010) as used by Wu et al. (2018) for instance-level discrimination, whereas we discriminate clean from noisy examples. The semi-supervised learning setup of GCNs (Kipf & Welling, 2017) uses a loss function that applies only to the labeled examples, and makes discrete predictions on unlabeled examples. In our case, all examples contribute to the loss but with different importance, while we infer real-valued class relevance for the noisy examples, to be used for subsequent learning. + +Function $F _ { \Theta }$ in (2) reduces to a Multi-Layer Perceptron (MLP) when the affinity matrix $A$ is zero, in which case all examples are disconnected. Using an MLP to perform cleaning would take each example into account independently of the others, while the GCN considers the collection of examples as a whole. MLP training is performed identically to GCN by minimizing (3). We compare the two alternatives in our experiments. + +![](images/d562245fb1728f832564e87b5ff05688b13eb6f98c403180ec263fd140ddad27.jpg) +Figure 2: Examples of clean images (left) for 1-shot classification, cumulative histogram of the predicted relevance for noisy images (middle), and representative noisy images (right), each having its position in the (descending) ranked list according to relevance and relevance value reported below. Test accuracy without and with additional data using class prototypes (6) is shown next to class names. + +# 5 LEARNING A CLASSIFIER WITH FEW CLEAN AND MANY NOISY EXAMPLES + +Our cleaning process applies when the clean labeled examples are few, but assumes a feature extractor $g _ { \theta }$ . That is, representation learning, label cleaning and classifier learning are decoupled. We follow few-shot learning in that we learn the representation by supervised classification on a set of base classes, obtaining $g _ { \theta }$ , and then solving new classification tasks on a distinct set of novel classes. In these new tasks, we assume few clean and many noisy labels as specified in Section 3, perform GCN-based cleaning as described in Section 4, and learn a classifier by weighing examples according to class relevance. Representation and classifier learning are described below. + +# 5.1 COSINE-SIMILARITY BASED CLASSIFIER + +We use a cosine-similarity based classifier (Gidaris & Komodakis, 2018; Qi et al., 2018), or cosine classifier for short. Given classes $C$ with $| C | = K$ , each class $c \in C$ is represented by a learnable parameter $\mathbf { w } _ { c } \in \mathbb { R } ^ { d }$ . The prediction of example $x \in \mathcal { X }$ is the class $c$ of maximum cosine similarity $\hat { \mathbf { w } } _ { c } ^ { \top } \hat { g } _ { \boldsymbol { \theta } } ( x ) ^ { 1 }$ + +$$ +\pi _ { \boldsymbol { \theta } , W } ( x ) = \arg \operatorname* { m a x } _ { c } \hat { \mathbf { w } } _ { c } ^ { \top } \hat { g } _ { \boldsymbol { \theta } } ( x ) , +$$ + +where $W = [ \mathbf { w } _ { 1 } , \ j . . . , \mathbf { w } _ { K } ] \in \mathbb { R } ^ { d \times K }$ . + +# 5.2 REPRESENTATION LEARNING: BASE CLASSES + +We are given a set $X _ { B } \subset { \mathcal { X } }$ of examples, each having a clean label in a set of base classes $C _ { B }$ with $| C _ { B } | = K _ { B }$ . These data are used to learn a feature representation, i.e. a feature extractor $g _ { \theta }$ , by learning a $K _ { B }$ -way base-class classifier for unseen data in $\mathcal { X }$ . The parameters $\theta$ of the feature extractor and $W _ { B }$ of the classifier are jointly learned by minimizing the cross entropy loss + +$$ +{ \cal L } _ { \mathcal B } ( C _ { \mathcal B } , X _ { \mathcal B } ; \boldsymbol { \theta } , W _ { \mathcal B } ) = - \sum _ { c \in C _ { \mathcal B } } \frac { 1 } { | X _ { \mathcal B } ^ { c } | } \sum _ { x \in X _ { \mathcal B } ^ { c } } \log ( \sigma ( s \hat { W } _ { \mathcal B } ^ { \top } \hat { g } _ { \boldsymbol \theta } ( x ) ) _ { c } ) , +$$ + +where $\sigma : \mathbb { R } ^ { K } \mathbb { R } ^ { K }$ is the softmax function with $\pmb { \sigma } ( \mathbf { a } ) _ { c } = e ^ { a _ { c } } / \sum _ { j \in C } e ^ { a _ { j } }$ for $\mathbf { a } \in \mathbb { R } ^ { K }$ , $s$ is a learnable scale parameter and $\hat { W } _ { \mathcal { B } } = [ \hat { \mathbf { w } } _ { 1 } , \hdots , \hat { \mathbf { w } } _ { K _ { \mathcal { B } } } ] \in \mathbb { R } ^ { d \times K _ { \mathcal { B } } }$ . Learning and inference are + +performed on base classes by $L _ { B } ( C _ { B } , X _ { B } ; \theta , W _ { B } )$ (5) and $\pi _ { \boldsymbol { \theta } , W _ { B } }$ (4), respectively. As a result, learned feature extractor parameters $\theta$ are used for base or novel classes, while the classifier parameters $W _ { B }$ can be used for base class or all-class classification, as discussed below. + +# 5.3 NEW CLASSIFICATION TASKS: NOVEL OR ALL CLASSES + +Each new task is related to a set of novel classes $C _ { \mathcal { L } }$ , disjoint from $C _ { B }$ . The goal is to learn a $K _ { \mathcal { L } }$ -way novel-class classifier or a $K _ { A }$ -way classifier on all classes $C _ { A } = C _ { B } \cup C _ { \mathcal { L } }$ for unseen data in $\mathcal { X }$ , where $K _ { \mathcal { A } } = K _ { B } + K _ { \mathcal { L } }$ . Unlike the typical few-shot learning task, each novel class contains few clean and many noise examples. + +Prior to learning classifiers for novel classes, training examples $x _ { i } \in X _ { \mathcal { Z } } ^ { c }$ are weighted by their relevance $r ( x _ { i } )$ to class $c$ . For a noisy example $x _ { i } \in X _ { \mathcal { E } } ^ { c }$ , we define $r ( x _ { i } ) = F _ { \Theta } ( \tilde { A } , V ) _ { i }$ where $F _ { \Theta } ( \tilde { A } , V )$ is the output vector of the GCN, while for a clean example $x _ { i } \in X _ { \mathcal { L } } ^ { c }$ we fix $r ( x _ { i } ) = 1$ Note that optimizing (3) does not guarantee $F _ { \Theta } ( \tilde { A } , V ) _ { i } = 1$ for clean examples $x _ { i } \in X _ { \mathcal { L } } ^ { c }$ . We define $\textstyle r ( X ) = \sum _ { x \in X } r ( x )$ for any set $X \subset { \mathcal { X } }$ . + +We first assume that we no longer have access to examples of base classes in new classification tasks and consider two different classifiers, class prototypes and cosine-similarity based classifier. Then, this assumption is dropped and the classifier and feature representation are learned jointly by fine-tuning the entire network. + +Class prototypes. For each novel class $c \in C _ { \mathcal { L } }$ , we define prototype ${ \bf w } _ { c }$ by + +$$ +\mathbf { w } _ { c } = \frac { 1 } { r ( X _ { \mathcal { E } } ^ { c } ) } \sum _ { x \in X _ { \mathcal { E } } ^ { c } } r ( x ) g _ { \theta } ( x ) . +$$ + +Prototypes are fixed vectors, not learnable parameters. Collecting them into matrix $\begin{array} { r l } { W _ { \mathcal { L } } } & { { } = } \end{array}$ $[ \mathbf { w } _ { 1 } , \dots , \mathbf { w } _ { K _ { \mathcal { L } } } ] \in \mathbb { R } ^ { d \times K _ { \mathcal { L } } }$ , $K _ { \mathcal { L } }$ -way prediction on novel classes is made by classifier $\pi _ { \boldsymbol { \theta } , W _ { \mathcal { L } } }$ (4), while $K _ { A }$ -way prediction on all (base and novel) classes by $\pi _ { \boldsymbol { \theta } , W _ { A } }$ , where $W _ { \mathcal { A } } = [ W _ { B } , W _ { \mathcal { L } } ]$ and $W _ { B }$ is learned according to $L _ { B } ( C _ { B } , X _ { B } ; \theta , W _ { B } )$ (5) and then kept fixed. + +Cosine classifier learning. Similarly to Section 5.2, given clean and noisy novel-class examples $X _ { \mathcal { E } }$ , we learn a parametric cosine classifier with parameters $W _ { \mathcal { L } } = [ \mathbf { w } _ { 1 } , \dots , \mathbf { w } _ { K _ { \mathcal { L } } } ] \in \mathbb { R } ^ { d \times K _ { \mathcal { L } } }$ by minimizing the weighted cross entropy loss $L _ { \mathcal { L } } ( C _ { \mathcal { L } } , X _ { \mathcal { E } } ; \theta , W _ { \mathcal { L } } )$ over $W _ { \mathcal { L } }$ , where + +$$ +L _ { \mathcal { L } } ( C _ { \mathcal { L } } , X _ { \mathcal { E } } ; \theta , W _ { \mathcal { L } } ) = - \sum _ { c \in C _ { \mathcal { L } } } \frac { 1 } { r ( X _ { \mathcal { E } } ^ { c } ) } \sum _ { x \in X _ { \mathcal { E } } ^ { c } } r ( x ) \log ( \sigma ( s \hat { W } _ { \mathcal { L } } ^ { \top } \hat { g } _ { \theta } ( x ) ) _ { c } ) , +$$ + +while the parameters $\theta$ of the feature extractor are fixed. The scale parameter $s$ is also fixed to the value obtained during base class learning. Prediction on novel only or all classes is then made as in the previous case. + +Deep network fine-tuning. We now drop the assumption that base class examples are not accessible and, given all examples $X _ { \mathcal { A } } = X _ { \mathcal { B } } \cup X _ { \mathcal { E } }$ , we jointly learn the parameters $\theta$ of the feature extractor and $W _ { \mathcal { A } } = ( W _ { B } , W _ { \mathcal { L } } )$ of the $K _ { A }$ -way cosine classifier for all classes by minimizing loss function + +$$ +L _ { A } ( C _ { A } , X _ { A } ; \theta , W _ { A } ) = L _ { B } ( C _ { B } , X _ { B } ; \theta , W _ { B } ) + L _ { \mathcal { L } } ( C _ { \mathcal { L } } , X _ { \mathcal { E } } ; \theta , W _ { \mathcal { L } } ) . +$$ + +Note that, due to overfitting on the few available examples, such learning is avoided in a few-shot learning setup. In a few cases, it takes the form of fine-tuning including all base class data (Qi et al., 2018), or only lasts for a few iterations when the base class data is not accessible (Finn et al., 2017). + +# 6 EXPERIMENTS + +# 6.1 EXPERIMENTAL SETUP + +Datasets and task setup. We extend the Low-Shot ImageNet benchmark introduced by Hariharan & Girshick (2017) by assuming many noisy examples for novel classes, in addition to the few clean ones. In this benchmark, the 1000 ImageNet classes (Russakovsky et al., 2015) are split into 389 base classes and 611 novel classes. The validation set contains 193 base and 300 novel classes, and the test set the remaining 196 base and 311 novel classes. The standard benchmark includes $k$ -shot classification, i.e. classification on $k$ clean examples per class, which we extend to $k$ clean and many noisy examples per class, with $k \in \{ 1 , 2 , 5 , 1 0 , 2 0 \}$ . Similar to Hariharan & Girshick (2017) we perform 5 tasks, each drawing a subset of $k$ clean examples per class. We report the average top-5 accuracy over the 5 tasks on novel or all classes of the test set. + +We use the YFCC100M dataset (Thomee et al., 2016) as a source for additional data with noisy labels. It contains approximatively 100M images collected from Flickr. Each image comes with a text description obtained from the user title and caption. We use the text description to obtain images with noisy labels. as discussed in Section 3. This process results in very different numbers of additional examples per class, with a minimum of zero for classes maillot and missile, and a maximum of 620,142 for class church/church building. + +Representation and classifier learning. In most experiments, we use ResNet-10 (He et al., 2016) as feature extractor as in Gidaris & Komodakis (2018). Classification for novel classes is performed with class prototypes (6), cosine classifier learning (7) or deep network fine-tuning (8). Hyper-parameters such as batch size and number of epochs, are tuned on the validation set. Possible values are 2048, 4096, and 8192 for batchsize and 10, 30 and 50 for number of epochs. The learning rate starts from 0.1 and is reduced to 0.001 at the end of training with cosine annealing (Loshchilov & Hutter, 2017). + +We handle the imbalance of the noisy set by normalizing by $r ( X _ { c } )$ in (7). Prototypes (6) are used to initialize $W _ { \mathcal { L } }$ of cosine classifier in (7), and the learned $W _ { \mathcal { L } }$ is used to initialize the corresponding part of $W _ { A }$ when fine-tuning the network by (8). In the latter case, we train all layers for 10 epochs with learning rate 0.01. We ignore examples $x _ { i }$ with relevance $r ( x _ { i } ) < 0 . 1$ to reduce the complexity when fine-tuning the network. + +We also report results with ResNet-50 as feature extractor, using the model trained on base classes by Hariharan & Girshick (2017). Following Douze et al. (2018), we apply PCA to the features to reduce their dimensionality to 256. Base classes are represented by class prototypes (6) in this case. + +GCN training is performed with Adam optimizer and a learning rate of 0.1 for 100 iterations. We use dropout with probability 0.5. The dimensionality of the input descriptors is $d = 5 1 2$ for ResNet-10 and $d = 2 5 6$ for ResNet-50 (after PCA). Dimensionality of the internal representation in (1) is $m = 1 6$ . The affinity matrix is constructed with reciprocal top-50 nearest neighbors. + +Baselines. We implement and evaluate several baseline methods. $\beta$ -cleaning assigns $r ( x _ { i } ) = \beta$ to all additional examples. We report results for $\beta = 1 . 0$ (unit relevance score) and $\beta ^ { * }$ , the optimal $\beta$ for all $k$ obtained on the validation set. $M L P$ , discussed in Section 4, learns a nonlinear mapping to assign relevance, but does not propagate over the graph. Label Propagation (LP) (Zhou et al., 2003a) propagates information by a linear operation. It solves the linear system $( I - \alpha D ^ { - 1 / 2 } A D ^ { - 1 / 2 } ) \mathbf { r } _ { c } =$ $\mathbf { y } _ { c }$ (Iscen et al., 2017) for each class $c$ , where $D$ is the degree matrix of $A$ , $\alpha = 0 . 9$ and $\mathbf { y } _ { c } \in \mathbb { R } ^ { N }$ is a $k$ -hot binary vector indicating the labeled examples of class $c$ . Relevance $r ( x _ { i } )$ is then the $i$ -th element $( \mathbf { r } _ { c } ) _ { i }$ of the solution. + +# 6.2 EXPERIMENTAL RESULTS + +The impact of importance weight $\lambda$ is measured on the validation set and the best performing value is used on the test set for each value of $k$ . Results are shown in Appendix A. The larger the value of $\lambda$ , the more the loss encourages noisy examples to be classified as negatives. As a consequence, large (small) $\lambda$ results in smaller (larger) relevance, on average, for noisy examples. The optimal $\lambda$ per value of $k$ suggests that the fewer the clean examples the larger the need for additional ones. + +Comparison with baselines using additional data is presented in Table 1. The use of additional data is mostly harmful for $\beta$ -weighting except for 1 and 2-shot. MLP offers improvements in most cases, implying that it manages to appropriately downweigh irrelevant examples. The consistent improvement of our method compared to MLP, especially large for small $k$ , suggests that it is beneficial to incorporate relations, with the affinity matrix $A$ modeling the structure of the feature space. LP is a classic approach that also uses $A$ but is a linear operation with no parameters, and is inferior to our method. The gain of cleaning ( $\beta = 1$ vs. ours) ranges from $11 \%$ to $20 \%$ . + +Table 1: Comparison with baselines using noisy examples. We report top-5 accuracy on novel classes with classification by class prototypes (6). + +
Methodk=1251020
FEW CLEAN EXAMPLES
Class proto.Gidaris & Komodakis (2018)45.3±0.6557.1±0.3769.3±0.3274.8±0.2077.8±0.24
FEW CLEAN & MANY NOISY EXAMPLES
β-weighting,β=156.1±0.0656.4±0.0857.1±0.0557.7±0.0858.7±0.06
β-weighting,β*55.6±0.2458.3±0.1463.4±0.2567.5±0.3471.0±0.22
Label Propagation Zhou et al. (2003a)62.6±0.3567.0±0.4174.6±0.3076.3±0.2377.7±0.18
MLP63.6±0.4168.8±0.4273.9±0.2575.6±0.2177.6±0.21
Ours67.8±0.1070.9±0.3073.7±0.1776.1±0.1278.2±0.14
+ +
METHODNoVEL CLASSESALL CLASSES
k=1251020k=1251020
RESNET-10 -FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)39.354.4 66.371.273.949.561.069.772.974.6
Logistic reg.w/H(Wang et al.,2018b)40.750.8 62.069.376.552.259.467.672.876.9
PMN w/H(Wang et al., 2018b)45.857.8 69.074.377.457.664.771.975.277.5
Class proto.(Gidaris & Komodakis,2018)45.3±0.6557.1±0.37 69.3±0.32 74.8±0.20 77.8±0.2457.0±0.3664.7±0.1672.5±0.1875.8±0.1677.4±0.19 58.1±0.4865.2±0.15 72.9±0.25 76.6±0.18 78.8±0.16
Class proto.w/At.(Gidaris& Komodakis,2018) 45.8±0.74 57.4±0.38 69.6±0.27 75.0±0.29 78.2±0.23
RESNET-1O -FEW CLEAN& MANY NOISY EXAMPLES
Ours -class proto.(6) Ours-cosine (7)67.8±0.10 70.9±0.30 73.7±0.20 76.1±0.16 78.2±0.1470.3±0.05 72.1±0.18 74.1±0.12 75.6±0.13 76.9±0.09
Ours - fine-tune (8)73.2±0.14 75.3±0.25 75.6±0.24 78.5±0.32 80.7±0.26 74.6±0.13 76.6±0.26 78.2±0.23 80.9±0.34 82.9±0.2071.9±0.0774.0±0.2376.5±0.1678.3±0.2380.2±0.18 76.0±0.1077.3±0.1378.7±0.1980.7±0.2582.2±0.14
RESNET-50 - FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)
PMN w/H(Wang et al.,2018b)49.6 64.0 54.7 66.874.4 77.478.1 81.480.0 83.861.4 65.771.4 73.578.0 80.280.0 82.881.1 84.5
RESNET-5O-FEW CLEAN&MANYUNLABELED EXAMPLES
Diffusion (Douze et al., 2018) Diffusion -logistic (Douze et al.,2018)63.6±0.61 69.5±0.60 75.2±0.40 78.5±0.34 80.8±0.18=-
64.0±0.7071.1±0.82 79.7±0.3883.9±0.10 86.3±0.17 RESNET-5O-FEWCLEAN&MANY NOISY EXAMPLES
Ours - class proto.(6)
Ours - cosine (7)78.0±0.38 80.2±0.3380.9±0.17 83.7±0.1985.7±0.1169.7±0.44 73.7±0.56 77.0±0.20 79.9±0.30 81.9±0.2977.6±0.26 79.1±0.20 79.9±0.09 82.1±0.22 83.8±0.1173.8±0.33 76.6±0.36 78.9±0.19 80.8±0.21 82.2±0.14
Ours - fine-tune (8)80.8±0.25 83.0±0.23 83.8±0.39 86.4±0.23 88.5±0.2081.6±0.20 83.2±0.16 84.3±0.23 86.2±0.17 87.8±0.03
+ +Table 2: Comparison to the state of the art on the Low-shot ImageNet benchmark. We report top-5 accuracy on novel and all classes. We use class prototypes (6), cosine classifier learning (7) and deep network fine-tuning (8) for classification with our GCN-based data addition method. + +Comparison with the state of the art is presented in Table 2. We significantly improve the performance by using additional data and cleaning compared to a number of different approaches, including the work by Gidaris & Komodakis (2018), which is our starting point. As expected, the gain is more pronounced for small $k$ , reaching more than $20 \%$ improvement for 1-shot novel accuracy. + +Closest to ours is the work of Douze et al. (2018), who use the same experimental setup and the same additional data, but without filtering by text and using noisy labels. We outperform their approach in all cases, while requiring much less computation: offline, we construct a separate small graph per class rather than a single graph over the entire 100M collection; online, we perform inference by cosine similarity to one prototype per class or a learned classifier rather than iterative diffusion on the entire collection. Note that by ignoring examples that are not given any noisy label, we are only using a tiny fraction of the 100M collection: in particular, only 3,744,994 images for the 311-class test split of the Low-shot ImageNet benchmark. In contrast to Douze et al. (2018), additional data brings improvement even at 20-shot with classifier learning or network fine-tuning. Most importantly, our approach does not require the entire 100M collection at inference. + +# 7 CONCLUSIONS + +In this paper we have introduced a new method for assigning class relevance to noisy images obtained by textual queries with class names. Our approach leverages one or a few labeled images per class and relies on a graph convolutional network (GCN) to propagate visual information from the labeled images to the noisy ones. The GCN is a binary classifier discriminating clean from noisy examples using a weighted binary cross-entropy loss function and inferring “clean” probability as a relevance measure for that class. Experimental results show that using noisy images weighted by this relevance measure significantly improves the classification accuracy. + +# REFERENCES + +Michael M Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Vandergheynst. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34(4):18–42, 2017. + +Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann Lecun. Spectral networks and locally connected networks on graphs. In ICLR, 2014. + +Wei-Yu Chen, Yen-Cheng Liu, Zsolt Kira, Yu-Chiang Frank Wang, and Jia-Bin Huang. A closer look at few-shot classification. 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In CVPR, July 2017. + +Ricardo Vilalta and Youssef Drissi. A perspective view and survey of meta-learning. Artificial intelligence review, 18(2):77–95, 2002. + +Oriol Vinyals, Charles Blundell, Tim Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In NeurIPS, 2016. + +Xiaolong Wang and Abhinav Gupta. Unsupervised learning of visual representations using videos. In CVPR, 2015. + +Yisen Wang, Weiyang Liu, Xingjun Ma, James Bailey, Hongyuan Zha, Le Song, and Shu-Tao Xia. Iterative learning with open-set noisy labels. In CVPR, 2018a. + +Yu-Xiong Wang, Ross Girshick, Martial Hebert, and Bharath Hariharan. Low-shot learning from imaginary data. In CVPR, 2018b. + +Jason Weston, Frédéric Ratle, and Ronan Collobert. Deep learning via semi-supervised embedding. In ICML, 2008. + +Zhirong Wu, Yuanjun Xiong, Stella Yu, and Dahua Lin. Unsupervised feature learning via non-parametric instance-level discrimination. CVPR, 2018. + +Dengyong Zhou, Olivier Bousquet, Thomas Navin Lal, Jason Weston, and Bernhard Schölkopf. Learning with local and global consistency. In NeurIPS, 2003a. + +Dengyong Zhou, Jason Weston, Arthur Gretton, Olivier Bousquet, and Bernhard Schölkopf. Ranking on data manifolds. In NeurIPS, 2003b. + +![](images/d75584f0f38734e1f60b1588c31cd97d99ef2d5781c2f1c3f7deaf0d2182f12d.jpg) +Figure 3: (a) Number of additional images per class $c$ sampled from YFCC-100M for all novel classes of Low-Shot ImageNet. (b) Number of classes per group, when $| X _ { \mathcal { Z } } ^ { c } |$ is sampled logarithmically into groups. (c) Accuracy improvement $\Delta$ Acc (difference of accuracy between our method with noisy examples and the baseline without noisy examples) for prototype classifier, for same groups as in (b). + +# A APPENDIX + +Noisy data statistics. We present statistics about the noisy examples of novel classes and the improvements of our method per class. Figure 3 (a) shows that the noisy examples for novel classes are long tailed (in log scale). There is a significant number of classes where we end up with less than 1000 extra examples, but we improve nevertheless; see Figure 3 (c). A small exception is 4 very rare classes out of 311, with around 3 additional images per class (leftmost bin in Figure 3 (b) and (c)). Note that in real world applications, one could use more resources like web queries for additional data. + +Impact of importance weight $\lambda ,$ . We present the impact of $\lambda$ (3) for different values of $k$ in the validation set of the extended Low-shot ImageNet benchmark in Figure 4. + +![](images/75723f011baed9f80d3665089fcfcb5e9cf2105defc23b6659f2c9df36dc3084.jpg) +Figure 4: Impact of $\lambda$ on the validation set of the extended Low-shot ImageNet benchmark with YFCC-100M for noisy examples using class prototypes (6). \ No newline at end of file diff --git a/parse/train/Bke6vTVYwH/Bke6vTVYwH_content_list.json b/parse/train/Bke6vTVYwH/Bke6vTVYwH_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a34da47ed3d1bfb2f41c7112d51e879aaf11c0ee --- /dev/null +++ b/parse/train/Bke6vTVYwH/Bke6vTVYwH_content_list.json @@ -0,0 +1,1575 @@ +[ + { + "type": "text", + "text": "GRAPH CONVOLUTIONAL NETWORKS FOR LEARNING WITH FEW CLEAN AND MANY NOISY LABELS ", + "text_level": 1, + "bbox": [ + 174, + 101, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work we consider the problem of learning a classifier from noisy labels when a few clean labeled examples are given. The structure of clean and noisy data is modeled by a graph per class and Graph Convolutional Networks (GCN) are used to predict class relevance of noisy examples. For each class, the GCN is treated as a binary classifier learning to discriminate clean from noisy examples using a weighted binary cross-entropy loss function, and then the GCN-inferred “clean” probability is exploited as a relevance measure. Each noisy example is weighted by its relevance when learning a classifier for the end task. We evaluate our method on an extended version of a few-shot learning problem, where the few clean examples of novel classes are supplemented with additional noisy data. Experimental results show that our GCN-based cleaning process significantly improves the classification accuracy over not cleaning the noisy data and standard few-shot classification where only few clean examples are used. The proposed GCN-based method outperforms the transductive approach (Douze et al., 2018) that is using the same additional data without labels. ", + "bbox": [ + 233, + 263, + 766, + 472 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 496, + 336, + 512 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "State-of-the-art deep learning methods require a large amount of manually labeled data. The need for supervision may be reduced by decoupling representation learning from the end task and/or using additional training data that are unlabeled, weakly labeled (with noisy labels), or belong to different domains or classes. Example approaches are transfer learning (Wang & Gupta, 2015), unsupervised representation learning (Wang & Gupta, 2015), semi-supervised learning (Weston et al., 2008), learning from noisy labels (Joulin et al., 2016) and few-shot learning (Snell et al., 2017). ", + "bbox": [ + 174, + 526, + 825, + 611 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Learning from noisy labels allows using large-scale data and labels from the web without human annotation effort. Most work focuses on learning the representation jointly with the end task, assuming there is still a considerable amount of clean labeled data (Patrini et al., 2017; Lee et al., 2018; Li et al., 2017). However, for a number of classes only very few or even no clean labeled examples might be available at the representation learning stage. Few-shot learning limits the labeled data to very few on the end task, while the representation is learned on a large training set of different classes (Hariharan & Girshick, 2017; Snell et al., 2017; Vinyals et al., 2016). Nevertheless, in many situations, more data with noisy labels are available or can be acquired for the end task. ", + "bbox": [ + 174, + 617, + 825, + 728 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "One interesting mix of few-shot learning with additional large-scale data is the work of Douze et al. (2018), where labels are propagated from few clean labeled examples to a large-scale collection. This collection is unlabeled and actually contains data of many more classes than the end task. Their method overall improves the classification accuracy, but at an additional computational cost; it is a transductive method, i.e., instead of learning a parametric classifier, the large-scale collection is still necessary at inference. ", + "bbox": [ + 174, + 734, + 825, + 819 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we learn a classifier from few clean labeled examples and additional weakly labeled data, while the representation is learned on different classes, as in few-shot learning. We assume the class names are known, and we use them to search an existing large collection of images with textual description. The result is a set of images with potentially relevant, but noisy labels. As shown in Figure 1, we clean this data using a graph convolutional network (GCN) (Kipf & Welling, 2017), which learns to predict a class relevance score per image based on the source (clean vs. noisy) of its connections in the graph. Both the clean and the noisy images are then used to learn a classifier, where the noisy examples are weighted by relevance. Unlike most existing work, our method operates independently per class and applies when clean labeled examples are few or even one per class. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/7eed9d45df54f8a4f9857ca30afe31d8710a4404ec6cdacce1d0af912c397be2.jpg", + "image_caption": [ + "Figure 1: Overview of our cleaning approach for 1-shot learning with noisy examples. We use the class name admiral to crawl noisy images from web and create an adjacency graph based on visual similarity. We then assign a relevance score to each noisy example with a graph convolutional network (GCN). Relevance scores are displayed next to the images. " + ], + "image_footnote": [], + "bbox": [ + 204, + 87, + 782, + 228 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 311, + 821, + 339 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We make the following contributions: ", + "bbox": [ + 176, + 347, + 419, + 361 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We learn a classifier on a large-scale weakly-labeled collection jointly with only few clean labeled examples. \n• To our knowledge, we are the first to use a GCN to clean noisy data: we cast a GCN as a binary classifier learning to discriminate clean from noisy data, and we use its inferred probabilities for the “clean” class as a relevance score per example. \nWe apply our method to a few-shot learning benchmark and show significant improvement in accuracy, while outperforming the method by Douze et al. (2018) using the same large-scale collection of data. ", + "bbox": [ + 215, + 367, + 825, + 479 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 174, + 500, + 341, + 516 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Learning with noisy labels is often concerned with estimating or learning a transition matrix (Natarajan et al., 2013; Patrini et al., 2017; Sukhbaatar et al., 2014) or knowledge graph (Li et al., 2017) between labels and correcting the loss function, which does not apply in our case since the classes in the noisy data are unknown. Most recent work on learning from large-scale weakly-labeled data focuses on learning the representation e.g. by metric learning (Lee et al., 2018; Wang et al., 2018a), bootstrapping (Reed et al., 2015), or distillation (Li et al., 2017). In our case however, since the clean labeled examples are few, we need to keep the representation mostly fixed. ", + "bbox": [ + 173, + 536, + 826, + 633 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Dealing with the noise, e.g. by thresholding (Lee et al., 2018), outlier detection (Wang et al., 2018a) or reweighting (Liu & Tao, 2015), is applicable while the representation is learned, based e.g. on the gradient of the loss (Ren et al., 2018b). In contrast, the relatively-shallow GCN that we propose effectively decouples reweighting from both representation learning and classifier learning. Learning to clean the noisy labels (Veit et al., 2017) typically assumes adequate human verified labels for training, which again is not the case in this work. ", + "bbox": [ + 173, + 640, + 825, + 724 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Few-shot learning. Meta-learning (Vilalta & Drissi, 2002) refers to learning at two levels, where generic knowledge is acquired before adapting to more specific tasks. In few-shot learning, this translates to learning on a set of base classes how to learn from few examples on a distinct set of novel classes without overfitting. For instance, optimization meta-learning (Finn et al., 2017; 2018; Ravi & Larochelle, 2017) amounts to learning a model that is easy to fine-tune in few steps. In our work, we study an extension of few-shot learning where more data are available on novel classes, reducing the risk of overfitting when fine-tuning the model. Metric learning approaches learn how to compare queries for instance to few examples (Vinyals et al., 2016) or to the corresponding class prototypes (Snell et al., 2017). Hariharan & Girshick (2017) and Wang et al. (2018b) learn how to generate novel-class examples, which is not needed when more data are actually available. ", + "bbox": [ + 174, + 734, + 825, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Gidaris & Komodakis (2018) learn on base classes a simpler cosine similarity-based parametric classifier, or simply cosine classifier, without meta-learning. The same classifier has been introduced independently by Qi et al. (2018), who further fine-tune the network, assuming access to the base class training set. A recent survey (Chen et al., 2019) confirms the superiority of the cosine classifier to previous work including meta-learning (Finn et al., 2017). We use the cosine classifier in this work, both for base and novel classes. All of the above use only the few labeled examples of the novel classes. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Making use of unlabeled data has been little explored in few-shot learning until recently. Ren et al. (2018a) introduce a semi-supervised few-shot classification task, where some labels are unknown. Liu et al. (2019) follow the same semi-supervised setup, but use graph-based label propagation (LP) (Zhou et al., 2003a) for classification and consider jointly all test images. These methods assume a meta-learning scenario, where only small-scale data is available at each training episode; arguably, such a small amount of data limits the representation adaptation and generalization to unseen data. Similarly, Rohrbach et al. (2013) use label propagation in a transductive setting, but at a larger scale assuming that all examples come from a set of known classes. Douze et al. (2018) extend to even larger scale, leveraging 100M unlabeled images in a graph without using additional text information. We focus on the latter large-scale scenario using the same 100M dataset. However, we filter by text to obtain noisy labels and follow an inductive approach by training a classifier for novel classes, such that the 100M collection is not needed at inference. ", + "bbox": [ + 173, + 166, + 826, + 333 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Graph neural networks are generalizations of convolutional networks to non-Euclidean spaces (Bronstein et al., 2017). Early spectral methods (Bruna et al., 2014; Henaff et al., 2015) have been succeeded by Chebyshev polynomial approximations (Defferrard et al., 2016), which avoid the high computational cost of computing eigenvectors. Graph convolutional networks (GCN) (Kipf & Welling, 2017) provide a further simplification by a first-order approximation of graph filtering and are applied to semi-supervised (Kipf & Welling, 2017) and subsequently to few-shot learning (Garcia & Bruna, 2018). In Kipf & Welling (2017), the loss function is applied to labeled examples to make predictions on unlabeled ones. Similarly in Garcia & Bruna (2018), GCNs make predictions on novel class examples. Gidaris & Komodakis (2019) use Graph Neural Networks as denoising autoencoders to generate class weights for novel classes. In contrast, we cast GCNs as binary classifiers discriminating clean from noisy examples: we apply a loss function to all examples, and then use the inferred probabilities as a class relevance measure, effectively cleaning the data. ", + "bbox": [ + 173, + 351, + 825, + 517 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our counter-intuitive objective of treating all noisy examples as negative can be compared to treating each example as a different class in instance-level discrimination (Wu et al., 2018). In fact, our loss function is similar to noise-contrastive estimation (NCE) (Gutmann & Hyvärinen, 2010) used in that work. According to our experiments, our GCN-based classifier outperforms classical LP (Zhou et al., 2003a) used for a similar purpose by Rohrbach et al. (2013). ", + "bbox": [ + 174, + 525, + 825, + 594 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PROBLEM FORMULATION ", + "text_level": 1, + "bbox": [ + 176, + 632, + 415, + 647 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We consider a space $\\mathcal { X }$ of examples. We are given a set $X _ { \\mathcal { L } } \\subset \\mathcal { X }$ of examples, each having a clean (manually verified) label in a set $C _ { \\mathcal { L } }$ of classes with $| C _ { \\mathcal { L } } | = K _ { \\mathcal { L } }$ . For any set $X \\subset { \\mathcal { X } }$ , we denote by $X ^ { c }$ its subset of examples having a label in class $c$ . We assume that the number $| X _ { \\mathcal { L } } ^ { c } |$ of examples labeled in each class $c \\in C _ { \\mathcal { L } }$ is only $k$ , typically in $\\{ 1 , 2 , 5 , 1 0 , 2 0 \\}$ . We are also given an additional set $X _ { \\mathcal { Z } } ^ { c }$ of examples, each with a set of noisy labels in $C _ { \\mathcal { L } }$ . The extended set of examples for class $c$ is now $X _ { \\mathcal { E } } ^ { c } = X _ { \\mathcal { L } } ^ { c } \\cup X _ { \\mathcal { Z } } ^ { c }$ . Examples or sets of examples having clean (noisy) labels are referred to as clean (noisy) as well. The goal is to train a classifier, using the additional noisy set in order to improve the accuracy compared to only using the small clean set. ", + "bbox": [ + 174, + 671, + 825, + 784 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We assume that we are given a feature extractor $g _ { \\theta } : \\mathcal { X } \\mathbb { R } ^ { d }$ , mapping an example to a $d .$ -dimensional vector. For instance, when examples are images, the feature extractor is typically a convolutional neural network (CNN) and $\\theta$ are the parameters of all layers. ", + "bbox": [ + 176, + 790, + 823, + 833 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work, we assume that the noisy set $X z$ is collected via web crawling with examples that are images accompanied with free-form text description and/or user tags originating from community photo collections. To make use of text data, we assume that the names of classes in $C _ { \\mathcal { L } }$ are given. An example in $X z$ is given a label in class $c \\in C _ { \\mathcal { L } }$ if its textual information contains the name of class $c$ ; it may then have none, one or more labels. In this way, we automatically infer labels for $X z$ without human effort, which are however noisy. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We perform cleaning by predicting a class relevance measure for each noisy example in $X _ { \\mathcal { Z } } ^ { c }$ , independently per class $c \\in C _ { \\mathcal { L } }$ . To simplify notation, we drop superscript $c$ where possible in this subsection and we denote $X _ { \\mathcal { E } } ^ { c }$ by $\\left\\{ x _ { 1 } , \\ldots , x _ { k } , x _ { k + 1 } , \\ldots , x _ { N } \\right\\}$ , where $X _ { \\mathcal { L } } ^ { c } ~ = ~ \\{ \\bar { x _ { 1 } } , \\ldots , x _ { k } \\}$ and $X _ { \\mathcal { Z } } ^ { c } = \\{ x _ { k + 1 } , \\ldots , x _ { N } \\}$ . The features of these examples are similarly represented by matrix $V = [ \\ b { \\mathrm { v } } _ { 1 } , \\ b { \\mathrm { ~ . ~ . ~ . ~ } } , \\ b { \\mathrm { v } } _ { k } , \\ b { \\mathrm { v } } _ { k + 1 } , \\ b { \\mathrm { ~ . ~ . ~ . ~ } } , \\ b { \\mathrm { v } } _ { N } ] \\in \\mathbb { R } ^ { d \\times N }$ , where $\\mathbf { v } _ { i } = g _ { \\theta } ( x _ { i } )$ for $i = 1 , \\ldots , N$ . ", + "bbox": [ + 174, + 132, + 825, + 207 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We construct an affinity matrix $A \\in \\mathbb { R } ^ { N \\times N }$ with elements $a _ { i j } = [ \\mathbf { v } _ { i } ^ { \\top } \\mathbf { v } _ { j } ] _ { + }$ if examples $\\mathbf { v } _ { i }$ and $\\mathbf { v } _ { j }$ are reciprocal nearest neighbors in $X _ { \\mathcal { E } } ^ { c }$ and 0 otherwise. Matrix $A$ has zero diagonal, but self-connections are added and then $A$ is normalized as $\\tilde { A } = D ^ { - 1 } ( A + I )$ with $D = \\mathrm { d i a g } ( ( A + I ) \\mathbf { 1 } )$ being the degree matrix of $A + I$ and 1 the all-ones vector. ", + "bbox": [ + 174, + 209, + 825, + 270 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Graph convolutional networks (GCNs) (Kipf & Welling, 2017) are formed by a sequence of layers. Each layer is a function $f _ { \\Theta } : \\mathbb { R } ^ { \\hat { N } \\times N } \\times \\mathbb { R } ^ { l \\times \\hat { N } } \\mathbb { R } ^ { n \\times \\hat { N } }$ of the form ", + "bbox": [ + 171, + 276, + 825, + 305 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/13b3539b42a92529ba0733c4bb034de201e83b06236457e097f398a2fd35d52b.jpg", + "text": "$$\nf _ { \\Theta } ( \\tilde { A } , Z ) = h ( \\Theta ^ { \\top } Z \\tilde { A } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 416, + 310, + 578, + 329 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $Z \\in \\mathbb { R } ^ { l \\times N }$ represents the input features, $\\boldsymbol { \\Theta } \\in \\mathbb { R } ^ { l \\times n }$ holds the parameters of the layer to be learned, and $h$ is a nonlinear activation function. Function $f _ { \\Theta }$ maps $l$ -dimensional input features to $n$ -dimensional output features. ", + "bbox": [ + 174, + 333, + 825, + 376 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this work we consider a two layer GCN with a scalar output per example. This network is a function $F _ { \\Theta } : \\mathbb { R } ^ { N \\times N } \\times \\mathbb { R } ^ { d \\times N } \\bar { \\mathbb { R } } ^ { N }$ given by ", + "bbox": [ + 176, + 382, + 821, + 411 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/aa8f3059c25a8e88d03f34ad4a60198132485042f75fe3825f0e8940f8ab17b9.jpg", + "text": "$$\nF _ { \\Theta } ( \\tilde { A } , V ) = \\sigma ( \\Theta _ { 2 } ^ { \\top } [ \\Theta _ { 1 } ^ { \\top } V \\tilde { A } ] _ { + } \\tilde { A } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 416, + 611, + 435 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\Theta = \\{ \\Theta _ { 1 } , \\Theta _ { 2 } \\}$ , $\\Theta _ { 1 } \\in \\mathbb { R } ^ { d \\times m }$ , $\\Theta _ { 2 } \\in \\mathbb { R } ^ { m \\times 1 }$ , $[ \\cdot ] _ { + }$ is the positive part or ReLU function (Nair & Hinton, 2010) and $\\bar { \\sigma ( } x ) = ( 1 + e ^ { - x } ) ^ { - 1 }$ for $x \\in \\mathbb { R }$ is the sigmoid function. Function $F _ { \\Theta }$ performs feature propagation through the affinity matrix in an analogy to classical graph-based propagation methods for classification (Zhou et al., 2003a) or search (Zhou et al., 2003b). ", + "bbox": [ + 174, + 440, + 825, + 500 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The output $F _ { \\Theta } ( \\tilde { A } , V )$ is a vector of length $N$ , with element $F _ { \\Theta } ( { \\tilde { A } } , V ) _ { i }$ in $[ 0 , 1 ]$ representing a relevance value of example $x _ { i }$ for class $c$ . To learn the parameters $\\Theta$ , we treat the GCN as a binary classifier where target output 1 corresponds to clean examples and 0 to noisy. In particular, we minimize the loss function ", + "bbox": [ + 174, + 503, + 825, + 560 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d837015149b4cf45be34663d466e1cbfb336b844aff7679816550bc5ca1f8182.jpg", + "text": "$$\nL _ { \\mathcal { G } } ( V , \\tilde { A } ; \\Theta ) = - \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\log \\left( F _ { \\Theta } ( \\tilde { A } , V ) _ { i } \\right) - \\frac { \\lambda } { N - k } \\sum _ { i = k + 1 } ^ { N } \\log \\left( 1 - F _ { \\Theta } ( \\tilde { A } , V ) _ { i } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 228, + 563, + 767, + 608 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This is a binary cross-entropy loss function where noisy examples are given an importance weight $\\lambda$ . Given the propagation on the nearest neighbor graph, and depending on the relative importance $\\lambda$ of the second term, noisy examples that are strongly connected to clean ones are still expected to receive high class relevance, while noisy examples that are not relevant to the current class are expected to get a class relevance near zero. ", + "bbox": [ + 173, + 612, + 825, + 683 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The impact of parameter $\\lambda$ is validated in Section 6, where we show that the fewer the available clean images are (smaller $k$ ) the smaller the importance weight should be. As is standard practice for GCNs in classification (Kipf & Welling, 2017), training is performed in batches of size $N$ , that is the entire set of features. ", + "bbox": [ + 174, + 689, + 825, + 746 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Figure 2 shows examples of clean images, corresponding noisy ones and the predicted relevance. Thanks to the visual similarity to the clean image, we can use relevance to resolve cases of polysemy, e.g. black widow (spider) vs. black widow (superhero), or cases like pineapple vs. pineapple juice. ", + "bbox": [ + 176, + 752, + 823, + 795 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Discussion. Loss function (3) is similar to noise-contrastive estimation (NCE) (Gutmann & Hyvärinen, 2010) as used by Wu et al. (2018) for instance-level discrimination, whereas we discriminate clean from noisy examples. The semi-supervised learning setup of GCNs (Kipf & Welling, 2017) uses a loss function that applies only to the labeled examples, and makes discrete predictions on unlabeled examples. In our case, all examples contribute to the loss but with different importance, while we infer real-valued class relevance for the noisy examples, to be used for subsequent learning. ", + "bbox": [ + 173, + 804, + 825, + 888 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Function $F _ { \\Theta }$ in (2) reduces to a Multi-Layer Perceptron (MLP) when the affinity matrix $A$ is zero, in which case all examples are disconnected. Using an MLP to perform cleaning would take each example into account independently of the others, while the GCN considers the collection of examples as a whole. MLP training is performed identically to GCN by minimizing (3). We compare the two alternatives in our experiments. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/d562245fb1728f832564e87b5ff05688b13eb6f98c403180ec263fd140ddad27.jpg", + "image_caption": [ + "Figure 2: Examples of clean images (left) for 1-shot classification, cumulative histogram of the predicted relevance for noisy images (middle), and representative noisy images (right), each having its position in the (descending) ranked list according to relevance and relevance value reported below. Test accuracy without and with additional data using class prototypes (6) is shown next to class names. " + ], + "image_footnote": [], + "bbox": [ + 205, + 94, + 790, + 309 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 383, + 825, + 426 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 LEARNING A CLASSIFIER WITH FEW CLEAN AND MANY NOISY EXAMPLES", + "text_level": 1, + "bbox": [ + 173, + 446, + 820, + 462 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our cleaning process applies when the clean labeled examples are few, but assumes a feature extractor $g _ { \\theta }$ . That is, representation learning, label cleaning and classifier learning are decoupled. We follow few-shot learning in that we learn the representation by supervised classification on a set of base classes, obtaining $g _ { \\theta }$ , and then solving new classification tasks on a distinct set of novel classes. In these new tasks, we assume few clean and many noisy labels as specified in Section 3, perform GCN-based cleaning as described in Section 4, and learn a classifier by weighing examples according to class relevance. Representation and classifier learning are described below. ", + "bbox": [ + 173, + 477, + 825, + 575 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.1 COSINE-SIMILARITY BASED CLASSIFIER ", + "text_level": 1, + "bbox": [ + 174, + 592, + 493, + 606 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use a cosine-similarity based classifier (Gidaris & Komodakis, 2018; Qi et al., 2018), or cosine classifier for short. Given classes $C$ with $| C | = K$ , each class $c \\in C$ is represented by a learnable parameter $\\mathbf { w } _ { c } \\in \\mathbb { R } ^ { d }$ . The prediction of example $x \\in \\mathcal { X }$ is the class $c$ of maximum cosine similarity $\\hat { \\mathbf { w } } _ { c } ^ { \\top } \\hat { g } _ { \\boldsymbol { \\theta } } ( x ) ^ { 1 }$ ", + "bbox": [ + 173, + 616, + 825, + 674 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/40e6a0f31bb15db55ac4c34edda9ebe293e0f22aac7359518e37b8c95fccd4b7.jpg", + "text": "$$\n\\pi _ { \\boldsymbol { \\theta } , W } ( x ) = \\arg \\operatorname* { m a x } _ { c } \\hat { \\mathbf { w } } _ { c } ^ { \\top } \\hat { g } _ { \\boldsymbol { \\theta } } ( x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 393, + 671, + 601, + 695 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $W = [ \\mathbf { w } _ { 1 } , \\ j . . . , \\mathbf { w } _ { K } ] \\in \\mathbb { R } ^ { d \\times K }$ . ", + "bbox": [ + 174, + 699, + 416, + 715 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.2 REPRESENTATION LEARNING: BASE CLASSES ", + "text_level": 1, + "bbox": [ + 173, + 731, + 529, + 746 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We are given a set $X _ { B } \\subset { \\mathcal { X } }$ of examples, each having a clean label in a set of base classes $C _ { B }$ with $| C _ { B } | = K _ { B }$ . These data are used to learn a feature representation, i.e. a feature extractor $g _ { \\theta }$ , by learning a $K _ { B }$ -way base-class classifier for unseen data in $\\mathcal { X }$ . The parameters $\\theta$ of the feature extractor and $W _ { B }$ of the classifier are jointly learned by minimizing the cross entropy loss ", + "bbox": [ + 174, + 756, + 826, + 814 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ae61401e91b7b7148fa78affb508b0916a0836a3c71827c7dbf2006dd9558870.jpg", + "text": "$$\n{ \\cal L } _ { \\mathcal B } ( C _ { \\mathcal B } , X _ { \\mathcal B } ; \\boldsymbol { \\theta } , W _ { \\mathcal B } ) = - \\sum _ { c \\in C _ { \\mathcal B } } \\frac { 1 } { | X _ { \\mathcal B } ^ { c } | } \\sum _ { x \\in X _ { \\mathcal B } ^ { c } } \\log ( \\sigma ( s \\hat { W } _ { \\mathcal B } ^ { \\top } \\hat { g } _ { \\boldsymbol \\theta } ( x ) ) _ { c } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 282, + 818, + 712, + 858 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\sigma : \\mathbb { R } ^ { K } \\mathbb { R } ^ { K }$ is the softmax function with $\\pmb { \\sigma } ( \\mathbf { a } ) _ { c } = e ^ { a _ { c } } / \\sum _ { j \\in C } e ^ { a _ { j } }$ for $\\mathbf { a } \\in \\mathbb { R } ^ { K }$ , $s$ is a learnable scale parameter and $\\hat { W } _ { \\mathcal { B } } = [ \\hat { \\mathbf { w } } _ { 1 } , \\hdots , \\hat { \\mathbf { w } } _ { K _ { \\mathcal { B } } } ] \\in \\mathbb { R } ^ { d \\times K _ { \\mathcal { B } } }$ . Learning and inference are ", + "bbox": [ + 174, + 864, + 825, + 901 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "performed on base classes by $L _ { B } ( C _ { B } , X _ { B } ; \\theta , W _ { B } )$ (5) and $\\pi _ { \\boldsymbol { \\theta } , W _ { B } }$ (4), respectively. As a result, learned feature extractor parameters $\\theta$ are used for base or novel classes, while the classifier parameters $W _ { B }$ can be used for base class or all-class classification, as discussed below. ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.3 NEW CLASSIFICATION TASKS: NOVEL OR ALL CLASSES ", + "text_level": 1, + "bbox": [ + 173, + 162, + 596, + 176 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Each new task is related to a set of novel classes $C _ { \\mathcal { L } }$ , disjoint from $C _ { B }$ . The goal is to learn a $K _ { \\mathcal { L } }$ -way novel-class classifier or a $K _ { A }$ -way classifier on all classes $C _ { A } = C _ { B } \\cup C _ { \\mathcal { L } }$ for unseen data in $\\mathcal { X }$ , where $K _ { \\mathcal { A } } = K _ { B } + K _ { \\mathcal { L } }$ . Unlike the typical few-shot learning task, each novel class contains few clean and many noise examples. ", + "bbox": [ + 174, + 188, + 826, + 244 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Prior to learning classifiers for novel classes, training examples $x _ { i } \\in X _ { \\mathcal { Z } } ^ { c }$ are weighted by their relevance $r ( x _ { i } )$ to class $c$ . For a noisy example $x _ { i } \\in X _ { \\mathcal { E } } ^ { c }$ , we define $r ( x _ { i } ) = F _ { \\Theta } ( \\tilde { A } , V ) _ { i }$ where $F _ { \\Theta } ( \\tilde { A } , V )$ is the output vector of the GCN, while for a clean example $x _ { i } \\in X _ { \\mathcal { L } } ^ { c }$ we fix $r ( x _ { i } ) = 1$ Note that optimizing (3) does not guarantee $F _ { \\Theta } ( \\tilde { A } , V ) _ { i } = 1$ for clean examples $x _ { i } \\in X _ { \\mathcal { L } } ^ { c }$ . We define $\\textstyle r ( X ) = \\sum _ { x \\in X } r ( x )$ for any set $X \\subset { \\mathcal { X } }$ . ", + "bbox": [ + 173, + 251, + 825, + 330 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We first assume that we no longer have access to examples of base classes in new classification tasks and consider two different classifiers, class prototypes and cosine-similarity based classifier. Then, this assumption is dropped and the classifier and feature representation are learned jointly by fine-tuning the entire network. ", + "bbox": [ + 174, + 335, + 825, + 392 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Class prototypes. For each novel class $c \\in C _ { \\mathcal { L } }$ , we define prototype ${ \\bf w } _ { c }$ by ", + "bbox": [ + 173, + 401, + 666, + 417 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/9d641c429881b24c6da69fdf3b5c2603b24182d5a2b4595d7fba7984967050ee.jpg", + "text": "$$\n\\mathbf { w } _ { c } = \\frac { 1 } { r ( X _ { \\mathcal { E } } ^ { c } ) } \\sum _ { x \\in X _ { \\mathcal { E } } ^ { c } } r ( x ) g _ { \\theta } ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 424, + 601, + 463 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Prototypes are fixed vectors, not learnable parameters. Collecting them into matrix $\\begin{array} { r l } { W _ { \\mathcal { L } } } & { { } = } \\end{array}$ $[ \\mathbf { w } _ { 1 } , \\dots , \\mathbf { w } _ { K _ { \\mathcal { L } } } ] \\in \\mathbb { R } ^ { d \\times K _ { \\mathcal { L } } }$ , $K _ { \\mathcal { L } }$ -way prediction on novel classes is made by classifier $\\pi _ { \\boldsymbol { \\theta } , W _ { \\mathcal { L } } }$ (4), while $K _ { A }$ -way prediction on all (base and novel) classes by $\\pi _ { \\boldsymbol { \\theta } , W _ { A } }$ , where $W _ { \\mathcal { A } } = [ W _ { B } , W _ { \\mathcal { L } } ]$ and $W _ { B }$ is learned according to $L _ { B } ( C _ { B } , X _ { B } ; \\theta , W _ { B } )$ (5) and then kept fixed. ", + "bbox": [ + 174, + 469, + 825, + 526 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Cosine classifier learning. Similarly to Section 5.2, given clean and noisy novel-class examples $X _ { \\mathcal { E } }$ , we learn a parametric cosine classifier with parameters $W _ { \\mathcal { L } } = [ \\mathbf { w } _ { 1 } , \\dots , \\mathbf { w } _ { K _ { \\mathcal { L } } } ] \\in \\mathbb { R } ^ { d \\times K _ { \\mathcal { L } } }$ by minimizing the weighted cross entropy loss $L _ { \\mathcal { L } } ( C _ { \\mathcal { L } } , X _ { \\mathcal { E } } ; \\theta , W _ { \\mathcal { L } } )$ over $W _ { \\mathcal { L } }$ , where ", + "bbox": [ + 174, + 535, + 825, + 579 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/9d9cc6a68fc31b95d6d72f3677cd4daa097ce6fca886266a36bbf086a956fc14.jpg", + "text": "$$\nL _ { \\mathcal { L } } ( C _ { \\mathcal { L } } , X _ { \\mathcal { E } } ; \\theta , W _ { \\mathcal { L } } ) = - \\sum _ { c \\in C _ { \\mathcal { L } } } \\frac { 1 } { r ( X _ { \\mathcal { E } } ^ { c } ) } \\sum _ { x \\in X _ { \\mathcal { E } } ^ { c } } r ( x ) \\log ( \\sigma ( s \\hat { W } _ { \\mathcal { L } } ^ { \\top } \\hat { g } _ { \\theta } ( x ) ) _ { c } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 263, + 585, + 735, + 626 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "while the parameters $\\theta$ of the feature extractor are fixed. The scale parameter $s$ is also fixed to the value obtained during base class learning. Prediction on novel only or all classes is then made as in the previous case. ", + "bbox": [ + 174, + 632, + 825, + 674 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Deep network fine-tuning. We now drop the assumption that base class examples are not accessible and, given all examples $X _ { \\mathcal { A } } = X _ { \\mathcal { B } } \\cup X _ { \\mathcal { E } }$ , we jointly learn the parameters $\\theta$ of the feature extractor and $W _ { \\mathcal { A } } = ( W _ { B } , W _ { \\mathcal { L } } )$ of the $K _ { A }$ -way cosine classifier for all classes by minimizing loss function ", + "bbox": [ + 173, + 684, + 825, + 728 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/379610d627fb0113437714afc09f9e9a7331a93b4bb191ea3d1aaf2e77db7ced.jpg", + "text": "$$\nL _ { A } ( C _ { A } , X _ { A } ; \\theta , W _ { A } ) = L _ { B } ( C _ { B } , X _ { B } ; \\theta , W _ { B } ) + L _ { \\mathcal { L } } ( C _ { \\mathcal { L } } , X _ { \\mathcal { E } } ; \\theta , W _ { \\mathcal { L } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 271, + 733, + 725, + 752 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note that, due to overfitting on the few available examples, such learning is avoided in a few-shot learning setup. In a few cases, it takes the form of fine-tuning including all base class data (Qi et al., 2018), or only lasts for a few iterations when the base class data is not accessible (Finn et al., 2017). ", + "bbox": [ + 174, + 757, + 825, + 800 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 820, + 326, + 837 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6.1 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 174, + 852, + 374, + 866 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Datasets and task setup. We extend the Low-Shot ImageNet benchmark introduced by Hariharan & Girshick (2017) by assuming many noisy examples for novel classes, in addition to the few clean ones. In this benchmark, the 1000 ImageNet classes (Russakovsky et al., 2015) are split into 389 base classes and 611 novel classes. The validation set contains 193 base and 300 novel classes, and the test set the remaining 196 base and 311 novel classes. The standard benchmark includes $k$ -shot classification, i.e. classification on $k$ clean examples per class, which we extend to $k$ clean and many noisy examples per class, with $k \\in \\{ 1 , 2 , 5 , 1 0 , 2 0 \\}$ . Similar to Hariharan & Girshick (2017) we perform 5 tasks, each drawing a subset of $k$ clean examples per class. We report the average top-5 accuracy over the 5 tasks on novel or all classes of the test set. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 186 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use the YFCC100M dataset (Thomee et al., 2016) as a source for additional data with noisy labels. It contains approximatively 100M images collected from Flickr. Each image comes with a text description obtained from the user title and caption. We use the text description to obtain images with noisy labels. as discussed in Section 3. This process results in very different numbers of additional examples per class, with a minimum of zero for classes maillot and missile, and a maximum of 620,142 for class church/church building. ", + "bbox": [ + 174, + 194, + 825, + 279 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Representation and classifier learning. In most experiments, we use ResNet-10 (He et al., 2016) as feature extractor as in Gidaris & Komodakis (2018). Classification for novel classes is performed with class prototypes (6), cosine classifier learning (7) or deep network fine-tuning (8). Hyper-parameters such as batch size and number of epochs, are tuned on the validation set. Possible values are 2048, 4096, and 8192 for batchsize and 10, 30 and 50 for number of epochs. The learning rate starts from 0.1 and is reduced to 0.001 at the end of training with cosine annealing (Loshchilov & Hutter, 2017). ", + "bbox": [ + 174, + 290, + 825, + 375 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We handle the imbalance of the noisy set by normalizing by $r ( X _ { c } )$ in (7). Prototypes (6) are used to initialize $W _ { \\mathcal { L } }$ of cosine classifier in (7), and the learned $W _ { \\mathcal { L } }$ is used to initialize the corresponding part of $W _ { A }$ when fine-tuning the network by (8). In the latter case, we train all layers for 10 epochs with learning rate 0.01. We ignore examples $x _ { i }$ with relevance $r ( x _ { i } ) < 0 . 1$ to reduce the complexity when fine-tuning the network. ", + "bbox": [ + 174, + 381, + 825, + 450 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We also report results with ResNet-50 as feature extractor, using the model trained on base classes by Hariharan & Girshick (2017). Following Douze et al. (2018), we apply PCA to the features to reduce their dimensionality to 256. Base classes are represented by class prototypes (6) in this case. ", + "bbox": [ + 174, + 457, + 825, + 500 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "GCN training is performed with Adam optimizer and a learning rate of 0.1 for 100 iterations. We use dropout with probability 0.5. The dimensionality of the input descriptors is $d = 5 1 2$ for ResNet-10 and $d = 2 5 6$ for ResNet-50 (after PCA). Dimensionality of the internal representation in (1) is $m = 1 6$ . The affinity matrix is constructed with reciprocal top-50 nearest neighbors. ", + "bbox": [ + 174, + 506, + 825, + 563 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Baselines. We implement and evaluate several baseline methods. $\\beta$ -cleaning assigns $r ( x _ { i } ) = \\beta$ to all additional examples. We report results for $\\beta = 1 . 0$ (unit relevance score) and $\\beta ^ { * }$ , the optimal $\\beta$ for all $k$ obtained on the validation set. $M L P$ , discussed in Section 4, learns a nonlinear mapping to assign relevance, but does not propagate over the graph. Label Propagation (LP) (Zhou et al., 2003a) propagates information by a linear operation. It solves the linear system $( I - \\alpha D ^ { - 1 / 2 } A D ^ { - 1 / 2 } ) \\mathbf { r } _ { c } =$ $\\mathbf { y } _ { c }$ (Iscen et al., 2017) for each class $c$ , where $D$ is the degree matrix of $A$ , $\\alpha = 0 . 9$ and $\\mathbf { y } _ { c } \\in \\mathbb { R } ^ { N }$ is a $k$ -hot binary vector indicating the labeled examples of class $c$ . Relevance $r ( x _ { i } )$ is then the $i$ -th element $( \\mathbf { r } _ { c } ) _ { i }$ of the solution. ", + "bbox": [ + 174, + 575, + 825, + 689 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.2 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 710, + 390, + 724 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The impact of importance weight $\\lambda$ is measured on the validation set and the best performing value is used on the test set for each value of $k$ . Results are shown in Appendix A. The larger the value of $\\lambda$ , the more the loss encourages noisy examples to be classified as negatives. As a consequence, large (small) $\\lambda$ results in smaller (larger) relevance, on average, for noisy examples. The optimal $\\lambda$ per value of $k$ suggests that the fewer the clean examples the larger the need for additional ones. ", + "bbox": [ + 174, + 743, + 825, + 814 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison with baselines using additional data is presented in Table 1. The use of additional data is mostly harmful for $\\beta$ -weighting except for 1 and 2-shot. MLP offers improvements in most cases, implying that it manages to appropriately downweigh irrelevant examples. The consistent improvement of our method compared to MLP, especially large for small $k$ , suggests that it is beneficial to incorporate relations, with the affinity matrix $A$ modeling the structure of the feature space. LP is a classic approach that also uses $A$ but is a linear operation with no parameters, and is inferior to our method. The gain of cleaning ( $\\beta = 1$ vs. ours) ranges from $11 \\%$ to $20 \\%$ . ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/a459708462385f825bec2f32d3a41cf64b24997f7906997386decb7226e7f454.jpg", + "table_caption": [ + "Table 1: Comparison with baselines using noisy examples. We report top-5 accuracy on novel classes with classification by class prototypes (6). " + ], + "table_footnote": [], + "table_body": "
Methodk=1251020
FEW CLEAN EXAMPLES
Class proto.Gidaris & Komodakis (2018)45.3±0.6557.1±0.3769.3±0.3274.8±0.2077.8±0.24
FEW CLEAN & MANY NOISY EXAMPLES
β-weighting,β=156.1±0.0656.4±0.0857.1±0.0557.7±0.0858.7±0.06
β-weighting,β*55.6±0.2458.3±0.1463.4±0.2567.5±0.3471.0±0.22
Label Propagation Zhou et al. (2003a)62.6±0.3567.0±0.4174.6±0.3076.3±0.2377.7±0.18
MLP63.6±0.4168.8±0.4273.9±0.2575.6±0.2177.6±0.21
Ours67.8±0.1070.9±0.3073.7±0.1776.1±0.1278.2±0.14
", + "bbox": [ + 225, + 101, + 761, + 224 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/2b93b35a59e17c23022787514572c2915798f4e6c89bd4a24615df6a04cefb60.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
METHODNoVEL CLASSESALL CLASSES
k=1251020k=1251020
RESNET-10 -FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)39.354.4 66.371.273.949.561.069.772.974.6
Logistic reg.w/H(Wang et al.,2018b)40.750.8 62.069.376.552.259.467.672.876.9
PMN w/H(Wang et al., 2018b)45.857.8 69.074.377.457.664.771.975.277.5
Class proto.(Gidaris & Komodakis,2018)45.3±0.6557.1±0.37 69.3±0.32 74.8±0.20 77.8±0.2457.0±0.3664.7±0.1672.5±0.1875.8±0.1677.4±0.19 58.1±0.4865.2±0.15 72.9±0.25 76.6±0.18 78.8±0.16
Class proto.w/At.(Gidaris& Komodakis,2018) 45.8±0.74 57.4±0.38 69.6±0.27 75.0±0.29 78.2±0.23
RESNET-1O -FEW CLEAN& MANY NOISY EXAMPLES
Ours -class proto.(6) Ours-cosine (7)67.8±0.10 70.9±0.30 73.7±0.20 76.1±0.16 78.2±0.1470.3±0.05 72.1±0.18 74.1±0.12 75.6±0.13 76.9±0.09
Ours - fine-tune (8)73.2±0.14 75.3±0.25 75.6±0.24 78.5±0.32 80.7±0.26 74.6±0.13 76.6±0.26 78.2±0.23 80.9±0.34 82.9±0.2071.9±0.0774.0±0.2376.5±0.1678.3±0.2380.2±0.18 76.0±0.1077.3±0.1378.7±0.1980.7±0.2582.2±0.14
RESNET-50 - FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)
PMN w/H(Wang et al.,2018b)49.6 64.0 54.7 66.874.4 77.478.1 81.480.0 83.861.4 65.771.4 73.578.0 80.280.0 82.881.1 84.5
RESNET-5O-FEW CLEAN&MANYUNLABELED EXAMPLES
Diffusion (Douze et al., 2018) Diffusion -logistic (Douze et al.,2018)63.6±0.61 69.5±0.60 75.2±0.40 78.5±0.34 80.8±0.18=-
64.0±0.7071.1±0.82 79.7±0.3883.9±0.10 86.3±0.17 RESNET-5O-FEWCLEAN&MANY NOISY EXAMPLES
Ours - class proto.(6)
Ours - cosine (7)78.0±0.38 80.2±0.3380.9±0.17 83.7±0.1985.7±0.1169.7±0.44 73.7±0.56 77.0±0.20 79.9±0.30 81.9±0.2977.6±0.26 79.1±0.20 79.9±0.09 82.1±0.22 83.8±0.1173.8±0.33 76.6±0.36 78.9±0.19 80.8±0.21 82.2±0.14
Ours - fine-tune (8)80.8±0.25 83.0±0.23 83.8±0.39 86.4±0.23 88.5±0.2081.6±0.20 83.2±0.16 84.3±0.23 86.2±0.17 87.8±0.03
", + "bbox": [ + 178, + 262, + 820, + 512 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 2: Comparison to the state of the art on the Low-shot ImageNet benchmark. We report top-5 accuracy on novel and all classes. We use class prototypes (6), cosine classifier learning (7) and deep network fine-tuning (8) for classification with our GCN-based data addition method. ", + "bbox": [ + 176, + 517, + 821, + 559 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Comparison with the state of the art is presented in Table 2. We significantly improve the performance by using additional data and cleaning compared to a number of different approaches, including the work by Gidaris & Komodakis (2018), which is our starting point. As expected, the gain is more pronounced for small $k$ , reaching more than $20 \\%$ improvement for 1-shot novel accuracy. ", + "bbox": [ + 174, + 565, + 825, + 621 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Closest to ours is the work of Douze et al. (2018), who use the same experimental setup and the same additional data, but without filtering by text and using noisy labels. We outperform their approach in all cases, while requiring much less computation: offline, we construct a separate small graph per class rather than a single graph over the entire 100M collection; online, we perform inference by cosine similarity to one prototype per class or a learned classifier rather than iterative diffusion on the entire collection. Note that by ignoring examples that are not given any noisy label, we are only using a tiny fraction of the 100M collection: in particular, only 3,744,994 images for the 311-class test split of the Low-shot ImageNet benchmark. In contrast to Douze et al. (2018), additional data brings improvement even at 20-shot with classifier learning or network fine-tuning. Most importantly, our approach does not require the entire 100M collection at inference. ", + "bbox": [ + 174, + 628, + 825, + 767 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 787, + 328, + 804 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this paper we have introduced a new method for assigning class relevance to noisy images obtained by textual queries with class names. Our approach leverages one or a few labeled images per class and relies on a graph convolutional network (GCN) to propagate visual information from the labeled images to the noisy ones. The GCN is a binary classifier discriminating clean from noisy examples using a weighted binary cross-entropy loss function and inferring “clean” probability as a relevance measure for that class. Experimental results show that using noisy images weighted by this relevance measure significantly improves the classification accuracy. 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", + "bbox": [ + 171, + 888, + 823, + 916 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/d75584f0f38734e1f60b1588c31cd97d99ef2d5781c2f1c3f7deaf0d2182f12d.jpg", + "image_caption": [ + "Figure 3: (a) Number of additional images per class $c$ sampled from YFCC-100M for all novel classes of Low-Shot ImageNet. (b) Number of classes per group, when $| X _ { \\mathcal { Z } } ^ { c } |$ is sampled logarithmically into groups. (c) Accuracy improvement $\\Delta$ Acc (difference of accuracy between our method with noisy examples and the baseline without noisy examples) for prototype classifier, for same groups as in (b). " + ], + "image_footnote": [], + "bbox": [ + 184, + 99, + 828, + 250 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 342, + 297, + 357 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Noisy data statistics. We present statistics about the noisy examples of novel classes and the improvements of our method per class. Figure 3 (a) shows that the noisy examples for novel classes are long tailed (in log scale). There is a significant number of classes where we end up with less than 1000 extra examples, but we improve nevertheless; see Figure 3 (c). A small exception is 4 very rare classes out of 311, with around 3 additional images per class (leftmost bin in Figure 3 (b) and (c)). Note that in real world applications, one could use more resources like web queries for additional data. ", + "bbox": [ + 173, + 376, + 825, + 474 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Impact of importance weight $\\lambda ,$ . We present the impact of $\\lambda$ (3) for different values of $k$ in the validation set of the extended Low-shot ImageNet benchmark in Figure 4. ", + "bbox": [ + 171, + 497, + 825, + 526 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/75723f011baed9f80d3665089fcfcb5e9cf2105defc23b6659f2c9df36dc3084.jpg", + "image_caption": [ + "Figure 4: Impact of $\\lambda$ on the validation set of the extended Low-shot ImageNet benchmark with YFCC-100M for noisy examples using class prototypes (6). " + ], + "image_footnote": [], + "bbox": [ + 321, + 554, + 676, + 656 + ], + "page_idx": 10 + } +] \ No newline at end of file diff --git a/parse/train/Bke6vTVYwH/Bke6vTVYwH_middle.json b/parse/train/Bke6vTVYwH/Bke6vTVYwH_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..2f57c046480d3f5867c000977e1864d2563401ac --- /dev/null +++ b/parse/train/Bke6vTVYwH/Bke6vTVYwH_middle.json @@ -0,0 +1,33758 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 80, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 99 + ], + "score": 1.0, + "content": "GRAPH CONVOLUTIONAL NETWORKS FOR LEARNING", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 102, + 436, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 436, + 118 + ], + "score": 1.0, + "content": "WITH FEW CLEAN AND MANY NOISY LABELS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 209, + 469, + 374 + ], + "lines": [ + { + "bbox": [ + 141, + 209, + 469, + 222 + ], + "spans": [ + { + "bbox": [ + 141, + 209, + 469, + 222 + ], + "score": 1.0, + "content": "In this work we consider the problem of learning a classifier from noisy labels when", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "score": 1.0, + "content": "a few clean labeled examples are given. The structure of clean and noisy data is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 232, + 470, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 232, + 470, + 244 + ], + "score": 1.0, + "content": "modeled by a graph per class and Graph Convolutional Networks (GCN) are used", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "score": 1.0, + "content": "to predict class relevance of noisy examples. 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Each noisy example is weighted by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "score": 1.0, + "content": "its relevance when learning a classifier for the end task. We evaluate our method on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 469, + 309 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 469, + 309 + ], + "score": 1.0, + "content": "an extended version of a few-shot learning problem, where the few clean examples", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 469, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 320 + ], + "score": 1.0, + "content": "of novel classes are supplemented with additional noisy data. 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The proposed GCN-based method outperforms", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 352, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 352, + 469, + 365 + ], + "score": 1.0, + "content": "the transductive approach (Douze et al., 2018) that is using the same additional", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 364, + 222, + 375 + ], + "spans": [ + { + "bbox": [ + 142, + 364, + 222, + 375 + ], + "score": 1.0, + "content": "data without labels.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "State-of-the-art deep learning methods require a large amount of manually labeled data. The need for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "supervision may be reduced by decoupling representation learning from the end task and/or using", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "additional training data that are unlabeled, weakly labeled (with noisy labels), or belong to different", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "domains or classes. Example approaches are transfer learning (Wang & Gupta, 2015), unsupervised", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 507, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 507, + 474 + ], + "score": 1.0, + "content": "representation learning (Wang & Gupta, 2015), semi-supervised learning (Weston et al., 2008),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 472, + 461, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 461, + 484 + ], + "score": 1.0, + "content": "learning from noisy labels (Joulin et al., 2016) and few-shot learning (Snell et al., 2017).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "Learning from noisy labels allows using large-scale data and labels from the web without human", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "annotation effort. Most work focuses on learning the representation jointly with the end task, assuming", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "there is still a considerable amount of clean labeled data (Patrini et al., 2017; Lee et al., 2018; Li", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "et al., 2017). However, for a number of classes only very few or even no clean labeled examples", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "might be available at the representation learning stage. Few-shot learning limits the labeled data", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 556 + ], + "score": 1.0, + "content": "to very few on the end task, while the representation is learned on a large training set of different", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "classes (Hariharan & Girshick, 2017; Snell et al., 2017; Vinyals et al., 2016). 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This", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "collection is unlabeled and actually contains data of many more classes than the end task. 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We assume", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "the class names are known, and we use them to search an existing large collection of images with", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "textual description. The result is a set of images with potentially relevant, but noisy labels. 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The structure of clean and noisy data is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 232, + 470, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 232, + 470, + 244 + ], + "score": 1.0, + "content": "modeled by a graph per class and Graph Convolutional Networks (GCN) are used", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "score": 1.0, + "content": "to predict class relevance of noisy examples. For each class, the GCN is treated", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 253, + 470, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 253, + 470, + 266 + ], + "score": 1.0, + "content": "as a binary classifier learning to discriminate clean from noisy examples using a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 264, + 471, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 264, + 471, + 277 + ], + "score": 1.0, + "content": "weighted binary cross-entropy loss function, and then the GCN-inferred “clean”", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 469, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 469, + 288 + ], + "score": 1.0, + "content": "probability is exploited as a relevance measure. Each noisy example is weighted by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "score": 1.0, + "content": "its relevance when learning a classifier for the end task. We evaluate our method on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 469, + 309 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 469, + 309 + ], + "score": 1.0, + "content": "an extended version of a few-shot learning problem, where the few clean examples", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 469, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 320 + ], + "score": 1.0, + "content": "of novel classes are supplemented with additional noisy data. Experimental results", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 319, + 469, + 331 + ], + "spans": [ + { + "bbox": [ + 141, + 319, + 469, + 331 + ], + "score": 1.0, + "content": "show that our GCN-based cleaning process significantly improves the classification", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 469, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 469, + 342 + ], + "score": 1.0, + "content": "accuracy over not cleaning the noisy data and standard few-shot classification where", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 341, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 341, + 469, + 354 + ], + "score": 1.0, + "content": "only few clean examples are used. The proposed GCN-based method outperforms", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 352, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 352, + 469, + 365 + ], + "score": 1.0, + "content": "the transductive approach (Douze et al., 2018) that is using the same additional", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 364, + 222, + 375 + ], + "spans": [ + { + "bbox": [ + 142, + 364, + 222, + 375 + ], + "score": 1.0, + "content": "data without labels.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 209, + 471, + 375 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "State-of-the-art deep learning methods require a large amount of manually labeled data. The need for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "supervision may be reduced by decoupling representation learning from the end task and/or using", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "additional training data that are unlabeled, weakly labeled (with noisy labels), or belong to different", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "domains or classes. Example approaches are transfer learning (Wang & Gupta, 2015), unsupervised", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 507, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 507, + 474 + ], + "score": 1.0, + "content": "representation learning (Wang & Gupta, 2015), semi-supervised learning (Weston et al., 2008),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 472, + 461, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 461, + 484 + ], + "score": 1.0, + "content": "learning from noisy labels (Joulin et al., 2016) and few-shot learning (Snell et al., 2017).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 417, + 507, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "Learning from noisy labels allows using large-scale data and labels from the web without human", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "annotation effort. Most work focuses on learning the representation jointly with the end task, assuming", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "there is still a considerable amount of clean labeled data (Patrini et al., 2017; Lee et al., 2018; Li", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "et al., 2017). However, for a number of classes only very few or even no clean labeled examples", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "might be available at the representation learning stage. Few-shot learning limits the labeled data", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 556 + ], + "score": 1.0, + "content": "to very few on the end task, while the representation is learned on a large training set of different", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "classes (Hariharan & Girshick, 2017; Snell et al., 2017; Vinyals et al., 2016). Nevertheless, in many", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 567, + 456, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 456, + 578 + ], + "score": 1.0, + "content": "situations, more data with noisy labels are available or can be acquired for the end task.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 489, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "One interesting mix of few-shot learning with additional large-scale data is the work of Douze et al.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "(2018), where labels are propagated from few clean labeled examples to a large-scale collection. This", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "collection is unlabeled and actually contains data of many more classes than the end task. Their", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "method overall improves the classification accuracy, but at an additional computational cost; it is a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "transductive method, i.e., instead of learning a parametric classifier, the large-scale collection is still", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 639, + 199, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 199, + 650 + ], + "score": 1.0, + "content": "necessary at inference.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 583, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "In this work, we learn a classifier from few clean labeled examples and additional weakly labeled", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "data, while the representation is learned on different classes, as in few-shot learning. We assume", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "the class names are known, and we use them to search an existing large collection of images with", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "textual description. The result is a set of images with potentially relevant, but noisy labels. As shown", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "score": 1.0, + "content": "in Figure 1, we clean this data using a graph convolutional network (GCN) (Kipf & Welling, 2017),", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "which learns to predict a class relevance score per image based on the source (clean vs. noisy) of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "its connections in the graph. 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In our case however, since the clean", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 491, + 406, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 406, + 504 + ], + "score": 1.0, + "content": "labeled examples are few, we need to keep the representation mostly fixed.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 425, + 507, + 504 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 508, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 506, + 520 + ], + "score": 1.0, + "content": "Dealing with the noise, e.g. by thresholding (Lee et al., 2018), outlier detection (Wang et al., 2018a)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "or reweighting (Liu & Tao, 2015), is applicable while the representation is learned, based e.g. on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "the gradient of the loss (Ren et al., 2018b). In contrast, the relatively-shallow GCN that we propose", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "effectively decouples reweighting from both representation learning and classifier learning. Learning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "to clean the noisy labels (Veit et al., 2017) typically assumes adequate human verified labels for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 563, + 305, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 305, + 575 + ], + "score": 1.0, + "content": "training, which again is not the case in this work.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 508, + 506, + 575 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "Few-shot learning. Meta-learning (Vilalta & Drissi, 2002) refers to learning at two levels, where", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "generic knowledge is acquired before adapting to more specific tasks. In few-shot learning, this", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "translates to learning on a set of base classes how to learn from few examples on a distinct set of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "novel classes without overfitting. For instance, optimization meta-learning (Finn et al., 2017; 2018;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "Ravi & Larochelle, 2017) amounts to learning a model that is easy to fine-tune in few steps. In our", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 637, + 507, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 507, + 651 + ], + "score": 1.0, + "content": "work, we study an extension of few-shot learning where more data are available on novel classes,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "reducing the risk of overfitting when fine-tuning the model. Metric learning approaches learn how to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "compare queries for instance to few examples (Vinyals et al., 2016) or to the corresponding class", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 104, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "prototypes (Snell et al., 2017). Hariharan & Girshick (2017) and Wang et al. (2018b) learn how to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 681, + 470, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 470, + 694 + ], + "score": 1.0, + "content": "generate novel-class examples, which is not needed when more data are actually available.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 583, + 507, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Gidaris & Komodakis (2018) learn on base classes a simpler cosine similarity-based parametric", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "classifier, or simply cosine classifier, without meta-learning. The same classifier has been introduced", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "independently by Qi et al. (2018), who further fine-tune the network, assuming access to the base", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "class training set. A recent survey (Chen et al., 2019) confirms the superiority of the cosine classifier", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "score": 1.0, + "content": "to previous work including meta-learning (Finn et al., 2017). We use the cosine classifier in this work,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "both for base and novel classes. All of the above use only the few labeled examples of the novel", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 140, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 140, + 128 + ], + "score": 1.0, + "content": "classes.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 43, + "bbox_fs": [ + 106, + 699, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "class training set. A recent survey (Chen et al., 2019) confirms the superiority of the cosine classifier", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "score": 1.0, + "content": "to previous work including meta-learning (Finn et al., 2017). We use the cosine classifier in this work,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "both for base and novel classes. All of the above use only the few labeled examples of the novel", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 140, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 140, + 128 + ], + "score": 1.0, + "content": "classes.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 506, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "Making use of unlabeled data has been little explored in few-shot learning until recently. Ren et al.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "(2018a) introduce a semi-supervised few-shot classification task, where some labels are unknown.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Liu et al. (2019) follow the same semi-supervised setup, but use graph-based label propagation", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "(LP) (Zhou et al., 2003a) for classification and consider jointly all test images. These methods assume", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "a meta-learning scenario, where only small-scale data is available at each training episode; arguably,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 200 + ], + "score": 1.0, + "content": "such a small amount of data limits the representation adaptation and generalization to unseen data.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "Similarly, Rohrbach et al. (2013) use label propagation in a transductive setting, but at a larger scale", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "assuming that all examples come from a set of known classes. Douze et al. (2018) extend to even", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 232 + ], + "score": 1.0, + "content": "larger scale, leveraging 100M unlabeled images in a graph without using additional text information.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "We focus on the latter large-scale scenario using the same 100M dataset. However, we filter by text", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "to obtain noisy labels and follow an inductive approach by training a classifier for novel classes, such", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 314, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 314, + 265 + ], + "score": 1.0, + "content": "that the 100M collection is not needed at inference.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 278, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 292 + ], + "score": 1.0, + "content": "Graph neural networks are generalizations of convolutional networks to non-Euclidean", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "spaces (Bronstein et al., 2017). Early spectral methods (Bruna et al., 2014; Henaff et al., 2015) have", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "been succeeded by Chebyshev polynomial approximations (Defferrard et al., 2016), which avoid the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "high computational cost of computing eigenvectors. Graph convolutional networks (GCN) (Kipf &", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "score": 1.0, + "content": "Welling, 2017) provide a further simplification by a first-order approximation of graph filtering and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "are applied to semi-supervised (Kipf & Welling, 2017) and subsequently to few-shot learning (Garcia", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 505, + 357 + ], + "score": 1.0, + "content": "& Bruna, 2018). In Kipf & Welling (2017), the loss function is applied to labeled examples to make", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "score": 1.0, + "content": "predictions on unlabeled ones. Similarly in Garcia & Bruna (2018), GCNs make predictions on novel", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "class examples. Gidaris & Komodakis (2019) use Graph Neural Networks as denoising autoencoders", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "to generate class weights for novel classes. In contrast, we cast GCNs as binary classifiers discrimi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 400 + ], + "score": 1.0, + "content": "nating clean from noisy examples: we apply a loss function to all examples, and then use the inferred", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 400, + 394, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 394, + 412 + ], + "score": 1.0, + "content": "probabilities as a class relevance measure, effectively cleaning the data.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 430 + ], + "score": 1.0, + "content": "Our counter-intuitive objective of treating all noisy examples as negative can be compared to treating", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "each example as a different class in instance-level discrimination (Wu et al., 2018). In fact, our loss", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "function is similar to noise-contrastive estimation (NCE) (Gutmann & Hyvärinen, 2010) used in that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "score": 1.0, + "content": "work. According to our experiments, our GCN-based classifier outperforms classical LP (Zhou et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 459, + 350, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 350, + 473 + ], + "score": 1.0, + "content": "2003a) used for a similar purpose by Rohrbach et al. 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We are given a set", + "type": "text" + }, + { + "bbox": [ + 331, + 534, + 368, + 544 + ], + "score": 0.91, + "content": "X _ { \\mathcal { L } } \\subset \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "of examples, each having a clean", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 238, + 558 + ], + "score": 1.0, + "content": "(manually verified) label in a set", + "type": "text" + }, + { + "bbox": [ + 238, + 545, + 252, + 555 + ], + "score": 0.9, + "content": "C _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 543, + 315, + 558 + ], + "score": 1.0, + "content": "of classes with", + "type": "text" + }, + { + "bbox": [ + 316, + 545, + 362, + 556 + ], + "score": 0.92, + "content": "| C _ { \\mathcal { L } } | = K _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 543, + 414, + 558 + ], + "score": 1.0, + "content": ". For any set", + "type": "text" + }, + { + "bbox": [ + 414, + 545, + 446, + 555 + ], + "score": 0.9, + "content": "X \\subset { \\mathcal { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 543, + 506, + 558 + ], + "score": 1.0, + "content": ", we denote by", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 107, + 556, + 121, + 565 + ], + "score": 0.86, + "content": "X ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 555, + 306, + 568 + ], + "score": 1.0, + "content": "its subset of examples having a label in class", + "type": "text" + }, + { + "bbox": [ + 306, + 557, + 312, + 565 + ], + "score": 0.68, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 555, + 431, + 568 + ], + "score": 1.0, + "content": ". 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We are also given an additional", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 120, + 589 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 577, + 136, + 589 + ], + "score": 0.89, + "content": "X _ { \\mathcal { Z } } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 577, + 323, + 589 + ], + "score": 1.0, + "content": "of examples, each with a set of noisy labels in", + "type": "text" + }, + { + "bbox": [ + 323, + 578, + 337, + 588 + ], + "score": 0.88, + "content": "C _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 577, + 498, + 589 + ], + "score": 1.0, + "content": ". The extended set of examples for class", + "type": "text" + }, + { + "bbox": [ + 498, + 580, + 504, + 587 + ], + "score": 0.61, + "content": "c", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 136, + 601 + ], + "score": 1.0, + "content": "is now", + "type": "text" + }, + { + "bbox": [ + 136, + 588, + 205, + 600 + ], + "score": 0.92, + "content": "X _ { \\mathcal { E } } ^ { c } = X _ { \\mathcal { L } } ^ { c } \\cup X _ { \\mathcal { Z } } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 587, + 506, + 601 + ], + "score": 1.0, + "content": ". Examples or sets of examples having clean (noisy) labels are referred to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "as clean (noisy) as well. The goal is to train a classifier, using the additional noisy set in order to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 369, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 369, + 622 + ], + "score": 1.0, + "content": "improve the accuracy compared to only using the small clean set.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 108, + 626, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 290, + 640 + ], + "score": 1.0, + "content": "We assume that we are given a feature extractor", + "type": "text" + }, + { + "bbox": [ + 291, + 626, + 345, + 638 + ], + "score": 0.92, + "content": "g _ { \\theta } : \\mathcal { X } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 625, + 446, + 640 + ], + "score": 1.0, + "content": ", mapping an example to a", + "type": "text" + }, + { + "bbox": [ + 447, + 627, + 453, + 637 + ], + "score": 0.8, + "content": "d .", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 625, + 506, + 640 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "vector. 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To make use of text data, we assume that the names of classes in", + "type": "text" + }, + { + "bbox": [ + 434, + 688, + 448, + 699 + ], + "score": 0.9, + "content": "C _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "are given. An", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 151, + 712 + ], + "score": 1.0, + "content": "example in", + "type": "text" + }, + { + "bbox": [ + 152, + 699, + 168, + 710 + ], + "score": 0.9, + "content": "X z", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 699, + 262, + 712 + ], + "score": 1.0, + "content": "is given a label in class", + "type": "text" + }, + { + "bbox": [ + 262, + 700, + 292, + 710 + ], + "score": 0.9, + "content": "c \\in C _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 699, + 496, + 712 + ], + "score": 1.0, + "content": "if its textual information contains the name of class", + "type": "text" + }, + { + "bbox": [ + 497, + 701, + 502, + 709 + ], + "score": 0.67, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 699, + 506, + 712 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 455, + 722 + ], + "score": 1.0, + "content": "it may then have none, one or more labels. In this way, we automatically infer labels for", + "type": "text" + }, + { + "bbox": [ + 455, + 710, + 471, + 721 + ], + "score": 0.9, + "content": "X z", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "without", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 267, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 267, + 733 + ], + "score": 1.0, + "content": "human effort, which are however noisy.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 506, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "Making use of unlabeled data has been little explored in few-shot learning until recently. Ren et al.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "(2018a) introduce a semi-supervised few-shot classification task, where some labels are unknown.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Liu et al. (2019) follow the same semi-supervised setup, but use graph-based label propagation", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "(LP) (Zhou et al., 2003a) for classification and consider jointly all test images. These methods assume", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "a meta-learning scenario, where only small-scale data is available at each training episode; arguably,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 200 + ], + "score": 1.0, + "content": "such a small amount of data limits the representation adaptation and generalization to unseen data.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "Similarly, Rohrbach et al. (2013) use label propagation in a transductive setting, but at a larger scale", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "assuming that all examples come from a set of known classes. Douze et al. (2018) extend to even", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 232 + ], + "score": 1.0, + "content": "larger scale, leveraging 100M unlabeled images in a graph without using additional text information.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "We focus on the latter large-scale scenario using the same 100M dataset. However, we filter by text", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "to obtain noisy labels and follow an inductive approach by training a classifier for novel classes, such", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 314, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 314, + 265 + ], + "score": 1.0, + "content": "that the 100M collection is not needed at inference.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 132, + 506, + 265 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 278, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 292 + ], + "score": 1.0, + "content": "Graph neural networks are generalizations of convolutional networks to non-Euclidean", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "spaces (Bronstein et al., 2017). Early spectral methods (Bruna et al., 2014; Henaff et al., 2015) have", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "been succeeded by Chebyshev polynomial approximations (Defferrard et al., 2016), which avoid the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "high computational cost of computing eigenvectors. Graph convolutional networks (GCN) (Kipf &", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "score": 1.0, + "content": "Welling, 2017) provide a further simplification by a first-order approximation of graph filtering and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "are applied to semi-supervised (Kipf & Welling, 2017) and subsequently to few-shot learning (Garcia", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 505, + 357 + ], + "score": 1.0, + "content": "& Bruna, 2018). In Kipf & Welling (2017), the loss function is applied to labeled examples to make", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "score": 1.0, + "content": "predictions on unlabeled ones. Similarly in Garcia & Bruna (2018), GCNs make predictions on novel", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "class examples. Gidaris & Komodakis (2019) use Graph Neural Networks as denoising autoencoders", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "to generate class weights for novel classes. In contrast, we cast GCNs as binary classifiers discrimi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 400 + ], + "score": 1.0, + "content": "nating clean from noisy examples: we apply a loss function to all examples, and then use the inferred", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 400, + 394, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 394, + 412 + ], + "score": 1.0, + "content": "probabilities as a class relevance measure, effectively cleaning the data.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 278, + 506, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 430 + ], + "score": 1.0, + "content": "Our counter-intuitive objective of treating all noisy examples as negative can be compared to treating", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "each example as a different class in instance-level discrimination (Wu et al., 2018). In fact, our loss", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "function is similar to noise-contrastive estimation (NCE) (Gutmann & Hyvärinen, 2010) used in that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "score": 1.0, + "content": "work. According to our experiments, our GCN-based classifier outperforms classical LP (Zhou et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 459, + 350, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 350, + 473 + ], + "score": 1.0, + "content": "2003a) used for a similar purpose by Rohrbach et al. 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We are given a set", + "type": "text" + }, + { + "bbox": [ + 331, + 534, + 368, + 544 + ], + "score": 0.91, + "content": "X _ { \\mathcal { L } } \\subset \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "of examples, each having a clean", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 238, + 558 + ], + "score": 1.0, + "content": "(manually verified) label in a set", + "type": "text" + }, + { + "bbox": [ + 238, + 545, + 252, + 555 + ], + "score": 0.9, + "content": "C _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 543, + 315, + 558 + ], + "score": 1.0, + "content": "of classes with", + "type": "text" + }, + { + "bbox": [ + 316, + 545, + 362, + 556 + ], + "score": 0.92, + "content": "| C _ { \\mathcal { L } } | = K _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 543, + 414, + 558 + ], + "score": 1.0, + "content": ". For any set", + "type": "text" + }, + { + "bbox": [ + 414, + 545, + 446, + 555 + ], + "score": 0.9, + "content": "X \\subset { \\mathcal { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 543, + 506, + 558 + ], + "score": 1.0, + "content": ", we denote by", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 107, + 556, + 121, + 565 + ], + "score": 0.86, + "content": "X ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 555, + 306, + 568 + ], + "score": 1.0, + "content": "its subset of examples having a label in class", + "type": "text" + }, + { + "bbox": [ + 306, + 557, + 312, + 565 + ], + "score": 0.68, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 555, + 431, + 568 + ], + "score": 1.0, + "content": ". 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We are also given an additional", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 120, + 589 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 577, + 136, + 589 + ], + "score": 0.89, + "content": "X _ { \\mathcal { Z } } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 577, + 323, + 589 + ], + "score": 1.0, + "content": "of examples, each with a set of noisy labels in", + "type": "text" + }, + { + "bbox": [ + 323, + 578, + 337, + 588 + ], + "score": 0.88, + "content": "C _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 577, + 498, + 589 + ], + "score": 1.0, + "content": ". The extended set of examples for class", + "type": "text" + }, + { + "bbox": [ + 498, + 580, + 504, + 587 + ], + "score": 0.61, + "content": "c", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 136, + 601 + ], + "score": 1.0, + "content": "is now", + "type": "text" + }, + { + "bbox": [ + 136, + 588, + 205, + 600 + ], + "score": 0.92, + "content": "X _ { \\mathcal { E } } ^ { c } = X _ { \\mathcal { L } } ^ { c } \\cup X _ { \\mathcal { Z } } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 587, + 506, + 601 + ], + "score": 1.0, + "content": ". Examples or sets of examples having clean (noisy) labels are referred to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "as clean (noisy) as well. The goal is to train a classifier, using the additional noisy set in order to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 369, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 369, + 622 + ], + "score": 1.0, + "content": "improve the accuracy compared to only using the small clean set.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 532, + 506, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 626, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 290, + 640 + ], + "score": 1.0, + "content": "We assume that we are given a feature extractor", + "type": "text" + }, + { + "bbox": [ + 291, + 626, + 345, + 638 + ], + "score": 0.92, + "content": "g _ { \\theta } : \\mathcal { X } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 625, + 446, + 640 + ], + "score": 1.0, + "content": ", mapping an example to a", + "type": "text" + }, + { + "bbox": [ + 447, + 627, + 453, + 637 + ], + "score": 0.8, + "content": "d .", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 625, + 506, + 640 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "vector. For instance, when examples are images, the feature extractor is typically a convolutional", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 648, + 352, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 216, + 662 + ], + "score": 1.0, + "content": "neural network (CNN) and", + "type": "text" + }, + { + "bbox": [ + 217, + 649, + 223, + 659 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 648, + 352, + 662 + ], + "score": 1.0, + "content": "are the parameters of all layers.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 625, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 276, + 678 + ], + "score": 1.0, + "content": "In this work, we assume that the noisy set", + "type": "text" + }, + { + "bbox": [ + 276, + 666, + 292, + 677 + ], + "score": 0.9, + "content": "X z", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "is collected via web crawling with examples that are", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "images accompanied with free-form text description and/or user tags originating from community", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 433, + 700 + ], + "score": 1.0, + "content": "photo collections. 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An", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 151, + 712 + ], + "score": 1.0, + "content": "example in", + "type": "text" + }, + { + "bbox": [ + 152, + 699, + 168, + 710 + ], + "score": 0.9, + "content": "X z", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 699, + 262, + 712 + ], + "score": 1.0, + "content": "is given a label in class", + "type": "text" + }, + { + "bbox": [ + 262, + 700, + 292, + 710 + ], + "score": 0.9, + "content": "c \\in C _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 699, + 496, + 712 + ], + "score": 1.0, + "content": "if its textual information contains the name of class", + "type": "text" + }, + { + "bbox": [ + 497, + 701, + 502, + 709 + ], + "score": 0.67, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 699, + 506, + 712 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 455, + 722 + ], + "score": 1.0, + "content": "it may then have none, one or more labels. In this way, we automatically infer labels for", + "type": "text" + }, + { + "bbox": [ + 455, + 710, + 471, + 721 + ], + "score": 0.9, + "content": "X z", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "without", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 267, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 267, + 733 + ], + "score": 1.0, + "content": "human effort, which are however noisy.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 665, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 472, + 119 + ], + "score": 1.0, + "content": "We perform cleaning by predicting a class relevance measure for each noisy example in", + "type": "text" + }, + { + "bbox": [ + 472, + 106, + 488, + 118 + ], + "score": 0.89, + "content": "X _ { \\mathcal { Z } } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 105, + 506, + 119 + ], + "score": 1.0, + "content": ", in-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 200, + 130 + ], + "score": 1.0, + "content": "dependently per class", + "type": "text" + }, + { + "bbox": [ + 200, + 117, + 235, + 128 + ], + "score": 0.93, + "content": "c \\in C _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 117, + 421, + 130 + ], + "score": 1.0, + "content": ". 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The scale parameter", + "type": "text" + }, + { + "bbox": [ + 421, + 504, + 428, + 511 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "is also fixed to the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 513, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 523 + ], + "score": 1.0, + "content": "value obtained during base class learning. 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We extend the Low-Shot ImageNet benchmark introduced by Hariharan &", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "Girshick (2017) by assuming many noisy examples for novel classes, in addition to the few clean", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "ones. In this benchmark, the 1000 ImageNet classes (Russakovsky et al., 2015) are split into 389", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "base classes and 611 novel classes. The validation set contains 193 base and 300 novel classes, and", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 477, + 105 + ], + "score": 1.0, + "content": "the test set the remaining 196 base and 311 novel classes. 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Similar to Hariharan & Girshick (2017) we", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 277, + 139 + ], + "score": 1.0, + "content": "perform 5 tasks, each drawing a subset of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 277, + 127, + 284, + 136 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 284, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "clean examples per class. 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The validation set contains 193 base and 300 novel classes, and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 477, + 105 + ], + "score": 1.0, + "content": "the test set the remaining 196 base and 311 novel classes. The standard benchmark includes", + "type": "text" + }, + { + "bbox": [ + 477, + 94, + 484, + 104 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "-shot", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 244, + 117 + ], + "score": 1.0, + "content": "classification, i.e. classification on", + "type": "text" + }, + { + "bbox": [ + 244, + 105, + 251, + 114 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 104, + 433, + 117 + ], + "score": 1.0, + "content": "clean examples per class, which we extend to", + "type": "text" + }, + { + "bbox": [ + 433, + 105, + 440, + 114 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "clean and many", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 235, + 128 + ], + "score": 1.0, + "content": "noisy examples per class, with", + "type": "text" + }, + { + "bbox": [ + 235, + 115, + 318, + 127 + ], + "score": 0.91, + "content": "k \\in \\{ 1 , 2 , 5 , 1 0 , 2 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 115, + 505, + 128 + ], + "score": 1.0, + "content": ". Similar to Hariharan & Girshick (2017) we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 277, + 139 + ], + "score": 1.0, + "content": "perform 5 tasks, each drawing a subset of", + "type": "text" + }, + { + "bbox": [ + 277, + 127, + 284, + 136 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "clean examples per class. We report the average top-5", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 357, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 357, + 149 + ], + "score": 1.0, + "content": "accuracy over the 5 tasks on novel or all classes of the test set.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "We use the YFCC100M dataset (Thomee et al., 2016) as a source for additional data with noisy", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "labels. It contains approximatively 100M images collected from Flickr. Each image comes with a text", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "description obtained from the user title and caption. We use the text description to obtain images with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "noisy labels. as discussed in Section 3. This process results in very different numbers of additional", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "examples per class, with a minimum of zero for classes maillot and missile, and a maximum of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 275, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 275, + 222 + ], + "score": 1.0, + "content": "620,142 for class church/church building.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "Representation and classifier learning. In most experiments, we use ResNet-10 (He et al., 2016) as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "feature extractor as in Gidaris & Komodakis (2018). Classification for novel classes is performed with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "class prototypes (6), cosine classifier learning (7) or deep network fine-tuning (8). Hyper-parameters", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "such as batch size and number of epochs, are tuned on the validation set. Possible values are 2048,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "4096, and 8192 for batchsize and 10, 30 and 50 for number of epochs. The learning rate starts from", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "0.1 and is reduced to 0.001 at the end of training with cosine annealing (Loshchilov & Hutter, 2017).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 346, + 315 + ], + "score": 1.0, + "content": "We handle the imbalance of the noisy set by normalizing by", + "type": "text" + }, + { + "bbox": [ + 346, + 302, + 372, + 314 + ], + "score": 0.91, + "content": "r ( X _ { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "in (7). Prototypes (6) are used to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 144, + 326 + ], + "score": 1.0, + "content": "initialize", + "type": "text" + }, + { + "bbox": [ + 145, + 313, + 161, + 324 + ], + "score": 0.88, + "content": "W _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 312, + 332, + 326 + ], + "score": 1.0, + "content": "of cosine classifier in (7), and the learned", + "type": "text" + }, + { + "bbox": [ + 333, + 314, + 349, + 324 + ], + "score": 0.9, + "content": "W _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "is used to initialize the corresponding", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 135, + 338 + ], + "score": 1.0, + "content": "part of", + "type": "text" + }, + { + "bbox": [ + 136, + 325, + 153, + 336 + ], + "score": 0.89, + "content": "W _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 323, + 506, + 338 + ], + "score": 1.0, + "content": "when fine-tuning the network by (8). In the latter case, we train all layers for 10 epochs", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 283, + 348 + ], + "score": 1.0, + "content": "with learning rate 0.01. We ignore examples", + "type": "text" + }, + { + "bbox": [ + 284, + 337, + 294, + 346 + ], + "score": 0.86, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 335, + 355, + 348 + ], + "score": 1.0, + "content": "with relevance", + "type": "text" + }, + { + "bbox": [ + 355, + 335, + 404, + 347 + ], + "score": 0.93, + "content": "r ( x _ { i } ) < 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "to reduce the complexity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 228, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 228, + 358 + ], + "score": 1.0, + "content": "when fine-tuning the network.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "We also report results with ResNet-50 as feature extractor, using the model trained on base classes", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "by Hariharan & Girshick (2017). Following Douze et al. (2018), we apply PCA to the features to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "reduce their dimensionality to 256. Base classes are represented by class prototypes (6) in this case.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "GCN training is performed with Adam optimizer and a learning rate of 0.1 for 100 iterations. We use", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 410, + 425 + ], + "score": 1.0, + "content": "dropout with probability 0.5. The dimensionality of the input descriptors is", + "type": "text" + }, + { + "bbox": [ + 410, + 413, + 444, + 423 + ], + "score": 0.89, + "content": "d = 5 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "for ResNet-10", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 124, + 436 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 424, + 163, + 434 + ], + "score": 0.89, + "content": "d = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "for ResNet-50 (after PCA). Dimensionality of the internal representation in (1) is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 435, + 446, + 448 + ], + "spans": [ + { + "bbox": [ + 107, + 435, + 140, + 445 + ], + "score": 0.89, + "content": "m = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 435, + 446, + 448 + ], + "score": 1.0, + "content": ". The affinity matrix is constructed with reciprocal top-50 nearest neighbors.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 372, + 469 + ], + "score": 1.0, + "content": "Baselines. We implement and evaluate several baseline methods.", + "type": "text" + }, + { + "bbox": [ + 372, + 457, + 380, + 468 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 456, + 451, + 469 + ], + "score": 1.0, + "content": "-cleaning assigns", + "type": "text" + }, + { + "bbox": [ + 452, + 456, + 493, + 469 + ], + "score": 0.92, + "content": "r ( x _ { i } ) = \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 291, + 479 + ], + "score": 1.0, + "content": "all additional examples. 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It solves the linear system", + "type": "text" + }, + { + "bbox": [ + 391, + 500, + 505, + 513 + ], + "score": 0.89, + "content": "( I - \\alpha D ^ { - 1 / 2 } A D ^ { - 1 / 2 } ) \\mathbf { r } _ { c } =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 511, + 504, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 118, + 524 + ], + "score": 0.83, + "content": "\\mathbf { y } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 511, + 256, + 525 + ], + "score": 1.0, + "content": "(Iscen et al., 2017) for each class", + "type": "text" + }, + { + "bbox": [ + 256, + 515, + 262, + 523 + ], + "score": 0.67, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 511, + 293, + 525 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 293, + 513, + 303, + 522 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 511, + 399, + 525 + ], + "score": 1.0, + "content": "is the degree matrix of", + "type": "text" + }, + { + "bbox": [ + 399, + 513, + 407, + 522 + ], + "score": 0.64, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 511, + 411, + 525 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 412, + 513, + 446, + 523 + ], + "score": 0.82, + "content": "\\alpha = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 511, + 464, + 525 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 465, + 513, + 504, + 524 + ], + "score": 0.87, + "content": "\\mathbf { y } _ { c } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 123, + 535 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 123, + 524, + 129, + 533 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 523, + 366, + 535 + ], + "score": 1.0, + "content": "-hot binary vector indicating the labeled examples of class", + "type": "text" + }, + { + "bbox": [ + 366, + 525, + 371, + 533 + ], + "score": 0.69, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 523, + 420, + 535 + ], + "score": 1.0, + "content": ". 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Results are shown in Appendix A. The larger the value of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 610, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 107, + 612, + 114, + 622 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 610, + 505, + 625 + ], + "score": 1.0, + "content": ", the more the loss encourages noisy examples to be classified as negatives. As a consequence, large", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 137, + 636 + ], + "score": 1.0, + "content": "(small)", + "type": "text" + }, + { + "bbox": [ + 138, + 623, + 145, + 632 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 621, + 481, + 636 + ], + "score": 1.0, + "content": "results in smaller (larger) relevance, on average, for noisy examples. 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The use of additional", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 213, + 679 + ], + "score": 1.0, + "content": "data is mostly harmful for", + "type": "text" + }, + { + "bbox": [ + 214, + 666, + 221, + 677 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "-weighting except for 1 and 2-shot. MLP offers improvements in most", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "cases, implying that it manages to appropriately downweigh irrelevant examples. The consistent", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 417, + 700 + ], + "score": 1.0, + "content": "improvement of our method compared to MLP, especially large for small", + "type": "text" + }, + { + "bbox": [ + 417, + 688, + 424, + 698 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 689, + 505, + 700 + ], + "score": 1.0, + "content": ", suggests that it is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 344, + 711 + ], + "score": 1.0, + "content": "beneficial to incorporate relations, with the affinity matrix", + "type": "text" + }, + { + "bbox": [ + 344, + 699, + 353, + 709 + ], + "score": 0.63, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "modeling the structure of the feature", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 289, + 722 + ], + "score": 1.0, + "content": "space. LP is a classic approach that also uses", + "type": "text" + }, + { + "bbox": [ + 290, + 711, + 298, + 720 + ], + "score": 0.81, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "but is a linear operation with no parameters, and is", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 457, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 289, + 734 + ], + "score": 1.0, + "content": "inferior to our method. The gain of cleaning (", + "type": "text" + }, + { + "bbox": [ + 289, + 721, + 314, + 732 + ], + "score": 0.8, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 720, + 402, + 734 + ], + "score": 1.0, + "content": "vs. ours) ranges from", + "type": "text" + }, + { + "bbox": [ + 403, + 721, + 422, + 731 + ], + "score": 0.87, + "content": "11 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 720, + 433, + 734 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 433, + 721, + 453, + 731 + ], + "score": 0.88, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 720, + 457, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 83, + 505, + 149 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "We use the YFCC100M dataset (Thomee et al., 2016) as a source for additional data with noisy", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "labels. It contains approximatively 100M images collected from Flickr. Each image comes with a text", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "description obtained from the user title and caption. We use the text description to obtain images with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "noisy labels. as discussed in Section 3. This process results in very different numbers of additional", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "examples per class, with a minimum of zero for classes maillot and missile, and a maximum of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 275, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 275, + 222 + ], + "score": 1.0, + "content": "620,142 for class church/church building.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 153, + 506, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "Representation and classifier learning. In most experiments, we use ResNet-10 (He et al., 2016) as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "feature extractor as in Gidaris & Komodakis (2018). Classification for novel classes is performed with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "class prototypes (6), cosine classifier learning (7) or deep network fine-tuning (8). Hyper-parameters", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "such as batch size and number of epochs, are tuned on the validation set. Possible values are 2048,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "4096, and 8192 for batchsize and 10, 30 and 50 for number of epochs. The learning rate starts from", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "0.1 and is reduced to 0.001 at the end of training with cosine annealing (Loshchilov & Hutter, 2017).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 230, + 506, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 346, + 315 + ], + "score": 1.0, + "content": "We handle the imbalance of the noisy set by normalizing by", + "type": "text" + }, + { + "bbox": [ + 346, + 302, + 372, + 314 + ], + "score": 0.91, + "content": "r ( X _ { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "in (7). Prototypes (6) are used to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 144, + 326 + ], + "score": 1.0, + "content": "initialize", + "type": "text" + }, + { + "bbox": [ + 145, + 313, + 161, + 324 + ], + "score": 0.88, + "content": "W _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 312, + 332, + 326 + ], + "score": 1.0, + "content": "of cosine classifier in (7), and the learned", + "type": "text" + }, + { + "bbox": [ + 333, + 314, + 349, + 324 + ], + "score": 0.9, + "content": "W _ { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "is used to initialize the corresponding", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 135, + 338 + ], + "score": 1.0, + "content": "part of", + "type": "text" + }, + { + "bbox": [ + 136, + 325, + 153, + 336 + ], + "score": 0.89, + "content": "W _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 323, + 506, + 338 + ], + "score": 1.0, + "content": "when fine-tuning the network by (8). In the latter case, we train all layers for 10 epochs", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 283, + 348 + ], + "score": 1.0, + "content": "with learning rate 0.01. We ignore examples", + "type": "text" + }, + { + "bbox": [ + 284, + 337, + 294, + 346 + ], + "score": 0.86, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 335, + 355, + 348 + ], + "score": 1.0, + "content": "with relevance", + "type": "text" + }, + { + "bbox": [ + 355, + 335, + 404, + 347 + ], + "score": 0.93, + "content": "r ( x _ { i } ) < 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "to reduce the complexity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 228, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 228, + 358 + ], + "score": 1.0, + "content": "when fine-tuning the network.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 302, + 506, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "We also report results with ResNet-50 as feature extractor, using the model trained on base classes", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "by Hariharan & Girshick (2017). Following Douze et al. (2018), we apply PCA to the features to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "reduce their dimensionality to 256. Base classes are represented by class prototypes (6) in this case.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 363, + 506, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "GCN training is performed with Adam optimizer and a learning rate of 0.1 for 100 iterations. We use", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 410, + 425 + ], + "score": 1.0, + "content": "dropout with probability 0.5. The dimensionality of the input descriptors is", + "type": "text" + }, + { + "bbox": [ + 410, + 413, + 444, + 423 + ], + "score": 0.89, + "content": "d = 5 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "for ResNet-10", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 124, + 436 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 424, + 163, + 434 + ], + "score": 0.89, + "content": "d = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "for ResNet-50 (after PCA). Dimensionality of the internal representation in (1) is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 435, + 446, + 448 + ], + "spans": [ + { + "bbox": [ + 107, + 435, + 140, + 445 + ], + "score": 0.89, + "content": "m = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 435, + 446, + 448 + ], + "score": 1.0, + "content": ". The affinity matrix is constructed with reciprocal top-50 nearest neighbors.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 401, + 505, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 372, + 469 + ], + "score": 1.0, + "content": "Baselines. We implement and evaluate several baseline methods.", + "type": "text" + }, + { + "bbox": [ + 372, + 457, + 380, + 468 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 456, + 451, + 469 + ], + "score": 1.0, + "content": "-cleaning assigns", + "type": "text" + }, + { + "bbox": [ + 452, + 456, + 493, + 469 + ], + "score": 0.92, + "content": "r ( x _ { i } ) = \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 291, + 479 + ], + "score": 1.0, + "content": "all additional examples. 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Methodk=1251020
FEW CLEAN EXAMPLES
Class proto.Gidaris & Komodakis (2018)45.3±0.6557.1±0.3769.3±0.3274.8±0.2077.8±0.24
FEW CLEAN & MANY NOISY EXAMPLES
β-weighting,β=156.1±0.0656.4±0.0857.1±0.0557.7±0.0858.7±0.06
β-weighting,β*55.6±0.2458.3±0.1463.4±0.2567.5±0.3471.0±0.22
Label Propagation Zhou et al. (2003a)62.6±0.3567.0±0.4174.6±0.3076.3±0.2377.7±0.18
MLP63.6±0.4168.8±0.4273.9±0.2575.6±0.2177.6±0.21
Ours67.8±0.1070.9±0.3073.7±0.1776.1±0.1278.2±0.14
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METHODNoVEL CLASSESALL CLASSES
k=1251020k=1251020
RESNET-10 -FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)39.354.4 66.371.273.949.561.069.772.974.6
Logistic reg.w/H(Wang et al.,2018b)40.750.8 62.069.376.552.259.467.672.876.9
PMN w/H(Wang et al., 2018b)45.857.8 69.074.377.457.664.771.975.277.5
Class proto.(Gidaris & Komodakis,2018)45.3±0.6557.1±0.37 69.3±0.32 74.8±0.20 77.8±0.2457.0±0.3664.7±0.1672.5±0.1875.8±0.1677.4±0.19 58.1±0.4865.2±0.15 72.9±0.25 76.6±0.18 78.8±0.16
Class proto.w/At.(Gidaris& Komodakis,2018) 45.8±0.74 57.4±0.38 69.6±0.27 75.0±0.29 78.2±0.23
RESNET-1O -FEW CLEAN& MANY NOISY EXAMPLES
Ours -class proto.(6) Ours-cosine (7)67.8±0.10 70.9±0.30 73.7±0.20 76.1±0.16 78.2±0.1470.3±0.05 72.1±0.18 74.1±0.12 75.6±0.13 76.9±0.09
Ours - fine-tune (8)73.2±0.14 75.3±0.25 75.6±0.24 78.5±0.32 80.7±0.26 74.6±0.13 76.6±0.26 78.2±0.23 80.9±0.34 82.9±0.2071.9±0.0774.0±0.2376.5±0.1678.3±0.2380.2±0.18 76.0±0.1077.3±0.1378.7±0.1980.7±0.2582.2±0.14
RESNET-50 - FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)
PMN w/H(Wang et al.,2018b)49.6 64.0 54.7 66.874.4 77.478.1 81.480.0 83.861.4 65.771.4 73.578.0 80.280.0 82.881.1 84.5
RESNET-5O-FEW CLEAN&MANYUNLABELED EXAMPLES
Diffusion (Douze et al., 2018) Diffusion -logistic (Douze et al.,2018)63.6±0.61 69.5±0.60 75.2±0.40 78.5±0.34 80.8±0.18=-
64.0±0.7071.1±0.82 79.7±0.3883.9±0.10 86.3±0.17 RESNET-5O-FEWCLEAN&MANY NOISY EXAMPLES
Ours - class proto.(6)
Ours - cosine (7)78.0±0.38 80.2±0.3380.9±0.17 83.7±0.1985.7±0.1169.7±0.44 73.7±0.56 77.0±0.20 79.9±0.30 81.9±0.2977.6±0.26 79.1±0.20 79.9±0.09 82.1±0.22 83.8±0.1173.8±0.33 76.6±0.36 78.9±0.19 80.8±0.21 82.2±0.14
Ours - fine-tune (8)80.8±0.25 83.0±0.23 83.8±0.39 86.4±0.23 88.5±0.2081.6±0.20 83.2±0.16 84.3±0.23 86.2±0.17 87.8±0.03
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Methodk=1251020
FEW CLEAN EXAMPLES
Class proto.Gidaris & Komodakis (2018)45.3±0.6557.1±0.3769.3±0.3274.8±0.2077.8±0.24
FEW CLEAN & MANY NOISY EXAMPLES
β-weighting,β=156.1±0.0656.4±0.0857.1±0.0557.7±0.0858.7±0.06
β-weighting,β*55.6±0.2458.3±0.1463.4±0.2567.5±0.3471.0±0.22
Label Propagation Zhou et al. (2003a)62.6±0.3567.0±0.4174.6±0.3076.3±0.2377.7±0.18
MLP63.6±0.4168.8±0.4273.9±0.2575.6±0.2177.6±0.21
Ours67.8±0.1070.9±0.3073.7±0.1776.1±0.1278.2±0.14
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METHODNoVEL CLASSESALL CLASSES
k=1251020k=1251020
RESNET-10 -FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)39.354.4 66.371.273.949.561.069.772.974.6
Logistic reg.w/H(Wang et al.,2018b)40.750.8 62.069.376.552.259.467.672.876.9
PMN w/H(Wang et al., 2018b)45.857.8 69.074.377.457.664.771.975.277.5
Class proto.(Gidaris & Komodakis,2018)45.3±0.6557.1±0.37 69.3±0.32 74.8±0.20 77.8±0.2457.0±0.3664.7±0.1672.5±0.1875.8±0.1677.4±0.19 58.1±0.4865.2±0.15 72.9±0.25 76.6±0.18 78.8±0.16
Class proto.w/At.(Gidaris& Komodakis,2018) 45.8±0.74 57.4±0.38 69.6±0.27 75.0±0.29 78.2±0.23
RESNET-1O -FEW CLEAN& MANY NOISY EXAMPLES
Ours -class proto.(6) Ours-cosine (7)67.8±0.10 70.9±0.30 73.7±0.20 76.1±0.16 78.2±0.1470.3±0.05 72.1±0.18 74.1±0.12 75.6±0.13 76.9±0.09
Ours - fine-tune (8)73.2±0.14 75.3±0.25 75.6±0.24 78.5±0.32 80.7±0.26 74.6±0.13 76.6±0.26 78.2±0.23 80.9±0.34 82.9±0.2071.9±0.0774.0±0.2376.5±0.1678.3±0.2380.2±0.18 76.0±0.1077.3±0.1378.7±0.1980.7±0.2582.2±0.14
RESNET-50 - FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)
PMN w/H(Wang et al.,2018b)49.6 64.0 54.7 66.874.4 77.478.1 81.480.0 83.861.4 65.771.4 73.578.0 80.280.0 82.881.1 84.5
RESNET-5O-FEW CLEAN&MANYUNLABELED EXAMPLES
Diffusion (Douze et al., 2018) Diffusion -logistic (Douze et al.,2018)63.6±0.61 69.5±0.60 75.2±0.40 78.5±0.34 80.8±0.18=-
64.0±0.7071.1±0.82 79.7±0.3883.9±0.10 86.3±0.17 RESNET-5O-FEWCLEAN&MANY NOISY EXAMPLES
Ours - class proto.(6)
Ours - cosine (7)78.0±0.38 80.2±0.3380.9±0.17 83.7±0.1985.7±0.1169.7±0.44 73.7±0.56 77.0±0.20 79.9±0.30 81.9±0.2977.6±0.26 79.1±0.20 79.9±0.09 82.1±0.22 83.8±0.1173.8±0.33 76.6±0.36 78.9±0.19 80.8±0.21 82.2±0.14
Ours - fine-tune (8)80.8±0.25 83.0±0.23 83.8±0.39 86.4±0.23 88.5±0.2081.6±0.20 83.2±0.16 84.3±0.23 86.2±0.17 87.8±0.03
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METHODNoVEL CLASSESALL CLASSES
k=1251020k=1251020
RESNET-10 -FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)39.354.4 66.371.273.949.561.069.772.974.6
Logistic reg.w/H(Wang et al.,2018b)40.750.8 62.069.376.552.259.467.672.876.9
PMN w/H(Wang et al., 2018b)45.857.8 69.074.377.457.664.771.975.277.5
Class proto.(Gidaris & Komodakis,2018)45.3±0.6557.1±0.37 69.3±0.32 74.8±0.20 77.8±0.2457.0±0.3664.7±0.1672.5±0.1875.8±0.1677.4±0.19 58.1±0.4865.2±0.15 72.9±0.25 76.6±0.18 78.8±0.16
Class proto.w/At.(Gidaris& Komodakis,2018) 45.8±0.74 57.4±0.38 69.6±0.27 75.0±0.29 78.2±0.23
RESNET-1O -FEW CLEAN& MANY NOISY EXAMPLES
Ours -class proto.(6) Ours-cosine (7)67.8±0.10 70.9±0.30 73.7±0.20 76.1±0.16 78.2±0.1470.3±0.05 72.1±0.18 74.1±0.12 75.6±0.13 76.9±0.09
Ours - fine-tune (8)73.2±0.14 75.3±0.25 75.6±0.24 78.5±0.32 80.7±0.26 74.6±0.13 76.6±0.26 78.2±0.23 80.9±0.34 82.9±0.2071.9±0.0774.0±0.2376.5±0.1678.3±0.2380.2±0.18 76.0±0.1077.3±0.1378.7±0.1980.7±0.2582.2±0.14
RESNET-50 - FEW CLEAN EXAMPLES
Proto.-Nets (Snell et al., 2017)
PMN w/H(Wang et al.,2018b)49.6 64.0 54.7 66.874.4 77.478.1 81.480.0 83.861.4 65.771.4 73.578.0 80.280.0 82.881.1 84.5
RESNET-5O-FEW CLEAN&MANYUNLABELED EXAMPLES
Diffusion (Douze et al., 2018) Diffusion -logistic (Douze et al.,2018)63.6±0.61 69.5±0.60 75.2±0.40 78.5±0.34 80.8±0.18=-
64.0±0.7071.1±0.82 79.7±0.3883.9±0.10 86.3±0.17 RESNET-5O-FEWCLEAN&MANY NOISY EXAMPLES
Ours - class proto.(6)
Ours - cosine (7)78.0±0.38 80.2±0.3380.9±0.17 83.7±0.1985.7±0.1169.7±0.44 73.7±0.56 77.0±0.20 79.9±0.30 81.9±0.2977.6±0.26 79.1±0.20 79.9±0.09 82.1±0.22 83.8±0.1173.8±0.33 76.6±0.36 78.9±0.19 80.8±0.21 82.2±0.14
Ours - fine-tune (8)80.8±0.25 83.0±0.23 83.8±0.39 86.4±0.23 88.5±0.2081.6±0.20 83.2±0.16 84.3±0.23 86.2±0.17 87.8±0.03
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Methodk=1251020
FEW CLEAN EXAMPLES
Class proto.Gidaris & Komodakis (2018)45.3±0.6557.1±0.3769.3±0.3274.8±0.2077.8±0.24
FEW CLEAN & MANY NOISY EXAMPLES
β-weighting,β=156.1±0.0656.4±0.0857.1±0.0557.7±0.0858.7±0.06
β-weighting,β*55.6±0.2458.3±0.1463.4±0.2567.5±0.3471.0±0.22
Label Propagation Zhou et al. (2003a)62.6±0.3567.0±0.4174.6±0.3076.3±0.2377.7±0.18
MLP63.6±0.4168.8±0.4273.9±0.2575.6±0.2177.6±0.21
Ours67.8±0.1070.9±0.3073.7±0.1776.1±0.1278.2±0.14
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"text": "" + } + ], + "page_info": { + "page_no": 10, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/Hk2aImxAb/Hk2aImxAb.md b/parse/train/Hk2aImxAb/Hk2aImxAb.md new file mode 100644 index 0000000000000000000000000000000000000000..ed26ab57bde259ae669c8489c148f654a9379984 --- /dev/null +++ b/parse/train/Hk2aImxAb/Hk2aImxAb.md @@ -0,0 +1,241 @@ +# MULTI-SCALE DENSE NETWORKS FOR RESOURCE EFFICIENT IMAGE CLASSIFICATION + +Gao Huang Cornell University + +Danlu Chen Fudan University + +Tianhong Li Tsinghua University + +Felix Wu Cornell University + +Laurens van der Maaten Facebook AI Research + +Kilian Weinberger Cornell University + +# ABSTRACT + +In this paper we investigate image classification with computational resource limits at test time. Two such settings are: 1. anytime classification, where the network’s prediction for a test example is progressively updated, facilitating the output of a prediction at any time; and 2. budgeted batch classification, where a fixed amount of computation is available to classify a set of examples that can be spent unevenly across “easier” and “harder” inputs. In contrast to most prior work, such as the popular Viola and Jones algorithm, our approach is based on convolutional neural networks. We train multiple classifiers with varying resource demands, which we adaptively apply during test time. To maximally re-use computation between the classifiers, we incorporate them as early-exits into a single deep convolutional neural network and inter-connect them with dense connectivity. To facilitate high quality classification early on, we use a two-dimensional multi-scale network architecture that maintains coarse and fine level features all-throughout the network. Experiments on three image-classification tasks demonstrate that our framework substantially improves the existing state-of-the-art in both settings. + +# 1 INTRODUCTION + +Recent years have witnessed a surge in demand for applications of visual object recognition, for instance, in self-driving cars (Bojarski et al., 2016) and content-based image search (Wan et al., 2014). This demand has in part been fueled through the promise generated by the astonishing progress of convolutional networks (CNNs) on visual object recognition benchmark competition datasets, such as ILSVRC (Deng et al., 2009) and COCO (Lin et al., 2014), where state-of-the-art models may have even surpassed human-level performance (He et al., 2015; 2016). + +However, the requirements of such competitions differ from realworld applications, which tend to incentivize resource-hungry models with high computational demands at inference time. For example, the COCO 2016 competition was won by a large ensemble of computationally intensive $\mathrm { \dot { C } N N s ^ { 1 } }$ — a model likely far too computationally expensive for any resource-aware application. Although much smaller models would also obtain decent error, very large, computationally intensive models seem necessary to correctly classify the hard examples that make up the bulk of the remaining misclassifications of modern algorithms. To illustrate this point, Figure 1 shows two images of horses. The left image depicts a horse in canonical pose and is easy to classify, whereas the right image is taken from a rare viewpoint and is likely in the tail of the data distribution. Computationally intensive models are needed to classify such tail examples correctly, but are wasteful when applied to canonical images such as the left one. + +![](images/2e2e080304ad2c7be1e5890d21d32ed57f0cf575c439df5f869a545d4f732d05.jpg) +Figure 1: Two images containing a horse. The left image is canonical and easy to detect even with a small model, whereas the right image requires a computationally more expensive network architecture. (Copyright Pixel Addict and Doyle (CC BY-ND 2.0).) + +In real-world applications, computation directly translates into power consumption, which should be minimized for environmental and economical reasons, and is a scarce commodity on mobile devices. This begs the question: why do we choose between either wasting computational resources by applying an unnecessarily computationally expensive model to easy images, or making mistakes by using an efficient model that fails to recognize difficult images? Ideally, our systems should automatically use small networks when test images are easy or computational resources limited, and use big networks when test images are hard or computation is abundant. + +Such systems would be beneficial in at least two settings with computational constraints at testtime: anytime prediction, where the network can be forced to output a prediction at any given point in time; and budgeted batch classification, where a fixed computational budget is shared across a large set of examples which can be spent unevenly across “easy” and “hard” examples. A practical use-case of anytime prediction is in mobile apps on Android devices: in 2015, there existed 24, 093 distinct Android devices2, each with its own distinct computational limitations. It is infeasible to train a different network that processes video frame-by-frame at a fixed framerate for each of these devices. Instead, you would like to train a single network that maximizes accuracy on all these devices, within the computational constraints of that device. The budget batch classification setting is ubiquitous in large-scale machine learning applications. Search engines, social media companies, on-line advertising agencies, all must process large volumes of data on limited hardware resources. For example, as of 2010, Google Image Search had over 10 Billion images indexed3, which has likely grown to over 1 Trillion since. Even if a new model to process these images is only 1/10s slower per image, this additional cost would add 3170 years of CPU time. In the budget batch classification setting, companies can improve the average accuracy by reducing the amount of computation spent on “easy” cases to save up computation for “hard” cases. + +Motivated by prior work in computer vision on resource-efficient recognition (Viola & Jones, 2001), we aim to develop CNNs that “slice” the computation and process these slices one-by-one, stopping the evaluation once the CPU time is depleted or the classification sufficiently certain (through “early exits”). Unfortunately, the architecture of CNNs is inherently at odds with the introduction of early exits. CNNs learn the data representation and the classifier jointly, which leads to two problems with early exits: 1. The features in the last layer are extracted directly to be used by the classifier, whereas earlier features are not. The inherent dilemma is that different kinds of features need to be extracted depending on how many layers are left until the classification. 2. The features in different layers of the network may have different scale. Typically, the first layers of a deep nets operate on a fine scale (to extract low-level features), whereas later layers transition (through pooling or strided convolution) to coarse scales that allow global context to enter the classifier. Both scales are needed but happen at different places in the network. + +We propose a novel network architecture that addresses both of these problems through careful design changes, allowing for resource-efficient image classification. Our network uses a cascade of intermediate classifiers throughout the network. The first problem, of classifiers altering the internal representation, is addressed through the introduction of dense connectivity (Huang et al., 2017). By connecting all layers to all classifiers, features are no longer dominated by the most imminent earlyexit and the trade-off between early or later classification can be performed elegantly as part of the loss function. The second problem, the lack of coarse-scale features in early layers, is addressed by adopting a multi-scale network structure. At each layer we produce features of all scales (fine-tocoarse), which facilitates good classification early on but also extracts low-level features that only become useful after several more layers of processing. Our network architecture is illustrated in Figure 2, and we refer to it as Multi-Scale DenseNet (MSDNet). + +We evaluate MSDNets on three image-classification datasets. In the anytime classification setting, we show that it is possible to provide the ability to output a prediction at any time while maintain high accuracies throughout. In the budget batch classification setting we show that MSDNets can be effectively used to adapt the amount of computation to the difficulty of the example to be classified, which allows us to reduce the computational requirements of our models drastically whilst performing on par with state-of-the-art CNNs in terms of overall classification accuracy. To our knowledge this is the first deep learning architecture of its kind that allows dynamic resource adaptation with a single model and obtains competitive results throughout. + +![](images/393456a645a070145e34d48f149ac9a7ed180043707a693bf4407c8f3be79371.jpg) +Figure 2: Illustration of the first four layers of an MSDNet with three scales. The horizontal direction corresponds to the layer direction (depth) of the network. The vertical direction corresponds to the scale of the feature maps. Horizontal arrows indicate a regular convolution operation, whereas diagonal and vertical arrows indicate a strided convolution operation. Classifiers only operate on feature maps at the coarsest scale. Connections across more than one layer are not drawn explicitly: they are implicit through recursive concatenations. + +# 2 RELATED WORK + +We briefly review related prior work on computation-efficient networks, memory-efficient networks, and resource-sensitive machine learning, from which our network architecture draws inspiration. + +Computation-efficient networks. Most prior work on (convolutional) networks that are computationally efficient at test time focuses on reducing model size after training. In particular, many studies propose to prune weights (LeCun et al., 1989; Hassibi et al., 1993; Li et al., 2017) or quantize weights (Hubara et al., 2016; Rastegari et al., 2016) during or after training. These approaches are generally effective because deep networks often have a substantial number of redundant weights that can be pruned or quantized without sacrificing (and sometimes even improving) performance. Prior work also studies approaches that directly learn compact models with less parameter redundancy. For example, the knowledge-distillation method (Bucilua et al., 2006; Hinton et al., 2014) trains small student networks to reproduce the output of a much larger teacher network or ensemble. Our work differs from those approaches in that we train a single model that trades off computation for accuracy at test time without any re-training or finetuning. Indeed, weight pruning and knowledge distillation can be used in combination with our approach, and may lead to further improvements. + +Resource-efficient machine learning. Various prior studies explore computationally efficient variants of traditional machine-learning models (Viola & Jones, 2001; Grubb & Bagnell, 2012; Karayev et al., 2014; Trapeznikov & Saligrama, 2013; Xu et al., 2012; 2013; Nan et al., 2015; Wang et al., 2015). Most of these studies focus on how to incorporate the computational requirements of computing particular features in the training of machine-learning models such as (gradient-boosted) decision trees. Whilst our study is certainly inspired by these results, the architecture we explore differs substantially: most prior work exploits characteristics of machine-learning models (such as decision trees) that do not apply to deep networks. Our work is possibly most closely related to recent work on FractalNets (Larsson et al., 2017), which can perform anytime prediction by progressively evaluating subnetworks of the full network. FractalNets differ from our work in that they are not explicitly optimized for computation efficiency and consequently our experiments show that MSDNets substantially outperform FractalNets. Our dynamic evaluation strategy for reducing batch computational cost is closely related to the the adaptive computation time approach (Graves, 2016; Figurnov et al., 2016), and the recently proposed method of adaptively evaluating neural networks (Bolukbasi et al., 2017). Different from these works, our method adopts a specially designed network with multiple classifiers, which are jointly optimized during training and can directly output confidence scores to control the evaluation process for each test example. The adaptive computation time method (Graves, 2016) and its extension (Figurnov et al., 2016) also perform adaptive evaluation on test examples to save batch computational cost, but focus on skipping units rather than layers. In (Odena et al., 2017), a “composer”model is trained to construct the evaluation network from a set of sub-modules for each test example. By contrast, our work uses a single CNN with multiple intermediate classifiers that is trained end-to-end. The Feedback Networks (Zamir et al., 2016) enable early predictions by making predictions in a recurrent fashion, which heavily shares parameters among classifiers, but is less efficient in sharing computation. + +Related network architectures. Our network architecture borrows elements from neural fabrics (Saxena & Verbeek, 2016) and others (Zhou et al., 2015; Jacobsen et al., 2017; Ke et al., 2016) + +![](images/6c59242daaab86116b0562f24208342d867eb57b7d0db805e434894dcf77ee2c.jpg) +Figure 3: Relative accuracy of the intermediate classifier (left) and the final classifier (right) when introducing a single intermediate classifier at different layers in a ResNet, DenseNet and MSDNet. All experiments were performed on the CIFAR-100 dataset. Higher is better. + +to rapidly construct a low-resolution feature map that is amenable to classification, whilst also maintaining feature maps of higher resolution that are essential for obtaining high classification accuracy. Our design differs from the neural fabrics (Saxena & Verbeek, 2016) substantially in that MSDNets have a reduced number of scales and no sparse channel connectivity or up-sampling paths. MSDNets are at least one order of magnitude more efficient and typically more accurate — for example, an MSDNet with less than 1 million parameters obtains a test error below $7 . 0 \%$ on CIFAR-10 (Krizhevsky & Hinton, 2009), whereas Saxena & Verbeek (2016) report $7 . 4 3 \%$ with over 20 million parameters. We use the same feature-concatenation approach as DenseNets (Huang et al., 2017), which allows us to bypass features optimized for early classifiers in later layers of the network. Our architecture is related to deeply supervised networks (Lee et al., 2015) in that it incorporates classifiers at multiple layers throughout the network. In contrast to all these prior architectures, our network is specifically designed to operate in resource-aware settings. + +# 3 PROBLEM SETUP + +We consider two settings that impose computational constraints at prediction time. + +Anytime prediction. In the anytime prediction setting (Grubb & Bagnell, 2012), there is a finite computational budget $B > 0$ available for each test example $\mathbf { x }$ . The computational budget is nondeterministic, and varies per test instance. It is determined by the occurrence of an event that requires the model to output a prediction immediately. We assume that the budget is drawn from some joint distribution $P ( \mathbf { x } , B )$ . In some applications $P ( B )$ may be independent of $P ( \mathbf { x } )$ and can be estimated. For example, if the event is governed by a Poisson process, $P ( B )$ is an exponential distribution. We denote the loss of a model $f ( \mathbf { x } )$ that has to produce a prediction for instance $\mathbf { x }$ within budget $B$ by $L ( f ( \mathbf { x } ) , B )$ . The goal of an anytime learner is to minimize the expected loss under the budget distribution: $L ( f ) = \mathbb { E } \left[ L ( f ( \mathbf { x } ) , B ) \right] _ { P ( \mathbf { x } , B ) } .$ Here, $L ( \cdot )$ denotes a suitable loss function. As is common in the empirical risk minimization framework, the expectation under $P ( \mathbf { x } , B )$ may be estimated by an average over samples from $P ( \mathbf { x } , B )$ . + +Budgeted batch classification. In the budgeted batch classification setting, the model needs to classify a set of examples $\mathcal { D } _ { t e s t } = \{ \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { M } \}$ within a finite computational budget $B > 0$ that is known in advance. The learner aims to minimize the loss across all examples in $\mathcal { D } _ { t e s t }$ within a cumulative cost bounded by $B$ , which we denote by $L ( f ( \mathcal { D } _ { t e s t } ) , B )$ for some suitable loss function $L ( \cdot )$ . It can potentially do so by spending less than BM computation on classifying an “easy” example whilst using more than $\textstyle { \frac { B } { M } }$ computation on classifying a “difficult” example. Therefore, the budget considered here is a soft constraint when we have a large batch of testing samples. + +# 4 MULTI-SCALE DENSE CONVOLUTIONAL NETWORKS + +A straightforward solution to the two problems introduced in Section 3 is to train multiple networks of increasing capacity, and sequentially evaluate them at test time (as in Bolukbasi et al. (2017)). In the anytime setting the evaluation can be stopped at any point and the most recent prediction is returned. In the batch setting, the evaluation is stopped prematurely the moment a network classifies the test sample with sufficient confidence. When the resources are so limited that the execution is terminated after the first network, this approach is optimal because the first network is trained for exactly this computational budget without compromises. However, in both settings, this scenario is rare. In the more common scenario where some test samples can require more processing time than others the approach is far from optimal because previously learned features are never re-used across the different networks. + +An alternative solution is to build a deep network with a cascade of classifiers operating on the features of internal layers: in such a network features computed for an earlier classifier can be re-used by later classifiers. However, na¨ıvely attaching intermediate early-exit classifiers to a stateof-the-art deep network leads to poor performance. + +There are two reasons why intermediate early-exit classifiers hurt the performance of deep neural networks: early classifiers lack coarse-level features and classifiers throughout interfere with the feature generation process. In this section we investigate these effects empirically (see Figure 3) and, in response to our findings, propose the MSDNet architecture illustrated in Figure 2. + +Problem: The lack of coarse-level features. Traditional neural networks learn features of fine scale in early layers and coarse scale in later layers (through repeated convolution, pooling, and strided convolution). Coarse scale features in the final layers are important to classify the content of the whole image into a single class. Early layers lack coarse-level features and early-exit classifiers attached to these layers will likely yield unsatisfactory high error rates. To illustrate this point, we attached4 intermediate classifiers to varying layers of a ResNet (He et al., 2016) and a DenseNet (Huang et al., 2017) on the CIFAR-100 dataset (Krizhevsky & Hinton, 2009). The blue and red dashed lines in the left plot of Figure 3 show the relative accuracies of these classifiers. All three plots gives rise to a clear trend: the accuracy of a classifier is highly correlated with its position within the network. Particularly in the case of the ResNet (blue line), one can observe a visible “staircase” pattern, with big improvements after the 2nd and 4th classifiers — located right after pooling layers. + +Solution: Multi-scale feature maps. To address this issue, MSDNets maintain a feature representation at multiple scales throughout the network, and all the classifiers only use the coarse-level features. The feature maps at a particular layer5 and scale are computed by concatenating the results of one or two convolutions: 1. the result of a regular convolution applied on the same-scale features from the previous layer (horizontal connections) and, if possible, 2. the result of a strided convolution applied on the finer-scale feature map from the previous layer (diagonal connections). The horizontal connections preserve and progress high-resolution information, which facilitates the construction of high-quality coarse features in later layers. The vertical connections produce coarse features throughout that are amenable to classification. The dashed black line in Figure 3 shows that MSDNets substantially increase the accuracy of early classifiers. + +Problem: Early classifiers interfere with later classifiers. The right plot of Figure 3 shows the accuracies of the final classifier as a function of the location of a single intermediate classifier, relative to the accuracy of a network without intermediate classifiers. The results show that the introduction of an intermediate classifier harms the final ResNet classifier (blue line), reducing its accuracy by up to $7 \%$ . We postulate that this accuracy degradation in the ResNet may be caused by the intermediate classifier influencing the early features to be optimized for the short-term and not for the final layers. This improves the accuracy of the immediate classifier but collapses information required to generate high quality features in later layers. This effect becomes more pronounced when the first classifier is attached to an earlier layer. + +Solution: Dense connectivity. By contrast, the DenseNet (red line) suffers much less from this effect. Dense connectivity (Huang et al., 2017) connects each layer with all subsequent layers and allows later layers to bypass features optimized for the short-term, to maintain the high accuracy of the final classifier. If an earlier layer collapses information to generate short-term features, the lost information can be recovered through the direct connection to its preceding layer. The final classifier’s performance becomes (more or less) independent of the location of the intermediate classifier. As far as we know, this is the first paper that discovers that dense connectivity is an important element to early-exit classifiers in deep networks, and we make it an integral design choice in MSDNets. + +![](images/8de8c69d9d36a341a0373469c5e2695c92df31ad7bca8cfbfd76709c742d677e.jpg) +Figure 4: The output $\mathbf { x } _ { \ell } ^ { s }$ of layer $\ell$ at the $s ^ { \mathrm { t h } }$ scale in a MSDNet. Herein, [. . . ] denotes the concatenation operator, $h _ { \ell } ^ { s } ( \cdot )$ a regular convolution transformation, and $\tilde { h } _ { \ell } ^ { s } ( \cdot )$ a strided convolutional. Note that the outputs of $h _ { \ell } ^ { s }$ and $\tilde { h } _ { \ell } ^ { s }$ have the same feature map size; their outputs are concatenated along the channel dimension. + +# 4.1 THE MSDNET ARCHITECTURE + +The MSDNet architecture is illustrated in Figure 2. We present its main components below. Additional details on the architecture are presented in Appendix A. + +First layer. The first layer $\ell = 1 $ ) is unique as it includes vertical connections in Figure 2. Its main purpose is to “seed” representations on all $S$ scales. One could view its vertical layout as a miniature “S-layers” convolutional network $S { = } 3$ in Figure 2). Let us denote the output feature maps at layer $\ell$ and scale $s$ as $\mathbf { x } _ { \ell } ^ { s }$ and the original input image as $\mathbf { x } _ { 0 } ^ { 1 }$ . Feature maps at coarser scales are obtained via down-sampling. The output $\mathbf { x } _ { 1 } ^ { s }$ of the first layer is formally given in the top row of Figure 4. + +Subsequent layers. Following Huang et al. (2017), the output feature maps $\mathbf { x } _ { \ell } ^ { s }$ produced at subsequent layers, $\ell > 1$ , and scales, $s$ , are a concatenation of transformed feature maps from all previous feature maps of scale $s$ and $s - 1$ (if $s > 1$ ). Formally, the $\ell \cdot$ -th layer of our network outputs a set of features at $S$ scales $\big \{ \mathbf { x } _ { \ell } ^ { 1 } , \dots , \mathbf { x } _ { \ell } ^ { S } \big \}$ , given in the last row of Figure 4. + +Classifiers. The classifiers in MSDNets also follow the dense connectivity pattern within the coarsest scale, $S$ , i.e., the classifier at layer $\ell$ uses all the features $\left[ \mathbf { x } _ { 1 } ^ { S } , \ldots , \mathbf { x } _ { \ell } ^ { S } \right]$ . Each classifier consists of two convolutional layers, followed by one average pooling layer and one linear layer. In practice, we only attach classifiers to some of the intermediate layers, and we let $f _ { k } ( \cdot )$ denote the $k ^ { \mathrm { { t h } } }$ classifier. During testing in the anytime setting we propagate the input through the network until the budget is exhausted and output the most recent prediction. In the batch budget setting at test time, an example traverses the network and exits after classifier $f _ { k }$ if its prediction confidence (we use the maximum value of the softmax probability as a confidence measure) exceeds a pre-determined threshold $\theta _ { k }$ . Before training, we compute the computational cost, $C _ { k }$ , required to process the network up to the $k ^ { \mathrm { { t h } } }$ classifier. We denote by $0 < q \le 1$ a fixed exit probability that a sample that reaches a classifier will obtain a classification with sufficient confidence to exit. We assume that $q$ is constant across all layers, which allows us to compute the probability that a sample exits at classifier $k$ as: $q _ { k } = z ( 1 - q ) ^ { \bar { k ^ { - 1 } } q }$ , where $z$ is a normalizing constant that ensures that $\begin{array} { r } { \sum _ { k } p ( q _ { k } ) = 1 } \end{array}$ . At test time, we need to ensure that the overall cost of classifying all samples in $\mathcal { D } _ { t e s t }$ does not exceed our budget $B$ (in expectation). This gives rise to the constraint $\begin{array} { r } { \left| \mathcal { D } _ { t e s t } \right| \sum _ { k } q _ { k } C _ { k } \le B } \end{array}$ . We can solve this constraint for $q$ and determine the thresholds $\theta _ { k }$ on a validation set in such a way that approximately $| \mathcal { D } _ { t e s t } | q _ { k }$ validation samples exit at the $k ^ { \mathrm { { t h } } }$ classifier. + +Loss functions. During training we use cross entropy loss functions $L ( f _ { k } )$ for all classifiers and minimize a weighted cumulative loss: $\begin{array} { r } { \frac { 1 } { | \mathcal { D } | } \sum _ { ( \mathbf { x } , y ) \in \mathcal { D } } \mathbf { \tilde { \sum } } _ { k } w _ { k } L ( f _ { k } ) } \end{array}$ . Herein, $\mathcal { D }$ denotes the training set and $w _ { k } \ge 0$ the weight of the $k$ -th classifier. If the budget distribution $P ( B )$ is known, we can use the weights $w _ { k }$ to incorporate our prior knowledge about the budget $B$ in the learning. Empirically, we find that using the same weight for all loss functions (i.e., setting $\forall k : w _ { k } = 1$ ) works well in practice. + +Network reduction and lazy evaluation. There are two straightforward ways to further reduce the computational requirements of MSDNets. First, it is inefficient to maintain all the finer scales until the last layer of the network. One simple strategy to reduce the size of the network is by splitting it into $S$ blocks along the depth dimension, and only keeping the coarsest $( S - i + 1 )$ scales in the $i ^ { \mathrm { t h } }$ block (a schematic layout of this structure is shown in Figure 9). This reduces computational cost for both training and testing. Every time a scale is removed from the network, we add a transition layer between the two blocks that merges the concatenated features using a $1 \times 1$ convolution and cuts the number of channels in half before feeding the fine-scale features into the coarser scale via a strided convolution (this is similar to the DenseNet-BC architecture of Huang et al. (2017)). Second, since a classifier at layer $\ell$ only uses features from the coarsest scale, the finer feature maps in layer $\ell$ (and some of the finer feature maps in the previous $S - 2$ layers) do not influence the prediction of that classifier. Therefore, we group the computation in “diagonal blocks” such that we only propagate the example along paths that are required for the evaluation of the next classifier. This minimizes unnecessary computations when we need to stop because the computational budget is exhausted. We call this strategy lazy evaluation. + +# 5 EXPERIMENTS + +We evaluate the effectiveness of our approach on three image classification datasets, i.e., the CIFAR10, CIFAR-100 (Krizhevsky & Hinton, 2009) and ILSVRC 2012 (ImageNet; Deng et al. (2009)) datasets. Code to reproduce all results is available at https://anonymous-url. Details on architectural configurations of MSDNets are described in Appendix A. + +Datasets. The two CIFAR datasets contain 50, 000 training and 10, 000 test images of $3 2 \times 3 2$ pixels; we hold out 5, 000 training images as a validation set. The datasets comprise 10 and 100 classes, respectively. We follow He et al. (2016) and apply standard data-augmentation techniques to the training images: images are zero-padded with 4 pixels on each side, and then randomly cropped to produce $3 2 \times 3 2$ images. Images are flipped horizontally with probability 0.5, and normalized by subtracting channel means and dividing by channel standard deviations. The ImageNet dataset comprises 1, 000 classes, with a total of 1.2 million training images and 50,000 validation images. We hold out 50,000 images from the training set to estimate the confidence threshold for classifiers in MSDNet. We adopt the data augmentation scheme of He et al. (2016) at training time; at test time, we classify a $2 2 4 \times 2 2 4$ center crop of images that were resized to $2 5 6 \times 2 5 6$ pixels. + +Training Details. We train all models using the framework of Gross & Wilber (2016). On the two CIFAR datasets, all models (including all baselines) are trained using stochastic gradient descent (SGD) with mini-batch size 64. We use Nesterov momentum with a momentum weight of 0.9 without dampening, and a weight decay of $1 0 ^ { - 4 }$ . All models are trained for 300 epochs, with an initial learning rate of 0.1, which is divided by a factor 10 after 150 and 225 epochs. We apply the same optimization scheme to the ImageNet dataset, except that we increase the mini-batch size to 256, and all the models are trained for 90 epochs with learning rate drops after 30 and 60 epochs. + +# 5.1 ANYTIME PREDICTION + +In the anytime prediction setting, the model maintains a progressively updated distribution over classes, and it can be forced to output its most up-to-date prediction at an arbitrary time. + +Baselines. There exist several baseline approaches for anytime prediction: FractalNets (Larsson et al., 2017), deeply supervised networks (Lee et al., 2015), and ensembles of deep networks of varying or identical sizes. FractalNets allow for multiple evaluation paths during inference time, which vary in computation time. In the anytime setting, paths are evaluated in order of increasing computation. In our result figures, we replicate the FractalNet results reported in the original paper (Larsson et al., 2017) for reference. Deeply supervised networks introduce multiple early-exit classifiers throughout a network, which are applied on the features of the particular layer they are attached to. Instead of using the original model proposed in Lee et al. (2015), we use the more competitive ResNet and DenseNet architectures (referred to as DenseNet- $B C$ in Huang et al. (2017)) as the base networks in our experiments with deeply supervised networks. We refer to these as $R e s N e t ^ { M C }$ and DenseNetMC, where $M C$ stands for multiple classifiers. Both networks require about $1 . 3 \times 1 0 ^ { 8 }$ FLOPs when fully evaluated; the detailed network configurations are presented in the supplementary material. In addition, we include ensembles of ResNets and DenseNets of varying or identical sizes. At test time, the networks are evaluated sequentially (in ascending order of network size) to obtain predictions for the test data. All predictions are averaged over the evaluated classifiers. On + +![](images/0e595351824a020ac424bb128829df21d92c0ecca2b1cea17c58c737e1308b61.jpg) +Figure 5: Accuracy (top-1) of anytime prediction models as a function of computational budget on the ImageNet (left) and CIFAR-100 (right) datasets. Higher is better. + +ImageNet, we compare MSDNet against a highly competitive ensemble of ResNets and DenseNets, with depth varying from 10 layers to 50 layers, and 36 layers to 121 layers, respectively. + +Anytime prediction results are presented in Figure 5. The left plot shows the top-1 classification accuracy on the ImageNet validation set. Here, for all budgets in our evaluation, the accuracy of MSDNet substantially outperforms the ResNets and DenseNets ensemble. In particular, when the budget ranges from $0 . 1 \times \bar { 1 0 } ^ { 1 0 }$ to $0 . 3 \times 1 0 ^ { 1 0 }$ FLOPs, MSDNet achieves $\sim 4 \% - 8 \%$ higher accuracy. + +We evaluate more baselines on CIFAR-100 (and CIFAR-10; see supplementary materials). We observe that MSDNet substantially outperforms ResNetsMC and DenseNetsMC at any computational budget within our range. This is due to the fact that after just a few layers, MSDNets have produced low-resolution feature maps that are much more suitable for classification than the high-resolution feature maps in the early layers of ResNets or DenseNets. MSDNet also outperforms the other baselines for nearly all computational budgets, although it performs on par with ensembles when the budget is very small. In the extremely low-budget regime, ensembles have an advantage because their predictions are performed by the first (small) network, which is optimized exclusively for the low budget. However, the accuracy of ensembles does not increase nearly as fast when the budget is increased. The MSDNet outperforms the ensemble as soon as the latter needs to evaluate a second model: unlike MSDNets, this forces the ensemble to repeat the computation of similar low-level features repeatedly. Ensemble accuracies saturate rapidly when all networks are shallow. + +# 5.2 BUDGETED BATCH CLASSIFICATION + +In budgeted batch classification setting, the predictive model receives a batch of $M$ instances and a computational budget $B$ for classifying all $M$ instances. In this setting, we use dynamic evaluation: we perform early-exiting of “easy” examples at early classifiers whilst propagating “hard” examples through the entire network, using the procedure described in Section 4. + +Baselines. On ImageNet, we compare the dynamically evaluated MSDNet with five ResNets (He et al., 2016) and five DenseNets (Huang et al., 2017), AlexNet (Krizhevsky et al., 2012), and GoogleLeNet (Szegedy et al., 2015); see the supplementary material for details. We also evaluate an ensemble of the five ResNets that uses exactly the same dynamic-evaluation procedure as MSDNets at test time: “easy” images are only propagated through the smallest ResNet-10, whereas “hard” images are classified by all five ResNet models (predictions are averaged across all evaluated networks in the ensemble). We classify batches of $M = 1 2 8$ images. + +On CIFAR-100, we compare MSDNet with several highly competitive baselines, including ResNets (He et al., 2016), DenseNets (Huang et al., 2017) of varying sizes, Stochastic Depth Networks (Huang et al., 2016), Wide ResNets (Zagoruyko & Komodakis, 2016) and FractalNets (Larsson et al., 2017). We also compare MSDNet to the $\mathrm { R e s N e t ^ { M C } }$ and DenseNetMC models that were used in Section 5.1, using dynamic evaluation at test time. We denote these baselines as $R e s N e t ^ { M C }$ / DenseNetMC with early-exits. To prevent the result plots from becoming too cluttered, we present CIFAR-100 results with dynamically evaluated ensembles in the supplementary material. We classify batches of $M = 2 5 6$ images at test time. + +Budgeted batch classification results on ImageNet are shown in the left panel of Figure 7. We trained three MSDNets with different depths, each of which covers a different range of computational budgets. We plot the performance of each MSDNet as a gray curve; we select the best model for each budget based on its accuracy on the validation set, and plot the corresponding accuracy as a black curve. The plot shows that the predictions of MSDNets with dynamic evaluation are substantially more accurate than those of ResNets and DenseNets that use the same amount of computation. For instance, with an average budget of $1 . 7 \times 1 0 ^ { 9 }$ FLOPs, MSDNet achieves a top-1 accuracy of ${ \sim } 7 5 \%$ , which is ${ \sim } 6 \%$ higher than that achieved by a ResNet with the same number of FLOPs. Compared to the computationally efficient DenseNets, MSDNet uses $\sim 2 - 3 \times$ times fewer FLOPs to achieve the same classification accuracy. Moreover, MSDNet with dynamic evaluation allows for very precise tuning of the computational budget that is consumed, which is not possible with individual ResNet or DenseNet models. The ensemble of ResNets or DenseNets with dynamic evaluation performs on par with or worse than their individual counterparts (but they do allow for setting the computational budget very precisely). + +![](images/047b8902b777ad1edf01cc1798508fdcb86d98ce8b5ec9a2971efe96719d19ec.jpg) +Figure 7: Accuracy (top-1) of budgeted batch classification models as a function of average computational budget per image the on ImageNet (left) and CIFAR-100 (right) datasets. Higher is better. + +The right panel of Figure 7 shows our results on CIFAR-100. The results show that MSDNets consistently outperform all baselines across all budgets. Notably, MSDNet performs on par with a 110- layer ResNet using only 1/10th of the computational budget and it is up to $\sim 5$ times more efficient than DenseNets, Stochastic Depth Networks, Wide ResNets, and FractalNets. Similar to results in the anytime-prediction setting, MSDNet substantially outperform ResNetsMC and DenseNets $_ { M C }$ with multiple intermediate classifiers, which provides further evidence that the coarse features in the MSDNet are important for high performance in earlier layers. + +Visualization. To illustrate the ability of our approach to reduce the computational requirements for classifying “easy” examples, we show twelve randomly sampled test images from two ImageNet classes in Figure 6. The top row shows “easy” examples that were correctly classified and exited by the first classifier. The bottom row shows “hard” examples that would have been incorrectly classified by the first classifier but were passed on because its uncertainty was too high. The figure suggests that early classifiers recognize prototypical class examples, whereas the last classifier recognizes non-typical images. + +![](images/105e212787f02ba3ad81fe470cd4782e898e2825cb6b2147833e9bbcf9aa8bfb.jpg) +Figure 6: Sampled images from the ImageNet classes Red wine and Volcano. Top row: images exited from the first classifier of a MSDNet with correct prediction; Bottom row: images failed to be correctly classified at the first classifier but were correctly predicted and exited at the last layer. + +# 5.3 MORE COMPUTATIONALLY EFFICIENT DENSENETS + +Here, we discuss an interesting finding during our exploration of the MSDNet architecture. We found that following the DenseNet structure to design our network, i.e., by keeping the number of output channels (or growth rate) the same at all scales, did not lead to optimal results in terms of the accuracy-speed trade-off. The main reason for this is that compared to network architectures like ResNets, the DenseNet structure tends to apply more filters on the high-resolution feature maps in the network. This helps to reduce the number of parameters in the model, but at the same time, it greatly increases the computational cost. We tried to modify DenseNets by doubling the growth rate after each transition layer, so that more filters are applied to low-resolution feature maps. It turns out that the resulting network, which we denote as DenseNet\*, significantly outperform the original DenseNet in terms of computational efficiency. + +![](images/f82f1fa9b6ae7df691196495c61b67a76cd3e63f7fdf3cb3be3f5fd5bc49b88b.jpg) +Figure 8: Test accuracy of DenseNet\* on CIFAR-100 under the anytime learning setting (left) and the budgeted batch setting (right). + +We experimented with DenseNet\* in our two settings with test time budget constraints. The left panel of Figure 8 shows the anytime prediction performance of an ensemble of DenseNets\* of varying depths. It outperforms the ensemble of original DenseNets of varying depth by a large margin, but is still slightly worse than MSDNets. In the budgeted batch budget setting, DenseNet\* also leads to significantly higher accuracy over its counterpart under all budgets, but is still substantially outperformed by MSDNets. + +# 6 CONCLUSION + +We presented the MSDNet, a novel convolutional network architecture, optimized to incorporate CPU budgets at test-time. Our design is based on two high-level design principles, to generate and maintain coarse level features throughout the network and to inter-connect the layers with dense connectivity. The former allows us to introduce intermediate classifiers even at early layers and the latter ensures that these classifiers do not interfere with each other. The final design is a two dimensional array of horizontal and vertical layers, which decouples depth and feature coarseness. Whereas in traditional convolutional networks features only become coarser with increasing depth, the MSDNet generates features of all resolutions from the first layer on and maintains them throughout. The result is an architecture with an unprecedented range of efficiency. A single network can outperform all competitive baselines on an impressive range of computational budgets ranging from highly limited CPU constraints to almost unconstrained settings. + +As future work we plan to investigate the use of resource-aware deep architectures beyond object classification, e.g. image segmentation (Long et al., 2015). 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Figure 9 gives an illustration of the reduced network. The convolutional layer functions in the first layer, $h _ { 1 } ^ { s }$ , denote a sequence of $3 { \times } 3$ convolutions (Conv), batch normalization (BN; Ioffe & Szegedy (2015)), and rectified linear unit (ReLU) activation. In the computation of $\tilde { h } _ { 1 } ^ { s }$ , down-sampling is performed by applying convolutions using strides that are powers of two. For subsequent feature layers, the transformations $h _ { \ell } ^ { s }$ and $\tilde { h } _ { \ell } ^ { s }$ are defined following the design in DenseNets (Huang et al., 2017): Conv $( 1 \times 1 )$ -BN-ReLU-Conv $\left( 3 \times 3 \right)$ -BN-ReLU. We set the number of output channels of the three scales to 6, 12, and 24, respectively. Each classifier has two down-sampling convolutional layers with 128 dimensional $3 \times 3$ filters, followed by a $2 \times 2$ average pooling layer and a linear layer. + +The MSDNet used for ImageNet has four scales, respectively producing 16, 32, 64, and 64 feature maps at each layer. The network reduction is also applied to reduce computational cost. The original images are first transformed by a $7 \times 7$ convolution and a $3 \times 3$ max pooling (both with stride 2), before entering the first layer of MSDNets. The classifiers have the same structure as those used for the CIFAR datasets, except that the number of output channels of each convolutional layer is set to be equal to the number of its input channels. + +![](images/d0de6b68dce803aca2a16de6affb943796e7f0bf1bb1eac9cef6c06e5c471db7.jpg) +Figure 9: Illustration of an MSDNet with network reduction. The network has $S = 3$ scales, and it is divided into three blocks, which maintain a decreasing number of scales. A transition layer is placed between two contiguous blocks. + +Network architecture for anytime prediction. The MSDNet used in our anytime-prediction experiments has 24 layers (each layer corresponds to a column in Fig. 1 of the main paper), using the reduced network with transition layers as described in Section 4. The classifiers operate on the output of the $2 \times ( i { + } 1 ) ^ { \mathrm { t h } }$ layers, with $i = 1 , \ldots , 1 1$ . On ImageNet, we use MSDNets with four scales, and the $i ^ { \mathrm { { t h } } }$ classifier operates on the $( k \times i + 3 ) ^ { \mathrm { t h } }$ layer (with $i = 1 , \ldots , 5$ ), where $k = 4 , 6$ and 7. For simplicity, the losses of all the classifiers are weighted equally during training. + +Network architecture for budgeted batch setting. The MSDNets used here for the two CIFAR datasets have depths ranging from 10 to 36 layers, using the reduced network with transition layers as described in Section 4. The $k ^ { \mathrm { { t h } } }$ classifier is attached to the $( \sum _ { i = 1 } ^ { k } i ) ^ { \mathrm { t h } }$ layer. The MSDNets used for ImageNet are the same as those described for the anytime learning setting. + +$\mathbf { R e s N e t } ^ { \mathbf { M C } }$ and DenseNetMC. The ResNetMC has 62 layers, with 10 residual blocks at each spatial resolution (for three resolutions): we train early-exit classifiers on the output of the $4 ^ { \mathrm { t h } }$ and $8 ^ { \mathrm { t h } }$ residual blocks at each resolution, producing a total of 6 intermediate classifiers (plus the final classification layer). The DenseNetMC consists of 52 layers with three dense blocks and each of them has 16 layers. The six intermediate classifiers are attached to the $6 ^ { \mathrm { { t h } } }$ and $1 2 ^ { \mathrm { t h } }$ layer in each block, also with dense connections to all previous layers in that block. + +# B ADDITIONAL RESULTS + +# B.1 ABLATION STUDY + +We perform additional experiments to shed light on the contributions of the three main components of MSDNet, viz., multi-scale feature maps, dense connectivity, and intermediate classifiers. + +We start from an MSDNet with six intermediate classifiers and remove the three main components one at a time. To make our comparisons fair, we keep the computational costs of the full networks similar, at around $3 . 0 \times 1 0 ^ { 8 }$ FLOPs, by adapting the network width, i.e., number of output channels at each layer. After removing all the three components in an MSDNet, we obtain a regular VGG-like convolutional network. We show the classification accuracy of all classifiers in a model in the left panel of Figure 10. Several observations can be made: 1. the dense connectivity is crucial for the performance of MSDNet and removing it hurts the overall accuracy drastically (orange vs. black curve); 2. removing multi-scale convolution hurts the accuracy only in the lower budget regions, which is consistent with our mo + +![](images/36de99964e9e6ab4f6f1967fffee5d8d4816d0686cfae2751a7e8cef27def4c9.jpg) +Figure 10: Ablation study (on CIFAR-100) of MSDNets that shows the effect of dense connectivity, multi-scale features, and intermediate classifiers. Higher is better. + +tivation that the multi-scale design introduces discriminative features early on; 3. the final canonical CNN (star) performs similarly as MSDNet under the specific budget that matches its evaluation cost exactly, but it is unsuited for varying budget constraints. The final CNN performs substantially better at its particular budget region than the model without dense connectivity (orange curve). This suggests that dense connectivity is particularly important in combination with multiple classifiers. + +# B.2 RESULTS ON CIFAR-10 + +For the CIFAR-10 dataset, we use the same MSDNets and baseline models as we used for CIFAR100, except that the networks used here have a 10-way fully connected layer at the end. The results under the anytime learning setting and the batch computational budget setting are shown in the left and right panel of Figure 11, respectively. Similar to what we have observed from the results on CIFAR-100 and ImageNet, MSDNets outperform all the baselines by a significant margin in both settings. As in the experiments presented in the main paper, ResNet and DenseNet models with multiple intermediate classifiers perform relatively poorly. + +![](images/adf6b406c38b10a1e3fedffbeafa7e030553b2ab7ceb4700cb0a1fb272d4ff92.jpg) +Figure 11: Classification accuracies on the CIFAR-10 dataset in the anytime-prediction setting (left) and the budgeted batch setting (right). \ No newline at end of file diff --git a/parse/train/Hk2aImxAb/Hk2aImxAb_content_list.json b/parse/train/Hk2aImxAb/Hk2aImxAb_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e1f3517b0a7c31b555d20498a6e1dcafed30c033 --- /dev/null +++ b/parse/train/Hk2aImxAb/Hk2aImxAb_content_list.json @@ -0,0 +1,1186 @@ +[ + { + "type": "text", + "text": "MULTI-SCALE DENSE NETWORKS FOR RESOURCE EFFICIENT IMAGE CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 99, + 794, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Gao Huang Cornell University ", + "bbox": [ + 184, + 170, + 305, + 199 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Danlu Chen Fudan University ", + "bbox": [ + 351, + 170, + 465, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tianhong Li Tsinghua University ", + "bbox": [ + 511, + 171, + 647, + 199 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Felix Wu Cornell University ", + "bbox": [ + 689, + 170, + 813, + 199 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Laurens van der Maaten Facebook AI Research ", + "bbox": [ + 183, + 219, + 357, + 248 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Kilian Weinberger Cornell University ", + "bbox": [ + 529, + 219, + 660, + 248 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 285, + 544, + 299 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper we investigate image classification with computational resource limits at test time. Two such settings are: 1. anytime classification, where the network’s prediction for a test example is progressively updated, facilitating the output of a prediction at any time; and 2. budgeted batch classification, where a fixed amount of computation is available to classify a set of examples that can be spent unevenly across “easier” and “harder” inputs. In contrast to most prior work, such as the popular Viola and Jones algorithm, our approach is based on convolutional neural networks. We train multiple classifiers with varying resource demands, which we adaptively apply during test time. To maximally re-use computation between the classifiers, we incorporate them as early-exits into a single deep convolutional neural network and inter-connect them with dense connectivity. To facilitate high quality classification early on, we use a two-dimensional multi-scale network architecture that maintains coarse and fine level features all-throughout the network. Experiments on three image-classification tasks demonstrate that our framework substantially improves the existing state-of-the-art in both settings. ", + "bbox": [ + 233, + 309, + 764, + 517 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 536, + 336, + 551 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent years have witnessed a surge in demand for applications of visual object recognition, for instance, in self-driving cars (Bojarski et al., 2016) and content-based image search (Wan et al., 2014). This demand has in part been fueled through the promise generated by the astonishing progress of convolutional networks (CNNs) on visual object recognition benchmark competition datasets, such as ILSVRC (Deng et al., 2009) and COCO (Lin et al., 2014), where state-of-the-art models may have even surpassed human-level performance (He et al., 2015; 2016). ", + "bbox": [ + 174, + 566, + 825, + 650 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, the requirements of such competitions differ from realworld applications, which tend to incentivize resource-hungry models with high computational demands at inference time. For example, the COCO 2016 competition was won by a large ensemble of computationally intensive $\\mathrm { \\dot { C } N N s ^ { 1 } }$ — a model likely far too computationally expensive for any resource-aware application. Although much smaller models would also obtain decent error, very large, computationally intensive models seem necessary to correctly classify the hard examples that make up the bulk of the remaining misclassifications of modern algorithms. To illustrate this point, Figure 1 shows two images of horses. The left image depicts a horse in canonical pose and is easy to classify, whereas the right image is taken from a rare viewpoint and is likely in the tail of the data distribution. Computationally intensive models are needed to classify such tail examples correctly, but are wasteful when applied to canonical images such as the left one. ", + "bbox": [ + 174, + 659, + 612, + 856 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/2e2e080304ad2c7be1e5890d21d32ed57f0cf575c439df5f869a545d4f732d05.jpg", + "image_caption": [ + "Figure 1: Two images containing a horse. The left image is canonical and easy to detect even with a small model, whereas the right image requires a computationally more expensive network architecture. (Copyright Pixel Addict and Doyle (CC BY-ND 2.0).) " + ], + "image_footnote": [], + "bbox": [ + 627, + 664, + 823, + 732 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 178, + 852, + 823, + 866 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In real-world applications, computation directly translates into power consumption, which should be minimized for environmental and economical reasons, and is a scarce commodity on mobile devices. This begs the question: why do we choose between either wasting computational resources by applying an unnecessarily computationally expensive model to easy images, or making mistakes by using an efficient model that fails to recognize difficult images? Ideally, our systems should automatically use small networks when test images are easy or computational resources limited, and use big networks when test images are hard or computation is abundant. ", + "bbox": [ + 178, + 873, + 820, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 172 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Such systems would be beneficial in at least two settings with computational constraints at testtime: anytime prediction, where the network can be forced to output a prediction at any given point in time; and budgeted batch classification, where a fixed computational budget is shared across a large set of examples which can be spent unevenly across “easy” and “hard” examples. A practical use-case of anytime prediction is in mobile apps on Android devices: in 2015, there existed 24, 093 distinct Android devices2, each with its own distinct computational limitations. It is infeasible to train a different network that processes video frame-by-frame at a fixed framerate for each of these devices. Instead, you would like to train a single network that maximizes accuracy on all these devices, within the computational constraints of that device. The budget batch classification setting is ubiquitous in large-scale machine learning applications. Search engines, social media companies, on-line advertising agencies, all must process large volumes of data on limited hardware resources. For example, as of 2010, Google Image Search had over 10 Billion images indexed3, which has likely grown to over 1 Trillion since. Even if a new model to process these images is only 1/10s slower per image, this additional cost would add 3170 years of CPU time. In the budget batch classification setting, companies can improve the average accuracy by reducing the amount of computation spent on “easy” cases to save up computation for “hard” cases. ", + "bbox": [ + 174, + 181, + 825, + 401 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Motivated by prior work in computer vision on resource-efficient recognition (Viola & Jones, 2001), we aim to develop CNNs that “slice” the computation and process these slices one-by-one, stopping the evaluation once the CPU time is depleted or the classification sufficiently certain (through “early exits”). Unfortunately, the architecture of CNNs is inherently at odds with the introduction of early exits. CNNs learn the data representation and the classifier jointly, which leads to two problems with early exits: 1. The features in the last layer are extracted directly to be used by the classifier, whereas earlier features are not. The inherent dilemma is that different kinds of features need to be extracted depending on how many layers are left until the classification. 2. The features in different layers of the network may have different scale. Typically, the first layers of a deep nets operate on a fine scale (to extract low-level features), whereas later layers transition (through pooling or strided convolution) to coarse scales that allow global context to enter the classifier. Both scales are needed but happen at different places in the network. ", + "bbox": [ + 174, + 410, + 825, + 575 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We propose a novel network architecture that addresses both of these problems through careful design changes, allowing for resource-efficient image classification. Our network uses a cascade of intermediate classifiers throughout the network. The first problem, of classifiers altering the internal representation, is addressed through the introduction of dense connectivity (Huang et al., 2017). By connecting all layers to all classifiers, features are no longer dominated by the most imminent earlyexit and the trade-off between early or later classification can be performed elegantly as part of the loss function. The second problem, the lack of coarse-scale features in early layers, is addressed by adopting a multi-scale network structure. At each layer we produce features of all scales (fine-tocoarse), which facilitates good classification early on but also extracts low-level features that only become useful after several more layers of processing. Our network architecture is illustrated in Figure 2, and we refer to it as Multi-Scale DenseNet (MSDNet). ", + "bbox": [ + 174, + 583, + 825, + 736 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We evaluate MSDNets on three image-classification datasets. In the anytime classification setting, we show that it is possible to provide the ability to output a prediction at any time while maintain high accuracies throughout. In the budget batch classification setting we show that MSDNets can be effectively used to adapt the amount of computation to the difficulty of the example to be classified, which allows us to reduce the computational requirements of our models drastically whilst performing on par with state-of-the-art CNNs in terms of overall classification accuracy. To our knowledge this is the first deep learning architecture of its kind that allows dynamic resource adaptation with a single model and obtains competitive results throughout. ", + "bbox": [ + 174, + 742, + 825, + 854 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/393456a645a070145e34d48f149ac9a7ed180043707a693bf4407c8f3be79371.jpg", + "image_caption": [ + "Figure 2: Illustration of the first four layers of an MSDNet with three scales. The horizontal direction corresponds to the layer direction (depth) of the network. The vertical direction corresponds to the scale of the feature maps. Horizontal arrows indicate a regular convolution operation, whereas diagonal and vertical arrows indicate a strided convolution operation. Classifiers only operate on feature maps at the coarsest scale. Connections across more than one layer are not drawn explicitly: they are implicit through recursive concatenations. " + ], + "image_footnote": [], + "bbox": [ + 181, + 83, + 826, + 213 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 318, + 344, + 334 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We briefly review related prior work on computation-efficient networks, memory-efficient networks, and resource-sensitive machine learning, from which our network architecture draws inspiration. ", + "bbox": [ + 176, + 347, + 823, + 376 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Computation-efficient networks. Most prior work on (convolutional) networks that are computationally efficient at test time focuses on reducing model size after training. In particular, many studies propose to prune weights (LeCun et al., 1989; Hassibi et al., 1993; Li et al., 2017) or quantize weights (Hubara et al., 2016; Rastegari et al., 2016) during or after training. These approaches are generally effective because deep networks often have a substantial number of redundant weights that can be pruned or quantized without sacrificing (and sometimes even improving) performance. Prior work also studies approaches that directly learn compact models with less parameter redundancy. For example, the knowledge-distillation method (Bucilua et al., 2006; Hinton et al., 2014) trains small student networks to reproduce the output of a much larger teacher network or ensemble. Our work differs from those approaches in that we train a single model that trades off computation for accuracy at test time without any re-training or finetuning. Indeed, weight pruning and knowledge distillation can be used in combination with our approach, and may lead to further improvements. ", + "bbox": [ + 174, + 382, + 825, + 549 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Resource-efficient machine learning. Various prior studies explore computationally efficient variants of traditional machine-learning models (Viola & Jones, 2001; Grubb & Bagnell, 2012; Karayev et al., 2014; Trapeznikov & Saligrama, 2013; Xu et al., 2012; 2013; Nan et al., 2015; Wang et al., 2015). Most of these studies focus on how to incorporate the computational requirements of computing particular features in the training of machine-learning models such as (gradient-boosted) decision trees. Whilst our study is certainly inspired by these results, the architecture we explore differs substantially: most prior work exploits characteristics of machine-learning models (such as decision trees) that do not apply to deep networks. Our work is possibly most closely related to recent work on FractalNets (Larsson et al., 2017), which can perform anytime prediction by progressively evaluating subnetworks of the full network. FractalNets differ from our work in that they are not explicitly optimized for computation efficiency and consequently our experiments show that MSDNets substantially outperform FractalNets. Our dynamic evaluation strategy for reducing batch computational cost is closely related to the the adaptive computation time approach (Graves, 2016; Figurnov et al., 2016), and the recently proposed method of adaptively evaluating neural networks (Bolukbasi et al., 2017). Different from these works, our method adopts a specially designed network with multiple classifiers, which are jointly optimized during training and can directly output confidence scores to control the evaluation process for each test example. The adaptive computation time method (Graves, 2016) and its extension (Figurnov et al., 2016) also perform adaptive evaluation on test examples to save batch computational cost, but focus on skipping units rather than layers. In (Odena et al., 2017), a “composer”model is trained to construct the evaluation network from a set of sub-modules for each test example. By contrast, our work uses a single CNN with multiple intermediate classifiers that is trained end-to-end. The Feedback Networks (Zamir et al., 2016) enable early predictions by making predictions in a recurrent fashion, which heavily shares parameters among classifiers, but is less efficient in sharing computation. ", + "bbox": [ + 173, + 555, + 825, + 888 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Related network architectures. Our network architecture borrows elements from neural fabrics (Saxena & Verbeek, 2016) and others (Zhou et al., 2015; Jacobsen et al., 2017; Ke et al., 2016) ", + "bbox": [ + 173, + 896, + 821, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/6c59242daaab86116b0562f24208342d867eb57b7d0db805e434894dcf77ee2c.jpg", + "image_caption": [ + "Figure 3: Relative accuracy of the intermediate classifier (left) and the final classifier (right) when introducing a single intermediate classifier at different layers in a ResNet, DenseNet and MSDNet. All experiments were performed on the CIFAR-100 dataset. Higher is better. " + ], + "image_footnote": [], + "bbox": [ + 187, + 93, + 792, + 265 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "to rapidly construct a low-resolution feature map that is amenable to classification, whilst also maintaining feature maps of higher resolution that are essential for obtaining high classification accuracy. Our design differs from the neural fabrics (Saxena & Verbeek, 2016) substantially in that MSDNets have a reduced number of scales and no sparse channel connectivity or up-sampling paths. MSDNets are at least one order of magnitude more efficient and typically more accurate — for example, an MSDNet with less than 1 million parameters obtains a test error below $7 . 0 \\%$ on CIFAR-10 (Krizhevsky & Hinton, 2009), whereas Saxena & Verbeek (2016) report $7 . 4 3 \\%$ with over 20 million parameters. We use the same feature-concatenation approach as DenseNets (Huang et al., 2017), which allows us to bypass features optimized for early classifiers in later layers of the network. Our architecture is related to deeply supervised networks (Lee et al., 2015) in that it incorporates classifiers at multiple layers throughout the network. In contrast to all these prior architectures, our network is specifically designed to operate in resource-aware settings. ", + "bbox": [ + 173, + 330, + 825, + 496 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 PROBLEM SETUP ", + "text_level": 1, + "bbox": [ + 176, + 512, + 348, + 529 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We consider two settings that impose computational constraints at prediction time. ", + "bbox": [ + 176, + 539, + 710, + 554 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Anytime prediction. In the anytime prediction setting (Grubb & Bagnell, 2012), there is a finite computational budget $B > 0$ available for each test example $\\mathbf { x }$ . The computational budget is nondeterministic, and varies per test instance. It is determined by the occurrence of an event that requires the model to output a prediction immediately. We assume that the budget is drawn from some joint distribution $P ( \\mathbf { x } , B )$ . In some applications $P ( B )$ may be independent of $P ( \\mathbf { x } )$ and can be estimated. For example, if the event is governed by a Poisson process, $P ( B )$ is an exponential distribution. We denote the loss of a model $f ( \\mathbf { x } )$ that has to produce a prediction for instance $\\mathbf { x }$ within budget $B$ by $L ( f ( \\mathbf { x } ) , B )$ . The goal of an anytime learner is to minimize the expected loss under the budget distribution: $L ( f ) = \\mathbb { E } \\left[ L ( f ( \\mathbf { x } ) , B ) \\right] _ { P ( \\mathbf { x } , B ) } .$ Here, $L ( \\cdot )$ denotes a suitable loss function. As is common in the empirical risk minimization framework, the expectation under $P ( \\mathbf { x } , B )$ may be estimated by an average over samples from $P ( \\mathbf { x } , B )$ . ", + "bbox": [ + 173, + 560, + 825, + 717 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Budgeted batch classification. In the budgeted batch classification setting, the model needs to classify a set of examples $\\mathcal { D } _ { t e s t } = \\{ \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { M } \\}$ within a finite computational budget $B > 0$ that is known in advance. The learner aims to minimize the loss across all examples in $\\mathcal { D } _ { t e s t }$ within a cumulative cost bounded by $B$ , which we denote by $L ( f ( \\mathcal { D } _ { t e s t } ) , B )$ for some suitable loss function $L ( \\cdot )$ . It can potentially do so by spending less than BM computation on classifying an “easy” example whilst using more than $\\textstyle { \\frac { B } { M } }$ computation on classifying a “difficult” example. Therefore, the budget considered here is a soft constraint when we have a large batch of testing samples. ", + "bbox": [ + 174, + 723, + 825, + 825 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 MULTI-SCALE DENSE CONVOLUTIONAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 840, + 643, + 857 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A straightforward solution to the two problems introduced in Section 3 is to train multiple networks of increasing capacity, and sequentially evaluate them at test time (as in Bolukbasi et al. (2017)). In the anytime setting the evaluation can be stopped at any point and the most recent prediction is returned. In the batch setting, the evaluation is stopped prematurely the moment a network classifies the test sample with sufficient confidence. When the resources are so limited that the execution is terminated after the first network, this approach is optimal because the first network is trained for exactly this computational budget without compromises. However, in both settings, this scenario is rare. In the more common scenario where some test samples can require more processing time than others the approach is far from optimal because previously learned features are never re-used across the different networks. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 186 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "An alternative solution is to build a deep network with a cascade of classifiers operating on the features of internal layers: in such a network features computed for an earlier classifier can be re-used by later classifiers. However, na¨ıvely attaching intermediate early-exit classifiers to a stateof-the-art deep network leads to poor performance. ", + "bbox": [ + 174, + 194, + 823, + 251 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "There are two reasons why intermediate early-exit classifiers hurt the performance of deep neural networks: early classifiers lack coarse-level features and classifiers throughout interfere with the feature generation process. In this section we investigate these effects empirically (see Figure 3) and, in response to our findings, propose the MSDNet architecture illustrated in Figure 2. ", + "bbox": [ + 174, + 257, + 825, + 313 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Problem: The lack of coarse-level features. Traditional neural networks learn features of fine scale in early layers and coarse scale in later layers (through repeated convolution, pooling, and strided convolution). Coarse scale features in the final layers are important to classify the content of the whole image into a single class. Early layers lack coarse-level features and early-exit classifiers attached to these layers will likely yield unsatisfactory high error rates. To illustrate this point, we attached4 intermediate classifiers to varying layers of a ResNet (He et al., 2016) and a DenseNet (Huang et al., 2017) on the CIFAR-100 dataset (Krizhevsky & Hinton, 2009). The blue and red dashed lines in the left plot of Figure 3 show the relative accuracies of these classifiers. All three plots gives rise to a clear trend: the accuracy of a classifier is highly correlated with its position within the network. Particularly in the case of the ResNet (blue line), one can observe a visible “staircase” pattern, with big improvements after the 2nd and 4th classifiers — located right after pooling layers. ", + "bbox": [ + 173, + 318, + 825, + 483 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Solution: Multi-scale feature maps. To address this issue, MSDNets maintain a feature representation at multiple scales throughout the network, and all the classifiers only use the coarse-level features. The feature maps at a particular layer5 and scale are computed by concatenating the results of one or two convolutions: 1. the result of a regular convolution applied on the same-scale features from the previous layer (horizontal connections) and, if possible, 2. the result of a strided convolution applied on the finer-scale feature map from the previous layer (diagonal connections). The horizontal connections preserve and progress high-resolution information, which facilitates the construction of high-quality coarse features in later layers. The vertical connections produce coarse features throughout that are amenable to classification. The dashed black line in Figure 3 shows that MSDNets substantially increase the accuracy of early classifiers. ", + "bbox": [ + 173, + 491, + 825, + 630 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Problem: Early classifiers interfere with later classifiers. The right plot of Figure 3 shows the accuracies of the final classifier as a function of the location of a single intermediate classifier, relative to the accuracy of a network without intermediate classifiers. The results show that the introduction of an intermediate classifier harms the final ResNet classifier (blue line), reducing its accuracy by up to $7 \\%$ . We postulate that this accuracy degradation in the ResNet may be caused by the intermediate classifier influencing the early features to be optimized for the short-term and not for the final layers. This improves the accuracy of the immediate classifier but collapses information required to generate high quality features in later layers. This effect becomes more pronounced when the first classifier is attached to an earlier layer. ", + "bbox": [ + 174, + 635, + 825, + 758 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Solution: Dense connectivity. By contrast, the DenseNet (red line) suffers much less from this effect. Dense connectivity (Huang et al., 2017) connects each layer with all subsequent layers and allows later layers to bypass features optimized for the short-term, to maintain the high accuracy of the final classifier. If an earlier layer collapses information to generate short-term features, the lost information can be recovered through the direct connection to its preceding layer. The final classifier’s performance becomes (more or less) independent of the location of the intermediate classifier. As far as we know, this is the first paper that discovers that dense connectivity is an important element to early-exit classifiers in deep networks, and we make it an integral design choice in MSDNets. ", + "bbox": [ + 173, + 766, + 825, + 849 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/8de8c69d9d36a341a0373469c5e2695c92df31ad7bca8cfbfd76709c742d677e.jpg", + "image_caption": [ + "Figure 4: The output $\\mathbf { x } _ { \\ell } ^ { s }$ of layer $\\ell$ at the $s ^ { \\mathrm { t h } }$ scale in a MSDNet. Herein, [. . . ] denotes the concatenation operator, $h _ { \\ell } ^ { s } ( \\cdot )$ a regular convolution transformation, and $\\tilde { h } _ { \\ell } ^ { s } ( \\cdot )$ a strided convolutional. Note that the outputs of $h _ { \\ell } ^ { s }$ and $\\tilde { h } _ { \\ell } ^ { s }$ have the same feature map size; their outputs are concatenated along the channel dimension. " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 826, + 226 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 295, + 825, + 337 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 THE MSDNET ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 357, + 418, + 371 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The MSDNet architecture is illustrated in Figure 2. We present its main components below. Additional details on the architecture are presented in Appendix A. ", + "bbox": [ + 174, + 383, + 820, + 412 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "First layer. The first layer $\\ell = 1 $ ) is unique as it includes vertical connections in Figure 2. Its main purpose is to “seed” representations on all $S$ scales. One could view its vertical layout as a miniature “S-layers” convolutional network $S { = } 3$ in Figure 2). Let us denote the output feature maps at layer $\\ell$ and scale $s$ as $\\mathbf { x } _ { \\ell } ^ { s }$ and the original input image as $\\mathbf { x } _ { 0 } ^ { 1 }$ . Feature maps at coarser scales are obtained via down-sampling. The output $\\mathbf { x } _ { 1 } ^ { s }$ of the first layer is formally given in the top row of Figure 4. ", + "bbox": [ + 174, + 419, + 825, + 489 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Subsequent layers. Following Huang et al. (2017), the output feature maps $\\mathbf { x } _ { \\ell } ^ { s }$ produced at subsequent layers, $\\ell > 1$ , and scales, $s$ , are a concatenation of transformed feature maps from all previous feature maps of scale $s$ and $s - 1$ (if $s > 1$ ). Formally, the $\\ell \\cdot$ -th layer of our network outputs a set of features at $S$ scales $\\big \\{ \\mathbf { x } _ { \\ell } ^ { 1 } , \\dots , \\mathbf { x } _ { \\ell } ^ { S } \\big \\}$ , given in the last row of Figure 4. ", + "bbox": [ + 174, + 496, + 825, + 553 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Classifiers. The classifiers in MSDNets also follow the dense connectivity pattern within the coarsest scale, $S$ , i.e., the classifier at layer $\\ell$ uses all the features $\\left[ \\mathbf { x } _ { 1 } ^ { S } , \\ldots , \\mathbf { x } _ { \\ell } ^ { S } \\right]$ . Each classifier consists of two convolutional layers, followed by one average pooling layer and one linear layer. In practice, we only attach classifiers to some of the intermediate layers, and we let $f _ { k } ( \\cdot )$ denote the $k ^ { \\mathrm { { t h } } }$ classifier. During testing in the anytime setting we propagate the input through the network until the budget is exhausted and output the most recent prediction. In the batch budget setting at test time, an example traverses the network and exits after classifier $f _ { k }$ if its prediction confidence (we use the maximum value of the softmax probability as a confidence measure) exceeds a pre-determined threshold $\\theta _ { k }$ . Before training, we compute the computational cost, $C _ { k }$ , required to process the network up to the $k ^ { \\mathrm { { t h } } }$ classifier. We denote by $0 < q \\le 1$ a fixed exit probability that a sample that reaches a classifier will obtain a classification with sufficient confidence to exit. We assume that $q$ is constant across all layers, which allows us to compute the probability that a sample exits at classifier $k$ as: $q _ { k } = z ( 1 - q ) ^ { \\bar { k ^ { - 1 } } q }$ , where $z$ is a normalizing constant that ensures that $\\begin{array} { r } { \\sum _ { k } p ( q _ { k } ) = 1 } \\end{array}$ . At test time, we need to ensure that the overall cost of classifying all samples in $\\mathcal { D } _ { t e s t }$ does not exceed our budget $B$ (in expectation). This gives rise to the constraint $\\begin{array} { r } { \\left| \\mathcal { D } _ { t e s t } \\right| \\sum _ { k } q _ { k } C _ { k } \\le B } \\end{array}$ . We can solve this constraint for $q$ and determine the thresholds $\\theta _ { k }$ on a validation set in such a way that approximately $| \\mathcal { D } _ { t e s t } | q _ { k }$ validation samples exit at the $k ^ { \\mathrm { { t h } } }$ classifier. ", + "bbox": [ + 173, + 559, + 825, + 794 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Loss functions. During training we use cross entropy loss functions $L ( f _ { k } )$ for all classifiers and minimize a weighted cumulative loss: $\\begin{array} { r } { \\frac { 1 } { | \\mathcal { D } | } \\sum _ { ( \\mathbf { x } , y ) \\in \\mathcal { D } } \\mathbf { \\tilde { \\sum } } _ { k } w _ { k } L ( f _ { k } ) } \\end{array}$ . Herein, $\\mathcal { D }$ denotes the training set and $w _ { k } \\ge 0$ the weight of the $k$ -th classifier. If the budget distribution $P ( B )$ is known, we can use the weights $w _ { k }$ to incorporate our prior knowledge about the budget $B$ in the learning. Empirically, we find that using the same weight for all loss functions (i.e., setting $\\forall k : w _ { k } = 1$ ) works well in practice. ", + "bbox": [ + 174, + 801, + 825, + 888 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Network reduction and lazy evaluation. There are two straightforward ways to further reduce the computational requirements of MSDNets. First, it is inefficient to maintain all the finer scales until the last layer of the network. One simple strategy to reduce the size of the network is by splitting it into $S$ blocks along the depth dimension, and only keeping the coarsest $( S - i + 1 )$ scales in the $i ^ { \\mathrm { t h } }$ block (a schematic layout of this structure is shown in Figure 9). This reduces computational cost for both training and testing. Every time a scale is removed from the network, we add a transition layer between the two blocks that merges the concatenated features using a $1 \\times 1$ convolution and cuts the number of channels in half before feeding the fine-scale features into the coarser scale via a strided convolution (this is similar to the DenseNet-BC architecture of Huang et al. (2017)). Second, since a classifier at layer $\\ell$ only uses features from the coarsest scale, the finer feature maps in layer $\\ell$ (and some of the finer feature maps in the previous $S - 2$ layers) do not influence the prediction of that classifier. Therefore, we group the computation in “diagonal blocks” such that we only propagate the example along paths that are required for the evaluation of the next classifier. This minimizes unnecessary computations when we need to stop because the computational budget is exhausted. We call this strategy lazy evaluation. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 284 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 304, + 326, + 320 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate the effectiveness of our approach on three image classification datasets, i.e., the CIFAR10, CIFAR-100 (Krizhevsky & Hinton, 2009) and ILSVRC 2012 (ImageNet; Deng et al. (2009)) datasets. Code to reproduce all results is available at https://anonymous-url. Details on architectural configurations of MSDNets are described in Appendix A. ", + "bbox": [ + 174, + 334, + 823, + 390 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Datasets. The two CIFAR datasets contain 50, 000 training and 10, 000 test images of $3 2 \\times 3 2$ pixels; we hold out 5, 000 training images as a validation set. The datasets comprise 10 and 100 classes, respectively. We follow He et al. (2016) and apply standard data-augmentation techniques to the training images: images are zero-padded with 4 pixels on each side, and then randomly cropped to produce $3 2 \\times 3 2$ images. Images are flipped horizontally with probability 0.5, and normalized by subtracting channel means and dividing by channel standard deviations. The ImageNet dataset comprises 1, 000 classes, with a total of 1.2 million training images and 50,000 validation images. We hold out 50,000 images from the training set to estimate the confidence threshold for classifiers in MSDNet. We adopt the data augmentation scheme of He et al. (2016) at training time; at test time, we classify a $2 2 4 \\times 2 2 4$ center crop of images that were resized to $2 5 6 \\times 2 5 6$ pixels. ", + "bbox": [ + 174, + 393, + 825, + 532 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Training Details. We train all models using the framework of Gross & Wilber (2016). On the two CIFAR datasets, all models (including all baselines) are trained using stochastic gradient descent (SGD) with mini-batch size 64. We use Nesterov momentum with a momentum weight of 0.9 without dampening, and a weight decay of $1 0 ^ { - 4 }$ . All models are trained for 300 epochs, with an initial learning rate of 0.1, which is divided by a factor 10 after 150 and 225 epochs. We apply the same optimization scheme to the ImageNet dataset, except that we increase the mini-batch size to 256, and all the models are trained for 90 epochs with learning rate drops after 30 and 60 epochs. ", + "bbox": [ + 174, + 537, + 825, + 635 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 ANYTIME PREDICTION ", + "text_level": 1, + "bbox": [ + 174, + 652, + 374, + 666 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the anytime prediction setting, the model maintains a progressively updated distribution over classes, and it can be forced to output its most up-to-date prediction at an arbitrary time. ", + "bbox": [ + 173, + 680, + 821, + 708 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Baselines. There exist several baseline approaches for anytime prediction: FractalNets (Larsson et al., 2017), deeply supervised networks (Lee et al., 2015), and ensembles of deep networks of varying or identical sizes. FractalNets allow for multiple evaluation paths during inference time, which vary in computation time. In the anytime setting, paths are evaluated in order of increasing computation. In our result figures, we replicate the FractalNet results reported in the original paper (Larsson et al., 2017) for reference. Deeply supervised networks introduce multiple early-exit classifiers throughout a network, which are applied on the features of the particular layer they are attached to. Instead of using the original model proposed in Lee et al. (2015), we use the more competitive ResNet and DenseNet architectures (referred to as DenseNet- $B C$ in Huang et al. (2017)) as the base networks in our experiments with deeply supervised networks. We refer to these as $R e s N e t ^ { M C }$ and DenseNetMC, where $M C$ stands for multiple classifiers. Both networks require about $1 . 3 \\times 1 0 ^ { 8 }$ FLOPs when fully evaluated; the detailed network configurations are presented in the supplementary material. In addition, we include ensembles of ResNets and DenseNets of varying or identical sizes. At test time, the networks are evaluated sequentially (in ascending order of network size) to obtain predictions for the test data. All predictions are averaged over the evaluated classifiers. On ", + "bbox": [ + 174, + 715, + 825, + 922 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/0e595351824a020ac424bb128829df21d92c0ecca2b1cea17c58c737e1308b61.jpg", + "image_caption": [ + "Figure 5: Accuracy (top-1) of anytime prediction models as a function of computational budget on the ImageNet (left) and CIFAR-100 (right) datasets. Higher is better. " + ], + "image_footnote": [], + "bbox": [ + 189, + 92, + 810, + 261 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "ImageNet, we compare MSDNet against a highly competitive ensemble of ResNets and DenseNets, with depth varying from 10 layers to 50 layers, and 36 layers to 121 layers, respectively. ", + "bbox": [ + 173, + 314, + 821, + 342 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Anytime prediction results are presented in Figure 5. The left plot shows the top-1 classification accuracy on the ImageNet validation set. Here, for all budgets in our evaluation, the accuracy of MSDNet substantially outperforms the ResNets and DenseNets ensemble. In particular, when the budget ranges from $0 . 1 \\times \\bar { 1 0 } ^ { 1 0 }$ to $0 . 3 \\times 1 0 ^ { 1 0 }$ FLOPs, MSDNet achieves $\\sim 4 \\% - 8 \\%$ higher accuracy. ", + "bbox": [ + 174, + 349, + 825, + 405 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We evaluate more baselines on CIFAR-100 (and CIFAR-10; see supplementary materials). We observe that MSDNet substantially outperforms ResNetsMC and DenseNetsMC at any computational budget within our range. This is due to the fact that after just a few layers, MSDNets have produced low-resolution feature maps that are much more suitable for classification than the high-resolution feature maps in the early layers of ResNets or DenseNets. MSDNet also outperforms the other baselines for nearly all computational budgets, although it performs on par with ensembles when the budget is very small. In the extremely low-budget regime, ensembles have an advantage because their predictions are performed by the first (small) network, which is optimized exclusively for the low budget. However, the accuracy of ensembles does not increase nearly as fast when the budget is increased. The MSDNet outperforms the ensemble as soon as the latter needs to evaluate a second model: unlike MSDNets, this forces the ensemble to repeat the computation of similar low-level features repeatedly. Ensemble accuracies saturate rapidly when all networks are shallow. ", + "bbox": [ + 174, + 411, + 825, + 579 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 BUDGETED BATCH CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 178, + 590, + 462, + 603 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In budgeted batch classification setting, the predictive model receives a batch of $M$ instances and a computational budget $B$ for classifying all $M$ instances. In this setting, we use dynamic evaluation: we perform early-exiting of “easy” examples at early classifiers whilst propagating “hard” examples through the entire network, using the procedure described in Section 4. ", + "bbox": [ + 174, + 609, + 825, + 665 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Baselines. On ImageNet, we compare the dynamically evaluated MSDNet with five ResNets (He et al., 2016) and five DenseNets (Huang et al., 2017), AlexNet (Krizhevsky et al., 2012), and GoogleLeNet (Szegedy et al., 2015); see the supplementary material for details. We also evaluate an ensemble of the five ResNets that uses exactly the same dynamic-evaluation procedure as MSDNets at test time: “easy” images are only propagated through the smallest ResNet-10, whereas “hard” images are classified by all five ResNet models (predictions are averaged across all evaluated networks in the ensemble). We classify batches of $M = 1 2 8$ images. ", + "bbox": [ + 173, + 672, + 825, + 770 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "On CIFAR-100, we compare MSDNet with several highly competitive baselines, including ResNets (He et al., 2016), DenseNets (Huang et al., 2017) of varying sizes, Stochastic Depth Networks (Huang et al., 2016), Wide ResNets (Zagoruyko & Komodakis, 2016) and FractalNets (Larsson et al., 2017). We also compare MSDNet to the $\\mathrm { R e s N e t ^ { M C } }$ and DenseNetMC models that were used in Section 5.1, using dynamic evaluation at test time. We denote these baselines as $R e s N e t ^ { M C }$ / DenseNetMC with early-exits. To prevent the result plots from becoming too cluttered, we present CIFAR-100 results with dynamically evaluated ensembles in the supplementary material. We classify batches of $M = 2 5 6$ images at test time. ", + "bbox": [ + 173, + 776, + 825, + 888 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Budgeted batch classification results on ImageNet are shown in the left panel of Figure 7. We trained three MSDNets with different depths, each of which covers a different range of computational budgets. We plot the performance of each MSDNet as a gray curve; we select the best model for each budget based on its accuracy on the validation set, and plot the corresponding accuracy as a black curve. The plot shows that the predictions of MSDNets with dynamic evaluation are substantially more accurate than those of ResNets and DenseNets that use the same amount of computation. For instance, with an average budget of $1 . 7 \\times 1 0 ^ { 9 }$ FLOPs, MSDNet achieves a top-1 accuracy of ${ \\sim } 7 5 \\%$ , which is ${ \\sim } 6 \\%$ higher than that achieved by a ResNet with the same number of FLOPs. Compared to the computationally efficient DenseNets, MSDNet uses $\\sim 2 - 3 \\times$ times fewer FLOPs to achieve the same classification accuracy. Moreover, MSDNet with dynamic evaluation allows for very precise tuning of the computational budget that is consumed, which is not possible with individual ResNet or DenseNet models. The ensemble of ResNets or DenseNets with dynamic evaluation performs on par with or worse than their individual counterparts (but they do allow for setting the computational budget very precisely). ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/047b8902b777ad1edf01cc1798508fdcb86d98ce8b5ec9a2971efe96719d19ec.jpg", + "image_caption": [ + "Figure 7: Accuracy (top-1) of budgeted batch classification models as a function of average computational budget per image the on ImageNet (left) and CIFAR-100 (right) datasets. Higher is better. " + ], + "image_footnote": [], + "bbox": [ + 189, + 93, + 807, + 262 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 320, + 825, + 488 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The right panel of Figure 7 shows our results on CIFAR-100. The results show that MSDNets consistently outperform all baselines across all budgets. Notably, MSDNet performs on par with a 110- layer ResNet using only 1/10th of the computational budget and it is up to $\\sim 5$ times more efficient than DenseNets, Stochastic Depth Networks, Wide ResNets, and FractalNets. Similar to results in the anytime-prediction setting, MSDNet substantially outperform ResNetsMC and DenseNets $_ { M C }$ with multiple intermediate classifiers, which provides further evidence that the coarse features in the MSDNet are important for high performance in earlier layers. ", + "bbox": [ + 173, + 494, + 825, + 592 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Visualization. To illustrate the ability of our approach to reduce the computational requirements for classifying “easy” examples, we show twelve randomly sampled test images from two ImageNet classes in Figure 6. The top row shows “easy” examples that were correctly classified and exited by the first classifier. The bottom row shows “hard” examples that would have been incorrectly classified by the first classifier but were passed on because its uncertainty was too high. The figure suggests that early classifiers recognize prototypical class examples, whereas the last classifier recognizes non-typical images. ", + "bbox": [ + 174, + 599, + 496, + 779 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/105e212787f02ba3ad81fe470cd4782e898e2825cb6b2147833e9bbcf9aa8bfb.jpg", + "image_caption": [ + "Figure 6: Sampled images from the ImageNet classes Red wine and Volcano. Top row: images exited from the first classifier of a MSDNet with correct prediction; Bottom row: images failed to be correctly classified at the first classifier but were correctly predicted and exited at the last layer. " + ], + "image_footnote": [], + "bbox": [ + 509, + 604, + 821, + 693 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.3 MORE COMPUTATIONALLY EFFICIENT DENSENETS ", + "text_level": 1, + "bbox": [ + 174, + 799, + 570, + 814 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Here, we discuss an interesting finding during our exploration of the MSDNet architecture. We found that following the DenseNet structure to design our network, i.e., by keeping the number of output channels (or growth rate) the same at all scales, did not lead to optimal results in terms of the accuracy-speed trade-off. The main reason for this is that compared to network architectures like ResNets, the DenseNet structure tends to apply more filters on the high-resolution feature maps in the network. This helps to reduce the number of parameters in the model, but at the same time, it greatly increases the computational cost. We tried to modify DenseNets by doubling the growth rate after each transition layer, so that more filters are applied to low-resolution feature maps. It turns out that the resulting network, which we denote as DenseNet\\*, significantly outperform the original DenseNet in terms of computational efficiency. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/f82f1fa9b6ae7df691196495c61b67a76cd3e63f7fdf3cb3be3f5fd5bc49b88b.jpg", + "image_caption": [ + "Figure 8: Test accuracy of DenseNet\\* on CIFAR-100 under the anytime learning setting (left) and the budgeted batch setting (right). " + ], + "image_footnote": [], + "bbox": [ + 191, + 104, + 808, + 273 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 344, + 825, + 386 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We experimented with DenseNet\\* in our two settings with test time budget constraints. The left panel of Figure 8 shows the anytime prediction performance of an ensemble of DenseNets\\* of varying depths. It outperforms the ensemble of original DenseNets of varying depth by a large margin, but is still slightly worse than MSDNets. In the budgeted batch budget setting, DenseNet\\* also leads to significantly higher accuracy over its counterpart under all budgets, but is still substantially outperformed by MSDNets. ", + "bbox": [ + 174, + 392, + 825, + 477 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 488, + 318, + 505 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We presented the MSDNet, a novel convolutional network architecture, optimized to incorporate CPU budgets at test-time. Our design is based on two high-level design principles, to generate and maintain coarse level features throughout the network and to inter-connect the layers with dense connectivity. The former allows us to introduce intermediate classifiers even at early layers and the latter ensures that these classifiers do not interfere with each other. The final design is a two dimensional array of horizontal and vertical layers, which decouples depth and feature coarseness. Whereas in traditional convolutional networks features only become coarser with increasing depth, the MSDNet generates features of all resolutions from the first layer on and maintains them throughout. The result is an architecture with an unprecedented range of efficiency. A single network can outperform all competitive baselines on an impressive range of computational budgets ranging from highly limited CPU constraints to almost unconstrained settings. ", + "bbox": [ + 174, + 516, + 825, + 669 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "As future work we plan to investigate the use of resource-aware deep architectures beyond object classification, e.g. image segmentation (Long et al., 2015). Further, we intend to explore approaches that combine MSDNets with model compression (Chen et al., 2015; Han et al., 2015), spatially adaptive computation (Figurnov et al., 2016) and more efficient convolution operations (Chollet, 2016; Howard et al., 2017) to further improve computational efficiency. ", + "bbox": [ + 174, + 676, + 823, + 746 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 765, + 326, + 779 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The authors are supported in part by grants from the National Science Foundation ( III-1525919, IIS-1550179, IIS-1618134, S&AS 1724282, and CCF-1740822), the Office of Naval Research DOD (N00014-17-1-2175), and the Bill and Melinda Gates Foundation. ", + "bbox": [ + 176, + 790, + 821, + 832 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 856, + 285, + 871 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Mariusz Bojarski, Davide Del Testa, Daniel Dworakowski, Bernhard Firner, Beat Flepp, Prasoon Goyal, Lawrence D Jackel, Mathew Monfort, Urs Muller, Jiakai Zhang, et al. End to end learning for self-driving cars. arXiv preprint arXiv:1604.07316, 2016. 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Interlinked convolutional neural networks for face parsing. In International Symposium on Neural Networks, pp. 222–231. Springer, 2015. ", + "bbox": [ + 171, + 58, + 828, + 924 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 78, + 828, + 921 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A DETAILS OF MSDNET ARCHITECTURE AND BASELINE NETWORKS", + "text_level": 1, + "bbox": [ + 173, + 102, + 769, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We use MSDNet with three scales on the CIFAR datasets, and the network reduction method introduced in 4.1 is applied. Figure 9 gives an illustration of the reduced network. The convolutional layer functions in the first layer, $h _ { 1 } ^ { s }$ , denote a sequence of $3 { \\times } 3$ convolutions (Conv), batch normalization (BN; Ioffe & Szegedy (2015)), and rectified linear unit (ReLU) activation. In the computation of $\\tilde { h } _ { 1 } ^ { s }$ , down-sampling is performed by applying convolutions using strides that are powers of two. For subsequent feature layers, the transformations $h _ { \\ell } ^ { s }$ and $\\tilde { h } _ { \\ell } ^ { s }$ are defined following the design in DenseNets (Huang et al., 2017): Conv $( 1 \\times 1 )$ -BN-ReLU-Conv $\\left( 3 \\times 3 \\right)$ -BN-ReLU. We set the number of output channels of the three scales to 6, 12, and 24, respectively. Each classifier has two down-sampling convolutional layers with 128 dimensional $3 \\times 3$ filters, followed by a $2 \\times 2$ average pooling layer and a linear layer. ", + "bbox": [ + 173, + 135, + 825, + 279 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The MSDNet used for ImageNet has four scales, respectively producing 16, 32, 64, and 64 feature maps at each layer. The network reduction is also applied to reduce computational cost. The original images are first transformed by a $7 \\times 7$ convolution and a $3 \\times 3$ max pooling (both with stride 2), before entering the first layer of MSDNets. The classifiers have the same structure as those used for the CIFAR datasets, except that the number of output channels of each convolutional layer is set to be equal to the number of its input channels. ", + "bbox": [ + 174, + 285, + 825, + 369 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/d0de6b68dce803aca2a16de6affb943796e7f0bf1bb1eac9cef6c06e5c471db7.jpg", + "image_caption": [ + "Figure 9: Illustration of an MSDNet with network reduction. The network has $S = 3$ scales, and it is divided into three blocks, which maintain a decreasing number of scales. A transition layer is placed between two contiguous blocks. " + ], + "image_footnote": [], + "bbox": [ + 238, + 388, + 743, + 507 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Network architecture for anytime prediction. The MSDNet used in our anytime-prediction experiments has 24 layers (each layer corresponds to a column in Fig. 1 of the main paper), using the reduced network with transition layers as described in Section 4. The classifiers operate on the output of the $2 \\times ( i { + } 1 ) ^ { \\mathrm { t h } }$ layers, with $i = 1 , \\ldots , 1 1$ . On ImageNet, we use MSDNets with four scales, and the $i ^ { \\mathrm { { t h } } }$ classifier operates on the $( k \\times i + 3 ) ^ { \\mathrm { t h } }$ layer (with $i = 1 , \\ldots , 5$ ), where $k = 4 , 6$ and 7. For simplicity, the losses of all the classifiers are weighted equally during training. ", + "bbox": [ + 174, + 568, + 825, + 651 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Network architecture for budgeted batch setting. The MSDNets used here for the two CIFAR datasets have depths ranging from 10 to 36 layers, using the reduced network with transition layers as described in Section 4. The $k ^ { \\mathrm { { t h } } }$ classifier is attached to the $( \\sum _ { i = 1 } ^ { k } i ) ^ { \\mathrm { t h } }$ layer. The MSDNets used for ImageNet are the same as those described for the anytime learning setting. ", + "bbox": [ + 174, + 659, + 825, + 717 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$\\mathbf { R e s N e t } ^ { \\mathbf { M C } }$ and DenseNetMC. The ResNetMC has 62 layers, with 10 residual blocks at each spatial resolution (for three resolutions): we train early-exit classifiers on the output of the $4 ^ { \\mathrm { t h } }$ and $8 ^ { \\mathrm { t h } }$ residual blocks at each resolution, producing a total of 6 intermediate classifiers (plus the final classification layer). The DenseNetMC consists of 52 layers with three dense blocks and each of them has 16 layers. The six intermediate classifiers are attached to the $6 ^ { \\mathrm { { t h } } }$ and $1 2 ^ { \\mathrm { t h } }$ layer in each block, also with dense connections to all previous layers in that block. ", + "bbox": [ + 174, + 722, + 825, + 808 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 832, + 398, + 847 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.1 ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 176, + 866, + 341, + 880 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We perform additional experiments to shed light on the contributions of the three main components of MSDNet, viz., multi-scale feature maps, dense connectivity, and intermediate classifiers. ", + "bbox": [ + 173, + 892, + 823, + 921 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We start from an MSDNet with six intermediate classifiers and remove the three main components one at a time. To make our comparisons fair, we keep the computational costs of the full networks similar, at around $3 . 0 \\times 1 0 ^ { 8 }$ FLOPs, by adapting the network width, i.e., number of output channels at each layer. After removing all the three components in an MSDNet, we obtain a regular VGG-like convolutional network. We show the classification accuracy of all classifiers in a model in the left panel of Figure 10. Several observations can be made: 1. the dense connectivity is crucial for the performance of MSDNet and removing it hurts the overall accuracy drastically (orange vs. black curve); 2. removing multi-scale convolution hurts the accuracy only in the lower budget regions, which is consistent with our mo", + "bbox": [ + 174, + 104, + 496, + 338 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/36de99964e9e6ab4f6f1967fffee5d8d4816d0686cfae2751a7e8cef27def4c9.jpg", + "image_caption": [ + "Figure 10: Ablation study (on CIFAR-100) of MSDNets that shows the effect of dense connectivity, multi-scale features, and intermediate classifiers. Higher is better. " + ], + "image_footnote": [], + "bbox": [ + 516, + 104, + 818, + 267 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "tivation that the multi-scale design introduces discriminative features early on; 3. the final canonical CNN (star) performs similarly as MSDNet under the specific budget that matches its evaluation cost exactly, but it is unsuited for varying budget constraints. The final CNN performs substantially better at its particular budget region than the model without dense connectivity (orange curve). This suggests that dense connectivity is particularly important in combination with multiple classifiers. ", + "bbox": [ + 174, + 339, + 825, + 409 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.2 RESULTS ON CIFAR-10 ", + "text_level": 1, + "bbox": [ + 174, + 425, + 385, + 440 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For the CIFAR-10 dataset, we use the same MSDNets and baseline models as we used for CIFAR100, except that the networks used here have a 10-way fully connected layer at the end. The results under the anytime learning setting and the batch computational budget setting are shown in the left and right panel of Figure 11, respectively. Similar to what we have observed from the results on CIFAR-100 and ImageNet, MSDNets outperform all the baselines by a significant margin in both settings. As in the experiments presented in the main paper, ResNet and DenseNet models with multiple intermediate classifiers perform relatively poorly. ", + "bbox": [ + 173, + 452, + 825, + 549 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/adf6b406c38b10a1e3fedffbeafa7e030553b2ab7ceb4700cb0a1fb272d4ff92.jpg", + "image_caption": [ + "Figure 11: Classification accuracies on the CIFAR-10 dataset in the anytime-prediction setting (left) and the budgeted batch setting (right). 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To maximally re-use computation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 344, + 469, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 469, + 356 + ], + "score": 1.0, + "content": "between the classifiers, we incorporate them as early-exits into a single deep con-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 354, + 470, + 367 + ], + "spans": [ + { + "bbox": [ + 142, + 354, + 470, + 367 + ], + "score": 1.0, + "content": "volutional neural network and inter-connect them with dense connectivity. To fa-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 366, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 142, + 366, + 469, + 378 + ], + "score": 1.0, + "content": "cilitate high quality classification early on, we use a two-dimensional multi-scale", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 377, + 469, + 388 + ], + "spans": [ + { + "bbox": [ + 141, + 377, + 469, + 388 + ], + "score": 1.0, + "content": "network architecture that maintains coarse and fine level features all-throughout", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 142, + 388, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 388, + 469, + 399 + ], + "score": 1.0, + "content": "the network. Experiments on three image-classification tasks demonstrate that our", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 398, + 456, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 456, + 411 + ], + "score": 1.0, + "content": "framework substantially improves the existing state-of-the-art in both settings.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 425, + 206, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 208, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 208, + 441 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "Recent years have witnessed a surge in demand for applications of visual object recognition, for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "score": 1.0, + "content": "instance, in self-driving cars (Bojarski et al., 2016) and content-based image search (Wan et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 469, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 505, + 485 + ], + "score": 1.0, + "content": "2014). This demand has in part been fueled through the promise generated by the astonishing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "progress of convolutional networks (CNNs) on visual object recognition benchmark competition", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "datasets, such as ILSVRC (Deng et al., 2009) and COCO (Lin et al., 2014), where state-of-the-art", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 505, + 441, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 441, + 518 + ], + "score": 1.0, + "content": "models may have even surpassed human-level performance (He et al., 2015; 2016).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 375, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 521, + 376, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 376, + 533 + ], + "score": 1.0, + "content": "However, the requirements of such competitions differ from real-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 532, + 376, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 376, + 545 + ], + "score": 1.0, + "content": "world applications, which tend to incentivize resource-hungry mod-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 543, + 376, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 376, + 555 + ], + "score": 1.0, + "content": "els with high computational demands at inference time. For exam-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 554, + 377, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 377, + 567 + ], + "score": 1.0, + "content": "ple, the COCO 2016 competition was won by a large ensemble of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 564, + 377, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 212, + 578 + ], + "score": 1.0, + "content": "computationally intensive", + "type": "text" + }, + { + "bbox": [ + 212, + 564, + 242, + 576 + ], + "score": 0.42, + "content": "\\mathrm { \\dot { C } N N s ^ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 564, + 377, + 578 + ], + "score": 1.0, + "content": "— a model likely far too compu-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 576, + 376, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 376, + 589 + ], + "score": 1.0, + "content": "tationally expensive for any resource-aware application. Although", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 586, + 376, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 376, + 600 + ], + "score": 1.0, + "content": "much smaller models would also obtain decent error, very large,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 599, + 376, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 376, + 610 + ], + "score": 1.0, + "content": "computationally intensive models seem necessary to correctly clas-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 609, + 376, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 376, + 621 + ], + "score": 1.0, + "content": "sify the hard examples that make up the bulk of the remaining mis-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 618, + 376, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 376, + 633 + ], + "score": 1.0, + "content": "classifications of modern algorithms. 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Two such settings are: 1. anytime classification, where the net-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 267, + 470, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 280 + ], + "score": 1.0, + "content": "work’s prediction for a test example is progressively updated, facilitating the out-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "score": 1.0, + "content": "put of a prediction at any time; and 2. budgeted batch classification, where a fixed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 289, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 470, + 301 + ], + "score": 1.0, + "content": "amount of computation is available to classify a set of examples that can be spent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "score": 1.0, + "content": "unevenly across “easier” and “harder” inputs. In contrast to most prior work, such", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 311, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 322 + ], + "score": 1.0, + "content": "as the popular Viola and Jones algorithm, our approach is based on convolution-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 322, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 322, + 469, + 334 + ], + "score": 1.0, + "content": "al neural networks. We train multiple classifiers with varying resource demands,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "score": 1.0, + "content": "which we adaptively apply during test time. To maximally re-use computation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 344, + 469, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 469, + 356 + ], + "score": 1.0, + "content": "between the classifiers, we incorporate them as early-exits into a single deep con-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 354, + 470, + 367 + ], + "spans": [ + { + "bbox": [ + 142, + 354, + 470, + 367 + ], + "score": 1.0, + "content": "volutional neural network and inter-connect them with dense connectivity. To fa-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 366, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 142, + 366, + 469, + 378 + ], + "score": 1.0, + "content": "cilitate high quality classification early on, we use a two-dimensional multi-scale", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 377, + 469, + 388 + ], + "spans": [ + { + "bbox": [ + 141, + 377, + 469, + 388 + ], + "score": 1.0, + "content": "network architecture that maintains coarse and fine level features all-throughout", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 142, + 388, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 388, + 469, + 399 + ], + "score": 1.0, + "content": "the network. Experiments on three image-classification tasks demonstrate that our", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 398, + 456, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 456, + 411 + ], + "score": 1.0, + "content": "framework substantially improves the existing state-of-the-art in both settings.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 22, + "bbox_fs": [ + 141, + 245, + 470, + 411 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 425, + 206, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 208, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 208, + 441 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "Recent years have witnessed a surge in demand for applications of visual object recognition, for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "score": 1.0, + "content": "instance, in self-driving cars (Bojarski et al., 2016) and content-based image search (Wan et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 469, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 505, + 485 + ], + "score": 1.0, + "content": "2014). This demand has in part been fueled through the promise generated by the astonishing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "progress of convolutional networks (CNNs) on visual object recognition benchmark competition", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "datasets, such as ILSVRC (Deng et al., 2009) and COCO (Lin et al., 2014), where state-of-the-art", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 505, + 441, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 441, + 518 + ], + "score": 1.0, + "content": "models may have even surpassed human-level performance (He et al., 2015; 2016).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 449, + 506, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 375, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 521, + 376, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 376, + 533 + ], + "score": 1.0, + "content": "However, the requirements of such competitions differ from real-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 532, + 376, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 376, + 545 + ], + "score": 1.0, + "content": "world applications, which tend to incentivize resource-hungry mod-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 543, + 376, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 376, + 555 + ], + "score": 1.0, + "content": "els with high computational demands at inference time. For exam-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 554, + 377, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 377, + 567 + ], + "score": 1.0, + "content": "ple, the COCO 2016 competition was won by a large ensemble of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 564, + 377, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 212, + 578 + ], + "score": 1.0, + "content": "computationally intensive", + "type": "text" + }, + { + "bbox": [ + 212, + 564, + 242, + 576 + ], + "score": 0.42, + "content": "\\mathrm { \\dot { C } N N s ^ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 564, + 377, + 578 + ], + "score": 1.0, + "content": "— a model likely far too compu-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 576, + 376, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 376, + 589 + ], + "score": 1.0, + "content": "tationally expensive for any resource-aware application. Although", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 586, + 376, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 376, + 600 + ], + "score": 1.0, + "content": "much smaller models would also obtain decent error, very large,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 599, + 376, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 376, + 610 + ], + "score": 1.0, + "content": "computationally intensive models seem necessary to correctly clas-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 609, + 376, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 376, + 621 + ], + "score": 1.0, + "content": "sify the hard examples that make up the bulk of the remaining mis-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 618, + 376, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 376, + 633 + ], + "score": 1.0, + "content": "classifications of modern algorithms. To illustrate this point, Fig-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 631, + 376, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 376, + 643 + ], + "score": 1.0, + "content": "ure 1 shows two images of horses. The left image depicts a horse", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 642, + 376, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 376, + 654 + ], + "score": 1.0, + "content": "in canonical pose and is easy to classify, whereas the right image is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 653, + 377, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 377, + 664 + ], + "score": 1.0, + "content": "taken from a rare viewpoint and is likely in the tail of the data dis-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 664, + 376, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 376, + 676 + ], + "score": 1.0, + "content": "tribution. 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The left image is canon-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 382, + 606, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 382, + 606, + 505, + 617 + ], + "score": 1.0, + "content": "ical and easy to detect even with", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 383, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 383, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "a small model, whereas the right", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 382, + 627, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 382, + 627, + 505, + 638 + ], + "score": 1.0, + "content": "image requires a computational-", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 383, + 637, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 383, + 637, + 505, + 648 + ], + "score": 1.0, + "content": "ly more expensive network archi-", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 383, + 647, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 383, + 647, + 505, + 658 + ], + "score": 1.0, + "content": "tecture. (Copyright Pixel Addict", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 383, + 656, + 492, + 669 + ], + "spans": [ + { + "bbox": [ + 383, + 656, + 492, + 669 + ], + "score": 1.0, + "content": "and Doyle (CC BY-ND 2.0).)", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 56.5 + } + ], + "index": 54.0 + }, + { + "type": "text", + "bbox": [ + 109, + 675, + 504, + 686 + ], + "lines": [], + "index": 61, + "bbox_fs": [ + 106, + 673, + 505, + 688 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 109, + 692, + 502, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 505, + 704 + ], + "score": 1.0, + "content": "In real-world applications, computation directly translates into power consumption, which should", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 702, + 504, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 504, + 714 + ], + "score": 1.0, + "content": "be minimized for environmental and economical reasons, and is a scarce commodity on mobile", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "devices. This begs the question: why do we choose between either wasting computational resources", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "by applying an unnecessarily computationally expensive model to easy images, or making mistakes", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "by using an efficient model that fails to recognize difficult images? Ideally, our systems should", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "automatically use small networks when test images are easy or computational resources limited, and", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 396, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 396, + 140 + ], + "score": 1.0, + "content": "use big networks when test images are hard or computation is abundant.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 62.5, + "bbox_fs": [ + 106, + 691, + 505, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "devices. This begs the question: why do we choose between either wasting computational resources", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "by applying an unnecessarily computationally expensive model to easy images, or making mistakes", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "by using an efficient model that fails to recognize difficult images? Ideally, our systems should", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "automatically use small networks when test images are easy or computational resources limited, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 396, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 396, + 140 + ], + "score": 1.0, + "content": "use big networks when test images are hard or computation is abundant.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "Such systems would be beneficial in at least two settings with computational constraints at test-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "time: anytime prediction, where the network can be forced to output a prediction at any given point", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "in time; and budgeted batch classification, where a fixed computational budget is shared across a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "large set of examples which can be spent unevenly across “easy” and “hard” examples. A practical", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "use-case of anytime prediction is in mobile apps on Android devices: in 2015, there existed 24, 093", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "distinct Android devices2, each with its own distinct computational limitations. It is infeasible to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "train a different network that processes video frame-by-frame at a fixed framerate for each of these", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 504, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 504, + 232 + ], + "score": 1.0, + "content": "devices. Instead, you would like to train a single network that maximizes accuracy on all these de-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "vices, within the computational constraints of that device. The budget batch classification setting is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "ubiquitous in large-scale machine learning applications. Search engines, social media companies,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "on-line advertising agencies, all must process large volumes of data on limited hardware resources.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "For example, as of 2010, Google Image Search had over 10 Billion images indexed3, which has like-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "ly grown to over 1 Trillion since. Even if a new model to process these images is only 1/10s slower", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "per image, this additional cost would add 3170 years of CPU time. In the budget batch classification", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "setting, companies can improve the average accuracy by reducing the amount of computation spent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 335, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 335, + 320 + ], + "score": 1.0, + "content": "on “easy” cases to save up computation for “hard” cases.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "Motivated by prior work in computer vision on resource-efficient recognition (Viola & Jones, 2001),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 334, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 349 + ], + "score": 1.0, + "content": "we aim to develop CNNs that “slice” the computation and process these slices one-by-one, stopping", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 359 + ], + "score": 1.0, + "content": "the evaluation once the CPU time is depleted or the classification sufficiently certain (through “early", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "exits”). Unfortunately, the architecture of CNNs is inherently at odds with the introduction of early", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "exits. CNNs learn the data representation and the classifier jointly, which leads to two problems", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "with early exits: 1. The features in the last layer are extracted directly to be used by the classifier,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 392, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 107, + 392, + 505, + 401 + ], + "score": 1.0, + "content": "whereas earlier features are not. The inherent dilemma is that different kinds of features need to be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "extracted depending on how many layers are left until the classification. 2. The features in different", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "layers of the network may have different scale. 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In the anytime classification setting,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "we show that it is possible to provide the ability to output a prediction at any time while maintain", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "high accuracies throughout. 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A practical", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "use-case of anytime prediction is in mobile apps on Android devices: in 2015, there existed 24, 093", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "distinct Android devices2, each with its own distinct computational limitations. It is infeasible to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "train a different network that processes video frame-by-frame at a fixed framerate for each of these", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 504, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 504, + 232 + ], + "score": 1.0, + "content": "devices. Instead, you would like to train a single network that maximizes accuracy on all these de-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "vices, within the computational constraints of that device. The budget batch classification setting is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "ubiquitous in large-scale machine learning applications. Search engines, social media companies,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "on-line advertising agencies, all must process large volumes of data on limited hardware resources.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "For example, as of 2010, Google Image Search had over 10 Billion images indexed3, which has like-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "ly grown to over 1 Trillion since. Even if a new model to process these images is only 1/10s slower", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "per image, this additional cost would add 3170 years of CPU time. In the budget batch classification", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "setting, companies can improve the average accuracy by reducing the amount of computation spent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 335, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 335, + 320 + ], + "score": 1.0, + "content": "on “easy” cases to save up computation for “hard” cases.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 144, + 506, + 320 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "Motivated by prior work in computer vision on resource-efficient recognition (Viola & Jones, 2001),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 334, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 349 + ], + "score": 1.0, + "content": "we aim to develop CNNs that “slice” the computation and process these slices one-by-one, stopping", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 359 + ], + "score": 1.0, + "content": "the evaluation once the CPU time is depleted or the classification sufficiently certain (through “early", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "exits”). Unfortunately, the architecture of CNNs is inherently at odds with the introduction of early", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "exits. CNNs learn the data representation and the classifier jointly, which leads to two problems", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "with early exits: 1. The features in the last layer are extracted directly to be used by the classifier,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 392, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 107, + 392, + 505, + 401 + ], + "score": 1.0, + "content": "whereas earlier features are not. 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In the anytime classification setting,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "we show that it is possible to provide the ability to output a prediction at any time while maintain", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "high accuracies throughout. In the budget batch classification setting we show that MSDNets can be", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 622, + 504, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 504, + 633 + ], + "score": 1.0, + "content": "effectively used to adapt the amount of computation to the difficulty of the example to be classified,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "which allows us to reduce the computational requirements of our models drastically whilst perform-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "ing on par with state-of-the-art CNNs in terms of overall classification accuracy. To our knowledge", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "this is the first deep learning architecture of its kind that allows dynamic resource adaptation with a", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 666, + 334, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 334, + 678 + ], + "score": 1.0, + "content": "single model and obtains competitive results throughout.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 587, + 506, + 678 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 66, + 506, + 169 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 66, + 506, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 66, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 111, + 66, + 506, + 169 + ], + "score": 0.959, + "type": "image", + "image_path": "393456a645a070145e34d48f149ac9a7ed180043707a693bf4407c8f3be79371.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 66, + 506, + 100.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 100.33333333333334, + 506, + 134.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 134.66666666666669, + 506, + 169.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 187, + 505, + 238 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Figure 2: Illustration of the first four layers of an MSDNet with three scales. The horizontal direction cor-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 198, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 209 + ], + "score": 1.0, + "content": "responds to the layer direction (depth) of the network. The vertical direction corresponds to the scale of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "score": 1.0, + "content": "feature maps. Horizontal arrows indicate a regular convolution operation, whereas diagonal and vertical arrows", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 217, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 504, + 230 + ], + "score": 1.0, + "content": "indicate a strided convolution operation. Classifiers only operate on feature maps at the coarsest scale. Connec-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 228, + 499, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 499, + 239 + ], + "score": 1.0, + "content": "tions across more than one layer are not drawn explicitly: they are implicit through recursive concatenations.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 108, + 252, + 211, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 213, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 213, + 267 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 108, + 275, + 504, + 298 + ], + "lines": [ + { + "bbox": [ + 107, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 107, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "We briefly review related prior work on computation-efficient networks, memory-efficient networks,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 286, + 493, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 493, + 299 + ], + "score": 1.0, + "content": "and resource-sensitive machine learning, from which our network architecture draws inspiration.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Computation-efficient networks. Most prior work on (convolutional) networks that are compu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 314, + 504, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 504, + 327 + ], + "score": 1.0, + "content": "tationally efficient at test time focuses on reducing model size after training. In particular, many", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "studies propose to prune weights (LeCun et al., 1989; Hassibi et al., 1993; Li et al., 2017) or quan-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "tize weights (Hubara et al., 2016; Rastegari et al., 2016) during or after training. These approaches", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "are generally effective because deep networks often have a substantial number of redundant weights", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "that can be pruned or quantized without sacrificing (and sometimes even improving) performance.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "Prior work also studies approaches that directly learn compact models with less parameter redundan-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "cy. For example, the knowledge-distillation method (Bucilua et al., 2006; Hinton et al., 2014) trains", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "small student networks to reproduce the output of a much larger teacher network or ensemble. Our", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "work differs from those approaches in that we train a single model that trades off computation for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "accuracy at test time without any re-training or finetuning. Indeed, weight pruning and knowledge", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 423, + 498, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 498, + 436 + ], + "score": 1.0, + "content": "distillation can be used in combination with our approach, and may lead to further improvements.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "Resource-efficient machine learning. Various prior studies explore computationally efficient vari-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "ants of traditional machine-learning models (Viola & Jones, 2001; Grubb & Bagnell, 2012; Karayev", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "et al., 2014; Trapeznikov & Saligrama, 2013; Xu et al., 2012; 2013; Nan et al., 2015; Wang et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "2015). Most of these studies focus on how to incorporate the computational requirements of com-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "puting particular features in the training of machine-learning models such as (gradient-boosted)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "decision trees. Whilst our study is certainly inspired by these results, the architecture we explore", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "differs substantially: most prior work exploits characteristics of machine-learning models (such as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "decision trees) that do not apply to deep networks. Our work is possibly most closely related to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "recent work on FractalNets (Larsson et al., 2017), which can perform anytime prediction by pro-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "score": 1.0, + "content": "gressively evaluating subnetworks of the full network. FractalNets differ from our work in that they", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "are not explicitly optimized for computation efficiency and consequently our experiments show that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "MSDNets substantially outperform FractalNets. Our dynamic evaluation strategy for reducing batch", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "computational cost is closely related to the the adaptive computation time approach (Graves, 2016;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "Figurnov et al., 2016), and the recently proposed method of adaptively evaluating neural networks", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "(Bolukbasi et al., 2017). Different from these works, our method adopts a specially designed net-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "work with multiple classifiers, which are jointly optimized during training and can directly output", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "confidence scores to control the evaluation process for each test example. The adaptive computation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 505, + 641 + ], + "score": 1.0, + "content": "time method (Graves, 2016) and its extension (Figurnov et al., 2016) also perform adaptive eval-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "uation on test examples to save batch computational cost, but focus on skipping units rather than", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "layers. In (Odena et al., 2017), a “composer”model is trained to construct the evaluation network", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "from a set of sub-modules for each test example. By contrast, our work uses a single CNN with", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "multiple intermediate classifiers that is trained end-to-end. The Feedback Networks (Zamir et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "2016) enable early predictions by making predictions in a recurrent fashion, which heavily shares", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 693, + 399, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 399, + 705 + ], + "score": 1.0, + "content": "parameters among classifiers, but is less efficient in sharing computation.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 710, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "Related network architectures. Our network architecture borrows elements from neural fabrics", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "(Saxena & Verbeek, 2016) and others (Zhou et al., 2015; Jacobsen et al., 2017; Ke et al., 2016)", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 66, + 506, + 169 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 66, + 506, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 66, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 111, + 66, + 506, + 169 + ], + "score": 0.959, + "type": "image", + "image_path": "393456a645a070145e34d48f149ac9a7ed180043707a693bf4407c8f3be79371.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 66, + 506, + 100.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 100.33333333333334, + 506, + 134.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 134.66666666666669, + 506, + 169.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 187, + 505, + 238 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Figure 2: Illustration of the first four layers of an MSDNet with three scales. The horizontal direction cor-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 198, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 209 + ], + "score": 1.0, + "content": "responds to the layer direction (depth) of the network. The vertical direction corresponds to the scale of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "score": 1.0, + "content": "feature maps. Horizontal arrows indicate a regular convolution operation, whereas diagonal and vertical arrows", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 217, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 504, + 230 + ], + "score": 1.0, + "content": "indicate a strided convolution operation. Classifiers only operate on feature maps at the coarsest scale. Connec-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 228, + 499, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 499, + 239 + ], + "score": 1.0, + "content": "tions across more than one layer are not drawn explicitly: they are implicit through recursive concatenations.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 108, + 252, + 211, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 213, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 213, + 267 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 108, + 275, + 504, + 298 + ], + "lines": [ + { + "bbox": [ + 107, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 107, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "We briefly review related prior work on computation-efficient networks, memory-efficient networks,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 286, + 493, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 493, + 299 + ], + "score": 1.0, + "content": "and resource-sensitive machine learning, from which our network architecture draws inspiration.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 106, + 275, + 505, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Computation-efficient networks. Most prior work on (convolutional) networks that are compu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 314, + 504, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 504, + 327 + ], + "score": 1.0, + "content": "tationally efficient at test time focuses on reducing model size after training. In particular, many", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "studies propose to prune weights (LeCun et al., 1989; Hassibi et al., 1993; Li et al., 2017) or quan-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "tize weights (Hubara et al., 2016; Rastegari et al., 2016) during or after training. These approaches", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "are generally effective because deep networks often have a substantial number of redundant weights", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "that can be pruned or quantized without sacrificing (and sometimes even improving) performance.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "Prior work also studies approaches that directly learn compact models with less parameter redundan-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "cy. For example, the knowledge-distillation method (Bucilua et al., 2006; Hinton et al., 2014) trains", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "small student networks to reproduce the output of a much larger teacher network or ensemble. Our", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "work differs from those approaches in that we train a single model that trades off computation for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "accuracy at test time without any re-training or finetuning. Indeed, weight pruning and knowledge", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 423, + 498, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 498, + 436 + ], + "score": 1.0, + "content": "distillation can be used in combination with our approach, and may lead to further improvements.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 303, + 506, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "Resource-efficient machine learning. Various prior studies explore computationally efficient vari-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "ants of traditional machine-learning models (Viola & Jones, 2001; Grubb & Bagnell, 2012; Karayev", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "et al., 2014; Trapeznikov & Saligrama, 2013; Xu et al., 2012; 2013; Nan et al., 2015; Wang et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "2015). Most of these studies focus on how to incorporate the computational requirements of com-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "puting particular features in the training of machine-learning models such as (gradient-boosted)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "decision trees. Whilst our study is certainly inspired by these results, the architecture we explore", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "differs substantially: most prior work exploits characteristics of machine-learning models (such as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "decision trees) that do not apply to deep networks. Our work is possibly most closely related to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "recent work on FractalNets (Larsson et al., 2017), which can perform anytime prediction by pro-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "score": 1.0, + "content": "gressively evaluating subnetworks of the full network. FractalNets differ from our work in that they", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "are not explicitly optimized for computation efficiency and consequently our experiments show that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "MSDNets substantially outperform FractalNets. Our dynamic evaluation strategy for reducing batch", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "computational cost is closely related to the the adaptive computation time approach (Graves, 2016;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "Figurnov et al., 2016), and the recently proposed method of adaptively evaluating neural networks", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "(Bolukbasi et al., 2017). Different from these works, our method adopts a specially designed net-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "work with multiple classifiers, which are jointly optimized during training and can directly output", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "confidence scores to control the evaluation process for each test example. The adaptive computation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 505, + 641 + ], + "score": 1.0, + "content": "time method (Graves, 2016) and its extension (Figurnov et al., 2016) also perform adaptive eval-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "uation on test examples to save batch computational cost, but focus on skipping units rather than", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "layers. In (Odena et al., 2017), a “composer”model is trained to construct the evaluation network", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "from a set of sub-modules for each test example. By contrast, our work uses a single CNN with", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "multiple intermediate classifiers that is trained end-to-end. The Feedback Networks (Zamir et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "2016) enable early predictions by making predictions in a recurrent fashion, which heavily shares", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 693, + 399, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 399, + 705 + ], + "score": 1.0, + "content": "parameters among classifiers, but is less efficient in sharing computation.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 441, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 710, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "Related network architectures. Our network architecture borrows elements from neural fabrics", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "(Saxena & Verbeek, 2016) and others (Zhou et al., 2015; Jacobsen et al., 2017; Ke et al., 2016)", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5, + "bbox_fs": [ + 106, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 74, + 485, + 210 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 74, + 485, + 210 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 74, + 485, + 210 + ], + "spans": [ + { + "bbox": [ + 115, + 74, + 485, + 210 + ], + "score": 0.968, + "type": "image", + "image_path": "6c59242daaab86116b0562f24208342d867eb57b7d0db805e434894dcf77ee2c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 74, + 485, + 119.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 119.33333333333334, + 485, + 164.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 164.66666666666669, + 485, + 210.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 217, + 505, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "Figure 3: Relative accuracy of the intermediate classifier (left) and the final classifier (right) when introducing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "a single intermediate classifier at different layers in a ResNet, DenseNet and MSDNet. All experiments were", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 237, + 305, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 305, + 249 + ], + "score": 1.0, + "content": "performed on the CIFAR-100 dataset. Higher is better.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "to rapidly construct a low-resolution feature map that is amenable to classification, whilst also", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "maintaining feature maps of higher resolution that are essential for obtaining high classification", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "score": 1.0, + "content": "accuracy. Our design differs from the neural fabrics (Saxena & Verbeek, 2016) substantially in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "that MSDNets have a reduced number of scales and no sparse channel connectivity or up-sampling", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "paths. MSDNets are at least one order of magnitude more efficient and typically more accurate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 316, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 481, + 329 + ], + "score": 1.0, + "content": "— for example, an MSDNet with less than 1 million parameters obtains a test error below", + "type": "text" + }, + { + "bbox": [ + 482, + 316, + 504, + 327 + ], + "score": 0.86, + "content": "7 . 0 \\%", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 456, + 340 + ], + "score": 1.0, + "content": "on CIFAR-10 (Krizhevsky & Hinton, 2009), whereas Saxena & Verbeek (2016) report", + "type": "text" + }, + { + "bbox": [ + 456, + 327, + 484, + 338 + ], + "score": 0.88, + "content": "7 . 4 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 337, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 506, + 353 + ], + "score": 1.0, + "content": "over 20 million parameters. We use the same feature-concatenation approach as DenseNets (Huang", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 348, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 104, + 348, + 506, + 362 + ], + "score": 1.0, + "content": "et al., 2017), which allows us to bypass features optimized for early classifiers in later layers of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "the network. Our architecture is related to deeply supervised networks (Lee et al., 2015) in that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "it incorporates classifiers at multiple layers throughout the network. In contrast to all these prior", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 382, + 459, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 459, + 395 + ], + "score": 1.0, + "content": "architectures, our network is specifically designed to operate in resource-aware settings.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 406, + 213, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 215, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 215, + 421 + ], + "score": 1.0, + "content": "3 PROBLEM SETUP", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 427, + 435, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 438, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 438, + 441 + ], + "score": 1.0, + "content": "We consider two settings that impose computational constraints at prediction time.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "Anytime prediction. In the anytime prediction setting (Grubb & Bagnell, 2012), there is a finite", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 195, + 468 + ], + "score": 1.0, + "content": "computational budget", + "type": "text" + }, + { + "bbox": [ + 196, + 456, + 220, + 466 + ], + "score": 0.92, + "content": "B > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 455, + 346, + 468 + ], + "score": 1.0, + "content": "available for each test example", + "type": "text" + }, + { + "bbox": [ + 347, + 457, + 354, + 465 + ], + "score": 0.28, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 455, + 505, + 468 + ], + "score": 1.0, + "content": ". The computational budget is nonde-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "terministic, and varies per test instance. It is determined by the occurrence of an event that requires", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "the model to output a prediction immediately. 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Our design differs from the neural fabrics (Saxena & Verbeek, 2016) substantially in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "that MSDNets have a reduced number of scales and no sparse channel connectivity or up-sampling", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "paths. 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In some applications", + "type": "text" + }, + { + "bbox": [ + 278, + 489, + 302, + 500 + ], + "score": 0.92, + "content": "P ( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 488, + 394, + 501 + ], + "score": 1.0, + "content": "may be independent of", + "type": "text" + }, + { + "bbox": [ + 395, + 488, + 417, + 500 + ], + "score": 0.92, + "content": "P ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "and can be estimated.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 343, + 513 + ], + "score": 1.0, + "content": "For example, if the event is governed by a Poisson process,", + "type": "text" + }, + { + "bbox": [ + 343, + 499, + 367, + 511 + ], + "score": 0.92, + "content": "P ( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "is an exponential distribution. We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 509, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 214, + 524 + ], + "score": 1.0, + "content": "denote the loss of a model", + "type": "text" + }, + { + "bbox": [ + 215, + 510, + 235, + 522 + ], + "score": 0.91, + "content": "f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 509, + 415, + 524 + ], + "score": 1.0, + "content": "that has to produce a prediction for instance", + "type": "text" + }, + { + "bbox": [ + 415, + 512, + 423, + 520 + ], + "score": 0.43, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 509, + 482, + 524 + ], + "score": 1.0, + "content": "within budget", + "type": "text" + }, + { + "bbox": [ + 482, + 511, + 491, + 520 + ], + "score": 0.76, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 509, + 505, + 524 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 107, + 521, + 154, + 533 + ], + "score": 0.92, + "content": "L ( f ( \\mathbf { x } ) , B )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 521, + 505, + 534 + ], + "score": 1.0, + "content": ". The goal of an anytime learner is to minimize the expected loss under the budget dis-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 530, + 507, + 549 + ], + "spans": [ + { + "bbox": [ + 104, + 530, + 146, + 549 + ], + "score": 1.0, + "content": "tribution:", + "type": "text" + }, + { + "bbox": [ + 146, + 532, + 270, + 546 + ], + "score": 0.91, + "content": "L ( f ) = \\mathbb { E } \\left[ L ( f ( \\mathbf { x } ) , B ) \\right] _ { P ( \\mathbf { x } , B ) } .", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 530, + 297, + 549 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 297, + 532, + 315, + 544 + ], + "score": 0.93, + "content": "L ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 530, + 507, + 549 + ], + "score": 1.0, + "content": "denotes a suitable loss function. As is common", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 383, + 558 + ], + "score": 1.0, + "content": "in the empirical risk minimization framework, the expectation under", + "type": "text" + }, + { + "bbox": [ + 383, + 545, + 418, + 558 + ], + "score": 0.93, + "content": "P ( \\mathbf { x } , B )", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "may be estimated by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 556, + 267, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 228, + 569 + ], + "score": 1.0, + "content": "an average over samples from", + "type": "text" + }, + { + "bbox": [ + 228, + 556, + 263, + 568 + ], + "score": 0.93, + "content": "P ( \\mathbf { x } , B )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 556, + 267, + 569 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 444, + 507, + 569 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "Budgeted batch classification. In the budgeted batch classification setting, the model needs to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 214, + 596 + ], + "score": 1.0, + "content": "classify a set of examples", + "type": "text" + }, + { + "bbox": [ + 214, + 584, + 307, + 596 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { t e s t } = \\{ \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { M } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 584, + 459, + 596 + ], + "score": 1.0, + "content": "within a finite computational budget", + "type": "text" + }, + { + "bbox": [ + 459, + 585, + 486, + 595 + ], + "score": 0.9, + "content": "B > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 596, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 444, + 607 + ], + "score": 1.0, + "content": "is known in advance. The learner aims to minimize the loss across all examples in", + "type": "text" + }, + { + "bbox": [ + 445, + 596, + 468, + 606 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 596, + 505, + 607 + ], + "score": 1.0, + "content": "within a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 605, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 220, + 620 + ], + "score": 1.0, + "content": "cumulative cost bounded by", + "type": "text" + }, + { + "bbox": [ + 221, + 606, + 230, + 616 + ], + "score": 0.81, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 605, + 316, + 620 + ], + "score": 1.0, + "content": ", which we denote by", + "type": "text" + }, + { + "bbox": [ + 316, + 606, + 379, + 618 + ], + "score": 0.93, + "content": "L ( f ( \\mathcal { D } _ { t e s t } ) , B )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 605, + 506, + 620 + ], + "score": 1.0, + "content": "for some suitable loss function", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 616, + 507, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 125, + 631 + ], + "score": 0.88, + "content": "L ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 617, + 307, + 632 + ], + "score": 1.0, + "content": ". It can potentially do so by spending less than", + "type": "text" + }, + { + "bbox": [ + 304, + 616, + 507, + 633 + ], + "score": 1.0, + "content": "BM computation on classifying an “easy” example", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 102, + 625, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 102, + 625, + 201, + 650 + ], + "score": 1.0, + "content": "whilst using more than", + "type": "text" + }, + { + "bbox": [ + 202, + 630, + 213, + 644 + ], + "score": 0.89, + "content": "\\textstyle { \\frac { B } { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 625, + 505, + 650 + ], + "score": 1.0, + "content": "computation on classifying a “difficult” example. Therefore, the budget", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 116, + 641, + 447, + 655 + ], + "spans": [ + { + "bbox": [ + 116, + 641, + 447, + 655 + ], + "score": 1.0, + "content": "considered here is a soft constraint when we have a large batch of testing samples.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 102, + 572, + 507, + 655 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 666, + 394, + 679 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 394, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 394, + 681 + ], + "score": 1.0, + "content": "4 MULTI-SCALE DENSE CONVOLUTIONAL NETWORKS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "A straightforward solution to the two problems introduced in Section 3 is to train multiple networks", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "of increasing capacity, and sequentially evaluate them at test time (as in Bolukbasi et al. (2017)).", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "In the anytime setting the evaluation can be stopped at any point and the most recent prediction is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "returned. In the batch setting, the evaluation is stopped prematurely the moment a network classifies", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "the test sample with sufficient confidence. When the resources are so limited that the execution is", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "terminated after the first network, this approach is optimal because the first network is trained for", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "exactly this computational budget without compromises. However, in both settings, this scenario is", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "rare. In the more common scenario where some test samples can require more processing time than", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "others the approach is far from optimal because previously learned features are never re-used across", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 199, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 199, + 149 + ], + "score": 1.0, + "content": "the different networks.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 688, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "the test sample with sufficient confidence. When the resources are so limited that the execution is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "terminated after the first network, this approach is optimal because the first network is trained for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "exactly this computational budget without compromises. However, in both settings, this scenario is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "rare. In the more common scenario where some test samples can require more processing time than", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "others the approach is far from optimal because previously learned features are never re-used across", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 199, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 199, + 149 + ], + "score": 1.0, + "content": "the different networks.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 504, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "An alternative solution is to build a deep network with a cascade of classifiers operating on the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "features of internal layers: in such a network features computed for an earlier classifier can be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "re-used by later classifiers. However, na¨ıvely attaching intermediate early-exit classifiers to a state-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 312, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 312, + 200 + ], + "score": 1.0, + "content": "of-the-art deep network leads to poor performance.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "There are two reasons why intermediate early-exit classifiers hurt the performance of deep neural", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "networks: early classifiers lack coarse-level features and classifiers throughout interfere with the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "feature generation process. In this section we investigate these effects empirically (see Figure 3)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 465, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 465, + 249 + ], + "score": 1.0, + "content": "and, in response to our findings, propose the MSDNet architecture illustrated in Figure 2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "Problem: The lack of coarse-level features. Traditional neural networks learn features of fine", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "scale in early layers and coarse scale in later layers (through repeated convolution, pooling, and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "strided convolution). Coarse scale features in the final layers are important to classify the content", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "of the whole image into a single class. Early layers lack coarse-level features and early-exit clas-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "sifiers attached to these layers will likely yield unsatisfactory high error rates. To illustrate this", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 307, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 506, + 319 + ], + "score": 1.0, + "content": "point, we attached4 intermediate classifiers to varying layers of a ResNet (He et al., 2016) and a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "DenseNet (Huang et al., 2017) on the CIFAR-100 dataset (Krizhevsky & Hinton, 2009). The blue", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "and red dashed lines in the left plot of Figure 3 show the relative accuracies of these classifiers.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "All three plots gives rise to a clear trend: the accuracy of a classifier is highly correlated with its", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 350, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 362 + ], + "score": 1.0, + "content": "position within the network. Particularly in the case of the ResNet (blue line), one can observe a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "visible “staircase” pattern, with big improvements after the 2nd and 4th classifiers — located right", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 372, + 189, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 189, + 385 + ], + "score": 1.0, + "content": "after pooling layers.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "Solution: Multi-scale feature maps. To address this issue, MSDNets maintain a feature repre-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "sentation at multiple scales throughout the network, and all the classifiers only use the coarse-level", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 412, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 504, + 423 + ], + "score": 1.0, + "content": "features. The feature maps at a particular layer5 and scale are computed by concatenating the re-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "sults of one or two convolutions: 1. the result of a regular convolution applied on the same-scale", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "features from the previous layer (horizontal connections) and, if possible, 2. the result of a strided", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 445, + 504, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 504, + 456 + ], + "score": 1.0, + "content": "convolution applied on the finer-scale feature map from the previous layer (diagonal connections).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "The horizontal connections preserve and progress high-resolution information, which facilitates the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "construction of high-quality coarse features in later layers. The vertical connections produce coarse", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "score": 1.0, + "content": "features throughout that are amenable to classification. The dashed black line in Figure 3 shows that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 488, + 366, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 366, + 500 + ], + "score": 1.0, + "content": "MSDNets substantially increase the accuracy of early classifiers.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "Problem: Early classifiers interfere with later classifiers. The right plot of Figure 3 shows the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "accuracies of the final classifier as a function of the location of a single intermediate classifier,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "relative to the accuracy of a network without intermediate classifiers. The results show that the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "introduction of an intermediate classifier harms the final ResNet classifier (blue line), reducing its", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 546, + 504, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 179, + 559 + ], + "score": 1.0, + "content": "accuracy by up to", + "type": "text" + }, + { + "bbox": [ + 180, + 546, + 194, + 557 + ], + "score": 0.86, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 546, + 504, + 559 + ], + "score": 1.0, + "content": ". We postulate that this accuracy degradation in the ResNet may be caused by", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "the intermediate classifier influencing the early features to be optimized for the short-term and not", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "for the final layers. This improves the accuracy of the immediate classifier but collapses information", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "required to generate high quality features in later layers. This effect becomes more pronounced", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 589, + 320, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 320, + 603 + ], + "score": 1.0, + "content": "when the first classifier is attached to an earlier layer.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "Solution: Dense connectivity. By contrast, the DenseNet (red line) suffers much less from this", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "effect. Dense connectivity (Huang et al., 2017) connects each layer with all subsequent layers and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "allows later layers to bypass features optimized for the short-term, to maintain the high accuracy", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "score": 1.0, + "content": "of the final classifier. If an earlier layer collapses information to generate short-term features, the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "lost information can be recovered through the direct connection to its preceding layer. The final", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "classifier’s performance becomes (more or less) independent of the location of the intermediate", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 690, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "4We select six evenly spaced locations for each of the networks to introduce the intermediate classifier.", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "Both the ResNet and DenseNet have three resolution blocks; each block offers two tentative locations for the", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 710, + 442, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 442, + 722 + ], + "score": 1.0, + "content": "intermediate classifier. The loss of the intermediate and final classifiers are equally weighted.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 720, + 349, + 733 + ], + "spans": [ + { + "bbox": [ + 118, + 720, + 349, + 733 + ], + "score": 1.0, + "content": "5Here, we use the term “layer” to refer to a column in Figure 2.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "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" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 83, + 506, + 149 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 504, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "An alternative solution is to build a deep network with a cascade of classifiers operating on the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "features of internal layers: in such a network features computed for an earlier classifier can be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "re-used by later classifiers. However, na¨ıvely attaching intermediate early-exit classifiers to a state-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 312, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 312, + 200 + ], + "score": 1.0, + "content": "of-the-art deep network leads to poor performance.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 154, + 506, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "There are two reasons why intermediate early-exit classifiers hurt the performance of deep neural", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "networks: early classifiers lack coarse-level features and classifiers throughout interfere with the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "feature generation process. In this section we investigate these effects empirically (see Figure 3)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 465, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 465, + 249 + ], + "score": 1.0, + "content": "and, in response to our findings, propose the MSDNet architecture illustrated in Figure 2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 204, + 506, + 249 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "Problem: The lack of coarse-level features. Traditional neural networks learn features of fine", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "scale in early layers and coarse scale in later layers (through repeated convolution, pooling, and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "strided convolution). Coarse scale features in the final layers are important to classify the content", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "of the whole image into a single class. Early layers lack coarse-level features and early-exit clas-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "sifiers attached to these layers will likely yield unsatisfactory high error rates. To illustrate this", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 307, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 506, + 319 + ], + "score": 1.0, + "content": "point, we attached4 intermediate classifiers to varying layers of a ResNet (He et al., 2016) and a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "DenseNet (Huang et al., 2017) on the CIFAR-100 dataset (Krizhevsky & Hinton, 2009). The blue", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "and red dashed lines in the left plot of Figure 3 show the relative accuracies of these classifiers.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "All three plots gives rise to a clear trend: the accuracy of a classifier is highly correlated with its", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 350, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 362 + ], + "score": 1.0, + "content": "position within the network. Particularly in the case of the ResNet (blue line), one can observe a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "visible “staircase” pattern, with big improvements after the 2nd and 4th classifiers — located right", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 372, + 189, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 189, + 385 + ], + "score": 1.0, + "content": "after pooling layers.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 251, + 506, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "Solution: Multi-scale feature maps. To address this issue, MSDNets maintain a feature repre-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "sentation at multiple scales throughout the network, and all the classifiers only use the coarse-level", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 412, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 504, + 423 + ], + "score": 1.0, + "content": "features. The feature maps at a particular layer5 and scale are computed by concatenating the re-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "sults of one or two convolutions: 1. the result of a regular convolution applied on the same-scale", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "features from the previous layer (horizontal connections) and, if possible, 2. the result of a strided", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 445, + 504, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 504, + 456 + ], + "score": 1.0, + "content": "convolution applied on the finer-scale feature map from the previous layer (diagonal connections).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "The horizontal connections preserve and progress high-resolution information, which facilitates the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "construction of high-quality coarse features in later layers. The vertical connections produce coarse", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "score": 1.0, + "content": "features throughout that are amenable to classification. The dashed black line in Figure 3 shows that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 488, + 366, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 366, + 500 + ], + "score": 1.0, + "content": "MSDNets substantially increase the accuracy of early classifiers.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 388, + 506, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "Problem: Early classifiers interfere with later classifiers. The right plot of Figure 3 shows the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "accuracies of the final classifier as a function of the location of a single intermediate classifier,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "relative to the accuracy of a network without intermediate classifiers. The results show that the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "introduction of an intermediate classifier harms the final ResNet classifier (blue line), reducing its", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 546, + 504, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 179, + 559 + ], + "score": 1.0, + "content": "accuracy by up to", + "type": "text" + }, + { + "bbox": [ + 180, + 546, + 194, + 557 + ], + "score": 0.86, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 546, + 504, + 559 + ], + "score": 1.0, + "content": ". 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This effect becomes more pronounced", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 589, + 320, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 320, + 603 + ], + "score": 1.0, + "content": "when the first classifier is attached to an earlier layer.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 501, + 505, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "Solution: Dense connectivity. By contrast, the DenseNet (red line) suffers much less from this", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "effect. 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The classifiers in MSDNets also follow the dense connectivity pattern within the coars-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 451, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 145, + 469 + ], + "score": 1.0, + "content": "est scale,", + "type": "text" + }, + { + "bbox": [ + 146, + 455, + 154, + 464 + ], + "score": 0.78, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 451, + 261, + 469 + ], + "score": 1.0, + "content": ", i.e., the classifier at layer", + "type": "text" + }, + { + "bbox": [ + 261, + 455, + 267, + 464 + ], + "score": 0.67, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 451, + 351, + 469 + ], + "score": 1.0, + "content": "uses all the features", + "type": "text" + }, + { + "bbox": [ + 351, + 453, + 404, + 467 + ], + "score": 0.93, + "content": "\\left[ \\mathbf { x } _ { 1 } ^ { S } , \\ldots , \\mathbf { x } _ { \\ell } ^ { S } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 451, + 507, + 469 + ], + "score": 1.0, + "content": ". Each classifier consists", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "of two convolutional layers, followed by one average pooling layer and one linear layer. 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During testing in the anytime setting we propagate the input through the network until the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "budget is exhausted and output the most recent prediction. 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We denote by", + "type": "text" + }, + { + "bbox": [ + 287, + 542, + 333, + 553 + ], + "score": 0.92, + "content": "0 < q \\le 1", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "a fixed exit probability that a sample that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 488, + 565 + ], + "score": 1.0, + "content": "reaches a classifier will obtain a classification with sufficient confidence to exit. We assume that", + "type": "text" + }, + { + "bbox": [ + 489, + 554, + 495, + 564 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 564, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 575 + ], + "score": 1.0, + "content": "constant across all layers, which allows us to compute the probability that a sample exits at classifier", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 113, + 584 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 573, + 128, + 588 + ], + "score": 1.0, + "content": "as:", + "type": "text" + }, + { + "bbox": [ + 129, + 573, + 207, + 586 + ], + "score": 0.92, + "content": "q _ { k } = z ( 1 - q ) ^ { \\bar { k ^ { - 1 } } q }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 573, + 236, + 588 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 237, + 576, + 243, + 584 + ], + "score": 0.78, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 573, + 413, + 588 + ], + "score": 1.0, + "content": "is a normalizing constant that ensures that", + "type": "text" + }, + { + "bbox": [ + 414, + 574, + 472, + 587 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sum _ { k } p ( q _ { k } ) = 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 573, + 506, + 588 + ], + "score": 1.0, + "content": ". 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There are two straightforward ways to further reduce the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "computational requirements of MSDNets. First, it is inefficient to maintain all the finer scales until", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "the last layer of the network. 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Second, since", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 182, + 172 + ], + "score": 1.0, + "content": "a classifier at layer", + "type": "text" + }, + { + "bbox": [ + 182, + 160, + 189, + 169 + ], + "score": 0.69, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 159, + 478, + 172 + ], + "score": 1.0, + "content": "only uses features from the coarsest scale, the finer feature maps in layer", + "type": "text" + }, + { + "bbox": [ + 478, + 160, + 484, + 169 + ], + "score": 0.59, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "(and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 295, + 183 + ], + "score": 1.0, + "content": "some of the finer feature maps in the previous", + "type": "text" + }, + { + "bbox": [ + 296, + 171, + 318, + 181 + ], + "score": 0.87, + "content": "S - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "layers) do not influence the prediction of that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "classifier. 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This minimizes", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "unnecessary computations when we need to stop because the computational budget is exhausted.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 214, + 255, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 255, + 227 + ], + "score": 1.0, + "content": "We call this strategy lazy evaluation.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 241, + 200, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 201, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 201, + 256 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 504, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "score": 1.0, + "content": "We evaluate the effectiveness of our approach on three image classification datasets, i.e., the CIFAR-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "10, CIFAR-100 (Krizhevsky & Hinton, 2009) and ILSVRC 2012 (ImageNet; Deng et al. (2009))", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "score": 1.0, + "content": "datasets. Code to reproduce all results is available at https://anonymous-url. Details on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 298, + 390, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 390, + 310 + ], + "score": 1.0, + "content": "architectural configurations of MSDNets are described in Appendix A.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 312, + 505, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 448, + 325 + ], + "score": 1.0, + "content": "Datasets. The two CIFAR datasets contain 50, 000 training and 10, 000 test images of", + "type": "text" + }, + { + "bbox": [ + 448, + 313, + 475, + 323 + ], + "score": 0.86, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "pixels;", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 338 + ], + "score": 1.0, + "content": "we hold out 5, 000 training images as a validation set. The datasets comprise 10 and 100 classes,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "respectively. We follow He et al. 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We train all models using the framework of Gross & Wilber (2016). On the two", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "CIFAR datasets, all models (including all baselines) are trained using stochastic gradient descent", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "(SGD) with mini-batch size 64. 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There exist several baseline approaches for anytime prediction: FractalNets (Larsson", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "et al., 2017), deeply supervised networks (Lee et al., 2015), and ensembles of deep networks of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "score": 1.0, + "content": "varying or identical sizes. FractalNets allow for multiple evaluation paths during inference time,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "which vary in computation time. In the anytime setting, paths are evaluated in order of increasing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "computation. In our result figures, we replicate the FractalNet results reported in the original paper", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "(Larsson et al., 2017) for reference. Deeply supervised networks introduce multiple early-exit classi-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "fiers throughout a network, which are applied on the features of the particular layer they are attached", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "to. Instead of using the original model proposed in Lee et al. (2015), we use the more competitive", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 351, + 667 + ], + "score": 1.0, + "content": "ResNet and DenseNet architectures (referred to as DenseNet-", + "type": "text" + }, + { + "bbox": [ + 351, + 655, + 365, + 665 + ], + "score": 0.29, + "content": "B C", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "in Huang et al. (2017)) as the base", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 664, + 507, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 445, + 678 + ], + "score": 1.0, + "content": "networks in our experiments with deeply supervised networks. We refer to these as", + "type": "text" + }, + { + "bbox": [ + 446, + 665, + 487, + 676 + ], + "score": 0.4, + "content": "R e s N e t ^ { M C }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 664, + 507, + 678 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 674, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 674, + 190, + 690 + ], + "score": 1.0, + "content": "DenseNetMC, where", + "type": "text" + }, + { + "bbox": [ + 190, + 677, + 210, + 687 + ], + "score": 0.73, + "content": "M C", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 674, + 461, + 690 + ], + "score": 1.0, + "content": "stands for multiple classifiers. Both networks require about", + "type": "text" + }, + { + "bbox": [ + 462, + 676, + 504, + 687 + ], + "score": 0.9, + "content": "1 . 3 \\times 1 0 ^ { 8 }", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "FLOPs when fully evaluated; the detailed network configurations are presented in the supplemen-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "tary material. In addition, we include ensembles of ResNets and DenseNets of varying or identical", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 711, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 722 + ], + "score": 1.0, + "content": "sizes. At test time, the networks are evaluated sequentially (in ascending order of network size) to", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "obtain predictions for the test data. All predictions are averaged over the evaluated classifiers. On", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 45 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 225 + ], + "lines": [], + "index": 6, + "bbox_fs": [ + 104, + 83, + 506, + 227 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 241, + 200, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 201, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 201, + 256 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 504, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "score": 1.0, + "content": "We evaluate the effectiveness of our approach on three image classification datasets, i.e., the CIFAR-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "10, CIFAR-100 (Krizhevsky & Hinton, 2009) and ILSVRC 2012 (ImageNet; Deng et al. (2009))", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "score": 1.0, + "content": "datasets. Code to reproduce all results is available at https://anonymous-url. Details on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 298, + 390, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 390, + 310 + ], + "score": 1.0, + "content": "architectural configurations of MSDNets are described in Appendix A.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 265, + 506, + 310 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 312, + 505, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 448, + 325 + ], + "score": 1.0, + "content": "Datasets. The two CIFAR datasets contain 50, 000 training and 10, 000 test images of", + "type": "text" + }, + { + "bbox": [ + 448, + 313, + 475, + 323 + ], + "score": 0.86, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "pixels;", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 338 + ], + "score": 1.0, + "content": "we hold out 5, 000 training images as a validation set. The datasets comprise 10 and 100 classes,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "respectively. We follow He et al. (2016) and apply standard data-augmentation techniques to the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "training images: images are zero-padded with 4 pixels on each side, and then randomly cropped", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 153, + 369 + ], + "score": 1.0, + "content": "to produce", + "type": "text" + }, + { + "bbox": [ + 153, + 357, + 183, + 367 + ], + "score": 0.89, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "images. Images are flipped horizontally with probability 0.5, and normalized", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "by subtracting channel means and dividing by channel standard deviations. The ImageNet dataset", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 377, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 104, + 377, + 506, + 392 + ], + "score": 1.0, + "content": "comprises 1, 000 classes, with a total of 1.2 million training images and 50,000 validation images.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "We hold out 50,000 images from the training set to estimate the confidence threshold for classifiers", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 401, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 413 + ], + "score": 1.0, + "content": "in MSDNet. We adopt the data augmentation scheme of He et al. (2016) at training time; at test", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 411, + 462, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 183, + 424 + ], + "score": 1.0, + "content": "time, we classify a", + "type": "text" + }, + { + "bbox": [ + 183, + 411, + 223, + 422 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 411, + 393, + 424 + ], + "score": 1.0, + "content": "center crop of images that were resized to", + "type": "text" + }, + { + "bbox": [ + 393, + 411, + 433, + 422 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 411, + 462, + 424 + ], + "score": 1.0, + "content": "pixels.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 312, + 506, + 424 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "Training Details. We train all models using the framework of Gross & Wilber (2016). On the two", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "CIFAR datasets, all models (including all baselines) are trained using stochastic gradient descent", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "(SGD) with mini-batch size 64. We use Nesterov momentum with a momentum weight of 0.9", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 284, + 472 + ], + "score": 1.0, + "content": "without dampening, and a weight decay of", + "type": "text" + }, + { + "bbox": [ + 284, + 458, + 305, + 469 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 457, + 506, + 472 + ], + "score": 1.0, + "content": ". All models are trained for 300 epochs, with an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "score": 1.0, + "content": "initial learning rate of 0.1, which is divided by a factor 10 after 150 and 225 epochs. We apply the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "same optimization scheme to the ImageNet dataset, except that we increase the mini-batch size to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 491, + 496, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 496, + 504 + ], + "score": 1.0, + "content": "256, and all the models are trained for 90 epochs with learning rate drops after 30 and 60 epochs.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 425, + 506, + 504 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 517, + 229, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 231, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 231, + 529 + ], + "score": 1.0, + "content": "5.1 ANYTIME PREDICTION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 503, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "In the anytime prediction setting, the model maintains a progressively updated distribution over", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 550, + 459, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 459, + 563 + ], + "score": 1.0, + "content": "classes, and it can be forced to output its most up-to-date prediction at an arbitrary time.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 539, + 505, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "score": 1.0, + "content": "Baselines. There exist several baseline approaches for anytime prediction: FractalNets (Larsson", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "et al., 2017), deeply supervised networks (Lee et al., 2015), and ensembles of deep networks of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "score": 1.0, + "content": "varying or identical sizes. FractalNets allow for multiple evaluation paths during inference time,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "which vary in computation time. In the anytime setting, paths are evaluated in order of increasing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "computation. In our result figures, we replicate the FractalNet results reported in the original paper", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "(Larsson et al., 2017) for reference. 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We refer to these as", + "type": "text" + }, + { + "bbox": [ + 446, + 665, + 487, + 676 + ], + "score": 0.4, + "content": "R e s N e t ^ { M C }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 664, + 507, + 678 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 674, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 674, + 190, + 690 + ], + "score": 1.0, + "content": "DenseNetMC, where", + "type": "text" + }, + { + "bbox": [ + 190, + 677, + 210, + 687 + ], + "score": 0.73, + "content": "M C", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 674, + 461, + 690 + ], + "score": 1.0, + "content": "stands for multiple classifiers. Both networks require about", + "type": "text" + }, + { + "bbox": [ + 462, + 676, + 504, + 687 + ], + "score": 0.9, + "content": "1 . 3 \\times 1 0 ^ { 8 }", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "FLOPs when fully evaluated; the detailed network configurations are presented in the supplemen-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "tary material. In addition, we include ensembles of ResNets and DenseNets of varying or identical", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 711, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 722 + ], + "score": 1.0, + "content": "sizes. At test time, the networks are evaluated sequentially (in ascending order of network size) to", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "obtain predictions for the test data. All predictions are averaged over the evaluated classifiers. On", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 45, + "bbox_fs": [ + 104, + 566, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 73, + 496, + 207 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 73, + 496, + 207 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 73, + 496, + 207 + ], + "spans": [ + { + "bbox": [ + 116, + 73, + 496, + 207 + ], + "score": 0.97, + "type": "image", + "image_path": "0e595351824a020ac424bb128829df21d92c0ecca2b1cea17c58c737e1308b61.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 73, + 496, + 117.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 117.66666666666666, + 496, + 162.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 162.33333333333331, + 496, + 206.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 103, + 215, + 504, + 235 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "Figure 5: Accuracy (top-1) of anytime prediction models as a function of computational budget on the ImageNet", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 224, + 305, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 305, + 237 + ], + "score": 1.0, + "content": "(left) and CIFAR-100 (right) datasets. Higher is better.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 503, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "ImageNet, we compare MSDNet against a highly competitive ensemble of ResNets and DenseNets,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 259, + 460, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 460, + 273 + ], + "score": 1.0, + "content": "with depth varying from 10 layers to 50 layers, and 36 layers to 121 layers, respectively.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 277, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 504, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 504, + 289 + ], + "score": 1.0, + "content": "Anytime prediction results are presented in Figure 5. The left plot shows the top-1 classification", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "accuracy on the ImageNet validation set. Here, for all budgets in our evaluation, the accuracy of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "MSDNet substantially outperforms the ResNets and DenseNets ensemble. In particular, when the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 308, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 186, + 325 + ], + "score": 1.0, + "content": "budget ranges from", + "type": "text" + }, + { + "bbox": [ + 186, + 309, + 227, + 321 + ], + "score": 0.91, + "content": "0 . 1 \\times \\bar { 1 0 } ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 308, + 239, + 325 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 239, + 309, + 280, + 321 + ], + "score": 0.9, + "content": "0 . 3 \\times 1 0 ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 308, + 389, + 325 + ], + "score": 1.0, + "content": "FLOPs, MSDNet achieves", + "type": "text" + }, + { + "bbox": [ + 389, + 310, + 435, + 321 + ], + "score": 0.92, + "content": "\\sim 4 \\% - 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 308, + 505, + 325 + ], + "score": 1.0, + "content": "higher accuracy.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "score": 1.0, + "content": "We evaluate more baselines on CIFAR-100 (and CIFAR-10; see supplementary materials). We", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 335, + 507, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 507, + 351 + ], + "score": 1.0, + "content": "observe that MSDNet substantially outperforms ResNetsMC and DenseNetsMC at any computational", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "budget within our range. This is due to the fact that after just a few layers, MSDNets have produced", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "low-resolution feature maps that are much more suitable for classification than the high-resolution", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "feature maps in the early layers of ResNets or DenseNets. MSDNet also outperforms the other", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "baselines for nearly all computational budgets, although it performs on par with ensembles when", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "the budget is very small. In the extremely low-budget regime, ensembles have an advantage because", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "their predictions are performed by the first (small) network, which is optimized exclusively for the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "low budget. However, the accuracy of ensembles does not increase nearly as fast when the budget is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "increased. The MSDNet outperforms the ensemble as soon as the latter needs to evaluate a second", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "model: unlike MSDNets, this forces the ensemble to repeat the computation of similar low-level", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 448, + 462, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 462, + 460 + ], + "score": 1.0, + "content": "features repeatedly. Ensemble accuracies saturate rapidly when all networks are shallow.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 109, + 468, + 283, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 285, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 285, + 479 + ], + "score": 1.0, + "content": "5.2 BUDGETED BATCH CLASSIFICATION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 428, + 495 + ], + "score": 1.0, + "content": "In budgeted batch classification setting, the predictive model receives a batch of", + "type": "text" + }, + { + "bbox": [ + 429, + 484, + 441, + 493 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 484, + 506, + 495 + ], + "score": 1.0, + "content": "instances and a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 195, + 506 + ], + "score": 1.0, + "content": "computational budget", + "type": "text" + }, + { + "bbox": [ + 195, + 495, + 205, + 504 + ], + "score": 0.8, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 495, + 277, + 506 + ], + "score": 1.0, + "content": "for classifying all", + "type": "text" + }, + { + "bbox": [ + 278, + 495, + 290, + 505 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "instances. In this setting, we use dynamic evaluation:", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "we perform early-exiting of “easy” examples at early classifiers whilst propagating “hard” examples", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 516, + 393, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 393, + 529 + ], + "score": 1.0, + "content": "through the entire network, using the procedure described in Section 4.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "Baselines. On ImageNet, we compare the dynamically evaluated MSDNet with five ResNets (He", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "et al., 2016) and five DenseNets (Huang et al., 2017), AlexNet (Krizhevsky et al., 2012), and Google-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "LeNet (Szegedy et al., 2015); see the supplementary material for details. We also evaluate an ensem-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "ble of the five ResNets that uses exactly the same dynamic-evaluation procedure as MSDNets at test", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "time: “easy” images are only propagated through the smallest ResNet-10, whereas “hard” images", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "are classified by all five ResNet models (predictions are averaged across all evaluated networks in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 598, + 330, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 259, + 612 + ], + "score": 1.0, + "content": "the ensemble). 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We also compare MSDNet to the", + "type": "text" + }, + { + "bbox": [ + 298, + 648, + 340, + 659 + ], + "score": 0.27, + "content": "\\mathrm { R e s N e t ^ { M C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 647, + 507, + 662 + ], + "score": 1.0, + "content": "and DenseNetMC models that were used", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 659, + 507, + 671 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 456, + 671 + ], + "score": 1.0, + "content": "in Section 5.1, using dynamic evaluation at test time. We denote these baselines as", + "type": "text" + }, + { + "bbox": [ + 456, + 659, + 498, + 670 + ], + "score": 0.29, + "content": "R e s N e t ^ { M C }", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 659, + 507, + 671 + ], + "score": 1.0, + "content": "/", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 669, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 669, + 506, + 685 + ], + "score": 1.0, + "content": "DenseNetMC with early-exits. To prevent the result plots from becoming too cluttered, we present", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "score": 1.0, + "content": "CIFAR-100 results with dynamically evaluated ensembles in the supplementary material. We clas-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 693, + 285, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 167, + 705 + ], + "score": 1.0, + "content": "sify batches of", + "type": "text" + }, + { + "bbox": [ + 167, + 693, + 204, + 703 + ], + "score": 0.89, + "content": "M = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 693, + 285, + 705 + ], + "score": 1.0, + "content": "images at test time.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Budgeted batch classification results on ImageNet are shown in the left panel of Figure 7. We", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 504, + 733 + ], + "score": 1.0, + "content": "trained three MSDNets with different depths, each of which covers a different range of compu-", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 73, + 496, + 207 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 73, + 496, + 207 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 73, + 496, + 207 + ], + "spans": [ + { + "bbox": [ + 116, + 73, + 496, + 207 + ], + "score": 0.97, + "type": "image", + "image_path": "0e595351824a020ac424bb128829df21d92c0ecca2b1cea17c58c737e1308b61.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 73, + 496, + 117.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 117.66666666666666, + 496, + 162.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 162.33333333333331, + 496, + 206.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 103, + 215, + 504, + 235 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "Figure 5: Accuracy (top-1) of anytime prediction models as a function of computational budget on the ImageNet", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 224, + 305, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 305, + 237 + ], + "score": 1.0, + "content": "(left) and CIFAR-100 (right) datasets. Higher is better.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 503, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "ImageNet, we compare MSDNet against a highly competitive ensemble of ResNets and DenseNets,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 259, + 460, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 460, + 273 + ], + "score": 1.0, + "content": "with depth varying from 10 layers to 50 layers, and 36 layers to 121 layers, respectively.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 106, + 249, + 505, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 277, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 504, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 504, + 289 + ], + "score": 1.0, + "content": "Anytime prediction results are presented in Figure 5. The left plot shows the top-1 classification", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "accuracy on the ImageNet validation set. Here, for all budgets in our evaluation, the accuracy of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "MSDNet substantially outperforms the ResNets and DenseNets ensemble. In particular, when the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 308, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 186, + 325 + ], + "score": 1.0, + "content": "budget ranges from", + "type": "text" + }, + { + "bbox": [ + 186, + 309, + 227, + 321 + ], + "score": 0.91, + "content": "0 . 1 \\times \\bar { 1 0 } ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 308, + 239, + 325 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 239, + 309, + 280, + 321 + ], + "score": 0.9, + "content": "0 . 3 \\times 1 0 ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 308, + 389, + 325 + ], + "score": 1.0, + "content": "FLOPs, MSDNet achieves", + "type": "text" + }, + { + "bbox": [ + 389, + 310, + 435, + 321 + ], + "score": 0.92, + "content": "\\sim 4 \\% - 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 308, + 505, + 325 + ], + "score": 1.0, + "content": "higher accuracy.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 277, + 506, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "score": 1.0, + "content": "We evaluate more baselines on CIFAR-100 (and CIFAR-10; see supplementary materials). We", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 335, + 507, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 507, + 351 + ], + "score": 1.0, + "content": "observe that MSDNet substantially outperforms ResNetsMC and DenseNetsMC at any computational", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "budget within our range. This is due to the fact that after just a few layers, MSDNets have produced", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "low-resolution feature maps that are much more suitable for classification than the high-resolution", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "feature maps in the early layers of ResNets or DenseNets. MSDNet also outperforms the other", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "baselines for nearly all computational budgets, although it performs on par with ensembles when", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "the budget is very small. In the extremely low-budget regime, ensembles have an advantage because", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "their predictions are performed by the first (small) network, which is optimized exclusively for the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "low budget. However, the accuracy of ensembles does not increase nearly as fast when the budget is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "increased. The MSDNet outperforms the ensemble as soon as the latter needs to evaluate a second", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "model: unlike MSDNets, this forces the ensemble to repeat the computation of similar low-level", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 448, + 462, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 462, + 460 + ], + "score": 1.0, + "content": "features repeatedly. Ensemble accuracies saturate rapidly when all networks are shallow.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 325, + 507, + 460 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 468, + 283, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 285, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 285, + 479 + ], + "score": 1.0, + "content": "5.2 BUDGETED BATCH CLASSIFICATION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 428, + 495 + ], + "score": 1.0, + "content": "In budgeted batch classification setting, the predictive model receives a batch of", + "type": "text" + }, + { + "bbox": [ + 429, + 484, + 441, + 493 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 484, + 506, + 495 + ], + "score": 1.0, + "content": "instances and a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 195, + 506 + ], + "score": 1.0, + "content": "computational budget", + "type": "text" + }, + { + "bbox": [ + 195, + 495, + 205, + 504 + ], + "score": 0.8, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 495, + 277, + 506 + ], + "score": 1.0, + "content": "for classifying all", + "type": "text" + }, + { + "bbox": [ + 278, + 495, + 290, + 505 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "instances. In this setting, we use dynamic evaluation:", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "we perform early-exiting of “easy” examples at early classifiers whilst propagating “hard” examples", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 516, + 393, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 393, + 529 + ], + "score": 1.0, + "content": "through the entire network, using the procedure described in Section 4.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 484, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "Baselines. On ImageNet, we compare the dynamically evaluated MSDNet with five ResNets (He", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "et al., 2016) and five DenseNets (Huang et al., 2017), AlexNet (Krizhevsky et al., 2012), and Google-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "LeNet (Szegedy et al., 2015); see the supplementary material for details. We also evaluate an ensem-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "ble of the five ResNets that uses exactly the same dynamic-evaluation procedure as MSDNets at test", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "time: “easy” images are only propagated through the smallest ResNet-10, whereas “hard” images", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "are classified by all five ResNet models (predictions are averaged across all evaluated networks in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 598, + 330, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 259, + 612 + ], + "score": 1.0, + "content": "the ensemble). We classify batches of", + "type": "text" + }, + { + "bbox": [ + 259, + 599, + 297, + 609 + ], + "score": 0.9, + "content": "M = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 598, + 330, + 612 + ], + "score": 1.0, + "content": "images.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 532, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "On CIFAR-100, we compare MSDNet with several highly competitive baselines, including ResNet-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "s (He et al., 2016), DenseNets (Huang et al., 2017) of varying sizes, Stochastic Depth Networks", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "(Huang et al., 2016), Wide ResNets (Zagoruyko & Komodakis, 2016) and FractalNets (Larsson", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 647, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 104, + 647, + 298, + 662 + ], + "score": 1.0, + "content": "et al., 2017). We also compare MSDNet to the", + "type": "text" + }, + { + "bbox": [ + 298, + 648, + 340, + 659 + ], + "score": 0.27, + "content": "\\mathrm { R e s N e t ^ { M C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 647, + 507, + 662 + ], + "score": 1.0, + "content": "and DenseNetMC models that were used", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 659, + 507, + 671 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 456, + 671 + ], + "score": 1.0, + "content": "in Section 5.1, using dynamic evaluation at test time. We denote these baselines as", + "type": "text" + }, + { + "bbox": [ + 456, + 659, + 498, + 670 + ], + "score": 0.29, + "content": "R e s N e t ^ { M C }", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 659, + 507, + 671 + ], + "score": 1.0, + "content": "/", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 669, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 669, + 506, + 685 + ], + "score": 1.0, + "content": "DenseNetMC with early-exits. To prevent the result plots from becoming too cluttered, we present", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "score": 1.0, + "content": "CIFAR-100 results with dynamically evaluated ensembles in the supplementary material. We clas-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 693, + 285, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 167, + 705 + ], + "score": 1.0, + "content": "sify batches of", + "type": "text" + }, + { + "bbox": [ + 167, + 693, + 204, + 703 + ], + "score": 0.89, + "content": "M = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 693, + 285, + 705 + ], + "score": 1.0, + "content": "images at test time.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 104, + 616, + 507, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Budgeted batch classification results on ImageNet are shown in the left panel of Figure 7. We", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 504, + 733 + ], + "score": 1.0, + "content": "trained three MSDNets with different depths, each of which covers a different range of compu-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "tational budgets. We plot the performance of each MSDNet as a gray curve; we select the best", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 265, + 504, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 504, + 277 + ], + "score": 1.0, + "content": "model for each budget based on its accuracy on the validation set, and plot the corresponding ac-", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "curacy as a black curve. The plot shows that the predictions of MSDNets with dynamic evaluation", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "are substantially more accurate than those of ResNets and DenseNets that use the same amount of", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 325, + 311 + ], + "score": 1.0, + "content": "computation. For instance, with an average budget of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 325, + 298, + 363, + 309 + ], + "score": 0.92, + "content": "1 . 7 \\times 1 0 ^ { 9 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 363, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "FLOPs, MSDNet achieves a top-1", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 155, + 322 + ], + "score": 1.0, + "content": "accuracy of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 155, + 309, + 182, + 320 + ], + "score": 0.89, + "content": "{ \\sim } 7 5 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 183, + 309, + 223, + 322 + ], + "score": 1.0, + "content": ", which is", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 223, + 309, + 246, + 320 + ], + "score": 0.88, + "content": "{ \\sim } 6 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 246, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "higher than that achieved by a ResNet with the same number of", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 417, + 333 + ], + "score": 1.0, + "content": "FLOPs. Compared to the computationally efficient DenseNets, MSDNet uses", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 418, + 321, + 455, + 331 + ], + "score": 0.9, + "content": "\\sim 2 - 3 \\times", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 455, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "times fewer", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "FLOPs to achieve the same classification accuracy. Moreover, MSDNet with dynamic evaluation", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "allows for very precise tuning of the computational budget that is consumed, which is not possible", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 353, + 504, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 504, + 365 + ], + "score": 1.0, + "content": "with individual ResNet or DenseNet models. The ensemble of ResNets or DenseNets with dynamic", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "score": 1.0, + "content": "evaluation performs on par with or worse than their individual counterparts (but they do allow for", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 376, + 303, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 303, + 388 + ], + "score": 1.0, + "content": "setting the computational budget very precisely).", + "type": "text", + "cross_page": true + } + ], + "index": 16 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 74, + 494, + 208 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 74, + 494, + 208 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 74, + 494, + 208 + ], + "spans": [ + { + "bbox": [ + 116, + 74, + 494, + 208 + ], + "score": 0.971, + "type": "image", + "image_path": "047b8902b777ad1edf01cc1798508fdcb86d98ce8b5ec9a2971efe96719d19ec.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 74, + 494, + 118.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 118.66666666666666, + 494, + 163.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 163.33333333333331, + 494, + 207.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 219, + 503, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "Figure 7: Accuracy (top-1) of budgeted batch classification models as a function of average computational", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 229, + 432, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 432, + 241 + ], + "score": 1.0, + "content": "budget per image the on ImageNet (left) and CIFAR-100 (right) datasets. Higher is better.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "tational budgets. We plot the performance of each MSDNet as a gray curve; we select the best", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 265, + 504, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 504, + 277 + ], + "score": 1.0, + "content": "model for each budget based on its accuracy on the validation set, and plot the corresponding ac-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "curacy as a black curve. The plot shows that the predictions of MSDNets with dynamic evaluation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "are substantially more accurate than those of ResNets and DenseNets that use the same amount of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 325, + 311 + ], + "score": 1.0, + "content": "computation. For instance, with an average budget of", + "type": "text" + }, + { + "bbox": [ + 325, + 298, + 363, + 309 + ], + "score": 0.92, + "content": "1 . 7 \\times 1 0 ^ { 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "FLOPs, MSDNet achieves a top-1", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 155, + 322 + ], + "score": 1.0, + "content": "accuracy of", + "type": "text" + }, + { + "bbox": [ + 155, + 309, + 182, + 320 + ], + "score": 0.89, + "content": "{ \\sim } 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 309, + 223, + 322 + ], + "score": 1.0, + "content": ", which is", + "type": "text" + }, + { + "bbox": [ + 223, + 309, + 246, + 320 + ], + "score": 0.88, + "content": "{ \\sim } 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "higher than that achieved by a ResNet with the same number of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 417, + 333 + ], + "score": 1.0, + "content": "FLOPs. Compared to the computationally efficient DenseNets, MSDNet uses", + "type": "text" + }, + { + "bbox": [ + 418, + 321, + 455, + 331 + ], + "score": 0.9, + "content": "\\sim 2 - 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "times fewer", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "FLOPs to achieve the same classification accuracy. Moreover, MSDNet with dynamic evaluation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "allows for very precise tuning of the computational budget that is consumed, which is not possible", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 353, + 504, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 504, + 365 + ], + "score": 1.0, + "content": "with individual ResNet or DenseNet models. The ensemble of ResNets or DenseNets with dynamic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "score": 1.0, + "content": "evaluation performs on par with or worse than their individual counterparts (but they do allow for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 376, + 303, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 303, + 388 + ], + "score": 1.0, + "content": "setting the computational budget very precisely).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "The right panel of Figure 7 shows our results on CIFAR-100. The results show that MSDNets con-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 403, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 504, + 415 + ], + "score": 1.0, + "content": "sistently outperform all baselines across all budgets. Notably, MSDNet performs on par with a 110-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 406, + 426 + ], + "score": 1.0, + "content": "layer ResNet using only 1/10th of the computational budget and it is up to", + "type": "text" + }, + { + "bbox": [ + 406, + 415, + 422, + 424 + ], + "score": 0.83, + "content": "\\sim 5", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "times more efficient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 426, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 437 + ], + "score": 1.0, + "content": "than DenseNets, Stochastic Depth Networks, Wide ResNets, and FractalNets. Similar to results in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 435, + 504, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 488, + 448 + ], + "score": 1.0, + "content": "the anytime-prediction setting, MSDNet substantially outperform ResNetsMC and DenseNets", + "type": "text" + }, + { + "bbox": [ + 488, + 435, + 504, + 444 + ], + "score": 0.31, + "content": "_ { M C }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 446, + 504, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 504, + 459 + ], + "score": 1.0, + "content": "with multiple intermediate classifiers, which provides further evidence that the coarse features in the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 457, + 354, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 354, + 471 + ], + "score": 1.0, + "content": "MSDNet are important for high performance in earlier layers.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 304, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 305, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 305, + 489 + ], + "score": 1.0, + "content": "Visualization. To illustrate the ability of our ap-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 486, + 305, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 305, + 498 + ], + "score": 1.0, + "content": "proach to reduce the computational requirements", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 496, + 305, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 305, + 509 + ], + "score": 1.0, + "content": "for classifying “easy” examples, we show twelve", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 508, + 305, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 305, + 519 + ], + "score": 1.0, + "content": "randomly sampled test images from two Ima-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 519, + 305, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 305, + 531 + ], + "score": 1.0, + "content": "geNet classes in Figure 6. The top row shows", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 306, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 306, + 542 + ], + "score": 1.0, + "content": "“easy” examples that were correctly classified", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 541, + 306, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 306, + 552 + ], + "score": 1.0, + "content": "and exited by the first classifier. The bottom row", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 551, + 306, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 306, + 563 + ], + "score": 1.0, + "content": "shows “hard” examples that would have been in-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 563, + 305, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 305, + 574 + ], + "score": 1.0, + "content": "correctly classified by the first classifier but were", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 574, + 304, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 304, + 585 + ], + "score": 1.0, + "content": "passed on because its uncertainty was too high.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 584, + 305, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 305, + 597 + ], + "score": 1.0, + "content": "The figure suggests that early classifiers recog-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 595, + 306, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 306, + 608 + ], + "score": 1.0, + "content": "nize prototypical class examples, whereas the last", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 606, + 271, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 271, + 619 + ], + "score": 1.0, + "content": "classifier recognizes non-typical images.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 30 + }, + { + "type": "image", + "bbox": [ + 312, + 479, + 503, + 549 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 312, + 479, + 503, + 549 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 312, + 479, + 503, + 549 + ], + "spans": [ + { + "bbox": [ + 312, + 479, + 503, + 549 + ], + "score": 0.967, + "type": "image", + "image_path": "105e212787f02ba3ad81fe470cd4782e898e2825cb6b2147833e9bbcf9aa8bfb.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 312, + 479, + 503, + 493.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 312, + 493.0, + 503, + 507.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 312, + 507.0, + 503, + 521.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 312, + 521.0, + 503, + 535.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 312, + 535.0, + 503, + 549.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 312, + 552, + 504, + 612 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 311, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 311, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "Figure 6: Sampled images from the ImageNet classes", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 312, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 312, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "Red wine and Volcano. Top row: images exited from", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 311, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 311, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "the first classifier of a MSDNet with correct predic-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 311, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 311, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "tion; Bottom row: images failed to be correctly clas-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 312, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 312, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "sified at the first classifier but were correctly predict-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 312, + 601, + 421, + 613 + ], + "spans": [ + { + "bbox": [ + 312, + 601, + 421, + 613 + ], + "score": 1.0, + "content": "ed and exited at the last layer.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + } + ], + "index": 38.75 + }, + { + "type": "title", + "bbox": [ + 107, + 633, + 349, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 350, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 350, + 646 + ], + "score": 1.0, + "content": "5.3 MORE COMPUTATIONALLY EFFICIENT DENSENETS", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "Here, we discuss an interesting finding during our exploration of the MSDNet architecture. We", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "score": 1.0, + "content": "found that following the DenseNet structure to design our network, i.e., by keeping the number of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "output channels (or growth rate) the same at all scales, did not lead to optimal results in terms of the", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "accuracy-speed trade-off. The main reason for this is that compared to network architectures like", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "ResNets, the DenseNet structure tends to apply more filters on the high-resolution feature maps in", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "the network. This helps to reduce the number of parameters in the model, but at the same time, it", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "score": 1.0, + "content": "greatly increases the computational cost. 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Higher is better.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 387 + ], + "lines": [], + "index": 10.5, + "bbox_fs": [ + 105, + 254, + 506, + 388 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "The right panel of Figure 7 shows our results on CIFAR-100. The results show that MSDNets con-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 403, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 504, + 415 + ], + "score": 1.0, + "content": "sistently outperform all baselines across all budgets. 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To illustrate the ability of our ap-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 486, + 305, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 305, + 498 + ], + "score": 1.0, + "content": "proach to reduce the computational requirements", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 496, + 305, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 305, + 509 + ], + "score": 1.0, + "content": "for classifying “easy” examples, we show twelve", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 508, + 305, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 305, + 519 + ], + "score": 1.0, + "content": "randomly sampled test images from two Ima-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 519, + 305, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 305, + 531 + ], + "score": 1.0, + "content": "geNet classes in Figure 6. The top row shows", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 306, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 306, + 542 + ], + "score": 1.0, + "content": "“easy” examples that were correctly classified", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 541, + 306, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 306, + 552 + ], + "score": 1.0, + "content": "and exited by the first classifier. 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We", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "score": 1.0, + "content": "found that following the DenseNet structure to design our network, i.e., by keeping the number of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "output channels (or growth rate) the same at all scales, did not lead to optimal results in terms of the", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "accuracy-speed trade-off. The main reason for this is that compared to network architectures like", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "ResNets, the DenseNet structure tends to apply more filters on the high-resolution feature maps in", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "the network. This helps to reduce the number of parameters in the model, but at the same time, it", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "score": 1.0, + "content": "greatly increases the computational cost. 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It turns", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "out that the resulting network, which we denote as DenseNet*, significantly outperform the original", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 293, + 297, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 297, + 308 + ], + "score": 1.0, + "content": "DenseNet in terms of computational efficiency.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 311, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "We experimented with DenseNet* in our two settings with test time budget constraints. 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This", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 313, + 499, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 499, + 325 + ], + "score": 1.0, + "content": "suggests that dense connectivity is particularly important in combination with multiple classifiers.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 269, + 505, + 325 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 337, + 236, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 236, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 236, + 350 + ], + "score": 1.0, + "content": "B.2 RESULTS ON CIFAR-10", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 358, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "For the CIFAR-10 dataset, we use the same MSDNets and baseline models as we used for CIFAR-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "100, except that the networks used here have a 10-way fully connected layer at the end. The results", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "under the anytime learning setting and the batch computational budget setting are shown in the left", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "and right panel of Figure 11, respectively. Similar to what we have observed from the results on", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "CIFAR-100 and ImageNet, MSDNets outperform all the baselines by a significant margin in both", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "settings. As in the experiments presented in the main paper, ResNet and DenseNet models with", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 423, + 342, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 342, + 438 + ], + "score": 1.0, + "content": "multiple intermediate classifiers perform relatively poorly.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 358, + 506, + 438 + ] + }, + { + "type": "image", + "bbox": [ + 116, + 447, + 496, + 582 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 447, + 496, + 582 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 116, + 447, + 496, + 582 + ], + "spans": [ + { + "bbox": [ + 116, + 447, + 496, + 582 + ], + "score": 0.959, + "type": "image", + "image_path": "adf6b406c38b10a1e3fedffbeafa7e030553b2ab7ceb4700cb0a1fb272d4ff92.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 116, + 447, + 496, + 492.0 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 116, + 492.0, + 496, + 537.0 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 116, + 537.0, + 496, + 582.0 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 594, + 505, + 615 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "Figure 11: Classification accuracies on the CIFAR-10 dataset in the anytime-prediction setting (left) and the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 604, + 217, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 217, + 614 + ], + "score": 1.0, + "content": "budgeted batch setting (right).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "index": 46.25 + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git 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+tomg@cs.umd.edu + +# ABSTRACT + +Adversarial patch attacks are among one of the most practical threat models against real-world computer vision systems. This paper studies certified and empirical defenses against patch attacks. We begin with a set of experiments showing that most existing defenses, which work by pre-processing input images to mitigate adversarial patches, are easily broken by simple white-box adversaries. Motivated by this finding, we propose the first certified defense against patch attacks, and propose faster methods for its training. Furthermore, we experiment with different patch shapes for testing, obtaining surprisingly good robustness transfer across shapes, and present preliminary results on certified defense against sparse attacks. Our complete implementation can be found on: https://github.com/Ping-C/certifiedpatchdefense. + +# 1 INTRODUCTION + +Despite the great success of neural networks for vision problems, they are easily fooled by adversarial attacks in which the input to a machine learning model is modified with the goal of manipulating its output. Research in this area is largely focused on norm-bounded attack (Madry et al., 2017; Tramer\` & Boneh, 2019; Shafahi et al., 2019), where the adversary is allowed to perturb all pixels in an image provided that the $\ell _ { p }$ -norm of the perturbation is within prescribed bounds. Other adversarial models were also proposed, such as functional (Laidlaw & Feizi, 2019), rotation/translation (Engstrom et al., 2017), and Wasserstein (Wong et al., 2019), all of which allow modification to all pixels. + +Whole-image perturbations are unrealistic for modeling ”physical-world” attacks, in which a realworld object is modified to evade detection. A physical adversary usually modifies an object using stickers or paint. Because this object may only occupy a small portion of an image, the adversary can only manipulate a limited number of pixels. As such, the more practical patch attack model was proposed (Brown et al., 2017). In a patch attack, the adversary may only change the pixels in a confined region, but is otherwise free to choose the values yielding the strongest attack. The threat to real-world computer vision systems is well-demonstrated in recent literature where carefully crafted patches can fool a classifier with high reliability (Brown et al., 2017; Karmon et al., 2018), make objects invisible to an object detector (Wu et al., 2019; Lee & Kolter, 2019), or fool a face recognition system (Sharif et al., 2017). In light of such effective physical-world patch attacks, very few defenses are known to date. + +In this paper, we study principled defenses against patch attacks. We begin by looking at existing defenses in the literature that claim to be effective against patch attacks, including Local Gradient Smoothing (LGS) (Naseer et al., 2019) and Digital Watermarking (DW) (Hayes, 2018). Similar to what has been observed for whole-image attacks by (Athalye et al., 2018), we show that these patch defenses are easily broken by stronger adversaries. Concretely, we demonstrate successful white-box attacks, where the adversary designs an attack against a known model, including any pre-processing steps. To cope with such potentially stronger adversaries, we train a robust model that produces a lower-bound on adversarial accuracy. In particular, we propose the first certifiable defense against patch attacks by extending interval bound propagation (IBP) defenses (Gowal et al., 2018; Mirman et al., 2018). We also propose modifications to IBP training to make it faster in the patch setting. Furthermore, we study the generalization of certified patch defenses to patches of different shapes, and observe that robustness transfers well across different patch types. We also present preliminary results on certified defense against the stronger sparse attack model, where a fixed number of possibly non-adjacent pixels can be arbitrarily modified (Modas et al., 2019). + +# 2 PROBLEM SETUP + +We consider a white-box adversary that is allowed to choose the location of the patch (chosen from a set $\mathbb { L }$ of possible locations) and can modify pixels within the particular patch (chosen from the set $\mathbb { P }$ ) similar to (Karmon et al., 2018). An attack is successful if the adversary changes the classification of the network to a wrong label. In this paper, we are primarily interested in the patch attack robust accuracy (adversarial accuracy for short) as defined by + +$$ +\underset { x \sim X } { \mathbb { E } } \operatorname* { m i n } _ { p \in \mathbb { P } , l \in \mathbb { L } } \mathcal { X } [ f ( A ( x , p , l ) ; \theta ) = y ] , +$$ + +where the operator $A$ places the adversarial patch p on a given image $\mathbf { X }$ at location $l$ , f is a neural network with parameter $\theta$ , $X$ is a distribution of images, and $\mathcal { X }$ is a characteristic function that takes value 1 if its argument is true, and 0 otherwise. + +In this model, the strength of the adversary can vary depending on the set of possible patches allowed, and the type of perturbation allowed within the patch. In what follows, we assume the standard setup in which the adversary is allowed any perturbation that maintains pixel intensities in the range $[ 0 , 1 ]$ . Unless otherwise noted, we also assume the patch is restricted to a square of prescribed size. We consider two different options for the set $\mathbb { L }$ of possible patch locations. First, we consider a weak adversary that can only place patches at the corner of an image. We find that even this weak model is enough to break existing patch defenses. Then, we consider a stronger adversary with no restrictions on patch location, and use this model to evaluate our proposed defenses. Note that an adversary, when restricted to modify only a square patch at location $l$ in the image, has the freedom to modify any non-square subset of these pixels. In other words, a certified defense against square patch attacks also provably subverts any non-square patch attack that fits inside a small enough square. + +In general, calculating the adversarial accuracy (1) is intractable due to non-convexity. Common approaches try to approximate it by solving the inner minimization using a gradient-based method. However, in Section 3, we show that depending on how the minimization is solved, the upper bound could be very loose: a model may appear to be very robust, but fail when faced with a stronger attack. To side-step the arms race between attacks and defenses, in Section 4, we extend the work of (Gowal et al., 2018) and (Mirman et al., 2018) to train a network that produces a lower bound on adversarial accuracy. We will refer to approximations of the upper bound as empirical adversarial accuracy and the lower bound as certified accuracy. + +# 3 VULNERABILITY OF EXISTING DEFENSES + +We start by examining existing defense strategies that claim to be effective against patch attacks. Similar to what has been observed for whole-image attacks by Athalye et al. (2018), we show that these patch defenses can easily be broken by white-box attacks, where the adversary optimizes against a given model including any pre-processing steps. + +# 3.1 EXISTING DEFENSES + +Under our threat model, two defenses have been proposed that each use input transformations to detect and remove adversarial patches. + +The first defense is based on the observation that the gradient of the loss with respect to the input image often exhibits large values near the perturbed pixels. In (Hayes, 2018), the proposed digital watermarking (DW) approach exploits this behavior to detect unusually dense regions of large gradient entries using saliency maps, before masking them out in the image. Despite a $1 2 \%$ drop in accuracy on clean (non-adversarial) images, this defense method supposedly achieves an empirical adversarial accuracy of $6 3 \%$ for non-targeted patch attacks of size $4 2 \times 4 2$ ( $\dot { 2 } \%$ of the image pixels), using 400 randomly picked images from ImageNet (Deng et al., 2009) on VGG19 (Simonyan & Zisserman, 2014). + +Table 1: Empirical adversarial accuracy of ImageNet classifiers defended with Local Gradient Smoothing and Digital Watermarking. We consider two types of adversaries, one that takes the defense into account during backpropagation and one that does not + +
Patch Size
AttackDefense42 × 4252 × 5260 ×60
IFGSMLGS78%75%71%
IFGSM+LGSLGS14%5%3%
IFGSMDW56%49%45%
IFGSM+DWDW13%8%5%
+ +The second defense, Local Gradient Smoothing (LGS) by Naseer et al. (2019) is based on the empirical observation that pixel values tend to change sharply within these adversarial patches. In other words, the image gradients tend to be large within these adversarial patches. Note that the image gradient here differs from the gradient in Hayes (2018), the former is with respect the changes of adjacent pixel values and the later is with respect to the classification loss. Naseer et al. (2019) propose suppressing this adversarial noise by multiplying each pixel with one minus its image gradient as in (2). To make their methods more effective, Naseer et al. (2019) also pre-process the image gradient with a normalization and a thresholding step. + +$$ +{ \hat { x } } = x \odot ( 1 - \lambda g ( x ) ) . +$$ + +The $\lambda$ here is a smoothing hyper-parameter. Naseer et al. (2019) claim the best adversarial accuracy on ImageNet with respect to patch attacks among all of the defenses we studied. They also claim that their defense is resilient to Backward Pass Differential Approximation (BPDA) from Athalye et al. (2018), one of the most effective methods to attack models that include a non-differentiable operator as a pre-processing step. + +# 3.2 BREAKING EXISTING DEFENSES + +Using a similar setup as in (Hayes, 2018; Naseer et al., 2019), we are able to mostly replicate the reported empirical adversarial accuracy for Iterative Fast Gradient Sign Method (IFGSM), a common gradient based attack, but we show that when the pre-processing step is taken into account, the empirical adversarial accuracy on ImageNet quickly drops from $\bar { \sim } 7 0 \% ( \sim 5 0 \% )$ for LGS(DW) to levels around $\sim 1 0 \%$ as shown in Table 1. + +Specifically, we break DW (Hayes, 2018) by applying BPDA, in which the non-differentiable operator is approximated with an identity mapping during the backward pass. We break LGS (Naseer et al., 2019) by directly incorporating the smoothing step during backpropagation. Even though the windowing and thresholding steps are non-differentiable, the smoothing operator provides enough gradient information for the attack to be effective. + +To make sure that our evaluation is fair, we used the exact same models as Hayes (2018) (VGG19) and Szegedy et al. (2016) (Inception V3). We also consider a weaker set of attackers that can only attack the corners, the same as their setting. Further, we ensure that we were able to replicate their reported result under similar setting. + +# 4 CERTIFIED DEFENSES + +Given the ease with which these supposedly strong defenses are broken, it is natural to seek methods that can rigorously guarantee robustness of a given model to patch attacks. With such certifiable guarantees in hand, we need not worry about an adversary with a stronger optimizer, or a more clever algorithm for choosing patch locations. + +# 4.1 BACKGROUND ON CERTIFIED DEFENSES + +Certified defenses have been intensely studied with respect to norm-bounded attacks (Cohen et al., 2019; Wong & Kolter, 2017; Gowal et al., 2018; Mirman et al., 2018; Zhang et al., 2019b). In all of these methods, in addition to the prediction model, there is also a verifier. Given a model and an input, the verifier outputs a certificate if it is guaranteed that the image can not be adversarially perturbed. This is done by checking whether there exists any nearby image (within a prescribed $\ell _ { p }$ distance) with a different label than the image being classified. While theoretical bounds exist on the size of this distance that hold for any classifier (Shafahi et al., 2018), exactly computing bounds for a specific classifier and test image is hard. Alternatively, the verifier may output a lower bound on the distance to the nearest image of a different label. This latter distance is referred to as the certifiable radius. Most of these verifiers provide a rather loose bound on the certifiable radius. However, if the verifier is differentiable, then the network can be trained with a loss that promotes tightness of this bound. We use the term certificate training to refer to the process of training with a loss that promotes strong certificates. Interval bound propagation (IBP) (Mirman et al., 2018; Gowal et al., 2018) is a very simple verifier that uses layer-wise interval arithmetic to produce a certificate. Even though the IBP certificate is generally loose, after certificate training, it yields state-of-the-art certifiably-robust models for $l _ { \infty }$ -norm bounded attacks (Gowal et al., 2018; Zhang et al., 2019b). In this paper, we extend IBP to train certifiably-robust networks resilient to patch attacks. We first introduce some notation and basic algorithms for IBP training. + +Notation We represent a neural network with a series of transformations $h ^ { ( k ) }$ for each of its $k$ layers. We use $z ^ { ( k ) } \in \bar { \mathbb { R } } ^ { n _ { k } }$ to denote the output of layer $k$ , where $n _ { k }$ is the number of units in the $k ^ { t h }$ layer and $z ^ { ( 0 ) }$ stands for the input. Specifically, the network computes + +$$ +z ^ { ( k ) } = h ^ { ( k - 1 ) } ( z ^ { ( k - 1 ) } ) \forall k = 1 , \dots , K . +$$ + +Certification Problem To produce a certificate for an input $x _ { 0 }$ , we want to verify that the following condition is true with respect to all possible labels $y$ : + +$$ +( e _ { y _ { t r u e } } - e _ { y } ) ^ { T } z ^ { ( K ) } = { \bf m } _ { y } \geq 0 \qquad \forall z ^ { ( 0 ) } \in \mathbb { B } ( x _ { 0 } ) \qquad \forall y . +$$ + +Here, $e _ { i }$ is the $i ^ { t h }$ basis vector, and $\mathbf { m } _ { y }$ is called the margin following Wong & Kolter (2017). Note that $\mathbf { m } _ { y _ { t r u e } }$ is always equal to 0. The vector $\mathbf { m }$ contains all margins corresponding to all labels. $\mathbb { B } ( x _ { 0 } )$ is the constraint set over which the adversarial input image may range. In a conventional setting, this is an $\ell _ { \infty }$ ball around $x _ { 0 }$ . In the case of patch attack, the constraint set contains all images formed by applying a patch to $x _ { 0 }$ ; + +$$ +\mathbb { B } ( x _ { 0 } ) = \{ A ( x _ { 0 } , p , l ) | p \in \mathbb { P } { \mathrm { ~ a n d ~ } } l \in \mathbb { L } \} . +$$ + +The Basics of Interval Bound Propagation (IBP) We now describe how to produce certificates using interval bound propagation as in (Gowal et al., 2018). Suppose that for each component in $z ^ { ( k - \bar { 1 } ) }$ we have an interval containing all the values which this component reaches as $z ^ { ( 0 ) }$ ranges over the ball $\mathbb { B } ( x _ { 0 } )$ . If $z ^ { ( k ) } = h ^ { ( k ) } ( z ^ { \top } )$ is a linear (or convolutional) layer of the form ${ z } ^ { ( k ) ^ { - } } =$ $W ^ { ( k ) } z ^ { ( k - 1 ) } + b ^ { ( k ) }$ , then we can get an outer approximation of the reachable interval range of activations by the next layer $z ^ { ( k ) }$ using the formulas below + +$$ +\begin{array} { r l } & { \overline { { z } } ^ { ( k ) } = W ^ { ( k ) } \frac { \overline { { z } } ^ { ( k - 1 ) } + \underline { { z } } ^ { ( k - 1 ) } } { 2 } + | W ^ { ( k ) } | \frac { \overline { { z } } ^ { ( k - 1 ) } - \underline { { z } } ^ { ( k - 1 ) } } { 2 } + b ^ { ( k ) } , } \\ & { \underline { { z } } ^ { ( k ) } = W ^ { ( k ) } \frac { \overline { { z } } ^ { ( k - 1 ) } + \underline { { z } } ^ { ( k - 1 ) } } { 2 } - | W ^ { ( k ) } | \frac { \overline { { z } } ^ { ( k - 1 ) } - \underline { { z } } ^ { ( k - 1 ) } } { 2 } + b ^ { ( k ) } . } \end{array} +$$ + +Here $\overline { { z } } ^ { ( k - 1 ) }$ denotes the upper bound of each interval, $\underline { z } ^ { ( k - 1 ) }$ the lower bound, and $| W ^ { ( k ) } |$ the element-wise absolute value. Alternatively, if $h ^ { ( k ) } ( z ^ { ( k - 1 ) } )$ is an element-wise monotonic activation (e.g., a ReLU), then we can calculate the outer approximation of the reachable interval range of the next layer using the formulas below. + +$$ +\begin{array} { r } { \overline { z } ^ { ( k ) } = h ^ { ( k ) } ( \overline { z } ^ { ( k - 1 ) } ) } \\ { \underline { z } ^ { ( k ) } = h ^ { ( k ) } ( \underline { z } ^ { ( k - 1 ) } ) . } \end{array} +$$ + +When the feasible set $\mathbb { B } ( x _ { 0 } )$ represents a simple $\ell _ { \infty }$ attack, the range of possible $z ^ { ( 0 ) }$ values is trivially characterized by an interval bound $\overline { { z } } ^ { ( 0 ) }$ and $\underline { z } ^ { ( 0 ) }$ . Then, by iteratively applying the above rules, we can propagate intervals through the network and eventually get $\overline { { z } } ^ { ( K ) }$ and $\smash { \mathcal { Z } ^ { ( K ) } }$ . A certificate can then be given if we can show that (3) is always true for outputs in the range $\overline { { z } } ^ { ( K ) }$ and $\smash { \boldsymbol { \mathcal { Z } } ^ { ( K ) } }$ with respect to all possible labels. More specifically, we can check that the following holds for all $y$ + +$$ +\begin{array} { r } { \underline { { \mathbf { m } } } _ { y } = e _ { y _ { t r u e } } ^ { T } \underline { { z } } ^ { ( K ) } - e _ { y } ^ { T } \overline { { z } } ^ { ( K ) } = \underline { { z } } _ { y _ { t r u e } } ^ { ( K ) } - \overline { { z } } _ { y } ^ { ( K ) } \geq 0 \quad \forall y . } \end{array} +$$ + +Training for Interval Bound Propagation To train a network to produce accurate interval bounds, we simply replace standard logits with the $- \mathbf { m }$ vector in (3). Note that all elements of $\mathbf { m }$ need to be larger than zero to satisfy the conditions in (3), and mytrue is always equal to zero. Put simply, we would like mytrue to be the least of all margins. We can promote this condition by training with the loss function + +$$ +\mathrm { C e r t i f i c a t e \ L o s s } = \mathrm { C r o s s \ E n t r o p y \ L o s s } ( - \mathbf { m } , y ) . +$$ + +Unlike regular neural network training, stochastic gradient descent for minimizing equation 10 is unstable, and a range of tricks are necessary to stabilize IBP training (Gowal et al., 2018). The first trick is merging the last linear weight matrix with $( e _ { y } - e _ { y _ { t r u e } } )$ before calculating $- { \underline { { \mathbf { m } } } } _ { y }$ . This allows a tighter characterization of the interval bound that noticeably improves results. The second trick uses an “epsilon schedule” in which training begins with a perturbation radius of zero, and this radius is slowly increased over time until a sentinel value is reached. Finally, a mixed loss function containing both a standard natural loss and an IBP loss is used. + +In all of our experiments, we use the merging technique and the epsilon schedule, but we do not use a mixed loss function containing a natural loss as it does not increase our certificate performance. + +# 4.2 CERTIFYING AGAINST PATCH ATTACKS + +We can now describe the extension of IBP to patches. If we specify the patch location, one can represent the feasible set of images with a simple interval bound: for pixels within the patch, the upper and lower bound is equal to 1 and 0. For pixels outside of the patch, the upper and lower bounds are both equal to the original pixel value. By passing this bound through the network, we would be able to get msingle location and verify that they satisfy the conditions in (3). + +However, we have to consider not just a single location, but all possible locations $\mathbb { L }$ to give a certificate. To adapt the bound to all possible location, we pass each of the possible patches through the network, and take the worst case margin. More specifically, + +$$ +\mathbf { m } ^ { \mathrm { e s } } ( \mathbb { L } ) _ { y } = \operatorname* { m i n } _ { l \in \mathbb { L } } \mathbf { m } ^ { \mathrm { s i n g l e ~ p a t c h } } ( l ) _ { y } \forall y . +$$ + +Similar to regular IBP training, we simply use $\underline { { \mathbf { m } } } ^ { \mathrm { e s } } ( \mathbb { L } )$ to calculate the cross entropy loss for training and backpropagation, + +$$ +\operatorname { C e r t i f i c a t e \operatorname { L o s s } } = \operatorname { C r o s s \operatorname { E n t r o p y \operatorname { L o s s } } } ( - \underline { { \mathbf { m } } } ^ { \mathrm { e s } } ( \mathbb { L } ) , y ) . +$$ + +Unfortunately, the cost of producing this na¨ıve certificate increases quadratically with image size. Consider that a CIFAR-10 image is of size $3 2 \times 3 2$ , requiring over a thousand interval bounds, one for each possible patch location. To alleviate this problem, we propose two certificate training methods: Random Patch and Guided Patch, so that the number of forward passes does not scale with the dimension of the inputs. + +Random Patch Certificate Training In this method, we simply select a random set of patches out of the possible patches and pass them forward. A level of robustness is achieved even though a very small number of random patches are selected compared to the total number of possible patches + +$$ +\underline { { \mathbf { m } } } ^ { \mathrm { r a n d o m \ p a t c h e s } } ( \mathbb { L } ) _ { y } = \underline { { \mathbf { m } } } ^ { \mathrm { e s } } ( S ) _ { y } +$$ + +where $S$ is a random subset of $\mathbb { L }$ . Similarly, the random patch certificate loss is calculated as below. + +$$ +\mathrm { R a n d o m ~ P a t c h ~ C e r t i f i c a t e ~ L o s s } = \mathrm { C r o s s ~ E n t r o p y ~ L o s s } ( - \underline { { \mathbf { m } } } ^ { \mathrm { r a n d o m ~ p a t c h e s } } ( \mathbb { L } ) , y ) +$$ + +Guided Patch Certificate Training In this method, we propose using a U-net (Ronneberger et al., 2015) to predict msingle patch, and then randomly select a couple of locations based on the predicted $\mathbf { m } ^ { \mathrm { s i n g l e } \mathrm { p a t c h } }$ so that fewer patches need to be passed forward. + +Note that very few patches contribute to the worst case bound $\underline { { \mathbf { m } } } ^ { \mathrm { e s } }$ in (11). In fact, the number of patches that yield the worst case margins will be no more than the number of labels. If we know the worst-case patches beforehand, then we can simply select the few worst-case patches during training. + +We propose to use U-net as the number of locations and margins is very large. For a square patch of size $n \times n$ and an image of size $m \times m$ , the total number of possible locations is $( m - n + \mathbf { \hat { 1 } } ) ^ { 2 }$ , and for each location the number of margins is equal to the number of possible labels. + +$$ +\begin{array} { l } { { \bf { m } } ^ { \mathrm { { p r e d } } } = { \bf { U } } { \mathrm { - n e t } } ( \mathrm { i m a g e } ) } \\ { { \bf { d i m } } ( { \bf { m } } ^ { \mathrm { { p r e d } } } ) = ( m - n + 1 , m - n + 1 , \# \mathrm { o f } \mathrm { l a b e l s } ) . } \end{array} +$$ + +Given the U-net prediction of $\underline { { \mathbf { m } } } ^ { \mathrm { p r e d } }$ , we then randomly select a single patch for each label based on the softmax of the predicted $\underline { { \mathbf { m } } } ^ { \mathrm { p r e d } }$ . The number of selected patches is equal to the number of labels. After these patches are passed forward, the U-net is then updated with a mean-squared-error loss between the predicted margins $\underline { { \mathbf { m } } } ^ { \mathrm { p r e d } }$ and the actual margins $\underline { { \mathbf { m } } } ^ { \mathrm { a c t u a l } }$ . Note that only a few patches are selected at a time, so that the mean-squared-error only passes through the selected patches. + +$$ +\mathrm { U } \mathrm { - n e t } \mathrm { L o s s } = \mathbf { M } \mathrm { S E } ( \mathbf { \underline { { m } } } ^ { \mathrm { p r e d } } , \mathbf { \underline { { m } } } ^ { \mathrm { a c t u a l } } ) . +$$ + +The network is trained with the following loss: + +Guided Patch Certificate $\operatorname { L o s s } = \operatorname { C r o s s } \operatorname { E n t r o p y } \operatorname { L o s s } ( - \underline { { \mathbf { m } } } ^ { \mathrm { g u i d e d } \mathrm { p a t c h e s } } ( \mathbb { L } ) , y ) .$ + +Certification Process In all our experiments, we check that equation (3) is satisfied by iterating over all possible patches and forward-passing the interval bounds generated for each patch; this overhead is tolerable at evaluation time. + +# 4.3 CERTIFYING AGAINST SPARSE ATTACKS + +IBP can also be adapted to defend against sparse attack where the attacker is allowed to modify a fixed number $( k )$ of pixels that may not be adjacent to each other (Modas et al., 2019). The only modification is that we have to change the bound calculated from the first layer to + +$$ +\overline { { { z } } } _ { i } ^ { ( 1 ) } = W _ { i , : } ^ { ( 1 ) } z ^ { ( 0 ) } + | W _ { i , : } ^ { ( 1 ) } | _ { t o p _ { k } } ~ z _ { i } ^ { ( 1 ) } = W _ { i , : } ^ { ( 1 ) } z ^ { ( 0 ) } - | W _ { i , : } ^ { ( 1 ) } | _ { t o p _ { k } } ~ \forall i +$$ + +and apply equation (5) and (6) for the subsequent layers. Here, $( . ) _ { t o p _ { k } }$ is the sum of the largest $k$ elements in the vector. + +# 5 EXPERIMENTS + +In this section, we compare our certified defenses with exiting algorithms on two datasets and three model architectures of varying complexity. We consider a strong attack setting in which adversarial patches can appear anywhere in the image. Different training strategies for the certified defense are also compared, which shows a trade-off between performance and training efficiency. Furthermore, we evaluate the transferability of a model trained using square patches to other adversarial shapes, including shapes that do not fit in any certified square. The training and architectural details can be found in Appendix A.1. We also present preliminary results on certified defense against sparse attacks. + +# 5.1 COMPARISON AGAINST EXISTING DEFENSES + +In this section, we study the effectiveness of our proposed IBP certified models against an adversary that is allowed to place patches anywhere in the image, even on top of the salient object. If the patch is sufficiently small, and does not cover a large portion of the salient object, then the model should still classify correctly, and defense against the perturbation should be possible. + +In the best case, our IBP certified model is able to achieve $9 1 . 6 \%$ certified (Table 2) with respect to a $2 \times 2$ patch $( \sim . 5 \%$ of image pixels) adversary on MNIST. For more challenging cases, such as a 5 $\times 5$ ( $\sim 2 . 5 \%$ of image pixels) patch adversary on CIFAR-10, the certified adversarial accuracy is only $2 4 . 9 \%$ (Table 2). Even though these existing defenses appear to achieve better or comparable adversarial accuracy as our IBP certified model when faced with a weak adversary, when faced with a stronger adversary their adversarial accuracy dropped to levels below our certified accuracy for all cases that we analyzed. + +When evaluating existing defenses, we only report cases where non-trivial adversarial accuracy is achieved against a weaker adversary. We do not explore cases where LGS and DW perform so poorly that no meaningful comparison can be done. LGS and DW are highly dependent on hyperparameters to work effectively against naive attacks, and yet neither Naseer et al. (2019) nor Hayes (2018) proposed a way to learn these hyperparameters. By trial and error, we were able to increase the adversarial accuracy against a weaker adversary for some settings, but not all. In addition, we also notice a peculiar feature of DW: when we increase the adversarial accuracy, the clean accuracy degrades, sometimes so much that it is even lower than the empirical adversarial accuracy. This happens because DW always removes a patch from the prediction. When an adversarial patch is detected, it is likely to be removed, enabling correct prediction. On the other hand, when there are no adversarial patches, DW removes actual salient information, resulting in lower clean accuracy. + +Here we did not compare our results with adversarial training, because even though it produces some of the most adversarially robust models, it does not offer any guarantees on the empirical robust accuracy, and could still be decreased further with stronger attacks. For example, Wang et al. (2019) proposed a stronger attack that could find $47 \%$ more adversarial examples compared to gradient based method. Further, adversarial training on all possible patches would be even more expensive compared to certificate training, and is slightly beyond our computational budget. + +Compared to state-of-the-art certified models for CIFAR with $L _ { \infty }$ -perturbation, where Zhang et al. (2019a) proposed a deterministic algorithm that achieves clean accuracy of $3 4 . 0 \%$ , our clean accuracy for our most robust CIFAR $5 \times 5$ model is $4 7 . 8 \%$ when using a large model (Table 2). + +Table 2: Comparison of our IBP certified patch defense against existing defenses. Empirical adversarial accuracy is calculated for 400 random images in both datasets. All results are averaged over three different models. + +
DatasetPatch SizeAdversaryDefenseClean AccuracyAccuracyEmpirical Certified Adversarial Accuracy
MNIST2×2IFGSMNone98.4%80.1%
2×2IFGSMLGS97.4%90.0%
2×2IFGSM+LGSLGS97.4%60.7%
2×2IFGSMIBP98.5%93.9%91.6%
5×5IFGSMNone98.5%3.3%
CIFAR5×5IFGSMIBP92.9%66.1%62.0%
2×2IFGSMNone66.3%25.4%1
2×2IFGSMLGS64.9%31.3%
2×2IFGSM+LGSLGS64.9%24.2%
2×2IFGSMDW47.1%43.3%
2×2IFGSM+DWDW47.1%20.2%=
2×2IFGSMIBP48.6%45.2%41.6%
5×5IFGSMNone66.5%0.4%
5×5IFGSMLGS51.2% 51.2%22.11%
5×5IFGSM +LGSLGS DW45.3%0.5% 59.3%
5×5IFGSM
5×5IFGSM+DWDW45.3%15.6%
5×5IFGSMIBP33.9%29.1%24.9%
+ +# 5.2 COMPARISON OF TRAINING STRATEGIES + +We find that given a fixed architecture all-patch certificate training achieves the best certified accuracy. However, given a fixed computational budget, random and guided training significantly outperform all-patch training. Finally, guided-patch certificate training consistently outperforms random-patch certificate training by a slim margin, indicating that the U-net is learning how to predict the minimum margin m. + +In Table 3, we see that given a fixed architecture all-patch certificate training significantly outperforms both random-patch certificate training and guided-patch certificate training in terms of certified accuracy, outperforming the second best certified defenses in each task by $2 . 6 \%$ (MNIST, $2 \times 2$ ), $7 . 3 \%$ (MNIST, $5 \times 5$ ), $3 . 9 \%$ (CIFAR-10, $2 \times 2 )$ ), and $3 . 4 \%$ (CIFAR-10, $5 \times 5$ ). However, all-patch certificate training is very expensive, taking on average 4 to 15 times longer than guided-patch certificate training and over 30 to 70 times longer than random-patch certificate training. + +On the other hand, given a limited computational budget, random-patch and guided-patch training significantly outperforms all-patch training. Due to the efficiency of random-patch and guided-patch training, they scale much better to large architectures. By switching to a large architecture (5 layer wide convolutional network), we are able to boost the certified accuracy by over $10 \%$ compared to the best performing all-patch small model (Table 2). Note that we are unable to all-patch train the same architecture as it will take almost 15 days to complete, and is out of our computational budget. + +Guided-patch certificate training is slightly more expensive compared to random patch, due to overhead from the U-net architecture. However, given the 10 patches picked, guided-patch certificate training consistently outperforms random-patch certificate training, indicating that the U-net is learning how to predict the minimum margin m. + +Table 3: Trade-off between certified accuracy and training time for different strategies. The numbers next to training strategies indicate the number of patches used for estimating the lower bound during training. Most training times are measured on a single 2080Ti GPU, with the exception of all-patch training which is run on four 2080Ti GPUs. For that specific case, the training time is multiplied by 4 for fair comparison. See Appendix A.6 for more detailed statistics. \*indicates the performance of the best performing large model trained with either random or guided patch. Detailed performance of the large models can be found in Appendix A.5 + +
2×25×5
DatasetTraining StrategyClean AccuracyCertified AccuracyTraining Time(h)Clean AccuracyCertified AccuracyTraining Time(h)
MNISTAllPatch98.5%91.5%9.392.0%60.4%8.4
Random(1)98.5%82.9%0.296.9%24.1%0.4
Random(5)98.6%86.6%0.395.8%42.1%0.3
Random(10)98.6%87.7%0.395.6%49.6%0.3
CIFARGuided(10) All Patch98.6% 50.9%88.9% 39.9%2.2 56.495.0% 33.5%53.1% 22.0%2.6 45.8
Random(1)53.6%21.6%0.643.6%6.1%0.6
Random(5)52.9%32.3%0.739.0%14.6%0.7
Random(10)51.9%35.6%0.838.8%18.6%
Guided(10)52.4%36.0%3.737.9%18.8%0.8 3.7
Large Model*65.8%51.9%22.447.8%30.3%15.4
+ +# 5.3 EFFECTIVENESS AGAINST SPARSE ATTACK + +The IBP based method can also be used to defend against sparse attack, see Section 4.3. Its performance is reasonable compared to patch defense (e.g. $9 1 . 5 \%$ certified accuracy for $2 \times 2$ patch vs $9 0 . 8 \%$ for $\mathrm { k } { = } 4$ ), even though the sparse attack model is much stronger. For convolutional networks, we increase the size of the first convolutional layer (i.e. from $3 \times 3$ to $1 1 \times 1 1$ ) so the interval bounds calculated are tighter. However, despite the change, fully-connected network still performs much better. For example, the certified accuracy drops from $2 5 . 6 \%$ to $1 3 . 8 \%$ when we switch from fully-connected to convolutional network for CIFAR10 and drops from $9 0 . 8 \%$ to $7 5 . 9 \%$ for MNIST respectively. Detailed results are shown in the Appendix A.4 Table 7. + +Table 4 compares our approach with the state-of-the-art certified sparse defense (Random Ablation) Levine & Feizi (2019). We use their best model with the largest medium radii to certify against various levels of sparsity. As shown in the table, our method achieves higher certified accuracy on the MNIST dataset over all the sparse radii, but lower on CIFAR-10. It is worth noting that we are using a much smaller and simpler model (a fully-connected network) compared to Random Ablation, which uses ResNet-50. + +Table 4: Certified accuracy for sparse defenses with IBP and Random Ablation. + +
DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1IBP-sparse98.4%96.0%
4IBP-sparse97.8%90.8%
10IBP-sparse95.2%86.8%
1Random Ablation96.7%90.3%
4Random Ablation96.7%79.1%
10Random Ablation96.7%29.2%
CIFAR1IBP-sparse48.4%40.0%
4IBP-sparse42.2%31.2%
10IBP-sparse37.0%25.6%
1Random Ablation78.3%68.6%
4Random Ablation78.3%61.3%
10Random Ablation78.3%45.0%
+ +# 5.4 TRANSFERABILITY TO PATCHES OF DIFFERENT SHAPES + +Since real-world adversarial patches may not always be square, the robust transferability of the model to shapes other than the square is important. + +Therefore, we evaluate the robustness of the square-patch-trained model to adversarial patches of different shapes while fixing the number of pixels. In all these experiments, we evaluate the certified accuracy for our largest model, on both MNIST and CIFAR datasets. We evaluate the transferability to various shapes including rectangle, line, parallelogram, and diamond. With the exception of rectangles, all the shapes have the exact same pixel count as the patches used for training. For rectangles, we use multiple choices of width and length, obtaining some combinations with slightly more pixels, and the worst accuracy is reported in Table 5. The exact shapes used can be found in Appendix A.2. + +The certified accuracy of our models generalize surprisingly well to other shapes, losing no more than than $5 \%$ in most cases for MNIST and no more than $6 \%$ for CIFAR-10 (Table 5). The largest degradation of accuracy happens for rectangles and lines, and it is mostly because the rectangle considered has more pixels compared to the square, and the line has less overlaps. However, it is still interesting that the certificate even generalizes to a straight line, even though the model was never trained to be robust to lines. In the case of MNIST with small patch size, the certified accuracy even improves when transferred to lines. + +Table 5: Certified accuracy for square-patch trained model for different shapes + +
DatasetPixel CountSquareRectangleLineDiamondParallelogram
MNIST491.6%=92.5%91.6%92.3%
CIFAR1669.4%55.4%46.7%68.13%70.2%
2559.7%50.9%32.4%53.6%55.2%
450.8%=46.1%48.6%49.8%
1636.9%29.0%32.1%35.7%36.3%
2530.3%25.1%29.0%30.1%30.7%
+ +# 6 CONCLUSION AND FUTURE WORK + +After establishing the weakness of known defenses to patch attacks, we proposed the first certified defense against this model. We demonstrated the effectiveness of our defense on two datasets, and proposed strategies to speed up robust training. Finally, we established the robust transferability of trained certified models to different shapes. In its current form, the proposed certified defense is unlikely to scale to ImageNet, and we hope the presented experiments will encourage further work along this direction. + +# REFERENCES + +Anish Athalye, Nicholas Carlini, and David Wagner. Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. arXiv preprint arXiv:1802.00420, 2018. + +Tom B Brown, Dandelion Mane, Aurko Roy, Mart ´ ´ın Abadi, and Justin Gilmer. Adversarial patch. arXiv preprint arXiv:1712.09665, 2017. + +Jeremy M Cohen, Elan Rosenfeld, and J Zico Kolter. Certified adversarial robustness via randomized smoothing. arXiv preprint arXiv:1902.02918, 2019. + +Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248–255. Ieee, 2009. + +Logan Engstrom, Brandon Tran, Dimitris Tsipras, Ludwig Schmidt, and Aleksander Madry. A rotation and a translation suffice: Fooling cnns with simple transformations. arXiv preprint arXiv:1712.02779, 2017. + +Sven Gowal, Krishnamurthy Dvijotham, Robert Stanforth, Rudy Bunel, Chongli Qin, Jonathan Uesato, Timothy Mann, and Pushmeet Kohli. On the effectiveness of interval bound propagation for training verifiably robust models. arXiv preprint arXiv:1810.12715, 2018. + +Jamie Hayes. On visible adversarial perturbations & digital watermarking. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp. 1597–1604, 2018. + +Danny Karmon, Daniel Zoran, and Yoav Goldberg. Lavan: Localized and visible adversarial noise. arXiv preprint arXiv:1801.02608, 2018. + +Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. + +Cassidy Laidlaw and Soheil Feizi. Functional adversarial attacks, 2019. + +Mark Lee and Zico Kolter. On physical adversarial patches for object detection, 2019. + +Alexander Levine and Soheil Feizi. Robustness certificates for sparse adversarial attacks by randomized ablation. arXiv preprint arXiv:1911.09272, 2019. + +Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. arXiv preprint arXiv:1706.06083, 2017. + +Matthew Mirman, Timon Gehr, and Martin Vechev. Differentiable abstract interpretation for provably robust neural networks. In International Conference on Machine Learning, pp. 3575–3583, 2018. + +Apostolos Modas, Seyed-Mohsen Moosavi-Dezfooli, and Pascal Frossard. Sparsefool: a few pixels make a big difference. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 9087–9096, 2019. + +Muzammal Naseer, Salman Khan, and Fatih Porikli. Local gradients smoothing: Defense against localized adversarial attacks. In 2019 IEEE Winter Conference on Applications of Computer Vision (WACV), pp. 1300–1307. IEEE, 2019. + +Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computerassisted intervention, pp. 234–241. Springer, 2015. + +Ali Shafahi, W Ronny Huang, Christoph Studer, Soheil Feizi, and Tom Goldstein. Are adversarial examples inevitable? arXiv preprint arXiv:1809.02104, 2018. + +Ali Shafahi, Mahyar Najibi, Amin Ghiasi, Zheng Xu, John Dickerson, Christoph Studer, Larry S Davis, Gavin Taylor, and Tom Goldstein. Adversarial training for free! arXiv preprint arXiv:1904.12843, 2019. + +Mahmood Sharif, Sruti Bhagavatula, Lujo Bauer, and Michael K Reiter. Adversarial generative nets: Neural network attacks on state-of-the-art face recognition. arXiv preprint arXiv:1801.00349, 2017. + +Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. + +Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2818–2826, 2016. + +Florian Tramer and Dan Boneh. Adversarial training and robustness for multiple perturbations. \` arXiv preprint arXiv:1904.13000, 2019. + +Shiqi Wang, Yizheng Chen, Ahmed Abdou, and Suman Jana. Enhancing gradient-based attacks with symbolic intervals, 2019. + +Eric Wong and J Zico Kolter. Provable defenses against adversarial examples via the convex outer adversarial polytope. arXiv preprint arXiv:1711.00851, 2017. + +Eric Wong, Frank R Schmidt, and J Zico Kolter. Wasserstein adversarial examples via projected sinkhorn iterations. arXiv preprint arXiv:1902.07906, 2019. + +Zuxuan Wu, Ser-Nam Lim, Larry Davis, and Tom Goldstein. Making an invisibility cloak: Real world adversarial attacks on object detectors. arXiv preprint arXiv:1910.14667, 2019. + +Huan Zhang, Hongge Chen, Chaowei Xiao, Bo Li, Duane Boning, and Cho-Jui Hsieh. Towards stable and efficient training of verifiably robust neural networks, 2019a. + +Huan Zhang, Hongge Chen, Chaowei Xiao, Bo Li, Duane Boning, and Cho-Jui Hsieh. Towards stable and efficient training of verifiably robust neural networks. arXiv preprint arXiv:1906.06316, 2019b. + +# A APPENDIX + +# A.1 EXPERIMENTAL SETTINGS AND NETWORK STRUCTURE + +We evaluate the proposed certified patch defense on three neural networks: a multilayer perceptron (MLP) with one 255-neuron hidden layer, and two convolutional neural networks (CNN) with different depths. The small CNN has two convolutional layers (kernel size 4, stride 2) of 4 and 8 output channels each, and a fully connected layer with 256 neurons. The large CNN has four convolutional layers with kernel size (3, 4, 3, 4), stride (1, 2, 1, 2), output channels (4, 4, 8 ,8), and two fully connected layer with 256 neurons. We run experiments on two datasets, MNIST and CIFAR10, with two different patch sizes $2 \times 2$ and $5 \times 5$ . For all experiments, we are using Adam (Kingma & Ba, 2014) with a learning rate of $5 e - 4$ for MNIST and $1 e - 3$ for CIFAR10, and with no weight decay. We also adopt a warm-up schedule in all experiments like (Zhang et al., 2019b), where the input interval bounds start at zero and grow to [-1,1] after 61/121 epochs for MNIST/CIFAR10 respectively. We train the models for a total of 100/200 epochs for MNIST/CIFAR10, where in the first 61/121 epochs the learning rate is fixed and in the following epochs, we reduce the learning rate by one half every 10 epochs. + +In addition, following (Gowal et al., 2018), we further evaluate the CIFAR10 on a larger model which has 5 convolutional layers with kernel size 3 and a fully connected layer with 512 neurons. This deeper and wider model achieves the clean accuracy around $8 9 \%$ , and has 17M parameters in total. Table 8 in Appendix A.5 describes the full certified patch results for this large model. + +# A.2 SAMPLE SHAPES FOR GENERALIZATION EXPERIMENTS + +We demonstrate generalization to other patch shapes that were not considered in training, obtaining surprisingly good transfer in robust accuracy; see the figure below and the results in Table 5. + +![](images/9f216bf62fce941917ea81757e0b262fd04bf16bd11424b98f95987ef90455cd.jpg) +Figure 1: Examples of shapes with pixels number 4 and 25. From left to right are square, parallelogram, diamond and rectangle (line) respectively. + +# A.3 BOUND POOLING + +Besides random-patch certificate training and guided-patch certificate training, we also experimented with the idea of bound pooling. All-patch training is very expensive as bounds generated by each potential patch has to be forward passed through the complete network. Bound pooling partially relieves the problem be pooling the interval bounds in intermediate layers thus reducing the forward pass in subsequent layers. + +Specifically, given a set of patches $\mathbb { P }$ , the interval bounds in the ith layer are $\bar { Z } ^ { ( i ) } ( \mathbb { P } ) = \{ \bar { z } ^ { ( i ) } ( p ) : p \in$ $\mathbb { P } \}$ and $\underline { { Z } } ^ { ( i ) } \mathbb { P } = \{ \underline { { z } } ^ { ( i ) } ( p ) : p \in \mathbb { P } \}$ . We can reduce the number of interval bounds by partitioning $\mathbb { P }$ into $n$ subsets $\{ \mathbb { S } ^ { 1 } , . . . , \mathbb { S } ^ { n } \}$ and calculate a new set of bounds $\begin{array} { r } { \bar { Z } _ { p o o l } ^ { ( i ) } ( \mathbb { P } ) = \{ \operatorname* { m a x } _ { p \in \mathbb { S } _ { i } } \bar { z } ^ { ( i ) } ( p ) : i \in [ n ] \} } \end{array}$ and Z(i) ( $\begin{array} { r } { \underline { { Z } } _ { p o o l } ^ { ( i ) } ( \mathbb { P } ) = \{ \operatorname* { m i n } _ { p \in \mathbb { S } _ { i } } \underline { { z } } ^ { ( i ) } ( p ) : i \in [ n ] \} } \end{array}$ . Depending on how $\mathbb { P }$ is partitioned, the bound pooling would work differently. In our experiments, we always select adjacent patches for each $\mathbb { S } _ { i }$ with the assumption that adjacent patches tend to generate similar bounds thus resulting in tighter certificate. + +Bound pooling, similar to random- and guided- patch training, trades performance for efficiency compared to all-patch certificate training. However, the trade off is not as favorable compared to random-patch and guided-patch training. For example, in Table 6, Pooling 16 $( 4 \times 4 )$ patches in the first layer reduces training time by $3 5 \%$ while loosing $0 . 7 \%$ in performance (on MNIST $2 \times 2$ ), but a similar level of performance can be achieved with guided-patch training with almost $90 \%$ reduction in training time. The trade off becomes greater when the model becomes larger. Overall, bound pooling is still quite expensive, and cannot scale to larger models like random-patch or guided-patch training. + +Table 6: Comparing bound pooling with the guided-patch and random-patch training. Pool 4 means that the adjacent $4 \times 4$ patches (16 patches) are pooled together in the first layer. Pool 2-2 means that the adjacent $2 \times 2$ bounds are pooled together in the first layer and then another $2 \times 2$ bound pooling happens at the second layer. This is similar to $4 \times 4$ pooling except the pooling operation is distributed between the first and second layer. All experiments are performed on a 4-layer convolutional network. + +
2×25×5
Dataset TrainingStrategyClean Certified TrainingClean Certified Training
Accuracy Accuracy Time(h)Accuracy Accuracy Time(h)
MNIST All Patch98.5% 91.6% 20.190.0% 59.7% 16.3
Pool 2Pool 4Random(1)Random(5)Random(10)Guided(10)98.0% 91.1% 15.885.2% 54.2% 11.6
97.2% 89.9% 13.270.4% 38.3% 10.2
98.5% 81.9% 0.396.8% 24.8% 0.4
98.6% 86.5% 0.394.9% 42.0% 0.5
98.6% 87.5% 0.594.7% 50.4% 0.6
98.7% 88.9% 2.294.0% 53.2% 3.4
CIFAR All Patch49.6% 41.6% 22.534.0% 25.0% 18.6
Pool 2Pool448.1% 39.4% 17.332.4% 24.2% 14.5
Pool4Pool 2-244.9% 37.1% 16.328.3% 20.6% 13.6
45.0% 37.4% 16.525.3% 19.1% 13.8
Random(1)53.2% 32.4% 0.642.7% 11.0% 0.6
Random(5)Random(10)52.2% 39.5% 0.937.8% 19.6% 0.9
50.8% 38.6% 1.038.4% 21.9% 1.0
Guided(10)53.0% 39.8% 4.036.1% 23.0% 3.9
+ +# A.4 MULTI-PATCH SPARSE TRAINING + +Here we list the detailed certified accuracy for various sparsity levels and model architectures. + +Table 7: Certified accuracy for sparse defenses with varying sparsity $k$ and models on both MNIST and CIFAR10, where “Conv $c \times c ^ { \prime \prime }$ represents for the convolutional network with first layer kernel size c. + +
DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1mlp98.4%96.0%
4mlp97.8%90.8%
10mlp95.2%86.8%
1Conv3x397.0%88.3%
CIFAR4Conv3x392.7%75.9%
1mlp48.4%40.0%
4mlp42.2%31.2%
10mlp37.0%25.6%
1Conv11x1134.8%27.4%
4Conv11x1125.1%18.3%
10Conv11x1117.2%13.8%
1Conv13x1338.6%29.7%
4Conv13x1328.1%19.6%
10Conv13x1322.4%15.3%
+ +# A.5 TRAINING WITH LARGER MODELS + +Recall that all-patch training considers all possible patches during training, which can be too expensive for larger models and/or images. The proposed random- and guided-patch training methods aim to reduce the training cost by considering only a subset of patches; please see Section 4.2 for more details. + +Table 8: The random and guided training strategy could yield significantly stronger model compared to all-patch training given a fixed computational budget. The random and guided training strategy allows us to train a larger model that would be infeasible to train otherwise. The guided-patch large model is able to boost the certified accuracy by over $10 \%$ compared to the best performing all-patch small model. + +
DatasetPatch SizeTraining StrategyModelClean AccuracyCertified AccuracyTraining Time(h)
CIFAR2×2All Patchmlp50.8%35.5%9.1
2 layer conv52.4%42.6%10.7
4 layer conv49.6%41.6%22.5
5 layer conv (wide)1-~360.0
Random(10) Random(20)5 layer conv (wide) 5 layer conv (wide)64.7% 64.4%49.0%9.5
CIFAR 5×5Guided(10)5 layer conv (wide)66.5%50.8% 49.2%15.8 12.2
Guided(20) All Patch5 layer conv (wide)65.8%51.9%22.4
mlp31.1%18.8%7.1
2 layer conv35.5%22.3%8.7
4 layer conv.34.0%25.0%18.6
5 layer conv (wide)1、~360.0
Random(10) Random(20)5 layer conv (wide)48.6%29.9%9.4
5 layer conv (wide)47.8%30.3%15.4
Guided(10)5 layer conv (wide)48.4%29.0%12.4
Guided(20)5 layer conv (wide)47.6%29.6%23.8
+ +# A.6 DETAILED STATISTICS ON TRAINING STRATEGIES + +Here we list the detailed statistics for each training strategies for Table 3 + +Table 9: Detailed statistics for the comparison of training strategies - $2 \times 2$ + +
DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution98.63/%91.38%21.0
Random (1)4 layer convolution98.48%91.63%80.3
fully connected (255,10)98.46%91.47%9.8
2 layer convolution98.69%82.57%0.2
4 layer convolution98.45%81.87%0.3
Random (5)fully connected (255,10)98.48%84.32%0.2
2 layer convolution98.75%85.87%0.3
4 layer convolution98.57%86.50%0.3
Random (10)fully connected (255,10)98.62%87.32%0.2
2 layer convolution98.73%87.31%0.3
4 layer convolution98.63%87.54%0.5
Guided (10)fully connected (255,10)98.49%88.13%0.2
2 layer convolution98.60%88.49%2.3
4 layer convolution98.70%88.85%2.2
CIFAR All Patchfully connected (255,10)98.63%89.44%2.2
2 layer convolution52.42%42.57%42.6
4 layer convolution49.58%41.57%89.8
Random (1)fully connected (255,10)50.83%35.49%36.6
2 layer convolution54.93%29.13%0.6
4 layer convolution53.22%32.35%0.6
Random (5)fully connected (255,10)52.76%03.21%0.5
2 layer convolution54.15%37.30%0.6
4 layer convolution52.19%39.45%0.9
Random (10)fully connected (255,10)52.38%20.17%0.6
2 layer convolution53.08%39.32%0.7
4 layer convolution50.80%38.57%1.0
Guided (10)fully connected (255,10)51.90%28.97%0.6
2 layer convolution53.04%38.81%3.7
4 layer convolution52.97%39.84%4.0
fully connected (255,10)51.32%29.44%3.6
+ +Table 10: Detailed statistics for the comparison of training strategies - $5 \times 5$ + +
DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution91.88%59.59%28.4
4 layer convolution90.03%59.72%65.2
Random (1)fully connected (255,10)93.96%61.97%7.2
2 layer convolution96.27%18.57%0.2
4 layer convolution96.83%24.79%0.4
Random (5)fully connected (255,10)97.60%29.04%0.2
2 layer convolution95.82%38.47%0.2
4 layer convolution94.85%42.02%0.5
Random (10)fully connected (255,10)96.73%45.89%0.2
2 layer convolution95.55%46.13%0.3
4 layer convolution fully connected (255,10)94.76% 96.40%50.43% 52.30%0.6
Guided (10)2 layer convolution95.28%50.28%0.2 2.3
93.98%53.17%3.4
4 layer convolution95.82%55.89%2.2
fully connected (255,10)35.48%
CIFAR All Patch Random (1)2 layer convolution33.95%22.31% 24.96%34.8
4 layer convolution fully connected (255,10)31.05%18.78%74.4 28.4
45.71%07.14%0.6
2 layer convolution42.65%10.99%
4 layer convolution42.34%0.6
fully connected (255,10)00.10%0.5
Random (5)2 layer convolution42.85%17.29%0.6
4 layer convolution37.80%19.63%0.9
fully connected (255,10)36.23%06.99%0.6
Random (10)2 layer convolution41.90%21.40%0.7
4 layer convolution38.41%21.90%1.0
fully connected (255,10) Guided (10)36.04%12.46%0.6
2 layer convolution42.08%20.77%3.6
4 layer convolution fully connected (255,10)36.08% 35.51%23.01% 12.56%3.9 3.5
\ No newline at end of file diff --git a/parse/train/HyeaSkrYPH/HyeaSkrYPH_content_list.json b/parse/train/HyeaSkrYPH/HyeaSkrYPH_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..871b451d86534db8dd69d164aabf49318fae435c --- /dev/null +++ b/parse/train/HyeaSkrYPH/HyeaSkrYPH_content_list.json @@ -0,0 +1,1829 @@ +[ + { + "type": "text", + "text": "CERTIFIED DEFENSES FOR ADVERSARIAL PATCHES∗ ", + "text_level": 1, + "bbox": [ + 173, + 99, + 802, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ping-yeh Chiang†, Renkun Ni†, Ahmed Abdelkader, Chen Zhu University of Maryland, College Park {pchiang,rn9zm,akader,chenzhu}@cs.umd.edu ", + "bbox": [ + 184, + 143, + 607, + 188 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Christoph Studer Cornell University studer@cornell.edu ", + "bbox": [ + 635, + 146, + 813, + 186 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tom Goldstein \nUniversity of Maryland, College Park \ntomg@cs.umd.edu ", + "bbox": [ + 184, + 209, + 431, + 250 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 287, + 544, + 303 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Adversarial patch attacks are among one of the most practical threat models against real-world computer vision systems. This paper studies certified and empirical defenses against patch attacks. We begin with a set of experiments showing that most existing defenses, which work by pre-processing input images to mitigate adversarial patches, are easily broken by simple white-box adversaries. Motivated by this finding, we propose the first certified defense against patch attacks, and propose faster methods for its training. Furthermore, we experiment with different patch shapes for testing, obtaining surprisingly good robustness transfer across shapes, and present preliminary results on certified defense against sparse attacks. Our complete implementation can be found on: https://github.com/Ping-C/certifiedpatchdefense. ", + "bbox": [ + 233, + 318, + 766, + 470 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 496, + 336, + 512 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite the great success of neural networks for vision problems, they are easily fooled by adversarial attacks in which the input to a machine learning model is modified with the goal of manipulating its output. Research in this area is largely focused on norm-bounded attack (Madry et al., 2017; Tramer\\` & Boneh, 2019; Shafahi et al., 2019), where the adversary is allowed to perturb all pixels in an image provided that the $\\ell _ { p }$ -norm of the perturbation is within prescribed bounds. Other adversarial models were also proposed, such as functional (Laidlaw & Feizi, 2019), rotation/translation (Engstrom et al., 2017), and Wasserstein (Wong et al., 2019), all of which allow modification to all pixels. ", + "bbox": [ + 174, + 527, + 825, + 626 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Whole-image perturbations are unrealistic for modeling ”physical-world” attacks, in which a realworld object is modified to evade detection. A physical adversary usually modifies an object using stickers or paint. Because this object may only occupy a small portion of an image, the adversary can only manipulate a limited number of pixels. As such, the more practical patch attack model was proposed (Brown et al., 2017). In a patch attack, the adversary may only change the pixels in a confined region, but is otherwise free to choose the values yielding the strongest attack. The threat to real-world computer vision systems is well-demonstrated in recent literature where carefully crafted patches can fool a classifier with high reliability (Brown et al., 2017; Karmon et al., 2018), make objects invisible to an object detector (Wu et al., 2019; Lee & Kolter, 2019), or fool a face recognition system (Sharif et al., 2017). In light of such effective physical-world patch attacks, very few defenses are known to date. ", + "bbox": [ + 174, + 632, + 825, + 785 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we study principled defenses against patch attacks. We begin by looking at existing defenses in the literature that claim to be effective against patch attacks, including Local Gradient Smoothing (LGS) (Naseer et al., 2019) and Digital Watermarking (DW) (Hayes, 2018). Similar to what has been observed for whole-image attacks by (Athalye et al., 2018), we show that these patch defenses are easily broken by stronger adversaries. Concretely, we demonstrate successful white-box attacks, where the adversary designs an attack against a known model, including any pre-processing steps. To cope with such potentially stronger adversaries, we train a robust model that produces a lower-bound on adversarial accuracy. In particular, we propose the first certifiable defense against patch attacks by extending interval bound propagation (IBP) defenses (Gowal et al., 2018; Mirman et al., 2018). We also propose modifications to IBP training to make it faster in the patch setting. Furthermore, we study the generalization of certified patch defenses to patches of different shapes, and observe that robustness transfers well across different patch types. We also present preliminary results on certified defense against the stronger sparse attack model, where a fixed number of possibly non-adjacent pixels can be arbitrarily modified (Modas et al., 2019). ", + "bbox": [ + 174, + 791, + 825, + 876 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PROBLEM SETUP ", + "text_level": 1, + "bbox": [ + 176, + 234, + 348, + 252 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We consider a white-box adversary that is allowed to choose the location of the patch (chosen from a set $\\mathbb { L }$ of possible locations) and can modify pixels within the particular patch (chosen from the set $\\mathbb { P }$ ) similar to (Karmon et al., 2018). An attack is successful if the adversary changes the classification of the network to a wrong label. In this paper, we are primarily interested in the patch attack robust accuracy (adversarial accuracy for short) as defined by ", + "bbox": [ + 174, + 266, + 825, + 337 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/a0c0b8fdaf4c03e2875d72b7ae98100dc0f9e0040781d003defee5ac2d30b80f.jpg", + "text": "$$\n\\underset { x \\sim X } { \\mathbb { E } } \\operatorname* { m i n } _ { p \\in \\mathbb { P } , l \\in \\mathbb { L } } \\mathcal { X } [ f ( A ( x , p , l ) ; \\theta ) = y ] ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 343, + 619, + 368 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where the operator $A$ places the adversarial patch p on a given image $\\mathbf { X }$ at location $l$ , f is a neural network with parameter $\\theta$ , $X$ is a distribution of images, and $\\mathcal { X }$ is a characteristic function that takes value 1 if its argument is true, and 0 otherwise. ", + "bbox": [ + 174, + 375, + 825, + 416 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this model, the strength of the adversary can vary depending on the set of possible patches allowed, and the type of perturbation allowed within the patch. In what follows, we assume the standard setup in which the adversary is allowed any perturbation that maintains pixel intensities in the range $[ 0 , 1 ]$ . Unless otherwise noted, we also assume the patch is restricted to a square of prescribed size. We consider two different options for the set $\\mathbb { L }$ of possible patch locations. First, we consider a weak adversary that can only place patches at the corner of an image. We find that even this weak model is enough to break existing patch defenses. Then, we consider a stronger adversary with no restrictions on patch location, and use this model to evaluate our proposed defenses. Note that an adversary, when restricted to modify only a square patch at location $l$ in the image, has the freedom to modify any non-square subset of these pixels. In other words, a certified defense against square patch attacks also provably subverts any non-square patch attack that fits inside a small enough square. ", + "bbox": [ + 174, + 424, + 825, + 577 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In general, calculating the adversarial accuracy (1) is intractable due to non-convexity. Common approaches try to approximate it by solving the inner minimization using a gradient-based method. However, in Section 3, we show that depending on how the minimization is solved, the upper bound could be very loose: a model may appear to be very robust, but fail when faced with a stronger attack. To side-step the arms race between attacks and defenses, in Section 4, we extend the work of (Gowal et al., 2018) and (Mirman et al., 2018) to train a network that produces a lower bound on adversarial accuracy. We will refer to approximations of the upper bound as empirical adversarial accuracy and the lower bound as certified accuracy. ", + "bbox": [ + 174, + 583, + 826, + 695 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 VULNERABILITY OF EXISTING DEFENSES ", + "text_level": 1, + "bbox": [ + 174, + 715, + 553, + 732 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We start by examining existing defense strategies that claim to be effective against patch attacks. Similar to what has been observed for whole-image attacks by Athalye et al. (2018), we show that these patch defenses can easily be broken by white-box attacks, where the adversary optimizes against a given model including any pre-processing steps. ", + "bbox": [ + 174, + 747, + 825, + 804 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3.1 EXISTING DEFENSES ", + "text_level": 1, + "bbox": [ + 176, + 820, + 361, + 834 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Under our threat model, two defenses have been proposed that each use input transformations to detect and remove adversarial patches. ", + "bbox": [ + 176, + 845, + 823, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The first defense is based on the observation that the gradient of the loss with respect to the input image often exhibits large values near the perturbed pixels. In (Hayes, 2018), the proposed digital watermarking (DW) approach exploits this behavior to detect unusually dense regions of large gradient entries using saliency maps, before masking them out in the image. Despite a $1 2 \\%$ drop in accuracy on clean (non-adversarial) images, this defense method supposedly achieves an empirical adversarial accuracy of $6 3 \\%$ for non-targeted patch attacks of size $4 2 \\times 4 2$ ( $\\dot { 2 } \\%$ of the image pixels), using 400 randomly picked images from ImageNet (Deng et al., 2009) on VGG19 (Simonyan & Zisserman, 2014). ", + "bbox": [ + 176, + 881, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "table", + "img_path": "images/643c1861bcf8cdebdd5fec558151ec5ffcdf3a16ae36fca9652c576e12ae2e0d.jpg", + "table_caption": [ + "Table 1: Empirical adversarial accuracy of ImageNet classifiers defended with Local Gradient Smoothing and Digital Watermarking. We consider two types of adversaries, one that takes the defense into account during backpropagation and one that does not " + ], + "table_footnote": [], + "table_body": "
Patch Size
AttackDefense42 × 4252 × 5260 ×60
IFGSMLGS78%75%71%
IFGSM+LGSLGS14%5%3%
IFGSMDW56%49%45%
IFGSM+DWDW13%8%5%
", + "bbox": [ + 285, + 156, + 712, + 244 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 270, + 826, + 340 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The second defense, Local Gradient Smoothing (LGS) by Naseer et al. (2019) is based on the empirical observation that pixel values tend to change sharply within these adversarial patches. In other words, the image gradients tend to be large within these adversarial patches. Note that the image gradient here differs from the gradient in Hayes (2018), the former is with respect the changes of adjacent pixel values and the later is with respect to the classification loss. Naseer et al. (2019) propose suppressing this adversarial noise by multiplying each pixel with one minus its image gradient as in (2). To make their methods more effective, Naseer et al. (2019) also pre-process the image gradient with a normalization and a thresholding step. ", + "bbox": [ + 173, + 347, + 825, + 459 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dacb9233c630d057e61b3d43c8d6e60411d85925020064ef1856ce95cb2d16f8.jpg", + "text": "$$\n{ \\hat { x } } = x \\odot ( 1 - \\lambda g ( x ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 424, + 467, + 573, + 483 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The $\\lambda$ here is a smoothing hyper-parameter. Naseer et al. (2019) claim the best adversarial accuracy on ImageNet with respect to patch attacks among all of the defenses we studied. They also claim that their defense is resilient to Backward Pass Differential Approximation (BPDA) from Athalye et al. (2018), one of the most effective methods to attack models that include a non-differentiable operator as a pre-processing step. ", + "bbox": [ + 173, + 491, + 825, + 561 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 BREAKING EXISTING DEFENSES ", + "text_level": 1, + "bbox": [ + 178, + 578, + 441, + 593 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Using a similar setup as in (Hayes, 2018; Naseer et al., 2019), we are able to mostly replicate the reported empirical adversarial accuracy for Iterative Fast Gradient Sign Method (IFGSM), a common gradient based attack, but we show that when the pre-processing step is taken into account, the empirical adversarial accuracy on ImageNet quickly drops from $\\bar { \\sim } 7 0 \\% ( \\sim 5 0 \\% )$ for LGS(DW) to levels around $\\sim 1 0 \\%$ as shown in Table 1. ", + "bbox": [ + 174, + 604, + 825, + 674 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Specifically, we break DW (Hayes, 2018) by applying BPDA, in which the non-differentiable operator is approximated with an identity mapping during the backward pass. We break LGS (Naseer et al., 2019) by directly incorporating the smoothing step during backpropagation. Even though the windowing and thresholding steps are non-differentiable, the smoothing operator provides enough gradient information for the attack to be effective. ", + "bbox": [ + 174, + 681, + 825, + 752 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To make sure that our evaluation is fair, we used the exact same models as Hayes (2018) (VGG19) and Szegedy et al. (2016) (Inception V3). We also consider a weaker set of attackers that can only attack the corners, the same as their setting. Further, we ensure that we were able to replicate their reported result under similar setting. ", + "bbox": [ + 174, + 758, + 825, + 815 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 CERTIFIED DEFENSES ", + "text_level": 1, + "bbox": [ + 176, + 835, + 388, + 852 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given the ease with which these supposedly strong defenses are broken, it is natural to seek methods that can rigorously guarantee robustness of a given model to patch attacks. With such certifiable guarantees in hand, we need not worry about an adversary with a stronger optimizer, or a more clever algorithm for choosing patch locations. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4.1 BACKGROUND ON CERTIFIED DEFENSES ", + "text_level": 1, + "bbox": [ + 176, + 103, + 491, + 117 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Certified defenses have been intensely studied with respect to norm-bounded attacks (Cohen et al., 2019; Wong & Kolter, 2017; Gowal et al., 2018; Mirman et al., 2018; Zhang et al., 2019b). In all of these methods, in addition to the prediction model, there is also a verifier. Given a model and an input, the verifier outputs a certificate if it is guaranteed that the image can not be adversarially perturbed. This is done by checking whether there exists any nearby image (within a prescribed $\\ell _ { p }$ distance) with a different label than the image being classified. While theoretical bounds exist on the size of this distance that hold for any classifier (Shafahi et al., 2018), exactly computing bounds for a specific classifier and test image is hard. Alternatively, the verifier may output a lower bound on the distance to the nearest image of a different label. This latter distance is referred to as the certifiable radius. Most of these verifiers provide a rather loose bound on the certifiable radius. However, if the verifier is differentiable, then the network can be trained with a loss that promotes tightness of this bound. We use the term certificate training to refer to the process of training with a loss that promotes strong certificates. Interval bound propagation (IBP) (Mirman et al., 2018; Gowal et al., 2018) is a very simple verifier that uses layer-wise interval arithmetic to produce a certificate. Even though the IBP certificate is generally loose, after certificate training, it yields state-of-the-art certifiably-robust models for $l _ { \\infty }$ -norm bounded attacks (Gowal et al., 2018; Zhang et al., 2019b). In this paper, we extend IBP to train certifiably-robust networks resilient to patch attacks. We first introduce some notation and basic algorithms for IBP training. ", + "bbox": [ + 174, + 130, + 825, + 380 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Notation We represent a neural network with a series of transformations $h ^ { ( k ) }$ for each of its $k$ layers. We use $z ^ { ( k ) } \\in \\bar { \\mathbb { R } } ^ { n _ { k } }$ to denote the output of layer $k$ , where $n _ { k }$ is the number of units in the $k ^ { t h }$ layer and $z ^ { ( 0 ) }$ stands for the input. Specifically, the network computes ", + "bbox": [ + 174, + 387, + 826, + 433 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/38a6af5bab14c81bc65b7bef9be01f14a5bb4b97642c51ad7f33b7744213b048.jpg", + "text": "$$\nz ^ { ( k ) } = h ^ { ( k - 1 ) } ( z ^ { ( k - 1 ) } ) \\forall k = 1 , \\dots , K .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 439, + 640, + 458 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Certification Problem To produce a certificate for an input $x _ { 0 }$ , we want to verify that the following condition is true with respect to all possible labels $y$ : ", + "bbox": [ + 173, + 472, + 826, + 502 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1f9d5c8120ecf851be6c3fa5abf5008341345ccf925375410a3fec3f6230774b.jpg", + "text": "$$\n( e _ { y _ { t r u e } } - e _ { y } ) ^ { T } z ^ { ( K ) } = { \\bf m } _ { y } \\geq 0 \\qquad \\forall z ^ { ( 0 ) } \\in \\mathbb { B } ( x _ { 0 } ) \\qquad \\forall y .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 507, + 689, + 529 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, $e _ { i }$ is the $i ^ { t h }$ basis vector, and $\\mathbf { m } _ { y }$ is called the margin following Wong & Kolter (2017). Note that $\\mathbf { m } _ { y _ { t r u e } }$ is always equal to 0. The vector $\\mathbf { m }$ contains all margins corresponding to all labels. $\\mathbb { B } ( x _ { 0 } )$ is the constraint set over which the adversarial input image may range. In a conventional setting, this is an $\\ell _ { \\infty }$ ball around $x _ { 0 }$ . In the case of patch attack, the constraint set contains all images formed by applying a patch to $x _ { 0 }$ ; ", + "bbox": [ + 173, + 535, + 825, + 607 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2ea4cbd9d09807a5ce309163b0be61414fa8ffa9ac763bf3b36027fab0c675a2.jpg", + "text": "$$\n\\mathbb { B } ( x _ { 0 } ) = \\{ A ( x _ { 0 } , p , l ) | p \\in \\mathbb { P } { \\mathrm { ~ a n d ~ } } l \\in \\mathbb { L } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 364, + 613, + 632, + 631 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The Basics of Interval Bound Propagation (IBP) We now describe how to produce certificates using interval bound propagation as in (Gowal et al., 2018). Suppose that for each component in $z ^ { ( k - \\bar { 1 } ) }$ we have an interval containing all the values which this component reaches as $z ^ { ( 0 ) }$ ranges over the ball $\\mathbb { B } ( x _ { 0 } )$ . If $z ^ { ( k ) } = h ^ { ( k ) } ( z ^ { \\top } )$ is a linear (or convolutional) layer of the form ${ z } ^ { ( k ) ^ { - } } =$ $W ^ { ( k ) } z ^ { ( k - 1 ) } + b ^ { ( k ) }$ , then we can get an outer approximation of the reachable interval range of activations by the next layer $z ^ { ( k ) }$ using the formulas below ", + "bbox": [ + 173, + 643, + 825, + 734 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1a03423793112af4d36e70a00c1d8a1e7b0b9581ec976adf34c671d6d1020242.jpg", + "text": "$$\n\\begin{array} { r l } & { \\overline { { z } } ^ { ( k ) } = W ^ { ( k ) } \\frac { \\overline { { z } } ^ { ( k - 1 ) } + \\underline { { z } } ^ { ( k - 1 ) } } { 2 } + | W ^ { ( k ) } | \\frac { \\overline { { z } } ^ { ( k - 1 ) } - \\underline { { z } } ^ { ( k - 1 ) } } { 2 } + b ^ { ( k ) } , } \\\\ & { \\underline { { z } } ^ { ( k ) } = W ^ { ( k ) } \\frac { \\overline { { z } } ^ { ( k - 1 ) } + \\underline { { z } } ^ { ( k - 1 ) } } { 2 } - | W ^ { ( k ) } | \\frac { \\overline { { z } } ^ { ( k - 1 ) } - \\underline { { z } } ^ { ( k - 1 ) } } { 2 } + b ^ { ( k ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 739, + 710, + 810 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here $\\overline { { z } } ^ { ( k - 1 ) }$ denotes the upper bound of each interval, $\\underline { z } ^ { ( k - 1 ) }$ the lower bound, and $| W ^ { ( k ) } |$ the element-wise absolute value. Alternatively, if $h ^ { ( k ) } ( z ^ { ( k - 1 ) } )$ is an element-wise monotonic activation (e.g., a ReLU), then we can calculate the outer approximation of the reachable interval range of the next layer using the formulas below. ", + "bbox": [ + 173, + 814, + 825, + 875 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/cc84f47f77535e3848b5ecc1e3a75d1e9012e03c6831c3c6e1882607141dc290.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { z } ^ { ( k ) } = h ^ { ( k ) } ( \\overline { z } ^ { ( k - 1 ) } ) } \\\\ { \\underline { z } ^ { ( k ) } = h ^ { ( k ) } ( \\underline { z } ^ { ( k - 1 ) } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 428, + 880, + 568, + 922 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "When the feasible set $\\mathbb { B } ( x _ { 0 } )$ represents a simple $\\ell _ { \\infty }$ attack, the range of possible $z ^ { ( 0 ) }$ values is trivially characterized by an interval bound $\\overline { { z } } ^ { ( 0 ) }$ and $\\underline { z } ^ { ( 0 ) }$ . Then, by iteratively applying the above rules, we can propagate intervals through the network and eventually get $\\overline { { z } } ^ { ( K ) }$ and $\\smash { \\mathcal { Z } ^ { ( K ) } }$ . A certificate can then be given if we can show that (3) is always true for outputs in the range $\\overline { { z } } ^ { ( K ) }$ and $\\smash { \\boldsymbol { \\mathcal { Z } } ^ { ( K ) } }$ with respect to all possible labels. More specifically, we can check that the following holds for all $y$ ", + "bbox": [ + 173, + 102, + 825, + 179 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/25fc8a53f7effcf2cd2df99ffb77095bfce036a317c1d5ceee2a63e54835de6c.jpg", + "text": "$$\n\\begin{array} { r } { \\underline { { \\mathbf { m } } } _ { y } = e _ { y _ { t r u e } } ^ { T } \\underline { { z } } ^ { ( K ) } - e _ { y } ^ { T } \\overline { { z } } ^ { ( K ) } = \\underline { { z } } _ { y _ { t r u e } } ^ { ( K ) } - \\overline { { z } } _ { y } ^ { ( K ) } \\geq 0 \\quad \\forall y . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 315, + 185, + 681, + 207 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training for Interval Bound Propagation To train a network to produce accurate interval bounds, we simply replace standard logits with the $- \\mathbf { m }$ vector in (3). Note that all elements of $\\mathbf { m }$ need to be larger than zero to satisfy the conditions in (3), and mytrue is always equal to zero. Put simply, we would like mytrue to be the least of all margins. We can promote this condition by training with the loss function ", + "bbox": [ + 173, + 219, + 825, + 289 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/72be3c53136bd187f40252a50653129ebf5fd67da8e8f872b8121036efc17f7a.jpg", + "text": "$$\n\\mathrm { C e r t i f i c a t e \\ L o s s } = \\mathrm { C r o s s \\ E n t r o p y \\ L o s s } ( - \\mathbf { m } , y ) .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 296, + 658, + 314 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Unlike regular neural network training, stochastic gradient descent for minimizing equation 10 is unstable, and a range of tricks are necessary to stabilize IBP training (Gowal et al., 2018). The first trick is merging the last linear weight matrix with $( e _ { y } - e _ { y _ { t r u e } } )$ before calculating $- { \\underline { { \\mathbf { m } } } } _ { y }$ . This allows a tighter characterization of the interval bound that noticeably improves results. The second trick uses an “epsilon schedule” in which training begins with a perturbation radius of zero, and this radius is slowly increased over time until a sentinel value is reached. Finally, a mixed loss function containing both a standard natural loss and an IBP loss is used. ", + "bbox": [ + 173, + 327, + 825, + 426 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In all of our experiments, we use the merging technique and the epsilon schedule, but we do not use a mixed loss function containing a natural loss as it does not increase our certificate performance. ", + "bbox": [ + 173, + 431, + 823, + 462 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 CERTIFYING AGAINST PATCH ATTACKS ", + "text_level": 1, + "bbox": [ + 176, + 478, + 482, + 492 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We can now describe the extension of IBP to patches. If we specify the patch location, one can represent the feasible set of images with a simple interval bound: for pixels within the patch, the upper and lower bound is equal to 1 and 0. For pixels outside of the patch, the upper and lower bounds are both equal to the original pixel value. By passing this bound through the network, we would be able to get msingle location and verify that they satisfy the conditions in (3). ", + "bbox": [ + 174, + 503, + 825, + 575 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "However, we have to consider not just a single location, but all possible locations $\\mathbb { L }$ to give a certificate. To adapt the bound to all possible location, we pass each of the possible patches through the network, and take the worst case margin. More specifically, ", + "bbox": [ + 174, + 580, + 825, + 622 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a4e469b062571eede31119d89cf7867b9051bdc6b2d1f47c03973309a1cc4ae1.jpg", + "text": "$$\n\\mathbf { m } ^ { \\mathrm { e s } } ( \\mathbb { L } ) _ { y } = \\operatorname* { m i n } _ { l \\in \\mathbb { L } } \\mathbf { m } ^ { \\mathrm { s i n g l e ~ p a t c h } } ( l ) _ { y } \\forall y .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 630, + 619, + 655 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Similar to regular IBP training, we simply use $\\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } } ( \\mathbb { L } )$ to calculate the cross entropy loss for training and backpropagation, ", + "bbox": [ + 173, + 661, + 826, + 690 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/badccbe6659e45077e905312268debefca934cb3ec05550287f687b5304211fa.jpg", + "text": "$$\n\\operatorname { C e r t i f i c a t e \\operatorname { L o s s } } = \\operatorname { C r o s s \\operatorname { E n t r o p y \\operatorname { L o s s } } } ( - \\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } } ( \\mathbb { L } ) , y ) .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 698, + 676, + 715 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Unfortunately, the cost of producing this na¨ıve certificate increases quadratically with image size. Consider that a CIFAR-10 image is of size $3 2 \\times 3 2$ , requiring over a thousand interval bounds, one for each possible patch location. To alleviate this problem, we propose two certificate training methods: Random Patch and Guided Patch, so that the number of forward passes does not scale with the dimension of the inputs. ", + "bbox": [ + 174, + 728, + 825, + 800 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Random Patch Certificate Training In this method, we simply select a random set of patches out of the possible patches and pass them forward. A level of robustness is achieved even though a very small number of random patches are selected compared to the total number of possible patches ", + "bbox": [ + 174, + 806, + 825, + 848 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/44945ff3b0611e84f9e7c490e701923833d7e3bd6d9b926f7bcd4dee8156df28.jpg", + "text": "$$\n\\underline { { \\mathbf { m } } } ^ { \\mathrm { r a n d o m \\ p a t c h e s } } ( \\mathbb { L } ) _ { y } = \\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } } ( S ) _ { y }\n$$", + "text_format": "latex", + "bbox": [ + 395, + 856, + 599, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $S$ is a random subset of $\\mathbb { L }$ . Similarly, the random patch certificate loss is calculated as below. ", + "bbox": [ + 176, + 881, + 823, + 896 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/40631a922fd1252beda45689f839d46b44e018b350319daebbe860eeb763a5e7.jpg", + "text": "$$\n\\mathrm { R a n d o m ~ P a t c h ~ C e r t i f i c a t e ~ L o s s } = \\mathrm { C r o s s ~ E n t r o p y ~ L o s s } ( - \\underline { { \\mathbf { m } } } ^ { \\mathrm { r a n d o m ~ p a t c h e s } } ( \\mathbb { L } ) , y )\n$$", + "text_format": "latex", + "bbox": [ + 236, + 904, + 753, + 921 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Guided Patch Certificate Training In this method, we propose using a U-net (Ronneberger et al., 2015) to predict msingle patch, and then randomly select a couple of locations based on the predicted $\\mathbf { m } ^ { \\mathrm { s i n g l e } \\mathrm { p a t c h } }$ so that fewer patches need to be passed forward. ", + "bbox": [ + 173, + 103, + 823, + 146 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note that very few patches contribute to the worst case bound $\\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } }$ in (11). In fact, the number of patches that yield the worst case margins will be no more than the number of labels. If we know the worst-case patches beforehand, then we can simply select the few worst-case patches during training. ", + "bbox": [ + 174, + 152, + 823, + 195 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We propose to use U-net as the number of locations and margins is very large. For a square patch of size $n \\times n$ and an image of size $m \\times m$ , the total number of possible locations is $( m - n + \\mathbf { \\hat { 1 } } ) ^ { 2 }$ , and for each location the number of margins is equal to the number of possible labels. ", + "bbox": [ + 174, + 202, + 823, + 244 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6e2b722e690020b0cd598c157f790ae7ddb81b815c1fc7d409fbc09cdc855329.jpg", + "text": "$$\n\\begin{array} { l } { { \\bf { m } } ^ { \\mathrm { { p r e d } } } = { \\bf { U } } { \\mathrm { - n e t } } ( \\mathrm { i m a g e } ) } \\\\ { { \\bf { d i m } } ( { \\bf { m } } ^ { \\mathrm { { p r e d } } } ) = ( m - n + 1 , m - n + 1 , \\# \\mathrm { o f } \\mathrm { l a b e l s } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 325, + 248, + 671, + 289 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Given the U-net prediction of $\\underline { { \\mathbf { m } } } ^ { \\mathrm { p r e d } }$ , we then randomly select a single patch for each label based on the softmax of the predicted $\\underline { { \\mathbf { m } } } ^ { \\mathrm { p r e d } }$ . The number of selected patches is equal to the number of labels. After these patches are passed forward, the U-net is then updated with a mean-squared-error loss between the predicted margins $\\underline { { \\mathbf { m } } } ^ { \\mathrm { p r e d } }$ and the actual margins $\\underline { { \\mathbf { m } } } ^ { \\mathrm { a c t u a l } }$ . Note that only a few patches are selected at a time, so that the mean-squared-error only passes through the selected patches. ", + "bbox": [ + 173, + 300, + 826, + 371 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d2bc1ac4b92bf8f45f0b03b98a291aaf65de753c0cb1dc9cce77eef4b0b12473.jpg", + "text": "$$\n\\mathrm { U } \\mathrm { - n e t } \\mathrm { L o s s } = \\mathbf { M } \\mathrm { S E } ( \\mathbf { \\underline { { m } } } ^ { \\mathrm { p r e d } } , \\mathbf { \\underline { { m } } } ^ { \\mathrm { a c t u a l } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 376, + 616, + 393 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The network is trained with the following loss: ", + "bbox": [ + 173, + 398, + 480, + 412 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Guided Patch Certificate $\\operatorname { L o s s } = \\operatorname { C r o s s } \\operatorname { E n t r o p y } \\operatorname { L o s s } ( - \\underline { { \\mathbf { m } } } ^ { \\mathrm { g u i d e d } \\mathrm { p a t c h e s } } ( \\mathbb { L } ) , y ) .$ ", + "bbox": [ + 248, + 419, + 750, + 436 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Certification Process In all our experiments, we check that equation (3) is satisfied by iterating over all possible patches and forward-passing the interval bounds generated for each patch; this overhead is tolerable at evaluation time. ", + "bbox": [ + 173, + 440, + 826, + 483 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 CERTIFYING AGAINST SPARSE ATTACKS ", + "text_level": 1, + "bbox": [ + 176, + 500, + 490, + 513 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "IBP can also be adapted to defend against sparse attack where the attacker is allowed to modify a fixed number $( k )$ of pixels that may not be adjacent to each other (Modas et al., 2019). The only modification is that we have to change the bound calculated from the first layer to ", + "bbox": [ + 174, + 525, + 826, + 568 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/17a421e1da69fa4d200afc0f132582985c1be61c501fb76dbccbf41606d6b64e.jpg", + "text": "$$\n\\overline { { { z } } } _ { i } ^ { ( 1 ) } = W _ { i , : } ^ { ( 1 ) } z ^ { ( 0 ) } + | W _ { i , : } ^ { ( 1 ) } | _ { t o p _ { k } } ~ z _ { i } ^ { ( 1 ) } = W _ { i , : } ^ { ( 1 ) } z ^ { ( 0 ) } - | W _ { i , : } ^ { ( 1 ) } | _ { t o p _ { k } } ~ \\forall i\n$$", + "text_format": "latex", + "bbox": [ + 263, + 571, + 735, + 594 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "and apply equation (5) and (6) for the subsequent layers. Here, $( . ) _ { t o p _ { k } }$ is the sum of the largest $k$ elements in the vector. ", + "bbox": [ + 174, + 599, + 826, + 628 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 647, + 326, + 662 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we compare our certified defenses with exiting algorithms on two datasets and three model architectures of varying complexity. We consider a strong attack setting in which adversarial patches can appear anywhere in the image. Different training strategies for the certified defense are also compared, which shows a trade-off between performance and training efficiency. Furthermore, we evaluate the transferability of a model trained using square patches to other adversarial shapes, including shapes that do not fit in any certified square. The training and architectural details can be found in Appendix A.1. We also present preliminary results on certified defense against sparse attacks. ", + "bbox": [ + 173, + 678, + 825, + 790 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 COMPARISON AGAINST EXISTING DEFENSES ", + "text_level": 1, + "bbox": [ + 174, + 808, + 522, + 820 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we study the effectiveness of our proposed IBP certified models against an adversary that is allowed to place patches anywhere in the image, even on top of the salient object. If the patch is sufficiently small, and does not cover a large portion of the salient object, then the model should still classify correctly, and defense against the perturbation should be possible. ", + "bbox": [ + 174, + 832, + 825, + 888 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the best case, our IBP certified model is able to achieve $9 1 . 6 \\%$ certified (Table 2) with respect to a $2 \\times 2$ patch $( \\sim . 5 \\%$ of image pixels) adversary on MNIST. For more challenging cases, such as a 5 $\\times 5$ ( $\\sim 2 . 5 \\%$ of image pixels) patch adversary on CIFAR-10, the certified adversarial accuracy is only $2 4 . 9 \\%$ (Table 2). Even though these existing defenses appear to achieve better or comparable adversarial accuracy as our IBP certified model when faced with a weak adversary, when faced with a stronger adversary their adversarial accuracy dropped to levels below our certified accuracy for all cases that we analyzed. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "When evaluating existing defenses, we only report cases where non-trivial adversarial accuracy is achieved against a weaker adversary. We do not explore cases where LGS and DW perform so poorly that no meaningful comparison can be done. LGS and DW are highly dependent on hyperparameters to work effectively against naive attacks, and yet neither Naseer et al. (2019) nor Hayes (2018) proposed a way to learn these hyperparameters. By trial and error, we were able to increase the adversarial accuracy against a weaker adversary for some settings, but not all. In addition, we also notice a peculiar feature of DW: when we increase the adversarial accuracy, the clean accuracy degrades, sometimes so much that it is even lower than the empirical adversarial accuracy. This happens because DW always removes a patch from the prediction. When an adversarial patch is detected, it is likely to be removed, enabling correct prediction. On the other hand, when there are no adversarial patches, DW removes actual salient information, resulting in lower clean accuracy. ", + "bbox": [ + 174, + 180, + 825, + 333 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Here we did not compare our results with adversarial training, because even though it produces some of the most adversarially robust models, it does not offer any guarantees on the empirical robust accuracy, and could still be decreased further with stronger attacks. For example, Wang et al. (2019) proposed a stronger attack that could find $47 \\%$ more adversarial examples compared to gradient based method. Further, adversarial training on all possible patches would be even more expensive compared to certificate training, and is slightly beyond our computational budget. ", + "bbox": [ + 174, + 340, + 825, + 424 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Compared to state-of-the-art certified models for CIFAR with $L _ { \\infty }$ -perturbation, where Zhang et al. (2019a) proposed a deterministic algorithm that achieves clean accuracy of $3 4 . 0 \\%$ , our clean accuracy for our most robust CIFAR $5 \\times 5$ model is $4 7 . 8 \\%$ when using a large model (Table 2). ", + "bbox": [ + 174, + 431, + 825, + 473 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/c05cfeed2d4e9f23cc5ab7a5b77dd639884d8719c99778848ff0b07130af40b8.jpg", + "table_caption": [ + "Table 2: Comparison of our IBP certified patch defense against existing defenses. Empirical adversarial accuracy is calculated for 400 random images in both datasets. All results are averaged over three different models. " + ], + "table_footnote": [], + "table_body": "
DatasetPatch SizeAdversaryDefenseClean AccuracyAccuracyEmpirical Certified Adversarial Accuracy
MNIST2×2IFGSMNone98.4%80.1%
2×2IFGSMLGS97.4%90.0%
2×2IFGSM+LGSLGS97.4%60.7%
2×2IFGSMIBP98.5%93.9%91.6%
5×5IFGSMNone98.5%3.3%
CIFAR5×5IFGSMIBP92.9%66.1%62.0%
2×2IFGSMNone66.3%25.4%1
2×2IFGSMLGS64.9%31.3%
2×2IFGSM+LGSLGS64.9%24.2%
2×2IFGSMDW47.1%43.3%
2×2IFGSM+DWDW47.1%20.2%=
2×2IFGSMIBP48.6%45.2%41.6%
5×5IFGSMNone66.5%0.4%
5×5IFGSMLGS51.2% 51.2%22.11%
5×5IFGSM +LGSLGS DW45.3%0.5% 59.3%
5×5IFGSM
5×5IFGSM+DWDW45.3%15.6%
5×5IFGSMIBP33.9%29.1%24.9%
", + "bbox": [ + 197, + 537, + 800, + 833 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 COMPARISON OF TRAINING STRATEGIES ", + "text_level": 1, + "bbox": [ + 176, + 856, + 493, + 869 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We find that given a fixed architecture all-patch certificate training achieves the best certified accuracy. However, given a fixed computational budget, random and guided training significantly outperform all-patch training. Finally, guided-patch certificate training consistently outperforms random-patch certificate training by a slim margin, indicating that the U-net is learning how to predict the minimum margin m. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Table 3, we see that given a fixed architecture all-patch certificate training significantly outperforms both random-patch certificate training and guided-patch certificate training in terms of certified accuracy, outperforming the second best certified defenses in each task by $2 . 6 \\%$ (MNIST, $2 \\times 2$ ), $7 . 3 \\%$ (MNIST, $5 \\times 5$ ), $3 . 9 \\%$ (CIFAR-10, $2 \\times 2 )$ ), and $3 . 4 \\%$ (CIFAR-10, $5 \\times 5$ ). However, all-patch certificate training is very expensive, taking on average 4 to 15 times longer than guided-patch certificate training and over 30 to 70 times longer than random-patch certificate training. ", + "bbox": [ + 174, + 138, + 825, + 223 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "On the other hand, given a limited computational budget, random-patch and guided-patch training significantly outperforms all-patch training. Due to the efficiency of random-patch and guided-patch training, they scale much better to large architectures. By switching to a large architecture (5 layer wide convolutional network), we are able to boost the certified accuracy by over $10 \\%$ compared to the best performing all-patch small model (Table 2). Note that we are unable to all-patch train the same architecture as it will take almost 15 days to complete, and is out of our computational budget. ", + "bbox": [ + 174, + 229, + 825, + 313 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Guided-patch certificate training is slightly more expensive compared to random patch, due to overhead from the U-net architecture. However, given the 10 patches picked, guided-patch certificate training consistently outperforms random-patch certificate training, indicating that the U-net is learning how to predict the minimum margin m. ", + "bbox": [ + 174, + 319, + 825, + 376 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/0eb115ffa03813f8cc48e12d421e5fbc40bdd687872b5c2e1df4dfd6b768f92f.jpg", + "table_caption": [ + "Table 3: Trade-off between certified accuracy and training time for different strategies. The numbers next to training strategies indicate the number of patches used for estimating the lower bound during training. Most training times are measured on a single 2080Ti GPU, with the exception of all-patch training which is run on four 2080Ti GPUs. For that specific case, the training time is multiplied by 4 for fair comparison. See Appendix A.6 for more detailed statistics. \\*indicates the performance of the best performing large model trained with either random or guided patch. Detailed performance of the large models can be found in Appendix A.5 " + ], + "table_footnote": [], + "table_body": "
2×25×5
DatasetTraining StrategyClean AccuracyCertified AccuracyTraining Time(h)Clean AccuracyCertified AccuracyTraining Time(h)
MNISTAllPatch98.5%91.5%9.392.0%60.4%8.4
Random(1)98.5%82.9%0.296.9%24.1%0.4
Random(5)98.6%86.6%0.395.8%42.1%0.3
Random(10)98.6%87.7%0.395.6%49.6%0.3
CIFARGuided(10) All Patch98.6% 50.9%88.9% 39.9%2.2 56.495.0% 33.5%53.1% 22.0%2.6 45.8
Random(1)53.6%21.6%0.643.6%6.1%0.6
Random(5)52.9%32.3%0.739.0%14.6%0.7
Random(10)51.9%35.6%0.838.8%18.6%
Guided(10)52.4%36.0%3.737.9%18.8%0.8 3.7
Large Model*65.8%51.9%22.447.8%30.3%15.4
", + "bbox": [ + 173, + 501, + 838, + 700 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 EFFECTIVENESS AGAINST SPARSE ATTACK ", + "text_level": 1, + "bbox": [ + 174, + 723, + 511, + 738 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The IBP based method can also be used to defend against sparse attack, see Section 4.3. Its performance is reasonable compared to patch defense (e.g. $9 1 . 5 \\%$ certified accuracy for $2 \\times 2$ patch vs $9 0 . 8 \\%$ for $\\mathrm { k } { = } 4$ ), even though the sparse attack model is much stronger. For convolutional networks, we increase the size of the first convolutional layer (i.e. from $3 \\times 3$ to $1 1 \\times 1 1$ ) so the interval bounds calculated are tighter. However, despite the change, fully-connected network still performs much better. For example, the certified accuracy drops from $2 5 . 6 \\%$ to $1 3 . 8 \\%$ when we switch from fully-connected to convolutional network for CIFAR10 and drops from $9 0 . 8 \\%$ to $7 5 . 9 \\%$ for MNIST respectively. Detailed results are shown in the Appendix A.4 Table 7. ", + "bbox": [ + 173, + 750, + 825, + 861 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 4 compares our approach with the state-of-the-art certified sparse defense (Random Ablation) Levine & Feizi (2019). We use their best model with the largest medium radii to certify against various levels of sparsity. As shown in the table, our method achieves higher certified accuracy on the MNIST dataset over all the sparse radii, but lower on CIFAR-10. It is worth noting that we are using a much smaller and simpler model (a fully-connected network) compared to Random Ablation, which uses ResNet-50. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/db03a2cb6fdb10975ea7bf080ddad1ceeaf16035224cb90629be05a5c6fed02e.jpg", + "table_caption": [ + "Table 4: Certified accuracy for sparse defenses with IBP and Random Ablation. " + ], + "table_footnote": [], + "table_body": "
DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1IBP-sparse98.4%96.0%
4IBP-sparse97.8%90.8%
10IBP-sparse95.2%86.8%
1Random Ablation96.7%90.3%
4Random Ablation96.7%79.1%
10Random Ablation96.7%29.2%
CIFAR1IBP-sparse48.4%40.0%
4IBP-sparse42.2%31.2%
10IBP-sparse37.0%25.6%
1Random Ablation78.3%68.6%
4Random Ablation78.3%61.3%
10Random Ablation78.3%45.0%
", + "bbox": [ + 214, + 167, + 785, + 353 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.4 TRANSFERABILITY TO PATCHES OF DIFFERENT SHAPES ", + "text_level": 1, + "bbox": [ + 173, + 377, + 596, + 390 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Since real-world adversarial patches may not always be square, the robust transferability of the model to shapes other than the square is important. ", + "bbox": [ + 176, + 401, + 823, + 430 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Therefore, we evaluate the robustness of the square-patch-trained model to adversarial patches of different shapes while fixing the number of pixels. In all these experiments, we evaluate the certified accuracy for our largest model, on both MNIST and CIFAR datasets. We evaluate the transferability to various shapes including rectangle, line, parallelogram, and diamond. With the exception of rectangles, all the shapes have the exact same pixel count as the patches used for training. For rectangles, we use multiple choices of width and length, obtaining some combinations with slightly more pixels, and the worst accuracy is reported in Table 5. The exact shapes used can be found in Appendix A.2. ", + "bbox": [ + 173, + 436, + 825, + 547 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The certified accuracy of our models generalize surprisingly well to other shapes, losing no more than than $5 \\%$ in most cases for MNIST and no more than $6 \\%$ for CIFAR-10 (Table 5). The largest degradation of accuracy happens for rectangles and lines, and it is mostly because the rectangle considered has more pixels compared to the square, and the line has less overlaps. However, it is still interesting that the certificate even generalizes to a straight line, even though the model was never trained to be robust to lines. In the case of MNIST with small patch size, the certified accuracy even improves when transferred to lines. ", + "bbox": [ + 173, + 554, + 825, + 652 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/5a45d2fb484cb951e71d7c77fd43e0fc10e7fd8eb4c7d8f6baba9438a01d4c7d.jpg", + "table_caption": [ + "Table 5: Certified accuracy for square-patch trained model for different shapes " + ], + "table_footnote": [], + "table_body": "
DatasetPixel CountSquareRectangleLineDiamondParallelogram
MNIST491.6%=92.5%91.6%92.3%
CIFAR1669.4%55.4%46.7%68.13%70.2%
2559.7%50.9%32.4%53.6%55.2%
450.8%=46.1%48.6%49.8%
1636.9%29.0%32.1%35.7%36.3%
2530.3%25.1%29.0%30.1%30.7%
", + "bbox": [ + 210, + 688, + 782, + 790 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 173, + 809, + 495, + 825 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "After establishing the weakness of known defenses to patch attacks, we proposed the first certified defense against this model. We demonstrated the effectiveness of our defense on two datasets, and proposed strategies to speed up robust training. Finally, we established the robust transferability of trained certified models to different shapes. In its current form, the proposed certified defense is unlikely to scale to ImageNet, and we hope the presented experiments will encourage further work along this direction. 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", + "bbox": [ + 174, + 470, + 826, + 513 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1 EXPERIMENTAL SETTINGS AND NETWORK STRUCTURE ", + "text_level": 1, + "bbox": [ + 176, + 133, + 602, + 148 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We evaluate the proposed certified patch defense on three neural networks: a multilayer perceptron (MLP) with one 255-neuron hidden layer, and two convolutional neural networks (CNN) with different depths. The small CNN has two convolutional layers (kernel size 4, stride 2) of 4 and 8 output channels each, and a fully connected layer with 256 neurons. The large CNN has four convolutional layers with kernel size (3, 4, 3, 4), stride (1, 2, 1, 2), output channels (4, 4, 8 ,8), and two fully connected layer with 256 neurons. We run experiments on two datasets, MNIST and CIFAR10, with two different patch sizes $2 \\times 2$ and $5 \\times 5$ . For all experiments, we are using Adam (Kingma & Ba, 2014) with a learning rate of $5 e - 4$ for MNIST and $1 e - 3$ for CIFAR10, and with no weight decay. We also adopt a warm-up schedule in all experiments like (Zhang et al., 2019b), where the input interval bounds start at zero and grow to [-1,1] after 61/121 epochs for MNIST/CIFAR10 respectively. We train the models for a total of 100/200 epochs for MNIST/CIFAR10, where in the first 61/121 epochs the learning rate is fixed and in the following epochs, we reduce the learning rate by one half every 10 epochs. ", + "bbox": [ + 173, + 160, + 825, + 340 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In addition, following (Gowal et al., 2018), we further evaluate the CIFAR10 on a larger model which has 5 convolutional layers with kernel size 3 and a fully connected layer with 512 neurons. This deeper and wider model achieves the clean accuracy around $8 9 \\%$ , and has 17M parameters in total. Table 8 in Appendix A.5 describes the full certified patch results for this large model. ", + "bbox": [ + 174, + 347, + 825, + 402 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 SAMPLE SHAPES FOR GENERALIZATION EXPERIMENTS", + "text_level": 1, + "bbox": [ + 174, + 421, + 596, + 434 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We demonstrate generalization to other patch shapes that were not considered in training, obtaining surprisingly good transfer in robust accuracy; see the figure below and the results in Table 5. ", + "bbox": [ + 173, + 445, + 823, + 474 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/9f216bf62fce941917ea81757e0b262fd04bf16bd11424b98f95987ef90455cd.jpg", + "image_caption": [ + "Figure 1: Examples of shapes with pixels number 4 and 25. From left to right are square, parallelogram, diamond and rectangle (line) respectively. " + ], + "image_footnote": [], + "bbox": [ + 308, + 497, + 689, + 775 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.3 BOUND POOLING ", + "text_level": 1, + "bbox": [ + 176, + 103, + 338, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Besides random-patch certificate training and guided-patch certificate training, we also experimented with the idea of bound pooling. All-patch training is very expensive as bounds generated by each potential patch has to be forward passed through the complete network. Bound pooling partially relieves the problem be pooling the interval bounds in intermediate layers thus reducing the forward pass in subsequent layers. ", + "bbox": [ + 174, + 128, + 825, + 199 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Specifically, given a set of patches $\\mathbb { P }$ , the interval bounds in the ith layer are $\\bar { Z } ^ { ( i ) } ( \\mathbb { P } ) = \\{ \\bar { z } ^ { ( i ) } ( p ) : p \\in$ $\\mathbb { P } \\}$ and $\\underline { { Z } } ^ { ( i ) } \\mathbb { P } = \\{ \\underline { { z } } ^ { ( i ) } ( p ) : p \\in \\mathbb { P } \\}$ . We can reduce the number of interval bounds by partitioning $\\mathbb { P }$ into $n$ subsets $\\{ \\mathbb { S } ^ { 1 } , . . . , \\mathbb { S } ^ { n } \\}$ and calculate a new set of bounds $\\begin{array} { r } { \\bar { Z } _ { p o o l } ^ { ( i ) } ( \\mathbb { P } ) = \\{ \\operatorname* { m a x } _ { p \\in \\mathbb { S } _ { i } } \\bar { z } ^ { ( i ) } ( p ) : i \\in [ n ] \\} } \\end{array}$ and Z(i) ( $\\begin{array} { r } { \\underline { { Z } } _ { p o o l } ^ { ( i ) } ( \\mathbb { P } ) = \\{ \\operatorname* { m i n } _ { p \\in \\mathbb { S } _ { i } } \\underline { { z } } ^ { ( i ) } ( p ) : i \\in [ n ] \\} } \\end{array}$ . Depending on how $\\mathbb { P }$ is partitioned, the bound pooling would work differently. In our experiments, we always select adjacent patches for each $\\mathbb { S } _ { i }$ with the assumption that adjacent patches tend to generate similar bounds thus resulting in tighter certificate. ", + "bbox": [ + 173, + 207, + 825, + 306 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Bound pooling, similar to random- and guided- patch training, trades performance for efficiency compared to all-patch certificate training. However, the trade off is not as favorable compared to random-patch and guided-patch training. For example, in Table 6, Pooling 16 $( 4 \\times 4 )$ patches in the first layer reduces training time by $3 5 \\%$ while loosing $0 . 7 \\%$ in performance (on MNIST $2 \\times 2$ ), but a similar level of performance can be achieved with guided-patch training with almost $90 \\%$ reduction in training time. The trade off becomes greater when the model becomes larger. Overall, bound pooling is still quite expensive, and cannot scale to larger models like random-patch or guided-patch training. ", + "bbox": [ + 173, + 311, + 825, + 424 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/6cedc241deed68c58913df8996a321dcb92824b0a654c9ed4cb84781332740b1.jpg", + "table_caption": [ + "Table 6: Comparing bound pooling with the guided-patch and random-patch training. Pool 4 means that the adjacent $4 \\times 4$ patches (16 patches) are pooled together in the first layer. Pool 2-2 means that the adjacent $2 \\times 2$ bounds are pooled together in the first layer and then another $2 \\times 2$ bound pooling happens at the second layer. This is similar to $4 \\times 4$ pooling except the pooling operation is distributed between the first and second layer. All experiments are performed on a 4-layer convolutional network. " + ], + "table_footnote": [], + "table_body": "
2×25×5
Dataset TrainingStrategyClean Certified TrainingClean Certified Training
Accuracy Accuracy Time(h)Accuracy Accuracy Time(h)
MNIST All Patch98.5% 91.6% 20.190.0% 59.7% 16.3
Pool 2Pool 4Random(1)Random(5)Random(10)Guided(10)98.0% 91.1% 15.885.2% 54.2% 11.6
97.2% 89.9% 13.270.4% 38.3% 10.2
98.5% 81.9% 0.396.8% 24.8% 0.4
98.6% 86.5% 0.394.9% 42.0% 0.5
98.6% 87.5% 0.594.7% 50.4% 0.6
98.7% 88.9% 2.294.0% 53.2% 3.4
CIFAR All Patch49.6% 41.6% 22.534.0% 25.0% 18.6
Pool 2Pool448.1% 39.4% 17.332.4% 24.2% 14.5
Pool4Pool 2-244.9% 37.1% 16.328.3% 20.6% 13.6
45.0% 37.4% 16.525.3% 19.1% 13.8
Random(1)53.2% 32.4% 0.642.7% 11.0% 0.6
Random(5)Random(10)52.2% 39.5% 0.937.8% 19.6% 0.9
50.8% 38.6% 1.038.4% 21.9% 1.0
Guided(10)53.0% 39.8% 4.036.1% 23.0% 3.9
", + "bbox": [ + 173, + 520, + 828, + 775 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.4 MULTI-PATCH SPARSE TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 103, + 449, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Here we list the detailed certified accuracy for various sparsity levels and model architectures. ", + "bbox": [ + 173, + 130, + 787, + 145 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/a4aaae91eba1ccdf0daccfb7c8539b3eb04cd6dcb4465a68c0780caebe05faae.jpg", + "table_caption": [ + "Table 7: Certified accuracy for sparse defenses with varying sparsity $k$ and models on both MNIST and CIFAR10, where “Conv $c \\times c ^ { \\prime \\prime }$ represents for the convolutional network with first layer kernel size c. " + ], + "table_footnote": [], + "table_body": "
DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1mlp98.4%96.0%
4mlp97.8%90.8%
10mlp95.2%86.8%
1Conv3x397.0%88.3%
CIFAR4Conv3x392.7%75.9%
1mlp48.4%40.0%
4mlp42.2%31.2%
10mlp37.0%25.6%
1Conv11x1134.8%27.4%
4Conv11x1125.1%18.3%
10Conv11x1117.2%13.8%
1Conv13x1338.6%29.7%
4Conv13x1328.1%19.6%
10Conv13x1322.4%15.3%
", + "bbox": [ + 232, + 210, + 764, + 422 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.5 TRAINING WITH LARGER MODELS ", + "text_level": 1, + "bbox": [ + 176, + 454, + 454, + 469 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Recall that all-patch training considers all possible patches during training, which can be too expensive for larger models and/or images. The proposed random- and guided-patch training methods aim to reduce the training cost by considering only a subset of patches; please see Section 4.2 for more details. ", + "bbox": [ + 174, + 481, + 825, + 536 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/42fc43caae4763d84f07dee1ef9d2252f3eb9def932e7d51cdf8bbff610d1244.jpg", + "table_caption": [ + "Table 8: The random and guided training strategy could yield significantly stronger model compared to all-patch training given a fixed computational budget. The random and guided training strategy allows us to train a larger model that would be infeasible to train otherwise. The guided-patch large model is able to boost the certified accuracy by over $10 \\%$ compared to the best performing all-patch small model. " + ], + "table_footnote": [], + "table_body": "
DatasetPatch SizeTraining StrategyModelClean AccuracyCertified AccuracyTraining Time(h)
CIFAR2×2All Patchmlp50.8%35.5%9.1
2 layer conv52.4%42.6%10.7
4 layer conv49.6%41.6%22.5
5 layer conv (wide)1-~360.0
Random(10) Random(20)5 layer conv (wide) 5 layer conv (wide)64.7% 64.4%49.0%9.5
CIFAR 5×5Guided(10)5 layer conv (wide)66.5%50.8% 49.2%15.8 12.2
Guided(20) All Patch5 layer conv (wide)65.8%51.9%22.4
mlp31.1%18.8%7.1
2 layer conv35.5%22.3%8.7
4 layer conv.34.0%25.0%18.6
5 layer conv (wide)1、~360.0
Random(10) Random(20)5 layer conv (wide)48.6%29.9%9.4
5 layer conv (wide)47.8%30.3%15.4
Guided(10)5 layer conv (wide)48.4%29.0%12.4
Guided(20)5 layer conv (wide)47.6%29.6%23.8
", + "bbox": [ + 178, + 630, + 820, + 882 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.6 DETAILED STATISTICS ON TRAINING STRATEGIES ", + "text_level": 1, + "bbox": [ + 174, + 103, + 565, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Here we list the detailed statistics for each training strategies for Table 3 ", + "bbox": [ + 173, + 128, + 647, + 145 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/fc04836ff4526e02b899ef602f30c5f44a6e919fce529c81f7427fcc1fb2231a.jpg", + "table_caption": [ + "Table 9: Detailed statistics for the comparison of training strategies - $2 \\times 2$ " + ], + "table_footnote": [], + "table_body": "
DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution98.63/%91.38%21.0
Random (1)4 layer convolution98.48%91.63%80.3
fully connected (255,10)98.46%91.47%9.8
2 layer convolution98.69%82.57%0.2
4 layer convolution98.45%81.87%0.3
Random (5)fully connected (255,10)98.48%84.32%0.2
2 layer convolution98.75%85.87%0.3
4 layer convolution98.57%86.50%0.3
Random (10)fully connected (255,10)98.62%87.32%0.2
2 layer convolution98.73%87.31%0.3
4 layer convolution98.63%87.54%0.5
Guided (10)fully connected (255,10)98.49%88.13%0.2
2 layer convolution98.60%88.49%2.3
4 layer convolution98.70%88.85%2.2
CIFAR All Patchfully connected (255,10)98.63%89.44%2.2
2 layer convolution52.42%42.57%42.6
4 layer convolution49.58%41.57%89.8
Random (1)fully connected (255,10)50.83%35.49%36.6
2 layer convolution54.93%29.13%0.6
4 layer convolution53.22%32.35%0.6
Random (5)fully connected (255,10)52.76%03.21%0.5
2 layer convolution54.15%37.30%0.6
4 layer convolution52.19%39.45%0.9
Random (10)fully connected (255,10)52.38%20.17%0.6
2 layer convolution53.08%39.32%0.7
4 layer convolution50.80%38.57%1.0
Guided (10)fully connected (255,10)51.90%28.97%0.6
2 layer convolution53.04%38.81%3.7
4 layer convolution52.97%39.84%4.0
fully connected (255,10)51.32%29.44%3.6
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DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution91.88%59.59%28.4
4 layer convolution90.03%59.72%65.2
Random (1)fully connected (255,10)93.96%61.97%7.2
2 layer convolution96.27%18.57%0.2
4 layer convolution96.83%24.79%0.4
Random (5)fully connected (255,10)97.60%29.04%0.2
2 layer convolution95.82%38.47%0.2
4 layer convolution94.85%42.02%0.5
Random (10)fully connected (255,10)96.73%45.89%0.2
2 layer convolution95.55%46.13%0.3
4 layer convolution fully connected (255,10)94.76% 96.40%50.43% 52.30%0.6
Guided (10)2 layer convolution95.28%50.28%0.2 2.3
93.98%53.17%3.4
4 layer convolution95.82%55.89%2.2
fully connected (255,10)35.48%
CIFAR All Patch Random (1)2 layer convolution33.95%22.31% 24.96%34.8
4 layer convolution fully connected (255,10)31.05%18.78%74.4 28.4
45.71%07.14%0.6
2 layer convolution42.65%10.99%
4 layer convolution42.34%0.6
fully connected (255,10)00.10%0.5
Random (5)2 layer convolution42.85%17.29%0.6
4 layer convolution37.80%19.63%0.9
fully connected (255,10)36.23%06.99%0.6
Random (10)2 layer convolution41.90%21.40%0.7
4 layer convolution38.41%21.90%1.0
fully connected (255,10) Guided (10)36.04%12.46%0.6
2 layer convolution42.08%20.77%3.6
4 layer convolution fully connected (255,10)36.08% 35.51%23.01% 12.56%3.9 3.5
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Our complete implementation can be found on:", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 361, + 429, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 361, + 429, + 375 + ], + "score": 1.0, + "content": "https://github.com/Ping-C/certifiedpatchdefense.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 208, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 208, + 409 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "Despite the great success of neural networks for vision problems, they are easily fooled by adversarial", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "attacks in which the input to a machine learning model is modified with the goal of manipulating its", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "output. Research in this area is largely focused on norm-bounded attack (Madry et al., 2017; Tramer`", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "& Boneh, 2019; Shafahi et al., 2019), where the adversary is allowed to perturb all pixels in an image", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 176, + 474 + ], + "score": 1.0, + "content": "provided that the", + "type": "text" + }, + { + "bbox": [ + 176, + 462, + 186, + 474 + ], + "score": 0.88, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "-norm of the perturbation is within prescribed bounds. 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The threat to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "real-world computer vision systems is well-demonstrated in recent literature where carefully crafted", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "patches can fool a classifier with high reliability (Brown et al., 2017; Karmon et al., 2018), make", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "objects invisible to an object detector (Wu et al., 2019; Lee & Kolter, 2019), or fool a face recognition", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 600, + 504, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 504, + 612 + ], + "score": 1.0, + "content": "system (Sharif et al., 2017). In light of such effective physical-world patch attacks, very few defenses", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 611, + 182, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 182, + 622 + ], + "score": 1.0, + "content": "are known to date.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "score": 1.0, + "content": "In this paper, we study principled defenses against patch attacks. We begin by looking at existing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "defenses in the literature that claim to be effective against patch attacks, including Local Gradient", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "Smoothing (LGS) (Naseer et al., 2019) and Digital Watermarking (DW) (Hayes, 2018). Similar to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "what has been observed for whole-image attacks by (Athalye et al., 2018), we show that these patch", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "defenses are easily broken by stronger adversaries. 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This paper studies certified and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 274, + 469, + 287 + ], + "spans": [ + { + "bbox": [ + 141, + 274, + 469, + 287 + ], + "score": 1.0, + "content": "empirical defenses against patch attacks. We begin with a set of experiments", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 285, + 469, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 469, + 298 + ], + "score": 1.0, + "content": "showing that most existing defenses, which work by pre-processing input images", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 297, + 470, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 297, + 470, + 308 + ], + "score": 1.0, + "content": "to mitigate adversarial patches, are easily broken by simple white-box adver-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 307, + 470, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 470, + 320 + ], + "score": 1.0, + "content": "saries. Motivated by this finding, we propose the first certified defense against", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 318, + 471, + 331 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 471, + 331 + ], + "score": 1.0, + "content": "patch attacks, and propose faster methods for its training. Furthermore, we ex-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 329, + 471, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 329, + 471, + 342 + ], + "score": 1.0, + "content": "periment with different patch shapes for testing, obtaining surprisingly good ro-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 340, + 471, + 352 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 471, + 352 + ], + "score": 1.0, + "content": "bustness transfer across shapes, and present preliminary results on certified de-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 352, + 470, + 363 + ], + "spans": [ + { + "bbox": [ + 142, + 352, + 470, + 363 + ], + "score": 1.0, + "content": "fense against sparse attacks. Our complete implementation can be found on:", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 361, + 429, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 361, + 429, + 375 + ], + "score": 1.0, + "content": "https://github.com/Ping-C/certifiedpatchdefense.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16, + "bbox_fs": [ + 141, + 253, + 471, + 375 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 208, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 208, + 409 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "Despite the great success of neural networks for vision problems, they are easily fooled by adversarial", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "attacks in which the input to a machine learning model is modified with the goal of manipulating its", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "output. Research in this area is largely focused on norm-bounded attack (Madry et al., 2017; Tramer`", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "& Boneh, 2019; Shafahi et al., 2019), where the adversary is allowed to perturb all pixels in an image", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 176, + 474 + ], + "score": 1.0, + "content": "provided that the", + "type": "text" + }, + { + "bbox": [ + 176, + 462, + 186, + 474 + ], + "score": 0.88, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "-norm of the perturbation is within prescribed bounds. Other adversarial models", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 473, + 507, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 507, + 486 + ], + "score": 1.0, + "content": "were also proposed, such as functional (Laidlaw & Feizi, 2019), rotation/translation (Engstrom et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 461, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 461, + 497 + ], + "score": 1.0, + "content": "2017), and Wasserstein (Wong et al., 2019), all of which allow modification to all pixels.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 419, + 507, + 497 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 507, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 507, + 514 + ], + "score": 1.0, + "content": "Whole-image perturbations are unrealistic for modeling ”physical-world” attacks, in which a real-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "score": 1.0, + "content": "world object is modified to evade detection. A physical adversary usually modifies an object using", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 522, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 537 + ], + "score": 1.0, + "content": "stickers or paint. Because this object may only occupy a small portion of an image, the adversary", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "can only manipulate a limited number of pixels. As such, the more practical patch attack model", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "was proposed (Brown et al., 2017). In a patch attack, the adversary may only change the pixels in a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "confined region, but is otherwise free to choose the values yielding the strongest attack. The threat to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "real-world computer vision systems is well-demonstrated in recent literature where carefully crafted", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "patches can fool a classifier with high reliability (Brown et al., 2017; Karmon et al., 2018), make", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "objects invisible to an object detector (Wu et al., 2019; Lee & Kolter, 2019), or fool a face recognition", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 600, + 504, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 504, + 612 + ], + "score": 1.0, + "content": "system (Sharif et al., 2017). In light of such effective physical-world patch attacks, very few defenses", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 611, + 182, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 182, + 622 + ], + "score": 1.0, + "content": "are known to date.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 500, + 507, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "score": 1.0, + "content": "In this paper, we study principled defenses against patch attacks. We begin by looking at existing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "defenses in the literature that claim to be effective against patch attacks, including Local Gradient", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "Smoothing (LGS) (Naseer et al., 2019) and Digital Watermarking (DW) (Hayes, 2018). Similar to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "what has been observed for whole-image attacks by (Athalye et al., 2018), we show that these patch", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "defenses are easily broken by stronger adversaries. Concretely, we demonstrate successful white-box", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 681, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 696 + ], + "score": 1.0, + "content": "attacks, where the adversary designs an attack against a known model, including any pre-processing", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 83, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 94 + ], + "score": 1.0, + "content": "steps. To cope with such potentially stronger adversaries, we train a robust model that produces a", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "lower-bound on adversarial accuracy. In particular, we propose the first certifiable defense against", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "patch attacks by extending interval bound propagation (IBP) defenses (Gowal et al., 2018; Mirman", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 507, + 129 + ], + "score": 1.0, + "content": "et al., 2018). We also propose modifications to IBP training to make it faster in the patch setting.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 507, + 140 + ], + "score": 1.0, + "content": "Furthermore, we study the generalization of certified patch defenses to patches of different shapes,", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "and observe that robustness transfers well across different patch types. We also present preliminary", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "results on certified defense against the stronger sparse attack model, where a fixed number of possibly", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 380, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 380, + 172 + ], + "score": 1.0, + "content": "non-adjacent pixels can be arbitrarily modified (Modas et al., 2019).", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 626, + 506, + 696 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 94 + ], + "score": 1.0, + "content": "steps. To cope with such potentially stronger adversaries, we train a robust model that produces a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "lower-bound on adversarial accuracy. In particular, we propose the first certifiable defense against", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "patch attacks by extending interval bound propagation (IBP) defenses (Gowal et al., 2018; Mirman", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 507, + 129 + ], + "score": 1.0, + "content": "et al., 2018). We also propose modifications to IBP training to make it faster in the patch setting.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 507, + 140 + ], + "score": 1.0, + "content": "Furthermore, we study the generalization of certified patch defenses to patches of different shapes,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "and observe that robustness transfers well across different patch types. We also present preliminary", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "results on certified defense against the stronger sparse attack model, where a fixed number of possibly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 380, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 380, + 172 + ], + "score": 1.0, + "content": "non-adjacent pixels can be arbitrarily modified (Modas et al., 2019).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 186, + 213, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 214, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 214, + 202 + ], + "score": 1.0, + "content": "2 PROBLEM SETUP", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "We consider a white-box adversary that is allowed to choose the location of the patch (chosen from a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 119, + 236 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 223, + 128, + 233 + ], + "score": 0.56, + "content": "\\mathbb { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 222, + 494, + 236 + ], + "score": 1.0, + "content": "of possible locations) and can modify pixels within the particular patch (chosen from the set", + "type": "text" + }, + { + "bbox": [ + 494, + 224, + 501, + 233 + ], + "score": 0.66, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 222, + 506, + 236 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "score": 1.0, + "content": "similar to (Karmon et al., 2018). An attack is successful if the adversary changes the classification", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "score": 1.0, + "content": "of the network to a wrong label. In this paper, we are primarily interested in the patch attack robust", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 256, + 325, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 325, + 268 + ], + "score": 1.0, + "content": "accuracy (adversarial accuracy for short) as defined by", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 272, + 379, + 292 + ], + "lines": [ + { + "bbox": [ + 231, + 272, + 379, + 292 + ], + "spans": [ + { + "bbox": [ + 231, + 272, + 379, + 292 + ], + "score": 0.93, + "content": "\\underset { x \\sim X } { \\mathbb { E } } \\operatorname* { m i n } _ { p \\in \\mathbb { P } , l \\in \\mathbb { L } } \\mathcal { X } [ f ( A ( x , p , l ) ; \\theta ) = y ] ,", + "type": "interline_equation", + "image_path": "a0c0b8fdaf4c03e2875d72b7ae98100dc0f9e0040781d003defee5ac2d30b80f.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 231, + 272, + 379, + 292 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 185, + 309 + ], + "score": 1.0, + "content": "where the operator", + "type": "text" + }, + { + "bbox": [ + 186, + 298, + 195, + 307 + ], + "score": 0.75, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 297, + 391, + 309 + ], + "score": 1.0, + "content": "places the adversarial patch p on a given image", + "type": "text" + }, + { + "bbox": [ + 392, + 299, + 399, + 308 + ], + "score": 0.56, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 297, + 445, + 309 + ], + "score": 1.0, + "content": "at location", + "type": "text" + }, + { + "bbox": [ + 446, + 298, + 450, + 308 + ], + "score": 0.68, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 297, + 505, + 309 + ], + "score": 1.0, + "content": ", f is a neural", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 203, + 320 + ], + "score": 1.0, + "content": "network with parameter", + "type": "text" + }, + { + "bbox": [ + 203, + 309, + 209, + 318 + ], + "score": 0.7, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 309, + 213, + 320 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 213, + 309, + 223, + 318 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 309, + 348, + 320 + ], + "score": 1.0, + "content": "is a distribution of images, and", + "type": "text" + }, + { + "bbox": [ + 348, + 309, + 358, + 318 + ], + "score": 0.84, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "is a characteristic function that takes", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 320, + 296, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 296, + 331 + ], + "score": 1.0, + "content": "value 1 if its argument is true, and 0 otherwise.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "In this model, the strength of the adversary can vary depending on the set of possible patches allowed,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "and the type of perturbation allowed within the patch. In what follows, we assume the standard setup", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 482, + 371 + ], + "score": 1.0, + "content": "in which the adversary is allowed any perturbation that maintains pixel intensities in the range", + "type": "text" + }, + { + "bbox": [ + 482, + 358, + 503, + 370 + ], + "score": 0.4, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 358, + 506, + 371 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "Unless otherwise noted, we also assume the patch is restricted to a square of prescribed size. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 381, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 274, + 392 + ], + "score": 1.0, + "content": "consider two different options for the set", + "type": "text" + }, + { + "bbox": [ + 275, + 381, + 283, + 390 + ], + "score": 0.64, + "content": "\\mathbb { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 381, + 505, + 392 + ], + "score": 1.0, + "content": "of possible patch locations. First, we consider a weak", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "adversary that can only place patches at the corner of an image. We find that even this weak model is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "enough to break existing patch defenses. Then, we consider a stronger adversary with no restrictions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 412, + 507, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 507, + 426 + ], + "score": 1.0, + "content": "on patch location, and use this model to evaluate our proposed defenses. Note that an adversary,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 336, + 436 + ], + "score": 1.0, + "content": "when restricted to modify only a square patch at location", + "type": "text" + }, + { + "bbox": [ + 337, + 425, + 341, + 434 + ], + "score": 0.42, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 425, + 505, + 436 + ], + "score": 1.0, + "content": "in the image, has the freedom to modify", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "any non-square subset of these pixels. In other words, a certified defense against square patch attacks", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 447, + 464, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 464, + 459 + ], + "score": 1.0, + "content": "also provably subverts any non-square patch attack that fits inside a small enough square.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 506, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "In general, calculating the adversarial accuracy (1) is intractable due to non-convexity. Common", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 507, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 507, + 487 + ], + "score": 1.0, + "content": "approaches try to approximate it by solving the inner minimization using a gradient-based method.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "However, in Section 3, we show that depending on how the minimization is solved, the upper bound", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "could be very loose: a model may appear to be very robust, but fail when faced with a stronger attack.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "To side-step the arms race between attacks and defenses, in Section 4, we extend the work of (Gowal", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "score": 1.0, + "content": "et al., 2018) and (Mirman et al., 2018) to train a network that produces a lower bound on adversarial", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 529, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 506, + 541 + ], + "score": 1.0, + "content": "accuracy. We will refer to approximations of the upper bound as empirical adversarial accuracy and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 538, + 261, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 261, + 553 + ], + "score": 1.0, + "content": "the lower bound as certified accuracy.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 567, + 339, + 580 + ], + "lines": [ + { + "bbox": [ + 104, + 565, + 340, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 340, + 582 + ], + "score": 1.0, + "content": "3 VULNERABILITY OF EXISTING DEFENSES", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 592, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "We start by examining existing defense strategies that claim to be effective against patch attacks.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "Similar to what has been observed for whole-image attacks by Athalye et al. (2018), we show that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "score": 1.0, + "content": "these patch defenses can easily be broken by white-box attacks, where the adversary optimizes against", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 625, + 308, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 308, + 639 + ], + "score": 1.0, + "content": "a given model including any pre-processing steps.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "title", + "bbox": [ + 108, + 650, + 221, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 222, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 222, + 663 + ], + "score": 1.0, + "content": "3.1 EXISTING DEFENSES", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 108, + 670, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 684 + ], + "score": 1.0, + "content": "Under our threat model, two defenses have been proposed that each use input transformations to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 681, + 262, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 262, + 694 + ], + "score": 1.0, + "content": "detect and remove adversarial patches.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "The first defense is based on the observation that the gradient of the loss with respect to the input", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "image often exhibits large values near the perturbed pixels. In (Hayes, 2018), the proposed digital", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "watermarking (DW) approach exploits this behavior to detect unusually dense regions of large", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [], + "index": 3.5, + "bbox_fs": [ + 104, + 83, + 507, + 172 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 186, + 213, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 214, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 214, + 202 + ], + "score": 1.0, + "content": "2 PROBLEM SETUP", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "We consider a white-box adversary that is allowed to choose the location of the patch (chosen from a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 119, + 236 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 223, + 128, + 233 + ], + "score": 0.56, + "content": "\\mathbb { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 222, + 494, + 236 + ], + "score": 1.0, + "content": "of possible locations) and can modify pixels within the particular patch (chosen from the set", + "type": "text" + }, + { + "bbox": [ + 494, + 224, + 501, + 233 + ], + "score": 0.66, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 222, + 506, + 236 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "score": 1.0, + "content": "similar to (Karmon et al., 2018). An attack is successful if the adversary changes the classification", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "score": 1.0, + "content": "of the network to a wrong label. 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In what follows, we assume the standard setup", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 482, + 371 + ], + "score": 1.0, + "content": "in which the adversary is allowed any perturbation that maintains pixel intensities in the range", + "type": "text" + }, + { + "bbox": [ + 482, + 358, + 503, + 370 + ], + "score": 0.4, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 358, + 506, + 371 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "Unless otherwise noted, we also assume the patch is restricted to a square of prescribed size. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 381, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 274, + 392 + ], + "score": 1.0, + "content": "consider two different options for the set", + "type": "text" + }, + { + "bbox": [ + 275, + 381, + 283, + 390 + ], + "score": 0.64, + "content": "\\mathbb { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 381, + 505, + 392 + ], + "score": 1.0, + "content": "of possible patch locations. First, we consider a weak", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "adversary that can only place patches at the corner of an image. We find that even this weak model is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "enough to break existing patch defenses. Then, we consider a stronger adversary with no restrictions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 412, + 507, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 507, + 426 + ], + "score": 1.0, + "content": "on patch location, and use this model to evaluate our proposed defenses. Note that an adversary,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 336, + 436 + ], + "score": 1.0, + "content": "when restricted to modify only a square patch at location", + "type": "text" + }, + { + "bbox": [ + 337, + 425, + 341, + 434 + ], + "score": 0.42, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 425, + 505, + 436 + ], + "score": 1.0, + "content": "in the image, has the freedom to modify", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "any non-square subset of these pixels. In other words, a certified defense against square patch attacks", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 447, + 464, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 464, + 459 + ], + "score": 1.0, + "content": "also provably subverts any non-square patch attack that fits inside a small enough square.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 336, + 507, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 506, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "In general, calculating the adversarial accuracy (1) is intractable due to non-convexity. Common", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 507, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 507, + 487 + ], + "score": 1.0, + "content": "approaches try to approximate it by solving the inner minimization using a gradient-based method.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "However, in Section 3, we show that depending on how the minimization is solved, the upper bound", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "could be very loose: a model may appear to be very robust, but fail when faced with a stronger attack.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "To side-step the arms race between attacks and defenses, in Section 4, we extend the work of (Gowal", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "score": 1.0, + "content": "et al., 2018) and (Mirman et al., 2018) to train a network that produces a lower bound on adversarial", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 529, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 506, + 541 + ], + "score": 1.0, + "content": "accuracy. We will refer to approximations of the upper bound as empirical adversarial accuracy and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 538, + 261, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 261, + 553 + ], + "score": 1.0, + "content": "the lower bound as certified accuracy.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 463, + 507, + 553 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 567, + 339, + 580 + ], + "lines": [ + { + "bbox": [ + 104, + 565, + 340, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 340, + 582 + ], + "score": 1.0, + "content": "3 VULNERABILITY OF EXISTING DEFENSES", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 592, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "We start by examining existing defense strategies that claim to be effective against patch attacks.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "Similar to what has been observed for whole-image attacks by Athalye et al. (2018), we show that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "score": 1.0, + "content": "these patch defenses can easily be broken by white-box attacks, where the adversary optimizes against", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 625, + 308, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 308, + 639 + ], + "score": 1.0, + "content": "a given model including any pre-processing steps.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 592, + 506, + 639 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 650, + 221, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 222, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 222, + 663 + ], + "score": 1.0, + "content": "3.1 EXISTING DEFENSES", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 108, + 670, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 684 + ], + "score": 1.0, + "content": "Under our threat model, two defenses have been proposed that each use input transformations to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 681, + 262, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 262, + 694 + ], + "score": 1.0, + "content": "detect and remove adversarial patches.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5, + "bbox_fs": [ + 106, + 669, + 505, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "The first defense is based on the observation that the gradient of the loss with respect to the input", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "image often exhibits large values near the perturbed pixels. In (Hayes, 2018), the proposed digital", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "watermarking (DW) approach exploits this behavior to detect unusually dense regions of large", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 453, + 227 + ], + "score": 1.0, + "content": "gradient entries using saliency maps, before masking them out in the image. Despite a", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 453, + 214, + 473, + 226 + ], + "score": 0.89, + "content": "1 2 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 473, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "drop in", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "accuracy on clean (non-adversarial) images, this defense method supposedly achieves an empirical", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 235, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 200, + 250 + ], + "score": 1.0, + "content": "adversarial accuracy of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 200, + 236, + 220, + 247 + ], + "score": 0.89, + "content": "6 3 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 220, + 235, + 369, + 250 + ], + "score": 1.0, + "content": "for non-targeted patch attacks of size", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 369, + 237, + 403, + 247 + ], + "score": 0.89, + "content": "4 2 \\times 4 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 403, + 235, + 407, + 250 + ], + "score": 1.0, + "content": "(", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 407, + 236, + 422, + 247 + ], + "score": 0.85, + "content": "\\dot { 2 } \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 422, + 235, + 506, + 250 + ], + "score": 1.0, + "content": "of the image pixels),", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "using 400 randomly picked images from ImageNet (Deng et al., 2009) on VGG19 (Simonyan &", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 258, + 181, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 181, + 271 + ], + "score": 1.0, + "content": "Zisserman, 2014).", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 175, + 124, + 436, + 194 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 89, + 504, + 123 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "score": 1.0, + "content": "Table 1: Empirical adversarial accuracy of ImageNet classifiers defended with Local Gradient", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 505, + 113 + ], + "score": 1.0, + "content": "Smoothing and Digital Watermarking. We consider two types of adversaries, one that takes the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 111, + 376, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 376, + 124 + ], + "score": 1.0, + "content": "defense into account during backpropagation and one that does not", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 175, + 124, + 436, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 175, + 124, + 436, + 194 + ], + "spans": [ + { + "bbox": [ + 175, + 124, + 436, + 194 + ], + "score": 0.98, + "html": "
Patch Size
AttackDefense42 × 4252 × 5260 ×60
IFGSMLGS78%75%71%
IFGSM+LGSLGS14%5%3%
IFGSMDW56%49%45%
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They also claim that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "their defense is resilient to Backward Pass Differential Approximation (BPDA) from Athalye et al.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "(2018), one of the most effective methods to attack models that include a non-differentiable operator", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 433, + 206, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 206, + 447 + ], + "score": 1.0, + "content": "as a pre-processing step.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 109, + 458, + 270, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 271, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 271, + 471 + ], + "score": 1.0, + "content": "3.2 BREAKING EXISTING DEFENSES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "Using a similar setup as in (Hayes, 2018; Naseer et al., 2019), we are able to mostly replicate the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "score": 1.0, + "content": "reported empirical adversarial accuracy for Iterative Fast Gradient Sign Method (IFGSM), a common", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "gradient based attack, but we show that when the pre-processing step is taken into account, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 367, + 526 + ], + "score": 1.0, + "content": "empirical adversarial accuracy on ImageNet quickly drops from", + "type": "text" + }, + { + "bbox": [ + 367, + 512, + 434, + 524 + ], + "score": 0.91, + "content": "\\bar { \\sim } 7 0 \\% ( \\sim 5 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 511, + 506, + 526 + ], + "score": 1.0, + "content": "for LGS(DW) to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 277, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 162, + 536 + ], + "score": 1.0, + "content": "levels around", + "type": "text" + }, + { + "bbox": [ + 162, + 523, + 192, + 534 + ], + "score": 0.92, + "content": "\\sim 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 523, + 277, + 536 + ], + "score": 1.0, + "content": "as shown in Table 1.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "score": 1.0, + "content": "Specifically, we break DW (Hayes, 2018) by applying BPDA, in which the non-differentiable", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "operator is approximated with an identity mapping during the backward pass. 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Patch Size
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Naseer et al. (2019) claim the best adversarial accuracy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 401, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 412 + ], + "score": 1.0, + "content": "on ImageNet with respect to patch attacks among all of the defenses we studied. They also claim that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "their defense is resilient to Backward Pass Differential Approximation (BPDA) from Athalye et al.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "(2018), one of the most effective methods to attack models that include a non-differentiable operator", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 433, + 206, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 206, + 447 + ], + "score": 1.0, + "content": "as a pre-processing step.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 388, + 506, + 447 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 458, + 270, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 271, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 271, + 471 + ], + "score": 1.0, + "content": "3.2 BREAKING EXISTING DEFENSES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "Using a similar setup as in (Hayes, 2018; Naseer et al., 2019), we are able to mostly replicate the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "score": 1.0, + "content": "reported empirical adversarial accuracy for Iterative Fast Gradient Sign Method (IFGSM), a common", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "gradient based attack, but we show that when the pre-processing step is taken into account, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 367, + 526 + ], + "score": 1.0, + "content": "empirical adversarial accuracy on ImageNet quickly drops from", + "type": "text" + }, + { + "bbox": [ + 367, + 512, + 434, + 524 + ], + "score": 0.91, + "content": "\\bar { \\sim } 7 0 \\% ( \\sim 5 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 511, + 506, + 526 + ], + "score": 1.0, + "content": "for LGS(DW) to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 277, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 162, + 536 + ], + "score": 1.0, + "content": "levels around", + "type": "text" + }, + { + "bbox": [ + 162, + 523, + 192, + 534 + ], + "score": 0.92, + "content": "\\sim 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 523, + 277, + 536 + ], + "score": 1.0, + "content": "as shown in Table 1.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 480, + 506, + 536 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "score": 1.0, + "content": "Specifically, we break DW (Hayes, 2018) by applying BPDA, in which the non-differentiable", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "operator is approximated with an identity mapping during the backward pass. We break LGS (Naseer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "et al., 2019) by directly incorporating the smoothing step during backpropagation. Even though the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "windowing and thresholding steps are non-differentiable, the smoothing operator provides enough", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 584, + 307, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 307, + 597 + ], + "score": 1.0, + "content": "gradient information for the attack to be effective.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 541, + 506, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "To make sure that our evaluation is fair, we used the exact same models as Hayes (2018) (VGG19)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "and Szegedy et al. (2016) (Inception V3). We also consider a weaker set of attackers that can only", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 624, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 635 + ], + "score": 1.0, + "content": "attack the corners, the same as their setting. Further, we ensure that we were able to replicate their", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 634, + 254, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 254, + 647 + ], + "score": 1.0, + "content": "reported result under similar setting.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 601, + 506, + 647 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 662, + 238, + 675 + ], + "lines": [ + { + "bbox": [ + 104, + 660, + 239, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 660, + 239, + 677 + ], + "score": 1.0, + "content": "4 CERTIFIED DEFENSES", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Given the ease with which these supposedly strong defenses are broken, it is natural to seek methods", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "that can rigorously guarantee robustness of a given model to patch attacks. With such certifiable", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "guarantees in hand, we need not worry about an adversary with a stronger optimizer, or a more clever", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 265, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 265, + 732 + ], + "score": 1.0, + "content": "algorithm for choosing patch locations.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 301, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 302, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 302, + 95 + ], + "score": 1.0, + "content": "4.1 BACKGROUND ON CERTIFIED DEFENSES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 103, + 505, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 115 + ], + "score": 1.0, + "content": "Certified defenses have been intensely studied with respect to norm-bounded attacks (Cohen et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "2019; Wong & Kolter, 2017; Gowal et al., 2018; Mirman et al., 2018; Zhang et al., 2019b). In all", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 137 + ], + "score": 1.0, + "content": "of these methods, in addition to the prediction model, there is also a verifier. Given a model and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 137, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 148 + ], + "score": 1.0, + "content": "an input, the verifier outputs a certificate if it is guaranteed that the image can not be adversarially", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 146, + 504, + 162 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 493, + 162 + ], + "score": 1.0, + "content": "perturbed. This is done by checking whether there exists any nearby image (within a prescribed", + "type": "text" + }, + { + "bbox": [ + 494, + 147, + 504, + 159 + ], + "score": 0.87, + "content": "\\ell _ { p }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "distance) with a different label than the image being classified. While theoretical bounds exist on the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 182 + ], + "score": 1.0, + "content": "size of this distance that hold for any classifier (Shafahi et al., 2018), exactly computing bounds for a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "specific classifier and test image is hard. Alternatively, the verifier may output a lower bound on the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 203 + ], + "score": 1.0, + "content": "distance to the nearest image of a different label. This latter distance is referred to as the certifiable", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 202, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 505, + 213 + ], + "score": 1.0, + "content": "radius. Most of these verifiers provide a rather loose bound on the certifiable radius. However, if the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "verifier is differentiable, then the network can be trained with a loss that promotes tightness of this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "bound. We use the term certificate training to refer to the process of training with a loss that promotes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 235, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 247 + ], + "score": 1.0, + "content": "strong certificates. Interval bound propagation (IBP) (Mirman et al., 2018; Gowal et al., 2018) is a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "very simple verifier that uses layer-wise interval arithmetic to produce a certificate. Even though the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "score": 1.0, + "content": "IBP certificate is generally loose, after certificate training, it yields state-of-the-art certifiably-robust", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 153, + 280 + ], + "score": 1.0, + "content": "models for", + "type": "text" + }, + { + "bbox": [ + 153, + 268, + 165, + 279 + ], + "score": 0.89, + "content": "l _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "-norm bounded attacks (Gowal et al., 2018; Zhang et al., 2019b). In this paper, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "extend IBP to train certifiably-robust networks resilient to patch attacks. 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Specifically, the network computes", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 348, + 392, + 363 + ], + "lines": [ + { + "bbox": [ + 219, + 348, + 392, + 363 + ], + "spans": [ + { + "bbox": [ + 219, + 348, + 392, + 363 + ], + "score": 0.92, + "content": "z ^ { ( k ) } = h ^ { ( k - 1 ) } ( z ^ { ( k - 1 ) } ) \\forall k = 1 , \\dots , K .", + "type": "interline_equation", + "image_path": "38a6af5bab14c81bc65b7bef9be01f14a5bb4b97642c51ad7f33b7744213b048.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 219, + 348, + 392, + 363 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 506, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 345, + 389 + ], + "score": 1.0, + "content": "Certification Problem To produce a certificate for an input", + "type": "text" + }, + { + "bbox": [ + 345, + 377, + 357, + 386 + ], + "score": 0.86, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 372, + 506, + 389 + ], + "score": 1.0, + "content": ", we want to verify that the following", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 318, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 308, + 400 + ], + "score": 1.0, + "content": "condition is true with respect to all possible labels", + "type": "text" + }, + { + "bbox": [ + 308, + 388, + 315, + 398 + ], + "score": 0.74, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 385, + 318, + 400 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 402, + 422, + 419 + ], + "lines": [ + { + "bbox": [ + 189, + 402, + 422, + 419 + ], + "spans": [ + { + "bbox": [ + 189, + 402, + 422, + 419 + ], + "score": 0.89, + "content": "( e _ { y _ { t r u e } } - e _ { y } ) ^ { T } z ^ { ( K ) } = { \\bf m } _ { y } \\geq 0 \\qquad \\forall z ^ { ( 0 ) } \\in \\mathbb { B } ( x _ { 0 } ) \\qquad \\forall y .", + "type": "interline_equation", + "image_path": "1f9d5c8120ecf851be6c3fa5abf5008341345ccf925375410a3fec3f6230774b.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 189, + 402, + 422, + 419 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 131, + 437 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 427, + 140, + 436 + ], + "score": 0.83, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 424, + 165, + 437 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 165, + 424, + 178, + 435 + ], + "score": 0.9, + "content": "i ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 424, + 248, + 437 + ], + "score": 1.0, + "content": "basis vector, and", + "type": "text" + }, + { + "bbox": [ + 248, + 426, + 264, + 438 + ], + "score": 0.88, + "content": "\\mathbf { m } _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "is called the margin following Wong & Kolter (2017). Note", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 435, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 123, + 451 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 438, + 153, + 448 + ], + "score": 0.9, + "content": "\\mathbf { m } _ { y _ { t r u e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 435, + 279, + 451 + ], + "score": 1.0, + "content": "is always equal to 0. 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Alternatively, if", + "type": "text" + }, + { + "bbox": [ + 290, + 659, + 341, + 672 + ], + "score": 0.93, + "content": "h ^ { ( k ) } ( z ^ { ( k - 1 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 655, + 506, + 673 + ], + "score": 1.0, + "content": "is an element-wise monotonic activation", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "(e.g., a ReLU), then we can calculate the outer approximation of the reachable interval range of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 681, + 254, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 254, + 694 + ], + "score": 1.0, + "content": "next layer using the formulas below.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 697, + 348, + 731 + ], + "lines": [ + { + "bbox": [ + 262, + 697, + 348, + 731 + ], + "spans": [ + { + "bbox": [ + 262, + 697, + 348, + 731 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\overline { z } ^ { ( k ) } = h ^ { ( k ) } ( \\overline { z } ^ { ( k - 1 ) } ) } \\\\ { \\underline { z } ^ { ( k ) } = h ^ { ( k ) } ( \\underline { z } ^ { ( k - 1 ) } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "cc84f47f77535e3848b5ecc1e3a75d1e9012e03c6831c3c6e1882607141dc290.jpg" + } + ] + } + ], + "index": 45.5, + "virtual_lines": [ + { + "bbox": [ + 262, + 697, + 348, + 714.0 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 262, + 714.0, + 348, + 731.0 + ], + "spans": [], + "index": 46 + } + ] + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 301, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 302, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 302, + 95 + ], + "score": 1.0, + "content": "4.1 BACKGROUND ON CERTIFIED DEFENSES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 103, + 505, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 115 + ], + "score": 1.0, + "content": "Certified defenses have been intensely studied with respect to norm-bounded attacks (Cohen et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "2019; Wong & Kolter, 2017; Gowal et al., 2018; Mirman et al., 2018; Zhang et al., 2019b). In all", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 137 + ], + "score": 1.0, + "content": "of these methods, in addition to the prediction model, there is also a verifier. Given a model and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 137, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 148 + ], + "score": 1.0, + "content": "an input, the verifier outputs a certificate if it is guaranteed that the image can not be adversarially", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 146, + 504, + 162 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 493, + 162 + ], + "score": 1.0, + "content": "perturbed. This is done by checking whether there exists any nearby image (within a prescribed", + "type": "text" + }, + { + "bbox": [ + 494, + 147, + 504, + 159 + ], + "score": 0.87, + "content": "\\ell _ { p }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "distance) with a different label than the image being classified. While theoretical bounds exist on the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 182 + ], + "score": 1.0, + "content": "size of this distance that hold for any classifier (Shafahi et al., 2018), exactly computing bounds for a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "specific classifier and test image is hard. Alternatively, the verifier may output a lower bound on the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 203 + ], + "score": 1.0, + "content": "distance to the nearest image of a different label. This latter distance is referred to as the certifiable", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 202, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 505, + 213 + ], + "score": 1.0, + "content": "radius. Most of these verifiers provide a rather loose bound on the certifiable radius. However, if the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "verifier is differentiable, then the network can be trained with a loss that promotes tightness of this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "bound. We use the term certificate training to refer to the process of training with a loss that promotes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 235, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 247 + ], + "score": 1.0, + "content": "strong certificates. Interval bound propagation (IBP) (Mirman et al., 2018; Gowal et al., 2018) is a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "very simple verifier that uses layer-wise interval arithmetic to produce a certificate. Even though the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "score": 1.0, + "content": "IBP certificate is generally loose, after certificate training, it yields state-of-the-art certifiably-robust", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 153, + 280 + ], + "score": 1.0, + "content": "models for", + "type": "text" + }, + { + "bbox": [ + 153, + 268, + 165, + 279 + ], + "score": 0.89, + "content": "l _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "-norm bounded attacks (Gowal et al., 2018; Zhang et al., 2019b). In this paper, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "extend IBP to train certifiably-robust networks resilient to patch attacks. We first introduce some", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 289, + 295, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 295, + 303 + ], + "score": 1.0, + "content": "notation and basic algorithms for IBP training.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 103, + 506, + 303 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 506, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 507, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 394, + 320 + ], + "score": 1.0, + "content": "Notation We represent a neural network with a series of transformations", + "type": "text" + }, + { + "bbox": [ + 394, + 306, + 412, + 317 + ], + "score": 0.91, + "content": "h ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 306, + 470, + 320 + ], + "score": 1.0, + "content": "for each of its", + "type": "text" + }, + { + "bbox": [ + 471, + 308, + 477, + 317 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 306, + 507, + 320 + ], + "score": 1.0, + "content": "layers.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 317, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 137, + 332 + ], + "score": 1.0, + "content": "We use", + "type": "text" + }, + { + "bbox": [ + 137, + 318, + 183, + 329 + ], + "score": 0.92, + "content": "z ^ { ( k ) } \\in \\bar { \\mathbb { R } } ^ { n _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 317, + 301, + 332 + ], + "score": 1.0, + "content": "to denote the output of layer", + "type": "text" + }, + { + "bbox": [ + 301, + 320, + 308, + 330 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 317, + 338, + 332 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 338, + 321, + 350, + 330 + ], + "score": 0.86, + "content": "n _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 317, + 466, + 332 + ], + "score": 1.0, + "content": "is the number of units in the", + "type": "text" + }, + { + "bbox": [ + 467, + 319, + 481, + 329 + ], + "score": 0.88, + "content": "k ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 317, + 506, + 332 + ], + "score": 1.0, + "content": "layer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 366, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 123, + 345 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 330, + 140, + 341 + ], + "score": 0.9, + "content": "z ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 329, + 366, + 345 + ], + "score": 1.0, + "content": "stands for the input. Specifically, the network computes", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 306, + 507, + 345 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 348, + 392, + 363 + ], + "lines": [ + { + "bbox": [ + 219, + 348, + 392, + 363 + ], + "spans": [ + { + "bbox": [ + 219, + 348, + 392, + 363 + ], + "score": 0.92, + "content": "z ^ { ( k ) } = h ^ { ( k - 1 ) } ( z ^ { ( k - 1 ) } ) \\forall k = 1 , \\dots , K .", + "type": "interline_equation", + "image_path": "38a6af5bab14c81bc65b7bef9be01f14a5bb4b97642c51ad7f33b7744213b048.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 219, + 348, + 392, + 363 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 506, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 345, + 389 + ], + "score": 1.0, + "content": "Certification Problem To produce a certificate for an input", + "type": "text" + }, + { + "bbox": [ + 345, + 377, + 357, + 386 + ], + "score": 0.86, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 372, + 506, + 389 + ], + "score": 1.0, + "content": ", we want to verify that the following", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 318, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 308, + 400 + ], + "score": 1.0, + "content": "condition is true with respect to all possible labels", + "type": "text" + }, + { + "bbox": [ + 308, + 388, + 315, + 398 + ], + "score": 0.74, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 385, + 318, + 400 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 372, + 506, + 400 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 402, + 422, + 419 + ], + "lines": [ + { + "bbox": [ + 189, + 402, + 422, + 419 + ], + "spans": [ + { + "bbox": [ + 189, + 402, + 422, + 419 + ], + "score": 0.89, + "content": "( e _ { y _ { t r u e } } - e _ { y } ) ^ { T } z ^ { ( K ) } = { \\bf m } _ { y } \\geq 0 \\qquad \\forall z ^ { ( 0 ) } \\in \\mathbb { B } ( x _ { 0 } ) \\qquad \\forall y .", + "type": "interline_equation", + "image_path": "1f9d5c8120ecf851be6c3fa5abf5008341345ccf925375410a3fec3f6230774b.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 189, + 402, + 422, + 419 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 131, + 437 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 427, + 140, + 436 + ], + "score": 0.83, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 424, + 165, + 437 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 165, + 424, + 178, + 435 + ], + "score": 0.9, + "content": "i ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 424, + 248, + 437 + ], + "score": 1.0, + "content": "basis vector, and", + "type": "text" + }, + { + "bbox": [ + 248, + 426, + 264, + 438 + ], + "score": 0.88, + "content": "\\mathbf { m } _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "is called the margin following Wong & Kolter (2017). Note", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 435, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 123, + 451 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 438, + 153, + 448 + ], + "score": 0.9, + "content": "\\mathbf { m } _ { y _ { t r u e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 435, + 279, + 451 + ], + "score": 1.0, + "content": "is always equal to 0. The vector", + "type": "text" + }, + { + "bbox": [ + 279, + 438, + 290, + 446 + ], + "score": 0.33, + "content": "\\mathbf { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 435, + 479, + 451 + ], + "score": 1.0, + "content": "contains all margins corresponding to all labels.", + "type": "text" + }, + { + "bbox": [ + 479, + 437, + 505, + 448 + ], + "score": 0.9, + "content": "\\mathbb { B } ( x _ { 0 } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "is the constraint set over which the adversarial input image may range. In a conventional setting, this", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 127, + 471 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 127, + 459, + 141, + 469 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 457, + 190, + 471 + ], + "score": 1.0, + "content": "ball around", + "type": "text" + }, + { + "bbox": [ + 190, + 459, + 201, + 469 + ], + "score": 0.85, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 457, + 505, + 471 + ], + "score": 1.0, + "content": ". In the case of patch attack, the constraint set contains all images formed by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 468, + 201, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 185, + 483 + ], + "score": 1.0, + "content": "applying a patch to", + "type": "text" + }, + { + "bbox": [ + 185, + 471, + 196, + 480 + ], + "score": 0.86, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 468, + 201, + 483 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 424, + 506, + 483 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 486, + 387, + 500 + ], + "lines": [ + { + "bbox": [ + 223, + 486, + 387, + 500 + ], + "spans": [ + { + "bbox": [ + 223, + 486, + 387, + 500 + ], + "score": 0.93, + "content": "\\mathbb { B } ( x _ { 0 } ) = \\{ A ( x _ { 0 } , p , l ) | p \\in \\mathbb { P } { \\mathrm { ~ a n d ~ } } l \\in \\mathbb { L } \\} .", + "type": "interline_equation", + "image_path": "2ea4cbd9d09807a5ce309163b0be61414fa8ffa9ac763bf3b36027fab0c675a2.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 223, + 486, + 387, + 500 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "The Basics of Interval Bound Propagation (IBP) We now describe how to produce certificates", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "using interval bound propagation as in (Gowal et al., 2018). Suppose that for each component in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 530, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 134, + 544 + ], + "score": 0.92, + "content": "z ^ { ( k - \\bar { 1 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 530, + 457, + 548 + ], + "score": 1.0, + "content": "we have an interval containing all the values which this component reaches as", + "type": "text" + }, + { + "bbox": [ + 457, + 533, + 474, + 544 + ], + "score": 0.89, + "content": "z ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 530, + 506, + 548 + ], + "score": 1.0, + "content": "ranges", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 543, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 160, + 559 + ], + "score": 1.0, + "content": "over the ball", + "type": "text" + }, + { + "bbox": [ + 160, + 545, + 186, + 558 + ], + "score": 0.9, + "content": "\\mathbb { B } ( x _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 543, + 201, + 559 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 201, + 545, + 283, + 558 + ], + "score": 0.93, + "content": "z ^ { ( k ) } = h ^ { ( k ) } ( z ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 543, + 475, + 559 + ], + "score": 1.0, + "content": "is a linear (or convolutional) layer of the form", + "type": "text" + }, + { + "bbox": [ + 475, + 545, + 505, + 557 + ], + "score": 0.89, + "content": "{ z } ^ { ( k ) ^ { - } } =", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 555, + 507, + 573 + ], + "spans": [ + { + "bbox": [ + 107, + 557, + 185, + 569 + ], + "score": 0.91, + "content": "W ^ { ( k ) } z ^ { ( k - 1 ) } + b ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 555, + 507, + 573 + ], + "score": 1.0, + "content": ", then we can get an outer approximation of the reachable interval range of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 569, + 342, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 221, + 583 + ], + "score": 1.0, + "content": "activations by the next layer", + "type": "text" + }, + { + "bbox": [ + 221, + 569, + 238, + 580 + ], + "score": 0.93, + "content": "z ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 569, + 342, + 583 + ], + "score": 1.0, + "content": "using the formulas below", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 511, + 507, + 583 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 586, + 435, + 642 + ], + "lines": [ + { + "bbox": [ + 175, + 586, + 435, + 642 + ], + "spans": [ + { + "bbox": [ + 175, + 586, + 435, + 642 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\overline { { z } } ^ { ( k ) } = W ^ { ( k ) } \\frac { \\overline { { z } } ^ { ( k - 1 ) } + \\underline { { z } } ^ { ( k - 1 ) } } { 2 } + | W ^ { ( k ) } | \\frac { \\overline { { z } } ^ { ( k - 1 ) } - \\underline { { z } } ^ { ( k - 1 ) } } { 2 } + b ^ { ( k ) } , } \\\\ & { \\underline { { z } } ^ { ( k ) } = W ^ { ( k ) } \\frac { \\overline { { z } } ^ { ( k - 1 ) } + \\underline { { z } } ^ { ( k - 1 ) } } { 2 } - | W ^ { ( k ) } | \\frac { \\overline { { z } } ^ { ( k - 1 ) } - \\underline { { z } } ^ { ( k - 1 ) } } { 2 } + b ^ { ( k ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "1a03423793112af4d36e70a00c1d8a1e7b0b9581ec976adf34c671d6d1020242.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 175, + 586, + 435, + 604.6666666666666 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 175, + 604.6666666666666, + 435, + 623.3333333333333 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 175, + 623.3333333333333, + 435, + 641.9999999999999 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 104, + 644, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 104, + 644, + 129, + 661 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 645, + 157, + 657 + ], + "score": 0.92, + "content": "\\overline { { z } } ^ { ( k - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 644, + 336, + 661 + ], + "score": 1.0, + "content": "denotes the upper bound of each interval,", + "type": "text" + }, + { + "bbox": [ + 337, + 646, + 364, + 658 + ], + "score": 0.91, + "content": "\\underline { z } ^ { ( k - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 644, + 459, + 661 + ], + "score": 1.0, + "content": "the lower bound, and", + "type": "text" + }, + { + "bbox": [ + 460, + 646, + 488, + 659 + ], + "score": 0.92, + "content": "| W ^ { ( k ) } |", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 644, + 506, + 661 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 655, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 289, + 673 + ], + "score": 1.0, + "content": "element-wise absolute value. 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A certificate can then", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 507, + 134 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 384, + 134 + ], + "score": 1.0, + "content": "be given if we can show that (3) is always true for outputs in the range", + "type": "text" + }, + { + "bbox": [ + 385, + 118, + 405, + 129 + ], + "score": 0.9, + "content": "\\overline { { z } } ^ { ( K ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 115, + 423, + 134 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 118, + 443, + 131 + ], + "score": 0.92, + "content": "\\smash { \\boldsymbol { \\mathcal { Z } } ^ { ( K ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 115, + 507, + 134 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 130, + 443, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 437, + 143 + ], + "score": 1.0, + "content": "all possible labels. More specifically, we can check that the following holds for all", + "type": "text" + }, + { + "bbox": [ + 437, + 133, + 443, + 142 + ], + "score": 0.79, + "content": "y", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 147, + 417, + 164 + ], + "lines": [ + { + "bbox": [ + 193, + 147, + 417, + 164 + ], + "spans": [ + { + "bbox": [ + 193, + 147, + 417, + 164 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\underline { { \\mathbf { m } } } _ { y } = e _ { y _ { t r u e } } ^ { T } \\underline { { z } } ^ { ( K ) } - e _ { y } ^ { T } \\overline { { z } } ^ { ( K ) } = \\underline { { z } } _ { y _ { t r u e } } ^ { ( K ) } - \\overline { { z } } _ { y } ^ { ( K ) } \\geq 0 \\quad \\forall y . } \\end{array}", + "type": "interline_equation", + "image_path": "25fc8a53f7effcf2cd2df99ffb77095bfce036a317c1d5ceee2a63e54835de6c.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 193, + 147, + 417, + 164 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 174, + 505, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "Training for Interval Bound Propagation To train a network to produce accurate interval bounds,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 276, + 197 + ], + "score": 1.0, + "content": "we simply replace standard logits with the", + "type": "text" + }, + { + "bbox": [ + 276, + 186, + 295, + 196 + ], + "score": 0.7, + "content": "- \\mathbf { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 185, + 450, + 197 + ], + "score": 1.0, + "content": "vector in (3). Note that all elements of", + "type": "text" + }, + { + "bbox": [ + 450, + 186, + 461, + 196 + ], + "score": 0.37, + "content": "\\mathbf { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 185, + 505, + 197 + ], + "score": 1.0, + "content": "need to be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 192, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 344, + 212 + ], + "score": 1.0, + "content": "larger than zero to satisfy the conditions in (3), and mytrue", + "type": "text" + }, + { + "bbox": [ + 344, + 195, + 505, + 209 + ], + "score": 1.0, + "content": "is always equal to zero. Put simply, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 182, + 221 + ], + "score": 1.0, + "content": "would like mytrue", + "type": "text" + }, + { + "bbox": [ + 199, + 208, + 504, + 219 + ], + "score": 1.0, + "content": "to be the least of all margins. We can promote this condition by training with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 176, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 176, + 230 + ], + "score": 1.0, + "content": "the loss function", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 235, + 403, + 249 + ], + "lines": [ + { + "bbox": [ + 207, + 235, + 403, + 249 + ], + "spans": [ + { + "bbox": [ + 207, + 235, + 403, + 249 + ], + "score": 0.82, + "content": "\\mathrm { C e r t i f i c a t e \\ L o s s } = \\mathrm { C r o s s \\ E n t r o p y \\ L o s s } ( - \\mathbf { m } , y ) .", + "type": "interline_equation", + "image_path": "72be3c53136bd187f40252a50653129ebf5fd67da8e8f872b8121036efc17f7a.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 207, + 235, + 403, + 249 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 259, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "Unlike regular neural network training, stochastic gradient descent for minimizing equation 10 is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "unstable, and a range of tricks are necessary to stabilize IBP training (Gowal et al., 2018). The first", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 280, + 507, + 298 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 303, + 298 + ], + "score": 1.0, + "content": "trick is merging the last linear weight matrix with", + "type": "text" + }, + { + "bbox": [ + 303, + 282, + 356, + 294 + ], + "score": 0.91, + "content": "( e _ { y } - e _ { y _ { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 280, + 430, + 298 + ], + "score": 1.0, + "content": "before calculating", + "type": "text" + }, + { + "bbox": [ + 430, + 283, + 453, + 295 + ], + "score": 0.9, + "content": "- { \\underline { { \\mathbf { m } } } } _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 280, + 507, + 298 + ], + "score": 1.0, + "content": ". This allows", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "a tighter characterization of the interval bound that noticeably improves results. The second trick uses", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "an “epsilon schedule” in which training begins with a perturbation radius of zero, and this radius is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 314, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 328 + ], + "score": 1.0, + "content": "slowly increased over time until a sentinel value is reached. Finally, a mixed loss function containing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 326, + 315, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 315, + 337 + ], + "score": 1.0, + "content": "both a standard natural loss and an IBP loss is used.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 504, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "In all of our experiments, we use the merging technique and the epsilon schedule, but we do not use a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 490, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 490, + 367 + ], + "score": 1.0, + "content": "mixed loss function containing a natural loss as it does not increase our certificate performance.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 379, + 295, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 297, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 297, + 392 + ], + "score": 1.0, + "content": "4.2 CERTIFYING AGAINST PATCH ATTACKS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "We can now describe the extension of IBP to patches. If we specify the patch location, one can", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "represent the feasible set of images with a simple interval bound: for pixels within the patch, the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "upper and lower bound is equal to 1 and 0. For pixels outside of the patch, the upper and lower", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "bounds are both equal to the original pixel value. By passing this bound through the network, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 440, + 437, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 440, + 246, + 457 + ], + "score": 1.0, + "content": "would be able to get msingle location", + "type": "text" + }, + { + "bbox": [ + 242, + 442, + 437, + 456 + ], + "score": 1.0, + "content": "and verify that they satisfy the conditions in (3).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 420, + 473 + ], + "score": 1.0, + "content": "However, we have to consider not just a single location, but all possible locations", + "type": "text" + }, + { + "bbox": [ + 420, + 461, + 428, + 470 + ], + "score": 0.56, + "content": "\\mathbb { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 459, + 507, + 473 + ], + "score": 1.0, + "content": "to give a certificate.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "To adapt the bound to all possible location, we pass each of the possible patches through the network,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 481, + 309, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 309, + 495 + ], + "score": 1.0, + "content": "and take the worst case margin. More specifically,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 499, + 379, + 519 + ], + "lines": [ + { + "bbox": [ + 231, + 499, + 379, + 519 + ], + "spans": [ + { + "bbox": [ + 231, + 499, + 379, + 519 + ], + "score": 0.94, + "content": "\\mathbf { m } ^ { \\mathrm { e s } } ( \\mathbb { L } ) _ { y } = \\operatorname* { m i n } _ { l \\in \\mathbb { L } } \\mathbf { m } ^ { \\mathrm { s i n g l e ~ p a t c h } } ( l ) _ { y } \\forall y .", + "type": "interline_equation", + "image_path": "a4e469b062571eede31119d89cf7867b9051bdc6b2d1f47c03973309a1cc4ae1.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 231, + 499, + 379, + 519 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 506, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 290, + 539 + ], + "score": 1.0, + "content": "Similar to regular IBP training, we simply use", + "type": "text" + }, + { + "bbox": [ + 290, + 525, + 322, + 537 + ], + "score": 0.92, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } } ( \\mathbb { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 523, + 506, + 539 + ], + "score": 1.0, + "content": "to calculate the cross entropy loss for training", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 536, + 195, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 195, + 549 + ], + "score": 1.0, + "content": "and backpropagation,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 553, + 414, + 567 + ], + "lines": [ + { + "bbox": [ + 198, + 553, + 414, + 567 + ], + "spans": [ + { + "bbox": [ + 198, + 553, + 414, + 567 + ], + "score": 0.84, + "content": "\\operatorname { C e r t i f i c a t e \\operatorname { L o s s } } = \\operatorname { C r o s s \\operatorname { E n t r o p y \\operatorname { L o s s } } } ( - \\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } } ( \\mathbb { L } ) , y ) .", + "type": "interline_equation", + "image_path": "badccbe6659e45077e905312268debefca934cb3ec05550287f687b5304211fa.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 198, + 553, + 414, + 567 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 506, + 590 + ], + "score": 1.0, + "content": "Unfortunately, the cost of producing this na¨ıve certificate increases quadratically with image size.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 275, + 601 + ], + "score": 1.0, + "content": "Consider that a CIFAR-10 image is of size", + "type": "text" + }, + { + "bbox": [ + 275, + 589, + 303, + 600 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 589, + 506, + 601 + ], + "score": 1.0, + "content": ", requiring over a thousand interval bounds, one for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "each possible patch location. To alleviate this problem, we propose two certificate training methods:", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Random Patch and Guided Patch, so that the number of forward passes does not scale with the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 622, + 205, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 205, + 635 + ], + "score": 1.0, + "content": "dimension of the inputs.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "Random Patch Certificate Training In this method, we simply select a random set of patches out", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 649, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 664 + ], + "score": 1.0, + "content": "of the possible patches and pass them forward. A level of robustness is achieved even though a very", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 661, + 487, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 487, + 674 + ], + "score": 1.0, + "content": "small number of random patches are selected compared to the total number of possible patches", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 678, + 367, + 693 + ], + "lines": [ + { + "bbox": [ + 242, + 678, + 367, + 693 + ], + "spans": [ + { + "bbox": [ + 242, + 678, + 367, + 693 + ], + "score": 0.91, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { r a n d o m \\ p a t c h e s } } ( \\mathbb { L } ) _ { y } = \\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } } ( S ) _ { y }", + "type": "interline_equation", + "image_path": "44945ff3b0611e84f9e7c490e701923833d7e3bd6d9b926f7bcd4dee8156df28.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 242, + 678, + 367, + 693 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 710 + ], + "lines": [ + { + "bbox": [ + 106, + 696, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 696, + 133, + 711 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 699, + 141, + 708 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 696, + 229, + 711 + ], + "score": 1.0, + "content": "is a random subset of", + "type": "text" + }, + { + "bbox": [ + 229, + 699, + 237, + 708 + ], + "score": 0.48, + "content": "\\mathbb { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 696, + 506, + 711 + ], + "score": 1.0, + "content": ". Similarly, the random patch certificate loss is calculated as below.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 716, + 461, + 730 + ], + "lines": [ + { + "bbox": [ + 145, + 716, + 461, + 730 + ], + "spans": [ + { + "bbox": [ + 145, + 716, + 461, + 730 + ], + "score": 0.56, + "content": "\\mathrm { R a n d o m ~ P a t c h ~ C e r t i f i c a t e ~ L o s s } = \\mathrm { C r o s s ~ E n t r o p y ~ L o s s } ( - \\underline { { \\mathbf { m } } } ^ { \\mathrm { r a n d o m ~ p a t c h e s } } ( \\mathbb { L } ) , y )", + "type": "interline_equation", + "image_path": "40631a922fd1252beda45689f839d46b44e018b350319daebbe860eeb763a5e7.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 145, + 716, + 461, + 730 + ], + "spans": [], + "index": 44 + } + ] + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "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" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 191, + 95 + ], + "score": 1.0, + "content": "When the feasible set", + "type": "text" + }, + { + "bbox": [ + 191, + 82, + 217, + 95 + ], + "score": 0.92, + "content": "\\mathbb { B } ( x _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 81, + 294, + 95 + ], + "score": 1.0, + "content": "represents a simple", + "type": "text" + }, + { + "bbox": [ + 294, + 83, + 307, + 93 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 81, + 419, + 95 + ], + "score": 1.0, + "content": "attack, the range of possible", + "type": "text" + }, + { + "bbox": [ + 419, + 81, + 435, + 92 + ], + "score": 0.89, + "content": "z ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "values is trivially", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 248, + 108 + ], + "score": 1.0, + "content": "characterized by an interval bound", + "type": "text" + }, + { + "bbox": [ + 249, + 93, + 265, + 105 + ], + "score": 0.89, + "content": "\\overline { { z } } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 93, + 283, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 284, + 94, + 300, + 106 + ], + "score": 0.91, + "content": "\\underline { z } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 93, + 506, + 108 + ], + "score": 1.0, + "content": ". Then, by iteratively applying the above rules, we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 105, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 356, + 120 + ], + "score": 1.0, + "content": "can propagate intervals through the network and eventually get", + "type": "text" + }, + { + "bbox": [ + 356, + 106, + 376, + 117 + ], + "score": 0.9, + "content": "\\overline { { z } } ^ { ( K ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 105, + 394, + 120 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 394, + 106, + 414, + 118 + ], + "score": 0.88, + "content": "\\smash { \\mathcal { Z } ^ { ( K ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 105, + 506, + 120 + ], + "score": 1.0, + "content": ". A certificate can then", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 507, + 134 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 384, + 134 + ], + "score": 1.0, + "content": "be given if we can show that (3) is always true for outputs in the range", + "type": "text" + }, + { + "bbox": [ + 385, + 118, + 405, + 129 + ], + "score": 0.9, + "content": "\\overline { { z } } ^ { ( K ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 115, + 423, + 134 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 118, + 443, + 131 + ], + "score": 0.92, + "content": "\\smash { \\boldsymbol { \\mathcal { Z } } ^ { ( K ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 115, + 507, + 134 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 130, + 443, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 437, + 143 + ], + "score": 1.0, + "content": "all possible labels. More specifically, we can check that the following holds for all", + "type": "text" + }, + { + "bbox": [ + 437, + 133, + 443, + 142 + ], + "score": 0.79, + "content": "y", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 104, + 81, + 507, + 143 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 147, + 417, + 164 + ], + "lines": [ + { + "bbox": [ + 193, + 147, + 417, + 164 + ], + "spans": [ + { + "bbox": [ + 193, + 147, + 417, + 164 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\underline { { \\mathbf { m } } } _ { y } = e _ { y _ { t r u e } } ^ { T } \\underline { { z } } ^ { ( K ) } - e _ { y } ^ { T } \\overline { { z } } ^ { ( K ) } = \\underline { { z } } _ { y _ { t r u e } } ^ { ( K ) } - \\overline { { z } } _ { y } ^ { ( K ) } \\geq 0 \\quad \\forall y . } \\end{array}", + "type": "interline_equation", + "image_path": "25fc8a53f7effcf2cd2df99ffb77095bfce036a317c1d5ceee2a63e54835de6c.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 193, + 147, + 417, + 164 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 174, + 505, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "Training for Interval Bound Propagation To train a network to produce accurate interval bounds,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 276, + 197 + ], + "score": 1.0, + "content": "we simply replace standard logits with the", + "type": "text" + }, + { + "bbox": [ + 276, + 186, + 295, + 196 + ], + "score": 0.7, + "content": "- \\mathbf { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 185, + 450, + 197 + ], + "score": 1.0, + "content": "vector in (3). Note that all elements of", + "type": "text" + }, + { + "bbox": [ + 450, + 186, + 461, + 196 + ], + "score": 0.37, + "content": "\\mathbf { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 185, + 505, + 197 + ], + "score": 1.0, + "content": "need to be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 192, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 344, + 212 + ], + "score": 1.0, + "content": "larger than zero to satisfy the conditions in (3), and mytrue", + "type": "text" + }, + { + "bbox": [ + 344, + 195, + 505, + 209 + ], + "score": 1.0, + "content": "is always equal to zero. Put simply, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 182, + 221 + ], + "score": 1.0, + "content": "would like mytrue", + "type": "text" + }, + { + "bbox": [ + 199, + 208, + 504, + 219 + ], + "score": 1.0, + "content": "to be the least of all margins. We can promote this condition by training with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 176, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 176, + 230 + ], + "score": 1.0, + "content": "the loss function", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 174, + 506, + 230 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 235, + 403, + 249 + ], + "lines": [ + { + "bbox": [ + 207, + 235, + 403, + 249 + ], + "spans": [ + { + "bbox": [ + 207, + 235, + 403, + 249 + ], + "score": 0.82, + "content": "\\mathrm { C e r t i f i c a t e \\ L o s s } = \\mathrm { C r o s s \\ E n t r o p y \\ L o s s } ( - \\mathbf { m } , y ) .", + "type": "interline_equation", + "image_path": "72be3c53136bd187f40252a50653129ebf5fd67da8e8f872b8121036efc17f7a.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 207, + 235, + 403, + 249 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 259, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "Unlike regular neural network training, stochastic gradient descent for minimizing equation 10 is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "unstable, and a range of tricks are necessary to stabilize IBP training (Gowal et al., 2018). The first", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 280, + 507, + 298 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 303, + 298 + ], + "score": 1.0, + "content": "trick is merging the last linear weight matrix with", + "type": "text" + }, + { + "bbox": [ + 303, + 282, + 356, + 294 + ], + "score": 0.91, + "content": "( e _ { y } - e _ { y _ { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 280, + 430, + 298 + ], + "score": 1.0, + "content": "before calculating", + "type": "text" + }, + { + "bbox": [ + 430, + 283, + 453, + 295 + ], + "score": 0.9, + "content": "- { \\underline { { \\mathbf { m } } } } _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 280, + 507, + 298 + ], + "score": 1.0, + "content": ". This allows", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "a tighter characterization of the interval bound that noticeably improves results. The second trick uses", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "an “epsilon schedule” in which training begins with a perturbation radius of zero, and this radius is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 314, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 328 + ], + "score": 1.0, + "content": "slowly increased over time until a sentinel value is reached. Finally, a mixed loss function containing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 326, + 315, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 315, + 337 + ], + "score": 1.0, + "content": "both a standard natural loss and an IBP loss is used.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 260, + 507, + 337 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 504, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "In all of our experiments, we use the merging technique and the epsilon schedule, but we do not use a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 490, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 490, + 367 + ], + "score": 1.0, + "content": "mixed loss function containing a natural loss as it does not increase our certificate performance.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 342, + 506, + 367 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 379, + 295, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 297, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 297, + 392 + ], + "score": 1.0, + "content": "4.2 CERTIFYING AGAINST PATCH ATTACKS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "We can now describe the extension of IBP to patches. If we specify the patch location, one can", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "represent the feasible set of images with a simple interval bound: for pixels within the patch, the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "upper and lower bound is equal to 1 and 0. For pixels outside of the patch, the upper and lower", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "bounds are both equal to the original pixel value. By passing this bound through the network, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 440, + 437, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 440, + 246, + 457 + ], + "score": 1.0, + "content": "would be able to get msingle location", + "type": "text" + }, + { + "bbox": [ + 242, + 442, + 437, + 456 + ], + "score": 1.0, + "content": "and verify that they satisfy the conditions in (3).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 399, + 506, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 420, + 473 + ], + "score": 1.0, + "content": "However, we have to consider not just a single location, but all possible locations", + "type": "text" + }, + { + "bbox": [ + 420, + 461, + 428, + 470 + ], + "score": 0.56, + "content": "\\mathbb { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 459, + 507, + 473 + ], + "score": 1.0, + "content": "to give a certificate.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "To adapt the bound to all possible location, we pass each of the possible patches through the network,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 481, + 309, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 309, + 495 + ], + "score": 1.0, + "content": "and take the worst case margin. More specifically,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 459, + 507, + 495 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 499, + 379, + 519 + ], + "lines": [ + { + "bbox": [ + 231, + 499, + 379, + 519 + ], + "spans": [ + { + "bbox": [ + 231, + 499, + 379, + 519 + ], + "score": 0.94, + "content": "\\mathbf { m } ^ { \\mathrm { e s } } ( \\mathbb { L } ) _ { y } = \\operatorname* { m i n } _ { l \\in \\mathbb { L } } \\mathbf { m } ^ { \\mathrm { s i n g l e ~ p a t c h } } ( l ) _ { y } \\forall y .", + "type": "interline_equation", + "image_path": "a4e469b062571eede31119d89cf7867b9051bdc6b2d1f47c03973309a1cc4ae1.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 231, + 499, + 379, + 519 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 506, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 290, + 539 + ], + "score": 1.0, + "content": "Similar to regular IBP training, we simply use", + "type": "text" + }, + { + "bbox": [ + 290, + 525, + 322, + 537 + ], + "score": 0.92, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } } ( \\mathbb { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 523, + 506, + 539 + ], + "score": 1.0, + "content": "to calculate the cross entropy loss for training", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 536, + 195, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 195, + 549 + ], + "score": 1.0, + "content": "and backpropagation,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 523, + 506, + 549 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 553, + 414, + 567 + ], + "lines": [ + { + "bbox": [ + 198, + 553, + 414, + 567 + ], + "spans": [ + { + "bbox": [ + 198, + 553, + 414, + 567 + ], + "score": 0.84, + "content": "\\operatorname { C e r t i f i c a t e \\operatorname { L o s s } } = \\operatorname { C r o s s \\operatorname { E n t r o p y \\operatorname { L o s s } } } ( - \\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } } ( \\mathbb { L } ) , y ) .", + "type": "interline_equation", + "image_path": "badccbe6659e45077e905312268debefca934cb3ec05550287f687b5304211fa.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 198, + 553, + 414, + 567 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 506, + 590 + ], + "score": 1.0, + "content": "Unfortunately, the cost of producing this na¨ıve certificate increases quadratically with image size.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 275, + 601 + ], + "score": 1.0, + "content": "Consider that a CIFAR-10 image is of size", + "type": "text" + }, + { + "bbox": [ + 275, + 589, + 303, + 600 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 589, + 506, + 601 + ], + "score": 1.0, + "content": ", requiring over a thousand interval bounds, one for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "each possible patch location. To alleviate this problem, we propose two certificate training methods:", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Random Patch and Guided Patch, so that the number of forward passes does not scale with the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 622, + 205, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 205, + 635 + ], + "score": 1.0, + "content": "dimension of the inputs.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 578, + 506, + 635 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "Random Patch Certificate Training In this method, we simply select a random set of patches out", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 649, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 664 + ], + "score": 1.0, + "content": "of the possible patches and pass them forward. A level of robustness is achieved even though a very", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 661, + 487, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 487, + 674 + ], + "score": 1.0, + "content": "small number of random patches are selected compared to the total number of possible patches", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 639, + 505, + 674 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 678, + 367, + 693 + ], + "lines": [ + { + "bbox": [ + 242, + 678, + 367, + 693 + ], + "spans": [ + { + "bbox": [ + 242, + 678, + 367, + 693 + ], + "score": 0.91, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { r a n d o m \\ p a t c h e s } } ( \\mathbb { L } ) _ { y } = \\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } } ( S ) _ { y }", + "type": "interline_equation", + "image_path": "44945ff3b0611e84f9e7c490e701923833d7e3bd6d9b926f7bcd4dee8156df28.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 242, + 678, + 367, + 693 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 710 + ], + "lines": [ + { + "bbox": [ + 106, + 696, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 696, + 133, + 711 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 699, + 141, + 708 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 696, + 229, + 711 + ], + "score": 1.0, + "content": "is a random subset of", + "type": "text" + }, + { + "bbox": [ + 229, + 699, + 237, + 708 + ], + "score": 0.48, + "content": "\\mathbb { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 696, + 506, + 711 + ], + "score": 1.0, + "content": ". Similarly, the random patch certificate loss is calculated as below.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 106, + 696, + 506, + 711 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 716, + 461, + 730 + ], + "lines": [ + { + "bbox": [ + 145, + 716, + 461, + 730 + ], + "spans": [ + { + "bbox": [ + 145, + 716, + 461, + 730 + ], + "score": 0.56, + "content": "\\mathrm { R a n d o m ~ P a t c h ~ C e r t i f i c a t e ~ L o s s } = \\mathrm { C r o s s ~ E n t r o p y ~ L o s s } ( - \\underline { { \\mathbf { m } } } ^ { \\mathrm { r a n d o m ~ p a t c h e s } } ( \\mathbb { L } ) , y )", + "type": "interline_equation", + "image_path": "40631a922fd1252beda45689f839d46b44e018b350319daebbe860eeb763a5e7.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 145, + 716, + 461, + 730 + ], + "spans": [], + "index": 44 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Guided Patch Certificate Training In this method, we propose using a U-net (Ronneberger et al.,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 90, + 507, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 90, + 507, + 107 + ], + "score": 1.0, + "content": "2015) to predict msingle patch, and then randomly select a couple of locations based on the predicted", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 349, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 151, + 117 + ], + "score": 0.62, + "content": "\\mathbf { m } ^ { \\mathrm { s i n g l e } \\mathrm { p a t c h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 102, + 349, + 117 + ], + "score": 1.0, + "content": "so that fewer patches need to be passed forward.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 121, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 361, + 133 + ], + "score": 1.0, + "content": "Note that very few patches contribute to the worst case bound", + "type": "text" + }, + { + "bbox": [ + 361, + 122, + 378, + 133 + ], + "score": 0.89, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 121, + 506, + 133 + ], + "score": 1.0, + "content": "in (11). In fact, the number of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "patches that yield the worst case margins will be no more than the number of labels. If we know the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 157 + ], + "score": 1.0, + "content": "worst-case patches beforehand, then we can simply select the few worst-case patches during training.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 504, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "We propose to use U-net as the number of locations and margins is very large. For a square patch of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 124, + 183 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 124, + 172, + 150, + 182 + ], + "score": 0.9, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 170, + 234, + 183 + ], + "score": 1.0, + "content": "and an image of size", + "type": "text" + }, + { + "bbox": [ + 234, + 172, + 265, + 182 + ], + "score": 0.9, + "content": "m \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 170, + 428, + 183 + ], + "score": 1.0, + "content": ", the total number of possible locations is", + "type": "text" + }, + { + "bbox": [ + 428, + 171, + 484, + 183 + ], + "score": 0.92, + "content": "( m - n + \\mathbf { \\hat { 1 } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 170, + 505, + 183 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 434, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 434, + 195 + ], + "score": 1.0, + "content": "for each location the number of margins is equal to the number of possible labels.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 197, + 411, + 229 + ], + "lines": [ + { + "bbox": [ + 199, + 197, + 411, + 229 + ], + "spans": [ + { + "bbox": [ + 199, + 197, + 411, + 229 + ], + "score": 0.86, + "content": "\\begin{array} { l } { { \\bf { m } } ^ { \\mathrm { { p r e d } } } = { \\bf { U } } { \\mathrm { - n e t } } ( \\mathrm { i m a g e } ) } \\\\ { { \\bf { d i m } } ( { \\bf { m } } ^ { \\mathrm { { p r e d } } } ) = ( m - n + 1 , m - n + 1 , \\# \\mathrm { o f } \\mathrm { l a b e l s } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6e2b722e690020b0cd598c157f790ae7ddb81b815c1fc7d409fbc09cdc855329.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 197, + 411, + 213.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 199, + 213.0, + 411, + 229.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 238, + 506, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 226, + 253 + ], + "score": 1.0, + "content": "Given the U-net prediction of", + "type": "text" + }, + { + "bbox": [ + 226, + 239, + 250, + 250 + ], + "score": 0.86, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { p r e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 236, + 506, + 253 + ], + "score": 1.0, + "content": ", we then randomly select a single patch for each label based on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 246, + 507, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 246, + 219, + 264 + ], + "score": 1.0, + "content": "the softmax of the predicted", + "type": "text" + }, + { + "bbox": [ + 219, + 250, + 244, + 261 + ], + "score": 0.84, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { p r e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 246, + 507, + 264 + ], + "score": 1.0, + "content": ". The number of selected patches is equal to the number of labels.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "score": 1.0, + "content": "After these patches are passed forward, the U-net is then updated with a mean-squared-error loss", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 269, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 228, + 286 + ], + "score": 1.0, + "content": "between the predicted margins", + "type": "text" + }, + { + "bbox": [ + 228, + 272, + 252, + 284 + ], + "score": 0.87, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { p r e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 269, + 344, + 286 + ], + "score": 1.0, + "content": "and the actual margins", + "type": "text" + }, + { + "bbox": [ + 344, + 272, + 372, + 284 + ], + "score": 0.88, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { a c t u a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 269, + 506, + 286 + ], + "score": 1.0, + "content": ". Note that only a few patches are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 471, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 471, + 296 + ], + "score": 1.0, + "content": "selected at a time, so that the mean-squared-error only passes through the selected patches.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 298, + 377, + 312 + ], + "lines": [ + { + "bbox": [ + 232, + 298, + 377, + 312 + ], + "spans": [ + { + "bbox": [ + 232, + 298, + 377, + 312 + ], + "score": 0.88, + "content": "\\mathrm { U } \\mathrm { - n e t } \\mathrm { L o s s } = \\mathbf { M } \\mathrm { S E } ( \\mathbf { \\underline { { m } } } ^ { \\mathrm { p r e d } } , \\mathbf { \\underline { { m } } } ^ { \\mathrm { a c t u a l } } ) .", + "type": "interline_equation", + "image_path": "d2bc1ac4b92bf8f45f0b03b98a291aaf65de753c0cb1dc9cce77eef4b0b12473.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 232, + 298, + 377, + 312 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 294, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 295, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 295, + 329 + ], + "score": 1.0, + "content": "The network is trained with the following loss:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 152, + 332, + 459, + 346 + ], + "lines": [ + { + "bbox": [ + 150, + 327, + 459, + 349 + ], + "spans": [ + { + "bbox": [ + 150, + 327, + 251, + 349 + ], + "score": 1.0, + "content": "Guided Patch Certificate", + "type": "text" + }, + { + "bbox": [ + 252, + 332, + 459, + 346 + ], + "score": 0.76, + "content": "\\operatorname { L o s s } = \\operatorname { C r o s s } \\operatorname { E n t r o p y } \\operatorname { L o s s } ( - \\underline { { \\mathbf { m } } } ^ { \\mathrm { g u i d e d } \\mathrm { p a t c h e s } } ( \\mathbb { L } ) , y ) .", + "type": "inline_equation", + "image_path": "2ec67db4a3aaf1d4cb78cb7b9f077e9d5d7458b681f5306b9c92a1a65c4d146b.jpg" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 506, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 362 + ], + "score": 1.0, + "content": "Certification Process In all our experiments, we check that equation (3) is satisfied by iterating over", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "all possible patches and forward-passing the interval bounds generated for each patch; this overhead", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 371, + 229, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 229, + 383 + ], + "score": 1.0, + "content": "is tolerable at evaluation time.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 396, + 300, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 301, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 301, + 408 + ], + "score": 1.0, + "content": "4.3 CERTIFYING AGAINST SPARSE ATTACKS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 506, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "IBP can also be adapted to defend against sparse attack where the attacker is allowed to modify a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 164, + 440 + ], + "score": 1.0, + "content": "fixed number", + "type": "text" + }, + { + "bbox": [ + 164, + 428, + 177, + 439 + ], + "score": 0.79, + "content": "( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "of pixels that may not be adjacent to each other (Modas et al., 2019). The only", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 439, + 434, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 434, + 451 + ], + "score": 1.0, + "content": "modification is that we have to change the bound calculated from the first layer to", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 453, + 450, + 471 + ], + "lines": [ + { + "bbox": [ + 161, + 453, + 450, + 471 + ], + "spans": [ + { + "bbox": [ + 161, + 453, + 450, + 471 + ], + "score": 0.9, + "content": "\\overline { { { z } } } _ { i } ^ { ( 1 ) } = W _ { i , : } ^ { ( 1 ) } z ^ { ( 0 ) } + | W _ { i , : } ^ { ( 1 ) } | _ { t o p _ { k } } ~ z _ { i } ^ { ( 1 ) } = W _ { i , : } ^ { ( 1 ) } z ^ { ( 0 ) } - | W _ { i , : } ^ { ( 1 ) } | _ { t o p _ { k } } ~ \\forall i", + "type": "interline_equation", + "image_path": "17a421e1da69fa4d200afc0f132582985c1be61c501fb76dbccbf41606d6b64e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 161, + 453, + 450, + 471 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 506, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 504, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 367, + 489 + ], + "score": 1.0, + "content": "and apply equation (5) and (6) for the subsequent layers. Here,", + "type": "text" + }, + { + "bbox": [ + 368, + 474, + 394, + 487 + ], + "score": 0.92, + "content": "( . ) _ { t o p _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 473, + 497, + 489 + ], + "score": 1.0, + "content": "is the sum of the largest", + "type": "text" + }, + { + "bbox": [ + 497, + 476, + 504, + 484 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 485, + 199, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 199, + 499 + ], + "score": 1.0, + "content": "elements in the vector.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 513, + 200, + 525 + ], + "lines": [ + { + "bbox": [ + 104, + 511, + 201, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 201, + 528 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 549 + ], + "score": 1.0, + "content": "In this section, we compare our certified defenses with exiting algorithms on two datasets and three", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "model architectures of varying complexity. We consider a strong attack setting in which adversarial", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "patches can appear anywhere in the image. Different training strategies for the certified defense are", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "also compared, which shows a trade-off between performance and training efficiency. Furthermore,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 581, + 507, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 507, + 595 + ], + "score": 1.0, + "content": "we evaluate the transferability of a model trained using square patches to other adversarial shapes,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "including shapes that do not fit in any certified square. The training and architectural details can", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 618 + ], + "score": 1.0, + "content": "be found in Appendix A.1. We also present preliminary results on certified defense against sparse", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 614, + 139, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 139, + 626 + ], + "score": 1.0, + "content": "attacks.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 640, + 320, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 320, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 320, + 651 + ], + "score": 1.0, + "content": "5.1 COMPARISON AGAINST EXISTING DEFENSES", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "In this section, we study the effectiveness of our proposed IBP certified models against an adversary", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "that is allowed to place patches anywhere in the image, even on top of the salient object. 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For more challenging cases, such as a 5", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Guided Patch Certificate Training In this method, we propose using a U-net (Ronneberger et al.,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 90, + 507, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 90, + 507, + 107 + ], + "score": 1.0, + "content": "2015) to predict msingle patch, and then randomly select a couple of locations based on the predicted", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 349, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 151, + 117 + ], + "score": 0.62, + "content": "\\mathbf { m } ^ { \\mathrm { s i n g l e } \\mathrm { p a t c h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 102, + 349, + 117 + ], + "score": 1.0, + "content": "so that fewer patches need to be passed forward.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 104, + 81, + 507, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 121, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 361, + 133 + ], + "score": 1.0, + "content": "Note that very few patches contribute to the worst case bound", + "type": "text" + }, + { + "bbox": [ + 361, + 122, + 378, + 133 + ], + "score": 0.89, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { e s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 121, + 506, + 133 + ], + "score": 1.0, + "content": "in (11). In fact, the number of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "patches that yield the worst case margins will be no more than the number of labels. If we know the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 157 + ], + "score": 1.0, + "content": "worst-case patches beforehand, then we can simply select the few worst-case patches during training.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 121, + 506, + 157 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 504, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "We propose to use U-net as the number of locations and margins is very large. For a square patch of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 124, + 183 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 124, + 172, + 150, + 182 + ], + "score": 0.9, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 170, + 234, + 183 + ], + "score": 1.0, + "content": "and an image of size", + "type": "text" + }, + { + "bbox": [ + 234, + 172, + 265, + 182 + ], + "score": 0.9, + "content": "m \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 170, + 428, + 183 + ], + "score": 1.0, + "content": ", the total number of possible locations is", + "type": "text" + }, + { + "bbox": [ + 428, + 171, + 484, + 183 + ], + "score": 0.92, + "content": "( m - n + \\mathbf { \\hat { 1 } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 170, + 505, + 183 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 434, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 434, + 195 + ], + "score": 1.0, + "content": "for each location the number of margins is equal to the number of possible labels.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 159, + 506, + 195 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 197, + 411, + 229 + ], + "lines": [ + { + "bbox": [ + 199, + 197, + 411, + 229 + ], + "spans": [ + { + "bbox": [ + 199, + 197, + 411, + 229 + ], + "score": 0.86, + "content": "\\begin{array} { l } { { \\bf { m } } ^ { \\mathrm { { p r e d } } } = { \\bf { U } } { \\mathrm { - n e t } } ( \\mathrm { i m a g e } ) } \\\\ { { \\bf { d i m } } ( { \\bf { m } } ^ { \\mathrm { { p r e d } } } ) = ( m - n + 1 , m - n + 1 , \\# \\mathrm { o f } \\mathrm { l a b e l s } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6e2b722e690020b0cd598c157f790ae7ddb81b815c1fc7d409fbc09cdc855329.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 197, + 411, + 213.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 199, + 213.0, + 411, + 229.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 238, + 506, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 226, + 253 + ], + "score": 1.0, + "content": "Given the U-net prediction of", + "type": "text" + }, + { + "bbox": [ + 226, + 239, + 250, + 250 + ], + "score": 0.86, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { p r e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 236, + 506, + 253 + ], + "score": 1.0, + "content": ", we then randomly select a single patch for each label based on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 246, + 507, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 246, + 219, + 264 + ], + "score": 1.0, + "content": "the softmax of the predicted", + "type": "text" + }, + { + "bbox": [ + 219, + 250, + 244, + 261 + ], + "score": 0.84, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { p r e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 246, + 507, + 264 + ], + "score": 1.0, + "content": ". The number of selected patches is equal to the number of labels.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "score": 1.0, + "content": "After these patches are passed forward, the U-net is then updated with a mean-squared-error loss", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 269, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 228, + 286 + ], + "score": 1.0, + "content": "between the predicted margins", + "type": "text" + }, + { + "bbox": [ + 228, + 272, + 252, + 284 + ], + "score": 0.87, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { p r e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 269, + 344, + 286 + ], + "score": 1.0, + "content": "and the actual margins", + "type": "text" + }, + { + "bbox": [ + 344, + 272, + 372, + 284 + ], + "score": 0.88, + "content": "\\underline { { \\mathbf { m } } } ^ { \\mathrm { a c t u a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 269, + 506, + 286 + ], + "score": 1.0, + "content": ". Note that only a few patches are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 471, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 471, + 296 + ], + "score": 1.0, + "content": "selected at a time, so that the mean-squared-error only passes through the selected patches.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 236, + 507, + 296 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 298, + 377, + 312 + ], + "lines": [ + { + "bbox": [ + 232, + 298, + 377, + 312 + ], + "spans": [ + { + "bbox": [ + 232, + 298, + 377, + 312 + ], + "score": 0.88, + "content": "\\mathrm { U } \\mathrm { - n e t } \\mathrm { L o s s } = \\mathbf { M } \\mathrm { S E } ( \\mathbf { \\underline { { m } } } ^ { \\mathrm { p r e d } } , \\mathbf { \\underline { { m } } } ^ { \\mathrm { a c t u a l } } ) .", + "type": "interline_equation", + "image_path": "d2bc1ac4b92bf8f45f0b03b98a291aaf65de753c0cb1dc9cce77eef4b0b12473.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 232, + 298, + 377, + 312 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 294, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 295, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 295, + 329 + ], + "score": 1.0, + "content": "The network is trained with the following loss:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 314, + 295, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 152, + 332, + 459, + 346 + ], + "lines": [ + { + "bbox": [ + 150, + 327, + 459, + 349 + ], + "spans": [ + { + "bbox": [ + 150, + 327, + 251, + 349 + ], + "score": 1.0, + "content": "Guided Patch Certificate", + "type": "text" + }, + { + "bbox": [ + 252, + 332, + 459, + 346 + ], + "score": 0.76, + "content": "\\operatorname { L o s s } = \\operatorname { C r o s s } \\operatorname { E n t r o p y } \\operatorname { L o s s } ( - \\underline { { \\mathbf { m } } } ^ { \\mathrm { g u i d e d } \\mathrm { p a t c h e s } } ( \\mathbb { L } ) , y ) .", + "type": "inline_equation", + "image_path": "2ec67db4a3aaf1d4cb78cb7b9f077e9d5d7458b681f5306b9c92a1a65c4d146b.jpg" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 150, + 327, + 459, + 349 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 506, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 362 + ], + "score": 1.0, + "content": "Certification Process In all our experiments, we check that equation (3) is satisfied by iterating over", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "all possible patches and forward-passing the interval bounds generated for each patch; this overhead", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 371, + 229, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 229, + 383 + ], + "score": 1.0, + "content": "is tolerable at evaluation time.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 348, + 506, + 383 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 396, + 300, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 301, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 301, + 408 + ], + "score": 1.0, + "content": "4.3 CERTIFYING AGAINST SPARSE ATTACKS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 506, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "IBP can also be adapted to defend against sparse attack where the attacker is allowed to modify a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 164, + 440 + ], + "score": 1.0, + "content": "fixed number", + "type": "text" + }, + { + "bbox": [ + 164, + 428, + 177, + 439 + ], + "score": 0.79, + "content": "( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "of pixels that may not be adjacent to each other (Modas et al., 2019). The only", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 439, + 434, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 434, + 451 + ], + "score": 1.0, + "content": "modification is that we have to change the bound calculated from the first layer to", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 417, + 505, + 451 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 453, + 450, + 471 + ], + "lines": [ + { + "bbox": [ + 161, + 453, + 450, + 471 + ], + "spans": [ + { + "bbox": [ + 161, + 453, + 450, + 471 + ], + "score": 0.9, + "content": "\\overline { { { z } } } _ { i } ^ { ( 1 ) } = W _ { i , : } ^ { ( 1 ) } z ^ { ( 0 ) } + | W _ { i , : } ^ { ( 1 ) } | _ { t o p _ { k } } ~ z _ { i } ^ { ( 1 ) } = W _ { i , : } ^ { ( 1 ) } z ^ { ( 0 ) } - | W _ { i , : } ^ { ( 1 ) } | _ { t o p _ { k } } ~ \\forall i", + "type": "interline_equation", + "image_path": "17a421e1da69fa4d200afc0f132582985c1be61c501fb76dbccbf41606d6b64e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 161, + 453, + 450, + 471 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 506, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 504, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 367, + 489 + ], + "score": 1.0, + "content": "and apply equation (5) and (6) for the subsequent layers. Here,", + "type": "text" + }, + { + "bbox": [ + 368, + 474, + 394, + 487 + ], + "score": 0.92, + "content": "( . ) _ { t o p _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 473, + 497, + 489 + ], + "score": 1.0, + "content": "is the sum of the largest", + "type": "text" + }, + { + "bbox": [ + 497, + 476, + 504, + 484 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 485, + 199, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 199, + 499 + ], + "score": 1.0, + "content": "elements in the vector.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 473, + 504, + 499 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 513, + 200, + 525 + ], + "lines": [ + { + "bbox": [ + 104, + 511, + 201, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 201, + 528 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 549 + ], + "score": 1.0, + "content": "In this section, we compare our certified defenses with exiting algorithms on two datasets and three", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "model architectures of varying complexity. We consider a strong attack setting in which adversarial", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "patches can appear anywhere in the image. Different training strategies for the certified defense are", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "also compared, which shows a trade-off between performance and training efficiency. Furthermore,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 581, + 507, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 507, + 595 + ], + "score": 1.0, + "content": "we evaluate the transferability of a model trained using square patches to other adversarial shapes,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "including shapes that do not fit in any certified square. The training and architectural details can", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 618 + ], + "score": 1.0, + "content": "be found in Appendix A.1. We also present preliminary results on certified defense against sparse", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 614, + 139, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 139, + 626 + ], + "score": 1.0, + "content": "attacks.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 538, + 507, + 626 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 640, + 320, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 320, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 320, + 651 + ], + "score": 1.0, + "content": "5.1 COMPARISON AGAINST EXISTING DEFENSES", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "In this section, we study the effectiveness of our proposed IBP certified models against an adversary", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "that is allowed to place patches anywhere in the image, even on top of the salient object. If the patch", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 104, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "is sufficiently small, and does not cover a large portion of the salient object, then the model should", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 693, + 421, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 421, + 705 + ], + "score": 1.0, + "content": "still classify correctly, and defense against the perturbation should be possible.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 659, + 505, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 337, + 722 + ], + "score": 1.0, + "content": "In the best case, our IBP certified model is able to achieve", + "type": "text" + }, + { + "bbox": [ + 337, + 710, + 364, + 720 + ], + "score": 0.86, + "content": "9 1 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 708, + 506, + 722 + ], + "score": 1.0, + "content": "certified (Table 2) with respect to a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 126, + 731 + ], + "score": 0.84, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 720, + 154, + 733 + ], + "score": 1.0, + "content": "patch", + "type": "text" + }, + { + "bbox": [ + 154, + 721, + 182, + 731 + ], + "score": 0.83, + "content": "( \\sim . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "of image pixels) adversary on MNIST. For more challenging cases, such as a 5", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 123, + 93 + ], + "score": 0.73, + "content": "\\times 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 124, + 82, + 129, + 95 + ], + "score": 1.0, + "content": "(", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 129, + 83, + 163, + 93 + ], + "score": 0.65, + "content": "\\sim 2 . 5 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 163, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "of image pixels) patch adversary on CIFAR-10, the certified adversarial accuracy is", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 127, + 107 + ], + "score": 1.0, + "content": "only", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 127, + 94, + 154, + 104 + ], + "score": 0.85, + "content": "2 4 . 9 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 155, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "(Table 2). Even though these existing defenses appear to achieve better or comparable", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "adversarial accuracy as our IBP certified model when faced with a weak adversary, when faced with", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "a stronger adversary their adversarial accuracy dropped to levels below our certified accuracy for all", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 202, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 202, + 139 + ], + "score": 1.0, + "content": "cases that we analyzed.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 708, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 123, + 93 + ], + "score": 0.73, + "content": "\\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 82, + 129, + 95 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 129, + 83, + 163, + 93 + ], + "score": 0.65, + "content": "\\sim 2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "of image pixels) patch adversary on CIFAR-10, the certified adversarial accuracy is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 127, + 107 + ], + "score": 1.0, + "content": "only", + "type": "text" + }, + { + "bbox": [ + 127, + 94, + 154, + 104 + ], + "score": 0.85, + "content": "2 4 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "(Table 2). 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This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "happens because DW always removes a patch from the prediction. When an adversarial patch is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "detected, it is likely to be removed, enabling correct prediction. On the other hand, when there are no", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 484, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 484, + 266 + ], + "score": 1.0, + "content": "adversarial patches, DW removes actual salient information, resulting in lower clean accuracy.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "Here we did not compare our results with adversarial training, because even though it produces some", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "of the most adversarially robust models, it does not offer any guarantees on the empirical robust", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "accuracy, and could still be decreased further with stronger attacks. 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Empirical adversar-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "ial accuracy is calculated for 400 random images in both datasets. All results are averaged over three", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 415, + 176, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 176, + 426 + ], + "score": 1.0, + "content": "different models.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "table_body", + "bbox": [ + 121, + 426, + 490, + 660 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 426, + 490, + 660 + ], + "spans": [ + { + "bbox": [ + 121, + 426, + 490, + 660 + ], + "score": 0.986, + "html": "
DatasetPatch SizeAdversaryDefenseClean AccuracyAccuracyEmpirical Certified Adversarial Accuracy
MNIST2×2IFGSMNone98.4%80.1%
2×2IFGSMLGS97.4%90.0%
2×2IFGSM+LGSLGS97.4%60.7%
2×2IFGSMIBP98.5%93.9%91.6%
5×5IFGSMNone98.5%3.3%
CIFAR5×5IFGSMIBP92.9%66.1%62.0%
2×2IFGSMNone66.3%25.4%1
2×2IFGSMLGS64.9%31.3%
2×2IFGSM+LGSLGS64.9%24.2%
2×2IFGSMDW47.1%43.3%
2×2IFGSM+DWDW47.1%20.2%=
2×2IFGSMIBP48.6%45.2%41.6%
5×5IFGSMNone66.5%0.4%
5×5IFGSMLGS51.2% 51.2%22.11%
5×5IFGSM +LGSLGS DW45.3%0.5% 59.3%
5×5IFGSM
5×5IFGSM+DWDW45.3%15.6%
5×5IFGSMIBP33.9%29.1%24.9%
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Finally, guided-patch certificate training consistently outperforms random-patch", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 505, + 139 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "When evaluating existing defenses, we only report cases where non-trivial adversarial accuracy is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "achieved against a weaker adversary. We do not explore cases where LGS and DW perform so poorly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "that no meaningful comparison can be done. LGS and DW are highly dependent on hyperparameters", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "to work effectively against naive attacks, and yet neither Naseer et al. (2019) nor Hayes (2018)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "proposed a way to learn these hyperparameters. By trial and error, we were able to increase the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "adversarial accuracy against a weaker adversary for some settings, but not all. In addition, we also", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 207, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 223 + ], + "score": 1.0, + "content": "notice a peculiar feature of DW: when we increase the adversarial accuracy, the clean accuracy", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "degrades, sometimes so much that it is even lower than the empirical adversarial accuracy. This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "happens because DW always removes a patch from the prediction. When an adversarial patch is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "detected, it is likely to be removed, enabling correct prediction. On the other hand, when there are no", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 484, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 484, + 266 + ], + "score": 1.0, + "content": "adversarial patches, DW removes actual salient information, resulting in lower clean accuracy.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 144, + 506, + 266 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "Here we did not compare our results with adversarial training, because even though it produces some", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "of the most adversarially robust models, it does not offer any guarantees on the empirical robust", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "accuracy, and could still be decreased further with stronger attacks. For example, Wang et al. (2019)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 280, + 315 + ], + "score": 1.0, + "content": "proposed a stronger attack that could find", + "type": "text" + }, + { + "bbox": [ + 281, + 303, + 301, + 313 + ], + "score": 0.87, + "content": "47 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "more adversarial examples compared to gradient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "based method. Further, adversarial training on all possible patches would be even more expensive", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 433, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 433, + 338 + ], + "score": 1.0, + "content": "compared to certificate training, and is slightly beyond our computational budget.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 270, + 506, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 507, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 356, + 354 + ], + "score": 1.0, + "content": "Compared to state-of-the-art certified models for CIFAR with", + "type": "text" + }, + { + "bbox": [ + 357, + 342, + 372, + 353 + ], + "score": 0.89, + "content": "L _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 341, + 507, + 354 + ], + "score": 1.0, + "content": "-perturbation, where Zhang et al.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 399, + 366 + ], + "score": 1.0, + "content": "(2019a) proposed a deterministic algorithm that achieves clean accuracy of", + "type": "text" + }, + { + "bbox": [ + 399, + 352, + 426, + 363 + ], + "score": 0.89, + "content": "3 4 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 351, + 506, + 366 + ], + "score": 1.0, + "content": ", our clean accuracy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 363, + 452, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 216, + 375 + ], + "score": 1.0, + "content": "for our most robust CIFAR", + "type": "text" + }, + { + "bbox": [ + 217, + 363, + 240, + 374 + ], + "score": 0.88, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 363, + 277, + 375 + ], + "score": 1.0, + "content": "model is", + "type": "text" + }, + { + "bbox": [ + 278, + 363, + 305, + 374 + ], + "score": 0.88, + "content": "4 7 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 363, + 452, + 375 + ], + "score": 1.0, + "content": "when using a large model (Table 2).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 341, + 507, + 375 + ] + }, + { + "type": "table", + "bbox": [ + 121, + 426, + 490, + 660 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 393, + 505, + 426 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 392, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 507, + 406 + ], + "score": 1.0, + "content": "Table 2: Comparison of our IBP certified patch defense against existing defenses. Empirical adversar-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "ial accuracy is calculated for 400 random images in both datasets. All results are averaged over three", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 415, + 176, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 176, + 426 + ], + "score": 1.0, + "content": "different models.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "table_body", + "bbox": [ + 121, + 426, + 490, + 660 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 426, + 490, + 660 + ], + "spans": [ + { + "bbox": [ + 121, + 426, + 490, + 660 + ], + "score": 0.986, + "html": "
DatasetPatch SizeAdversaryDefenseClean AccuracyAccuracyEmpirical Certified Adversarial Accuracy
MNIST2×2IFGSMNone98.4%80.1%
2×2IFGSMLGS97.4%90.0%
2×2IFGSM+LGSLGS97.4%60.7%
2×2IFGSMIBP98.5%93.9%91.6%
5×5IFGSMNone98.5%3.3%
CIFAR5×5IFGSMIBP92.9%66.1%62.0%
2×2IFGSMNone66.3%25.4%1
2×2IFGSMLGS64.9%31.3%
2×2IFGSM+LGSLGS64.9%24.2%
2×2IFGSMDW47.1%43.3%
2×2IFGSM+DWDW47.1%20.2%=
2×2IFGSMIBP48.6%45.2%41.6%
5×5IFGSMNone66.5%0.4%
5×5IFGSMLGS51.2% 51.2%22.11%
5×5IFGSM +LGSLGS DW45.3%0.5% 59.3%
5×5IFGSM
5×5IFGSM+DWDW45.3%15.6%
5×5IFGSMIBP33.9%29.1%24.9%
", + "type": "table", + "image_path": "c05cfeed2d4e9f23cc5ab7a5b77dd639884d8719c99778848ff0b07130af40b8.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 121, + 426, + 490, + 504.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 121, + 504.0, + 490, + 582.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 121, + 582.0, + 490, + 660.0 + ], + "spans": [], + "index": 30 + } + ] + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 678, + 302, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 303, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 303, + 691 + ], + "score": 1.0, + "content": "5.2 COMPARISON OF TRAINING STRATEGIES", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 507, + 713 + ], + "score": 1.0, + "content": "We find that given a fixed architecture all-patch certificate training achieves the best certified accuracy.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "However, given a fixed computational budget, random and guided training significantly outperform", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "all-patch training. Finally, guided-patch certificate training consistently outperforms random-patch", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "certificate training by a slim margin, indicating that the U-net is learning how to predict the minimum", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 152, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 152, + 108 + ], + "score": 1.0, + "content": "margin m.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 696, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "certificate training by a slim margin, indicating that the U-net is learning how to predict the minimum", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 152, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 152, + 108 + ], + "score": 1.0, + "content": "margin m.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "In Table 3, we see that given a fixed architecture all-patch certificate training significantly outperforms", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "both random-patch certificate training and guided-patch certificate training in terms of certified", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 411, + 145 + ], + "score": 1.0, + "content": "accuracy, outperforming the second best certified defenses in each task by", + "type": "text" + }, + { + "bbox": [ + 411, + 132, + 434, + 143 + ], + "score": 0.84, + "content": "2 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 132, + 474, + 145 + ], + "score": 1.0, + "content": "(MNIST,", + "type": "text" + }, + { + "bbox": [ + 474, + 132, + 500, + 143 + ], + "score": 0.87, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 129, + 154 + ], + "score": 0.84, + "content": "7 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 142, + 168, + 156 + ], + "score": 1.0, + "content": "(MNIST,", + "type": "text" + }, + { + "bbox": [ + 168, + 144, + 193, + 154 + ], + "score": 0.87, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 142, + 199, + 156 + ], + "score": 1.0, + "content": "),", + "type": "text" + }, + { + "bbox": [ + 199, + 144, + 222, + 154 + ], + "score": 0.84, + "content": "3 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 142, + 273, + 156 + ], + "score": 1.0, + "content": "(CIFAR-10,", + "type": "text" + }, + { + "bbox": [ + 273, + 144, + 299, + 155 + ], + "score": 0.67, + "content": "2 \\times 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 142, + 321, + 156 + ], + "score": 1.0, + "content": "), and", + "type": "text" + }, + { + "bbox": [ + 321, + 143, + 344, + 154 + ], + "score": 0.86, + "content": "3 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 142, + 394, + 156 + ], + "score": 1.0, + "content": "(CIFAR-10,", + "type": "text" + }, + { + "bbox": [ + 395, + 144, + 420, + 154 + ], + "score": 0.73, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "). However, all-patch", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "certificate training is very expensive, taking on average 4 to 15 times longer than guided-patch", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 461, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 461, + 180 + ], + "score": 1.0, + "content": "certificate training and over 30 to 70 times longer than random-patch certificate training.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "score": 1.0, + "content": "On the other hand, given a limited computational budget, random-patch and guided-patch training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "significantly outperforms all-patch training. Due to the efficiency of random-patch and guided-patch", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "training, they scale much better to large architectures. 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Note that we are unable to all-patch train the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "same architecture as it will take almost 15 days to complete, and is out of our computational budget.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "Guided-patch certificate training is slightly more expensive compared to random patch, due to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "overhead from the U-net architecture. 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The numbers", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 104, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "next to training strategies indicate the number of patches used for estimating the lower bound during", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "training. Most training times are measured on a single 2080Ti GPU, with the exception of all-patch", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "training which is run on four 2080Ti GPUs. 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2×25×5
DatasetTraining StrategyClean AccuracyCertified AccuracyTraining Time(h)Clean AccuracyCertified AccuracyTraining Time(h)
MNISTAllPatch98.5%91.5%9.392.0%60.4%8.4
Random(1)98.5%82.9%0.296.9%24.1%0.4
Random(5)98.6%86.6%0.395.8%42.1%0.3
Random(10)98.6%87.7%0.395.6%49.6%0.3
CIFARGuided(10) All Patch98.6% 50.9%88.9% 39.9%2.2 56.495.0% 33.5%53.1% 22.0%2.6 45.8
Random(1)53.6%21.6%0.643.6%6.1%0.6
Random(5)52.9%32.3%0.739.0%14.6%0.7
Random(10)51.9%35.6%0.838.8%18.6%
Guided(10)52.4%36.0%3.737.9%18.8%0.8 3.7
Large Model*65.8%51.9%22.447.8%30.3%15.4
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Its", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 338, + 618 + ], + "score": 1.0, + "content": "performance is reasonable compared to patch defense (e.g.", + "type": "text" + }, + { + "bbox": [ + 338, + 605, + 365, + 615 + ], + "score": 0.86, + "content": "9 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 605, + 450, + 618 + ], + "score": 1.0, + "content": "certified accuracy for", + "type": "text" + }, + { + "bbox": [ + 450, + 605, + 470, + 615 + ], + "score": 0.87, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "patch vs", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 134, + 627 + ], + "score": 0.87, + "content": "9 0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 615, + 148, + 628 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 149, + 617, + 166, + 627 + ], + "score": 0.8, + "content": "\\mathrm { k } { = } 4", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "), even though the sparse attack model is much stronger. For convolutional networks,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 365, + 639 + ], + "score": 1.0, + "content": "we increase the size of the first convolutional layer (i.e. from", + "type": "text" + }, + { + "bbox": [ + 366, + 627, + 391, + 637 + ], + "score": 0.9, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 627, + 403, + 639 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 404, + 627, + 439, + 638 + ], + "score": 0.9, + "content": "1 1 \\times 1 1", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 627, + 505, + 639 + ], + "score": 1.0, + "content": ") so the interval", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "bounds calculated are tighter. 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Detailed results are shown in the Appendix A.4 Table 7.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "score": 1.0, + "content": "Table 4 compares our approach with the state-of-the-art certified sparse defense (Random Abla-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "tion) Levine & Feizi (2019). We use their best model with the largest medium radii to certify against", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "various levels of sparsity. 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2×25×5
DatasetTraining StrategyClean AccuracyCertified AccuracyTraining Time(h)Clean AccuracyCertified AccuracyTraining Time(h)
MNISTAllPatch98.5%91.5%9.392.0%60.4%8.4
Random(1)98.5%82.9%0.296.9%24.1%0.4
Random(5)98.6%86.6%0.395.8%42.1%0.3
Random(10)98.6%87.7%0.395.6%49.6%0.3
CIFARGuided(10) All Patch98.6% 50.9%88.9% 39.9%2.2 56.495.0% 33.5%53.1% 22.0%2.6 45.8
Random(1)53.6%21.6%0.643.6%6.1%0.6
Random(5)52.9%32.3%0.739.0%14.6%0.7
Random(10)51.9%35.6%0.838.8%18.6%
Guided(10)52.4%36.0%3.737.9%18.8%0.8 3.7
Large Model*65.8%51.9%22.447.8%30.3%15.4
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Its", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 338, + 618 + ], + "score": 1.0, + "content": "performance is reasonable compared to patch defense (e.g.", + "type": "text" + }, + { + "bbox": [ + 338, + 605, + 365, + 615 + ], + "score": 0.86, + "content": "9 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 605, + 450, + 618 + ], + "score": 1.0, + "content": "certified accuracy for", + "type": "text" + }, + { + "bbox": [ + 450, + 605, + 470, + 615 + ], + "score": 0.87, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "patch vs", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 134, + 627 + ], + "score": 0.87, + "content": "9 0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 615, + 148, + 628 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 149, + 617, + 166, + 627 + ], + "score": 0.8, + "content": "\\mathrm { k } { = } 4", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "), even though the sparse attack model is much stronger. For convolutional networks,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 365, + 639 + ], + "score": 1.0, + "content": "we increase the size of the first convolutional layer (i.e. from", + "type": "text" + }, + { + "bbox": [ + 366, + 627, + 391, + 637 + ], + "score": 0.9, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 627, + 403, + 639 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 404, + 627, + 439, + 638 + ], + "score": 0.9, + "content": "1 1 \\times 1 1", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 627, + 505, + 639 + ], + "score": 1.0, + "content": ") so the interval", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "bounds calculated are tighter. However, despite the change, fully-connected network still performs", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 348, + 661 + ], + "score": 1.0, + "content": "much better. For example, the certified accuracy drops from", + "type": "text" + }, + { + "bbox": [ + 349, + 649, + 376, + 659 + ], + "score": 0.88, + "content": "2 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 649, + 388, + 661 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 388, + 649, + 415, + 659 + ], + "score": 0.88, + "content": "1 3 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "when we switch from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 391, + 673 + ], + "score": 1.0, + "content": "fully-connected to convolutional network for CIFAR10 and drops from", + "type": "text" + }, + { + "bbox": [ + 391, + 660, + 418, + 671 + ], + "score": 0.88, + "content": "9 0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 659, + 429, + 673 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 430, + 660, + 457, + 671 + ], + "score": 0.86, + "content": "7 5 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "for MNIST", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 386, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 386, + 684 + ], + "score": 1.0, + "content": "respectively. Detailed results are shown in the Appendix A.4 Table 7.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 593, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "score": 1.0, + "content": "Table 4 compares our approach with the state-of-the-art certified sparse defense (Random Abla-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "tion) Levine & Feizi (2019). We use their best model with the largest medium radii to certify against", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "various levels of sparsity. As shown in the table, our method achieves higher certified accuracy on", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "the MNIST dataset over all the sparse radii, but lower on CIFAR-10. It is worth noting that we are", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "score": 1.0, + "content": "using a much smaller and simpler model (a fully-connected network) compared to Random Ablation,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 200, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 200, + 105 + ], + "score": 1.0, + "content": "which uses ResNet-50.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 687, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "score": 1.0, + "content": "using a much smaller and simpler model (a fully-connected network) compared to Random Ablation,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 200, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 200, + 105 + ], + "score": 1.0, + "content": "which uses ResNet-50.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table", + "bbox": [ + 131, + 133, + 481, + 280 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 144, + 120, + 465, + 132 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 119, + 466, + 133 + ], + "spans": [ + { + "bbox": [ + 145, + 119, + 466, + 133 + ], + "score": 1.0, + "content": "Table 4: Certified accuracy for sparse defenses with IBP and Random Ablation.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "table_body", + "bbox": [ + 131, + 133, + 481, + 280 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 131, + 133, + 481, + 280 + ], + "spans": [ + { + "bbox": [ + 131, + 133, + 481, + 280 + ], + "score": 0.985, + "html": "
DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1IBP-sparse98.4%96.0%
4IBP-sparse97.8%90.8%
10IBP-sparse95.2%86.8%
1Random Ablation96.7%90.3%
4Random Ablation96.7%79.1%
10Random Ablation96.7%29.2%
CIFAR1IBP-sparse48.4%40.0%
4IBP-sparse42.2%31.2%
10IBP-sparse37.0%25.6%
1Random Ablation78.3%68.6%
4Random Ablation78.3%61.3%
10Random Ablation78.3%45.0%
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In all these experiments, we evaluate the certified", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "accuracy for our largest model, on both MNIST and CIFAR datasets. We evaluate the transferability", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 380, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 392 + ], + "score": 1.0, + "content": "to various shapes including rectangle, line, parallelogram, and diamond. With the exception of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "rectangles, all the shapes have the exact same pixel count as the patches used for training. For", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 400, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 506, + 415 + ], + "score": 1.0, + "content": "rectangles, we use multiple choices of width and length, obtaining some combinations with slightly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "more pixels, and the worst accuracy is reported in Table 5. The exact shapes used can be found in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 423, + 168, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 168, + 435 + ], + "score": 1.0, + "content": "Appendix A.2.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "The certified accuracy of our models generalize surprisingly well to other shapes, losing no more", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 146, + 464 + ], + "score": 1.0, + "content": "than than", + "type": "text" + }, + { + "bbox": [ + 146, + 451, + 161, + 462 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 450, + 339, + 464 + ], + "score": 1.0, + "content": "in most cases for MNIST and no more than", + "type": "text" + }, + { + "bbox": [ + 340, + 452, + 355, + 462 + ], + "score": 0.86, + "content": "6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "for CIFAR-10 (Table 5). The largest", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 463, + 504, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 504, + 474 + ], + "score": 1.0, + "content": "degradation of accuracy happens for rectangles and lines, and it is mostly because the rectangle", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 474, + 504, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 504, + 485 + ], + "score": 1.0, + "content": "considered has more pixels compared to the square, and the line has less overlaps. However, it is still", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "interesting that the certificate even generalizes to a straight line, even though the model was never", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "trained to be robust to lines. In the case of MNIST with small patch size, the certified accuracy even", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 506, + 249, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 249, + 518 + ], + "score": 1.0, + "content": "improves when transferred to lines.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "table", + "bbox": [ + 129, + 545, + 479, + 626 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 146, + 532, + 462, + 544 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 148, + 531, + 463, + 546 + ], + "spans": [ + { + "bbox": [ + 148, + 531, + 463, + 546 + ], + "score": 1.0, + "content": "Table 5: Certified accuracy for square-patch trained model for different shapes", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 129, + 545, + 479, + 626 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 129, + 545, + 479, + 626 + ], + "spans": [ + { + "bbox": [ + 129, + 545, + 479, + 626 + ], + "score": 0.984, + "html": "
DatasetPixel CountSquareRectangleLineDiamondParallelogram
MNIST491.6%=92.5%91.6%92.3%
CIFAR1669.4%55.4%46.7%68.13%70.2%
2559.7%50.9%32.4%53.6%55.2%
450.8%=46.1%48.6%49.8%
1636.9%29.0%32.1%35.7%36.3%
2530.3%25.1%29.0%30.1%30.7%
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DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1IBP-sparse98.4%96.0%
4IBP-sparse97.8%90.8%
10IBP-sparse95.2%86.8%
1Random Ablation96.7%90.3%
4Random Ablation96.7%79.1%
10Random Ablation96.7%29.2%
CIFAR1IBP-sparse48.4%40.0%
4IBP-sparse42.2%31.2%
10IBP-sparse37.0%25.6%
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The exact shapes used can be found in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 423, + 168, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 168, + 435 + ], + "score": 1.0, + "content": "Appendix A.2.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 346, + 506, + 435 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "The certified accuracy of our models generalize surprisingly well to other shapes, losing no more", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 146, + 464 + ], + "score": 1.0, + "content": "than than", + "type": "text" + }, + { + "bbox": [ + 146, + 451, + 161, + 462 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 450, + 339, + 464 + ], + "score": 1.0, + "content": "in most cases for MNIST and no more than", + "type": "text" + }, + { + "bbox": [ + 340, + 452, + 355, + 462 + ], + "score": 0.86, + "content": "6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "for CIFAR-10 (Table 5). The largest", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 463, + 504, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 504, + 474 + ], + "score": 1.0, + "content": "degradation of accuracy happens for rectangles and lines, and it is mostly because the rectangle", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 474, + 504, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 504, + 485 + ], + "score": 1.0, + "content": "considered has more pixels compared to the square, and the line has less overlaps. However, it is still", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "interesting that the certificate even generalizes to a straight line, even though the model was never", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "trained to be robust to lines. 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We also adopt a warm-up schedule in all experiments like (Zhang et al., 2019b), where the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "input interval bounds start at zero and grow to [-1,1] after 61/121 epochs for MNIST/CIFAR10", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 104, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "respectively. We train the models for a total of 100/200 epochs for MNIST/CIFAR10, where in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "score": 1.0, + "content": "first 61/121 epochs the learning rate is fixed and in the following epochs, we reduce the learning rate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 258, + 223, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 223, + 271 + ], + "score": 1.0, + "content": "by one half every 10 epochs.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 126, + 506, + 271 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 504, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 504, + 286 + ], + "score": 1.0, + "content": "In addition, following (Gowal et al., 2018), we further evaluate the CIFAR10 on a larger model which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "has 5 convolutional layers with kernel size 3 and a fully connected layer with 512 neurons. This", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 349, + 309 + ], + "score": 1.0, + "content": "deeper and wider model achieves the clean accuracy around", + "type": "text" + }, + { + "bbox": [ + 349, + 297, + 369, + 308 + ], + "score": 0.88, + "content": "8 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 298, + 506, + 309 + ], + "score": 1.0, + "content": ", and has 17M parameters in total.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 449, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 449, + 320 + ], + "score": 1.0, + "content": "Table 8 in Appendix A.5 describes the full certified patch results for this large model.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 275, + 506, + 320 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 334, + 365, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 365, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 365, + 345 + ], + "score": 1.0, + "content": "A.2 SAMPLE SHAPES FOR GENERALIZATION EXPERIMENTS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 504, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 367 + ], + "score": 1.0, + "content": "We demonstrate generalization to other patch shapes that were not considered in training, obtaining", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 365, + 477, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 477, + 376 + ], + "score": 1.0, + "content": "surprisingly good transfer in robust accuracy; see the figure below and the results in Table 5.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 352, + 506, + 376 + ] + }, + { + "type": "image", + "bbox": [ + 189, + 394, + 422, + 614 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 189, + 394, + 422, + 614 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 189, + 394, + 422, + 614 + ], + "spans": [ + { + "bbox": [ + 189, + 394, + 422, + 614 + ], + "score": 0.969, + "type": "image", + "image_path": "9f216bf62fce941917ea81757e0b262fd04bf16bd11424b98f95987ef90455cd.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 394, + 422, + 407.75 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 189, + 407.75, + 422, + 421.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 189, + 421.5, + 422, + 435.25 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 189, + 435.25, + 422, + 449.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 189, + 449.0, + 422, + 462.75 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 189, + 462.75, + 422, + 476.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 189, + 476.5, + 422, + 490.25 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 189, + 490.25, + 422, + 504.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 189, + 504.0, + 422, + 517.75 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 189, + 517.75, + 422, + 531.5 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 189, + 531.5, + 422, + 545.25 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 189, + 545.25, + 422, + 559.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 189, + 559.0, + 422, + 572.75 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 189, + 572.75, + 422, + 586.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 189, + 586.5, + 422, + 600.25 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 189, + 600.25, + 422, + 614.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 637, + 505, + 660 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 636, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 651 + ], + "score": 1.0, + "content": "Figure 1: Examples of shapes with pixels number 4 and 25. From left to right are square, parallelo-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 648, + 301, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 301, + 661 + ], + "score": 1.0, + "content": "gram, diamond and rectangle (line) respectively.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + } + ], + "index": 34.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 207, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 208, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 208, + 96 + ], + "score": 1.0, + "content": "A.3 BOUND POOLING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "score": 1.0, + "content": "Besides random-patch certificate training and guided-patch certificate training, we also experimented", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "with the idea of bound pooling. All-patch training is very expensive as bounds generated by each", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "potential patch has to be forward passed through the complete network. Bound pooling partially", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 136, + 504, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 504, + 148 + ], + "score": 1.0, + "content": "relieves the problem be pooling the interval bounds in intermediate layers thus reducing the forward", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 211, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 211, + 159 + ], + "score": 1.0, + "content": "pass in subsequent layers.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 242, + 178 + ], + "score": 1.0, + "content": "Specifically, given a set of patches", + "type": "text" + }, + { + "bbox": [ + 242, + 165, + 249, + 175 + ], + "score": 0.74, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 163, + 403, + 178 + ], + "score": 1.0, + "content": ", the interval bounds in the ith layer are", + "type": "text" + }, + { + "bbox": [ + 403, + 164, + 505, + 177 + ], + "score": 0.93, + "content": "\\bar { Z } ^ { ( i ) } ( \\mathbb { P } ) = \\{ \\bar { z } ^ { ( i ) } ( p ) : p \\in", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 175, + 504, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 119, + 190 + ], + "score": 0.84, + "content": "\\mathbb { P } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 175, + 137, + 192 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 137, + 176, + 244, + 190 + ], + "score": 0.92, + "content": "\\underline { { Z } } ^ { ( i ) } \\mathbb { P } = \\{ \\underline { { z } } ^ { ( i ) } ( p ) : p \\in \\mathbb { P } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 175, + 496, + 192 + ], + "score": 1.0, + "content": ". 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Depending on how", + "type": "text" + }, + { + "bbox": [ + 364, + 208, + 372, + 217 + ], + "score": 0.76, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 205, + 506, + 222 + ], + "score": 1.0, + "content": "is partitioned, the bound pooling", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 458, + 232 + ], + "score": 1.0, + "content": "would work differently. 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The trade off becomes greater when the model becomes larger. Overall, bound", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "pooling is still quite expensive, and cannot scale to larger models like random-patch or guided-patch", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 142, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 142, + 339 + ], + "score": 1.0, + "content": "training.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "table", + "bbox": [ + 106, + 412, + 507, + 614 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 355, + 506, + 411 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "Table 6: Comparing bound pooling with the guided-patch and random-patch training. Pool 4 means", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 173, + 379 + ], + "score": 1.0, + "content": "that the adjacent", + "type": "text" + }, + { + "bbox": [ + 173, + 367, + 197, + 376 + ], + "score": 0.89, + "content": "4 \\times 4", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "patches (16 patches) are pooled together in the first layer. 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2×25×5
Dataset TrainingStrategyClean Certified TrainingClean Certified Training
Accuracy Accuracy Time(h)Accuracy Accuracy Time(h)
MNIST All Patch98.5% 91.6% 20.190.0% 59.7% 16.3
Pool 2Pool 4Random(1)Random(5)Random(10)Guided(10)98.0% 91.1% 15.885.2% 54.2% 11.6
97.2% 89.9% 13.270.4% 38.3% 10.2
98.5% 81.9% 0.396.8% 24.8% 0.4
98.6% 86.5% 0.394.9% 42.0% 0.5
98.6% 87.5% 0.594.7% 50.4% 0.6
98.7% 88.9% 2.294.0% 53.2% 3.4
CIFAR All Patch49.6% 41.6% 22.534.0% 25.0% 18.6
Pool 2Pool448.1% 39.4% 17.332.4% 24.2% 14.5
Pool4Pool 2-244.9% 37.1% 16.328.3% 20.6% 13.6
45.0% 37.4% 16.525.3% 19.1% 13.8
Random(1)53.2% 32.4% 0.642.7% 11.0% 0.6
Random(5)Random(10)52.2% 39.5% 0.937.8% 19.6% 0.9
50.8% 38.6% 1.038.4% 21.9% 1.0
Guided(10)53.0% 39.8% 4.036.1% 23.0% 3.9
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In our experiments, we always select adjacent patches for each", + "type": "text" + }, + { + "bbox": [ + 458, + 220, + 469, + 231 + ], + "score": 0.87, + "content": "\\mathbb { S } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "with the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "assumption that adjacent patches tend to generate similar bounds thus resulting in tighter certificate.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 101, + 163, + 506, + 243 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 247, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "score": 1.0, + "content": "Bound pooling, similar to random- and guided- patch training, trades performance for efficiency", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "compared to all-patch certificate training. 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2×25×5
Dataset TrainingStrategyClean Certified TrainingClean Certified Training
Accuracy Accuracy Time(h)Accuracy Accuracy Time(h)
MNIST All Patch98.5% 91.6% 20.190.0% 59.7% 16.3
Pool 2Pool 4Random(1)Random(5)Random(10)Guided(10)98.0% 91.1% 15.885.2% 54.2% 11.6
97.2% 89.9% 13.270.4% 38.3% 10.2
98.5% 81.9% 0.396.8% 24.8% 0.4
98.6% 86.5% 0.394.9% 42.0% 0.5
98.6% 87.5% 0.594.7% 50.4% 0.6
98.7% 88.9% 2.294.0% 53.2% 3.4
CIFAR All Patch49.6% 41.6% 22.534.0% 25.0% 18.6
Pool 2Pool448.1% 39.4% 17.332.4% 24.2% 14.5
Pool4Pool 2-244.9% 37.1% 16.328.3% 20.6% 13.6
45.0% 37.4% 16.525.3% 19.1% 13.8
Random(1)53.2% 32.4% 0.642.7% 11.0% 0.6
Random(5)Random(10)52.2% 39.5% 0.937.8% 19.6% 0.9
50.8% 38.6% 1.038.4% 21.9% 1.0
Guided(10)53.0% 39.8% 4.036.1% 23.0% 3.9
", + "type": "table", + "image_path": "6cedc241deed68c58913df8996a321dcb92824b0a654c9ed4cb84781332740b1.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 106, + 412, + 507, + 479.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 106, + 479.3333333333333, + 507, + 546.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 106, + 546.6666666666666, + 507, + 614.0 + ], + "spans": [], + "index": 27 + } + ] + } + ], + "index": 24.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 275, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 276, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 276, + 95 + ], + "score": 1.0, + "content": "A.4 MULTI-PATCH SPARSE TRAINING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 482, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 483, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 483, + 116 + ], + "score": 1.0, + "content": "Here we list the detailed certified accuracy for various sparsity levels and model architectures.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "table", + "bbox": [ + 142, + 167, + 468, + 335 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 133, + 505, + 166 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 133, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 382, + 147 + ], + "score": 1.0, + "content": "Table 7: Certified accuracy for sparse defenses with varying sparsity", + "type": "text" + }, + { + "bbox": [ + 383, + 135, + 389, + 144 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 133, + 505, + 147 + ], + "score": 1.0, + "content": "and models on both MNIST", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 146, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 223, + 156 + ], + "score": 1.0, + "content": "and CIFAR10, where “Conv", + "type": "text" + }, + { + "bbox": [ + 223, + 146, + 250, + 156 + ], + "score": 0.88, + "content": "c \\times c ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 146, + 505, + 156 + ], + "score": 1.0, + "content": "represents for the convolutional network with first layer kernel", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 156, + 135, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 135, + 168 + ], + "score": 1.0, + "content": "size c.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "table_body", + "bbox": [ + 142, + 167, + 468, + 335 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 142, + 167, + 468, + 335 + ], + "spans": [ + { + "bbox": [ + 142, + 167, + 468, + 335 + ], + "score": 0.983, + "html": "
DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1mlp98.4%96.0%
4mlp97.8%90.8%
10mlp95.2%86.8%
1Conv3x397.0%88.3%
CIFAR4Conv3x392.7%75.9%
1mlp48.4%40.0%
4mlp42.2%31.2%
10mlp37.0%25.6%
1Conv11x1134.8%27.4%
4Conv11x1125.1%18.3%
10Conv11x1117.2%13.8%
1Conv13x1338.6%29.7%
4Conv13x1328.1%19.6%
10Conv13x1322.4%15.3%
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DatasetPatch SizeTraining StrategyModelClean AccuracyCertified AccuracyTraining Time(h)
CIFAR2×2All Patchmlp50.8%35.5%9.1
2 layer conv52.4%42.6%10.7
4 layer conv49.6%41.6%22.5
5 layer conv (wide)1-~360.0
Random(10) Random(20)5 layer conv (wide) 5 layer conv (wide)64.7% 64.4%49.0%9.5
CIFAR 5×5Guided(10)5 layer conv (wide)66.5%50.8% 49.2%15.8 12.2
Guided(20) All Patch5 layer conv (wide)65.8%51.9%22.4
mlp31.1%18.8%7.1
2 layer conv35.5%22.3%8.7
4 layer conv.34.0%25.0%18.6
5 layer conv (wide)1、~360.0
Random(10) Random(20)5 layer conv (wide)48.6%29.9%9.4
5 layer conv (wide)47.8%30.3%15.4
Guided(10)5 layer conv (wide)48.4%29.0%12.4
Guided(20)5 layer conv (wide)47.6%29.6%23.8
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DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1mlp98.4%96.0%
4mlp97.8%90.8%
10mlp95.2%86.8%
1Conv3x397.0%88.3%
CIFAR4Conv3x392.7%75.9%
1mlp48.4%40.0%
4mlp42.2%31.2%
10mlp37.0%25.6%
1Conv11x1134.8%27.4%
4Conv11x1125.1%18.3%
10Conv11x1117.2%13.8%
1Conv13x1338.6%29.7%
4Conv13x1328.1%19.6%
10Conv13x1322.4%15.3%
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DatasetPatch SizeTraining StrategyModelClean AccuracyCertified AccuracyTraining Time(h)
CIFAR2×2All Patchmlp50.8%35.5%9.1
2 layer conv52.4%42.6%10.7
4 layer conv49.6%41.6%22.5
5 layer conv (wide)1-~360.0
Random(10) Random(20)5 layer conv (wide) 5 layer conv (wide)64.7% 64.4%49.0%9.5
CIFAR 5×5Guided(10)5 layer conv (wide)66.5%50.8% 49.2%15.8 12.2
Guided(20) All Patch5 layer conv (wide)65.8%51.9%22.4
mlp31.1%18.8%7.1
2 layer conv35.5%22.3%8.7
4 layer conv.34.0%25.0%18.6
5 layer conv (wide)1、~360.0
Random(10) Random(20)5 layer conv (wide)48.6%29.9%9.4
5 layer conv (wide)47.8%30.3%15.4
Guided(10)5 layer conv (wide)48.4%29.0%12.4
Guided(20)5 layer conv (wide)47.6%29.6%23.8
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DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution98.63/%91.38%21.0
Random (1)4 layer convolution98.48%91.63%80.3
fully connected (255,10)98.46%91.47%9.8
2 layer convolution98.69%82.57%0.2
4 layer convolution98.45%81.87%0.3
Random (5)fully connected (255,10)98.48%84.32%0.2
2 layer convolution98.75%85.87%0.3
4 layer convolution98.57%86.50%0.3
Random (10)fully connected (255,10)98.62%87.32%0.2
2 layer convolution98.73%87.31%0.3
4 layer convolution98.63%87.54%0.5
Guided (10)fully connected (255,10)98.49%88.13%0.2
2 layer convolution98.60%88.49%2.3
4 layer convolution98.70%88.85%2.2
CIFAR All Patchfully connected (255,10)98.63%89.44%2.2
2 layer convolution52.42%42.57%42.6
4 layer convolution49.58%41.57%89.8
Random (1)fully connected (255,10)50.83%35.49%36.6
2 layer convolution54.93%29.13%0.6
4 layer convolution53.22%32.35%0.6
Random (5)fully connected (255,10)52.76%03.21%0.5
2 layer convolution54.15%37.30%0.6
4 layer convolution52.19%39.45%0.9
Random (10)fully connected (255,10)52.38%20.17%0.6
2 layer convolution53.08%39.32%0.7
4 layer convolution50.80%38.57%1.0
Guided (10)fully connected (255,10)51.90%28.97%0.6
2 layer convolution53.04%38.81%3.7
4 layer convolution52.97%39.84%4.0
fully connected (255,10)51.32%29.44%3.6
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DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution98.63/%91.38%21.0
Random (1)4 layer convolution98.48%91.63%80.3
fully connected (255,10)98.46%91.47%9.8
2 layer convolution98.69%82.57%0.2
4 layer convolution98.45%81.87%0.3
Random (5)fully connected (255,10)98.48%84.32%0.2
2 layer convolution98.75%85.87%0.3
4 layer convolution98.57%86.50%0.3
Random (10)fully connected (255,10)98.62%87.32%0.2
2 layer convolution98.73%87.31%0.3
4 layer convolution98.63%87.54%0.5
Guided (10)fully connected (255,10)98.49%88.13%0.2
2 layer convolution98.60%88.49%2.3
4 layer convolution98.70%88.85%2.2
CIFAR All Patchfully connected (255,10)98.63%89.44%2.2
2 layer convolution52.42%42.57%42.6
4 layer convolution49.58%41.57%89.8
Random (1)fully connected (255,10)50.83%35.49%36.6
2 layer convolution54.93%29.13%0.6
4 layer convolution53.22%32.35%0.6
Random (5)fully connected (255,10)52.76%03.21%0.5
2 layer convolution54.15%37.30%0.6
4 layer convolution52.19%39.45%0.9
Random (10)fully connected (255,10)52.38%20.17%0.6
2 layer convolution53.08%39.32%0.7
4 layer convolution50.80%38.57%1.0
Guided (10)fully connected (255,10)51.90%28.97%0.6
2 layer convolution53.04%38.81%3.7
4 layer convolution52.97%39.84%4.0
fully connected (255,10)51.32%29.44%3.6
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DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution91.88%59.59%28.4
4 layer convolution90.03%59.72%65.2
Random (1)fully connected (255,10)93.96%61.97%7.2
2 layer convolution96.27%18.57%0.2
4 layer convolution96.83%24.79%0.4
Random (5)fully connected (255,10)97.60%29.04%0.2
2 layer convolution95.82%38.47%0.2
4 layer convolution94.85%42.02%0.5
Random (10)fully connected (255,10)96.73%45.89%0.2
2 layer convolution95.55%46.13%0.3
4 layer convolution fully connected (255,10)94.76% 96.40%50.43% 52.30%0.6
Guided (10)2 layer convolution95.28%50.28%0.2 2.3
93.98%53.17%3.4
4 layer convolution95.82%55.89%2.2
fully connected (255,10)35.48%
CIFAR All Patch Random (1)2 layer convolution33.95%22.31% 24.96%34.8
4 layer convolution fully connected (255,10)31.05%18.78%74.4 28.4
45.71%07.14%0.6
2 layer convolution42.65%10.99%
4 layer convolution42.34%0.6
fully connected (255,10)00.10%0.5
Random (5)2 layer convolution42.85%17.29%0.6
4 layer convolution37.80%19.63%0.9
fully connected (255,10)36.23%06.99%0.6
Random (10)2 layer convolution41.90%21.40%0.7
4 layer convolution38.41%21.90%1.0
fully connected (255,10) Guided (10)36.04%12.46%0.6
2 layer convolution42.08%20.77%3.6
4 layer convolution fully connected (255,10)36.08% 35.51%23.01% 12.56%3.9 3.5
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DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution91.88%59.59%28.4
4 layer convolution90.03%59.72%65.2
Random (1)fully connected (255,10)93.96%61.97%7.2
2 layer convolution96.27%18.57%0.2
4 layer convolution96.83%24.79%0.4
Random (5)fully connected (255,10)97.60%29.04%0.2
2 layer convolution95.82%38.47%0.2
4 layer convolution94.85%42.02%0.5
Random (10)fully connected (255,10)96.73%45.89%0.2
2 layer convolution95.55%46.13%0.3
4 layer convolution fully connected (255,10)94.76% 96.40%50.43% 52.30%0.6
Guided (10)2 layer convolution95.28%50.28%0.2 2.3
93.98%53.17%3.4
4 layer convolution95.82%55.89%2.2
fully connected (255,10)35.48%
CIFAR All Patch Random (1)2 layer convolution33.95%22.31% 24.96%34.8
4 layer convolution fully connected (255,10)31.05%18.78%74.4 28.4
45.71%07.14%0.6
2 layer convolution42.65%10.99%
4 layer convolution42.34%0.6
fully connected (255,10)00.10%0.5
Random (5)2 layer convolution42.85%17.29%0.6
4 layer convolution37.80%19.63%0.9
fully connected (255,10)36.23%06.99%0.6
Random (10)2 layer convolution41.90%21.40%0.7
4 layer convolution38.41%21.90%1.0
fully connected (255,10) Guided (10)36.04%12.46%0.6
2 layer convolution42.08%20.77%3.6
4 layer convolution fully connected (255,10)36.08% 35.51%23.01% 12.56%3.9 3.5
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Patch Size
AttackDefense42 × 4252 × 5260 ×60
IFGSMLGS78%75%71%
IFGSM+LGSLGS14%5%3%
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DatasetPatch SizeAdversaryDefenseClean AccuracyAccuracyEmpirical Certified Adversarial Accuracy
MNIST2×2IFGSMNone98.4%80.1%
2×2IFGSMLGS97.4%90.0%
2×2IFGSM+LGSLGS97.4%60.7%
2×2IFGSMIBP98.5%93.9%91.6%
5×5IFGSMNone98.5%3.3%
CIFAR5×5IFGSMIBP92.9%66.1%62.0%
2×2IFGSMNone66.3%25.4%1
2×2IFGSMLGS64.9%31.3%
2×2IFGSM+LGSLGS64.9%24.2%
2×2IFGSMDW47.1%43.3%
2×2IFGSM+DWDW47.1%20.2%=
2×2IFGSMIBP48.6%45.2%41.6%
5×5IFGSMNone66.5%0.4%
5×5IFGSMLGS51.2% 51.2%22.11%
5×5IFGSM +LGSLGS DW45.3%0.5% 59.3%
5×5IFGSM
5×5IFGSM+DWDW45.3%15.6%
5×5IFGSMIBP33.9%29.1%24.9%
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2×25×5
DatasetTraining StrategyClean AccuracyCertified AccuracyTraining Time(h)Clean AccuracyCertified AccuracyTraining Time(h)
MNISTAllPatch98.5%91.5%9.392.0%60.4%8.4
Random(1)98.5%82.9%0.296.9%24.1%0.4
Random(5)98.6%86.6%0.395.8%42.1%0.3
Random(10)98.6%87.7%0.395.6%49.6%0.3
CIFARGuided(10) All Patch98.6% 50.9%88.9% 39.9%2.2 56.495.0% 33.5%53.1% 22.0%2.6 45.8
Random(1)53.6%21.6%0.643.6%6.1%0.6
Random(5)52.9%32.3%0.739.0%14.6%0.7
Random(10)51.9%35.6%0.838.8%18.6%
Guided(10)52.4%36.0%3.737.9%18.8%0.8 3.7
Large Model*65.8%51.9%22.447.8%30.3%15.4
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DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1IBP-sparse98.4%96.0%
4IBP-sparse97.8%90.8%
10IBP-sparse95.2%86.8%
1Random Ablation96.7%90.3%
4Random Ablation96.7%79.1%
10Random Ablation96.7%29.2%
CIFAR1IBP-sparse48.4%40.0%
4IBP-sparse42.2%31.2%
10IBP-sparse37.0%25.6%
1Random Ablation78.3%68.6%
4Random Ablation78.3%61.3%
10Random Ablation78.3%45.0%
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DatasetPixel CountSquareRectangleLineDiamondParallelogram
MNIST491.6%=92.5%91.6%92.3%
CIFAR1669.4%55.4%46.7%68.13%70.2%
2559.7%50.9%32.4%53.6%55.2%
450.8%=46.1%48.6%49.8%
1636.9%29.0%32.1%35.7%36.3%
2530.3%25.1%29.0%30.1%30.7%
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2×25×5
Dataset TrainingStrategyClean Certified TrainingClean Certified Training
Accuracy Accuracy Time(h)Accuracy Accuracy Time(h)
MNIST All Patch98.5% 91.6% 20.190.0% 59.7% 16.3
Pool 2Pool 4Random(1)Random(5)Random(10)Guided(10)98.0% 91.1% 15.885.2% 54.2% 11.6
97.2% 89.9% 13.270.4% 38.3% 10.2
98.5% 81.9% 0.396.8% 24.8% 0.4
98.6% 86.5% 0.394.9% 42.0% 0.5
98.6% 87.5% 0.594.7% 50.4% 0.6
98.7% 88.9% 2.294.0% 53.2% 3.4
CIFAR All Patch49.6% 41.6% 22.534.0% 25.0% 18.6
Pool 2Pool448.1% 39.4% 17.332.4% 24.2% 14.5
Pool4Pool 2-244.9% 37.1% 16.328.3% 20.6% 13.6
45.0% 37.4% 16.525.3% 19.1% 13.8
Random(1)53.2% 32.4% 0.642.7% 11.0% 0.6
Random(5)Random(10)52.2% 39.5% 0.937.8% 19.6% 0.9
50.8% 38.6% 1.038.4% 21.9% 1.0
Guided(10)53.0% 39.8% 4.036.1% 23.0% 3.9
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DatasetPatch SizeTraining StrategyModelClean AccuracyCertified AccuracyTraining Time(h)
CIFAR2×2All Patchmlp50.8%35.5%9.1
2 layer conv52.4%42.6%10.7
4 layer conv49.6%41.6%22.5
5 layer conv (wide)1-~360.0
Random(10) Random(20)5 layer conv (wide) 5 layer conv (wide)64.7% 64.4%49.0%9.5
CIFAR 5×5Guided(10)5 layer conv (wide)66.5%50.8% 49.2%15.8 12.2
Guided(20) All Patch5 layer conv (wide)65.8%51.9%22.4
mlp31.1%18.8%7.1
2 layer conv35.5%22.3%8.7
4 layer conv.34.0%25.0%18.6
5 layer conv (wide)1、~360.0
Random(10) Random(20)5 layer conv (wide)48.6%29.9%9.4
5 layer conv (wide)47.8%30.3%15.4
Guided(10)5 layer conv (wide)48.4%29.0%12.4
Guided(20)5 layer conv (wide)47.6%29.6%23.8
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DatasetSparsity (k)ModelClean AccuracyCertified Accuracy
MNIST1mlp98.4%96.0%
4mlp97.8%90.8%
10mlp95.2%86.8%
1Conv3x397.0%88.3%
CIFAR4Conv3x392.7%75.9%
1mlp48.4%40.0%
4mlp42.2%31.2%
10mlp37.0%25.6%
1Conv11x1134.8%27.4%
4Conv11x1125.1%18.3%
10Conv11x1117.2%13.8%
1Conv13x1338.6%29.7%
4Conv13x1328.1%19.6%
10Conv13x1322.4%15.3%
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DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution98.63/%91.38%21.0
Random (1)4 layer convolution98.48%91.63%80.3
fully connected (255,10)98.46%91.47%9.8
2 layer convolution98.69%82.57%0.2
4 layer convolution98.45%81.87%0.3
Random (5)fully connected (255,10)98.48%84.32%0.2
2 layer convolution98.75%85.87%0.3
4 layer convolution98.57%86.50%0.3
Random (10)fully connected (255,10)98.62%87.32%0.2
2 layer convolution98.73%87.31%0.3
4 layer convolution98.63%87.54%0.5
Guided (10)fully connected (255,10)98.49%88.13%0.2
2 layer convolution98.60%88.49%2.3
4 layer convolution98.70%88.85%2.2
CIFAR All Patchfully connected (255,10)98.63%89.44%2.2
2 layer convolution52.42%42.57%42.6
4 layer convolution49.58%41.57%89.8
Random (1)fully connected (255,10)50.83%35.49%36.6
2 layer convolution54.93%29.13%0.6
4 layer convolution53.22%32.35%0.6
Random (5)fully connected (255,10)52.76%03.21%0.5
2 layer convolution54.15%37.30%0.6
4 layer convolution52.19%39.45%0.9
Random (10)fully connected (255,10)52.38%20.17%0.6
2 layer convolution53.08%39.32%0.7
4 layer convolution50.80%38.57%1.0
Guided (10)fully connected (255,10)51.90%28.97%0.6
2 layer convolution53.04%38.81%3.7
4 layer convolution52.97%39.84%4.0
fully connected (255,10)51.32%29.44%3.6
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DatasetTraining StrategiesModel ArchitectureClean AccuracyCertified AccuracyTraining Time
MNISTAll Patch2 layer convolution91.88%59.59%28.4
4 layer convolution90.03%59.72%65.2
Random (1)fully connected (255,10)93.96%61.97%7.2
2 layer convolution96.27%18.57%0.2
4 layer convolution96.83%24.79%0.4
Random (5)fully connected (255,10)97.60%29.04%0.2
2 layer convolution95.82%38.47%0.2
4 layer convolution94.85%42.02%0.5
Random (10)fully connected (255,10)96.73%45.89%0.2
2 layer convolution95.55%46.13%0.3
4 layer convolution fully connected (255,10)94.76% 96.40%50.43% 52.30%0.6
Guided (10)2 layer convolution95.28%50.28%0.2 2.3
93.98%53.17%3.4
4 layer convolution95.82%55.89%2.2
fully connected (255,10)35.48%
CIFAR All Patch Random (1)2 layer convolution33.95%22.31% 24.96%34.8
4 layer convolution fully connected (255,10)31.05%18.78%74.4 28.4
45.71%07.14%0.6
2 layer convolution42.65%10.99%
4 layer convolution42.34%0.6
fully connected (255,10)00.10%0.5
Random (5)2 layer convolution42.85%17.29%0.6
4 layer convolution37.80%19.63%0.9
fully connected (255,10)36.23%06.99%0.6
Random (10)2 layer convolution41.90%21.40%0.7
4 layer convolution38.41%21.90%1.0
fully connected (255,10) Guided (10)36.04%12.46%0.6
2 layer convolution42.08%20.77%3.6
4 layer convolution fully connected (255,10)36.08% 35.51%23.01% 12.56%3.9 3.5
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Recent approaches for this task either apply a fixed policy on a state-of-the art Transformer model, or a learnable monotonic attention on a weaker recurrent neural network-based structure. In this paper, we propose a new attention mechanism, Monotonic Multihead Attention (MMA), which extends the monotonic attention mechanism to multihead attention. We also introduce two novel and interpretable approaches for latency control that are specifically designed for multiple attention heads. We apply MMA to the simultaneous machine translation task and demonstrate better latency-quality tradeoffs compared to MILk, the previous state-of-the-art approach. We analyze how the latency controls affect the attention span and we study the relationship between the speed of a head and the layer it belongs to. Finally, we motivate the introduction of our model by analyzing the effect of the number of decoder layers and heads on quality and latency.1 + +# 1 INTRODUCTION + +Simultaneous machine translation adds the capability of a live interpreter to machine translation: a simultaneous model starts generating a translation before it has finished reading the entire source sentence. Such models are useful in any situation where translation needs to be done in real time. For example, simultaneous models can translate live video captions or facilitate conversations between people speaking different languages. In a usual translation model, the encoder first reads the entire sentence, then the decoder writes the target sentence. On the other hand, a simultaneous neural machine translation model alternates between reading the input and writing the output using either a fixed or learned policy. + +Monotonic attention mechanisms fall into the flexible policy category, in which the policies are automatically learned from data. Recent work exploring monotonic attention variants for simultaneous translation include: hard monotonic attention (Raffel et al., 2017), monotonic chunkwise attention (MoChA) (Chiu & Raffel, 2018) and monotonic infinite lookback attention (MILk) (Arivazhagan et al., 2019). MILk in particular has shown better quality/latency trade-offs than fixed policy approaches, such as wait- $k$ (Ma et al., 2019) or wait-if-\* (Cho & Esipova, 2016) policies. MILk also outperforms hard monotonic attention and MoChA; while the other two monotonic attention mechanisms only consider a fixed window, MILk computes a softmax attention over all previous encoder states, which may be the key to its improved latency-quality tradeoffs. These monotonic attention approaches also provide a closed-form expression for the expected alignment between source and target tokens. + +However, monotonic attention-based models, including the state-of-the-art MILk, were built on top of RNN-based models. RNN-based models have been outperformed by the recent state-of-the-art Transformer model (Vaswani et al., 2017), which features multiple encoder-decoder attention layers and multihead attention at each layer. + +We thus propose monotonic multihead attention (MMA), which combines the high translation quality from multilayer multihead attention and low latency from monotonic attention. We propose two variants, Hard MMA (MMA-H) and Infinite Lookback MMA (MMA-IL). MMA-H is designed with streaming systems in mind where the attention span must be limited. MMA-IL emphasizes the quality of the translation system. We also propose two novel latency regularization methods. The first encourages the model to be faster by directly minimizing the average latency. The second encourages the attention heads to maintain similar positions, preventing the latency from being dominated by a single or a few heads. + +The main contributions of this paper are: (1) A novel monotonic attention mechanism, monotonic multihead attention, which enables the Transformer model to perform online decoding. This model leverages the power of the Transformer and the efficiency of monotonic attention. (2) Better latency/quality tradeoffs compared to the MILk model, the previous state-of-the-art, on two standard translation benchmarks, IWSLT15 English-Vietnamese (En-Vi) and WMT15 German-English (DeEn). (3) Analyses on how our model is able to control the attention span and on the relationship between the speed of a head and the layer it belongs to. We motivate the design of our model with an ablation study on the number of decoder layers and the number of decoder heads. + +# 2 MONOTONIC MULTIHEAD ATTENTION MODEL + +In this section, we review the monotonic attention-based approaches in RNN-based encoder-decoder models. We then introduce the two types of Monotonic Multihead Attention (MMA) for Transformer models: MMA-H and MMA-IL. Finally, we introduce strategies to control latency and coverage. + +# 2.1 MONOTONIC ATTENTION + +The hard monotonic attention mechanism (Raffel et al., 2017) was first introduced in order to achieve online linear time decoding for RNN-based encoder-decoder models. We denote the input sequence as $\mathbf { x } = \{ x _ { 1 } , . . . , x _ { T } \}$ , and the corresponding encoder states as $\mathbf { m } = \{ m _ { 1 } , . . . , m _ { T } \}$ , with $T$ being the length of the source sequence. The model generates a target sequence $\mathbf { y } = \{ y _ { 1 } , . . . , y _ { U } \}$ with $U$ being the length of the target sequence. At the $i$ -th decoding step, the decoder only attends to one encoder state $m _ { t _ { i } }$ with $t _ { i } = j$ . When generating a new target token $y _ { i }$ , the decoder chooses whether to move one step forward or to stay at the current position based on a Bernoulli selection probability $p _ { i , j }$ , so that $t _ { i } \ \geq \ t _ { i - 1 }$ . Denoting the decoder state at the $i$ -th position, starting from $j = t _ { i - 1 } , t _ { i - 1 } + 1 , t _ { i - 1 } + 2 , . . . _ $ , this process can be calculated as follows: 2 + +$$ +\begin{array} { l c l } { e _ { i , j } } & { = } & { \mathrm { M o n o t o n i c E n e r g y } ( s _ { i - 1 } , m _ { j } ) } \\ { p _ { i , j } } & { = } & { \mathrm { S i g m o i d } \left( e _ { i , j } \right) } \\ { z _ { i , j } } & { \sim } & { \mathrm { B e r n o u l l i } ( p _ { i , j } ) } \end{array} +$$ + +When $z _ { i , j } = 1$ , we set $t _ { i } = j$ and start generating a target token $y _ { i }$ ; otherwise, we set $t _ { i } = j + 1$ and repeat the process. During training, an expected alignment $_ { \pmb { \alpha } }$ is introduced to replace the softmax attention. It can be calculated in a recurrent manner, shown in Equation 4: + +$$ +\begin{array} { l } { \displaystyle \alpha _ { i , j } = p _ { i , j } \sum _ { k = 1 } ^ { j } \left( \alpha _ { i - 1 , k } \prod _ { l = k } ^ { j - 1 } \left( 1 - p _ { i , l } \right) \right) } \\ { \displaystyle = p _ { i , j } \left( \left( 1 - p _ { i , j - 1 } \right) \frac { \alpha _ { i , j - 1 } } { p _ { i , j - 1 } } + \alpha _ { i - 1 , j } \right) } \end{array} +$$ + +Raffel et al. (2017) also introduce a closed-form parallel solution for the recurrence relation in Equation 5: + +$$ +\alpha _ { i , : } = p _ { i , : } \mathrm { c u m p r o d } ( 1 - p _ { i , : } ) \mathrm { c u m s u m } \left( \frac { \alpha _ { i - 1 , : } } { \mathrm { c u m p r o d } ( 1 - p _ { i , : } ) } \right) +$$ + +where $\begin{array} { r } { \mathtt { c u m p r o d } ( x ) = [ 1 , x _ { 1 } , x _ { 1 } x _ { 2 } , . . . , \prod _ { i = 1 } ^ { | x | - 1 } x _ { i } ] } \end{array}$ and $\mathsf { c u m s u m } ( { \pmb x } ) = [ x _ { 1 } , x _ { 1 } + x _ { 2 } , . . . , \sum _ { i = 1 } ^ { | { \pmb x } | } x _ { i } ]$ In practice, the denominator in Equation 5 is clamped into a range of [, 1] to avoid numerical instabilities introduced by cumprod. Although this monotonic attention mechanism achieves online linear time decoding, the decoder can only attend to one encoder state. This limitation can diminish translation quality as there may be insufficient information for reordering. + +Moreover, the model lacks a mechanism to adjust latency based on different requirements at decoding time. To address these issues, Chiu & Raffel (2018) introduce Monotonic Chunkwise Attention (MoChA), which allows the decoder to apply softmax attention to a fixed-length subsequence of encoder states. Alternatively, Arivazhagan et al. (2019) introduce Monotonic Infinite Lookback Attention (MILk) which allows the decoder to access encoder states from the beginning of the source sequence. The expected attention for the MILk model is defined in Equation 6. + +$$ +\beta _ { i , j } = \sum _ { k = j } ^ { | x | } \left( \frac { \alpha _ { i , k } \exp ( u _ { i , j } ) } { \sum _ { l = 1 } ^ { k } \exp ( u _ { i , l } ) } \right) +$$ + +# 2.2 MONOTONIC MULTIHEAD ATTENTION + +Previous monotonic attention approaches are based on RNN encoder-decoder models with a single attention and haven’t explored the power of the Transformer model. 3 The Transformer architecture (Vaswani et al., 2017) has recently become the state-of-the-art for machine translation (Barrault et al., 2019). An important feature of the Transformer is the use of a separate multihead attention module at each layer. Thus, we propose a new approach, Monotonic Multihead Attention (MMA), which combines the expressive power of multihead attention and the low latency of monotonic attention. + +Multihead attention allows each decoder layer to have multiple heads, where each head can compute a different attention distribution. Given queries $Q$ , keys $K$ and values $V$ , multihead attention MultiHead $( Q , K , V )$ is defined in Equation 7. + +$$ +\begin{array} { r } { \begin{array} { l } { \mathrm { M u l t i H e a d } ( Q , K , V ) = \mathrm { C o n c a t } ( \mathrm { h e a d } _ { 1 } , . . . , \mathrm { h e a d } _ { H } ) W ^ { O } } \\ { \mathrm { w h e r e ~ h e a d } _ { h } = \mathrm { A t t e n t i o n } \left( Q W _ { h } ^ { Q } , K W _ { h } ^ { K } , V W _ { h } ^ { V } , \right) } \end{array} } \end{array} +$$ + +The attention function is the scaled dot-product attention, defined in Equation 8: + +$$ +{ \mathrm { A t t e n t i o n } } ( Q , K , V ) = { \mathrm { S o f t m a x } } \left( { \frac { Q K ^ { T } } { \sqrt { d _ { k } } } } \right) V +$$ + +There are three applications of multihead attention in the Transformer model: + +1. The Encoder contains self-attention layers where all of the queries, keys and values come from previous layers. +2. The Decoder contains self-attention layers that allow each position in the decoder to attend to all positions in the decoder up to and including that position. +3. The Encoder-Decoder attention contains multihead attention layers where queries come from the previous decoder layer and the keys and values come from the output of the encoder. Every decoder layer has a separate encoder-decoder attention. + +For MMA, we assign each head to operate as a separate monotonic attention in encoder-decoder attention. + +For a transformer with $L$ decoder layers and $H$ attention heads per layer, we define the selection process of the $h$ -th head encoder-decoder attention in the $l$ -th decoder layer as + +$$ +\begin{array} { l c l } { e _ { i , j } ^ { { l , h } } } & { = } & { \left( \frac { m _ { j } W _ { { l , h } } ^ { K } ( s _ { i - 1 } W _ { { l , h } } ^ { Q } ) ^ { T } } { \sqrt { d _ { k } } } \right) _ { i , j } } \\ { p _ { i , j } ^ { { l , h } } } & { = } & { \mathrm { S i g m o i d } ( e _ { i , j } ) } \\ { z _ { i , j } ^ { { l , h } } } & { \sim } & { \mathrm { B e r n o u l l i } ( p _ { i , j } ) } \end{array} +$$ + +![](images/d597335b4efb8a0b81899a74e4bee9cd4514669123bca34eef86fbf7c1b7c310.jpg) +Figure 1: Monotonic Attention (Left) versus Monotonic Multihead Attention (Right). + +where $W _ { l , h }$ is the input projection matrix, $d _ { k }$ is the dimension of the attention head. We make the selection process independent for each head in each layer. We then investigate two types of MMA, MMA-H(ard) and MMA-IL(infinite lookback). For MMA-H, we use Equation 4 in order to calculate the expected alignment for each layer each head, given $p _ { i , j } ^ { l , h }$ . For MMA-IL, we calculate the softmax energy for each head as follows: + +$$ +u _ { i , j } ^ { l , h } = \mathrm { S o f t E n e r g y } = \left( \frac { m _ { j } \hat { W } _ { l , h } ^ { K } ( s _ { i - 1 } \hat { W } _ { l , h } ^ { Q } ) ^ { T } } { \sqrt { d _ { k } } } \right) _ { i , j } +$$ + +and then use Equation 6 to calculate the expected attention. Each attention head in MMA-H hardattends to one encoder state. On the other hand, each attention head in MMA-IL can attend to all previous encoder states. Thus, MMA-IL allows the model to leverage more information for translation, but MMA-H may be better suited for streaming systems with stricter efficiency requirements. Finally, our models use unidirectional encoders: the encoder self-attention can only attend to previous states, which is also required for simultaneous translation. + +At inference time, our decoding strategy is shown in Algorithm 1. For each $l , h$ , at decoding step $i$ , we apply the sampling processes discussed in subsection 2.1 individually and set the encoder step at $t _ { i } ^ { l , \bar { h } }$ . Then a hard alignment or partial softmax attention from encoder states, shown in Equation 13, will be retrieved to feed into the decoder to generate the $i$ -th token. The model will write a new target token only after all the attentions have decided to write. In other words, the heads that have decided to write must wait until the others have finished reading. + +$$ +\begin{array}{c} \begin{array} { r l } & { \qquad c _ { i } ^ { l } = \mathrm { C o n c a t } ( c _ { i } ^ { l , 1 } , c _ { i } ^ { l , 2 } , . . . , c _ { i } ^ { l , H } ) } \\ & { \qquad \mathrm \quad \mathbf { \ " } } \\ & { \qquad \mathbf { \ " } \mathbf { \ " } e _ { i } ^ { l , h } = f _ { \mathrm { c o n t e x t } } ( \boldsymbol { h } , t _ { i } ^ { l , h } ) = \displaystyle \left\{ \sum _ { j = 1 } ^ { m _ { t _ { i } ^ { l , h } } } \frac { \mathrm { e x p } \left( u _ { i , j } ^ { l , h } \right) } { \sum _ { j = 1 } ^ { t _ { i } ^ { l , h } } \mathrm { e x p } \left( u _ { i , j } ^ { l , h } \right) } m _ { j } \quad \mathrm { M M A \mathrm { - } I L } \right.} \end{array} \end{array} +$$ + +Figure 1 illustrates a comparison between our model and the monotonic model with one attention head. Compared with the monotonic model, the MMA model is able to set attention to different positions so that it can still attend to previous states while reading each new token. Each head can adjust its speed on-the-fly. Some heads read new inputs, while the others can stay in the past to retain the source history information. Even with the hard alignment variant (MMA-H), the model is still able to preserve the history information by setting heads to past states. In contrast, the hard monotonic model, which only has one head, loses the previous information at the attention layer. + +# 2.3 LATENCY CONTROL + +Effective simultaneous machine translation must balance quality and latency. At a high level, latency measures how many source tokens the model has read until a translation is generated. The model we have introduced in subsection 2.2 is not able to control latency on its own. While MMA allows simultaneous translation by having a read or write schedule for each head, the overall latency is determined by the fastest head, i.e. the head that reads the most. It is possible that a head always reads new input without producing output, which would result in the maximum possible latency. Note that the attention behaviors in MMA-H and MMA-IL can be different. In MMA-IL, a head reaching the end of the sentence will provide the model with maximum information about the source sentence. On the other hand, in the case of MMA-H, reaching the end of sentence for a head only + +Algorithm 1 MMA monotonic decoding. Because each head is independent, we compute line 3 to 16 in parallel + +Input: ${ \bf { \delta } } _ { \bf { \mathcal { X } } } =$ source tokens, $h =$ encoder states, $i = 1 , j = 1 , t _ { 0 } ^ { l , h } = 1 , y _ { 0 } =$ StartOfSequence. + +
1:while yi-1 ≠ EndOfSequence do
2:tmax =1
3:h = empty sequence
4:forl←1toLdo
5:forh←1to Hdo
6:forj←ttoacldo
7:l,h Pi,j, = Sigmoid (MonotonicEnergy(Si-1,mj ))
8:ifpi l,h > 0.5 then
9:=j
10:l,h =fcontext(h,t,h）
11:Ci Break
12:else
13:if j>tmax then
14:Read token x j
15:Calculate state h j and append to h
16:tmax =j
17:c = Concat(c,1,,cH）
18:s =DecoderLayer'(s-1,s)
19:yi = Output(s)
20:
+ +gives a hard alignment to the end-of-sentence token, which provides very little information to the decoder. Furthermore, it is possible that an MMA-H attention head stays at the beginning of sentence without moving forward. Such a head would not cause latency issues but would degrade the model quality since the decoder would not have any information about the input. In addition, this behavior is not suited for streaming systems. + +To address these issues, we introduce two latency control methods. The first one is weighted average latency, shown in Equation 14: + +$$ +g _ { i } ^ { W } = \frac { \exp ( g _ { i } ^ { l , h } ) } { \sum _ { l = 1 } ^ { L } \sum _ { h = 1 } ^ { H } \exp ( g _ { i } ^ { l , h } ) } g _ { i } ^ { l , h } +$$ + +where $\begin{array} { r } { g _ { i } ^ { l , h } = \sum _ { j = 1 } ^ { | x | } j \alpha _ { i , j } } \end{array}$ . Then we calculate the latency loss with a differentiable latency metric $\mathcal { C }$ + +$$ +\begin{array} { r c l } { { L _ { a v g } } } & { { = } } & { { { \mathcal { C } } \left( g ^ { W } \right) } } \end{array} +$$ + +Like Arivazhagan et al. (2019), we use the Differentiable Average Lagging. It is important to note that, unlike the original latency augmented training in Arivazhagan et al. (2019), Equation 15 is not the expected latency metric given $\mathcal { C }$ , but weighted average $\mathcal { C }$ on all the attentions. The real expected latency is $\hat { \pmb g } = \mathrm { m a x } _ { l , h } \left( \pmb g ^ { l , h } \right)$ instead of $\bar { \pmb { g } }$ , but using this directly would only affect the speed of the fastest head. Equation 15 can control every head in a way that the faster heads will be automatically assigned to larger weights and slower heads will also be moderately regularized. For MMA-H models, we found that the latency of are mainly due to outliers that skip almost every token. The weighted average latency loss is not sufficient to control the outliers. We therefore introduce the head divergence loss, the average variance of expected delays at each step, defined in Equation 16: + +$$ +\begin{array} { c c l } { { { \cal L } _ { v a r } } } & { { = } } & { { \displaystyle \frac { 1 } { L H } \sum _ { l = 1 } ^ { L } \sum _ { h = 1 } ^ { H } \left( g _ { i } ^ { l , h } - \bar { g } _ { i } \right) ^ { 2 } } } \end{array} +$$ + +where $\begin{array} { r } { \bar { g _ { i } } = \frac { 1 } { L H } \sum g _ { i } } \end{array}$ The final objective function is presented in Equation 17: + +$$ +L ( \theta ) = - \log ( \pmb { y } \mid \pmb { x } ; \theta ) + \lambda _ { a v g } L _ { a v g } + \lambda _ { v a r } L _ { v a r } +$$ + +where $\lambda _ { a v g } , \lambda _ { v a r }$ are hyperparameters that control both losses. Intuitively, while $\lambda _ { a v g }$ controls the overall speed, $\lambda _ { v a r }$ controls the divergence of the heads. Combining these two losses, we are able to dynamically control the range of attention heads so that we can control the latency and the reading buffer. For MMA-IL model, we only use $L _ { a v g }$ ; for MMA-H we only use $L _ { v a r }$ . + +# 3 EXPERIMENTAL SETUP + +# 3.1 EVALUATION METRICS + +We evaluate our model using quality and latency. For translation quality, we use tokenized BLEU 4 for IWSLT15 En-Vi and detokenized BLEU with SacreBLEU (Post, 2018) for WMT15 De-En. For latency, we use three different recent metrics, Average Proportion (AP) (Cho & Esipova, 2016), Average Lagging (AL) (Ma et al., 2019) and Differentiable Average Lagging (DAL) (Arivazhagan et al., 2019) 5. We remind the reader of the metric definitions in Appendix A.2. + +# 3.2 DATASETS + +Table 1: Number of sentences in each split. + +
DatasetTrainValidationTest
IWSLT15 En-Vi133k12681553
WMT15 De-En4.5M30002169
+ +Table 2: Offline model performance with unidirectional encoder and greedy decoding. + +
DatasetRNNTransformer
IWSLT15 En-Vi25.6628.7
WMT15 De-En28.4 (Arivazhagan et al., 2019)32.3
+ +
DatasetBeam SearchBidirectional EncoderUnidirectional Encoder
WMT15 De-En132.632.3
433.033.0
IWSLT15 En-Vi128.729.4
1028.829.5
+ +Table 3: Effect of using a unidirectional encoder and greedy decoding to BLEU score. + +We evaluate our method on two standard machine translation datasets, IWSLT14 En-Vi and WMT15 De-En. Statistics of the datasets can be found in Table 1. For each dataset, we apply tokenization with the Moses (Koehn et al., 2007) tokenizer and preserve casing. + +IWSLT15 English-Vietnamese TED talks from IWSLT 2015 Evaluation Campaign (Cettolo et al., 2016). We follow the settings from Luong & Manning (2015) and Raffel et al. (2017). We replace words with frequency less than 5 by $< u n k >$ . We use tst2012 as a validation set tst2013 as a test set. + +WMT15 German-English We follow the setting from Arivazhagan et al. (2019). We apply byte pair encoding (BPE) (Sennrich et al., 2016) jointly on the source and target to construct a shared vocabulary with 32K symbols. We use newstest2013 as validation set and newstest2015 as test set. + +# 3.3 MODELS + +We evaluate MMA-H and MMA-IL models on both datasets. The MILK model we evaluate on IWSLT15 En-Vi is based on Luong et al. (2015) rather than RNMT $^ +$ (Chen et al., 2018). In general, our offline models use unidirectional encoders, i.e. the encoder self-attention can only attend to previous states, and greedy decoding. We report offline model performance in Table 2 and the effect of using unidirectional encoders and greedy decoding in Table 3. For MMA models, we replace the encoder-decoder layers with MMA and keep other hyperparameter settings the same as the offline model. Detailed hyperparameter settings can be found in subsection A.1. We use the Fairseq library (Ott et al., 2019) for our implementation. + +# 4 RESULTS + +In this section, we present the main results of our model in terms of latency-quality tradeoffs, ablation studies and analyses. In the first study, we analyze the effect of the variance loss on the attention span. Then, we study the effect of the number of decoder layers and decoder heads on quality and latency. We also provide a case study for the behavior of attention heads in an example. Finally, we study the relationship between the rank of an attention head and the layer it belongs to. + +![](images/e122b7036362526ca9df24fd1d502cd25f56d35adb6686383751be035f195e06.jpg) +Figure 2: Latency-quality tradeoffs for MILk (Arivazhagan et al., 2019) and MMA on IWSLT15 En-Vi and WMT15 De-En. Black dashed line indicates the unidirectional offline transformer model with greedy search. + +# 4.1 LATENCY-QUALITY TRADEOFFS + +We plot the quality-latency curves for MMA-H and MMA-IL in Figure 2. The BLEU and latency scores on the test sets are generated by setting a latency range and selecting the checkpoint with best BLEU score on the validation set. We use differentiable average lagging (Arivazhagan et al., 2019) when setting the latency range. We find that for a given latency, our models obtain a better translation quality. While MMA-IL tends to have a decrease in quality as the latency decreases, MMA-H has a small gain in quality as latency decreases: a larger latency does not necessarily mean an increase in source information available to the model. In fact, the large latency is from the outlier attention heads, which skip the entire source sentence and point to the end of the sentence. The outliers not only increase the latency but they also do not provide useful information. We introduce the attention variance loss to eliminate the outliers, as such a loss makes the attention heads focus on the current context for translating the new target token. + +It is interesting to observe that MMA-H has a better latency-quality tradeoff than $\mathrm { M I L K } ^ { 7 }$ even though each head only attends to only one state. Although MMA-H is not yet able to handle an arbitrarily long input (without resorting to segmenting the input), since both encoder and decoder self-attention have an infinite lookback, that model represents a good step in that direction. + +# 4.2 ATTENTION SPAN + +In subsection 2.3, we introduced the attention variance loss to MMA-H in order to prevent outlier attention heads from increasing the latency or increasing the attention span. We have already evaluated the effectiveness of this method on latency in subsection 4.1. We also want to measure the difference between the fastest and slowest heads at each decoding step. We define the average + +attention span in Equation 18: + +$$ +\bar { S } = \frac { 1 } { \vert \pmb { y } \vert } \left( \sum _ { i } ^ { \vert \pmb { y } \vert } \underset { l , h } { \operatorname* { m a x } } t _ { i } ^ { l , h } - \underset { l , h } { \operatorname* { m i n } } t _ { i } ^ { l , h } \right) +$$ + +It estimates the reading buffer we need for streaming translation. We show the relation between the average attention span versus $\lambda _ { v a r }$ in Figure 3. As expected, the average attention span is reduced as we increase $\lambda _ { v a r }$ . + +![](images/0f751c2faf86a11373cba7cf525d322dc182947f9c6a18f187c49c68d5fad146.jpg) +Figure 3: Effect of $\lambda _ { v a r }$ on the average attention span. The variance loss works as intended by reducing the span with higher weights. + +# 4.3 EFFECT ON NUMBER OF LAYERS AND NUMBER OF HEADS + +One motivation to introduce MMA is to adapt the Transformer, which is the current state-of-the-art model for machine translation, to online decoding. Important features of the Transformer architecture include having a separate attention layer for each decoder layer block and multihead attention. In this section, we test the effect of these two components on the offline, MMA-H, and MMA-IL models from a quality and latency perspective. We report quality as measured by detokenized BLEU and latency as measured by DAL on the WMT13 validation set in Figure 4. We set $\lambda _ { a v g } = 0 . 2$ for MMA-IL and $\lambda _ { v a r } = 0 . 2$ for MMA-H. + +The offline model benefits from having more than one decoder layer. In the case of 1 decoder layer, increasing the number of attention heads is beneficial but in the case of 3 and 6 decoder layers, we do not see much benefit from using more than 2 heads. The best performance is obtained for 3 layers and 2 heads (6 effective heads). The MMA-IL model behaves similarly to the offline model, and the best performance is observed with 6 layers and 4 heads (24 effective heads). For MMA-H, with 1 layer, performance improves with more heads. With 3 layers, the single-head setting is the most effective (3 effective heads). Finally, with 6 layers, the best performance is reached with 16 heads (96 effective heads). + +The general trend we observe is that performance improves as we increase the number of effective heads, either from multiple layers or multihead attention, up to a certain point, then either plateaus or degrades. This motivates the introduction of the MMA model. + +We also note that latency increases with the number of effective attention heads. This is due to having fixed loss weights: when more heads are involved, we should increase $\lambda _ { v a r }$ or $\lambda _ { a v g }$ to better control latency. + +# 4.4 ATTENTION BEHAVIORS + +We characterize attention behaviors by providing a running example of MMA-H and MMA-IL, shown in Figure 5. Each curve represents the path that an attention head goes through at inference time. For MMA-H, shown in Figure 5a, we found that when the source and target tokens have the same order, the attention heads behave linearly and the distance between fastest head and slowest head is small. For example, this can be observed from partial sentence pair “I also didn’t know that” and target tokens “Tôi cũng không biết rằng”, which have the same order. However, when the source tokens and target tokens have different orders, such as “the second step” and “bước (step) thứ hai (second)”, the model will generate “bước (step)” first and some heads will stay in the past to retain the information for later reordered translation “thứ hai (second)”. We can also see that the attention heads have a near-diagonal trajectory, which is appropriate for streaming inputs. + +![](images/587b8944db8cce31706c574a36a9ec6d40a868064c34c383e996803aed29b0b0.jpg) +Figure 4: Effect of the number of decoder attention heads and the number of decoder attention layers on quality and latency, reported on the WMT13 validation set. + +The behavior of the heads in MMA-IL models is shown in Figure 5b. Notice that we remove the partial softmax alignment in this figure. We don’t expect streaming capability for MMA-IL: some heads stop at early position of the source sentence to retain the history information. Moreover, because MMA-IL has more information when generating a new target token, it tends to produce translations with better quality. In this example, the MMA-IL model has a better translation on “isolate the victim” than MMA-H (“là cô lập nạn nhân” vs “là tách biệt nạn nhân”) + +![](images/f9caf5f55023339b16fb95b389bc65cc48ec79600f2fc76d230026679d0627e7.jpg) +Figure 5: Running examples on IWSLT15 English-Vietnamese dataset + +# 4.5 RANK OF THE HEADS + +In Figure 6, we calculate the average and standard deviation of rank of each head when generating every target token. For MMA-IL, we find that heads in lower layers tend to have higher rank and are thus slower. However, in MMA-H, the difference of the average rank are smaller. Furthermore, the standard deviation is very large which means that the order of the heads in MMA-H changes frequently over the inference process. + +![](images/945370a5e1d53aa3ee0f21fc5e5f3110a0171fb97a8e1797f201cb0464090bba.jpg) +Figure 6: The average rank of attention heads during inference on IWSLT15 En-Vi. Error bars indicate the standard deviation. L indicates the layer number and $\mathrm { H }$ indicates the head number. + +# 5 RELATED WORK + +Recent work on simultaneous machine translation falls into three categories. In the first one, models use a rule-based policy for reading input and writing output. Cho & Esipova (2016) propose a WaitIf-\* policy to enable an offline model to decode simultaneously. Ma et al. (2019) propose a wait- $k$ policy where the model first reads $k$ tokens, then alternates between read and write actions. Dalvi et al. (2018) propose an incremental decoding method, also based on a rule-based schedule. In the second category, a flexible policy is learnt from data. Grissom II et al. (2014) introduce a Markov chain to phrase-based machine translation models for simultaneous machine translation, in which they apply reinforcement learning to learn the read-write policy based on states. Gu et al. (2017) introduce an agent which learns to make decisions on when to translate from the interaction with a pre-trained offline neural machine translation model. Luo et al. (2017) used continuous rewards policy gradient for online alignments for speech recognition. Lawson et al. (2018) proposed a hard alignment with variational inference for online decoding. Alinejad et al. (2018) propose a new operation ”predict” which predicts future source tokens. Zheng et al. (2019b) introduce a restricted dynamic oracle and restricted imitation learning for simultaneous translation. Zheng et al. (2019a) train the agent with an action sequence from labels that are generated based on the rank of the gold target word given partial input. Models from the last category leverage monotonic attention and replace the softmax attention with an expected attention calculated from a stepwise Bernoulli selection probability. Raffel et al. (2017) first introduce the concept of monotonic attention for online linear time decoding, where the attention only attends to one encoder state at a time. Chiu & Raffel (2018) extended that work to let the model attend to a chunk of encoder state. Arivazhagan et al. (2019) also make use of the monotonic attention but introduce an infinite lookback to improve the translation quality. + +# 6 CONCLUSION + +In this paper, we propose two variants of the monotonic multihead attention model for simultaneous machine translation. By introducing two new targeted loss terms which allow us to control both latency and attention span, we are able to leverage the power of the Transformer architecture to achieve better quality-latency trade-offs than the previous state-of-the-art model. We also present detailed ablation studies demonstrating the efficacy and rationale of our approach. By introducing these stronger simultaneous sequence-to-sequence models, we hope to facilitate important applications, such as high-quality real-time interpretation between human speakers. + +# REFERENCES + +Ashkan Alinejad, Maryam Siahbani, and Anoop Sarkar. Prediction improves simultaneous neural machine translation. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pp. 3022–3027, 2018. + +Naveen Arivazhagan, Colin Cherry, Wolfgang Macherey, Chung-Cheng Chiu, Semih Yavuz, Ruoming Pang, Wei Li, and Colin Raffel. Monotonic infinite lookback attention for simultaneous machine translation. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pp. 1313–1323, Florence, Italy, July 2019. Association for Computational Linguistics. URL https://www.aclweb.org/anthology/P19-1126. + +Lo¨ıc Barrault, Ondˇrej Bojar, Marta R. Costa-jussa, Christian Federmann, Mark Fishel, Yvette \` Graham, Barry Haddow, Matthias Huck, Philipp Koehn, Shervin Malmasi, Christof Monz, Mathias Muller, Santanu Pal, Matt Post, and Marcos Zampieri. Findings of the 2019 con- ¨ ference on machine translation (WMT19). In Proceedings of the Fourth Conference on Machine Translation (Volume 2: Shared Task Papers, Day 1), pp. 1–61, Florence, Italy, August 2019. Association for Computational Linguistics. doi: 10.18653/v1/W19-5301. URL https://www.aclweb.org/anthology/W19-5301. + +Mauro Cettolo, Niehues Jan, Stuker Sebastian, Luisa Bentivogli, Roldano Cattoni, and Marcello ¨ Federico. The iwslt 2016 evaluation campaign. In International Workshop on Spoken Language Translation, 2016. + +Mia Xu Chen, Orhan Firat, Ankur Bapna, Melvin Johnson, Wolfgang Macherey, George Foster, Llion Jones, Mike Schuster, Noam Shazeer, Niki Parmar, et al. The best of both worlds: Combining recent advances in neural machine translation. In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 76–86, 2018. + +Chung-Cheng Chiu and Colin Raffel. Monotonic chunkwise attention. 2018. URL https:// openreview.net/pdf?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ Hko85plCW. + +Kyunghyun Cho and Masha Esipova. Can neural machine translation do simultaneous translation? arXiv preprint arXiv:1606.02012, 2016. + +Fahim Dalvi, Nadir Durrani, Hassan Sajjad, and Stephan Vogel. Incremental decoding and training methods for simultaneous translation in neural machine translation. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers), pp. 493–499, New Orleans, Louisiana, June 2018. Association for Computational Linguistics. doi: 10.18653/v1/N18-2079. URL https://www.aclweb.org/anthology/N18-2079. + +Alvin Grissom II, He He, Jordan Boyd-Graber, John Morgan, and Hal Daume III. Don’t until the ´ final verb wait: Reinforcement learning for simultaneous machine translation. In Proceedings of the 2014 Conference on empirical methods in natural language processing (EMNLP), pp. 1342– 1352, 2014. + +Jiatao Gu, Graham Neubig, Kyunghyun Cho, and Victor OK Li. Learning to translate in real-time with neural machine translation. In 15th Conference of the European Chapter of the Association for Computational Linguistics, EACL 2017, pp. 1053–1062. Association for Computational Linguistics (ACL), 2017. + +Philipp Koehn, Hieu Hoang, Alexandra Birch, Chris Callison-Burch, Marcello Federico, Nicola Bertoldi, Brooke Cowan, Wade Shen, Christine Moran, Richard Zens, Chris Dyer, Ondˇrej Bojar, Alexandra Constantin, and Evan Herbst. Moses: Open source toolkit for statistical machine translation. In Proceedings of the 45th Annual Meeting of the Association for Computational Linguistics Companion Volume Proceedings of the Demo and Poster Sessions, pp. 177–180, Prague, Czech Republic, June 2007. Association for Computational Linguistics. URL https://www.aclweb.org/anthology/P07-2045. + +Dieterich Lawson, Chung-Cheng Chiu, George Tucker, Colin Raffel, Kevin Swersky, and Navdeep Jaitly. Learning hard alignments with variational inference. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5799–5803. IEEE, 2018. + +Yuping Luo, Chung-Cheng Chiu, Navdeep Jaitly, and Ilya Sutskever. Learning online alignments with continuous rewards policy gradient. In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2801–2805. IEEE, 2017. + +Minh-Thang Luong and Christopher D Manning. Stanford neural machine translation systems for spoken language domains. 2015. + +Minh-Thang Luong, Hieu Pham, and Christopher D Manning. Effective approaches to attentionbased neural machine translation. arXiv preprint arXiv:1508.04025, 2015. + +Mingbo Ma, Liang Huang, Hao Xiong, Renjie Zheng, Kaibo Liu, Baigong Zheng, Chuanqiang Zhang, Zhongjun He, Hairong Liu, Xing Li, Hua Wu, and Haifeng Wang. STACL: Simultaneous translation with implicit anticipation and controllable latency using prefix-to-prefix framework. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pp. 3025–3036, Florence, Italy, July 2019. Association for Computational Linguistics. URL https://www.aclweb.org/anthology/P19-1289. + +Myle Ott, Sergey Edunov, Alexei Baevski, Angela Fan, Sam Gross, Nathan Ng, David Grangier, and Michael Auli. fairseq: A fast, extensible toolkit for sequence modeling. In Proceedings of NAACL-HLT 2019: Demonstrations, 2019. + +Matt Post. A call for clarity in reporting BLEU scores. In Proceedings of the Third Conference on Machine Translation: Research Papers, pp. 186–191, Belgium, Brussels, October 2018. Association for Computational Linguistics. URL https://www.aclweb.org/anthology/ W18-6319. + +Colin Raffel, Minh-Thang Luong, Peter J Liu, Ron J Weiss, and Douglas Eck. Online and linear-time attention by enforcing monotonic alignments. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 2837–2846. JMLR. org, 2017. + +Rico Sennrich, Barry Haddow, and Alexandra Birch. Neural machine translation of rare words with subword units. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 1715–1725, 2016. + +Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998–6008, 2017. + +Baigong Zheng, Renjie Zheng, Mingbo Ma, and Liang Huang. Simpler and faster learning of adaptive policies for simultaneous translation. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pp. 1349–1354, Hong Kong, China, November 2019a. Association for Computational Linguistics. doi: 10.18653/v1/D19-1137. URL https://www.aclweb.org/anthology/D19-1137. + +Baigong Zheng, Renjie Zheng, Mingbo Ma, and Liang Huang. Simultaneous translation with flexible policy via restricted imitation learning. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pp. 5816–5822, Florence, Italy, July 2019b. Association for Computational Linguistics. doi: 10.18653/v1/P19-1582. URL https://www.aclweb. org/anthology/P19-1582. + +# A APPENDIX + +# A.1 HYPERPARAMETERS + +The hyperparameters we used for offline and monotonic transformer models are defined in Table 4. + +# A.2 LATENCY METRICS DEFINITIONS + +Given the delays $\mathbf { g } = \{ g _ { 1 } , g _ { 2 } , . . . , g _ { | \mathbf { y } | } \}$ of generating each target token, AP, AL and DAL are defined in Table 5. + +Table 4: Offline and monotonic models hyperparameters. + +
HyperparameterWMT15 German-EnglishIWSLT English-Vietnamese
encoder embed dim1024512
encoder ffn embed dim40961024
encoder attention heads164
encoder layers6
decoder embed dim1024512
decoder ffn embed dim40961024
decoder attention heads164
decoder layers6
dropout0.3
optimizeradam
adam-β(0.9,0.98)
clip-norm0.0
lr lr scheduler0.0005
inverse sqrt
warmup-updates4000
warmup-init-lr1e-07
label-smoothing0.1
max tokens3584×8×8×216000
+ +Table 5: The calculation of latency metrics, given source $_ { \textbf { \em x } }$ , target $\textbf { { y } }$ and delays $\pmb { g }$ + +
Latency Metric
Average Proportiony 1 M gi lacllyl i=1
Average Lagging1 T i-1 gi 1y1/ac T i=1 where T = arg maxi(gi = |xl)
1 ly Differentiable Average Lagginglyl i-1 M gi lyl//ac| i=1 gi i=0 where g' = ly max(gi, 9i-1 i<0 [x
+ +# A.3 DETAILED RESULTS + +We provide the detailed results in Figure 2 as Table 6 and Table 7. + +# A.4 THRESHOLD OF READING ACTION + +We explore a simple method that can adjust system’s latency at inference time without training new models. In Algorithm 1 line 8, 0.5 was used as an threshold. One can set different threshold $p$ during the inference time to control the latency. We run the pilot experiments on IWSLT15 En-Vi dataset and the results are shown as Table 8. Although this method doesn’t require training new model, it dramatically hurts the translation quality. + +Table 6: Detailed results for MMA-H and MMA-IL on WMT15 DeEn + +
BLEU AP AL DAL
Xavg MMA-IL
0.05 30.7 0.78 10.91 12.64
0.1 30.5 0.70 7.42 8.82
0.2 30.1 0.63 5.17 6.41
0.3 30.3 0.60 4.18 5.35
0.4 29.2 0.59 3.75 4.90
0.5 26.7 0.59 3.69 4.83
0.75 25.5 0.58 3.40 4.46
1.0 25.1 0.56 3.00 4.03
Xvar MMA-H
0.1 28.5 0.74 8.94 10.83
0.2 28.9 0.69 6.82 8.622
0.3 29.2 0.64 5.45 7.03
0.4 28.5 0.59 3.90 5.21
0.5 28.5 0.59 3.88 5.19
0.6 29.6 0.56 3.13 4.32
0.7 29.1 0.56 2.93 4.10
+ +Table 7: Detailed results for MILk, MMA-H and MMA-IL on IWSLT15 En-Vi + +
BLEUAPALDAL
入MILk
0.124.620.715.937.19
0.224.680.674.905.97
0.324.310.654.455.43
0.423.730.644.285.24
XaugMMA-IL
0.0228.280.767.098.29
0.0428.330.705.446.57
0.128.420.674.635.65
0.228.470.633.574.44
0.327.90.592.983.81
0.427.730.582.683.46
XuarMMA-H
0.0227.260.777.528.71
0.15.226.31
27.680.693.81
0.228.060.634.84
0.427.790.623.574.59
0.827.950.603.224.19
+ +# A.5 AVERAGE LOSS FOR MMA-H + +We explore applying a simple average instead of a weighted average loss to MMA-H. The results are shown in Figure 7 and Table 9. We find that even with very large weights, we are unable to reduce the overall latency. In addition, we find that the weighted average loss severely affects the translation quality negatively. On the other hand, the divergence loss we propose in Equation 16 can efficiently reduce the latency while retaining relatively good translation quality for MMA-H models. + +
Reading ThresholdWeighted Average Latency Loss
pBLEUAPALDALLavgBLEUAPALDAL
0.525.50.7923877.136738.271870.0225.50.7923877.136738.27187
0.425.250.737495.720036.858120.0425.680.7281075.528566.61744
0.323.060.6973984.880876.033420.324.90.6027032.900543.68039
0.218.370.6782984.710995.946360.225.30.6369143.545774.38623
0.18.730.6964525.52257.204390.125.480.6844244.579015.54102
+ +Table 8: Comparison between setting threshold for reading action and weighted average latency loss. + +![](images/aea53beea98c86b1baeb911bfa9ca93df3d5c23c9dfd3fe4f079a502cf059346.jpg) +Figure 7: Effect average loss, weighted average loss and variance loss on MMA-H on WMT15 DeEn development set. +Table 9: Detailed numbers on average loss, weighted average loss and head divergence loss on WMT15 De-En development set + +
BLEUAPALDAL
XaugAverage
0.0528.50.86258114.384717.3702
0.127.80.85543513.97417.02
0.2280.83532412.853115.908
0.428.10.81940811.981614.9763
1.028.20.81060911.752814.6695
2.028.10.80025811.176314.0761
8.028.40.80643911.528914.6431
XaugWeighted Average
0.0228.240.77392210.210912.2274
0.0424.350.6858347.067168.64069
0.067.800.87582516.204619.0892
0.089.510.573723.920116.1421
0.19.780.5565853.30075.46142
XvarDivergence
0.127.350.7360258.7096810.5253
0.227.640.6814916.639148.3856
0.327.370.66236.049027.71922
0.427.620.6381885.316726.86834
0.527.500.6257594.930446.38998
1.027.10.5821943.648644.90997
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Recent approaches for this task either apply a fixed policy on a state-of-the art Transformer model, or a learnable monotonic attention on a weaker recurrent neural network-based structure. In this paper, we propose a new attention mechanism, Monotonic Multihead Attention (MMA), which extends the monotonic attention mechanism to multihead attention. We also introduce two novel and interpretable approaches for latency control that are specifically designed for multiple attention heads. We apply MMA to the simultaneous machine translation task and demonstrate better latency-quality tradeoffs compared to MILk, the previous state-of-the-art approach. We analyze how the latency controls affect the attention span and we study the relationship between the speed of a head and the layer it belongs to. Finally, we motivate the introduction of our model by analyzing the effect of the number of decoder layers and heads on quality and latency.1 ", + "bbox": [ + 233, + 295, + 764, + 489 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 512, + 336, + 529 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Simultaneous machine translation adds the capability of a live interpreter to machine translation: a simultaneous model starts generating a translation before it has finished reading the entire source sentence. Such models are useful in any situation where translation needs to be done in real time. For example, simultaneous models can translate live video captions or facilitate conversations between people speaking different languages. In a usual translation model, the encoder first reads the entire sentence, then the decoder writes the target sentence. On the other hand, a simultaneous neural machine translation model alternates between reading the input and writing the output using either a fixed or learned policy. ", + "bbox": [ + 174, + 542, + 825, + 655 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Monotonic attention mechanisms fall into the flexible policy category, in which the policies are automatically learned from data. Recent work exploring monotonic attention variants for simultaneous translation include: hard monotonic attention (Raffel et al., 2017), monotonic chunkwise attention (MoChA) (Chiu & Raffel, 2018) and monotonic infinite lookback attention (MILk) (Arivazhagan et al., 2019). MILk in particular has shown better quality/latency trade-offs than fixed policy approaches, such as wait- $k$ (Ma et al., 2019) or wait-if-\\* (Cho & Esipova, 2016) policies. MILk also outperforms hard monotonic attention and MoChA; while the other two monotonic attention mechanisms only consider a fixed window, MILk computes a softmax attention over all previous encoder states, which may be the key to its improved latency-quality tradeoffs. These monotonic attention approaches also provide a closed-form expression for the expected alignment between source and target tokens. ", + "bbox": [ + 174, + 661, + 825, + 814 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, monotonic attention-based models, including the state-of-the-art MILk, were built on top of RNN-based models. RNN-based models have been outperformed by the recent state-of-the-art Transformer model (Vaswani et al., 2017), which features multiple encoder-decoder attention layers and multihead attention at each layer. ", + "bbox": [ + 174, + 821, + 825, + 877 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We thus propose monotonic multihead attention (MMA), which combines the high translation quality from multilayer multihead attention and low latency from monotonic attention. We propose two variants, Hard MMA (MMA-H) and Infinite Lookback MMA (MMA-IL). MMA-H is designed with streaming systems in mind where the attention span must be limited. MMA-IL emphasizes the quality of the translation system. We also propose two novel latency regularization methods. The first encourages the model to be faster by directly minimizing the average latency. The second encourages the attention heads to maintain similar positions, preventing the latency from being dominated by a single or a few heads. ", + "bbox": [ + 174, + 103, + 825, + 214 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main contributions of this paper are: (1) A novel monotonic attention mechanism, monotonic multihead attention, which enables the Transformer model to perform online decoding. This model leverages the power of the Transformer and the efficiency of monotonic attention. (2) Better latency/quality tradeoffs compared to the MILk model, the previous state-of-the-art, on two standard translation benchmarks, IWSLT15 English-Vietnamese (En-Vi) and WMT15 German-English (DeEn). (3) Analyses on how our model is able to control the attention span and on the relationship between the speed of a head and the layer it belongs to. We motivate the design of our model with an ablation study on the number of decoder layers and the number of decoder heads. ", + "bbox": [ + 174, + 222, + 825, + 333 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 MONOTONIC MULTIHEAD ATTENTION MODEL ", + "text_level": 1, + "bbox": [ + 174, + 352, + 591, + 369 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we review the monotonic attention-based approaches in RNN-based encoder-decoder models. We then introduce the two types of Monotonic Multihead Attention (MMA) for Transformer models: MMA-H and MMA-IL. Finally, we introduce strategies to control latency and coverage. ", + "bbox": [ + 174, + 383, + 825, + 426 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 MONOTONIC ATTENTION ", + "text_level": 1, + "bbox": [ + 174, + 441, + 390, + 457 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The hard monotonic attention mechanism (Raffel et al., 2017) was first introduced in order to achieve online linear time decoding for RNN-based encoder-decoder models. We denote the input sequence as $\\mathbf { x } = \\{ x _ { 1 } , . . . , x _ { T } \\}$ , and the corresponding encoder states as $\\mathbf { m } = \\{ m _ { 1 } , . . . , m _ { T } \\}$ , with $T$ being the length of the source sequence. The model generates a target sequence $\\mathbf { y } = \\{ y _ { 1 } , . . . , y _ { U } \\}$ with $U$ being the length of the target sequence. At the $i$ -th decoding step, the decoder only attends to one encoder state $m _ { t _ { i } }$ with $t _ { i } = j$ . When generating a new target token $y _ { i }$ , the decoder chooses whether to move one step forward or to stay at the current position based on a Bernoulli selection probability $p _ { i , j }$ , so that $t _ { i } \\ \\geq \\ t _ { i - 1 }$ . Denoting the decoder state at the $i$ -th position, starting from $j = t _ { i - 1 } , t _ { i - 1 } + 1 , t _ { i - 1 } + 2 , . . . _ $ , this process can be calculated as follows: 2 ", + "bbox": [ + 173, + 467, + 825, + 595 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/1e00e4ac01ca17148a42234276fd2bfe3915f021f95dc26ea66ee0c9a33d3eed.jpg", + "text": "$$\n\\begin{array} { l c l } { e _ { i , j } } & { = } & { \\mathrm { M o n o t o n i c E n e r g y } ( s _ { i - 1 } , m _ { j } ) } \\\\ { p _ { i , j } } & { = } & { \\mathrm { S i g m o i d } \\left( e _ { i , j } \\right) } \\\\ { z _ { i , j } } & { \\sim } & { \\mathrm { B e r n o u l l i } ( p _ { i , j } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 366, + 598, + 630, + 652 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "When $z _ { i , j } = 1$ , we set $t _ { i } = j$ and start generating a target token $y _ { i }$ ; otherwise, we set $t _ { i } = j + 1$ and repeat the process. During training, an expected alignment $_ { \\pmb { \\alpha } }$ is introduced to replace the softmax attention. It can be calculated in a recurrent manner, shown in Equation 4: ", + "bbox": [ + 174, + 655, + 825, + 696 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/9fcfcb3331ff46aa76bca6824b0c78a69ca6768a0880249b79da62fe5c67246e.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\alpha _ { i , j } = p _ { i , j } \\sum _ { k = 1 } ^ { j } \\left( \\alpha _ { i - 1 , k } \\prod _ { l = k } ^ { j - 1 } \\left( 1 - p _ { i , l } \\right) \\right) } \\\\ { \\displaystyle = p _ { i , j } \\left( \\left( 1 - p _ { i , j - 1 } \\right) \\frac { \\alpha _ { i , j - 1 } } { p _ { i , j - 1 } } + \\alpha _ { i - 1 , j } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 699, + 645, + 780 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Raffel et al. (2017) also introduce a closed-form parallel solution for the recurrence relation in Equation 5: ", + "bbox": [ + 171, + 781, + 825, + 809 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/f0260ef1810c1eebfba3c0000f312b7b966bc53ade5ae602fbde952ecf9512af.jpg", + "text": "$$\n\\alpha _ { i , : } = p _ { i , : } \\mathrm { c u m p r o d } ( 1 - p _ { i , : } ) \\mathrm { c u m s u m } \\left( \\frac { \\alpha _ { i - 1 , : } } { \\mathrm { c u m p r o d } ( 1 - p _ { i , : } ) } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 287, + 806, + 709, + 842 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\mathtt { c u m p r o d } ( x ) = [ 1 , x _ { 1 } , x _ { 1 } x _ { 2 } , . . . , \\prod _ { i = 1 } ^ { | x | - 1 } x _ { i } ] } \\end{array}$ and $\\mathsf { c u m s u m } ( { \\pmb x } ) = [ x _ { 1 } , x _ { 1 } + x _ { 2 } , . . . , \\sum _ { i = 1 } ^ { | { \\pmb x } | } x _ { i } ]$ In practice, the denominator in Equation 5 is clamped into a range of [\u000f, 1] to avoid numerical instabilities introduced by cumprod. Although this monotonic attention mechanism achieves online linear time decoding, the decoder can only attend to one encoder state. This limitation can diminish translation quality as there may be insufficient information for reordering. ", + "bbox": [ + 174, + 844, + 825, + 888 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Moreover, the model lacks a mechanism to adjust latency based on different requirements at decoding time. To address these issues, Chiu & Raffel (2018) introduce Monotonic Chunkwise Attention (MoChA), which allows the decoder to apply softmax attention to a fixed-length subsequence of encoder states. Alternatively, Arivazhagan et al. (2019) introduce Monotonic Infinite Lookback Attention (MILk) which allows the decoder to access encoder states from the beginning of the source sequence. The expected attention for the MILk model is defined in Equation 6. ", + "bbox": [ + 173, + 138, + 825, + 223 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/78bfc0bec32bf57998374bb3d9fd4320e604813236b365050f7ae485aaf76c8b.jpg", + "text": "$$\n\\beta _ { i , j } = \\sum _ { k = j } ^ { | x | } \\left( \\frac { \\alpha _ { i , k } \\exp ( u _ { i , j } ) } { \\sum _ { l = 1 } ^ { k } \\exp ( u _ { i , l } ) } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 395, + 227, + 602, + 272 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 MONOTONIC MULTIHEAD ATTENTION ", + "text_level": 1, + "bbox": [ + 176, + 286, + 480, + 301 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Previous monotonic attention approaches are based on RNN encoder-decoder models with a single attention and haven’t explored the power of the Transformer model. 3 The Transformer architecture (Vaswani et al., 2017) has recently become the state-of-the-art for machine translation (Barrault et al., 2019). An important feature of the Transformer is the use of a separate multihead attention module at each layer. Thus, we propose a new approach, Monotonic Multihead Attention (MMA), which combines the expressive power of multihead attention and the low latency of monotonic attention. ", + "bbox": [ + 173, + 311, + 825, + 410 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Multihead attention allows each decoder layer to have multiple heads, where each head can compute a different attention distribution. Given queries $Q$ , keys $K$ and values $V$ , multihead attention MultiHead $( Q , K , V )$ is defined in Equation 7. ", + "bbox": [ + 174, + 416, + 823, + 459 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/de1d1097b118ce2087ff6b092be4c6925950224ebff552137d1e4295e9316fbe.jpg", + "text": "$$\n\\begin{array} { r } { \\begin{array} { l } { \\mathrm { M u l t i H e a d } ( Q , K , V ) = \\mathrm { C o n c a t } ( \\mathrm { h e a d } _ { 1 } , . . . , \\mathrm { h e a d } _ { H } ) W ^ { O } } \\\\ { \\mathrm { w h e r e ~ h e a d } _ { h } = \\mathrm { A t t e n t i o n } \\left( Q W _ { h } ^ { Q } , K W _ { h } ^ { K } , V W _ { h } ^ { V } , \\right) } \\end{array} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 463, + 691, + 512 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The attention function is the scaled dot-product attention, defined in Equation 8: ", + "bbox": [ + 174, + 512, + 699, + 526 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0d323037e45d49bd45cbb4027ffc444fe6c4f1f38ef8e2f5e64f180bada2e268.jpg", + "text": "$$\n{ \\mathrm { A t t e n t i o n } } ( Q , K , V ) = { \\mathrm { S o f t m a x } } \\left( { \\frac { Q K ^ { T } } { \\sqrt { d _ { k } } } } \\right) V\n$$", + "text_format": "latex", + "bbox": [ + 351, + 530, + 647, + 565 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "There are three applications of multihead attention in the Transformer model: ", + "bbox": [ + 173, + 566, + 681, + 582 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. The Encoder contains self-attention layers where all of the queries, keys and values come from previous layers. \n2. The Decoder contains self-attention layers that allow each position in the decoder to attend to all positions in the decoder up to and including that position. \n3. The Encoder-Decoder attention contains multihead attention layers where queries come from the previous decoder layer and the keys and values come from the output of the encoder. Every decoder layer has a separate encoder-decoder attention. ", + "bbox": [ + 212, + 590, + 825, + 699 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For MMA, we assign each head to operate as a separate monotonic attention in encoder-decoder attention. ", + "bbox": [ + 173, + 708, + 821, + 737 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For a transformer with $L$ decoder layers and $H$ attention heads per layer, we define the selection process of the $h$ -th head encoder-decoder attention in the $l$ -th decoder layer as ", + "bbox": [ + 173, + 743, + 823, + 772 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/06a1000b3fd8419e67564e93aa659485bec477830abd46b839218bb915b6434b.jpg", + "text": "$$\n\\begin{array} { l c l } { e _ { i , j } ^ { { l , h } } } & { = } & { \\left( \\frac { m _ { j } W _ { { l , h } } ^ { K } ( s _ { i - 1 } W _ { { l , h } } ^ { Q } ) ^ { T } } { \\sqrt { d _ { k } } } \\right) _ { i , j } } \\\\ { p _ { i , j } ^ { { l , h } } } & { = } & { \\mathrm { S i g m o i d } ( e _ { i , j } ) } \\\\ { z _ { i , j } ^ { { l , h } } } & { \\sim } & { \\mathrm { B e r n o u l l i } ( p _ { i , j } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 775, + 625, + 866 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/d597335b4efb8a0b81899a74e4bee9cd4514669123bca34eef86fbf7c1b7c310.jpg", + "image_caption": [ + "Figure 1: Monotonic Attention (Left) versus Monotonic Multihead Attention (Right). " + ], + "image_footnote": [], + "bbox": [ + 269, + 97, + 730, + 181 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $W _ { l , h }$ is the input projection matrix, $d _ { k }$ is the dimension of the attention head. We make the selection process independent for each head in each layer. We then investigate two types of MMA, MMA-H(ard) and MMA-IL(infinite lookback). For MMA-H, we use Equation 4 in order to calculate the expected alignment for each layer each head, given $p _ { i , j } ^ { l , h }$ . For MMA-IL, we calculate the softmax energy for each head as follows: ", + "bbox": [ + 174, + 229, + 825, + 303 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/76dbee94d92d243f0b5d5234e42d63c1c42aef73bfbee0d4196977dd708667e3.jpg", + "text": "$$\nu _ { i , j } ^ { l , h } = \\mathrm { S o f t E n e r g y } = \\left( \\frac { m _ { j } \\hat { W } _ { l , h } ^ { K } ( s _ { i - 1 } \\hat { W } _ { l , h } ^ { Q } ) ^ { T } } { \\sqrt { d _ { k } } } \\right) _ { i , j }\n$$", + "text_format": "latex", + "bbox": [ + 316, + 309, + 647, + 356 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "and then use Equation 6 to calculate the expected attention. Each attention head in MMA-H hardattends to one encoder state. On the other hand, each attention head in MMA-IL can attend to all previous encoder states. Thus, MMA-IL allows the model to leverage more information for translation, but MMA-H may be better suited for streaming systems with stricter efficiency requirements. Finally, our models use unidirectional encoders: the encoder self-attention can only attend to previous states, which is also required for simultaneous translation. ", + "bbox": [ + 173, + 361, + 825, + 445 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "At inference time, our decoding strategy is shown in Algorithm 1. For each $l , h$ , at decoding step $i$ , we apply the sampling processes discussed in subsection 2.1 individually and set the encoder step at $t _ { i } ^ { l , \\bar { h } }$ . Then a hard alignment or partial softmax attention from encoder states, shown in Equation 13, will be retrieved to feed into the decoder to generate the $i$ -th token. The model will write a new target token only after all the attentions have decided to write. In other words, the heads that have decided to write must wait until the others have finished reading. ", + "bbox": [ + 174, + 452, + 825, + 537 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/55a64c3eb31107e09be4521a10e290bda8cf5aa3c978766c5de794153806eee4.jpg", + "text": "$$\n\\begin{array}{c} \\begin{array} { r l } & { \\qquad c _ { i } ^ { l } = \\mathrm { C o n c a t } ( c _ { i } ^ { l , 1 } , c _ { i } ^ { l , 2 } , . . . , c _ { i } ^ { l , H } ) } \\\\ & { \\qquad \\mathrm \\quad \\mathbf { \\ \" } } \\\\ & { \\qquad \\mathbf { \\ \" } \\mathbf { \\ \" } e _ { i } ^ { l , h } = f _ { \\mathrm { c o n t e x t } } ( \\boldsymbol { h } , t _ { i } ^ { l , h } ) = \\displaystyle \\left\\{ \\sum _ { j = 1 } ^ { m _ { t _ { i } ^ { l , h } } } \\frac { \\mathrm { e x p } \\left( u _ { i , j } ^ { l , h } \\right) } { \\sum _ { j = 1 } ^ { t _ { i } ^ { l , h } } \\mathrm { e x p } \\left( u _ { i , j } ^ { l , h } \\right) } m _ { j } \\quad \\mathrm { M M A \\mathrm { - } I L } \\right.} \\end{array} \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 542, + 733, + 643 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Figure 1 illustrates a comparison between our model and the monotonic model with one attention head. Compared with the monotonic model, the MMA model is able to set attention to different positions so that it can still attend to previous states while reading each new token. Each head can adjust its speed on-the-fly. Some heads read new inputs, while the others can stay in the past to retain the source history information. Even with the hard alignment variant (MMA-H), the model is still able to preserve the history information by setting heads to past states. In contrast, the hard monotonic model, which only has one head, loses the previous information at the attention layer. ", + "bbox": [ + 173, + 656, + 825, + 755 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 LATENCY CONTROL ", + "text_level": 1, + "bbox": [ + 176, + 772, + 352, + 786 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Effective simultaneous machine translation must balance quality and latency. At a high level, latency measures how many source tokens the model has read until a translation is generated. The model we have introduced in subsection 2.2 is not able to control latency on its own. While MMA allows simultaneous translation by having a read or write schedule for each head, the overall latency is determined by the fastest head, i.e. the head that reads the most. It is possible that a head always reads new input without producing output, which would result in the maximum possible latency. Note that the attention behaviors in MMA-H and MMA-IL can be different. In MMA-IL, a head reaching the end of the sentence will provide the model with maximum information about the source sentence. On the other hand, in the case of MMA-H, reaching the end of sentence for a head only ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 MMA monotonic decoding. Because each head is independent, we compute line 3 to 16 in parallel ", + "bbox": [ + 169, + 103, + 825, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Input: ${ \\bf { \\delta } } _ { \\bf { \\mathcal { X } } } =$ source tokens, $h =$ encoder states, $i = 1 , j = 1 , t _ { 0 } ^ { l , h } = 1 , y _ { 0 } =$ StartOfSequence. ", + "bbox": [ + 174, + 137, + 735, + 154 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/d3ce67f84553fcb769a11cd9c9b6bc0f2cc38f43189af848cae8c2b15b180a38.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
1:while yi-1 ≠ EndOfSequence do
2:tmax =1
3:h = empty sequence
4:forl←1toLdo
5:forh←1to Hdo
6:forj←ttoacldo
7:l,h Pi,j, = Sigmoid (MonotonicEnergy(Si-1,mj ))
8:ifpi l,h > 0.5 then
9:=j
10:l,h =fcontext(h,t,h）
11:Ci Break
12:else
13:if j>tmax then
14:Read token x j
15:Calculate state h j and append to h
16:tmax =j
17:c = Concat(c,1,,cH）
18:s =DecoderLayer'(s-1,s)
19:yi = Output(s)
20:
", + "bbox": [ + 176, + 148, + 830, + 430 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "gives a hard alignment to the end-of-sentence token, which provides very little information to the decoder. Furthermore, it is possible that an MMA-H attention head stays at the beginning of sentence without moving forward. Such a head would not cause latency issues but would degrade the model quality since the decoder would not have any information about the input. In addition, this behavior is not suited for streaming systems. ", + "bbox": [ + 173, + 453, + 825, + 523 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To address these issues, we introduce two latency control methods. The first one is weighted average latency, shown in Equation 14: ", + "bbox": [ + 173, + 530, + 823, + 559 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/dbc1c8df2033f3a8bb69ca4c0b77443bdc70f12f5d733fc36d65bd4af90e8f1b.jpg", + "text": "$$\ng _ { i } ^ { W } = \\frac { \\exp ( g _ { i } ^ { l , h } ) } { \\sum _ { l = 1 } ^ { L } \\sum _ { h = 1 } ^ { H } \\exp ( g _ { i } ^ { l , h } ) } g _ { i } ^ { l , h }\n$$", + "text_format": "latex", + "bbox": [ + 385, + 561, + 611, + 603 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { g _ { i } ^ { l , h } = \\sum _ { j = 1 } ^ { | x | } j \\alpha _ { i , j } } \\end{array}$ . Then we calculate the latency loss with a differentiable latency metric $\\mathcal { C }$ ", + "bbox": [ + 171, + 608, + 821, + 625 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6fb597bba8952ce63096825ef7fd9a9f8875bf203ed1269e09e958a57183435b.jpg", + "text": "$$\n\\begin{array} { r c l } { { L _ { a v g } } } & { { = } } & { { { \\mathcal { C } } \\left( g ^ { W } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 431, + 628, + 565, + 650 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Like Arivazhagan et al. (2019), we use the Differentiable Average Lagging. It is important to note that, unlike the original latency augmented training in Arivazhagan et al. (2019), Equation 15 is not the expected latency metric given $\\mathcal { C }$ , but weighted average $\\mathcal { C }$ on all the attentions. The real expected latency is $\\hat { \\pmb g } = \\mathrm { m a x } _ { l , h } \\left( \\pmb g ^ { l , h } \\right)$ instead of $\\bar { \\pmb { g } }$ , but using this directly would only affect the speed of the fastest head. Equation 15 can control every head in a way that the faster heads will be automatically assigned to larger weights and slower heads will also be moderately regularized. For MMA-H models, we found that the latency of are mainly due to outliers that skip almost every token. The weighted average latency loss is not sufficient to control the outliers. We therefore introduce the head divergence loss, the average variance of expected delays at each step, defined in Equation 16: ", + "bbox": [ + 173, + 651, + 825, + 777 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d133be55bb9c06500803d6386f5b6eccc9edfd580fd7b016a14f265bd0e6ae51.jpg", + "text": "$$\n\\begin{array} { c c l } { { { \\cal L } _ { v a r } } } & { { = } } & { { \\displaystyle \\frac { 1 } { L H } \\sum _ { l = 1 } ^ { L } \\sum _ { h = 1 } ^ { H } \\left( g _ { i } ^ { l , h } - \\bar { g } _ { i } \\right) ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 372, + 780, + 624, + 825 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\bar { g _ { i } } = \\frac { 1 } { L H } \\sum g _ { i } } \\end{array}$ The final objective function is presented in Equation 17: ", + "bbox": [ + 171, + 828, + 683, + 844 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4b3bedce76971cb30dbdf17af15e5198dd875394ef3d36c4eb5806191c3ef503.jpg", + "text": "$$\nL ( \\theta ) = - \\log ( \\pmb { y } \\mid \\pmb { x } ; \\theta ) + \\lambda _ { a v g } L _ { a v g } + \\lambda _ { v a r } L _ { v a r }\n$$", + "text_format": "latex", + "bbox": [ + 338, + 847, + 660, + 864 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\lambda _ { a v g } , \\lambda _ { v a r }$ are hyperparameters that control both losses. Intuitively, while $\\lambda _ { a v g }$ controls the overall speed, $\\lambda _ { v a r }$ controls the divergence of the heads. Combining these two losses, we are able to dynamically control the range of attention heads so that we can control the latency and the reading buffer. For MMA-IL model, we only use $L _ { a v g }$ ; for MMA-H we only use $L _ { v a r }$ . ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 102, + 398, + 118 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 EVALUATION METRICS ", + "text_level": 1, + "bbox": [ + 176, + 132, + 375, + 147 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate our model using quality and latency. For translation quality, we use tokenized BLEU 4 for IWSLT15 En-Vi and detokenized BLEU with SacreBLEU (Post, 2018) for WMT15 De-En. For latency, we use three different recent metrics, Average Proportion (AP) (Cho & Esipova, 2016), Average Lagging (AL) (Ma et al., 2019) and Differentiable Average Lagging (DAL) (Arivazhagan et al., 2019) 5. We remind the reader of the metric definitions in Appendix A.2. ", + "bbox": [ + 174, + 159, + 825, + 229 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 246, + 287, + 260 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/a4adcb998588a1deeeefc79d3a485af2e00f0490e81384c1c82b26e2507977b3.jpg", + "table_caption": [ + "Table 1: Number of sentences in each split. " + ], + "table_footnote": [], + "table_body": "
DatasetTrainValidationTest
IWSLT15 En-Vi133k12681553
WMT15 De-En4.5M30002169
", + "bbox": [ + 348, + 273, + 647, + 327 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/5a20b82351116b7130fbb23c91d9e020a3379820bc239f4a9c9b9c1744ddac41.jpg", + "table_caption": [ + "Table 2: Offline model performance with unidirectional encoder and greedy decoding. " + ], + "table_footnote": [], + "table_body": "
DatasetRNNTransformer
IWSLT15 En-Vi25.6628.7
WMT15 De-En28.4 (Arivazhagan et al., 2019)32.3
", + "bbox": [ + 289, + 352, + 702, + 406 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/3318ac05024d09cb9e614ffff33ca527e2945c2fe72be0d076ab05741cb6c556.jpg", + "table_caption": [], + "table_footnote": [ + "Table 3: Effect of using a unidirectional encoder and greedy decoding to BLEU score. " + ], + "table_body": "
DatasetBeam SearchBidirectional EncoderUnidirectional Encoder
WMT15 De-En132.632.3
433.033.0
IWSLT15 En-Vi128.729.4
1028.829.5
", + "bbox": [ + 236, + 431, + 758, + 510 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate our method on two standard machine translation datasets, IWSLT14 En-Vi and WMT15 De-En. Statistics of the datasets can be found in Table 1. For each dataset, we apply tokenization with the Moses (Koehn et al., 2007) tokenizer and preserve casing. ", + "bbox": [ + 176, + 540, + 823, + 583 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "IWSLT15 English-Vietnamese TED talks from IWSLT 2015 Evaluation Campaign (Cettolo et al., 2016). We follow the settings from Luong & Manning (2015) and Raffel et al. (2017). We replace words with frequency less than 5 by $< u n k >$ . We use tst2012 as a validation set tst2013 as a test set. ", + "bbox": [ + 174, + 593, + 821, + 636 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "WMT15 German-English We follow the setting from Arivazhagan et al. (2019). We apply byte pair encoding (BPE) (Sennrich et al., 2016) jointly on the source and target to construct a shared vocabulary with 32K symbols. We use newstest2013 as validation set and newstest2015 as test set. ", + "bbox": [ + 174, + 638, + 821, + 681 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 MODELS ", + "text_level": 1, + "bbox": [ + 174, + 698, + 276, + 712 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate MMA-H and MMA-IL models on both datasets. The MILK model we evaluate on IWSLT15 En-Vi is based on Luong et al. (2015) rather than RNMT $^ +$ (Chen et al., 2018). In general, our offline models use unidirectional encoders, i.e. the encoder self-attention can only attend to previous states, and greedy decoding. We report offline model performance in Table 2 and the effect of using unidirectional encoders and greedy decoding in Table 3. For MMA models, we replace the encoder-decoder layers with MMA and keep other hyperparameter settings the same as the offline model. Detailed hyperparameter settings can be found in subsection A.1. We use the Fairseq library (Ott et al., 2019) for our implementation. ", + "bbox": [ + 174, + 723, + 825, + 835 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 281, + 117 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we present the main results of our model in terms of latency-quality tradeoffs, ablation studies and analyses. In the first study, we analyze the effect of the variance loss on the attention span. Then, we study the effect of the number of decoder layers and decoder heads on quality and latency. We also provide a case study for the behavior of attention heads in an example. Finally, we study the relationship between the rank of an attention head and the layer it belongs to. ", + "bbox": [ + 174, + 136, + 825, + 205 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e122b7036362526ca9df24fd1d502cd25f56d35adb6686383751be035f195e06.jpg", + "image_caption": [ + "Figure 2: Latency-quality tradeoffs for MILk (Arivazhagan et al., 2019) and MMA on IWSLT15 En-Vi and WMT15 De-En. Black dashed line indicates the unidirectional offline transformer model with greedy search. " + ], + "image_footnote": [], + "bbox": [ + 174, + 224, + 808, + 465 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 LATENCY-QUALITY TRADEOFFS ", + "text_level": 1, + "bbox": [ + 176, + 545, + 437, + 559 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We plot the quality-latency curves for MMA-H and MMA-IL in Figure 2. The BLEU and latency scores on the test sets are generated by setting a latency range and selecting the checkpoint with best BLEU score on the validation set. We use differentiable average lagging (Arivazhagan et al., 2019) when setting the latency range. We find that for a given latency, our models obtain a better translation quality. While MMA-IL tends to have a decrease in quality as the latency decreases, MMA-H has a small gain in quality as latency decreases: a larger latency does not necessarily mean an increase in source information available to the model. In fact, the large latency is from the outlier attention heads, which skip the entire source sentence and point to the end of the sentence. The outliers not only increase the latency but they also do not provide useful information. We introduce the attention variance loss to eliminate the outliers, as such a loss makes the attention heads focus on the current context for translating the new target token. ", + "bbox": [ + 174, + 573, + 825, + 726 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "It is interesting to observe that MMA-H has a better latency-quality tradeoff than $\\mathrm { M I L K } ^ { 7 }$ even though each head only attends to only one state. Although MMA-H is not yet able to handle an arbitrarily long input (without resorting to segmenting the input), since both encoder and decoder self-attention have an infinite lookback, that model represents a good step in that direction. ", + "bbox": [ + 174, + 733, + 825, + 789 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 ATTENTION SPAN ", + "text_level": 1, + "bbox": [ + 176, + 809, + 338, + 824 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In subsection 2.3, we introduced the attention variance loss to MMA-H in order to prevent outlier attention heads from increasing the latency or increasing the attention span. We have already evaluated the effectiveness of this method on latency in subsection 4.1. We also want to measure the difference between the fastest and slowest heads at each decoding step. We define the average ", + "bbox": [ + 174, + 837, + 825, + 892 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "attention span in Equation 18: ", + "bbox": [ + 174, + 103, + 370, + 118 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/abf4018244aad5899baee17fe91fedebf02e95cb4d0f3e9cc679c572b65b9cc5.jpg", + "text": "$$\n\\bar { S } = \\frac { 1 } { \\vert \\pmb { y } \\vert } \\left( \\sum _ { i } ^ { \\vert \\pmb { y } \\vert } \\underset { l , h } { \\operatorname* { m a x } } t _ { i } ^ { l , h } - \\underset { l , h } { \\operatorname* { m i n } } t _ { i } ^ { l , h } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 375, + 126, + 620, + 176 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "It estimates the reading buffer we need for streaming translation. We show the relation between the average attention span versus $\\lambda _ { v a r }$ in Figure 3. As expected, the average attention span is reduced as we increase $\\lambda _ { v a r }$ . ", + "bbox": [ + 174, + 183, + 825, + 227 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/0f751c2faf86a11373cba7cf525d322dc182947f9c6a18f187c49c68d5fad146.jpg", + "image_caption": [ + "Figure 3: Effect of $\\lambda _ { v a r }$ on the average attention span. The variance loss works as intended by reducing the span with higher weights. " + ], + "image_footnote": [], + "bbox": [ + 289, + 239, + 710, + 351 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 EFFECT ON NUMBER OF LAYERS AND NUMBER OF HEADS ", + "text_level": 1, + "bbox": [ + 174, + 428, + 611, + 440 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "One motivation to introduce MMA is to adapt the Transformer, which is the current state-of-the-art model for machine translation, to online decoding. Important features of the Transformer architecture include having a separate attention layer for each decoder layer block and multihead attention. In this section, we test the effect of these two components on the offline, MMA-H, and MMA-IL models from a quality and latency perspective. We report quality as measured by detokenized BLEU and latency as measured by DAL on the WMT13 validation set in Figure 4. We set $\\lambda _ { a v g } = 0 . 2$ for MMA-IL and $\\lambda _ { v a r } = 0 . 2$ for MMA-H. ", + "bbox": [ + 174, + 453, + 825, + 550 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The offline model benefits from having more than one decoder layer. In the case of 1 decoder layer, increasing the number of attention heads is beneficial but in the case of 3 and 6 decoder layers, we do not see much benefit from using more than 2 heads. The best performance is obtained for 3 layers and 2 heads (6 effective heads). The MMA-IL model behaves similarly to the offline model, and the best performance is observed with 6 layers and 4 heads (24 effective heads). For MMA-H, with 1 layer, performance improves with more heads. With 3 layers, the single-head setting is the most effective (3 effective heads). Finally, with 6 layers, the best performance is reached with 16 heads (96 effective heads). ", + "bbox": [ + 174, + 558, + 825, + 669 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The general trend we observe is that performance improves as we increase the number of effective heads, either from multiple layers or multihead attention, up to a certain point, then either plateaus or degrades. This motivates the introduction of the MMA model. ", + "bbox": [ + 176, + 676, + 821, + 718 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also note that latency increases with the number of effective attention heads. This is due to having fixed loss weights: when more heads are involved, we should increase $\\lambda _ { v a r }$ or $\\lambda _ { a v g }$ to better control latency. ", + "bbox": [ + 174, + 724, + 825, + 767 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 ATTENTION BEHAVIORS ", + "text_level": 1, + "bbox": [ + 176, + 786, + 380, + 800 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We characterize attention behaviors by providing a running example of MMA-H and MMA-IL, shown in Figure 5. Each curve represents the path that an attention head goes through at inference time. For MMA-H, shown in Figure 5a, we found that when the source and target tokens have the same order, the attention heads behave linearly and the distance between fastest head and slowest head is small. For example, this can be observed from partial sentence pair “I also didn’t know that” and target tokens “Tôi cũng không biết rằng”, which have the same order. However, when the source tokens and target tokens have different orders, such as “the second step” and “bước (step) thứ hai (second)”, the model will generate “bước (step)” first and some heads will stay in the past to retain the information for later reordered translation “thứ hai (second)”. We can also see that the attention heads have a near-diagonal trajectory, which is appropriate for streaming inputs. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/587b8944db8cce31706c574a36a9ec6d40a868064c34c383e996803aed29b0b0.jpg", + "image_caption": [ + "Figure 4: Effect of the number of decoder attention heads and the number of decoder attention layers on quality and latency, reported on the WMT13 validation set. " + ], + "image_footnote": [], + "bbox": [ + 189, + 103, + 810, + 343 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 412, + 821, + 441 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The behavior of the heads in MMA-IL models is shown in Figure 5b. Notice that we remove the partial softmax alignment in this figure. We don’t expect streaming capability for MMA-IL: some heads stop at early position of the source sentence to retain the history information. Moreover, because MMA-IL has more information when generating a new target token, it tends to produce translations with better quality. In this example, the MMA-IL model has a better translation on “isolate the victim” than MMA-H (“là cô lập nạn nhân” vs “là tách biệt nạn nhân”) ", + "bbox": [ + 174, + 448, + 825, + 532 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/f9caf5f55023339b16fb95b389bc65cc48ec79600f2fc76d230026679d0627e7.jpg", + "image_caption": [ + "Figure 5: Running examples on IWSLT15 English-Vietnamese dataset " + ], + "image_footnote": [], + "bbox": [ + 240, + 550, + 758, + 767 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.5 RANK OF THE HEADS ", + "text_level": 1, + "bbox": [ + 176, + 827, + 362, + 840 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In Figure 6, we calculate the average and standard deviation of rank of each head when generating every target token. For MMA-IL, we find that heads in lower layers tend to have higher rank and are thus slower. However, in MMA-H, the difference of the average rank are smaller. Furthermore, the standard deviation is very large which means that the order of the heads in MMA-H changes frequently over the inference process. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/945370a5e1d53aa3ee0f21fc5e5f3110a0171fb97a8e1797f201cb0464090bba.jpg", + "image_caption": [ + "Figure 6: The average rank of attention heads during inference on IWSLT15 En-Vi. Error bars indicate the standard deviation. L indicates the layer number and $\\mathrm { H }$ indicates the head number. " + ], + "image_footnote": [], + "bbox": [ + 238, + 101, + 754, + 299 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 361, + 344, + 377 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Recent work on simultaneous machine translation falls into three categories. In the first one, models use a rule-based policy for reading input and writing output. Cho & Esipova (2016) propose a WaitIf-\\* policy to enable an offline model to decode simultaneously. Ma et al. (2019) propose a wait- $k$ policy where the model first reads $k$ tokens, then alternates between read and write actions. Dalvi et al. (2018) propose an incremental decoding method, also based on a rule-based schedule. In the second category, a flexible policy is learnt from data. Grissom II et al. (2014) introduce a Markov chain to phrase-based machine translation models for simultaneous machine translation, in which they apply reinforcement learning to learn the read-write policy based on states. Gu et al. (2017) introduce an agent which learns to make decisions on when to translate from the interaction with a pre-trained offline neural machine translation model. Luo et al. (2017) used continuous rewards policy gradient for online alignments for speech recognition. Lawson et al. (2018) proposed a hard alignment with variational inference for online decoding. Alinejad et al. (2018) propose a new operation ”predict” which predicts future source tokens. Zheng et al. (2019b) introduce a restricted dynamic oracle and restricted imitation learning for simultaneous translation. Zheng et al. (2019a) train the agent with an action sequence from labels that are generated based on the rank of the gold target word given partial input. Models from the last category leverage monotonic attention and replace the softmax attention with an expected attention calculated from a stepwise Bernoulli selection probability. Raffel et al. (2017) first introduce the concept of monotonic attention for online linear time decoding, where the attention only attends to one encoder state at a time. Chiu & Raffel (2018) extended that work to let the model attend to a chunk of encoder state. Arivazhagan et al. (2019) also make use of the monotonic attention but introduce an infinite lookback to improve the translation quality. ", + "bbox": [ + 173, + 392, + 825, + 698 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 718, + 318, + 734 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we propose two variants of the monotonic multihead attention model for simultaneous machine translation. By introducing two new targeted loss terms which allow us to control both latency and attention span, we are able to leverage the power of the Transformer architecture to achieve better quality-latency trade-offs than the previous state-of-the-art model. We also present detailed ablation studies demonstrating the efficacy and rationale of our approach. By introducing these stronger simultaneous sequence-to-sequence models, we hope to facilitate important applications, such as high-quality real-time interpretation between human speakers. 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", + "bbox": [ + 174, + 685, + 825, + 755 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 781, + 299, + 796 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1 HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 176, + 811, + 361, + 827 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The hyperparameters we used for offline and monotonic transformer models are defined in Table 4. ", + "bbox": [ + 173, + 838, + 818, + 853 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 LATENCY METRICS DEFINITIONS ", + "text_level": 1, + "bbox": [ + 176, + 868, + 450, + 883 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Given the delays $\\mathbf { g } = \\{ g _ { 1 } , g _ { 2 } , . . . , g _ { | \\mathbf { y } | } \\}$ of generating each target token, AP, AL and DAL are defined in Table 5. ", + "bbox": [ + 174, + 895, + 823, + 922 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/786fa73e10dfc49d9f8315c3ad84f607dc02200090f3349dded387e08a2387cf.jpg", + "table_caption": [ + "Table 4: Offline and monotonic models hyperparameters. " + ], + "table_footnote": [], + "table_body": "
HyperparameterWMT15 German-EnglishIWSLT English-Vietnamese
encoder embed dim1024512
encoder ffn embed dim40961024
encoder attention heads164
encoder layers6
decoder embed dim1024512
decoder ffn embed dim40961024
decoder attention heads164
decoder layers6
dropout0.3
optimizeradam
adam-β(0.9,0.98)
clip-norm0.0
lr lr scheduler0.0005
inverse sqrt
warmup-updates4000
warmup-init-lr1e-07
label-smoothing0.1
max tokens3584×8×8×216000
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Latency Metric
Average Proportiony 1 M gi lacllyl i=1
Average Lagging1 T i-1 gi 1y1/ac T i=1 where T = arg maxi(gi = |xl)
1 ly Differentiable Average Lagginglyl i-1 M gi lyl//ac| i=1 gi i=0 where g' = ly max(gi, 9i-1 i<0 [x
", + "bbox": [ + 232, + 434, + 766, + 670 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.3 DETAILED RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 738, + 354, + 752 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We provide the detailed results in Figure 2 as Table 6 and Table 7. ", + "bbox": [ + 174, + 770, + 606, + 785 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.4 THRESHOLD OF READING ACTION ", + "text_level": 1, + "bbox": [ + 176, + 820, + 452, + 835 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We explore a simple method that can adjust system’s latency at inference time without training new models. In Algorithm 1 line 8, 0.5 was used as an threshold. One can set different threshold $p$ during the inference time to control the latency. We run the pilot experiments on IWSLT15 En-Vi dataset and the results are shown as Table 8. Although this method doesn’t require training new model, it dramatically hurts the translation quality. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/a524a579244b5b105b551d4c29c7040c9b89dc48418684ab944b7caf945bb225.jpg", + "table_caption": [ + "Table 6: Detailed results for MMA-H and MMA-IL on WMT15 DeEn " + ], + "table_footnote": [], + "table_body": "
BLEU AP AL DAL
Xavg MMA-IL
0.05 30.7 0.78 10.91 12.64
0.1 30.5 0.70 7.42 8.82
0.2 30.1 0.63 5.17 6.41
0.3 30.3 0.60 4.18 5.35
0.4 29.2 0.59 3.75 4.90
0.5 26.7 0.59 3.69 4.83
0.75 25.5 0.58 3.40 4.46
1.0 25.1 0.56 3.00 4.03
Xvar MMA-H
0.1 28.5 0.74 8.94 10.83
0.2 28.9 0.69 6.82 8.622
0.3 29.2 0.64 5.45 7.03
0.4 28.5 0.59 3.90 5.21
0.5 28.5 0.59 3.88 5.19
0.6 29.6 0.56 3.13 4.32
0.7 29.1 0.56 2.93 4.10
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BLEUAPALDAL
入MILk
0.124.620.715.937.19
0.224.680.674.905.97
0.324.310.654.455.43
0.423.730.644.285.24
XaugMMA-IL
0.0228.280.767.098.29
0.0428.330.705.446.57
0.128.420.674.635.65
0.228.470.633.574.44
0.327.90.592.983.81
0.427.730.582.683.46
XuarMMA-H
0.0227.260.777.528.71
0.15.226.31
27.680.693.81
0.228.060.634.84
0.427.790.623.574.59
0.827.950.603.224.19
", + "bbox": [ + 366, + 422, + 632, + 734 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.5 AVERAGE LOSS FOR MMA-H ", + "text_level": 1, + "bbox": [ + 176, + 784, + 423, + 799 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We explore applying a simple average instead of a weighted average loss to MMA-H. The results are shown in Figure 7 and Table 9. We find that even with very large weights, we are unable to reduce the overall latency. In addition, we find that the weighted average loss severely affects the translation quality negatively. On the other hand, the divergence loss we propose in Equation 16 can efficiently reduce the latency while retaining relatively good translation quality for MMA-H models. ", + "bbox": [ + 174, + 810, + 825, + 881 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/4d553d6e69ce5de1583cfa97a411d692404d8e6b604846e59cf235e6f7445988.jpg", + "table_caption": [], + "table_footnote": [ + "Table 8: Comparison between setting threshold for reading action and weighted average latency loss. " + ], + "table_body": "
Reading ThresholdWeighted Average Latency Loss
pBLEUAPALDALLavgBLEUAPALDAL
0.525.50.7923877.136738.271870.0225.50.7923877.136738.27187
0.425.250.737495.720036.858120.0425.680.7281075.528566.61744
0.323.060.6973984.880876.033420.324.90.6027032.900543.68039
0.218.370.6782984.710995.946360.225.30.6369143.545774.38623
0.18.730.6964525.52257.204390.125.480.6844244.579015.54102
", + "bbox": [ + 173, + 111, + 844, + 231 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/aea53beea98c86b1baeb911bfa9ca93df3d5c23c9dfd3fe4f079a502cf059346.jpg", + "image_caption": [ + "Figure 7: Effect average loss, weighted average loss and variance loss on MMA-H on WMT15 DeEn development set. ", + "Table 9: Detailed numbers on average loss, weighted average loss and head divergence loss on WMT15 De-En development set " + ], + "image_footnote": [], + "bbox": [ + 184, + 286, + 810, + 453 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/4e530cf2f50274c42af2cc82e308e69bcf9358bdf52a0d674618eb44495a7b1b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
BLEUAPALDAL
XaugAverage
0.0528.50.86258114.384717.3702
0.127.80.85543513.97417.02
0.2280.83532412.853115.908
0.428.10.81940811.981614.9763
1.028.20.81060911.752814.6695
2.028.10.80025811.176314.0761
8.028.40.80643911.528914.6431
XaugWeighted Average
0.0228.240.77392210.210912.2274
0.0424.350.6858347.067168.64069
0.067.800.87582516.204619.0892
0.089.510.573723.920116.1421
0.19.780.5565853.30075.46142
XvarDivergence
0.127.350.7360258.7096810.5253
0.227.640.6814916.639148.3856
0.327.370.66236.049027.71922
0.427.620.6381885.316726.86834
0.527.500.6257594.930446.38998
1.027.10.5821943.648644.90997
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Such models are useful in any situation where translation needs to be done in real time. For", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 464, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 504, + 476 + ], + "score": 1.0, + "content": "example, simultaneous models can translate live video captions or facilitate conversations between", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "people speaking different languages. In a usual translation model, the encoder first reads the entire", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "sentence, then the decoder writes the target sentence. 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In this", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "score": 1.0, + "content": "paper, we propose a new attention mechanism, Monotonic Multihead Attention", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 469, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 469, + 301 + ], + "score": 1.0, + "content": "(MMA), which extends the monotonic attention mechanism to multihead atten-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "score": 1.0, + "content": "tion. 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Such models are useful in any situation where translation needs to be done in real time. For", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 464, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 504, + 476 + ], + "score": 1.0, + "content": "example, simultaneous models can translate live video captions or facilitate conversations between", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "people speaking different languages. In a usual translation model, the encoder first reads the entire", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "sentence, then the decoder writes the target sentence. 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Recent work exploring monotonic attention variants for simultaneous", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "translation include: hard monotonic attention (Raffel et al., 2017), monotonic chunkwise attention", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "(MoChA) (Chiu & Raffel, 2018) and monotonic infinite lookback attention (MILk) (Arivazhagan", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 567, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 505, + 582 + ], + "score": 1.0, + "content": "et al., 2019). MILk in particular has shown better quality/latency trade-offs than fixed policy ap-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 200, + 591 + ], + "score": 1.0, + "content": "proaches, such as wait-", + "type": "text" + }, + { + "bbox": [ + 200, + 580, + 207, + 589 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "(Ma et al., 2019) or wait-if-* (Cho & Esipova, 2016) policies. MILk also", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "outperforms hard monotonic attention and MoChA; while the other two monotonic attention mech-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 602, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 505, + 613 + ], + "score": 1.0, + "content": "anisms only consider a fixed window, MILk computes a softmax attention over all previous encoder", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "states, which may be the key to its improved latency-quality tradeoffs. These monotonic attention", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "approaches also provide a closed-form expression for the expected alignment between source and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 635, + 163, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 163, + 646 + ], + "score": 1.0, + "content": "target tokens.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 524, + 506, + 646 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 651, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 104, + 649, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 104, + 649, + 506, + 664 + ], + "score": 1.0, + "content": "However, monotonic attention-based models, including the state-of-the-art MILk, were built on top", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "of RNN-based models. RNN-based models have been outperformed by the recent state-of-the-art", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 672, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 686 + ], + "score": 1.0, + "content": "Transformer model (Vaswani et al., 2017), which features multiple encoder-decoder attention layers", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 684, + 258, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 258, + 696 + ], + "score": 1.0, + "content": "and multihead attention at each layer.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 104, + 649, + 506, + 696 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "We thus propose monotonic multihead attention (MMA), which combines the high translation qual-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "ity from multilayer multihead attention and low latency from monotonic attention. We propose two", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "variants, Hard MMA (MMA-H) and Infinite Lookback MMA (MMA-IL). MMA-H is designed with", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "streaming systems in mind where the attention span must be limited. MMA-IL emphasizes the qual-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "ity of the translation system. We also propose two novel latency regularization methods. The first", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "encourages the model to be faster by directly minimizing the average latency. The second encour-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "ages the attention heads to maintain similar positions, preventing the latency from being dominated", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 215, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 215, + 171 + ], + "score": 1.0, + "content": "by a single or a few heads.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "The main contributions of this paper are: (1) A novel monotonic attention mechanism, monotonic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "multihead attention, which enables the Transformer model to perform online decoding. This model", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "leverages the power of the Transformer and the efficiency of monotonic attention. (2) Better la-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "tency/quality tradeoffs compared to the MILk model, the previous state-of-the-art, on two standard", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 504, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 504, + 231 + ], + "score": 1.0, + "content": "translation benchmarks, IWSLT15 English-Vietnamese (En-Vi) and WMT15 German-English (De-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "En). (3) Analyses on how our model is able to control the attention span and on the relationship", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "between the speed of a head and the layer it belongs to. We motivate the design of our model with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 445, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 445, + 265 + ], + "score": 1.0, + "content": "an ablation study on the number of decoder layers and the number of decoder heads.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 279, + 362, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 365, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 365, + 294 + ], + "score": 1.0, + "content": "2 MONOTONIC MULTIHEAD ATTENTION MODEL", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "In this section, we review the monotonic attention-based approaches in RNN-based encoder-decoder", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "models. We then introduce the two types of Monotonic Multihead Attention (MMA) for Transformer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 495, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 495, + 340 + ], + "score": 1.0, + "content": "models: MMA-H and MMA-IL. Finally, we introduce strategies to control latency and coverage.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 107, + 350, + 239, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 240, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 240, + 363 + ], + "score": 1.0, + "content": "2.1 MONOTONIC ATTENTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 472 + ], + "lines": [ + { + "bbox": [ + 107, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 107, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "The hard monotonic attention mechanism (Raffel et al., 2017) was first introduced in order to achieve", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 380, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 396 + ], + "score": 1.0, + "content": "online linear time decoding for RNN-based encoder-decoder models. We denote the input sequence", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 392, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 392, + 118, + 407 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 393, + 191, + 405 + ], + "score": 0.93, + "content": "\\mathbf { x } = \\{ x _ { 1 } , . . . , x _ { T } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 392, + 361, + 407 + ], + "score": 1.0, + "content": ", and the corresponding encoder states as", + "type": "text" + }, + { + "bbox": [ + 362, + 393, + 443, + 405 + ], + "score": 0.92, + "content": "\\mathbf { m } = \\{ m _ { 1 } , . . . , m _ { T } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 392, + 469, + 407 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 470, + 394, + 478, + 403 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 392, + 506, + 407 + ], + "score": 1.0, + "content": "being", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 411, + 418 + ], + "score": 1.0, + "content": "the length of the source sequence. The model generates a target sequence", + "type": "text" + }, + { + "bbox": [ + 411, + 405, + 483, + 416 + ], + "score": 0.93, + "content": "\\mathbf { y } = \\{ y _ { 1 } , . . . , y _ { U } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 403, + 506, + 418 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 115, + 425 + ], + "score": 0.77, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 414, + 312, + 428 + ], + "score": 1.0, + "content": "being the length of the target sequence. 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We propose two", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "variants, Hard MMA (MMA-H) and Infinite Lookback MMA (MMA-IL). MMA-H is designed with", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "streaming systems in mind where the attention span must be limited. MMA-IL emphasizes the qual-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "ity of the translation system. We also propose two novel latency regularization methods. The first", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "encourages the model to be faster by directly minimizing the average latency. The second encour-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "ages the attention heads to maintain similar positions, preventing the latency from being dominated", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 215, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 215, + 171 + ], + "score": 1.0, + "content": "by a single or a few heads.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 505, + 171 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "The main contributions of this paper are: (1) A novel monotonic attention mechanism, monotonic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "multihead attention, which enables the Transformer model to perform online decoding. This model", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "leverages the power of the Transformer and the efficiency of monotonic attention. (2) Better la-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "tency/quality tradeoffs compared to the MILk model, the previous state-of-the-art, on two standard", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 504, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 504, + 231 + ], + "score": 1.0, + "content": "translation benchmarks, IWSLT15 English-Vietnamese (En-Vi) and WMT15 German-English (De-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "En). (3) Analyses on how our model is able to control the attention span and on the relationship", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "between the speed of a head and the layer it belongs to. We motivate the design of our model with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 445, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 445, + 265 + ], + "score": 1.0, + "content": "an ablation study on the number of decoder layers and the number of decoder heads.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 176, + 505, + 265 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 279, + 362, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 365, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 365, + 294 + ], + "score": 1.0, + "content": "2 MONOTONIC MULTIHEAD ATTENTION MODEL", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "In this section, we review the monotonic attention-based approaches in RNN-based encoder-decoder", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "models. We then introduce the two types of Monotonic Multihead Attention (MMA) for Transformer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 495, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 495, + 340 + ], + "score": 1.0, + "content": "models: MMA-H and MMA-IL. Finally, we introduce strategies to control latency and coverage.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 304, + 505, + 340 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 350, + 239, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 240, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 240, + 363 + ], + "score": 1.0, + "content": "2.1 MONOTONIC ATTENTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 472 + ], + "lines": [ + { + "bbox": [ + 107, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 107, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "The hard monotonic attention mechanism (Raffel et al., 2017) was first introduced in order to achieve", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 380, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 396 + ], + "score": 1.0, + "content": "online linear time decoding for RNN-based encoder-decoder models. We denote the input sequence", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 392, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 392, + 118, + 407 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 393, + 191, + 405 + ], + "score": 0.93, + "content": "\\mathbf { x } = \\{ x _ { 1 } , . . . , x _ { T } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 392, + 361, + 407 + ], + "score": 1.0, + "content": ", and the corresponding encoder states as", + "type": "text" + }, + { + "bbox": [ + 362, + 393, + 443, + 405 + ], + "score": 0.92, + "content": "\\mathbf { m } = \\{ m _ { 1 } , . . . , m _ { T } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 392, + 469, + 407 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 470, + 394, + 478, + 403 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 392, + 506, + 407 + ], + "score": 1.0, + "content": "being", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 411, + 418 + ], + "score": 1.0, + "content": "the length of the source sequence. 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Given queries", + "type": "text" + }, + { + "bbox": [ + 320, + 342, + 329, + 353 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 341, + 354, + 354 + ], + "score": 1.0, + "content": ", keys", + "type": "text" + }, + { + "bbox": [ + 354, + 342, + 365, + 352 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 341, + 411, + 354 + ], + "score": 1.0, + "content": "and values", + "type": "text" + }, + { + "bbox": [ + 411, + 342, + 420, + 352 + ], + "score": 0.8, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 341, + 506, + 354 + ], + "score": 1.0, + "content": ", multihead attention", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 352, + 293, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 150, + 364 + ], + "score": 1.0, + "content": "MultiHead", + "type": "text" + }, + { + "bbox": [ + 150, + 352, + 192, + 365 + ], + "score": 0.93, + "content": "( Q , K , V )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 353, + 293, + 364 + ], + "score": 1.0, + "content": "is defined in Equation 7.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 330, + 506, + 365 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 367, + 423, + 406 + ], + "lines": [ + { + "bbox": [ + 183, + 367, + 423, + 406 + ], + "spans": [ + { + "bbox": [ + 183, + 367, + 423, + 406 + ], + "score": 0.49, + "content": "\\begin{array} { r } { \\begin{array} { l } { \\mathrm { M u l t i H e a d } ( Q , K , V ) = \\mathrm { C o n c a t } ( \\mathrm { h e a d } _ { 1 } , . . . , \\mathrm { h e a d } _ { H } ) W ^ { O } } \\\\ { \\mathrm { w h e r e ~ h e a d } _ { h } = \\mathrm { A t t e n t i o n } \\left( Q W _ { h } ^ { Q } , K W _ { h } ^ { K } , V W _ { h } ^ { V } , \\right) } \\end{array} } \\end{array}", + "type": "interline_equation", + "image_path": "de1d1097b118ce2087ff6b092be4c6925950224ebff552137d1e4295e9316fbe.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 182, + 367, + 423, + 380.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 182, + 380.0, + 423, + 393.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 182, + 393.0, + 423, + 406.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 428, + 417 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 429, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 429, + 419 + ], + "score": 1.0, + "content": "The attention function is the scaled dot-product attention, defined in Equation 8:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 405, + 429, + 419 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 420, + 396, + 448 + ], + "lines": [ + { + "bbox": [ + 215, + 420, + 396, + 448 + ], + "spans": [ + { + "bbox": [ + 215, + 420, + 396, + 448 + ], + "score": 0.92, + "content": "{ \\mathrm { A t t e n t i o n } } ( Q , K , V ) = { \\mathrm { S o f t m a x } } \\left( { \\frac { Q K ^ { T } } { \\sqrt { d _ { k } } } } \\right) V", + "type": "interline_equation", + "image_path": "0d323037e45d49bd45cbb4027ffc444fe6c4f1f38ef8e2f5e64f180bada2e268.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 420, + 396, + 434.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 215, + 434.0, + 396, + 448.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 449, + 417, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 418, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 418, + 462 + ], + "score": 1.0, + "content": "There are three applications of multihead attention in the Transformer model:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 448, + 418, + 462 + ] + }, + { + "type": "list", + "bbox": [ + 130, + 468, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 129, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 129, + 468, + 505, + 482 + ], + "score": 1.0, + "content": "1. 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1 } W _ { { l , h } } ^ { Q } ) ^ { T } } { \\sqrt { d _ { k } } } \\right) _ { i , j } } \\\\ { p _ { i , j } ^ { { l , h } } } & { = } & { \\mathrm { S i g m o i d } ( e _ { i , j } ) } \\\\ { z _ { i , j } ^ { { l , h } } } & { \\sim } & { \\mathrm { B e r n o u l l i } ( p _ { i , j } ) } \\end{array}", + "type": "interline_equation", + "image_path": "06a1000b3fd8419e67564e93aa659485bec477830abd46b839218bb915b6434b.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 226, + 614, + 383, + 628.4 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 226, + 628.4, + 383, + 642.8 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 226, + 642.8, + 383, + 657.1999999999999 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 226, + 657.1999999999999, + 383, + 671.5999999999999 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 226, + 671.5999999999999, + 383, + 685.9999999999999 + ], + "spans": [], + "index": 43 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 165, + 77, + 447, + 144 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 165, + 77, + 447, + 144 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 77, + 447, + 144 + ], + "spans": [ + { + "bbox": [ + 165, + 77, + 447, + 144 + ], + "score": 0.958, + "type": "image", + "image_path": "d597335b4efb8a0b81899a74e4bee9cd4514669123bca34eef86fbf7c1b7c310.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 165, + 77, + 447, + 99.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 165, + 99.33333333333333, + 447, + 121.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 165, + 121.66666666666666, + 447, + 144.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 150, + 150, + 460, + 161 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 149, + 461, + 163 + ], + "spans": [ + { + "bbox": [ + 149, + 149, + 461, + 163 + ], + "score": 1.0, + "content": "Figure 1: Monotonic Attention (Left) versus Monotonic Multihead Attention (Right).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 134, + 196 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 183, + 155, + 195 + ], + "score": 0.91, + "content": "W _ { l , h }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 182, + 284, + 196 + ], + "score": 1.0, + "content": "is the input projection matrix,", + "type": "text" + }, + { + "bbox": [ + 285, + 183, + 296, + 194 + ], + "score": 0.87, + "content": "d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 182, + 505, + 196 + ], + "score": 1.0, + "content": "is the dimension of the attention head. We make", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 193, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 206 + ], + "score": 1.0, + "content": "the selection process independent for each head in each layer. We then investigate two types of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "MMA, MMA-H(ard) and MMA-IL(infinite lookback). For MMA-H, we use Equation 4 in order to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 216, + 507, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 216, + 370, + 234 + ], + "score": 1.0, + "content": "calculate the expected alignment for each layer each head, given", + "type": "text" + }, + { + "bbox": [ + 370, + 216, + 387, + 231 + ], + "score": 0.92, + "content": "p _ { i , j } ^ { l , h }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 216, + 507, + 234 + ], + "score": 1.0, + "content": ". For MMA-IL, we calculate", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 229, + 287, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 287, + 241 + ], + "score": 1.0, + "content": "the softmax energy for each head as follows:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 245, + 396, + 282 + ], + "lines": [ + { + "bbox": [ + 194, + 245, + 396, + 282 + ], + "spans": [ + { + "bbox": [ + 194, + 245, + 396, + 282 + ], + "score": 0.94, + "content": "u _ { i , j } ^ { l , h } = \\mathrm { S o f t E n e r g y } = \\left( \\frac { m _ { j } \\hat { W } _ { l , h } ^ { K } ( s _ { i - 1 } \\hat { W } _ { l , h } ^ { Q } ) ^ { T } } { \\sqrt { d _ { k } } } \\right) _ { i , j }", + "type": "interline_equation", + "image_path": "76dbee94d92d243f0b5d5234e42d63c1c42aef73bfbee0d4196977dd708667e3.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 245, + 396, + 263.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 194, + 263.5, + 396, + 282.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "score": 1.0, + "content": "and then use Equation 6 to calculate the expected attention. Each attention head in MMA-H hard-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 298, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 308 + ], + "score": 1.0, + "content": "attends to one encoder state. On the other hand, each attention head in MMA-IL can attend to all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "previous encoder states. Thus, MMA-IL allows the model to leverage more information for transla-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 319, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 504, + 331 + ], + "score": 1.0, + "content": "tion, but MMA-H may be better suited for streaming systems with stricter efficiency requirements.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 330, + 504, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 504, + 343 + ], + "score": 1.0, + "content": "Finally, our models use unidirectional encoders: the encoder self-attention can only attend to previ-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 343, + 356, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 356, + 353 + ], + "score": 1.0, + "content": "ous states, which is also required for simultaneous translation.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 410, + 371 + ], + "score": 1.0, + "content": "At inference time, our decoding strategy is shown in Algorithm 1. 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Then a hard alignment or partial softmax attention from encoder states, shown in Equation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 347, + 405 + ], + "score": 1.0, + "content": "13, will be retrieved to feed into the decoder to generate the", + "type": "text" + }, + { + "bbox": [ + 348, + 394, + 352, + 403 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "-th token. The model will write a new", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "target token only after all the attentions have decided to write. In other words, the heads that have", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 413, + 366, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 366, + 428 + ], + "score": 1.0, + "content": "decided to write must wait until the others have finished reading.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 430, + 449, + 510 + ], + "lines": [ + { + "bbox": [ + 181, + 430, + 449, + 510 + ], + "spans": [ + { + "bbox": [ + 181, + 430, + 449, + 510 + ], + "score": 0.87, + "content": "\\begin{array}{c} \\begin{array} { r l } & { \\qquad c _ { i } ^ { l } = \\mathrm { C o n c a t } ( c _ { i } ^ { l , 1 } , c _ { i } ^ { l , 2 } , . . . , c _ { i } ^ { l , H } ) } \\\\ & { \\qquad \\mathrm \\quad \\mathbf { \\ \" } } \\\\ & { \\qquad \\mathbf { \\ \" } \\mathbf { \\ \" } e _ { i } ^ { l , h } = f _ { \\mathrm { c o n t e x t } } ( \\boldsymbol { h } , t _ { i } ^ { l , h } ) = \\displaystyle \\left\\{ \\sum _ { j = 1 } ^ { m _ { t _ { i } ^ { l , h } } } \\frac { \\mathrm { e x p } \\left( u _ { i , j } ^ { l , h } \\right) } { \\sum _ { j = 1 } ^ { t _ { i } ^ { l , h } } \\mathrm { e x p } \\left( u _ { i , j } ^ { l , h } \\right) } m _ { j } \\quad \\mathrm { M M A \\mathrm { - } I L } \\right.} \\end{array} \\end{array}", + "type": "interline_equation", + "image_path": "55a64c3eb31107e09be4521a10e290bda8cf5aa3c978766c5de794153806eee4.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 181, + 430, + 449, + 456.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 181, + 456.6666666666667, + 449, + 483.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 181, + 483.33333333333337, + 449, + 510.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "Figure 1 illustrates a comparison between our model and the monotonic model with one attention", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "score": 1.0, + "content": "head. Compared with the monotonic model, the MMA model is able to set attention to different", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "positions so that it can still attend to previous states while reading each new token. Each head can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 554, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 566 + ], + "score": 1.0, + "content": "adjust its speed on-the-fly. Some heads read new inputs, while the others can stay in the past to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "retain the source history information. Even with the hard alignment variant (MMA-H), the model", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 576, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 587 + ], + "score": 1.0, + "content": "is still able to preserve the history information by setting heads to past states. In contrast, the hard", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 587, + 494, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 494, + 599 + ], + "score": 1.0, + "content": "monotonic model, which only has one head, loses the previous information at the attention layer.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 612, + 216, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 218, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 218, + 625 + ], + "score": 1.0, + "content": "2.3 LATENCY CONTROL", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "Effective simultaneous machine translation must balance quality and latency. At a high level, latency", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "measures how many source tokens the model has read until a translation is generated. The model", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "we have introduced in subsection 2.2 is not able to control latency on its own. While MMA allows", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "simultaneous translation by having a read or write schedule for each head, the overall latency is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "determined by the fastest head, i.e. the head that reads the most. It is possible that a head always", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "reads new input without producing output, which would result in the maximum possible latency.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 700, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 710 + ], + "score": 1.0, + "content": "Note that the attention behaviors in MMA-H and MMA-IL can be different. In MMA-IL, a head", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "reaching the end of the sentence will provide the model with maximum information about the source", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 504, + 733 + ], + "score": 1.0, + "content": "sentence. On the other hand, in the case of MMA-H, reaching the end of sentence for a head only", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 165, + 77, + 447, + 144 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 165, + 77, + 447, + 144 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 77, + 447, + 144 + ], + "spans": [ + { + "bbox": [ + 165, + 77, + 447, + 144 + ], + "score": 0.958, + "type": "image", + "image_path": "d597335b4efb8a0b81899a74e4bee9cd4514669123bca34eef86fbf7c1b7c310.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 165, + 77, + 447, + 99.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 165, + 99.33333333333333, + 447, + 121.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 165, + 121.66666666666666, + 447, + 144.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 150, + 150, + 460, + 161 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 149, + 461, + 163 + ], + "spans": [ + { + "bbox": [ + 149, + 149, + 461, + 163 + ], + "score": 1.0, + "content": "Figure 1: Monotonic Attention (Left) versus Monotonic Multihead Attention (Right).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 134, + 196 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 183, + 155, + 195 + ], + "score": 0.91, + "content": "W _ { l , h }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 182, + 284, + 196 + ], + "score": 1.0, + "content": "is the input projection matrix,", + "type": "text" + }, + { + "bbox": [ + 285, + 183, + 296, + 194 + ], + "score": 0.87, + "content": "d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 182, + 505, + 196 + ], + "score": 1.0, + "content": "is the dimension of the attention head. We make", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 193, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 206 + ], + "score": 1.0, + "content": "the selection process independent for each head in each layer. We then investigate two types of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "MMA, MMA-H(ard) and MMA-IL(infinite lookback). For MMA-H, we use Equation 4 in order to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 216, + 507, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 216, + 370, + 234 + ], + "score": 1.0, + "content": "calculate the expected alignment for each layer each head, given", + "type": "text" + }, + { + "bbox": [ + 370, + 216, + 387, + 231 + ], + "score": 0.92, + "content": "p _ { i , j } ^ { l , h }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 216, + 507, + 234 + ], + "score": 1.0, + "content": ". For MMA-IL, we calculate", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 229, + 287, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 287, + 241 + ], + "score": 1.0, + "content": "the softmax energy for each head as follows:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 182, + 507, + 241 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 245, + 396, + 282 + ], + "lines": [ + { + "bbox": [ + 194, + 245, + 396, + 282 + ], + "spans": [ + { + "bbox": [ + 194, + 245, + 396, + 282 + ], + "score": 0.94, + "content": "u _ { i , j } ^ { l , h } = \\mathrm { S o f t E n e r g y } = \\left( \\frac { m _ { j } \\hat { W } _ { l , h } ^ { K } ( s _ { i - 1 } \\hat { W } _ { l , h } ^ { Q } ) ^ { T } } { \\sqrt { d _ { k } } } \\right) _ { i , j }", + "type": "interline_equation", + "image_path": "76dbee94d92d243f0b5d5234e42d63c1c42aef73bfbee0d4196977dd708667e3.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 245, + 396, + 263.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 194, + 263.5, + 396, + 282.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "score": 1.0, + "content": "and then use Equation 6 to calculate the expected attention. Each attention head in MMA-H hard-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 298, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 308 + ], + "score": 1.0, + "content": "attends to one encoder state. On the other hand, each attention head in MMA-IL can attend to all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "previous encoder states. Thus, MMA-IL allows the model to leverage more information for transla-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 319, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 504, + 331 + ], + "score": 1.0, + "content": "tion, but MMA-H may be better suited for streaming systems with stricter efficiency requirements.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 330, + 504, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 504, + 343 + ], + "score": 1.0, + "content": "Finally, our models use unidirectional encoders: the encoder self-attention can only attend to previ-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 343, + 356, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 356, + 353 + ], + "score": 1.0, + "content": "ous states, which is also required for simultaneous translation.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 286, + 505, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 410, + 371 + ], + "score": 1.0, + "content": "At inference time, our decoding strategy is shown in Algorithm 1. For each", + "type": "text" + }, + { + "bbox": [ + 411, + 358, + 425, + 370 + ], + "score": 0.86, + "content": "l , h", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 357, + 496, + 371 + ], + "score": 1.0, + "content": ", at decoding step", + "type": "text" + }, + { + "bbox": [ + 497, + 359, + 501, + 369 + ], + "score": 0.59, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 357, + 506, + 371 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "score": 1.0, + "content": "we apply the sampling processes discussed in subsection 2.1 individually and set the encoder step", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 101, + 376, + 509, + 398 + ], + "spans": [ + { + "bbox": [ + 101, + 376, + 116, + 398 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 380, + 131, + 394 + ], + "score": 0.9, + "content": "t _ { i } ^ { l , \\bar { h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 376, + 509, + 398 + ], + "score": 1.0, + "content": ". Then a hard alignment or partial softmax attention from encoder states, shown in Equation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 347, + 405 + ], + "score": 1.0, + "content": "13, will be retrieved to feed into the decoder to generate the", + "type": "text" + }, + { + "bbox": [ + 348, + 394, + 352, + 403 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "-th token. The model will write a new", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "target token only after all the attentions have decided to write. In other words, the heads that have", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 413, + 366, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 366, + 428 + ], + "score": 1.0, + "content": "decided to write must wait until the others have finished reading.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 101, + 357, + 509, + 428 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 430, + 449, + 510 + ], + "lines": [ + { + "bbox": [ + 181, + 430, + 449, + 510 + ], + "spans": [ + { + "bbox": [ + 181, + 430, + 449, + 510 + ], + "score": 0.87, + "content": "\\begin{array}{c} \\begin{array} { r l } & { \\qquad c _ { i } ^ { l } = \\mathrm { C o n c a t } ( c _ { i } ^ { l , 1 } , c _ { i } ^ { l , 2 } , . . . , c _ { i } ^ { l , H } ) } \\\\ & { \\qquad \\mathrm \\quad \\mathbf { \\ \" } } \\\\ & { \\qquad \\mathbf { \\ \" } \\mathbf { \\ \" } e _ { i } ^ { l , h } = f _ { \\mathrm { c o n t e x t } } ( \\boldsymbol { h } , t _ { i } ^ { l , h } ) = \\displaystyle \\left\\{ \\sum _ { j = 1 } ^ { m _ { t _ { i } ^ { l , h } } } \\frac { \\mathrm { e x p } \\left( u _ { i , j } ^ { l , h } \\right) } { \\sum _ { j = 1 } ^ { t _ { i } ^ { l , h } } \\mathrm { e x p } \\left( u _ { i , j } ^ { l , h } \\right) } m _ { j } \\quad \\mathrm { M M A \\mathrm { - } I L } \\right.} \\end{array} \\end{array}", + "type": "interline_equation", + "image_path": "55a64c3eb31107e09be4521a10e290bda8cf5aa3c978766c5de794153806eee4.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 181, + 430, + 449, + 456.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 181, + 456.6666666666667, + 449, + 483.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 181, + 483.33333333333337, + 449, + 510.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "Figure 1 illustrates a comparison between our model and the monotonic model with one attention", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "score": 1.0, + "content": "head. Compared with the monotonic model, the MMA model is able to set attention to different", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "positions so that it can still attend to previous states while reading each new token. Each head can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 554, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 566 + ], + "score": 1.0, + "content": "adjust its speed on-the-fly. Some heads read new inputs, while the others can stay in the past to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "retain the source history information. Even with the hard alignment variant (MMA-H), the model", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 576, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 587 + ], + "score": 1.0, + "content": "is still able to preserve the history information by setting heads to past states. In contrast, the hard", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 587, + 494, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 494, + 599 + ], + "score": 1.0, + "content": "monotonic model, which only has one head, loses the previous information at the attention layer.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 520, + 506, + 599 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 612, + 216, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 218, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 218, + 625 + ], + "score": 1.0, + "content": "2.3 LATENCY CONTROL", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "Effective simultaneous machine translation must balance quality and latency. At a high level, latency", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "measures how many source tokens the model has read until a translation is generated. The model", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "we have introduced in subsection 2.2 is not able to control latency on its own. While MMA allows", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "simultaneous translation by having a read or write schedule for each head, the overall latency is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "determined by the fastest head, i.e. the head that reads the most. It is possible that a head always", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "reads new input without producing output, which would result in the maximum possible latency.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 700, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 710 + ], + "score": 1.0, + "content": "Note that the attention behaviors in MMA-H and MMA-IL can be different. In MMA-IL, a head", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "reaching the end of the sentence will provide the model with maximum information about the source", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 504, + 733 + ], + "score": 1.0, + "content": "sentence. On the other hand, in the case of MMA-H, reaching the end of sentence for a head only", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 632, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 506, + 95 + ], + "score": 1.0, + "content": "Algorithm 1 MMA monotonic decoding. Because each head is independent, we compute line 3 to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 163, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 163, + 106 + ], + "score": 1.0, + "content": "16 in parallel", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 109, + 450, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 450, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 135, + 124 + ], + "score": 1.0, + "content": "Input:", + "type": "text" + }, + { + "bbox": [ + 135, + 112, + 154, + 120 + ], + "score": 0.74, + "content": "{ \\bf { \\delta } } _ { \\bf { \\mathcal { X } } } =", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 108, + 207, + 124 + ], + "score": 1.0, + "content": "source tokens,", + "type": "text" + }, + { + "bbox": [ + 207, + 111, + 225, + 120 + ], + "score": 0.75, + "content": "h =", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 108, + 280, + 124 + ], + "score": 1.0, + "content": "encoder states,", + "type": "text" + }, + { + "bbox": [ + 280, + 109, + 385, + 122 + ], + "score": 0.78, + "content": "i = 1 , j = 1 , t _ { 0 } ^ { l , h } = 1 , y _ { 0 } =", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 108, + 450, + 124 + ], + "score": 1.0, + "content": "StartOfSequence.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "table", + "bbox": [ + 108, + 118, + 508, + 341 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 118, + 508, + 341 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 118, + 508, + 341 + ], + "spans": [ + { + "bbox": [ + 108, + 118, + 508, + 341 + ], + "score": 0.342, + "html": "
1:while yi-1 ≠ EndOfSequence do
2:tmax =1
3:h = empty sequence
4:forl←1toLdo
5:forh←1to Hdo
6:forj←ttoacldo
7:l,h Pi,j, = Sigmoid (MonotonicEnergy(Si-1,mj ))
8:ifpi l,h > 0.5 then
9:=j
10:l,h =fcontext(h,t,h）
11:Ci Break
12:else
13:if j>tmax then
14:Read token x j
15:Calculate state h j and append to h
16:tmax =j
17:c = Concat(c,1,,cH）
18:s =DecoderLayer'(s-1,s)
19:yi = Output(s)
20:
", + "type": "table", + "image_path": "d3ce67f84553fcb769a11cd9c9b6bc0f2cc38f43189af848cae8c2b15b180a38.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 108, + 118, + 508, + 192.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 192.33333333333331, + 508, + 266.66666666666663 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 108, + 266.66666666666663, + 508, + 340.99999999999994 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 359, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "gives a hard alignment to the end-of-sentence token, which provides very little information to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "decoder. Furthermore, it is possible that an MMA-H attention head stays at the beginning of sentence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 381, + 504, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 504, + 393 + ], + "score": 1.0, + "content": "without moving forward. Such a head would not cause latency issues but would degrade the model", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "score": 1.0, + "content": "quality since the decoder would not have any information about the input. In addition, this behavior", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 403, + 249, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 249, + 416 + ], + "score": 1.0, + "content": "is not suited for streaming systems.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 435 + ], + "score": 1.0, + "content": "To address these issues, we introduce two latency control methods. The first one is weighted average", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 431, + 232, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 232, + 444 + ], + "score": 1.0, + "content": "latency, shown in Equation 14:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 445, + 374, + 478 + ], + "lines": [ + { + "bbox": [ + 236, + 445, + 374, + 478 + ], + "spans": [ + { + "bbox": [ + 236, + 445, + 374, + 478 + ], + "score": 0.95, + "content": "g _ { i } ^ { W } = \\frac { \\exp ( g _ { i } ^ { l , h } ) } { \\sum _ { l = 1 } ^ { L } \\sum _ { h = 1 } ^ { H } \\exp ( g _ { i } ^ { l , h } ) } g _ { i } ^ { l , h }", + "type": "interline_equation", + "image_path": "dbc1c8df2033f3a8bb69ca4c0b77443bdc70f12f5d733fc36d65bd4af90e8f1b.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 236, + 445, + 374, + 461.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 236, + 461.5, + 374, + 478.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 482, + 503, + 495 + ], + "lines": [ + { + "bbox": [ + 104, + 477, + 501, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 133, + 500 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 480, + 210, + 497 + ], + "score": 0.94, + "content": "\\begin{array} { r } { g _ { i } ^ { l , h } = \\sum _ { j = 1 } ^ { | x | } j \\alpha _ { i , j } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 477, + 494, + 500 + ], + "score": 1.0, + "content": ". Then we calculate the latency loss with a differentiable latency metric", + "type": "text" + }, + { + "bbox": [ + 494, + 484, + 501, + 493 + ], + "score": 0.77, + "content": "\\mathcal { C }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 498, + 346, + 515 + ], + "lines": [ + { + "bbox": [ + 264, + 498, + 346, + 515 + ], + "spans": [ + { + "bbox": [ + 264, + 498, + 346, + 515 + ], + "score": 0.91, + "content": "\\begin{array} { r c l } { { L _ { a v g } } } & { { = } } & { { { \\mathcal { C } } \\left( g ^ { W } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "6fb597bba8952ce63096825ef7fd9a9f8875bf203ed1269e09e958a57183435b.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 264, + 498, + 346, + 515 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "Like Arivazhagan et al. (2019), we use the Differentiable Average Lagging. It is important to note", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "score": 1.0, + "content": "that, unlike the original latency augmented training in Arivazhagan et al. (2019), Equation 15 is not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 242, + 551 + ], + "score": 1.0, + "content": "the expected latency metric given", + "type": "text" + }, + { + "bbox": [ + 242, + 539, + 249, + 549 + ], + "score": 0.74, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 538, + 339, + 551 + ], + "score": 1.0, + "content": ", but weighted average", + "type": "text" + }, + { + "bbox": [ + 340, + 539, + 347, + 549 + ], + "score": 0.77, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "on all the attentions. The real expected", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 546, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 147, + 564 + ], + "score": 1.0, + "content": "latency is", + "type": "text" + }, + { + "bbox": [ + 147, + 549, + 223, + 562 + ], + "score": 0.9, + "content": "\\hat { \\pmb g } = \\mathrm { m a x } _ { l , h } \\left( \\pmb g ^ { l , h } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 546, + 266, + 564 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 266, + 551, + 273, + 561 + ], + "score": 0.81, + "content": "\\bar { \\pmb { g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 546, + 506, + 564 + ], + "score": 1.0, + "content": ", but using this directly would only affect the speed of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "fastest head. Equation 15 can control every head in a way that the faster heads will be automatically", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "score": 1.0, + "content": "assigned to larger weights and slower heads will also be moderately regularized. For MMA-H", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 582, + 504, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 504, + 594 + ], + "score": 1.0, + "content": "models, we found that the latency of are mainly due to outliers that skip almost every token. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "weighted average latency loss is not sufficient to control the outliers. We therefore introduce the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 604, + 501, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 501, + 617 + ], + "score": 1.0, + "content": "head divergence loss, the average variance of expected delays at each step, defined in Equation 16:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 618, + 382, + 654 + ], + "lines": [ + { + "bbox": [ + 228, + 618, + 382, + 654 + ], + "spans": [ + { + "bbox": [ + 228, + 618, + 382, + 654 + ], + "score": 0.94, + "content": "\\begin{array} { c c l } { { { \\cal L } _ { v a r } } } & { { = } } & { { \\displaystyle \\frac { 1 } { L H } \\sum _ { l = 1 } ^ { L } \\sum _ { h = 1 } ^ { H } \\left( g _ { i } ^ { l , h } - \\bar { g } _ { i } \\right) ^ { 2 } } } \\end{array}", + "type": "interline_equation", + "image_path": "d133be55bb9c06500803d6386f5b6eccc9edfd580fd7b016a14f265bd0e6ae51.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 228, + 618, + 382, + 636.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 228, + 636.0, + 382, + 654.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 656, + 418, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 421, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 133, + 672 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 656, + 192, + 670 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\bar { g _ { i } } = \\frac { 1 } { L H } \\sum g _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 655, + 421, + 672 + ], + "score": 1.0, + "content": "The final objective function is presented in Equation 17:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 671, + 404, + 685 + ], + "lines": [ + { + "bbox": [ + 207, + 671, + 404, + 685 + ], + "spans": [ + { + "bbox": [ + 207, + 671, + 404, + 685 + ], + "score": 0.87, + "content": "L ( \\theta ) = - \\log ( \\pmb { y } \\mid \\pmb { x } ; \\theta ) + \\lambda _ { a v g } L _ { a v g } + \\lambda _ { v a r } L _ { v a r }", + "type": "interline_equation", + "image_path": "4b3bedce76971cb30dbdf17af15e5198dd875394ef3d36c4eb5806191c3ef503.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 207, + 671, + 404, + 685 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 133, + 701 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 688, + 178, + 699 + ], + "score": 0.28, + "content": "\\lambda _ { a v g } , \\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 687, + 433, + 701 + ], + "score": 1.0, + "content": "are hyperparameters that control both losses. Intuitively, while", + "type": "text" + }, + { + "bbox": [ + 434, + 688, + 454, + 700 + ], + "score": 0.91, + "content": "\\lambda _ { a v g }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "controls the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 163, + 712 + ], + "score": 1.0, + "content": "overall speed,", + "type": "text" + }, + { + "bbox": [ + 164, + 700, + 183, + 710 + ], + "score": 0.92, + "content": "\\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "controls the divergence of the heads. Combining these two losses, we are able to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "dynamically control the range of attention heads so that we can control the latency and the reading", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 718, + 425, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 718, + 271, + 735 + ], + "score": 1.0, + "content": "buffer. For MMA-IL model, we only use", + "type": "text" + }, + { + "bbox": [ + 272, + 721, + 293, + 733 + ], + "score": 0.9, + "content": "L _ { a v g }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 718, + 399, + 735 + ], + "score": 1.0, + "content": "; for MMA-H we only use", + "type": "text" + }, + { + "bbox": [ + 399, + 721, + 420, + 732 + ], + "score": 0.91, + "content": "L _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 718, + 425, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 506, + 95 + ], + "score": 1.0, + "content": "Algorithm 1 MMA monotonic decoding. Because each head is independent, we compute line 3 to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 163, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 163, + 106 + ], + "score": 1.0, + "content": "16 in parallel", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 80, + 506, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 109, + 450, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 450, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 135, + 124 + ], + "score": 1.0, + "content": "Input:", + "type": "text" + }, + { + "bbox": [ + 135, + 112, + 154, + 120 + ], + "score": 0.74, + "content": "{ \\bf { \\delta } } _ { \\bf { \\mathcal { X } } } =", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 108, + 207, + 124 + ], + "score": 1.0, + "content": "source tokens,", + "type": "text" + }, + { + "bbox": [ + 207, + 111, + 225, + 120 + ], + "score": 0.75, + "content": "h =", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 108, + 280, + 124 + ], + "score": 1.0, + "content": "encoder states,", + "type": "text" + }, + { + "bbox": [ + 280, + 109, + 385, + 122 + ], + "score": 0.78, + "content": "i = 1 , j = 1 , t _ { 0 } ^ { l , h } = 1 , y _ { 0 } =", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 108, + 450, + 124 + ], + "score": 1.0, + "content": "StartOfSequence.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 108, + 450, + 124 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 118, + 508, + 341 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 118, + 508, + 341 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 118, + 508, + 341 + ], + "spans": [ + { + "bbox": [ + 108, + 118, + 508, + 341 + ], + "score": 0.342, + "html": "
1:while yi-1 ≠ EndOfSequence do
2:tmax =1
3:h = empty sequence
4:forl←1toLdo
5:forh←1to Hdo
6:forj←ttoacldo
7:l,h Pi,j, = Sigmoid (MonotonicEnergy(Si-1,mj ))
8:ifpi l,h > 0.5 then
9:=j
10:l,h =fcontext(h,t,h）
11:Ci Break
12:else
13:if j>tmax then
14:Read token x j
15:Calculate state h j and append to h
16:tmax =j
17:c = Concat(c,1,,cH）
18:s =DecoderLayer'(s-1,s)
19:yi = Output(s)
20:
", + "type": "table", + "image_path": "d3ce67f84553fcb769a11cd9c9b6bc0f2cc38f43189af848cae8c2b15b180a38.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 108, + 118, + 508, + 192.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 192.33333333333331, + 508, + 266.66666666666663 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 108, + 266.66666666666663, + 508, + 340.99999999999994 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 359, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "gives a hard alignment to the end-of-sentence token, which provides very little information to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "decoder. Furthermore, it is possible that an MMA-H attention head stays at the beginning of sentence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 381, + 504, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 504, + 393 + ], + "score": 1.0, + "content": "without moving forward. Such a head would not cause latency issues but would degrade the model", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "score": 1.0, + "content": "quality since the decoder would not have any information about the input. In addition, this behavior", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 403, + 249, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 249, + 416 + ], + "score": 1.0, + "content": "is not suited for streaming systems.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 359, + 506, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 435 + ], + "score": 1.0, + "content": "To address these issues, we introduce two latency control methods. The first one is weighted average", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 431, + 232, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 232, + 444 + ], + "score": 1.0, + "content": "latency, shown in Equation 14:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 417, + 506, + 444 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 445, + 374, + 478 + ], + "lines": [ + { + "bbox": [ + 236, + 445, + 374, + 478 + ], + "spans": [ + { + "bbox": [ + 236, + 445, + 374, + 478 + ], + "score": 0.95, + "content": "g _ { i } ^ { W } = \\frac { \\exp ( g _ { i } ^ { l , h } ) } { \\sum _ { l = 1 } ^ { L } \\sum _ { h = 1 } ^ { H } \\exp ( g _ { i } ^ { l , h } ) } g _ { i } ^ { l , h }", + "type": "interline_equation", + "image_path": "dbc1c8df2033f3a8bb69ca4c0b77443bdc70f12f5d733fc36d65bd4af90e8f1b.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 236, + 445, + 374, + 461.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 236, + 461.5, + 374, + 478.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 482, + 503, + 495 + ], + "lines": [ + { + "bbox": [ + 104, + 477, + 501, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 133, + 500 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 480, + 210, + 497 + ], + "score": 0.94, + "content": "\\begin{array} { r } { g _ { i } ^ { l , h } = \\sum _ { j = 1 } ^ { | x | } j \\alpha _ { i , j } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 477, + 494, + 500 + ], + "score": 1.0, + "content": ". Then we calculate the latency loss with a differentiable latency metric", + "type": "text" + }, + { + "bbox": [ + 494, + 484, + 501, + 493 + ], + "score": 0.77, + "content": "\\mathcal { C }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 477, + 501, + 500 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 498, + 346, + 515 + ], + "lines": [ + { + "bbox": [ + 264, + 498, + 346, + 515 + ], + "spans": [ + { + "bbox": [ + 264, + 498, + 346, + 515 + ], + "score": 0.91, + "content": "\\begin{array} { r c l } { { L _ { a v g } } } & { { = } } & { { { \\mathcal { C } } \\left( g ^ { W } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "6fb597bba8952ce63096825ef7fd9a9f8875bf203ed1269e09e958a57183435b.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 264, + 498, + 346, + 515 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "Like Arivazhagan et al. (2019), we use the Differentiable Average Lagging. It is important to note", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "score": 1.0, + "content": "that, unlike the original latency augmented training in Arivazhagan et al. (2019), Equation 15 is not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 242, + 551 + ], + "score": 1.0, + "content": "the expected latency metric given", + "type": "text" + }, + { + "bbox": [ + 242, + 539, + 249, + 549 + ], + "score": 0.74, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 538, + 339, + 551 + ], + "score": 1.0, + "content": ", but weighted average", + "type": "text" + }, + { + "bbox": [ + 340, + 539, + 347, + 549 + ], + "score": 0.77, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "on all the attentions. The real expected", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 546, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 147, + 564 + ], + "score": 1.0, + "content": "latency is", + "type": "text" + }, + { + "bbox": [ + 147, + 549, + 223, + 562 + ], + "score": 0.9, + "content": "\\hat { \\pmb g } = \\mathrm { m a x } _ { l , h } \\left( \\pmb g ^ { l , h } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 546, + 266, + 564 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 266, + 551, + 273, + 561 + ], + "score": 0.81, + "content": "\\bar { \\pmb { g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 546, + 506, + 564 + ], + "score": 1.0, + "content": ", but using this directly would only affect the speed of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "fastest head. Equation 15 can control every head in a way that the faster heads will be automatically", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "score": 1.0, + "content": "assigned to larger weights and slower heads will also be moderately regularized. For MMA-H", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 582, + 504, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 504, + 594 + ], + "score": 1.0, + "content": "models, we found that the latency of are mainly due to outliers that skip almost every token. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "weighted average latency loss is not sufficient to control the outliers. We therefore introduce the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 604, + 501, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 501, + 617 + ], + "score": 1.0, + "content": "head divergence loss, the average variance of expected delays at each step, defined in Equation 16:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 516, + 506, + 617 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 618, + 382, + 654 + ], + "lines": [ + { + "bbox": [ + 228, + 618, + 382, + 654 + ], + "spans": [ + { + "bbox": [ + 228, + 618, + 382, + 654 + ], + "score": 0.94, + "content": "\\begin{array} { c c l } { { { \\cal L } _ { v a r } } } & { { = } } & { { \\displaystyle \\frac { 1 } { L H } \\sum _ { l = 1 } ^ { L } \\sum _ { h = 1 } ^ { H } \\left( g _ { i } ^ { l , h } - \\bar { g } _ { i } \\right) ^ { 2 } } } \\end{array}", + "type": "interline_equation", + "image_path": "d133be55bb9c06500803d6386f5b6eccc9edfd580fd7b016a14f265bd0e6ae51.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 228, + 618, + 382, + 636.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 228, + 636.0, + 382, + 654.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 656, + 418, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 421, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 133, + 672 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 656, + 192, + 670 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\bar { g _ { i } } = \\frac { 1 } { L H } \\sum g _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 655, + 421, + 672 + ], + "score": 1.0, + "content": "The final objective function is presented in Equation 17:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 655, + 421, + 672 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 671, + 404, + 685 + ], + "lines": [ + { + "bbox": [ + 207, + 671, + 404, + 685 + ], + "spans": [ + { + "bbox": [ + 207, + 671, + 404, + 685 + ], + "score": 0.87, + "content": "L ( \\theta ) = - \\log ( \\pmb { y } \\mid \\pmb { x } ; \\theta ) + \\lambda _ { a v g } L _ { a v g } + \\lambda _ { v a r } L _ { v a r }", + "type": "interline_equation", + "image_path": "4b3bedce76971cb30dbdf17af15e5198dd875394ef3d36c4eb5806191c3ef503.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 207, + 671, + 404, + 685 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 133, + 701 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 688, + 178, + 699 + ], + "score": 0.28, + "content": "\\lambda _ { a v g } , \\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 687, + 433, + 701 + ], + "score": 1.0, + "content": "are hyperparameters that control both losses. Intuitively, while", + "type": "text" + }, + { + "bbox": [ + 434, + 688, + 454, + 700 + ], + "score": 0.91, + "content": "\\lambda _ { a v g }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "controls the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 163, + 712 + ], + "score": 1.0, + "content": "overall speed,", + "type": "text" + }, + { + "bbox": [ + 164, + 700, + 183, + 710 + ], + "score": 0.92, + "content": "\\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "controls the divergence of the heads. Combining these two losses, we are able to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "dynamically control the range of attention heads so that we can control the latency and the reading", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 718, + 425, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 718, + 271, + 735 + ], + "score": 1.0, + "content": "buffer. For MMA-IL model, we only use", + "type": "text" + }, + { + "bbox": [ + 272, + 721, + 293, + 733 + ], + "score": 0.9, + "content": "L _ { a v g }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 718, + 399, + 735 + ], + "score": 1.0, + "content": "; for MMA-H we only use", + "type": "text" + }, + { + "bbox": [ + 399, + 721, + 420, + 732 + ], + "score": 0.91, + "content": "L _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 718, + 425, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 687, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 244, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "score": 1.0, + "content": "3 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 105, + 230, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 231, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 231, + 119 + ], + "score": 1.0, + "content": "3.1 EVALUATION METRICS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "We evaluate our model using quality and latency. For translation quality, we use tokenized BLEU 4", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "for IWSLT15 En-Vi and detokenized BLEU with SacreBLEU (Post, 2018) for WMT15 De-En. For", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "latency, we use three different recent metrics, Average Proportion (AP) (Cho & Esipova, 2016),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "Average Lagging (AL) (Ma et al., 2019) and Differentiable Average Lagging (DAL) (Arivazhagan", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 424, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 424, + 182 + ], + "score": 1.0, + "content": "et al., 2019) 5. 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DatasetTrainValidationTest
IWSLT15 En-Vi133k12681553
WMT15 De-En4.5M30002169
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DatasetRNNTransformer
IWSLT15 En-Vi25.6628.7
WMT15 De-En28.4 (Arivazhagan et al., 2019)32.3
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DatasetBeam SearchBidirectional EncoderUnidirectional Encoder
WMT15 De-En132.632.3
433.033.0
IWSLT15 En-Vi128.729.4
1028.829.5
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Statistics of the datasets can be found in Table 1. For each dataset, we apply tokenization", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 449, + 375, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 375, + 464 + ], + "score": 1.0, + "content": "with the Moses (Koehn et al., 2007) tokenizer and preserve casing.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 503, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "IWSLT15 English-Vietnamese TED talks from IWSLT 2015 Evaluation Campaign (Cettolo et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "2016). We follow the settings from Luong & Manning (2015) and Raffel et al. (2017). We replace", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 252, + 505 + ], + "score": 1.0, + "content": "words with frequency less than 5 by", + "type": "text" + }, + { + "bbox": [ + 252, + 493, + 283, + 503 + ], + "score": 0.69, + "content": "< u n k >", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 492, + 505, + 505 + ], + "score": 1.0, + "content": ". We use tst2012 as a validation set tst2013 as a test set.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 503, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "WMT15 German-English We follow the setting from Arivazhagan et al. (2019). We apply byte", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "pair encoding (BPE) (Sennrich et al., 2016) jointly on the source and target to construct a shared", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 529, + 503, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 503, + 541 + ], + "score": 1.0, + "content": "vocabulary with 32K symbols. We use newstest2013 as validation set and newstest2015 as test set.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 169, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 171, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 171, + 567 + ], + "score": 1.0, + "content": "3.3 MODELS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "We evaluate MMA-H and MMA-IL models on both datasets. The MILK model we evaluate on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 373, + 597 + ], + "score": 1.0, + "content": "IWSLT15 En-Vi is based on Luong et al. (2015) rather than RNMT", + "type": "text" + }, + { + "bbox": [ + 374, + 586, + 380, + 595 + ], + "score": 0.48, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "(Chen et al., 2018). In general,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "our offline models use unidirectional encoders, i.e. the encoder self-attention can only attend to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 608, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 619 + ], + "score": 1.0, + "content": "previous states, and greedy decoding. We report offline model performance in Table 2 and the effect", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "of using unidirectional encoders and greedy decoding in Table 3. For MMA models, we replace", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "score": 1.0, + "content": "the encoder-decoder layers with MMA and keep other hyperparameter settings the same as the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "offline model. Detailed hyperparameter settings can be found in subsection A.1. We use the Fairseq", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 651, + 302, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 302, + 663 + ], + "score": 1.0, + "content": "library (Ott et al., 2019) for our implementation.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 669, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 667, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 117, + 667, + 506, + 683 + ], + "score": 1.0, + "content": "4We acquire the data from https://nlp.stanford.edu/projects/nmt/, which is tokenized. We do not have the", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 679, + 408, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 408, + 691 + ], + "score": 1.0, + "content": "tokenizer which processed this data, thus we report tokenize d BLEU for IWSLT15", + "type": "text" + } + ] + }, + { + "bbox": [ + 117, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 117, + 689, + 505, + 703 + ], + "score": 1.0, + "content": "5Latency metrics are computed on BPE tokens for WMT15 De-En – consistent with Arivazhagan et al.", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 700, + 288, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 288, + 712 + ], + "score": 1.0, + "content": "(2019) – and on word tokens for IWSLT15 En-Vi.", + "type": "text" + } + ] + }, + { + "bbox": [ + 117, + 708, + 506, + 726 + ], + "spans": [ + { + "bbox": [ + 117, + 708, + 506, + 726 + ], + "score": 1.0, + "content": "6 Luong & Manning (2015) report a BLEU score of 23.0 but they didn’t mention what type of BLEU score", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "they used. This score is from our implementation on the data aquired from https://nlp.stanford.edu/projects/nmt/", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 244, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "score": 1.0, + "content": "3 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 105, + 230, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 231, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 231, + 119 + ], + "score": 1.0, + "content": "3.1 EVALUATION METRICS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "We evaluate our model using quality and latency. For translation quality, we use tokenized BLEU 4", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "for IWSLT15 En-Vi and detokenized BLEU with SacreBLEU (Post, 2018) for WMT15 De-En. For", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "latency, we use three different recent metrics, Average Proportion (AP) (Cho & Esipova, 2016),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "Average Lagging (AL) (Ma et al., 2019) and Differentiable Average Lagging (DAL) (Arivazhagan", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 424, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 424, + 182 + ], + "score": 1.0, + "content": "et al., 2019) 5. 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DatasetTrainValidationTest
IWSLT15 En-Vi133k12681553
WMT15 De-En4.5M30002169
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DatasetRNNTransformer
IWSLT15 En-Vi25.6628.7
WMT15 De-En28.4 (Arivazhagan et al., 2019)32.3
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DatasetBeam SearchBidirectional EncoderUnidirectional Encoder
WMT15 De-En132.632.3
433.033.0
IWSLT15 En-Vi128.729.4
1028.829.5
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Statistics of the datasets can be found in Table 1. For each dataset, we apply tokenization", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 449, + 375, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 375, + 464 + ], + "score": 1.0, + "content": "with the Moses (Koehn et al., 2007) tokenizer and preserve casing.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 428, + 505, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 503, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "IWSLT15 English-Vietnamese TED talks from IWSLT 2015 Evaluation Campaign (Cettolo et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "2016). We follow the settings from Luong & Manning (2015) and Raffel et al. (2017). We replace", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 252, + 505 + ], + "score": 1.0, + "content": "words with frequency less than 5 by", + "type": "text" + }, + { + "bbox": [ + 252, + 493, + 283, + 503 + ], + "score": 0.69, + "content": "< u n k >", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 492, + 505, + 505 + ], + "score": 1.0, + "content": ". We use tst2012 as a validation set tst2013 as a test set.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 470, + 505, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 503, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "WMT15 German-English We follow the setting from Arivazhagan et al. (2019). We apply byte", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "pair encoding (BPE) (Sennrich et al., 2016) jointly on the source and target to construct a shared", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 529, + 503, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 503, + 541 + ], + "score": 1.0, + "content": "vocabulary with 32K symbols. We use newstest2013 as validation set and newstest2015 as test set.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 506, + 505, + 541 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 169, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 171, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 171, + 567 + ], + "score": 1.0, + "content": "3.3 MODELS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "We evaluate MMA-H and MMA-IL models on both datasets. The MILK model we evaluate on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 373, + 597 + ], + "score": 1.0, + "content": "IWSLT15 En-Vi is based on Luong et al. (2015) rather than RNMT", + "type": "text" + }, + { + "bbox": [ + 374, + 586, + 380, + 595 + ], + "score": 0.48, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "(Chen et al., 2018). In general,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "our offline models use unidirectional encoders, i.e. the encoder self-attention can only attend to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 608, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 619 + ], + "score": 1.0, + "content": "previous states, and greedy decoding. We report offline model performance in Table 2 and the effect", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "of using unidirectional encoders and greedy decoding in Table 3. For MMA models, we replace", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "score": 1.0, + "content": "the encoder-decoder layers with MMA and keep other hyperparameter settings the same as the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "offline model. Detailed hyperparameter settings can be found in subsection A.1. We use the Fairseq", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 651, + 302, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 302, + 663 + ], + "score": 1.0, + "content": "library (Ott et al., 2019) for our implementation.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 574, + 506, + 663 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 172, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 174, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 174, + 96 + ], + "score": 1.0, + "content": "4 RESULTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 108, + 505, + 163 + ], + "lines": [ + { + "bbox": [ + 106, + 108, + 504, + 120 + ], + "spans": [ + { + "bbox": [ + 106, + 108, + 504, + 120 + ], + "score": 1.0, + "content": "In this section, we present the main results of our model in terms of latency-quality tradeoffs, abla-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 120, + 504, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 120, + 504, + 130 + ], + "score": 1.0, + "content": "tion studies and analyses. In the first study, we analyze the effect of the variance loss on the attention", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 505, + 143 + ], + "score": 1.0, + "content": "span. Then, we study the effect of the number of decoder layers and decoder heads on quality and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 506, + 154 + ], + "score": 1.0, + "content": "latency. We also provide a case study for the behavior of attention heads in an example. 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Black dashed line indicates the unidirectional offline transformer model with greedy search.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 8.25 + }, + { + "type": "title", + "bbox": [ + 108, + 432, + 268, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 270, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 270, + 445 + ], + "score": 1.0, + "content": "4.1 LATENCY-QUALITY TRADEOFFS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "score": 1.0, + "content": "We plot the quality-latency curves for MMA-H and MMA-IL in Figure 2. 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We show the relation between the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 227, + 169 + ], + "score": 1.0, + "content": "average attention span versus", + "type": "text" + }, + { + "bbox": [ + 227, + 157, + 247, + 168 + ], + "score": 0.91, + "content": "\\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "in Figure 3. As expected, the average attention span is reduced", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 167, + 191, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 166, + 181 + ], + "score": 1.0, + "content": "as we increase", + "type": "text" + }, + { + "bbox": [ + 167, + 168, + 186, + 178 + ], + "score": 0.91, + "content": "\\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 167, + 191, + 181 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "image", + "bbox": [ + 177, + 190, + 435, + 278 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 177, + 190, + 435, + 278 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 177, + 190, + 435, + 278 + ], + "spans": [ + { + "bbox": [ + 177, + 190, + 435, + 278 + ], + "score": 0.962, + "type": "image", + "image_path": "0f751c2faf86a11373cba7cf525d322dc182947f9c6a18f187c49c68d5fad146.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 177, + 190, + 435, + 219.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 177, + 219.33333333333334, + 435, + 248.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 177, + 248.66666666666669, + 435, + 278.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 288, + 503, + 309 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 180, + 300 + ], + "score": 1.0, + "content": "Figure 3: Effect of", + "type": "text" + }, + { + "bbox": [ + 180, + 289, + 198, + 298 + ], + "score": 0.92, + "content": "\\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "on the average attention span. The variance loss works as intended by reducing the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 298, + 201, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 201, + 310 + ], + "score": 1.0, + "content": "span with higher weights.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 8.25 + }, + { + "type": "title", + "bbox": [ + 107, + 339, + 374, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 375, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 375, + 351 + ], + "score": 1.0, + "content": "4.3 EFFECT ON NUMBER OF LAYERS AND NUMBER OF HEADS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 359, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "One motivation to introduce MMA is to adapt the Transformer, which is the current state-of-the-art", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "model for machine translation, to online decoding. Important features of the Transformer architec-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 382, + 504, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 504, + 393 + ], + "score": 1.0, + "content": "ture include having a separate attention layer for each decoder layer block and multihead attention.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 392, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 504, + 405 + ], + "score": 1.0, + "content": "In this section, we test the effect of these two components on the offline, MMA-H, and MMA-IL", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 403, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 504, + 415 + ], + "score": 1.0, + "content": "models from a quality and latency perspective. We report quality as measured by detokenized BLEU", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 442, + 428 + ], + "score": 1.0, + "content": "and latency as measured by DAL on the WMT13 validation set in Figure 4. We set", + "type": "text" + }, + { + "bbox": [ + 443, + 414, + 490, + 426 + ], + "score": 0.92, + "content": "\\lambda _ { a v g } = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 425, + 265, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 163, + 437 + ], + "score": 1.0, + "content": "MMA-IL and", + "type": "text" + }, + { + "bbox": [ + 164, + 425, + 210, + 437 + ], + "score": 0.93, + "content": "\\lambda _ { v a r } = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 425, + 265, + 437 + ], + "score": 1.0, + "content": "for MMA-H.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "The offline model benefits from having more than one decoder layer. In the case of 1 decoder layer,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "increasing the number of attention heads is beneficial but in the case of 3 and 6 decoder layers, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "score": 1.0, + "content": "do not see much benefit from using more than 2 heads. The best performance is obtained for 3 layers", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 475, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 486 + ], + "score": 1.0, + "content": "and 2 heads (6 effective heads). The MMA-IL model behaves similarly to the offline model, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 497 + ], + "score": 1.0, + "content": "the best performance is observed with 6 layers and 4 heads (24 effective heads). For MMA-H, with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "1 layer, performance improves with more heads. With 3 layers, the single-head setting is the most", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "score": 1.0, + "content": "effective (3 effective heads). Finally, with 6 layers, the best performance is reached with 16 heads", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 519, + 189, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 189, + 531 + ], + "score": 1.0, + "content": "(96 effective heads).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 108, + 536, + 503, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "The general trend we observe is that performance improves as we increase the number of effective", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "heads, either from multiple layers or multihead attention, up to a certain point, then either plateaus", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 558, + 368, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 368, + 569 + ], + "score": 1.0, + "content": "or degrades. This motivates the introduction of the MMA model.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "We also note that latency increases with the number of effective attention heads. This is due to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 416, + 599 + ], + "score": 1.0, + "content": "having fixed loss weights: when more heads are involved, we should increase", + "type": "text" + }, + { + "bbox": [ + 416, + 586, + 436, + 597 + ], + "score": 0.91, + "content": "\\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 584, + 448, + 599 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 448, + 586, + 469, + 598 + ], + "score": 0.91, + "content": "\\lambda _ { a v g }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 584, + 506, + 599 + ], + "score": 1.0, + "content": "to better", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 595, + 170, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 170, + 610 + ], + "score": 1.0, + "content": "control latency.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 108, + 623, + 233, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 234, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 234, + 635 + ], + "score": 1.0, + "content": "4.4 ATTENTION BEHAVIORS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "We characterize attention behaviors by providing a running example of MMA-H and MMA-IL,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "shown in Figure 5. Each curve represents the path that an attention head goes through at inference", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "time. For MMA-H, shown in Figure 5a, we found that when the source and target tokens have the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "same order, the attention heads behave linearly and the distance between fastest head and slowest", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "head is small. For example, this can be observed from partial sentence pair “I also didn’t know that”", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "and target tokens “Tôi cũng không biết rằng”, which have the same order. However, when the source", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "tokens and target tokens have different orders, such as “the second step” and “bước (step) thứ hai", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "(second)”, the model will generate “bước (step)” first and some heads will stay in the past to retain", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 227, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 228, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 228, + 95 + ], + "score": 1.0, + "content": "attention span in Equation 18:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 228, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 100, + 380, + 140 + ], + "lines": [ + { + "bbox": [ + 230, + 100, + 380, + 140 + ], + "spans": [ + { + "bbox": [ + 230, + 100, + 380, + 140 + ], + "score": 0.95, + "content": "\\bar { S } = \\frac { 1 } { \\vert \\pmb { y } \\vert } \\left( \\sum _ { i } ^ { \\vert \\pmb { y } \\vert } \\underset { l , h } { \\operatorname* { m a x } } t _ { i } ^ { l , h } - \\underset { l , h } { \\operatorname* { m i n } } t _ { i } ^ { l , h } \\right)", + "type": "interline_equation", + "image_path": "abf4018244aad5899baee17fe91fedebf02e95cb4d0f3e9cc679c572b65b9cc5.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 230, + 100, + 380, + 120.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 230, + 120.0, + 380, + 140.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "It estimates the reading buffer we need for streaming translation. 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As expected, the average attention span is reduced", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 167, + 191, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 166, + 181 + ], + "score": 1.0, + "content": "as we increase", + "type": "text" + }, + { + "bbox": [ + 167, + 168, + 186, + 178 + ], + "score": 0.91, + "content": "\\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 167, + 191, + 181 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 145, + 505, + 181 + ] + }, + { + "type": "image", + "bbox": [ + 177, + 190, + 435, + 278 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 177, + 190, + 435, + 278 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 177, + 190, + 435, + 278 + ], + "spans": [ + { + "bbox": [ + 177, + 190, + 435, + 278 + ], + "score": 0.962, + "type": "image", + "image_path": "0f751c2faf86a11373cba7cf525d322dc182947f9c6a18f187c49c68d5fad146.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 177, + 190, + 435, + 219.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 177, + 219.33333333333334, + 435, + 248.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 177, + 248.66666666666669, + 435, + 278.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 288, + 503, + 309 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 180, + 300 + ], + "score": 1.0, + "content": "Figure 3: Effect of", + "type": "text" + }, + { + "bbox": [ + 180, + 289, + 198, + 298 + ], + "score": 0.92, + "content": "\\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "on the average attention span. The variance loss works as intended by reducing the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 298, + 201, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 201, + 310 + ], + "score": 1.0, + "content": "span with higher weights.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 8.25 + }, + { + "type": "title", + "bbox": [ + 107, + 339, + 374, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 375, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 375, + 351 + ], + "score": 1.0, + "content": "4.3 EFFECT ON NUMBER OF LAYERS AND NUMBER OF HEADS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 359, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "One motivation to introduce MMA is to adapt the Transformer, which is the current state-of-the-art", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "model for machine translation, to online decoding. Important features of the Transformer architec-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 382, + 504, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 504, + 393 + ], + "score": 1.0, + "content": "ture include having a separate attention layer for each decoder layer block and multihead attention.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 392, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 504, + 405 + ], + "score": 1.0, + "content": "In this section, we test the effect of these two components on the offline, MMA-H, and MMA-IL", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 403, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 504, + 415 + ], + "score": 1.0, + "content": "models from a quality and latency perspective. We report quality as measured by detokenized BLEU", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 442, + 428 + ], + "score": 1.0, + "content": "and latency as measured by DAL on the WMT13 validation set in Figure 4. We set", + "type": "text" + }, + { + "bbox": [ + 443, + 414, + 490, + 426 + ], + "score": 0.92, + "content": "\\lambda _ { a v g } = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 425, + 265, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 163, + 437 + ], + "score": 1.0, + "content": "MMA-IL and", + "type": "text" + }, + { + "bbox": [ + 164, + 425, + 210, + 437 + ], + "score": 0.93, + "content": "\\lambda _ { v a r } = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 425, + 265, + 437 + ], + "score": 1.0, + "content": "for MMA-H.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 358, + 506, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "The offline model benefits from having more than one decoder layer. In the case of 1 decoder layer,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "increasing the number of attention heads is beneficial but in the case of 3 and 6 decoder layers, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "score": 1.0, + "content": "do not see much benefit from using more than 2 heads. The best performance is obtained for 3 layers", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 475, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 486 + ], + "score": 1.0, + "content": "and 2 heads (6 effective heads). The MMA-IL model behaves similarly to the offline model, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 497 + ], + "score": 1.0, + "content": "the best performance is observed with 6 layers and 4 heads (24 effective heads). For MMA-H, with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "1 layer, performance improves with more heads. With 3 layers, the single-head setting is the most", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "score": 1.0, + "content": "effective (3 effective heads). Finally, with 6 layers, the best performance is reached with 16 heads", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 519, + 189, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 189, + 531 + ], + "score": 1.0, + "content": "(96 effective heads).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 441, + 506, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 536, + 503, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "The general trend we observe is that performance improves as we increase the number of effective", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "heads, either from multiple layers or multihead attention, up to a certain point, then either plateaus", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 558, + 368, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 368, + 569 + ], + "score": 1.0, + "content": "or degrades. This motivates the introduction of the MMA model.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 535, + 505, + 569 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "We also note that latency increases with the number of effective attention heads. This is due to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 416, + 599 + ], + "score": 1.0, + "content": "having fixed loss weights: when more heads are involved, we should increase", + "type": "text" + }, + { + "bbox": [ + 416, + 586, + 436, + 597 + ], + "score": 0.91, + "content": "\\lambda _ { v a r }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 584, + 448, + 599 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 448, + 586, + 469, + 598 + ], + "score": 0.91, + "content": "\\lambda _ { a v g }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 584, + 506, + 599 + ], + "score": 1.0, + "content": "to better", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 595, + 170, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 170, + 610 + ], + "score": 1.0, + "content": "control latency.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 574, + 506, + 610 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 623, + 233, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 234, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 234, + 635 + ], + "score": 1.0, + "content": "4.4 ATTENTION BEHAVIORS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "We characterize attention behaviors by providing a running example of MMA-H and MMA-IL,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "shown in Figure 5. Each curve represents the path that an attention head goes through at inference", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "time. For MMA-H, shown in Figure 5a, we found that when the source and target tokens have the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "same order, the attention heads behave linearly and the distance between fastest head and slowest", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "head is small. For example, this can be observed from partial sentence pair “I also didn’t know that”", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "and target tokens “Tôi cũng không biết rằng”, which have the same order. However, when the source", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "tokens and target tokens have different orders, such as “the second step” and “bước (step) thứ hai", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "(second)”, the model will generate “bước (step)” first and some heads will stay in the past to retain", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 326, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 341 + ], + "score": 1.0, + "content": "the information for later reordered translation “thứ hai (second)”. We can also see that the attention", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 338, + 428, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 428, + 351 + ], + "score": 1.0, + "content": "heads have a near-diagonal trajectory, which is appropriate for streaming inputs.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 643, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 82, + 496, + 272 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 82, + 496, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 82, + 496, + 272 + ], + "spans": [ + { + "bbox": [ + 116, + 82, + 496, + 272 + ], + "score": 0.976, + "type": "image", + "image_path": "587b8944db8cce31706c574a36a9ec6d40a868064c34c383e996803aed29b0b0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 82, + 496, + 145.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 145.33333333333334, + 496, + 208.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 208.66666666666669, + 496, + 272.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 282, + 503, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 295 + ], + "score": 1.0, + "content": "Figure 4: Effect of the number of decoder attention heads and the number of decoder attention layers on quality", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 292, + 294, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 294, + 303 + ], + "score": 1.0, + "content": "and latency, reported on the WMT13 validation set.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 503, + 350 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 341 + ], + "score": 1.0, + "content": "the information for later reordered translation “thứ hai (second)”. We can also see that the attention", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 338, + 428, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 428, + 351 + ], + "score": 1.0, + "content": "heads have a near-diagonal trajectory, which is appropriate for streaming inputs.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "score": 1.0, + "content": "The behavior of the heads in MMA-IL models is shown in Figure 5b. Notice that we remove the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "partial softmax alignment in this figure. We don’t expect streaming capability for MMA-IL: some", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 376, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 506, + 391 + ], + "score": 1.0, + "content": "heads stop at early position of the source sentence to retain the history information. Moreover,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "because MMA-IL has more information when generating a new target token, it tends to produce", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "translations with better quality. In this example, the MMA-IL model has a better translation on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 411, + 439, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 439, + 422 + ], + "score": 1.0, + "content": "“isolate the victim” than MMA-H (“là cô lập nạn nhân” vs “là tách biệt nạn nhân”)", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "image", + "bbox": [ + 147, + 436, + 464, + 608 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 147, + 436, + 464, + 608 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 147, + 436, + 464, + 608 + ], + "spans": [ + { + "bbox": [ + 147, + 436, + 464, + 608 + ], + "score": 0.971, + "type": "image", + "image_path": "f9caf5f55023339b16fb95b389bc65cc48ec79600f2fc76d230026679d0627e7.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 147, + 436, + 464, + 493.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 147, + 493.3333333333333, + 464, + 550.6666666666666 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 147, + 550.6666666666666, + 464, + 608.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 176, + 616, + 434, + 628 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 176, + 615, + 435, + 630 + ], + "spans": [ + { + "bbox": [ + 176, + 615, + 435, + 630 + ], + "score": 1.0, + "content": "Figure 5: Running examples on IWSLT15 English-Vietnamese dataset", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + } + ], + "index": 15.0 + }, + { + "type": "title", + "bbox": [ + 108, + 655, + 222, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 223, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 223, + 667 + ], + "score": 1.0, + "content": "4.5 RANK OF THE HEADS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "In Figure 6, we calculate the average and standard deviation of rank of each head when generating", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "every target token. For MMA-IL, we find that heads in lower layers tend to have higher rank and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "are thus slower. However, in MMA-H, the difference of the average rank are smaller. 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HyperparameterWMT15 German-EnglishIWSLT English-Vietnamese
encoder embed dim1024512
encoder ffn embed dim40961024
encoder attention heads164
encoder layers6
decoder embed dim1024512
decoder ffn embed dim40961024
decoder attention heads164
decoder layers6
dropout0.3
optimizeradam
adam-β(0.9,0.98)
clip-norm0.0
lr lr scheduler0.0005
inverse sqrt
warmup-updates4000
warmup-init-lr1e-07
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Latency Metric
Average Proportiony 1 M gi lacllyl i=1
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HyperparameterWMT15 German-EnglishIWSLT English-Vietnamese
encoder embed dim1024512
encoder ffn embed dim40961024
encoder attention heads164
encoder layers6
decoder embed dim1024512
decoder ffn embed dim40961024
decoder attention heads164
decoder layers6
dropout0.3
optimizeradam
adam-β(0.9,0.98)
clip-norm0.0
lr lr scheduler0.0005
inverse sqrt
warmup-updates4000
warmup-init-lr1e-07
label-smoothing0.1
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Latency Metric
Average Proportiony 1 M gi lacllyl i=1
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BLEU AP AL DAL
Xavg MMA-IL
0.05 30.7 0.78 10.91 12.64
0.1 30.5 0.70 7.42 8.82
0.2 30.1 0.63 5.17 6.41
0.3 30.3 0.60 4.18 5.35
0.4 29.2 0.59 3.75 4.90
0.5 26.7 0.59 3.69 4.83
0.75 25.5 0.58 3.40 4.46
1.0 25.1 0.56 3.00 4.03
Xvar MMA-H
0.1 28.5 0.74 8.94 10.83
0.2 28.9 0.69 6.82 8.622
0.3 29.2 0.64 5.45 7.03
0.4 28.5 0.59 3.90 5.21
0.5 28.5 0.59 3.88 5.19
0.6 29.6 0.56 3.13 4.32
0.7 29.1 0.56 2.93 4.10
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BLEUAPALDAL
入MILk
0.124.620.715.937.19
0.224.680.674.905.97
0.324.310.654.455.43
0.423.730.644.285.24
XaugMMA-IL
0.0228.280.767.098.29
0.0428.330.705.446.57
0.128.420.674.635.65
0.228.470.633.574.44
0.327.90.592.983.81
0.427.730.582.683.46
XuarMMA-H
0.0227.260.777.528.71
0.15.226.31
27.680.693.81
0.228.060.634.84
0.427.790.623.574.59
0.827.950.603.224.19
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BLEU AP AL DAL
Xavg MMA-IL
0.05 30.7 0.78 10.91 12.64
0.1 30.5 0.70 7.42 8.82
0.2 30.1 0.63 5.17 6.41
0.3 30.3 0.60 4.18 5.35
0.4 29.2 0.59 3.75 4.90
0.5 26.7 0.59 3.69 4.83
0.75 25.5 0.58 3.40 4.46
1.0 25.1 0.56 3.00 4.03
Xvar MMA-H
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0.2 28.9 0.69 6.82 8.622
0.3 29.2 0.64 5.45 7.03
0.4 28.5 0.59 3.90 5.21
0.5 28.5 0.59 3.88 5.19
0.6 29.6 0.56 3.13 4.32
0.7 29.1 0.56 2.93 4.10
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BLEUAPALDAL
入MILk
0.124.620.715.937.19
0.224.680.674.905.97
0.324.310.654.455.43
0.423.730.644.285.24
XaugMMA-IL
0.0228.280.767.098.29
0.0428.330.705.446.57
0.128.420.674.635.65
0.228.470.633.574.44
0.327.90.592.983.81
0.427.730.582.683.46
XuarMMA-H
0.0227.260.777.528.71
0.15.226.31
27.680.693.81
0.228.060.634.84
0.427.790.623.574.59
0.827.950.603.224.19
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Reading ThresholdWeighted Average Latency Loss
pBLEUAPALDALLavgBLEUAPALDAL
0.525.50.7923877.136738.271870.0225.50.7923877.136738.27187
0.425.250.737495.720036.858120.0425.680.7281075.528566.61744
0.323.060.6973984.880876.033420.324.90.6027032.900543.68039
0.218.370.6782984.710995.946360.225.30.6369143.545774.38623
0.18.730.6964525.52257.204390.125.480.6844244.579015.54102
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BLEUAPALDAL
XaugAverage
0.0528.50.86258114.384717.3702
0.127.80.85543513.97417.02
0.2280.83532412.853115.908
0.428.10.81940811.981614.9763
1.028.20.81060911.752814.6695
2.028.10.80025811.176314.0761
8.028.40.80643911.528914.6431
XaugWeighted Average
0.0228.240.77392210.210912.2274
0.0424.350.6858347.067168.64069
0.067.800.87582516.204619.0892
0.089.510.573723.920116.1421
0.19.780.5565853.30075.46142
XvarDivergence
0.127.350.7360258.7096810.5253
0.227.640.6814916.639148.3856
0.327.370.66236.049027.71922
0.427.620.6381885.316726.86834
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BLEUAPALDAL
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1:while yi-1 ≠ EndOfSequence do
2:tmax =1
3:h = empty sequence
4:forl←1toLdo
5:forh←1to Hdo
6:forj←ttoacldo
7:l,h Pi,j, = Sigmoid (MonotonicEnergy(Si-1,mj ))
8:ifpi l,h > 0.5 then
9:=j
10:l,h =fcontext(h,t,h）
11:Ci Break
12:else
13:if j>tmax then
14:Read token x j
15:Calculate state h j and append to h
16:tmax =j
17:c = Concat(c,1,,cH）
18:s =DecoderLayer'(s-1,s)
19:yi = Output(s)
20:
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DatasetBeam SearchBidirectional EncoderUnidirectional Encoder
WMT15 De-En132.632.3
433.033.0
IWSLT15 En-Vi128.729.4
1028.829.5
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DatasetRNNTransformer
IWSLT15 En-Vi25.6628.7
WMT15 De-En28.4 (Arivazhagan et al., 2019)32.3
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DatasetTrainValidationTest
IWSLT15 En-Vi133k12681553
WMT15 De-En4.5M30002169
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HyperparameterWMT15 German-EnglishIWSLT English-Vietnamese
encoder embed dim1024512
encoder ffn embed dim40961024
encoder attention heads164
encoder layers6
decoder embed dim1024512
decoder ffn embed dim40961024
decoder attention heads164
decoder layers6
dropout0.3
optimizeradam
adam-β(0.9,0.98)
clip-norm0.0
lr lr scheduler0.0005
inverse sqrt
warmup-updates4000
warmup-init-lr1e-07
label-smoothing0.1
max tokens3584×8×8×216000
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Latency Metric
Average Proportiony 1 M gi lacllyl i=1
Average Lagging1 T i-1 gi 1y1/ac T i=1 where T = arg maxi(gi = |xl)
1 ly Differentiable Average Lagginglyl i-1 M gi lyl//ac| i=1 gi i=0 where g' = ly max(gi, 9i-1 i<0 [x
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BLEU AP AL DAL
Xavg MMA-IL
0.05 30.7 0.78 10.91 12.64
0.1 30.5 0.70 7.42 8.82
0.2 30.1 0.63 5.17 6.41
0.3 30.3 0.60 4.18 5.35
0.4 29.2 0.59 3.75 4.90
0.5 26.7 0.59 3.69 4.83
0.75 25.5 0.58 3.40 4.46
1.0 25.1 0.56 3.00 4.03
Xvar MMA-H
0.1 28.5 0.74 8.94 10.83
0.2 28.9 0.69 6.82 8.622
0.3 29.2 0.64 5.45 7.03
0.4 28.5 0.59 3.90 5.21
0.5 28.5 0.59 3.88 5.19
0.6 29.6 0.56 3.13 4.32
0.7 29.1 0.56 2.93 4.10
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BLEUAPALDAL
入MILk
0.124.620.715.937.19
0.224.680.674.905.97
0.324.310.654.455.43
0.423.730.644.285.24
XaugMMA-IL
0.0228.280.767.098.29
0.0428.330.705.446.57
0.128.420.674.635.65
0.228.470.633.574.44
0.327.90.592.983.81
0.427.730.582.683.46
XuarMMA-H
0.0227.260.777.528.71
0.15.226.31
27.680.693.81
0.228.060.634.84
0.427.790.623.574.59
0.827.950.603.224.19
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Reading ThresholdWeighted Average Latency Loss
pBLEUAPALDALLavgBLEUAPALDAL
0.525.50.7923877.136738.271870.0225.50.7923877.136738.27187
0.425.250.737495.720036.858120.0425.680.7281075.528566.61744
0.323.060.6973984.880876.033420.324.90.6027032.900543.68039
0.218.370.6782984.710995.946360.225.30.6369143.545774.38623
0.18.730.6964525.52257.204390.125.480.6844244.579015.54102
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BLEUAPALDAL
XaugAverage
0.0528.50.86258114.384717.3702
0.127.80.85543513.97417.02
0.2280.83532412.853115.908
0.428.10.81940811.981614.9763
1.028.20.81060911.752814.6695
2.028.10.80025811.176314.0761
8.028.40.80643911.528914.6431
XaugWeighted Average
0.0228.240.77392210.210912.2274
0.0424.350.6858347.067168.64069
0.067.800.87582516.204619.0892
0.089.510.573723.920116.1421
0.19.780.5565853.30075.46142
XvarDivergence
0.127.350.7360258.7096810.5253
0.227.640.6814916.639148.3856
0.327.370.66236.049027.71922
0.427.620.6381885.316726.86834
0.527.500.6257594.930446.38998
1.027.10.5821943.648644.90997
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+ "page_no": 14, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/SJeLIgBKPS/SJeLIgBKPS.md b/parse/train/SJeLIgBKPS/SJeLIgBKPS.md new file mode 100644 index 0000000000000000000000000000000000000000..88a4c03d43900bfa8679591e63fe20f1ba0f7e3e --- /dev/null +++ b/parse/train/SJeLIgBKPS/SJeLIgBKPS.md @@ -0,0 +1,2020 @@ +# GRADIENT DESCENT MAXIMIZES THE MARGIN OF HOMOGENEOUS NEURAL NETWORKS + +Kaifeng Lyu & Jian Li +Institute for Interdisciplinary Information Sciences +Tsinghua University +Beijing, China +vfleaking@gmail.com,lijian83@mail.tsinghua.edu.cn + +# ABSTRACT + +In this paper, we study the implicit regularization of the gradient descent algorithm in homogeneous neural networks, including fully-connected and convolutional neural networks with ReLU or LeakyReLU activations. In particular, we study the gradient descent or gradient flow (i.e., gradient descent with infinitesimal step size) optimizing the logistic loss or cross-entropy loss of any homogeneous model (possibly non-smooth), and show that if the training loss decreases below a certain threshold, then we can define a smoothed version of the normalized margin which increases over time. We also formulate a natural constrained optimization problem related to margin maximization, and prove that both the normalized margin and its smoothed version converge to the objective value at a KKT point of the optimization problem. Our results generalize the previous results for logistic regression with one-layer or multi-layer linear networks, and provide more quantitative convergence results with weaker assumptions than previous results for homogeneous smooth neural networks. We conduct several experiments to justify our theoretical finding on MNIST and CIFAR-10 datasets. Finally, as margin is closely related to robustness, we discuss potential benefits of training longer for improving the robustness of the model. + +# 1 INTRODUCTION + +A major open question in deep learning is why gradient descent or its variants, are biased towards solutions with good generalization performance on the test set. To achieve a better understanding, previous works have studied the implicit bias of gradient descent in different settings. One simple but insightful setting is linear logistic regression on linearly separable data. In this setting, the model is parameterized by a weight vector $\textbf { \em w }$ , and the class prediction for any data point $_ { \textbf { \em x } }$ is determined by the sign of ${ \pmb w } ^ { \top } { \pmb x }$ . Therefore, only the direction ${ w / \| \boldsymbol { w } \| _ { 2 } }$ is important for making prediction. Soudry et al. (2018a;b); Ji & Telgarsky (2018; 2019c); Nacson et al. (2019c) investigated this problem and proved that the direction of $\pmb { w }$ converges to the direction that maximizes the $L ^ { 2 }$ -margin while the norm of $\pmb { w }$ diverges to $+ \infty$ , if we train $\pmb { w }$ with (stochastic) gradient descent on logistic loss. Interestingly, this convergent direction is the same as that of any regularization path: any sequence of weight vectors $\{ { \pmb w } _ { t } \}$ such that every ${ \pmb w } _ { t }$ is a global minimum of the $L ^ { 2 }$ -regularized loss $\begin{array} { r } { \mathcal { L } ( \pmb { w } ) + \frac { \lambda _ { t } } { 2 } \| \pmb { w } \| _ { 2 } ^ { 2 } } \end{array}$ with $\lambda _ { t } \to 0$ (Rosset et al., 2004). Indeed, the trajectory of gradient descent is also pointwise close to a regularization path (Suggala et al., 2018). + +The aforementioned linear logistic regression can be viewed as a single-layer neural network. A natural and important question is to what extent gradient descent has similiar implicit bias for modern deep neural networks. For theoretical analysis, a natural candidate is to consider homogeneous neural networks. Here a neural network $\Phi$ is said to be (positively) homogeneous if there is a number $L > 0$ (called the order) such that the network output $\Phi ( \pmb \theta ; \pmb x )$ , where $\pmb { \theta }$ stands for the parameter and $_ { \textbf { \em x } }$ stands for the input, satisfies the following: + +$$ +\forall c > 0 : \Phi ( c \pmb \theta ; \pmb x ) = c ^ { L } \Phi ( \pmb \theta ; \pmb x ) \mathrm { f o r ~ a l l ~ } \pmb \theta \mathrm { ~ a n d ~ } \pmb x . +$$ + +It is important to note that many neural networks are homogeneous (Neyshabur et al., 2015a; Du et al., 2018). For example, deep fully-connected neural networks or deep CNNs with ReLU or + +LeakyReLU activations can be made homogeneous if we remove all the bias terms, and the order $L$ is exactly equal to the number of layers. + +In (Wei et al., 2019), it is shown that the regularization path does converge to the max-margin direction for homogeneous neural networks with cross-entropy or logistic loss. This result suggests that gradient descent or gradient flow may also converges to the max-margin direction by assuming homogeneity, and this is indeed true for some sub-classes of homogeneous neural networks. For gradient flow, this convergent direction is proven for linear fully-connected networks (Ji & Telgarsky, 2019a). For gradient descent on linear fully-connected and convolutional networks, (Gunasekar et al., 2018b) formulate a constrained optimization problem related to margin maximization and prove that gradient descent converges to the direction of a KKT point or even the max-margin direction, under various assumptions including the convergence of loss and gradient directions. In an independent work, (Nacson et al., 2019a) generalize the result in (Gunasekar et al., 2018b) to smooth homogeneous models (we will discuss this work in more details in Section 2). + +# 1.1 MAIN RESULTS + +In this paper, we identify a minimal set of assumptions for proving our theoretical results for homogeneous neural networks on classification tasks. Besides homogeneity, we make two additional assumptions: + +1. Exponential-type Loss Function. We require the loss function to have certain exponential tail (see Appendix A for the details). This assumption is not restrictive as it includes the most popular classfication losses: exponential loss, logistic loss and cross-entropy loss. +2. Separability. The neural network can separate the training data during training (i.e., the neural network can achieve $1 0 0 \%$ training accuracy)1. + +While the first assumption is natural, the second requires some explanation. In fact, we assume that at some time $t _ { 0 }$ , the training loss is smaller than a threshold, and the threshold here is chosen to be so small that the training accuracy is guaranteed to be $1 0 0 \%$ (e.g., for the logistic loss and cross-entropy loss, the threshold can be set to $\mathrm { l n } 2$ ). Empirically, state-of-the-art CNNs for image classification can even fit randomly labeled data easily (Zhang et al., 2017). Recent theoretical work on over-parameterized neural networks (Allen-Zhu et al., 2019; Zou et al., 2018) show that gradient descent can fit the training data if the width is large enough. Furthermore, in order to study the margin, ensuring the training data can be separated is inevitable; otherwise, there is no positive margin between the data and decision boundary. + +Our Contribution. Similar to linear models, for homogeneous models, only the direction of parameter $\pmb \theta$ is important for making predictions, and one can see that the margin $\gamma ( \pmb \theta )$ scales linearly with $\lVert \pmb { \theta } \rVert _ { 2 } ^ { L }$ , when fixing the direction of $\pmb \theta$ . To compare margins among $\pmb { \theta }$ in different directions, it makes sense to study the normalized margin, $\bar { \gamma } ( \pmb \theta ) : = \gamma ( \pmb \theta ) / \| \pmb \theta \| _ { 2 } ^ { L }$ . + +In this paper, we focus on the training dynamics of the network after $t _ { 0 }$ (recall that $t _ { 0 }$ is a time that the training loss is less than the threshold). Our theoretical results can answer the following questions regarding the normalized margin. + +First, how does the normalized margin change during training? The answer may seem complicated since one can easily come up with examples in which $\bar { \gamma }$ increases or decreases in a short time interval. However, we can show that the overall trend of the normalized margin is to increase in the following sense: there exists a smoothed version of the normalized margin, denoted as $\tilde { \gamma }$ , such that (1) $| \tilde { \gamma } - \bar { \tilde { \gamma } } | \to 0$ as $t \to \infty$ ; and $( 2 ) \tilde { \gamma }$ is non-decreasing for $t > t _ { 0 }$ . + +Second, how large is the normalized margin at convergence? To answer this question, we formulate a natural constrained optimization problem which aims to directly maximize the margin. We show that every limit point of $\{ \pmb \theta ( t ) / \| \pmb \theta ( t ) \| _ { 2 } : t > 0 \}$ is along the direction of a KKT point of the maxmargin problem. This indicates that gradient descent/gradient flow performs margin maximization implicitly in deep homogeneous networks. This result can be seen as a significant generalization of previous works (Soudry et al., 2018a;b; Ji & Telgarsky, $2 0 1 9 \mathrm { a }$ ; Gunasekar et al., 2018b) from linear classifiers to homogeneous classifiers. + +![](images/2e818eb552978f56e83f486e74331e74a063c35463a07ca1e458303572110c47.jpg) +Figure 1: (a) Training CNNs with and without bias on MNIST, using SGD with learning rate 0.01. The training loss (left) decreases over time, and the normalized margin (right) keeps increasing after the model is fitted, but the growth rate is slow $( \approx 1 . 8 \times 1 0 ^ { - 4 }$ after 10000 epochs). (b) Training CNNs with and without bias on MNIST, using SGD with the loss-based learning rate scheduler. The training loss (left) decreases exponentially over time $( < 1 0 ^ { - 8 0 0 }$ after 9000 epochs), and the normalized margin (right) increases rapidly after the model is fitted $( \approx 1 . 2 \times 1 0 ^ { - 3 }$ after 10000 epochs, $1 0 \times$ larger than that of SGD with learning rate 0.01). Experimental details are in Appendix $\mathbf { K }$ . + +As by-products of the above results, we derive tight asymptotic convergence/growth rates of the loss and weights. It is shown in (Soudry et al., 2018a;b; Ji & Telgarsky, 2018; 2019c) that the loss decreases at the rate of $O ( 1 / t )$ , the weight norm grows as $O ( \log t )$ for linear logistic regression. In this work, we generalize the result by showing that the loss decreases at the rate of $O ( 1 / ( t ( \log t ) ^ { 2 - 2 / L } ) )$ and the weight norm grows as $O ( ( \log t ) ^ { 1 / L } )$ for homogeneous neural networks with exponential loss, logistic loss, or cross-entropy loss. + +Experiments.2 The main practical implication of our theoretical result is that training longer can enlarge the normalized margin. To justify this claim empiricaly, we train CNNs on MNIST and CIFAR-10 with SGD (see Section K.1). Results on MNIST are presented in Figure 1. For constant step size, we can see that the normalized margin keeps increasing, but the growth rate is rather slow (because the gradient gets smaller and smaller). Inspired by our convergence results for gradient descent, we use a learning rate scheduling method which enlarges the learning rate according to the current training loss, then the training loss decreases exponentially faster and the normalized margin increases significantly faster as well. + +For feedforward neural networks with ReLU activation, the normalized margin on a training sample is closely related to the $L ^ { 2 }$ -robustness (the $L ^ { 2 }$ -distance from the training sample to the decision boundary). Indeed, the former divided by a Lipschitz constant is a lower bound for the latter. For example, the normalized margin is a lower bound for the $L ^ { 2 }$ -robustness on fully-connected networks with ReLU activation (see, e.g., Theorem 4 in (Sokolic et al., 2017)). This fact suggests that training longer may have potential benefits on improving the robustness of the model. In our experiments, we observe noticeable improvements of $L ^ { 2 }$ -robustness on both training and test sets (see Section K.2). + +# 2 RELATED WORK + +Implicit Bias in Training Linear Classifiers. For linear logistic regression on linearly separable data, Soudry et al. (2018a;b) showed that full-batch gradient descent converges in the direction of the max $L ^ { 2 }$ -margin solution of the corresponding hard-margin Support Vector Machine (SVM). Subsequent works extended this result in several ways: Nacson et al. (2019c) extended the results to the case of stochastic gradient descent; Gunasekar et al. (2018a) considered other optimization methods; Nacson et al. (2019b) considered other loss functions including those with poly-exponential tails; Ji & Telgarsky (2018; 2019c) characterized the convergence of weight direction without assuming separability; Ji & Telgarsky (2019b) proved a tighter convergence rate for the weight direction. + +Those results on linear logistic regression have been generalized to deep linear networks. Ji & Telgarsky (2019a) showed that the product of weights in a deep linear network with strictly decreasing loss converges in the direction of the max $L ^ { 2 }$ -margin solution. Gunasekar et al. (2018b) showed more general results for gradient descent on linear fully-connected and convolutional networks with exponential loss, under various assumptions on the convergence of the loss and gradient direction. + +Margin maximization phenomenon is also studied for boosting methods (Schapire et al., 1998; Schapire & Freund, 2012; Shalev-Shwartz & Singer, 2010; Telgarsky, 2013) and Normalized Perceptron (Ramdas & Pena, 2016). + +Implicit Bias in Training Nonlinear Classifiers. Soudry et al. (2018a) analyzed the case where there is only one trainable layer of a ReLU network. Xu et al. (2018) characterized the implicit bias for the model consisting of one single ReLU unit. Our work is closely related to a recent independent work by (Nacson et al., 2019a) which we discuss in details below. + +Comparison with (Nacson et al., 2019a). Very recently, (Nacson et al., 2019a) analyzed gradient descent for smooth homogeneous models and proved the convergence of parameter direction to a KKT point of the aforementioned max-margin problem. Compared with their work, our work adopt much weaker assumptions: (1) They assume the training loss converges to 0, but in our work we only require that the training loss is lower than a small threshold value at some time $t _ { 0 }$ (and we prove the exact convergence rate of the loss after $t _ { 0 }$ ); (2) They assume the convergence of parameter direction3, while we prove that KKT conditions hold for all limit points of $\{ \pmb \theta ( t ) \bar { / } \lVert \pmb \theta ( t ) \rVert _ { 2 } : t > 0 \}$ , without requiring any convergence assumption; (3) They assume the convergence of the direction of losses (the direction of the vector whose entries are loss values on every data point) and Linear Independence Constraint Qualification (LICQ) for the max-margin problem, while we do not need such assumptions. Besides the above differences in assumptions, we also prove the monotonicity of the normalized margin and provide tight convergence rate for training loss. We believe both results are interesting in their own right. + +Another technical difference is that their work analyzes discrete gradient descent on smooth homogeneous models (which fails to capture ReLU networks). In our work, we analyze both gradient descent on smooth homogeneous models and also gradient flow on homogeneous models which could be non-smooth. + +Other Works on Implicit Bias. Banburski et al. (2019) also studied the dynamics of gradient flow and among other things, provided mathematical insights to the implicit bias towards max margin solution for homogeneous networks. We note that their analysis of gradient flow decomposes the dynamics to the tangent component and radial component, which is similar to our proof of Theorem 4.1 in spirit. Wilson et al. (2017); Ali et al. (2019); Gunasekar et al. (2018a) showed that for the linear least-square problem gradient-based methods converge to the unique global minimum that is closest to the initialization in $L ^ { 2 }$ distance. Du et al. (2019); Jacot et al. (2018); Lee et al. (2019); Arora et al. (2019b) showed that over-parameterized neural networks of sufficient width (or infinite width) behave as linear models with Neural Tangent Kernel (NTK) with proper initialization and gradient descent converges linearly to a global minimum near the initial point. Other related works include (Ma et al., 2019; Gidel et al., 2019; Arora et al., $2 0 1 9 \mathrm { a }$ ; Suggala et al., 2018; Blanc et al., 2019; Neyshabur et al., 2015b;a). + +# 3 PRELIMINARIES + +Basic Notations. For any $N \in \mathbb N$ , let $[ N ] = \{ 1 , \dots , N \}$ . $\lVert \pmb { v } \rVert _ { 2 }$ denotes the $L ^ { 2 }$ -norm of a vector $\textbf { { v } }$ . The default base of log is $e$ . For a function $f : { \mathbb { R } ^ { d } } \to { \mathbb { R } }$ , $\nabla f ( { \pmb x } )$ stands for the gradient at $_ { \textbf { \em x } }$ if it exists. A function $f : X \to \mathbb { R } ^ { d }$ is $\mathcal { C } ^ { k }$ -smooth if $f$ is $k$ times continuously differentiable. A function $f : X \to \mathbb { R }$ is locally Lipschitz if for every $\pmb { x } \in X$ there exists a neighborhood $U$ of $_ { \textbf { \em x } }$ such that the restriction of $f$ on $U$ is Lipschitz continuous. + +Non-smooth Analysis. For a locally Lipschitz function $f : X \to \mathbb { R }$ , the Clarke’s subdifferential (Clarke, 1975; Clarke et al., 2008; Davis et al., 2020) at $\pmb { x } \in X$ is the convex set + +$$ +\partial ^ { \circ } f ( { \pmb x } ) : = \operatorname { c o n v } \left. \operatorname* { l i m } _ { k \to \infty } \nabla f ( { \pmb x } _ { k } ) : { \pmb x } _ { k } \to { \pmb x } , f \mathrm { ~ i s ~ d i f f e r e n t i a b l e ~ a t } { \pmb x } _ { k } \right. . +$$ + +For brevity, we say that a function $z : I \mathbb { R } ^ { d }$ on the interval $I$ is an arc if $_ z$ is absolutely continuous for any compact sub-interval of exists. Following the terminolo $I$ . For an arc in (Davis e $_ z$ , a $z ^ { \prime } ( t )$ (or 20), $\textstyle { \frac { d z } { d t } } ( t ) )$ stands for the derivative at y that a locally Lipschitz fu $t$ if ittion $f : { \mathbb { R } ^ { d } } \to { \mathbb { R } }$ admits a chain rule if for any arc $z : [ 0 , + \infty ) \to \mathbb { R } ^ { \bar { d } }$ , $\forall h \in \partial ^ { \circ } f ( \bar { z } ( t ) ) \bar { : } ( f \circ z ) ^ { \prime } ( t ) =$ $\langle h , z ^ { \prime } ( t ) \rangle$ holds for a.e. $t > 0$ (see also Appendix I). + +Binary Classification. Let $\Phi$ be a neural network, assumed to be parameterized by $\pmb { \theta }$ . The output of $\Phi$ on an input $\pmb { x } \in \mathbb { R } ^ { d _ { \mathbf { x } } }$ is a real number $\Phi ( \pmb \theta ; \pmb x )$ , and the sign of $\Phi ( \pmb \theta ; \pmb x )$ stands for the classification result. A dataset is denoted by $\mathcal { D } = \{ ( x _ { n } , y _ { n } ) : n \in [ N ] \}$ , where $\pmb { x } _ { n } \in \mathbb { R } ^ { d _ { \mathrm { { x } } } }$ stands for a data input and $y _ { n } \in \{ \pm 1 \}$ stands for the corresponding label. For a loss function $\ell : \mathbb { R } \to \mathbb { R }$ , we define the training loss of $\Phi$ on the dataset $\mathcal { D }$ to be $\begin{array} { r } { \mathcal { L } ( \pmb { \theta } ) : = \sum _ { n = 1 } ^ { N } \ell ( y _ { n } \Phi ( \pmb { \theta } ; \pmb { x } _ { n } ) ) } \end{array}$ . + +Gradient Descent. We consider the process of training this neural network $\Phi$ with either gradient descent or gradient flow. For gradient descent, we assume the training loss $\mathcal { L } ( \pmb \theta )$ is $\mathcal { C } ^ { 2 }$ -smooth and describe the gradient descnet process as $\pmb { \theta } ( t + 1 ) = \pmb { \theta } ( t ) - \eta ( t ) \nabla \mathcal { L } ( \pmb { \theta } ( \bar { t } ) )$ , where $\dot { \eta } ( t )$ is the learning rate at time $t$ and $\nabla \mathcal { L } ( \pmb \theta ( t ) )$ is the gradient of $\mathcal { L }$ at $\pmb \theta ( t )$ . + +Gradient Flow. For gradient flow, we do not assume the differentibility but only some regularity assumptions including locally Lipschitz. Gradient flow can be seen as gradient descent with infinitesimal step size. In this model, $\pmb \theta$ changes continuously with time, and the trajectory of parameter $\pmb { \theta }$ during training is an arc $\pmb \theta : [ 0 , + \infty ) \dot { } \mathbb R ^ { d } , t \mapsto \pmb \theta ( \dot { t } )$ that satisfies the differential inclusion $\begin{array} { r } { \frac { d { \pmb \theta } ( t ) } { d t } \in - \partial ^ { \circ } \mathcal { L } ( { \pmb \theta } ( t ) ) } \end{array}$ for a.e. $t \geq 0$ . The Clarke’s subdifferential $\partial ^ { \circ } { \mathcal { L } }$ is a natural generalization of the usual differential to non-differentiable functions. If $\mathcal { L } ( \pmb \theta )$ is actually a ${ \mathcal { C } } ^ { 1 }$ -smooth function, the above differential inclusion reduces to $\begin{array} { r } { \frac { d \pmb { \theta } ( t ) } { d t } = - \nabla \mathcal { L } ( \pmb { \theta } ( t ) ) } \end{array}$ for all $t \geq 0$ , which corresponds to the gradient flow with differential in the usual sense. + +# 4 GRADIENT DESCENT / GRADIENT FLOW ON HOMOGENEOUS MODEL + +In this section, we first state our results for gradient flow and gradient descent on homogeneous models with exponential loss $\ell ( q ) : = e ^ { - q }$ for simplicity of presentation. Due to space limit, we defer the more general results which hold for a large family of loss functions (including logistic loss and cross-entropy loss) to Appendix A, F and $\mathbf { G }$ . + +# 4.1 ASSUMPTIONS + +Gradient Flow. For gradient flow, we assume the following: + +(A1). (Regularity). For any fixed $_ { \textbf { \em x } }$ , $\Phi ( \cdot ; x )$ is locally Lipschitz and admits a chain rule; +(A2). (Homogeneity). There exists $L > 0$ such that $\forall \alpha > 0 : \Phi ( \alpha \pmb { \theta } ; \pmb { x } ) = \alpha ^ { L } \Phi ( \pmb { \theta } ; \pmb { x } )$ ; +(A3). (Exponential Loss). $\ell ( q ) = e ^ { - q }$ ; +(A4). (Separability). There exists a time $t _ { 0 }$ such that $\mathcal { L } ( \pmb { \theta } ( t _ { 0 } ) ) < 1$ . + +(A1) is a technical assumption about the regularity of the network output. As shown in (Davis et al., 2020), the output of almost every neural network admits a chain rule (as long as the neural network is composed by definable pieces in an o-minimal structure, e.g., ReLU, sigmoid, LeakyReLU). + +(A2) assumes the homogeneity, the main property we assume in this work. (A3), (A4) correspond to the two conditions introduced in Section 1. The exponential loss in (A3) is main focus of this section. (A4) is a separability assumption: the condition $\mathcal { L } ( \pmb \theta ( t _ { 0 } ) ) < 1$ ensures that $\ell ( y _ { n } \Phi ( \pmb \theta ( t _ { 0 } ) ; \pmb x _ { n } ) ) < 1$ for all $n \in [ N ]$ , and thus $y _ { n } \bar { \Phi } ( \pmb \theta ( t _ { 0 } ) ; \pmb x _ { n } ) > 0$ , meaning that $\Phi$ classifies every ${ \mathbf { \mathcal { x } } } _ { n }$ correctly. + +Gradient Descent. For gradient descent, we assume (A2), (A3), (A4) similarly as for gradient flow, and the following two assumptions (S1) and (S5). + +(S1). (Smoothness). For any fixed $_ { \textbf { \em x } }$ , $\Phi ( \cdot ; x )$ is $\mathcal { C } ^ { 2 }$ -smooth on $\mathbb { R } ^ { d } \backslash \{ \mathbf { 0 } \}$ . +(S5). (Learning rate condition, Informal). $\eta ( t ) = \eta _ { 0 }$ for a sufficiently small constant $\eta _ { 0 }$ . In fact, $\eta ( t )$ is even allowed to be as large as $\begin{array} { r } { \dot { O } ( \mathcal { L } ( \dot { t } ) ^ { - 1 } \mathrm { p o l y l o g } \frac { 1 } { \mathcal { L } ( t ) } ) } \end{array}$ ). See Appendix E.1 for the details. + +(S5) is natural since deep neural networks are usually trained with constant learning rates. (S1) ensures the smoothness of $\Phi$ , which is often assumed in the optimization literature in order to analyze gradient descent. While (S1) does not hold for neural networks with ReLU, it does hold for neural networks with smooth homogeneous activation such as the quadratic activation $\phi ( x ) : = x ^ { 2 }$ (Li et al., 2018b; Du & Lee, 2018) or powers of ReLU $\phi ( x ) : = \mathrm { R e L U } ( x ) ^ { \alpha }$ for $\alpha > 2$ (Zhong et al., 2017; Klusowski & Barron, 2018; Li et al., 2019). + +# 4.2 MAIN THEOREM: MONOTONICITY OF NORMALIZED MARGINS + +The margin for a single data point $( { \pmb x } _ { n } , y _ { n } )$ is defined to be $q _ { n } ( \pmb \theta ) : = y _ { n } \Phi ( \pmb \theta ; \pmb x _ { n } )$ , and the margin for the entire dataset is defined to be $q _ { \operatorname* { m i n } } ( \pmb { \theta } ) : = \operatorname* { m i n } _ { n \in [ N ] } q _ { n } ( \pmb { \theta } )$ . By homogenity, the margin $q _ { \mathrm { m i n } } ( \pmb { \theta } )$ scales linearly with $\lVert \pmb { \theta } \rVert _ { 2 } ^ { L }$ for any fixed direction since $q _ { \mathrm { m i n } } ( c \pmb \theta ) = c ^ { L } q _ { \mathrm { m i n } } ( \pmb \theta )$ . So we consider the normalized margin defined as below: + +$$ +\bar { \gamma } ( \pmb \theta ) : = q _ { \mathrm { m i n } } \left( \frac { \pmb \theta } { \lVert \pmb \theta \rVert _ { 2 } } \right) = \frac { q _ { \mathrm { m i n } } ( \pmb \theta ) } { \lVert \pmb \theta \rVert _ { 2 } ^ { L } } . +$$ + +We say $f$ is an $\epsilon$ -additive approximation for the normalized margin if $\bar { \gamma } \mathrm { ~ - ~ } \epsilon \le f \le \bar { \gamma }$ , and $c$ - multiplicative approximation if $c \bar { \gamma } \leq f \leq \bar { \gamma }$ . + +Gradient Flow. Our first result is on the overall trend of the normalized margin $\bar { \gamma } ( \pmb \theta ( t ) )$ . For both gradient flow and gradient descent, we identify a smoothed version of the normalized margin, and show that it is non-decreasing during training. More specifically, we have the following theorem for gradient flow. + +Theorem 4.1 (Corollary of Theorem A.7). Under assumptions $( A I ) ~ - ~ ( A 4 )$ , there exists an $O ( \| \pmb { \theta } \| _ { 2 } ^ { - L } )$ -additive approximation function $\widetilde \gamma ( \pmb \theta )$ for the normalized margin such that the following statements are true for gradient flow: + +1. For a.e. $t > t _ { 0 }$ , $\begin{array} { r } { \frac { d } { d t } \tilde { \gamma } ( \pmb { \theta } ( t ) ) \geq 0 } \end{array}$ ; + +$t > t _ { 0 }$ $\begin{array} { r } { \frac { d } { d t } \tilde { \gamma } ( \pmb { \theta } ( t ) ) > 0 } \end{array}$ ${ \frac { d } { d t } } { \frac { \pmb { \theta } ( t ) } { \| \pmb { \theta } ( t ) \| _ { 2 } } } = 0$ + +3. $\mathcal { L } ( \pmb \theta ( t ) ) 0$ and $\| \pmb \theta ( t ) \| _ { 2 } \to \infty$ as $t \to + \infty$ ; therefore, $| \bar { \gamma } ( \pmb \theta ( t ) ) - \tilde { \gamma } ( \pmb \theta ( t ) ) | 0$ . + +More concretely, the function $\tilde { \gamma } ( \pmb \theta )$ in Theorem 4.1 is defined as + +$$ +\widetilde { \gamma } ( \pmb \theta ) : = \frac { \log \frac { 1 } { \mathcal { L } ( \pmb \theta ) } } { \| \pmb \theta \| _ { 2 } ^ { L } } = \frac { - \log \left( \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( \pmb \theta ) } \right) } { \| \pmb \theta \| _ { 2 } ^ { L } } . +$$ + +Note that the only difference between $\bar { \gamma } ( \pmb \theta )$ and $\tilde { \gamma } ( \pmb \theta )$ is that the margin $q _ { \mathrm { m i n } } ( \pmb { \theta } )$ in $\bar { \gamma } ( \pmb \theta )$ is replaced by $\begin{array} { r } { \log \frac { 1 } { \mathcal { L } ( \pmb { \theta } ) } = - \mathrm { L S E } ( - q _ { 1 } ( \pmb { \theta } ) , \dots , - q _ { N } ( \pmb { \theta } ) ) } \end{array}$ , where $\operatorname { L S E } ( a _ { 1 } , \dots , a _ { N } ) = \log ( \exp ( a _ { 1 } ) +$ $\cdot \cdot \cdot + \exp ( a _ { N } ) )$ is the LogSumExp function. This is indeed a very natural idea, and previous works on linear models (e.g., (Telgarsky, 2013; Nacson et al., 2019b)) also approximate $q _ { \mathrm { m i n } }$ with LogSumExp in the analysis of margin. It is easy to see why $\widetilde \gamma ( \pmb \theta )$ is an $O ( \| \pmb { \theta } \| _ { 2 } ^ { - L } )$ -additive approximation for $\bar { \gamma } ( \pmb \theta )$ : $\begin{array} { r } { \mathrm { ~ : ~ } e ^ { a _ { \mathrm { m a x } } } \leq \sum _ { n = 1 } ^ { N } e ^ { a _ { n } } \leq N e ^ { a _ { \mathrm { m a x } } } } \end{array}$ holds for $a _ { \mathrm { { m a x } } } = \operatorname* { m a x } \{ a _ { 1 } , \dots , a _ { N } \}$ , so $a _ { \mathrm { m a x } } \leq \mathrm { L S E } ( a _ { 1 } , \ldots , a _ { N } ) \leq a _ { \mathrm { m a x } } \bar { + } \log N$ ; combining this with the definition of $\widetilde \gamma ( \pmb \theta )$ gives $\bar { \gamma } ( \pmb \theta ) - \lVert \pmb \theta \rVert _ { 2 } ^ { - L } \log N \leq \tilde { \gamma } ( \pmb \theta ) \leq \bar { \gamma } ( \pmb \theta )$ . + +Gradient Descent. For gradient descent, Theorem 4.1 holds similarly with a slightly different function $\hat { \gamma } ( \pmb \theta )$ that approximates $\bar { \gamma } ( \pmb \theta )$ multiplicatively rather than additively. + +Theorem 4.2 (Corollary of Theorem E.2). Under assumptions $( S I )$ , (A2) - (A4), (S5), there exists an $( 1 - O ( 1 / ( \log { \frac { 1 } { \mathcal { L } } } ) ) )$ -multiplicative approximation function $\hat { \gamma } ( \pmb \theta )$ for the normalized margin such that the following statements are true for gradient descent: + +Due to the discreteness of gradient descent, the explicit formula for $\hat { \gamma } ( \pmb \theta )$ is somewhat technical, and we refer the readers to Appendix E for full details. + +Convergence Rates. It is shown in Theorem 4.1, 4.2 that $\mathcal { L } ( \pmb \theta ( t ) ) 0$ and $\| \pmb \theta ( t ) \| _ { 2 } \to \infty$ . In fact, with a more refined analysis, we can prove tight loss convergence and weight growth rates using the monotonicity of normalized margins. + +Theorem 4.3 (Corollary of Theorem A.10 and E.5). For gradient flow under assumptions (A1) - (A4) or gradient descent under assumptions (S1), (A2) - (A4), (S5), we have the following tight bounds for training loss and weight norm: + +$$ +\mathcal { L } ( \pmb \theta ( t ) ) = \Theta \left( \frac { 1 } { T ( \log T ) ^ { 2 - 2 / L } } \right) \qquad a n d \qquad \| \pmb \theta ( t ) \| _ { 2 } = \Theta \left( ( \log T ) ^ { 1 / L } \right) , +$$ + +where $T = t$ for gradient flow and $\begin{array} { r } { T = \sum _ { \tau = t _ { 0 } } ^ { t - 1 } \eta ( \tau ) } \end{array}$ for gradient descent. + +# 4.3 MAIN THEOREM: CONVERGENCE TO KKT POINTS + +For gradient flow, $\tilde { \gamma }$ is upper-bounded by $\tilde { \gamma } \le \bar { \gamma } \le \operatorname* { s u p } \{ q _ { n } ( \pmb \theta ) : \lVert \pmb \theta \rVert _ { 2 } = 1 \}$ . Combining this with Theorem 4.1 and the monotone convergence theorem, it is not hard to see that $\scriptstyle \operatorname* { l i m } _ { t \to + \infty } { \bar { \gamma } } ( \pmb { \theta } ( t ) )$ and $\scriptstyle \operatorname* { l i m } _ { t \to + \infty } { \tilde { \gamma } } ( \pmb { \theta } ( t ) )$ exist and equal to the same value. Using a similar argument, we can draw the same conclusion for gradient descent. + +To understand the implicit regularization effect, a natural question arises: what optimality property does the limit of normalized margin have? To this end, we identify a natural constrained optimization problem related to margin maximization, and prove that $\pmb \theta ( t )$ directionally converges to its KKT points, as shown below. We note that we can extend this result to the finite time case, and show that gradient flow or gradient descent passes through an approximate KKT point after a certain amount of time. See Theorem A.9 in Appendix A and Theorem E.4 in Appendix E for the details. We will briefly review the definition of KKT points and approximate KKT points for a constraint optimization problem in Appendix C.1. + +Theorem 4.4 (Corollary of Theorem A.8 and E.3). For gradient flow under assumptions (A1) - (A4) or gradient descent under assumptions (S1), (A2) - (A4), (S5), any limit point $\bar { \pmb { \theta } }$ of $\left\{ \frac { \pmb \theta ( t ) } { \| \pmb \theta ( t ) \| _ { 2 } } : t \geq 0 \right\}$ is along the direction of a KKT point of the following constrained optimization problem $( P )$ : + +$$ +\operatorname* { m i n } \quad \frac { 1 } { 2 } \| \pmb { \theta } \| _ { 2 } ^ { 2 } \qquad \quad s . t . \quad q _ { n } ( \pmb { \theta } ) \geq 1 \qquad \forall n \in [ N ] +$$ + +That is, for any limit point $\bar { \theta }$ , there exists a scaling factor $\alpha > 0$ such that $\alpha \bar { \pmb { \theta } }$ satisfies Karush-KuhnTucker (KKT) conditions of $( P )$ . + +Minimizing (P) over its feasible region is equivalent to maximizing the normalized margin over all possible directions. The proof is as follows. Note that we only need to consider all feasible points $\pmb { \theta }$ with $q _ { \mathrm { m i n } } ( \pmb \theta ) > 0$ . For a fixed $\pmb { \theta }$ , $\alpha \pmb { \theta }$ is a feasible point of (P) iff $\alpha \ge q _ { \mathrm { m i n } } ( \pmb { \theta } ) ^ { - 1 / L }$ . Thus, the minimum objective value over all feasible points of (P) in the direction of $\pmb \theta$ is $\begin{array} { r l } { \frac { 1 } { 2 } \| \pmb { \theta } / q _ { \mathrm { m i n } } ( \pmb { \theta } ) ^ { 1 / L } \| _ { 2 } ^ { 2 } = } \end{array}$ $\frac { 1 } { 2 } \bar { \gamma } ( \pmb { \theta } ) ^ { - 2 / L }$ . Taking minimum over all possible directions, we can conclude that if the maximum normalized margin is $\bar { \gamma } _ { * }$ , then the minimum objective of (P) is $\frac { 1 } { 2 } \bar { \gamma } _ { * } ^ { - 2 / L }$ . + +It can be proved that (P) satisfies the Mangasarian-Fromovitz Constraint Qualification (MFCQ) (See Lemma C.7). Thus, KKT conditions are first-order necessary conditions for global optimality. For linear models, KKT conditions are also sufficient for ensuring global optimality; however, for deep homogeneous networks, $q _ { n } ( \pmb \theta )$ can be highly non-convex. Indeed, as gradient descent is a first-order optimization method, if we do not make further assumptions on $q _ { n } ( \pmb \theta )$ , then it is easy to construct examples that gradient descent does not lead to a normalized margin that is globally optimal. Thus, proving the convergence to KKT points is perhaps the best we can hope for in our setting, and it is an interesting future work to prove stronger convergence results with further natural assumptions. + +Moreover, we can prove the following corollary, which characterizes the optimality of the normalized margin using SVM with Neural Tangent Kernel (NTK, introduced in (Jacot et al., 2018)) defined at limit points. The proof is deferred to Appendix C.6. + +Corollary 4.5 (Corollary of Theorem 4.4). Assume (S1). Then for gradient flow under assumptions $( A 2 ) \textrm { - } ( A 4 )$ or gradient descent under assumptions (A2) - (A4), (S5), any limit point $\bar { \pmb { \theta } }$ of $\{ \pmb \theta ( t ) / \| \pmb \theta ( t ) \| _ { 2 } : t \geq 0 \}$ is along the max-margin direction for the hard-margin SVM with kernel + +$K _ { \bar { \theta } } ( x , x ^ { \prime } ) = \left. \nabla \Phi _ { x } ( \bar { \theta } ) , \nabla \Phi _ { x ^ { \prime } } ( \bar { \theta } ) \right.$ , where $\Phi _ { x } ( \pmb { \theta } ) : = \Phi ( \pmb { \theta } ; \pmb { x } )$ . That is, for some $\alpha > 0$ , $\alpha \bar { \pmb { \theta } }$ is the optimal solution for the following constrained optimization problem: + +$$ +\operatorname* { m i n } \quad \frac { 1 } { 2 } \| \pmb { \theta } \| _ { 2 } ^ { 2 } \qquad \quad s . t . \quad y _ { n } \left. \pmb { \theta } , \nabla \Phi _ { \pmb { x } _ { n } } ( \bar { \pmb { \theta } } ) \right. \geq 1 \qquad \forall n \in [ N ] +$$ + +If we assume $( A I )$ instead of $( S I )$ for gradient flow, then there exists a mapping $h ( x ) \in \partial ^ { \circ } \Phi _ { x } ( \bar { \theta } )$ such that the same conclusion holds for $K _ { \bar { \theta } } ( x , x ^ { \prime } ) = \langle h ( x ) , h ( x ^ { \prime } ) \rangle$ . + +# 4.4 OTHER MAIN RESULTS + +The above results can be extended to other settings as shown below. + +Other Binary Classification Loss. The results on exponential loss can be generalized to a much broader class of binary classification loss. The class includes the logistic loss which is one of the most popular loss functions, $\ell ( q ) = \log ( 1 + e ^ { - q } )$ . The function class also includes other losses with exponential tail, e.g., $\ell ( q ) = e ^ { - q ^ { 3 } } , \ell ( q ) = \log ( 1 + e ^ { - q ^ { 3 } } )$ . For all those loss functions, we can use its inverse function $\ell ^ { - 1 }$ to define the smoothed normalized margin as follows + +$$ +\tilde { \gamma } ( \pmb { \theta } ) = \frac { \ell ^ { - 1 } ( \mathcal { L } ( \pmb { \theta } ) ) } { \| \pmb { \theta } \| _ { 2 } ^ { L } } . +$$ + +Then all our results for gradient flow continue to hold (Appendix A). Using a similar modification, we can also extend it to gradient descent (Appendix F). + +Cross-entropy Loss. In multi-class classification, we can define $q _ { n }$ to be the difference between the classification score for the true label and the maximum score for the other labels, then the margin qmin := minn∈[N] qn and the normalized margin γ¯(θ) := qmin(θ)kθkL can be similarly defined as before. In Appendix G, we define the smoothed normalized margin for cross-entropy loss to be the same as that for logistic loss (See Remark A.4). Then we show that Theorem 4.1 and Theorem 4.4 still hold (but with a slightly different definition of (P)) for gradient flow, and we also extend the results to gradient descent. + +Multi-homogeneous Models. Some neural networks indeed possess a stronger property than homogeneity, which we call multi-homogeneity. For example, the output of a CNN (without bias terms) is 1-homogeneous with respect to the weights of each layer. In general, we say that a neural network $\Phi ( \pmb \theta ; \pmb x )$ with $\pmb \theta = ( \pmb w _ { 1 } , \dots , \pmb w _ { m } )$ is $\left( k _ { 1 } , \ldots , k _ { m } \right)$ -homogeneous if for any and any $c _ { 1 } , \ldots , c _ { m } > 0$ , we have $\begin{array} { r } { \Phi \big ( c _ { 1 } \pmb { w } _ { 1 } , \ldots , c _ { m } \pmb { w } _ { m } ; \pmb { x } \big ) = \prod _ { i = 1 } ^ { m } c _ { i } ^ { k _ { i } } \cdot \Phi \big ( \pmb { w } _ { 1 } , \ldots , \pmb { w } _ { m } ; \pmb { x } \big ) } \end{array}$ . In the previous example, an $L$ -layer CNN with layer weights $\pmb \theta = ( \pmb w _ { 1 } , \dots , \pmb w _ { L } )$ is $( 1 , \ldots , 1 )$ -homogeneous. + +easily see that that , so our previous analys $\left( k _ { 1 } , \ldots , k _ { m } \right)$ -homogeneity implies neous models still appli $L$ -homogeneity, where to multi-homogeneous $L \ =$ $\textstyle \sum _ { i = 1 } ^ { m } k _ { i }$ + +$$ +\bar { \gamma } ( \pmb { w } _ { 1 } , \ldots , \pmb { w } _ { m } ) : = q _ { \operatorname* { m i n } } ( \frac { \pmb { w } _ { 1 } } { \lVert \pmb { w } _ { 1 } \rVert _ { 2 } } , \allowbreak \ldots , \frac { \pmb { w } _ { m } } { \lVert \pmb { w } _ { m } \rVert _ { 2 } } ) = \frac { q _ { \operatorname* { m i n } } } { \prod _ { i = 1 } ^ { m } \lVert \pmb { w } _ { i } \rVert _ { 2 } ^ { k _ { i } } } . +$$ + +In this case, the smoothed approximation of $\bar { \gamma }$ for general binary classification loss (under some conditions) can be similarly defined for gradient flow: + +$$ +\widetilde { \gamma } ( \pmb { w } _ { 1 } , \dots , \pmb { w } _ { m } ) : = \frac { \ell ^ { - 1 } ( \pmb { \mathscr { L } } ) } { \prod _ { i = 1 } ^ { m } \| \pmb { w } _ { i } \| _ { 2 } ^ { k _ { i } } } , +$$ + +It can be shown that $\tilde { \gamma }$ is also non-decreasing during training when the loss is small enough (Appendix $_ \mathrm { H }$ ). In the case of cross-entropy loss, we can still define $\tilde { \gamma }$ by (5) while $\ell ( \cdot )$ is set to the logistic loss in the formula. + +# 5 PROOF SKETCH: GRADIENT FLOW ON HOMOGENEOUS MODEL WITH EXPONENTIAL LOSS + +In this section, we present a proof sketch in the case of gradient flow on homogeneous model with exponential loss to illustrate our proof ideas. Due to space limit, the proof for the main theorems on gradient flow and gradient descent in Section 4 are deferred to Appendix A and E respectively. + +For convenience, we introduce a few more notations for a $L$ -homogeneous neural network $\Phi ( \pmb \theta ; \pmb x )$ . Let $S ^ { d - 1 } = \{ \pmb \theta \in \mathbb { R } ^ { d } : \| \pmb \theta \| _ { 2 } = 1 \}$ be the set of $L ^ { 2 }$ -normalized parameters. Define $\rho : = \| \pmb \theta \| _ { 2 }$ and $\begin{array} { r } { \hat { \pmb \theta } : = \frac { \pmb \theta } { \Vert \pmb \theta \Vert _ { 2 } } \in \mathcal { S } ^ { d - 1 } } \end{array}$ to be the length and direction of $\pmb \theta$ . For both gradient descent and gradient flow, $\pmb \theta$ is a function of time $t$ . For convenience, we also view the functions of $\pmb \theta$ , including $\mathcal { L } ( \pmb { \theta } ) , q _ { n } ( \pmb { \theta } ) , q _ { \mathrm { m i n } } ( \pmb { \theta } )$ , as functions of $t$ . So we can write $\mathscr { L } ( t ) : = \mathscr { L } ( \pmb \theta ( t ) ) , q _ { n } ( t ) : =$ $q _ { n } ( \pmb \theta ( t ) ) , q _ { \mathrm { m i n } } ( t ) : = q _ { \mathrm { m i n } } ( \pmb \theta ( t ) )$ . + +Lemma 5.1 below is the key lemma in our proof. It decomposes the growth of the smoothed normalized margin into the ratio of two quantities related to the radial and tangential velocity components of $\pmb { \theta }$ respectively. We will give a proof sketch for this later in this section. We believe that this lemma is of independent interest. + +Lemma 5.1 (Corollary of Lemma B.1). For a.e. $t > t _ { 0 }$ , + +$$ +\frac { d } { d t } \log \rho > 0 \quad a n d \quad \frac { d } { d t } \log \tilde { \gamma } \geq L \left( \frac { d } { d t } \log \rho \right) ^ { - 1 } \left\| \frac { d \hat { \pmb { \theta } } } { d t } \right\| _ { 2 } ^ { 2 } . +$$ + +Using Lemma 5.1, the first two claims in Theorem 4.1 can be directly proved. For the third claim, we make use of the monotonicity of the margin to lower bound the gradient, and then show $\mathcal { L } \to 0$ and $\rho + \infty$ . Recall that $\tilde { \gamma }$ is an $O ( \rho ^ { - L } )$ -additive approximation for $\bar { \gamma }$ . So this proves the third claim. We defer the detailed proof to Appendix B. + +To show Theorem 4.4, we first change the time measure to $\log \rho$ , i.e., now we see $t$ as a function of $\log \rho$ . So the second inequality in Lemma 5.1 can be rewritten as $\begin{array} { r } { \frac { d \log \tilde { \gamma } } { d \log \rho } \geq L \| \frac { d \hat { \pmb \theta } } { d \log \rho } \| _ { 2 } ^ { 2 } } \end{array}$ . Integrating on both sides and noting that $\tilde { \gamma }$ is upper-bounded, we know that there must be many instant $\log \rho$ such that $\big \| \frac { d \hat { \pmb { \theta } } } { d \log \rho } \big \| _ { 2 }$ is small. By analyzing the landscape of training loss, we show that these points are “approximate” KKT points. Then we show that every convergent sub-sequence of $\{ { \hat { \pmb { \theta } } } ( t ) : t \geq 0 \}$ can be modified to be a sequence of “approximate” KKT points which converges to the same limit. Then we conclude the proof by applying a theorem from (Dutta et al., 2013) to show that the limit of this convergent sequence of “approximate” KKT points is a KKT point. We defer the detailed proof to Appendix C. + +Now we give a proof sketch for Lemma 5.1, in which we derive the formula of $\tilde { \gamma }$ step by step. In the proof, we obtain several clean close form formulas for several relevant quantities, by using the chain rule and Euler’s theorem for homogeneous functions extensively. + +Proof Sketch of Lemma 5.1. For ease of presentation, we ignore the regularity issues of taking derivatives in this proof sketch. We start from the equation $\begin{array} { r } { \frac { d \mathcal { L } } { d t } = - \left. \partial ^ { \circ } \mathcal { L } ( \pmb { \theta } ( t ) ) , \frac { d \pmb { \theta } } { d t } \right. = - \left\| \frac { d \pmb { \theta } } { d t } \right\| _ { 2 } ^ { 2 } } \end{array}$ which follows from the chain rule (see also Lemma I.3). Then we note that $\textstyle { \frac { d \theta } { d t } }$ can be decomposed into two parts: the radial component $\begin{array} { r } { \pmb { v } : = \hat { \pmb { \theta } } \hat { \pmb { \theta } } ^ { \top } \frac { d \pmb { \theta } } { d t } } \end{array}$ and the tangent component $\begin{array} { r } { \mathbf { \hat { \boldsymbol { u } } } : = ( \mathbf { \boldsymbol { I } } - \hat { \theta } \hat { \theta } ^ { \top } ) \frac { d \theta } { d t } } \end{array}$ . + +The radial component is easier to analyze. By the chain rule, $\begin{array} { r } { \| \pmb { v } \| _ { 2 } = \hat { \pmb { \theta } } ^ { \top } \frac { d \pmb { \theta } } { d t } = \frac { 1 } { \rho } \left. \pmb { \theta } , \frac { d \pmb { \theta } } { d t } \right. = \frac { 1 } { \rho } \cdot \frac { 1 } { 2 } \frac { d \rho ^ { 2 } } { d t } } \end{array}$ . Fo r 12 dρ2dt , we have an exact formula: + +$$ +{ \frac { 1 } { 2 } } { \frac { d \rho ^ { 2 } } { d t } } = \left. \theta , { \frac { d \theta } { d t } } \right. = \left. \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \partial ^ { \circ } q _ { n } , \theta \right. = L \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { n } . +$$ + +The last equality is due to $\langle \partial ^ { \circ } q _ { n } , \theta \rangle = L q _ { n }$ by homogeneity of $q _ { n }$ . This is sometimes called Euler’s theorem for homogeneous functions (see Theorem B.2). For differentiable $q _ { n }$ , it can be easily proved by taking the derivative over $c$ on both sides of $q _ { n } ( c \pmb { \theta } ) = c ^ { L } q _ { n } ( \pmb { \theta } )$ and letting $c = 1$ . + +With (6), we can lower bound $\textstyle { \frac { 1 } { 2 } } { \frac { d \rho ^ { 2 } } { d t } }$ by + +$$ +\frac { 1 } { 2 } \frac { d \rho ^ { 2 } } { d t } = L \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { n } \ge L \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { \operatorname* { m i n } } \ge L \cdot \mathcal { L } \log \frac { 1 } { \mathcal { L } } , +$$ + +where the last inequality uses the fact that $e ^ { - q _ { \mathrm { m i n } } } \leq { \mathcal { L } }$ . (7) also implies that $\textstyle { \frac { 1 } { 2 } } { \frac { d \rho ^ { 2 } } { d t } } > 0$ for $t > t _ { 0 }$ since $\mathcal { L } ( t _ { 0 } ) < 1$ and $\mathcal { L }$ is non-increasing. As $\begin{array} { r } { \frac { d } { d t } \log \rho = \frac { 1 } { 2 \rho ^ { 2 } } \frac { d \rho ^ { 2 } } { d t } } \end{array}$ , this also proves the first inequality of Lemma 5.1. + +Now, we have $\begin{array} { r } { \left\| \pmb { v } \right\| _ { 2 } ^ { 2 } = \frac { 1 } { \rho ^ { 2 } } \left( \frac { 1 } { 2 } \frac { d \rho ^ { 2 } } { d t } \right) ^ { 2 } = \frac { 1 } { 2 } \frac { d \rho ^ { 2 } } { d t } \cdot \frac { d } { d t } \log \rho } \end{array}$ on the one hand; on the other hand, by the chain rule we have $\begin{array} { r } { \frac { d \hat { \pmb { \theta } } } { d t } = \frac { 1 } { \rho ^ { 2 } } ( \rho \frac { d \pmb { \theta } } { d t } - \frac { d \rho } { d t } \pmb { \theta } ) = \frac { 1 } { \rho ^ { 2 } } ( \rho \frac { d \pmb { \theta } } { d t } - ( \hat { \pmb { \theta } } ^ { \top } \frac { d \pmb { \theta } } { d t } ) \pmb { \theta } ) = \frac { \pmb { u } } { \rho } } \end{array}$ . So we have + +$$ +- { \frac { d { \mathcal { L } } } { d t } } = \left\| { \frac { d \theta } { d t } } \right\| _ { 2 } ^ { 2 } = \left\| v \right\| _ { 2 } ^ { 2 } + \left\| u \right\| _ { 2 } ^ { 2 } = { \frac { 1 } { 2 } } { \frac { d \rho ^ { 2 } } { d t } } \cdot { \frac { d } { d t } } \log \rho + \rho ^ { 2 } \left\| { \frac { d \hat { \theta } } { d t } } \right\| _ { 2 } ^ { 2 } +$$ + +Dividing 12 dρ2dt on the leftmost and rightmost sides, we have + +$$ +- { \frac { d { \mathcal { L } } } { d t } } \cdot \left( { \frac { 1 } { 2 } } { \frac { d \rho ^ { 2 } } { d t } } \right) ^ { - 1 } = { \frac { d } { d t } } \log \rho + \left( { \frac { d } { d t } } \log \rho \right) ^ { - 1 } \left\| { \frac { d { \hat { \pmb { \theta } } } } { d t } } \right\| _ { 2 } ^ { 2 } . +$$ + +By $\begin{array} { r } { - \frac { d \mathcal { L } } { d t } \ge 0 } \end{array}$ and (7), the LHS is no greater than $\begin{array} { r } { - \frac { d \mathcal { L } } { d t } \cdot \left( L \cdot \mathcal { L } \log \frac { 1 } { \mathcal { L } } \right) ^ { - 1 } = \frac { 1 } { L } \log \log \frac { 1 } { \mathcal { L } } } \end{array}$ . Thus we dt 2 have $\begin{array} { r } { \frac { d } { d t } \log \log \frac { 1 } { \mathcal { L } } - L \frac { d } { d t } \log \rho \geq L \left( \frac { d } { d t } \log \rho \right) ^ { - 1 } \left. \frac { d \hat { \theta } } { d t } \right. _ { 2 } ^ { 2 } } \end{array}$ , where the LHS is exactly $\begin{array} { r } { \frac { d } { d t } \log \tilde { \gamma } } \end{array}$ . + +# 6 DISCUSSION AND FUTURE DIRECTIONS + +In this paper, we analyze the dynamics of gradient flow/descent of homogeneous neural networks under a minimal set of assumptions. The main technical contribution of our work is to prove rigorously that for gradient flow/descent, the normalized margin is increasing and converges to a KKT point of a natural max-margin problem. Our results leads to some natural further questions: + +• Can we generalize our results for gradient descent on smooth neural networks to nonsmooth ones? In the smooth case, we can lower bound the decrement of training loss by the gradient norm squared, multiplied by a factor related to learning rate. However, in the non-smooth case, no such inequality is known in the optimization literature. +• Can we make more structural assumptions on the neural network to prove stronger results? In this work, we use a minimal set of assumptions to show that the convergent direction of parameters is a KKT point. A potential research direction is to identify more key properties of modern neural networks and show that the normalized margin at convergence is locally or globally optimal (in terms of optimizing (P)). Can we extend our results to neural networks with bias terms? In our experiments, the normalized margin of the CNN with bias also increases during training despite that its output is non-homogeneous. It is very interesting (and technically challenging) to provide a rigorous proof for this fact. + +# ACKNOWLEDGMENTS + +The research is supported in part by the National Natural Science Foundation of China Grant 61822203, 61772297, 61632016, 61761146003and the Zhongguancun Haihua Institute for Frontier Information Technology and Turing AI Institute of Nanjing. We thank Liwei Wang for helpful suggestions on the connection between margin and robustness. We thank Sanjeev Arora, Tianle Cai, Simon Du, Jason D. Lee, Zhiyuan Li, Tengyu Ma, Ruosong Wang for helpful discussions. + +# REFERENCES + +Pierre-Antoine Absil, Robert Mahony, and Benjamin Andrews. Convergence of the iterates of descent methods for analytic cost functions. SIAM Journal on Optimization, 16(2):531–547, 2005. + +Alnur Ali, J. Zico Kolter, and Ryan J. Tibshirani. A continuous-time view of early stopping for least squares regression. 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Stochastic gradient descent optimizes over-parameterized deep relu networks. arXiv preprint arXiv:1811.08888, 2018. + +Guus Zoutendijk. Mathematical programming methods. 1976. + +# A RESULTS FOR GENERAL LOSS + +In this section, we state our results for a broad class of binary classification loss. A major consequence of this generalization is that the logistic loss, one of the most popular loss functions, $\ell ( \bar { q } ) = \log ( 1 + e ^ { - \bar { q } } )$ is included. The function class also includes other losses with exponential tail, e.g., $\ell ( q ) = e ^ { - q ^ { 3 } } , \ell ( q ) = \log ( 1 + e ^ { - q ^ { 3 } } )$ . + +# A.1 ASSUMPTIONS + +We first focus on gradient flow. We assume (A1), (A2) as we do for exponential loss. For (A3), (A4), we replace them with two weaker assumptions (B3), (B4). All the assumptions are listed below: + +(A1). (Regularity). For any fixed $_ { \textbf { \em x } }$ , $\Phi ( \cdot ; x )$ is locally Lipschitz and admits a chain rule; +(A2). (Homogeneity). There exists $L > 0$ such that $\forall \alpha > 0 : \Phi ( \alpha \pmb { \theta } ; \pmb { x } ) = \alpha ^ { L } \Phi ( \pmb { \theta } ; \pmb { x } )$ ; +(B3). The loss function $\ell ( q )$ can be expressed as $\ell ( q ) = e ^ { - f ( q ) }$ such that (B3.1). $f : \mathbb { R } \to \mathbb { R }$ is ${ \mathcal { C } } ^ { 1 }$ -smooth. (B3.2). $f ^ { \prime } ( q ) > 0$ for all $q \in \mathbb { R }$ . (B3.3). There exists $b _ { f } \ \geq \ 0$ such that $f ^ { \prime } ( q ) q$ is non-decreasing for $q \in ( b _ { f } , + \infty )$ , and $f ^ { \prime } ( q ) q \to + \infty$ as $q \to + \infty$ . (B3.4). Let $g \ : \ [ f ( b _ { f } ) , + \infty ) \ \ [ b _ { f } , + \infty )$ be the inverse function of $f$ on the domain $[ b _ { f } , + \infty )$ . There exists $b _ { g } ~ \overset { \cdot } { = } ~ \operatorname* { m a x } \{ 2 f ( b _ { f } ) , f ( 2 b _ { f } ) \} , K \geq 1$ such that $g ^ { \prime } ( x ) \leq$ $K g ^ { \prime } ( \theta x )$ and $f ^ { \prime } ( y ) ~ \leq ~ \breve { K } f ^ { \prime } ( \theta y )$ for all $\dot { x } \in ( \bar { b _ { g } } , + \infty ) , y \in ( g ( b _ { g } ) , + \infty )$ and $\theta \in [ 1 / 2 , 1 )$ . + +(B4). (Separability). There exists a time $t _ { 0 }$ such that $\mathcal { L } ( t _ { 0 } ) < e ^ { - f ( b _ { f } ) } = \ell ( b _ { f } )$ . + +(A1) and (A2) remain unchanged. (B3) is satisfied by exponential loss $\ell ( q ) = e ^ { - q }$ (with $f ( q ) = q )$ and logistic loss $\ell ( q ) = \log ( \bar { 1 } + e ^ { - q } )$ (with $f ( q ) \stackrel { \cdot } { = } - \log \log ( 1 + e ^ { - q } ) \stackrel { \cdot } { ) }$ . (B4) are essentially the same as (A4) but (B4) uses a threshold value that depends on the loss function. Assuming (B3), it is easy to see that (B4) ensures the separability of data since $\ell ( q _ { n } ) < e ^ { - f ( b _ { f } ) }$ implies $q _ { n } > b _ { f } \geq 0$ . For logistic loss, we can set $b _ { f } = 0$ (see Remark A.2). So the corresponding threshold value in (B4) is $\ell ( 0 ) = \log 2$ . + +Now we discuss each of the assumptions in (B3). (B3.1) is a natural assumption on smoothness. (B3.2) requires $\ell ( \cdot )$ to be monotone decreasing, which is also natural since $\ell ( \cdot )$ is used for binary classification. The rest of two assumptions in (B3) characterize the properties of $\ell ^ { \prime } ( q )$ when $q$ is large enough. (B3.3) is an assumption that appears naturally from the proof. For exponential loss, $f ^ { \prime } ( q ) q = q$ is always non-decreasing, so we can set $b _ { f } = 0$ . In (B3.4), the inverse function $g$ is defined. It is guaranteed by (B3.1) and (B3.2) that $g$ always exists and $g$ is also ${ \mathcal { C } } ^ { 1 }$ -smooth. Though (B3.4) looks very complicated, it essentially says that $f ^ { \prime } ( \Theta ( q ) ) = \Theta ( \bar { f } ^ { \prime } ( q ) ) , g ^ { \prime } ( \Theta ( q ) ) = \Theta ( g ^ { \prime } ( \bar { q } ) )$ as $q \infty$ . (B3.4) is indeed a technical assumption that enables us to asymptotically compare the loss or the length of gradient at different data points. It is possible to base our results on weaker assumptions than (B3.4), but we use (B3.4) for simplicity since it has already been satisfied by many loss functions such as the aforementioned examples. + +We summarize the corresponding $f , g$ and $b _ { f }$ for exponential loss and logistic loss below: + +Remark A.1. Exponential loss $\ell ( q ) = e ^ { - q }$ satisfies (B3) with + +$$ +f ( q ) = q \qquad f ^ { \prime } ( q ) = 1 \qquad g ( q ) = q \qquad g ^ { \prime } ( q ) = 1 \qquad b _ { f } = 0 \qquad \ell ( b _ { f } ) = 1 . +$$ + +Remark A.2. Logistic loss $\ell ( q ) = \log ( 1 + e ^ { - q } )$ satisfies $( B 3 )$ with + +$$ +\begin{array} { l l } { { f ( q ) = - \log \log ( 1 + e ^ { - q } ) = \Theta ( q ) ~ } } & { { ~ f ^ { \prime } ( q ) = \displaystyle \frac { e ^ { - q } } { ( 1 + e ^ { - q } ) \log ( 1 + e ^ { - q } ) } = \Theta ( 1 ) } } \\ { { { } } } & { { { } } } \\ { { g ( q ) = - \log ( e ^ { e ^ { - q } } - 1 ) = \Theta ( q ) ~ } } & { { ~ g ^ { \prime } ( q ) = \displaystyle \frac { e ^ { - q } } { e ^ { e ^ { - q } } - 1 } = \Theta ( 1 ) } } \\ { { { } } } & { { { } } } \\ { { b _ { f } = 0 ~ } } & { { ~ \ell ( b _ { f } ) = \log 2 . } } \end{array} +$$ + +The proof for Remark A.1 is trivial. For Remark A.2, we give a proof below. + +Proof for Remark A.2. By simple calculations, the formulas for $f ( q ) , f ^ { \prime } ( q ) , g ( q ) , g ^ { \prime } ( q )$ are correct. (B3.1) is trivial. $f ^ { \prime } ( q ) ~ = ~ \frac { e ^ { - q } } { ( 1 + e ^ { - q } ) \log ( 1 + e ^ { - q } ) } ~ > ~ 0$ , so (B3.2) is satisfied. For (B3.3), note that $\begin{array} { r } { f ^ { \prime } ( q ) q = \frac { q } { ( 1 + e ^ { q } ) \log ( 1 + e ^ { - q } ) } } \end{array}$ . The denominator is a decreasing function since + +$$ +{ \frac { d } { d q } } \left( ( 1 + e ^ { q } ) \log ( 1 + e ^ { - q } ) \right) = e ^ { q } \log ( 1 + e ^ { - q } ) - 1 < e ^ { q } \cdot e ^ { - q } - 1 = 0 . +$$ + +Thus, $f ^ { \prime } ( q ) q$ is a strictly increasing function on $\mathbb { R }$ . As $b _ { f }$ is required to be non-negative, we set $b _ { f } = 0$ . For proving that $f ^ { \prime } ( q ) q \to + \infty$ and (B4), we only need to notice that $\begin{array} { r } { f ^ { \prime } ( q ) \sim \frac { e ^ { - q } } { 1 \cdot e ^ { - q } } = 1 } \end{array}$ and $g ^ { \prime } ( x ) = \bar { 1 } / f ^ { \prime } ( \bar { g ( x ) } ) \sim 1$ . □ + +# A.2 SMOOTHED NORMALIZED MARGIN + +For a loss function $\ell ( \cdot )$ satisfying (B3), it is easy to see from (B3.2) that its inverse function $\ell ^ { - 1 } ( \cdot )$ must exist. For this kind of loss functions, we define the smoothed normalized margin as follows: + +Definition A.3. For a loss function $\ell ( \cdot )$ satisfying (B3), the smoothed normalized margin $\widetilde \gamma ( \pmb \theta )$ of $\pmb \theta$ is defined as + +$$ +\widetilde { \gamma } ( \pmb { \theta } ) : = \frac { \ell ^ { - 1 } ( \mathcal { L } ) } { \rho ^ { L } } = \frac { g ( \log \frac { 1 } { \mathcal { L } } ) } { \rho ^ { L } } = \frac { g \left( - \log \left( \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ( \pmb { \theta } ) ) } \right) \right) } { \rho ^ { L } } , +$$ + +where $\ell ^ { - 1 } ( \cdot )$ is the inverse function of $\ell ( \cdot )$ and $\rho : = \| \pmb \theta \| _ { 2 }$ + +Remark A.4. For logistic loss $\ell ( q ) = \log ( 1 + e ^ { - q } )$ , $\begin{array} { r } { \widetilde \gamma ( \pmb \theta ) = \rho ^ { - L } \log \frac { 1 } { \exp ( \mathcal L ) - 1 } } \end{array}$ ; for exponential loss $\ell ( q ) = e ^ { - q }$ , $\begin{array} { r } { \tilde { \gamma } ( \pmb { \theta } ) = \rho ^ { - L } \log \frac { 1 } { \mathcal { L } } } \end{array}$ , which is the same as (3). + +Now we give some insights on how well $\tilde { \gamma } ( \pmb \theta )$ approximates $\bar { \gamma } ( \pmb \theta )$ using a similar argument as in Section 4.2. Using the LogSumExp function, the smoothed normalized margin $\widetilde \gamma ( \pmb \theta )$ can also be written as + +$$ +\tilde { \gamma } ( \pmb \theta ) = \frac { g ( - \mathrm { L S E } ( - f ( q _ { 1 } ) , \dots , - f ( q _ { N } ) ) ) } { \rho ^ { L } } . +$$ + +LSE is a $( \log N )$ -additive approximation for max. So we can roughly approximate $\widetilde \gamma ( \pmb \theta )$ by + +$$ +\tilde { \gamma } ( \pmb \theta ) \approx \frac { g ( - \operatorname* { m a x } \{ - f ( q _ { 1 } ) , \dots , - f ( q _ { N } ) \} ) } { \rho ^ { L } } = \frac { g ( f ( q _ { \operatorname* { m i n } } ) ) } { \rho ^ { L } } = \bar { \gamma } ( \pmb \theta ) . +$$ + +Note that (B3.3) is crucial to make the above approximation reasonable. Similar to exponential loss, we can show the following lemma asserting that $\tilde { \gamma }$ is a good approximation of $\bar { \gamma }$ . + +Lemma A.5. Assuming $( B 3 ) ^ { 4 }$ , we have the following properties about the margin: + +(a) $\begin{array} { r } { f ( q _ { \mathrm { m i n } } ) - \log N \leq \log \frac { 1 } { \mathcal { L } } \leq f ( q _ { \mathrm { m i n } } ) . } \end{array}$ + +(b) $\begin{array} { r } { I f \log { \frac { 1 } { \mathcal { L } } } > f ( b _ { f } ) } \end{array}$ , then there exists $\xi \in \left( f ( q _ { \operatorname* { m i n } } ) - \log N , f ( q _ { \operatorname* { m i n } } ) \right) \cap \left( b _ { f } , + \infty \right)$ such that + +$$ +\bar { \gamma } - \frac { g ^ { \prime } ( \xi ) \log N } { \rho ^ { L } } \leq \tilde { \gamma } \leq \bar { \gamma } . +$$ + +(c) For a seuqnce of parameters $\{ \pmb \theta _ { m } \in \mathbb { R } ^ { d } : m \in \mathbb { N } \}$ , if $\mathcal { L } ( \pmb { \theta } _ { m } ) 0$ , then $\lvert \tilde { \gamma } ( \pmb { \theta } _ { m } ) - \bar { \gamma } ( \pmb { \theta } _ { m } ) \rvert 0$ . + +Proof. (a) can be easily deduced from $e ^ { - f ( q _ { \mathrm { m i n } } ) } \leq \mathcal { L } \leq N e ^ { - f ( q _ { \mathrm { m i n } } ) }$ . Combining $( a )$ and the monotonicity of $g ( \cdot )$ , we further have $g ( s ) \leq g ( \log \frac { 1 } { \mathcal { L } } ) \leq q _ { \operatorname* { m i n } }$ for $s : = \operatorname* { m a x } \{ f ( b _ { f } ) , f ( q _ { \operatorname* { m i n } } ) \ : -$ $\log N \}$ . By the mean value Theorem, there exists $\xi \in \widetilde { ( s , f ( q _ { \operatorname* { m i n } } ) ) } )$ such that $g ( s ) = { \dot { g } } ( f ( q _ { \mathrm { m i n } } ) ) -$ $g ^ { \prime } ( \xi ) ( f ( q _ { \mathrm { m i n } } ) - s ) \geq q _ { \mathrm { m i n } } - g ^ { \prime } ( \xi ) \log N$ . Dividing $\rho ^ { L }$ on each side of $q _ { \operatorname* { m i n } } - g ^ { \prime } ( \xi ) \log N \le$ $g ( \log { \frac { 1 } { \mathcal { L } } } ) \leq q _ { \operatorname* { m i n } }$ proves $( b )$ . + +Now we prove $( c )$ . Without loss of generality, we assume log $\frac { 1 } { \mathcal { L } ( \pmb { \theta } _ { m } ) } > f ( b _ { f } )$ for all $\pmb { \theta } _ { m }$ . It follows from $( b )$ that for every $\pmb { \theta } _ { m }$ there exists $\xi _ { m } \in ( f ( q _ { \operatorname* { m i n } } ( \theta _ { m } ) ) - \log N , f ( q _ { \operatorname* { m i n } } ( \theta _ { m } ) ) ) \cap ( b _ { f } , + \infty )$ such that + +$$ +\bar { \gamma } ( \pmb \theta _ { m } ) - ( g ^ { \prime } ( \xi _ { m } ) \log N ) / \rho ( \pmb \theta _ { m } ) ^ { L } \leq \widetilde \gamma ( \pmb \theta _ { m } ) \leq \bar { \gamma } ( \pmb \theta _ { m } ) . +$$ + +Note that $\begin{array} { r c l } { \xi _ { m } } & { \ge } & { f \big ( q _ { \mathrm { m i n } } ( \pmb \theta _ { m } ) \big ) - \log N \ge \log \frac { 1 } { \mathcal { L } ( \pmb \theta _ { m } ) } - \log N + \infty } \end{array}$ . So $\begin{array} { r l } { \frac { g ( \xi _ { m } ) } { g ^ { \prime } ( \xi _ { m } ) } } & { { } = } \end{array}$ $f ^ { \prime } ( g ( \xi _ { m } ) ) g ( \xi _ { m } ) \to + \infty$ by (B3.3). Also note that there exists a constant $B _ { 0 }$ such that $\bar { \gamma } ( \pmb { \theta } _ { m } ) \leq B _ { 0 }$ for all $m$ since $\bar { \gamma }$ is continuous on the unit sphere $S ^ { d - 1 }$ . So we have + +$$ +\frac { g ^ { \prime } ( \xi _ { m } ) } { \rho ( \theta _ { m } ) ^ { L } } = \frac { g ^ { \prime } ( \xi _ { m } ) } { g ( \xi _ { m } ) } \frac { g ( \xi _ { m } ) } { \rho ( \theta _ { m } ) ^ { L } } \le \frac { g ^ { \prime } ( \xi _ { m } ) } { g ( \xi _ { m } ) } \frac { q _ { \operatorname* { m i n } } ( \theta _ { m } ) } { \rho ( \theta _ { m } ) ^ { L } } = \frac { g ^ { \prime } ( \xi _ { m } ) } { g ( \xi _ { m } ) } \cdot \bar { \gamma } ( \theta _ { m } ) \le \frac { g ^ { \prime } ( \xi _ { m } ) } { g ( \xi _ { m } ) } \cdot B \to 0 , +$$ + +where the first inequality follows since $\xi _ { m } \leq f ( q _ { \operatorname* { m i n } } ( \pmb \theta _ { m } ) )$ . Together with (8), we have $\tilde { \gamma } ( \pmb { \theta } _ { m } ) -$ $\bar { \gamma } ( \pmb \theta _ { m } ) | ~ 0$ . + +Remark A.6. For exponential loss, we have already shown in Section 4.2 that $\tilde { \gamma } ( \pmb \theta )$ is an $O ( \rho ^ { - L } )$ - additive approximation for $\bar { \gamma } ( \pmb \theta )$ . For logistic loss, it follows easily from $g ^ { \prime } ( q ) = \Theta ( 1 )$ and $( b )$ of Lemma A.5 that $\tilde { \gamma } ( \pmb \theta )$ is an $O ( \rho ^ { - L } )$ -additive approximation for $\bar { \gamma } ( \pmb \theta )$ if $\mathcal { L }$ is sufficiently small. + +# A.3 THEOREMS + +Now we state our main theorems. For the monotonicity of the normalized margin, we have the following theorem. The proof is provided in Appendix B. + +Theorem A.7. Under assumptions (A1), (A2), $( B 3 ) ^ { 4 }$ , $( B 4 )$ , the following statements are true for gradient flow: + +1. For a.e. $t > t _ { 0 }$ , $\begin{array} { r } { \frac { d } { d t } \tilde { \gamma } ( \pmb { \theta } ( t ) ) \geq 0 } \end{array}$ ; +2. For a.e. t > t0, either ddt γ˜(θ(t)) > 0 or ddt θ(t)kθ(t)k2 ; +3. $\mathcal { L } ( \pmb \theta ( t ) ) 0$ and $\| \pmb \theta ( t ) \| _ { 2 } \to \infty$ as $t \to + \infty$ ; therefore, $| \bar { \gamma } ( \pmb \theta ( t ) ) - \tilde { \gamma } ( \pmb \theta ( t ) ) | 0 .$ . + +For the normalized margin at convergence, we have two theorems, one for infinite-time limiting case, and the other being a finite-time quantitative result. Their proofs can be found in Appendix $\textrm { C }$ As in the exponential loss case, we define the constrained optimization problem (P) as follows: + +$$ +\begin{array} { r l } { \operatorname* { m i n } } & { \frac { 1 } { 2 } \left\| \pmb { \theta } \right\| _ { 2 } ^ { 2 } } \\ { \mathrm { s . t . } } & { q _ { n } ( \pmb { \theta } ) \geq 1 \qquad \forall n \in [ N ] } \end{array} +$$ + +First, we show the directional convergence of $\pmb \theta ( t )$ to a KKT point of (P). + +Theorem A.8. Consider gradient flow under assumptions (A1), (A2), $( B 3 )$ , $( B 4 )$ . For every limit point $\bar { \pmb { \theta } }$ of $\left\{ { \hat { \pmb { \theta } } } ( t ) : t \geq 0 \right\}$ , $\bar { \theta } / q _ { \mathrm { m i n } } ( \bar { \theta } ) ^ { 1 / L }$ is a KKT point of $( P )$ . + +Second, we show that after finite time, gradient flow can pass through an approximate KKT point. + +Theorem A.9. Consider gradient flow under assumptions (A1), (A2), $( B 3 )$ , $( B 4 )$ . For any $\epsilon , \delta > 0$ , there exists $r : = \Theta ( \log \delta ^ { \bar { - } 1 } )$ and $\bar { \Delta } : = \Theta ( \epsilon ^ { - 2 } )$ such that $\theta / \dot { q } _ { \mathrm { m i n } } ( \theta ) ^ { 1 / L }$ is an $( \epsilon , \delta )$ -KKT point at some time $t _ { * }$ satisfying $\log \| \pmb { \theta } ( t _ { * } ) \| _ { 2 } \in ( r , r + \Delta )$ . + +For the definitions for KKT points and approximate KKT points, we refer the readers to Appendix C.1 for more details. + +With a refined analysis, we can also provide tight rates for loss convergence and weight growth. The proof is given in Appendix D. + +Theorem A.10. Under assumptions (A1), (A2), $( B 3 )$ , $( B 4 )$ , we have the following tight rates for loss convergence and weight growth: + +$$ +\mathcal { L } ( \pmb \theta ( t ) ) = \Theta \left( \frac { g ( \log t ) ^ { 2 / L } } { t ( \log t ) ^ { 2 } } \right) \quad a n d \quad \| \pmb \theta ( t ) \| _ { 2 } = \Theta ( g ( \log t ) ^ { 1 / L } ) . +$$ + +Applying Theorem A.10 to exponential loss and logistic loss, in which $g ( x ) = \Theta ( x )$ , we have the following corollary: + +Corollary A.11. If $\ell ( \cdot )$ is the exponential or logistic loss, then, + +$$ +\mathcal { L } ( \pmb \theta ( t ) ) = \Theta \left( \frac { 1 } { t ( \log t ) ^ { 2 - 2 / L } } \right) \quad a n d \quad \| \pmb \theta ( t ) \| _ { 2 } = \Theta ( ( \log t ) ^ { 1 / L } ) . +$$ + +# B MARGIN MONOTONICITY FOR GENERAL LOSS + +In this section, we consider gradient flow and prove Theorem A.7. We assume (A1), (A2), (B3), (B4) as mentioned in Appendix A. + +We follow the notations in Section 5 to define $\rho : = \| \pmb \theta \| _ { 2 }$ and $\begin{array} { r } { \hat { \pmb \theta } : = \frac { \pmb \theta } { \| \pmb \theta \| _ { 2 } } \in \mathcal { S } ^ { d - 1 } } \end{array}$ , and sometimes we view the functions of $\pmb { \theta }$ as functions of $t$ . + +# B.1 PROOF FOR PROPOSITION 1 AND 2 + +To prove the first two propositions, we generalize our key lemma (Lemma 5.1) to general loss. + +Lemma B.1. For $\tilde { \gamma }$ defined in Definition A.3, the following holds for all $t > t _ { 0 }$ , + +$$ +\frac { d } { d t } \log \rho > 0 \quad a n d \quad \frac { d } { d t } \log \tilde { \gamma } \geq L \left( \frac { d } { d t } \log \rho \right) ^ { - 1 } \left\| \frac { d \hat { \pmb { \theta } } } { d t } \right\| _ { 2 } ^ { 2 } . +$$ + +Before proving Lemma B.1, we review two important properties of homogeneous functions. Note that these two properties are usually shown for smooth functions. By considering Clarke’s subdifferential, we can generalize it to locally Lipschitz functions that admit chain rules: + +Theorem B.2. Let $F : \mathbb { R } ^ { d } \mathbb { R }$ be a locally Lipschitz function that admits a chain rule. If $F$ is $k$ -homogeneous, then + +(a) For all $\pmb { x } \in \mathbb { R } ^ { d }$ and $\alpha > 0$ , $\partial ^ { \circ } F ( \alpha x ) = \alpha ^ { k - 1 } \partial ^ { \circ } F ( x )$ That is, $\partial ^ { \circ } F ( \alpha x ) = \{ \alpha ^ { k - 1 } h : h \in \partial ^ { \circ } F ( x ) \} .$ . + +(b) (Euler’s Theorem for Homogeneous Functions). For all $\pmb { x } \in \mathbb { R } ^ { d }$ , + +$\langle { \pmb x } , \partial ^ { \circ } F ( { \pmb x } ) \rangle = k \cdot F ( { \pmb x } )$ That is, $\langle { \pmb x } , { \pmb h } \rangle = { \boldsymbol k } \cdot { \boldsymbol F } ( { \pmb x } )$ for all $\pmb { h } \in \partial ^ { \circ } F ( \pmb { x } )$ . + +Proof. Let $D$ be the set of points $_ { \textbf { \em x } }$ such that $F$ is differentiable at $_ { \textbf { \em x } }$ . According to the definition of Clarke’s subdifferential, for proving $( a )$ , it is sufficient to show that + +$$ +\{ \operatorname* { l i m } _ { k \to \infty } \nabla F ( \alpha { \pmb x } _ { k } ) : { \pmb x } _ { k } \to { \pmb x } , \alpha { \pmb x } _ { k } \in D \} = \{ \alpha ^ { k - 1 } \operatorname* { l i m } _ { k \to \infty } \nabla F ( { \pmb x } _ { k } ) : { \pmb x } _ { k } \to { \pmb x } , { \pmb x } _ { k } \in D \} +$$ + +Fix $\pmb { x } _ { k } \in D$ . Let $U$ be a neighborhood of $\scriptstyle { \mathbf { { \mathit { x } } } } _ { k }$ . By definition of homogeneity, for any $\pmb { h } \in \mathbb { R } ^ { d }$ and any $\pmb { y } \in U \setminus \{ \pmb { x } _ { k } \}$ , + +$$ +\frac { F ( \alpha y ) - F ( \alpha x _ { k } ) - \left. \alpha y - \alpha x _ { k } , \alpha ^ { k - 1 } h \right. } { \| \alpha y - \alpha x _ { k } \| _ { 2 } } = \alpha ^ { k - 1 } \cdot \frac { F ( y ) - F ( { \pmb x } _ { k } ) - \left. y - { \pmb x } _ { k } , h \right. } { \| { \pmb y } - { \pmb x } _ { k } \| _ { 2 } } . +$$ + +Taking limits ${ \pmb y } { \pmb x } _ { k }$ on both sides, we know that the LHS converges to 0 iff the RHS converges to 0. Then by definition of differetiability and gradient, $F$ is differentiable at $\alpha { \pmb x } _ { k }$ iff it is differentiable at $\scriptstyle { \mathbf { { \mathit { x } } } } _ { k }$ , and $\nabla F ( \alpha \mathbf { x } _ { k } ) = \alpha ^ { k - 1 } h$ iff $\nabla \tilde { F } ( \mathbf { \boldsymbol { x } } _ { k } ) = h$ . This proves (10). + +To prove $( b )$ , we fix $\textbf { \em x } \in \mathbb { R } ^ { d }$ . Let $z \ : \ [ 0 , + \infty ) \ \to \ \mathbb { R } ^ { d } , \alpha \ \mapsto \ \alpha x$ be an arc. By definition of homogeneity, $( F \circ z ) ( \alpha ) = \alpha ^ { k } F ( \pmb { x } )$ for $\alpha > 0$ . Taking derivative with respect to $\alpha$ on both sides (for differentiable points), we have + +$$ +\forall \pmb { h } \in \partial ^ { \circ } F ( \alpha \pmb { x } ) : \langle \pmb { x } , \pmb { h } \rangle = k \alpha ^ { k - 1 } F ( \pmb { x } ) +$$ + +holds for a.e. $\alpha > 0$ . Pick an arbitrary $\alpha > 0$ making (11) hold. Then by $( a )$ , (11) is equivalent to $\forall { \pmb { h } } \in \partial ^ { \circ } F ( { \pmb x } ) : \langle { \pmb x } , \alpha ^ { k - 1 } { \pmb h } \rangle = k \alpha ^ { k - 1 } F ( { \pmb x } )$ , which proves $( b )$ . □ + +Applying Theorem B.2 to homogeneous neural networks, we have the following corollary: + +Corollary B.3. Under the assumptions (A1) and (A2), for any $\pmb \theta \in \mathbb { R } ^ { d }$ and $\pmb { x } \in \mathbb { R } ^ { d _ { \mathbf { x } } }$ , + +$$ +\langle \pmb { \theta } , \partial ^ { \circ } \Phi _ { \pmb { x } } ( \pmb { \theta } ) \rangle = L \cdot \Phi _ { \pmb { x } } ( \pmb { \theta } ) , +$$ + +where $\Phi _ { x } ( \pmb { \theta } ) = \Phi ( \pmb { \theta } ; \pmb { x } )$ is the network output for a fixed input $_ { \textbf { \em x } }$ . + +Corollary B.3 can be used to derive an exact formula for the weight growth during training. + +Theorem B.4. For a.e. $t \geq 0$ , + +$$ +\frac { 1 } { 2 } \frac { d \rho ^ { 2 } } { d t } = L \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) q _ { n } . +$$ + +Proof. The proof idea is to use Corollary B.3 and chain rules (See Appendix I for chain rules in Clarke’s sense). Applying the chain rule on $t \mapsto \rho ^ { 2 } = \| \pmb \theta \| _ { 2 } ^ { 2 }$ yields $\begin{array} { r } { \frac { 1 } { 2 } \frac { d \rho ^ { 2 } } { d t } \ = \ - \langle \theta , h \rangle } \end{array}$ for all $h \in \partial ^ { \circ } { \mathcal { L } }$ and a.e. $t > 0$ . Then applying the chain rule on $\pmb \theta \mapsto \mathcal L$ , we have + +$$ +- \partial ^ { \circ } { \mathcal { L } } \subseteq \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) \partial ^ { \circ } q _ { n } = \left\{ \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) { h _ { n } } : { h _ { n } } \in \partial ^ { \circ } q _ { n } \right\} . +$$ + +By Corollary B.3, $\langle \pmb \theta , \pmb h _ { n } \rangle = L q _ { n }$ , and thus $\begin{array} { r } { \frac { 1 } { 2 } \frac { d \rho ^ { 2 } } { d t } = L \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) q _ { n } } \end{array}$ . + +For convenience, we define $\begin{array} { r } { \nu ( t ) : = \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) q _ { n } } \end{array}$ for all $t \geq 0$ . Then Theorem B.4 can be rephrased as $\begin{array} { r } { \frac { 1 } { 2 } \frac { d \rho ^ { 2 } } { d t } = L \nu ( t ) } \end{array}$ for a.e. $t \geq 0$ . + +Lemma B.5. For all $t > t _ { 0 }$ + +$$ +\nu ( t ) \geq \frac { g ( \log \frac { 1 } { \mathcal { L } } ) } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } \mathcal { L } . +$$ + +Proof. By Lemma A.5, $q _ { n } \geq g ( \log { \frac { 1 } { \mathcal { L } } } )$ for all $n \in [ N ]$ . Then by Assumption (B3), + +$$ +f ^ { \prime } ( q _ { n } ) q _ { n } \geq f ^ { \prime } ( g ( \log { \frac { 1 } { \mathcal { L } } } ) ) \cdot g ( \log { \frac { 1 } { \mathcal { L } } } ) = { \frac { g ( \log { \frac { 1 } { \mathcal { L } } } ) } { g ^ { \prime } ( \log { \frac { 1 } { \mathcal { L } } } ) } } . +$$ + +Combining this with the definitions of $\nu ( t )$ and $\mathcal { L }$ gives + +$$ +\begin{array} { r } { \nu ( t ) = \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) q _ { n } \geq \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } \displaystyle \frac { g ( \log \frac { 1 } { \mathscr { L } } ) } { g ^ { \prime } ( \log \frac { 1 } { \mathscr { L } } ) } = \displaystyle \frac { g ( \log \frac { 1 } { \mathscr { L } } ) } { g ^ { \prime } ( \log \frac { 1 } { \mathscr { L } } ) } \mathscr { L } . } \end{array} +$$ + +Proof for Lemma B.1. Note that $\begin{array} { r } { \frac { d } { d t } \log \rho = \frac { 1 } { 2 \rho ^ { 2 } } \frac { d \rho ^ { 2 } } { d t } = L \frac { \nu ( t ) } { \rho ^ { 2 } } } \end{array}$ by Theorem B.4. Then it simply follows from Lemma B.5 that $\begin{array} { r } { \frac { d } { d t } \log \rho > 0 } \end{array}$ for a.e. $t > t _ { 0 }$ . For the second inequality, we first prove that $\begin{array} { r } { \log \tilde { \gamma } = \log \left( \ell ^ { - 1 } ( \mathcal { L } ) / \rho ^ { L } \right) = \log \left( g ( \log \frac { 1 } { \mathcal { L } } ) / \rho ^ { L } \right) } \end{array}$ exists for all $t \geq t _ { 0 }$ . $\mathcal { L } ( t )$ is non-increasing with $t$ . So $\mathcal { L } ( t ) < e ^ { - f ( b _ { f } ) }$ for all $t \geq t _ { 0 }$ . This implies that (1) $\log { \frac { 1 } { \mathcal { L } } }$ is always in the domain of $g$ ; (2) $\rho > 0$ (otherwise $\mathcal { L } \geq N e ^ { - f ( 0 ) } > e ^ { - f ( b _ { f } ) }$ , contradicting (B4)). Therefore, $\begin{array} { r } { \widetilde { \gamma } : = g ( \log \frac { 1 } { \mathcal { L } } ) / \rho ^ { L } } \end{array}$ exists and is always positive for all $t \geq t _ { 0 }$ , which proves the existence of $\log { \tilde { \gamma } }$ . + +By the chain rule and Lemma B.5, we have + +$$ +\begin{array} { r l r } & { \frac { d } { d t } \log \tilde { \gamma } = \frac { d } { d t } \left( \log \left( g ( \log \frac { 1 } { \mathcal { L } } ) \right) - L \log \rho \right) = \frac { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } { g ( \log \frac { 1 } { \mathcal { L } } ) } \cdot \frac { 1 } { \mathcal { L } } \cdot \left( - \frac { d \mathcal { L } } { d t } \right) - L ^ { 2 } \cdot \frac { \nu ( t ) } { \rho ^ { 2 } } } & \\ & { \geq \frac { 1 } { \nu ( t ) } \cdot \left( - \frac { d \mathcal { L } } { d t } \right) - L ^ { 2 } \cdot \frac { \nu ( t ) } { \rho ^ { 2 } } } & \\ & { \geq \frac { 1 } { \nu ( t ) } \cdot \left( - \frac { d \mathcal { L } } { d t } - \frac { L ^ { 2 } \nu ( t ) ^ { 2 } } { \rho ^ { 2 } } \right) . } & \end{array} +$$ + +On the one hand, by Theorem B.4. $\begin{array} { r } { - \frac { d \mathcal { L } } { d t } = \left. \frac { d \pmb { \theta } } { d t } \right. _ { 2 } ^ { 2 } } \\ { \infty } \end{array}$ for a.e. se toget $t > 0$ by Lemma I.3; on the other hand, lds $\begin{array} { r } { L \nu ( t ) = \left. \theta , \frac { d \theta } { d t } \right. } \end{array}$ + +$$ +{ \frac { d } { d t } } \log { \tilde { \gamma } } \geq { \frac { 1 } { \nu ( t ) } } \left( \left\| { \frac { d \theta } { d t } } \right\| _ { 2 } ^ { 2 } - \left. { \hat { \theta } } , { \frac { d \theta } { d t } } \right. ^ { 2 } \right) = { \frac { 1 } { \nu ( t ) } } \left\| ( I - { \hat { \theta } } { \hat { \theta } } ^ { \top } ) { \frac { d \theta } { d t } } \right\| _ { 2 } ^ { 2 } . +$$ + +By the chain rule, $\begin{array} { r } { \frac { d \hat { \pmb { \theta } } } { d t } = \frac { 1 } { \rho } ( \pmb { I } - \hat { \pmb { \theta } } \hat { \pmb { \theta } } ^ { \top } ) \frac { d \pmb { \theta } } { d t } } \end{array}$ for a.e. $t > 0$ . So we have + +$$ +\frac { d } { d t } \log \tilde { \gamma } \geq \frac { \rho ^ { 2 } } { \nu ( t ) } \left\| \frac { d \hat { \pmb { \theta } } } { d t } \right\| _ { 2 } ^ { 2 } = L \left( \frac { d } { d t } \log \rho \right) ^ { - 1 } \left\| \frac { d \hat { \pmb { \theta } } } { d t } \right\| _ { 2 } ^ { 2 } . +$$ + +# B.2 PROOF FOR PROPOSITION 3 + +To prove the third proposition, we prove the following lemma to show that $\mathcal { L } 0$ by giving an upper bound for $\mathcal { L }$ . Since $\mathcal { L }$ can never be 0 for bounded $\rho$ , $\mathcal { L } \to 0$ directly implies $\rho + \infty$ . For showing $| \bar { \gamma } - \tilde { \gamma } | \to 0$ , we only need to apply $( c )$ in Lemma A.5, which shows this when $\mathcal { L } \to 0$ . + +Lemma B.6. For all $t > t _ { 0 }$ + +$$ +G ( 1 / \mathcal { L } ( t ) ) \geq L ^ { 2 } \widetilde { \gamma } ( t _ { 0 } ) ^ { 2 / L } ( t - t _ { 0 } ) \qquad f o r G ( x ) : = \int _ { 1 / \mathcal { L } ( t _ { 0 } ) } ^ { x } \frac { g ^ { \prime } ( \log u ) ^ { 2 } } { g ( \log u ) ^ { 2 - 2 / L } } d u . +$$ + +Therefore, $\mathcal { L } ( t ) 0$ and $\rho ( t ) \to + \infty$ as $t \to \infty$ . + +Proof for Lemma B.6. By Lemma I.3 and Theorem B.4, + +$$ +- { \frac { d { \mathcal { L } } } { d t } } = \left\| { \frac { d \pmb { \theta } } { d t } } \right\| _ { 2 } ^ { 2 } \geq \left. { \hat { \pmb { \theta } } } , { \frac { d \pmb { \theta } } { d t } } \right. ^ { 2 } = L ^ { 2 } \cdot { \frac { \nu ( t ) ^ { 2 } } { \rho ^ { 2 } } } +$$ + +Using Lemma B.5 to lower bound $\nu$ and replacing $\rho$ with $\left( g ( \log \frac { 1 } { \mathcal { L } } ) / \tilde { \gamma } \right) ^ { 1 / L }$ by the definition of $\tilde { \gamma }$ we have + +$$ +- \frac { d \mathcal { L } } { d t } \geq L ^ { 2 } \cdot \bigg ( \frac { g ( \log \frac { 1 } { \mathcal { L } } ) } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } \mathcal { L } \bigg ) ^ { 2 } \cdot \bigg ( \frac { \widetilde { \gamma } ( t ) } { g ( \log \frac { 1 } { \mathcal { L } } ) } \bigg ) ^ { 2 / L } \geq L ^ { 2 } \widetilde { \gamma } ( t _ { 0 } ) ^ { 2 / L } \cdot \frac { g ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 - 2 / L } } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 } } \mathcal { L } , +$$ + +where the last inequality uses the monotonicity of $\tilde { \gamma }$ . So the following holds for a.e. $t \geq t _ { 0 }$ + +$$ +\frac { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 } } { g ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 - 2 / L } } \cdot \frac { d } { d t } \frac { 1 } { \mathcal { L } } \ge L ^ { 2 } \tilde { \gamma } ( t _ { 0 } ) ^ { 2 / L } . +$$ + +Integrating on both sides from $t _ { 0 }$ to $t$ , we can conclude that + +$$ +G ( 1 / \mathcal { L } ) \geq L ^ { 2 } \tilde { \gamma } ( t _ { 0 } ) ^ { 2 / L } ( t - t _ { 0 } ) . +$$ + +Note that $1 / \mathcal { L }$ is non-decreasing. If $1 / \mathcal { L }$ does not grow to $+ \infty$ , then neither does $G ( 1 / \mathcal { L } )$ . But the RHS grows to $+ \infty$ , which leads to a contradiction. So $\mathcal { L } \to 0$ . + +To make $\mathcal { L } \to 0$ , $q _ { \mathrm { m i n } }$ must converge to $+ \infty$ . So $\rho \to + \infty$ . + +# C CONVERGENCE TO THE MAX-MARGIN SOLUTION + +In this section, we analyze the convergent direction of $\pmb { \theta }$ and prove Theorem A.8 and A.9, assuming (A1), (A2), (B3), (B4) as mentioned in Section A. + +We follow the notations in Section 5 to define $\rho : = \| \pmb \theta \| _ { 2 }$ and $\begin{array} { r } { \hat { \pmb \theta } : = \frac { \pmb \theta } { \| \pmb \theta \| _ { 2 } } \in \mathcal { S } ^ { d - 1 } } \end{array}$ , and sometimes we view the functions of $\pmb { \theta }$ as functions of $t$ . + +# C.1 PRELIMINARIES FOR KKT CONDITIONS + +We first review the definition of Karush-Kuhn-Tucker (KKT) conditions for non-smooth optimization problems following from (Dutta et al., 2013). + +Consider the following optimization problem (P) for $\pmb { x } \in \mathbb { R } ^ { d }$ : + +$$ +\begin{array} { r l } { \operatorname* { m i n } } & { { } f ( { \pmb x } ) } \\ { \mathrm { s . t . } } & { { } g _ { n } ( { \pmb x } ) \le 0 \qquad \forall n \in [ N ] } \end{array} +$$ + +where $f , g _ { 1 } , \ldots , g _ { n } : \mathbb { R } ^ { d } \to \mathbb { R }$ are locally Lipschitz functions. We say that $\pmb { x } \in \mathbb { R } ^ { d }$ is a feasible point of (P) if $_ { \textbf { \em x } }$ satisfies $g _ { n } ( { \pmb x } ) \le 0$ for all $n \in [ N ]$ . + +Definition C.1 (KKT Point). A feasible point $_ { \textbf { \em x } }$ of (P) is a KKT point if $_ { \textbf { \em x } }$ satisfies KKT conditions: there exists $\lambda _ { 1 } , \ldots , \lambda _ { N } \geq 0$ such that + +$$ +2 . \ \forall n \in [ N ] : \lambda _ { n } g _ { n } ( { \pmb x } ) = 0 . +$$ + +It is important to note that a global minimum of (P) may not be a KKT point, but under some regularity assumptions, the KKT conditions become a necessary condition for global optimality. The regularity condition we shall use in this paper is the non-smooth version of Mangasarian-Fromovitz Constraint Qualification (MFCQ) (see, e.g., the constraint qualification (C.Q.5) in (Giorgi et al., 2004)): + +Definition C.2 (MFCQ). For a feasible point $_ { \textbf { \em x } }$ of (P), (P) is said to satisfy MFCQ at $_ { \textbf { \em x } }$ if there exists $\pmb { v } \in \mathbb { R } ^ { d }$ such that for all $n \in [ N ]$ with $g _ { n } ( { \pmb x } ) = 0$ , + +$$ +\forall h \in \partial ^ { \circ } g _ { n } ( { \pmb x } ) : \langle h , { \pmb v } \rangle > 0 . +$$ + +Following from (Dutta et al., 2013), we define an approximate version of KKT point, as shown below. Note that this definition is essentially the modified $\epsilon$ -KKT point defined in their paper, but these two definitions differ in the following two ways: (1) First, in their paper, the subdifferential is allowed to be evaluated in a neighborhood of $_ { \textbf { \em x } }$ , so our definition is slightly stronger; (2) Second, their paper fixes $\delta = \epsilon ^ { 2 }$ , but in our definition we make them independent. + +Definition C.3 (Approximate KKT Point). For $\epsilon , \delta > 0$ , a feasible point $_ { \textbf { \em x } }$ of (P) is an $( \epsilon , \delta )$ -KKT point if there exists $\lambda _ { n } \geq 0 , \pmb { k } \in \partial ^ { \circ } f ( \pmb { x } ) , \pmb { h } _ { n } \in \partial ^ { \circ } g _ { n } ( \pmb { x } )$ for all $n \in [ N ]$ such that + +$$ +\begin{array} { r } { 1 . \ \left\| k + \sum _ { n \in [ N ] } \lambda _ { n } \pmb { h } _ { n } ( \pmb { x } ) \right\| _ { 2 } \leq \epsilon ; } \end{array} +$$ + +$$ +2 . \forall n \in [ N ] : \lambda _ { n } g _ { n } ( { \pmb x } ) \geq - \delta . +$$ + +As shown in (Dutta et al., 2013), $( \epsilon , \delta )$ -KKT point is an approximate version of KKT point in the sense that a series of $( \epsilon , \delta )$ -KKT points can converge to a KKT point. We restate their theorem in our setting: + +Theorem C.4 (Corollary of Theorem 3.6 in (Dutta et al., 2013)). Let $\{ \pmb { x } _ { k } \in \mathbb { R } ^ { d } : k \in \mathbb { N } \}$ be a sequence of feasible points of $( P )$ , $\{ \epsilon _ { k } > 0 : k \in \mathbb { N } \}$ and $\{ \delta _ { k } > 0 : k \in \tilde { \mathbb { N } } \}$ be two sequences. $\scriptstyle { \mathbf { { \mathit { x } } } } _ { k }$ is an $( \epsilon _ { k } , \delta _ { k } )$ -KKT point for every $k$ , and $\epsilon _ { k } 0 , \delta _ { k } 0$ . If $\mathbfit { x } _ { k } \to \mathbfit { x }$ as $k \to + \infty$ and MFCQ holds at $_ { \textbf { \em x } }$ , then $_ { \textbf { \em x } }$ is a KKT point of $( P )$ . + +# C.2 KKT CONDITIONS FOR (P) + +Recall that for a homogeneous neural network, the optimization problem (P) is defined as follows: + +$$ +\begin{array} { r l } { \operatorname* { m i n } } & { \frac { 1 } { 2 } \left\| \pmb { \theta } \right\| _ { 2 } ^ { 2 } } \\ { \mathrm { s . t . } } & { q _ { n } ( \pmb { \theta } ) \geq 1 \qquad \forall n \in [ N ] } \end{array} +$$ + +Using the terminologies and notations in Appendix C.1, the objective and constraints are $f ( { \pmb x } ) =$ $\begin{array} { r l } { { { \frac { 1 } { 2 } } \| \pmb { x } \| _ { 2 } ^ { 2 } } } \end{array}$ and $g _ { n } ( \pmb { x } ) = 1 - q _ { n } ( \pmb { x } )$ . The KKT points and approximate KKT points for (P) are defined as follows: + +Definition C.5 (KKT Point of (P)). A feasible point $\pmb \theta$ of (P) is a KKT point if there exist $\lambda _ { 1 } , \dots , \lambda _ { N } \ge 0$ such that + +1. $\begin{array} { r } { \pmb { \theta } - \sum _ { n = 1 } ^ { N } \lambda _ { n } \pmb { h } _ { n } = \mathbf { 0 } } \end{array}$ for some $h _ { 1 } , \ldots , h _ { N }$ satisfying $h _ { n } \in \partial ^ { \circ } q _ { n } ( \theta )$ +2. $\forall n \in [ N ] : \lambda _ { n } ( q _ { n } ( \pmb \theta ) - 1 ) = 0 .$ + +Definition C.6 (Approximate KKT Point of (P)). A feasible point $\pmb { \theta }$ of $( \mathrm { P } )$ is an $( \epsilon , \delta )$ -KKT point of (P) if there exists $\lambda _ { 1 } , \ldots , \lambda _ { N } \geq 0$ such that + +$$ +2 . \forall n \in [ N ] : \lambda _ { n } ( q _ { n } ( \pmb \theta ) - 1 ) \leq \delta . +$$ + +By the homogeneity of $q _ { n }$ , it is easy to see that (P) satisfies MFCQ, and thus KKT conditions are first-order necessary condition for global optimality. + +Lemma C.7. $( P )$ satisfies MFCQ at every feasible point $\pmb { \theta }$ . + +Proof. Take $\boldsymbol { v } : = \boldsymbol { \theta }$ . For all $n \in [ N ]$ satisfying $q _ { n } = 1$ , by homogeneity of $q _ { n }$ , + +$$ +\langle v , h \rangle = L q _ { n } ( \pmb { \theta } ) = L > 0 +$$ + +holds for any $\pmb { h } \in - \partial ^ { \circ } q _ { n } ( \pmb { \theta } ) = \partial ^ { \circ } \big ( 1 - q _ { n } ( \pmb { \theta } ) \big )$ . + +# C.3 KEY LEMMAS + +Define β(t) := $\begin{array} { r } { \beta ( t ) : = \frac { 1 } { \left\| \frac { d \theta } { d t } \right\| _ { 2 } } \left. \hat { \pmb { \theta } } , \frac { d \pmb { \theta } } { d t } \right. } \end{array}$ to be the cosine of the angle between $\pmb \theta$ and $\textstyle { \frac { d \theta } { d t } }$ . Here $\beta ( t )$ is only defined for a.e. $t > 0$ . Since $q _ { n }$ is locally Lipschitz, it can be shown that $q _ { n }$ is (globally) Lipschitz on the compact set $S ^ { d - 1 }$ , which is the unit sphere in $\mathbb { R } ^ { d }$ . Define + +$$ +\begin{array} { r l } & { B _ { 0 } : = \operatorname* { s u p } \{ \cfrac { q _ { n } } { \rho ^ { L } } : \theta \in \mathbb { R } ^ { d } \setminus \{ \mathbf { 0 } \} \} } \\ & { \quad = \operatorname* { s u p } \{ q _ { n } : \theta \in \mathcal { S } ^ { d - 1 } \} < \infty . } \\ & { B _ { 1 } : = \operatorname* { s u p } \{ \cfrac { \| h \| _ { 2 } } { \rho ^ { L - 1 } } : \theta \in \mathbb { R } ^ { d } \setminus \{ \mathbf { 0 } \} , h \in \partial ^ { \circ } q _ { n } , n \in [ N ] \} } \\ & { \quad = \operatorname* { s u p } \{ \| h \| _ { 2 } : \theta \in \mathcal { S } ^ { d - 1 } , \boldsymbol { h } \in \partial ^ { \circ } q _ { n } , n \in [ N ] \} < \infty . } \end{array} +$$ + +For showing Theorem A.8 and Theorem A.9, we first prove Lemma C.8. In light of this lemma, if we aim to show that $\pmb \theta$ is along the direction of an approximate KKT point, we only need to show $\beta 1$ (which makes $\epsilon \to 0$ ) and $\mathcal { L } \to 0$ (which makes $\delta 0$ ). + +Lemma C.8. Let $C _ { 1 } , C _ { 2 }$ be two constants defined as + +$$ +C _ { 1 } : = \frac { \sqrt { 2 } } { \tilde { \gamma } ( t _ { 0 } ) ^ { 1 / L } } , \qquad C _ { 2 } : = \frac { 2 e N K ^ { 2 } } { L \tilde { \gamma } ( t _ { 0 } ) ^ { 2 / L } } \left( \frac { B _ { 1 } } { \tilde { \gamma } ( t _ { 0 } ) } \right) ^ { \log _ { 2 } K } , +$$ + +where $K , b _ { g }$ are constants specified in (B3.4). $I f \log { \frac { 1 } { \mathcal { L } } } \geq b _ { g }$ at time $t > t _ { 0 }$ , then $\tilde { \pmb { \theta } } : = \pmb { \theta } / q _ { \mathrm { m i n } } ( \pmb { \theta } ) ^ { 1 / L }$ is an $( \epsilon , \delta )$ -KKT point of $( P )$ , where $\epsilon : = C _ { 1 } \sqrt { 1 - \beta }$ and $\delta : = C _ { 2 } / ( \log \frac { 1 } { \mathcal { L } } )$ . + +Proof. Let $\begin{array} { r } \begin{array} { r } { \begin{array} { r } { \boldsymbol { h } ( t ) : = \frac { d \theta } { d t } ( t ) } \end{array} } \end{array} \end{array}$ for a.e. $t > 0$ . By the chain rule, there exist $\pmb { h } _ { 1 } , \dots , \pmb { h } _ { N }$ such that $h _ { n } \in \partial ^ { \circ } q _ { n }$ and $\begin{array} { r } { \pmb { h } = \sum _ { \pmb { n } \in [ N ] } e ^ { - \pmb { f } ( \ v { q } _ { n } ) } f ^ { \prime } ( \ v { q } _ { n } ) \pmb { h } _ { n } } \end{array}$ . Let $\tilde { \pmb { h } } _ { n } : = \pmb { h } _ { n } / q _ { \mathrm { m i n } } ^ { 1 - 1 / L } \in \partial ^ { \circ } q _ { n } ( \tilde { \pmb { \theta } } )$ q1−1/Lmin ∈ ∂◦qn(θ˜) (Recall that θ˜ := θ/qmin(θ)1/L). Construct λn := q1−2/min $\lambda _ { n } : = q _ { \mathrm { m i n } } ^ { 1 - 2 / L } \rho \cdot e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) / \left\| h \right\| _ { 2 }$ . Now we only need to show + +$$ +\begin{array} { r l } & { \left\| \tilde { \theta } - \displaystyle \sum _ { n = 1 } ^ { N } \lambda _ { n } \tilde { h } _ { n } \right\| _ { 2 } ^ { 2 } \leq \frac { 2 } { \tilde { \gamma } ^ { 2 / L } } ( 1 - \beta ) } \\ & { \displaystyle \sum _ { n = 1 } ^ { N } \lambda _ { n } ( q _ { n } ( \tilde { \theta } ) - 1 ) \leq \left( \frac { 2 e N K ^ { 2 } } { L \tilde { \gamma } ^ { 2 / L } } \left( \frac { B _ { 1 } } { \tilde { \gamma } } \right) ^ { \log _ { 2 } K } \right) \frac { 1 } { f ( \tilde { \gamma } \rho ^ { L } ) } . } \end{array} +$$ + +Then $\tilde { \theta }$ can be shown to be an $( \epsilon , \delta )$ -KKT point by the monotonicity $\tilde { \gamma } ( t ) \geq \tilde { \gamma } ( t _ { 0 } )$ for $t > t _ { 0 }$ + +Proof of (12). From our construction, $\begin{array} { r } { \sum _ { n = 1 } ^ { N } \lambda _ { n } \tilde { \pmb { h } } _ { n } = q _ { \mathrm { m i n } } ^ { - 1 / L } \rho \pmb { h } / \left\| \pmb { h } \right\| _ { 2 } } \end{array}$ . So + +$$ +\left\| \tilde { \theta } - \sum _ { n = 1 } ^ { N } \lambda _ { n } \tilde { h } _ { n } \right\| _ { 2 } ^ { 2 } = q _ { \operatorname* { m i n } } ^ { - 2 / L } \rho ^ { 2 } \left\| \hat { \theta } - \frac { h } { \| h \| _ { 2 } } \right\| _ { 2 } ^ { 2 } = q _ { \operatorname* { m i n } } ^ { - 2 / L } \rho ^ { 2 } ( 2 - 2 \beta ) \leq \frac { 2 } { \tilde { \gamma } ^ { 2 / L } } ( 1 - \beta ) , +$$ + +where the last equality is by Lemma A.5. + +Proof for (13). According to our construction, + +$$ +\sum _ { n = 1 } ^ { N } \lambda _ { n } ( q _ { n } ( \tilde { \theta } ) - 1 ) = \frac { q _ { \operatorname* { m i n } } ^ { - 2 / L } \rho } { \| h \| _ { 2 } } \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) ( q _ { n } - q _ { \operatorname* { m i n } } ) . +$$ + +Note that $\| h \| _ { 2 } \geq \left. h , \hat { \theta } \right. = L \nu / \rho$ . By Lemma B.5 and Lemma D.1, we have + +$$ +\nu \geq \frac { g ( \log \frac { 1 } { \mathcal { L } } ) } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } \mathcal { L } \geq \frac { 1 } { 2 K } \log \frac { 1 } { \mathcal { L } } \cdot \mathcal { L } \geq \frac { 1 } { 2 K } f ( \widetilde { \gamma } \rho ^ { L } ) e ^ { - f ( q _ { \operatorname* { m i n } } ) } , +$$ + +where the last inequality uses $\begin{array} { r } { f ( \tilde { \gamma } \rho ^ { L } ) = \log \frac { 1 } { \mathcal { L } } } \end{array}$ and $\mathcal { L } \geq e ^ { - f ( q _ { \mathrm { m i n } } ) }$ . Combining these gives + +$$ +\sum _ { n = 1 } ^ { N } \lambda _ { n } ( q _ { n } ( \tilde { \theta } ) - 1 ) \leq \frac { 2 K q _ { \operatorname* { m i n } } ^ { - 2 / L } \rho ^ { 2 } } { L f ( \tilde { \gamma } \rho ^ { L } ) } \sum _ { n = 1 } ^ { N } e ^ { f ( q _ { \operatorname* { m i n } } ) - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) ( q _ { n } - q _ { \operatorname* { m i n } } ) . +$$ + +If $q _ { n } > q _ { \mathrm { m i n } }$ , then by the mean value theorem there exists $\xi _ { n } ~ \in ~ ( q _ { \operatorname* { m i n } } , q _ { n } )$ such that $f ( q _ { n } ) \mathrm { ~ - ~ }$ $f ( q _ { \mathrm { m i n } } ) = f ^ { \prime } ( \xi _ { n } ) ( q _ { n } - q _ { \mathrm { m i n } } )$ . By Assumption (B3.4), we know that $f ^ { \prime } ( q _ { n } ) \leq K ^ { \lceil \log _ { 2 } ( q _ { n } / \xi _ { n } ) \rceil } f ^ { \prime } ( \xi _ { n } )$ . Note that $\begin{array} { r } { \lceil \log _ { 2 } ( q _ { n } / \xi _ { n } ) \rceil \leq \log _ { 2 } ( 2 B _ { 1 } \rho ^ { L } / q _ { \mathrm { m i n } } ) \leq \log _ { 2 } ( 2 B _ { 1 } / \tilde { \gamma } ) . } \end{array}$ . Then we have + +$$ +\begin{array} { r l } & { \displaystyle \sum _ { n = 1 } ^ { N } \lambda _ { n } ( q _ { n } ( \tilde { \theta } ) - 1 ) \le \frac { 2 K q _ { \operatorname* { m i n } } ^ { - 2 / L } \rho ^ { 2 } } { L f ( \tilde { \gamma } \rho ^ { L } ) } K ^ { \log _ { 2 } ( 2 B _ { 1 } / \tilde { \gamma } ) } \displaystyle \sum _ { n \colon { q _ { n } \ne q _ { \operatorname* { m i n } } } } e ^ { - f ^ { \prime } ( \xi _ { n } ) ( q _ { n } - q _ { \operatorname* { m i n } } ) } f ^ { \prime } ( \xi _ { n } ) ( q _ { n } - q _ { \operatorname* { m i n } } ) } \\ & { \displaystyle \le \frac { 2 K \tilde { \gamma } ^ { - 2 / L } \rho ^ { 2 } } { L f ( \tilde { \gamma } \rho ^ { L } ) } K ^ { \log _ { 2 } ( 2 B _ { 1 } / \tilde { \gamma } ) } \cdot N e } \end{array} +$$ + +sec uality uses $q _ { \mathrm { m i n } } ^ { - 2 / L } \rho ^ { 2 } \le \tilde { \gamma } ^ { - 2 / L }$ b ma A.5 and the fact that the function $x \mapsto e ^ { - x } x$ $( 0 , + \infty )$ attains the maximum value $e$ $x = 1$ + +By Theorem A.7, we have already known that $\mathcal { L } 0$ . So it remains to bound $\beta ( t )$ . For this, we first prove the following lemma to bound the integral of $\beta ( t )$ . + +Lemma C.9. For all $t _ { 2 } > t _ { 1 } \geq t _ { 0 }$ , + +$$ +\int _ { t _ { 1 } } ^ { t _ { 2 } } \left( \beta ( \tau ) ^ { - 2 } - 1 \right) \cdot \frac { d } { d \tau } \log \rho ( \tau ) \cdot d \tau \leq \frac { 1 } { L } \log \frac { \tilde { \gamma } ( t _ { 2 } ) } { \tilde { \gamma } ( t _ { 1 } ) } . +$$ + +Proof. By Lemma B.4, $\begin{array} { r } { \frac { d } { d t } \log \rho = \frac 1 { 2 \rho ^ { 2 } } \frac { d \rho ^ { 2 } } { d t } = \frac { L \nu } { \rho ^ { 2 } } } \end{array}$ for a.e. $t > 0$ . By Lemma B.1, for a.e. $t \in ( t _ { 1 } , t _ { 2 } )$ + +$$ +\frac { d } { d t } \log \tilde { \gamma } \geq L \cdot \left. \frac { \rho ^ { 2 } } { L \nu } \frac { d \hat { \pmb \theta } } { d t } \right. _ { 2 } ^ { 2 } \cdot \frac { d } { d t } \log \rho . +$$ + +By the chain rule, $\begin{array} { r } { \frac { d \hat { \pmb { \theta } } } { d t } = \frac { 1 } { \rho } ( \pmb { I } - \hat { \pmb { \theta } } \hat { \pmb { \theta } } ^ { \top } ) \frac { d \pmb { \theta } } { d t } } \end{array}$ . So we have + +$$ +\left\| \frac { \rho ^ { 2 } } { L \nu } \frac { d \hat { \theta } } { d t } \right\| _ { 2 } ^ { 2 } = \left\| \frac { \rho } { L \nu ( t ) } ( I - \hat { \theta } \hat { \theta } ^ { \top } ) \frac { d \theta } { d t } \right\| _ { 2 } ^ { 2 } = \frac { \left\| \frac { d \theta } { d t } \right\| _ { 2 } ^ { 2 } - \left. \frac { d \theta } { d t } , \hat { \theta } \right. ^ { 2 } } { \left. \frac { d \theta } { d t } , \hat { \theta } \right. ^ { 2 } } = \beta ^ { - 2 } - 1 . +$$ + +where the last equality follows from the definition of $\beta$ . Combining 14 and 15, we have + +$$ +\frac { d } { d t } \log \tilde { \gamma } \geq L \left( \beta ^ { - 2 } - 1 \right) \cdot \frac { d } { d t } \log \rho . +$$ + +Integrating on both sides from $t _ { 1 }$ to $t _ { 2 }$ proves the lemma. + +A direct corollary of Lemma C.9 is the upper bound for the minimum $\beta ^ { 2 } - 1$ within a time interval: Corollary C.10. For all $t _ { 2 } > t _ { 1 } \ge t _ { 0 }$ , then there exists $t _ { * } \in ( t _ { 1 } , t _ { 2 } )$ such that + +$$ +\beta ( t _ { * } ) ^ { - 2 } - 1 \le \frac { 1 } { L } \cdot \frac { \log \tilde { \gamma } ( t _ { 2 } ) - \log \tilde { \gamma } ( t _ { 1 } ) } { \log \rho ( t _ { 2 } ) - \log \rho ( t _ { 1 } ) } . +$$ + +Proof. Denote the RHS as $C$ . Assume to the contrary that $\beta ( \tau ) ^ { - 2 } - 1 > C$ for a.e. $\tau \in ( t _ { 1 } , t _ { 2 } )$ . By Lemma B.1, $\log \rho ( \tau ) > 0$ for a.e. $\tau \in ( t _ { 1 } , t _ { 2 } )$ . Then by Lemma C.9, we have + +$$ +\frac { 1 } { L } \log \frac { \tilde { \gamma } ( t _ { 2 } ) } { \tilde { \gamma } ( t _ { 1 } ) } > \int _ { t _ { 1 } } ^ { t _ { 2 } } { C \cdot \frac { d } { d \tau } \log \rho ( \tau ) \cdot d \tau } = C \cdot ( \log \rho ( t _ { 2 } ) - \log \rho ( t _ { 1 } ) ) = \frac { 1 } { L } \log \frac { \tilde { \gamma } ( t _ { 2 } ) } { \tilde { \gamma } ( t _ { 1 } ) } , +$$ + +which leads to a contradiction. + +In the rest of this section, we present both asymptotic and non-asymptotic analyses for the directional convergence by using Corollary C.10 to bound $\beta ( t )$ . + +# C.4 ASYMPTOTIC ANALYSIS + +We first prove an auxiliary lemma which gives an upper bound for the change of $\hat { \pmb { \theta } }$ . + +Lemma C.11. For a.e. $t > t _ { 0 }$ , + +$$ +\left\| \frac { d \hat { \pmb \theta } } { d t } \right\| _ { 2 } \le \frac { B _ { 1 } } { L \tilde { \gamma } } \cdot \frac { d } { d t } \log \rho . +$$ + +Proof. Observe that $\begin{array} { r } { \left\| \frac { d \hat { \pmb \theta } } { d t } \right\| _ { 2 } = \frac { 1 } { \rho } \left\| ( \pmb I - \hat { \pmb \theta } \hat { \pmb \theta } ^ { \top } ) \frac { d \pmb \theta } { d t } \right\| _ { 2 } \leq \frac { 1 } { \rho } \left\| \frac { d \pmb \theta } { d t } \right\| _ { 2 } . } \end{array}$ . It is sufficient to bound $\left\| { \frac { d \theta } { d t } } \right\| _ { 2 }$ . By and the chain rule, there exists $\begin{array} { r } { \frac { d \theta } { d t } = \sum _ { n \in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) { h _ { n } ( t ) } } \end{array}$ $h _ { 1 } , \ldots , h _ { N } : [ 0 , + \infty ) \to \mathbb { R } ^ { d }$ . By definition of satisfying that for a.e. $B _ { 1 }$ , $\| h _ { n } \| _ { 2 } \leq B _ { 1 } \rho ^ { L - 1 }$ $t > 0$ for a.e. , $h _ { n } ( t ) \in \partial ^ { \circ } q _ { n }$ $t > 0$ . So + +$$ +\begin{array} { l } { \displaystyle \left\| \frac { d \pmb { \theta } } { d t } \right\| _ { 2 } \leq \sum _ { n \in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) \left\| \pmb { h } _ { n } \right\| _ { 2 } } \\ { \displaystyle \qquad \leq \sum _ { n \in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) q _ { n } \cdot \frac { 1 } { q _ { n } } \cdot B _ { 1 } \rho ^ { L - 1 } . } \end{array} +$$ + +Note that every summand is positive. By Lemma A.5, $q _ { n }$ is lower-bounded by $q _ { n } \geq q _ { \mathrm { m i n } } \geq$ $g ( \log { \frac { 1 } { \mathcal { L } } } )$ , so we can replace $q _ { n }$ with $g ( \log { \frac { 1 } { \mathcal { L } } } )$ in the above inequality. Combining with the fact that $\begin{array} { r } { \sum _ { n \in [ N ] } { \stackrel { \sim } { e } } ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) q _ { n } \quad } \end{array}$ is just $\nu$ , we have + +$$ +\left\| \frac { d \pmb { \theta } } { d t } \right\| _ { 2 } \leq \frac { \nu } { g ( \log \frac { 1 } { \mathcal { L } } ) } \cdot B _ { 1 } \rho ^ { L - 1 } = \frac { B _ { 1 } \nu } { \tilde { \gamma } \rho } . +$$ + +So we have $\begin{array} { r } { \left\| \frac { d \hat { \pmb \theta } } { d t } \right\| _ { 2 } \le \frac { 1 } { \rho } \left\| \frac { d \pmb \theta } { d t } \right\| _ { 2 } \le \frac { B _ { 1 } } { L \tilde { \gamma } } \cdot \frac { L \nu } { \rho ^ { 2 } } = \frac { B _ { 1 } } { L \tilde { \gamma } } \cdot \frac { d } { d t } \log \rho . } \end{array}$ + +To prove Theorem A.8, we consider each limit point $\bar { \theta } / q _ { \mathrm { m i n } } ( \bar { \theta } ) ^ { 1 / L }$ , and construct a series of approximate KKT points converging to it. Then $\bar { \theta } / q _ { \mathrm { m i n } } ( \bar { \theta } ) ^ { 1 / L }$ can be shown to be a KKT point by Theorem C.4. The following lemma ensures that such construction exists. + +Lemma C.12. For every limit point $\bar { \pmb { \theta } }$ of $\left\{ { \hat { \pmb { \theta } } } ( t ) : t \geq 0 \right\}$ , there exists a sequence of $\{ t _ { m } : m \in \mathbb { N } \}$ such that $t _ { m }$ $\cdot + \infty , \hat { \pmb { \theta } } ( t _ { m } ) \bar { \pmb { \theta } }$ , and $\beta ( t _ { m } ) 1$ . + +Proof. Let $\{ \epsilon _ { m } > 0 : m \in \mathbb { N } \}$ be an arbitrary sequence with $\epsilon _ { m } \to 0$ . Now we construct $\left\{ { { t } _ { m } } \right\}$ by induction. Suppose $t _ { 1 } < t _ { 2 } < \dots < t _ { m - 1 }$ have already been constructed. Since $\bar { \pmb { \theta } }$ is a limit point and $\tilde { \gamma } ( t ) \uparrow \tilde { \gamma } _ { \infty }$ (recall that $\tilde { \gamma } _ { \infty } : = \mathrm { l i m } _ { t + \infty } \tilde { \gamma } ( t ) )$ , there exists $s _ { m } > t _ { m - 1 }$ such that + +$$ +\left\| \hat { \pmb \theta } ( s _ { m } ) - \bar { \pmb \theta } \right\| _ { 2 } \leq \epsilon _ { m } \quad \mathrm { a n d } \quad \frac { 1 } { L } \log \frac { \widetilde { \gamma } _ { \infty } } { \widetilde { \gamma } ( s _ { m } ) } \leq \epsilon _ { m } ^ { 3 } . +$$ + +Let $s _ { m } ^ { \prime } > s _ { m }$ be a time such that $\log { \rho ( s _ { m } ^ { \prime } ) } = \log { \rho ( s _ { m } ) } + \epsilon _ { m }$ . According to Theorem A.7, $\log \rho \to + \infty$ , so $s _ { m } ^ { \prime }$ must exist. We construct $t _ { m } \in ( s _ { m } , s _ { m } ^ { \prime } )$ to be a time that $\bar { \beta } ( t _ { m } ) ^ { - 2 } - 1 \leq \epsilon _ { m } ^ { 2 }$ , where the existence can be shown by Corollary C.10. + +Now we show that this construction meets our requirement. It follows from $\beta ( t _ { m } ) ^ { - 2 } - 1 \leq \epsilon _ { m } ^ { 2 }$ that $\beta ( t _ { m } ) \geq 1 / \sqrt { 1 + \epsilon _ { m } ^ { 2 } } \to 1$ . By Lemma C.11, we also know that + +$$ +\biggl \| \hat { \pmb \theta } ( t _ { m } ) - \bar { \pmb \theta } \biggl \| _ { 2 } \leq \biggl \| \hat { \pmb \theta } ( t _ { m } ) - \hat { \pmb \theta } ( s _ { m } ) \biggr \| _ { 2 } + \biggl \| \hat { \pmb \theta } ( s _ { m } ) - \bar { \pmb \theta } \biggl \| _ { 2 } \leq \frac { B _ { 1 } } { L \tilde { \gamma } ( t _ { 0 } ) } \cdot \epsilon _ { m } + \epsilon _ { m } 0 . +$$ + +This completes the proof. + +Proof of Theorem A.8. Let $\tilde { \pmb { \theta } } : = \bar { \pmb { \theta } } / q _ { \mathrm { m i n } } ( \bar { \pmb { \theta } } ) ^ { 1 / L }$ for short. Let $\{ t _ { m } : m \in \mathbb { N } \}$ be the sequence constructed as in Lemma C.12. For each $t _ { m }$ , define $\epsilon ( t _ { m } )$ and $\delta ( t _ { m } )$ as in Lemma C.8. Then we know that $\pmb { \theta } ( t _ { m } ) / q _ { \mathrm { m i n } } ( t _ { m } ) ^ { 1 / L }$ is an $( \epsilon ( t _ { m } ) , \delta ( t _ { m } ) )$ -KKT point and $\pmb { \theta } ( t _ { m } ) / q _ { \mathrm { m i n } } ( t _ { m } ) ^ { 1 / L } \tilde { \pmb { \theta } } , \epsilon ( t _ { m } ) $ $0 , \delta ( t _ { m } ) \to 0$ . By Lemma C.7, (P) satisfies MFCQ. Applying Theorem C.4 proves the theorem. + +# C.5 NON-ASYMPTOTIC ANALYSIS + +Proof of Theorem A.9. Let C0 := 1L log γ˜∞γ˜(t0) . Without loss of generality, we assume $\begin{array} { r } { \epsilon < \frac { \sqrt { 6 } } { 2 } C _ { 1 } } \end{array}$ and $\delta < C _ { 2 } / f ( b _ { f } )$ . + +Let $t _ { 1 }$ be the time such that $\begin{array} { r } { \log \rho ( t _ { 1 } ) = \frac { 1 } { L } \log \frac { g ( C _ { 2 } \delta ^ { - 1 } ) } { \tilde { \gamma } ( t _ { 0 } ) } = \Theta ( \log \delta ^ { - 1 } ) } \end{array}$ g g(C2δ−1) = Θ(log δ−1) and t2 be the time such that $\begin{array} { r } { \log \rho ( t _ { 2 } ) - \log \rho ( t _ { 1 } ) = \frac { 1 } { 2 } C _ { 0 } C _ { 1 } ^ { 2 } \epsilon ^ { - 2 } = \Theta ( \epsilon ^ { - 2 } ) } \end{array}$ . By Corollary C.10, there exists $t _ { * } \in ( t _ { 1 } , t _ { 2 } )$ such that $\beta ( t _ { * } ) ^ { - 2 } - 1 \le 2 \epsilon ^ { 2 } C _ { 1 } ^ { - 2 }$ . + +Now we argue that $\tilde { \pmb { \theta } } ( t _ { * } )$ is an $( \epsilon , \delta )$ -KKT point. By Lemma C.8, we only need to show $C _ { 1 } \sqrt { 1 - \beta ( t _ { * } ) } \le \epsilon$ and $C _ { 2 } / f ( \tilde { \gamma } ( t _ { * } ) \rho ( t _ { * } ) ^ { L } ) \leq \delta$ . For the first inequality, by assumption $\epsilon < \frac { \sqrt { 6 } } { 2 } C _ { 1 }$ , we know that $\beta ( t _ { * } ) ^ { - 2 } - 1 < 3$ , which implies $| \beta ( t _ { * } ) | ~ < ~ \frac { 1 } { 2 }$ . Then we have $1 - \beta ( t _ { * } ) \ \leq$ 22C −2 · β(t∗)2 $\frac { \beta ( t _ { * } ) ^ { 2 } } { 1 + \beta ( t _ { * } ) } ~ \leq ~ \epsilon ^ { 2 } C _ { 1 } ^ { - 2 }$ . Therefore, $C _ { 1 } \sqrt { 1 - \beta ( t _ { * } ) } \le \epsilon$ holds. For the second inequality, $\begin{array} { r } { \tilde { \gamma } ( t _ { * } ) \rho ( t _ { * } ) ^ { L } \geq \tilde { \gamma } ( t _ { * } ) \cdot \frac { g ( C _ { 2 } \delta ^ { - 1 } ) } { \tilde { \gamma } ( t _ { 0 } ) } \geq g ( C _ { 2 } \delta ^ { - 1 } ) } \end{array}$ . Therefore, $C _ { 2 } / f ( \tilde { \gamma } ( t _ { * } ) \rho ( t _ { * } ) ^ { L } ) \leq \delta$ holds. □ + +# C.6 PROOF FOR COROLLARY 4.5 + +By the homogeneity of $q _ { n }$ , we can characterize KKT points using kernel SVM. + +Lemma C.13. If $\pmb { \theta } _ { \ast }$ is KKT point of $( P )$ , then there exists $h _ { n } \in \partial ^ { \circ } \Phi _ { x _ { n } } ( \theta _ { * } )$ for $n \in [ N ]$ such that $\scriptstyle { \frac { 1 } { L } } \theta _ { * }$ is an optimal solution for the following constrained optimization problem $( Q )$ : + +$$ +\begin{array} { r l } { \operatorname* { m i n } } & { \frac { 1 } { 2 } \left\| \pmb { \theta } \right\| _ { 2 } ^ { 2 } } \\ { s . t . } & { y _ { n } \left. \pmb { \theta } , \pmb { h } _ { n } \right. \geq 1 \qquad \forall n \in [ N ] } \end{array} +$$ + +Proof. It is easy to see that (Q) is a convex optimization problem. For $\begin{array} { r } { \pmb { \theta } = \frac { 2 } { L } \pmb { \theta } _ { * } } \end{array}$ , from Theorem B.2, we can see that $y _ { n _ { \cdot } } \langle \theta , h _ { n } \rangle = 2 q _ { n } ( \theta _ { \ast } ) \geq 2 > 1$ , which implies Slater’s condition. Thus, we only need to show that $\scriptstyle { \frac { 1 } { L } } \theta _ { * }$ satisfies KKT conditions for (Q). + +By the KKT conditions for $( \mathrm { P } )$ , we can construct $h _ { n } \in \partial ^ { \circ } q _ { n } ( \theta _ { * } )$ for $n \in [ N ]$ such that $\theta _ { * } \mathrm { ~ - ~ }$ $\textstyle \sum _ { n = 1 } ^ { N } \lambda _ { n } y _ { n } h _ { n } = \mathbf { 0 }$ for some $\lambda _ { 1 } , \dots , \lambda _ { N } \ge 0$ and $\lambda _ { n } ( q _ { n } ( \pmb { \theta } _ { * } ) - 1 ) = 0$ . Thus, $\scriptstyle { \frac { 1 } { L } } \theta _ { * }$ satisfies + +$$ +\begin{array} { r } { \frac { 1 } { L } \lambda _ { n } \left( y _ { n } \left. \frac { 1 } { L } \pmb { \theta } _ { * } , \pmb { h } _ { n } \right. - 1 \right) = \frac { 1 } { L } \lambda _ { n } \left( q _ { n } ( \pmb { \theta } ) - 1 \right) \geq 0 . } \end{array} +$$ + +So $\scriptstyle { \frac { 1 } { L } } \theta _ { * }$ satisfies KKT conditions for (Q). + +Now we prove Corollary 4.5 in Section 4.3. + +Proof. By Theorem A.8, every limit point $\bar { \pmb { \theta } }$ is along the direction of a KKT point of (P). Combining this with Lemma C.13, we know that every limit point $\bar { \pmb { \theta } }$ is also along the max-margin direction of (Q). + +For smooth models, $h _ { n }$ in $( \mathrm { Q } )$ is exactly the gradient $\nabla \Phi _ { x _ { n } } ( { \bar { \theta } } )$ . So, (Q) is the optimization problem for SVM with kernel $K _ { \bar { \theta } } ( x , x ^ { \prime } ) = \left. \nabla \Phi _ { x } ( \bar { \theta } ) , \nabla \Phi _ { x ^ { \prime } } ( \bar { \theta } ) \right.$ . For non-smooth models, we can construct an arbitrary function $h ( x ) \in \partial ^ { \circ } \Phi _ { x } ( \bar { \theta } )$ that ensures $\pmb { h } ( \pmb { x } _ { n } ) = \pmb { h } _ { n }$ . Then, (Q) is the optimization problem for SVM with kernel $K _ { \bar { \theta } } ( x , x ^ { \prime } ) = \langle h ( x ) , h ( x ^ { \prime } ) \rangle$ . □ + +# D TIGHT BOUNDS FOR LOSS CONVERGENCE AND WEIGHT GROWTH + +In this section, we give proof for Theorem A.10, which gives tight bounds for loss convergence and weight growth under Assumption (A1), (A2), (B3), (B4). + +# D.1 CONSEQUENCES OF (B3.4) + +Before proving Theorem A.10, we show some consequences of (B3.4). + +Lemma D.1. For $f ( \cdot )$ and $g ( \cdot )$ , we have + +1. For all $x \in [ b _ { g } , + \infty )$ , $\begin{array} { r } { \frac { g ( x ) } { g ^ { \prime } ( x ) } \in [ \frac { 1 } { 2 K } x , 2 K x ] , } \end{array}$ ; + +2. For all $y \in [ g ( b _ { g } ) , + \infty )$ , $\begin{array} { r } { \frac { f ( y ) } { f ^ { \prime } ( y ) } \in [ \frac { 1 } { 2 K } y , 2 K y ] . } \end{array}$ . + +Thus, $g ( x ) = \Theta ( x g ^ { \prime } ( x ) ) , f ( y ) = \Theta ( y f ^ { \prime } ( y ) ) .$ + +Proof. To prove Item 1, it is sufficient to show that + +$$ +\begin{array} { l } { \displaystyle { g ( x ) = b _ { f } + \int _ { f ( b _ { f } ) } ^ { x } g ^ { \prime } ( u ) d u \geq \int _ { x / 2 } ^ { x } g ^ { \prime } ( u ) d u } } \\ { \displaystyle { \qquad \geq ( x / 2 ) \cdot \frac { g ^ { \prime } ( x ) } { K } = \frac { 1 } { 2 K } \cdot x g ^ { \prime } ( x ) } . } \end{array} +$$ + +$$ +\begin{array} { l } { { \displaystyle x = f ( b _ { f } ) + \int _ { f ( b _ { f } ) } ^ { x } g ^ { \prime } ( u ) f ^ { \prime } ( g ( u ) ) d u \geq \frac { f ^ { \prime } ( g ( x ) ) } { K } \int _ { f ( g ( x ) / 2 ) } ^ { x } g ^ { \prime } ( u ) d u } } \\ { { \displaystyle \quad = ( g ( x ) / 2 ) \cdot \frac { f ^ { \prime } ( g ( x ) ) } { K } = \frac { 1 } { 2 K } \cdot \frac { g ( x ) } { g ^ { \prime } ( x ) } . } } \end{array} +$$ + +To prove Item 2, we only need to notice that Item 1 implies yf 0(y) = g(f(y))g0(f(y)) $\begin{array} { r } { y f ^ { \prime } ( y ) = \frac { g ( f ( y ) ) } { g ^ { \prime } ( f ( y ) ) } \in [ \frac { 1 } { 2 K } f ( y ) , 2 K f ( y ) ] } \end{array}$ for all $y \in [ g ( b _ { g } ) , + \infty )$ . □ + +Recall that (B3.4) directly implies that $f ^ { \prime } ( \Theta ( x ) ) = \Theta ( f ^ { \prime } ( x ) )$ and $g ^ { \prime } ( \Theta ( x ) ) = \Theta ( g ^ { \prime } ( x ) )$ . Combining this with Lemma D.1, we have the following corollary: + +Corollary D.2. $f ( \Theta ( x ) ) = \Theta ( f ( x ) )$ and $g ( \Theta ( x ) ) = \Theta ( g ( x ) ) .$ . + +Also, note that Lemma D.1 essentially shows that $( \log f ( x ) ) ^ { \prime } = \Theta ( 1 / x )$ and $( \log g ( x ) ) ^ { \prime } = \Theta ( 1 / x )$ . S $) \log f ( x ) = \Theta ( \log x )$ and $\log g ( x ) = \Theta ( \log x )$ , which means that $f$ and $g$ grow at most polynomially. + +Corollary D.3. $f ( x ) = x ^ { \Theta ( 1 ) }$ and $g ( x ) = x ^ { \Theta ( 1 ) }$ . + +# D.2 PROOF FOR THEOREM A.10 + +We follow the notations in Section 5 to define $\rho : = \| \pmb \theta \| _ { 2 }$ and $\begin{array} { r } { \hat { \pmb \theta } : = \frac { \pmb \theta } { \| \pmb \theta \| _ { 2 } } \in \mathcal { S } ^ { d - 1 } } \end{array}$ , and sometimes we view the functions of as functions of $t$ . And we use the notations $B _ { 0 } , B _ { 1 }$ from Appendix C.3. + +The key idea to prove Theorem A.10 is to utilize Lemma B.6, in which $\mathcal { L } ( t )$ is bounded from above by $\frac { 1 } { G ^ { - 1 } ( \Omega ( t ) ) }$ . So upper bounding $\mathcal { L } ( t )$ reduces to lower bounding $G ^ { - 1 }$ . In the following lemma, we obtain tight asymptotic bounds for $G ( \cdot )$ and $G ^ { - 1 } ( \cdot )$ : + +Lemma D.4. For function $G ( \cdot )$ defined in Lemma B.6 and its inverse function $G ^ { - 1 } ( \cdot )$ , we have the following bounds: + +$$ +G ( x ) = \Theta \left( { \frac { g ( \log x ) ^ { 2 / L } } { ( \log x ) ^ { 2 } } } x \right) \quad a n d \quad G ^ { - 1 } ( y ) = \Theta \left( { \frac { ( \log y ) ^ { 2 } } { g ( \log y ) ^ { 2 / L } } } y \right) . +$$ + +Proof. We first prove the bounds for $G ( x )$ , and then prove the bounds for $G ^ { - 1 } ( y )$ . + +Bounding for $G ( x )$ . Let $\begin{array} { r } { C _ { G } = \int _ { 1 / \mathcal { L } ( t _ { 0 } ) } ^ { \exp ( b _ { g } ) } \frac { g ^ { \prime } ( \log u ) ^ { 2 } } { g ( \log u ) ^ { 2 - 2 / L } } d u } \end{array}$ . For x ≥ exp(bg), + +$$ +\begin{array} { l } { { \displaystyle G ( x ) = C _ { G } + \int _ { \exp ( b _ { g } ) } ^ { x } \frac { g ^ { \prime } ( \log u ) ^ { 2 } } { g ( \log u ) ^ { 2 - 2 / L } } d u } } \\ { { \displaystyle \ = C _ { G } + \int _ { \exp ( b _ { g } ) } ^ { x } \left( \frac { g ^ { \prime } ( \log u ) \log u } { g ( \log u ) } \right) ^ { 2 } \frac { g ( \log u ) ^ { 2 / L } } { ( \log u ) ^ { 2 } } d u , } } \end{array} +$$ + +By Lemma D.1, + +$$ +G ( x ) \leq C _ { G } + 4 K ^ { 2 } g ( \log x ) ^ { 2 / L } \int _ { \exp ( b _ { g } ) } ^ { x } \frac { 1 } { ( \log u ) ^ { 2 } } d u = O \left( \frac { g ( \log x ) ^ { 2 / L } } { ( \log x ) ^ { 2 } } x \right) . +$$ + +On the other hand, for $x \ge \exp ( 2 b _ { g } )$ , we have + +$$ +G ( x ) \geq \frac { 1 } { 4 K ^ { 2 } } \int _ { \sqrt x } ^ { x } \frac { g ( \log u ) ^ { 2 / L } } { ( \log u ) ^ { 2 } } d u \geq \frac { g ( ( \log x ) / 2 ) ^ { 2 / L } } { 4 K ^ { 2 } } \int _ { \sqrt x } ^ { x } \frac { 1 } { ( \log u ) ^ { 2 } } d u = \Omega \left( \frac { g ( \log x ) ^ { 2 / L } } { ( \log x ) ^ { 2 } } x \right) . +$$ + +Bounding for $G ^ { - 1 } ( y )$ . Let $x = G ^ { - 1 } ( y )$ for $y \geq 0$ . $G ( x )$ always has a finite value whenever $x$ is finite. So $x \to + \infty$ when $y + \infty$ . According to the first part of the proof, we know that $\begin{array} { r } { y \ = \ \Theta \left( { \frac { g ( \log x ) ^ { 2 / L } } { ( \log x ) ^ { 2 } } } x \right) } \end{array}$ . Taking logarithm on both sides and using Corollary D.3, we have $\log y = \Theta ( \log x )$ . By Corollary D.2, $g ( \log y ) = g ( \Theta ( \log x ) ) = \Theta ( g ( \log x ) )$ . Therefore, + +$$ +{ \frac { ( \log y ) ^ { 2 } } { g ( \log y ) ^ { 2 / L } } } y = \Theta \left( { \frac { ( \log x ) ^ { 2 } } { g ( \log x ) ^ { 2 / L } } } y \right) = \Theta ( x ) . +$$ + +This implies that $\begin{array} { r } { x = \Theta \left( \frac { ( \log y ) ^ { 2 } } { g ( \log y ) ^ { 2 / L } } y \right) } \end{array}$ + +For other bounds, we derive them as follows. We first show that $\begin{array} { r } { g ( \log \frac { 1 } { \mathcal { L } } ) = \Theta ( \rho ^ { L } ) } \end{array}$ . With this equivalence, we derive an upper bound for the gradient at each time $t$ in terms of $\mathcal { L }$ , and take an integration to bound $\mathcal { L } ( t )$ from below. Now we have both lower and upper bounds for $\mathcal { L } ( t )$ . Plugging these two bounds to $\begin{array} { r } { g ( \log \frac { 1 } { \mathcal { L } } ) = \Theta ( \rho ^ { L } ) } \end{array}$ gives the lower and upper bounds for $\rho ( t )$ . + +Proof for Theorem A.10. We first prove the upper bound for $\mathcal { L }$ . Then we derive lower and upper bounds for $\rho$ in terms of $\mathcal { L }$ , and use these bounds to give a lower bound for $\mathcal { L }$ . Finally, we plug in the tight bounds for $\mathcal { L }$ to obtain the lower and upper bounds for $\rho$ in terms of $t$ . + +Upper Bounding $\mathcal { L }$ . By Lemma B.6, we have $\begin{array} { r } { \frac { 1 } { \mathcal { L } } \geq G ^ { - 1 } ( \Omega ( t ) ) } \end{array}$ . Using Lemma D.4, we have  (log t)2g(log t)2/L t, which completes the proof. + +Bounding $\rho$ in Terms of $\mathcal { L }$ . $\tilde { \gamma } ( t ) \ \geq \ \tilde { \gamma } ( t _ { 0 } )$ , so $\begin{array} { r } { \rho ^ { L } \ \leq \ \frac { 1 } { \tilde { \gamma } ( t _ { 0 } ) } g ( \log \frac { 1 } { \mathcal { L } } ) \ = \ O ( g ( \log \frac { 1 } { \mathcal { L } } ) ) } \end{array}$ . On the other hand, $g ( \log { \frac { 1 } { \mathcal { L } } } ) \leq q _ { \operatorname* { m i n } } \leq B _ { 0 } \rho ^ { L }$ . So $\begin{array} { r } { \rho ^ { L } = \Omega ( g ( \log \frac { 1 } { \mathcal { L } } ) ) } \end{array}$ . Therefore, we have the following relationship between $\rho ^ { L }$ and $g ( \log { \frac { 1 } { \mathcal { L } } } )$ : + +$$ +\rho ^ { L } = \Theta ( g ( \log \frac { 1 } { \mathcal { L } } ) ) . +$$ + +Lower Bounding $\mathcal { L }$ . Let $\pmb { h } _ { 1 } , \dots , \pmb { h } _ { N }$ be a set of vectors such that $\begin{array} { r } { h _ { n } \in { \frac { \partial q _ { n } } { \partial \theta } } } \end{array}$ and + +$$ +\frac { d \pmb { \theta } } { d t } = \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) \pmb { h } _ { n } . +$$ + +By (17) and Corollary D.2, $f ^ { \prime } ( q _ { n } ) \ = \ f ^ { \prime } ( { \cal O } ( \rho ^ { L } ) ) \ = \ f ^ { \prime } ( { \cal O } ( g ( \log \frac { 1 } { \cal L } ) ) ) \ = \ { \cal O } ( f ^ { \prime } ( g ( \log \frac { 1 } { \cal L } ) ) ) \ =$ $\begin{array} { r } { O ( 1 / g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) ) } \end{array}$ . Again by (17), we have $\| h _ { n } \| _ { 2 } \le B _ { 1 } \rho ^ { L - 1 } = O ( g ( \log { \frac { 1 } { \mathcal { L } } } ) ^ { 1 - 1 / L } )$ . Combining these two bounds together, it follows from Corollary D.2 that + +$$ +f ^ { \prime } ( q _ { n } ) \left\| h _ { n } \right\| _ { 2 } = O \left( \frac { g ( \log \frac { 1 } { \mathcal { L } } ) ^ { 1 - 1 / L } } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } \right) = O \left( \frac { \log \frac { 1 } { \mathcal { L } } } { g ( \log \frac { 1 } { \mathcal { L } } ) ^ { 1 / L } } \right) . +$$ + +Thus, + +$$ +\begin{array} { l } { \displaystyle - \frac { d \mathcal { L } } { d t } = \left\| \frac { d \theta } { d t } \right\| _ { 2 } ^ { 2 } = \left\| \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \prime } ( q _ { n } ) h _ { n } \right\| _ { 2 } ^ { 2 } } \\ { \displaystyle \qquad \leq \left( \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } \cdot \displaystyle \operatorname* { m a x } _ { n \in [ N ] } \left\{ f ^ { \prime } ( q _ { n } ) \left\| h _ { n } \right\| _ { 2 } \right\} \right) ^ { 2 } \leq \mathcal { L } ^ { 2 } \cdot O \left( \frac { ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 } } { g ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 / L } } \right) . } \end{array} +$$ + +By definition of $G ( \cdot )$ , this implies that there exists a constant $c$ such that $\begin{array} { r } { \frac { d } { d t } G ( \frac { 1 } { \mathcal { L } } ) \leq c } \end{array}$ for any $\mathcal { L }$ that is small enough. We can complete our proof by applying Lemma D.4. + +Bounding $\rho$ in Terms of $t$ . By (17) and the tight bound for $\mathcal { L } ( t )$ $, ~ \rho ^ { L } ~ = ~ \Theta ( g ( \log \frac { 1 } { \mathcal { L } } ) ) ~ =$ $\Theta ( g ( \Theta ( \log t ) ) )$ ). Using Corollary D.2, we can conclude that $\rho ^ { L } = \Theta ( g ( \log t ) )$ . □ + +# E GRADIENT DESCENT: SMOOTH HOMOGENEOUS MODELS WITH EXPONENTIAL LOSS + +In this section, we discretize our proof to prove similar results for gradient descent on smooth homogeneous models. As usual, the update rule of gradient descent is defined as + +$$ +\pmb \theta ( t + 1 ) = \pmb \theta ( t ) - \eta ( t ) \nabla \mathcal L ( t ) +$$ + +Here $\eta ( t )$ is the learning rate, and $\nabla \mathcal { L } ( t ) : = \nabla \mathcal { L } ( \pmb { \theta } ( t ) )$ is the gradient of $\mathcal { L }$ at $\pmb \theta ( t )$ . + +We first focus on the exponential loss. At the end of this section (Appendix F), we discuss how to extend the proof to general loss functions with a similar assumption as (B3). + +The main difficulty for discretizing our previous analysis comes from the fact that the original version of the smoothed normalized margin $\begin{array} { r } { \tilde { \gamma } ( \pmb { \theta } ) : = \rho ^ { - \tilde { L } } \log \frac { 1 } { \mathcal { L } } } \end{array}$ becomes less smooth when $\rho \to + \infty$ . Thus, if we take a Taylor expansion for $\tilde { \gamma } ( \pmb { \theta } ( t + 1 ) )$ from the point $\pmb \theta ( t )$ , although one can show that the first-order term is positive as in the gradient flow analysis, the second-order term is unlikely to be bounded during gradient descent with a constant step size. To get a smoothed version of the normalized margin that is monotone increasing, we need to define another one that is even smoother than $\tilde { \gamma }$ . + +Technically, recall that $\begin{array} { r } { \frac { d \mathcal { L } } { d t } = - \| \nabla \mathcal { L } \| _ { 2 } ^ { 2 } } \end{array}$ does not hold exactly for gradient descent. However, if the smoothness can be bounded by $s ( t )$ , then it is well-known that + +$$ +\mathcal { L } ( t + 1 ) - \mathcal { L } ( t ) \leq - \eta ( t ) ( 1 - s ( t ) \eta ( t ) ) \| \nabla \mathcal { L } \| _ { 2 } ^ { 2 } . +$$ + +By analyzing the landscape of $\mathcal { L }$ , one can easily find that the smoothness is bounded locally by $\dot { O ( } \mathcal { L } \cdot \mathrm { p o l y l o g } ( \frac { 1 } { \mathcal { L } } ) \big )$ . Thus, if we set $\eta ( t )$ to a constant or set it appropriately according to the loss, then this discretization error becomes negligible. Using this insight, we define the new smoothed normalized margin $\hat { \gamma }$ in a way that it increases slightly slower than $\tilde { \gamma }$ during training to cancel the effect of discretization error. + +# E.1 ASSUMPTIONS + +As stated in Section 4.1, we assume (A2), (A3), (A4) similarly as for gradient flow, and two additional assumptions (S1) and (S5). + +(S1). (Smoothness). For any fixed $_ { \textbf { \em x } }$ , $\Phi ( \cdot ; x )$ is $\mathcal { C } ^ { 2 }$ -smooth on $\mathbb { R } ^ { d } \backslash \{ \mathbf { 0 } \}$ . +(A2). (Homogeneity). There exists $L > 0$ such that $\forall \alpha > 0 : \Phi ( \alpha \pmb { \theta } ; \pmb { x } ) = \alpha ^ { L } \Phi ( \pmb { \theta } ; \pmb { x } ) ;$ ; +(A3). (Exponential Loss). $\ell ( q ) = e ^ { - q }$ ; +(A4). (Separability). There exists a time $t _ { 0 }$ such that $\mathcal { L } ( \pmb { \theta } ( t _ { 0 } ) ) < 1$ . +(S5). (Learing rate condition). $\begin{array} { r } { \sum _ { t \geq 0 } \eta ( t ) = + \infty } \end{array}$ and $\eta ( t ) \leq H ( \mathcal { L } ( \pmb { \theta } ( t ) ) )$ . + +Here $H ( \mathcal { L } )$ is a function of the current training loss. The explicit formula of $H ( \mathcal { L } )$ is given below: + +$$ +H ( x ) : = \frac { \mu ( x ) } { C _ { \eta } \kappa ( x ) } = \Theta \left( \frac { 1 } { x ( \log \frac { 1 } { x } ) ^ { 3 - 2 / L } } \right) , +$$ + +where $C _ { \eta }$ is a constant, and $\kappa ( \boldsymbol { x } ) , \mu ( \boldsymbol { x } )$ are two non-decreasing functions. For constant learning rate $\eta ( t ) = \eta _ { 0 }$ , (S5) is satisfied when $\eta _ { 0 }$ if sufficiently small. + +Roughly speaking, $C _ { \eta } \kappa ( \boldsymbol { x } )$ is an upper bound for the smoothness of $\mathcal { L }$ in a neighborhood of $\pmb \theta$ when $x = { \mathcal { L } } ( \theta )$ . And we set the learning rate $\eta ( t )$ to be the inverse of the smoothness multiplied by a factor $\dot { \mu ( x ) } = o ( 1 )$ . In our analysis, $\mu ( x )$ can be any non-decreasing function that maps $\left[ 0 , \mathcal { L } ( \dot { t } _ { 0 } ) \right]$ to $( 0 , 1 / 2 ]$ and makes the integral $\begin{array} { r } { \int _ { 0 } ^ { 1 / 2 } \mu ( x ) d x } \end{array}$ exist. But for simplicity, we define $\mu ( x )$ as + +$$ +\mu ( x ) : = \frac { \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } } { 2 \log \frac { 1 } { x } } = \Theta \left( \frac { 1 } { \log \frac { 1 } { x } } \right) . +$$ + +The value of $C _ { \eta }$ will be specified later. The definition of $\kappa ( x )$ depends on $L$ . For $0 < L \leq 1$ , we define $\kappa ( x )$ as + +$$ +\kappa ( x ) : = x \left( \log \frac { 1 } { x } \right) ^ { 2 - 2 / L } +$$ + +For $L > 1$ , we define $\kappa ( x )$ as + +$$ +\kappa ( x ) : = \left\{ \begin{array} { l l } { x \left( \log \frac { 1 } { x } \right) ^ { 2 - 2 / L } \quad } & { x \in ( 0 , e ^ { 2 / L - 2 } ] } \\ { \kappa _ { \operatorname* { m a x } } } & { x \in ( e ^ { 2 / L - 2 } , 1 ) } \end{array} \right. +$$ + +where $\kappa _ { \mathrm { m a x } } : = e ^ { ( 2 - 2 / L ) ( \ln ( 2 - 2 / L ) - 1 ) }$ . The specific meaning of $C _ { \eta } , \kappa ( x )$ and $\mu ( x )$ will become clear in our analysis. + +# E.2 SMOOTHED NORMALIZED MARGIN + +Now we define the smoothed normalized margins. As usual, we define $\begin{array} { r } { \widetilde \gamma ( \pmb { \theta } ) : = \frac { \log \frac { 1 } { \mathcal { L } } } { \rho ^ { L } } } \end{array}$ . At the same time, we also define + +$$ +\hat { \gamma } ( \pmb { \theta } ) : = \frac { e ^ { \phi ( \mathcal { L } ) } } { \rho ^ { L } } . +$$ + +Here $\phi : ( 0 , \mathcal { L } ( t _ { 0 } ) ] \to ( 0 , + \infty )$ is constructed as follows. Construct the first-order derivative of $\phi ( x )$ as + +$$ +\phi ^ { \prime } ( x ) = - \operatorname* { s u p } \left\{ \frac { 1 + 2 ( 1 + \lambda ( w ) / L ) \mu ( w ) } { w \log \frac { 1 } { w } } : w \in \left[ x , \mathcal { L } ( t _ { 0 } ) \right] \right\} , +$$ + +where $\begin{array} { r } { \lambda ( x ) : = ( \log \frac { 1 } { x } ) ^ { - 1 } } \end{array}$ . And then we set $\phi ( x )$ to be + +$$ +\phi ( x ) = \log \log { \frac { 1 } { x } } + \int _ { 0 } ^ { x } \left( \phi ^ { \prime } ( w ) + { \frac { 1 } { w \log { \frac { 1 } { w } } } } \right) d w . +$$ + +It can be verified that $\phi ( x )$ is well-defined and $\phi ^ { \prime } ( x )$ is indeed the first-order derivative of $\phi ( x )$ . +Moreover, we have the following relationship among $\hat { \gamma } , \tilde { \gamma }$ and $\bar { \gamma }$ . + +Lemma E.1. $\hat { \gamma } ( \pmb \theta )$ is well-defined for $\mathcal { L } ( \pmb { \theta } ) \leq \mathcal { L } ( t _ { 0 } )$ and has the following properties: + +(a) If $\mathcal { L } ( \pmb { \theta } ) \leq \mathcal { L } ( t _ { 0 } )$ , then $\hat { \gamma } ( \pmb \theta ) < \tilde { \gamma } ( \pmb \theta ) \leq \bar { \gamma } ( \pmb \theta ) .$ . + +$( b )$ Let $\{ \pmb \theta _ { m } \in \mathbb { R } ^ { d } : m \in \mathbb { N } \}$ be a sequence of parameters. If $\mathcal { L } ( \pmb { \theta } _ { m } ) 0$ , then + +$$ +\frac { \hat { \gamma } ( \pmb \theta _ { m } ) } { \bar { \gamma } ( \pmb \theta _ { m } ) } = 1 - O ( ( \log \frac { 1 } { \mathcal { L } ( \pmb \theta _ { m } ) } ) ^ { - 1 } ) 1 . +$$ + +Proof. First we verify that $\hat { \gamma }$ is well-defined. To see this, we only need to verify that + +$$ +I ( x ) : = \int _ { 0 } ^ { x } \left( \phi ^ { \prime } ( w ) + { \frac { 1 } { w \log { \frac { 1 } { w } } } } \right) d w +$$ + +exists for all $x \in ( 0 , \mathcal { L } ( t _ { 0 } ) ]$ , then it is trivial to see that $\phi ^ { \prime } ( w )$ is indeed the derivative of $\phi ( w )$ by I 0 ( x ) = φ 0 ( x ) + 1x log 1 . + +Note that $I ( x )$ exists for all $x \in ( 0 , \mathcal { L } ( t _ { 0 } ) ]$ as long as $I ( x )$ exists for a small enough $x > 0$ . By definition, it is easy to verify that $\begin{array} { r } { r ( w ) : = \frac { 1 + 2 ( 1 + \lambda ( w ) / L ) \mu ( w ) } { w \log \frac { 1 } { w } } } \end{array}$ is decreasing when $w$ is small enough. Thus, for a small enough $w > 0$ , we have + +$$ +- \phi ^ { \prime } ( w ) = r ( w ) = \frac { 1 + 2 ( 1 + \frac { 1 } { L } ( \log \frac { 1 } { w } ) ^ { - 1 } ) \frac { \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } } { 2 \log \frac { 1 } { w } } } { w \log \frac { 1 } { w } } = \frac { 1 } { w \log \frac { 1 } { w } } + \frac { \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } ( 1 + \frac { 1 } { L } ( \log \frac { 1 } { w } ) ^ { - 1 } ) } { w ( \log \frac { 1 } { w } ) ^ { 2 } } . +$$ + +So we have the following for small enough $x$ : + +$$ +\begin{array} { r } { I ( x ) = \displaystyle \int _ { 0 } ^ { x } - \frac { \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } ( 1 + \frac { 1 } { L } ( \log \frac { 1 } { w } ) ^ { - 1 } ) } { w ( \log \frac { 1 } { w } ) ^ { 2 } } d w = - \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } ( \frac { 1 } { \log \frac { 1 } { w } } + \frac { 1 } { 2 L ( \log \frac { 1 } { w } ) ^ { 2 } } ) _ { 0 } ^ { x } } \\ { \displaystyle = - \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } ( \frac { 1 } { \log \frac { 1 } { x } } + \frac { 1 } { 2 L ( \log \frac { 1 } { x } ) ^ { 2 } } ) . } \end{array} +$$ + +This proves the existence of $I ( x )$ . + +Now we prove (a). By Lemma A.5, $\tilde { \gamma } ( \pmb \theta ) \leq \tilde { \gamma } ( \pmb \theta )$ , so we only need to prove that $\hat { \gamma } _ { \perp } ( \pmb { \theta } ) < \tilde { \gamma } ( \pmb { \theta } )$ . To see this, note that for all $w \leq \mathcal { L } ( t _ { 0 } ) , r ( w ) > \frac { 1 } { w \log \frac { 1 } { w } }$ . So we have $- \bar { \phi } ^ { \prime } ( w ) > \frac { 1 } { w \log \frac { 1 } { w } }$ 1g 1w , and this implies $I ( x ) < 0$ for all $x \le \mathcal { L } ( t _ { 0 } )$ . Then it holds for all $\pmb \theta$ with $\mathcal { L } ( \pmb { \theta } ) \leq \mathcal { L } ( t _ { 0 } )$ that + +$$ +\frac { \widehat { \gamma } ( \pmb \theta ) } { \widetilde { \gamma } ( \pmb \theta ) } = \frac { e ^ { \phi ( \mathcal { L } ( \pmb \theta ) ) } } { \log \frac { 1 } { \mathcal { L } ( \pmb \theta ) } } = \frac { e ^ { \log \log \frac { 1 } { \mathcal { L } ( \pmb \theta ) } + I ( \mathcal { L } ( \pmb \theta ) ) } } { e ^ { \log \log \frac { 1 } { \mathcal { L } ( \pmb \theta ) } } } = e ^ { I ( \mathcal { L } ( \pmb \theta ) ) } < 0 . +$$ + +To prove (b), we combine (19) and (20), then for small enough $\mathcal { L } ( \pmb { \theta } _ { m } )$ , we have + +$$ +\begin{array} { r l } & { \frac { \hat { \gamma } ( \theta ) } { \hat { \gamma } ( \theta ) } = \frac { \hat { \gamma } ( \theta ) } { \hat { \gamma } ( \theta ) } \cdot \frac { \hat { \gamma } ( \theta ) } { \hat { \gamma } ( \theta ) } } \\ & { \quad \quad \quad = \exp \left( - \log \frac { 1 } { \mathcal { L } ( { t _ { 0 } } ) } \left( \frac { 1 } { \log \frac { 1 } { \mathcal { L } ( \theta _ { m } ) } } + \frac { 1 } { 2 L ( \log \frac { 1 } { \mathcal { L } ( \theta _ { m } ) } ) ^ { 2 } } \right) \right) \cdot \frac { \log \frac { 1 } { \mathcal { L } ( \theta _ { m } ) } } { q _ { \operatorname* { m i n } } ( \theta _ { m } ) } } \\ & { \quad \quad \quad = \exp \left( - \mathcal { O } \left( \frac { 1 } { \log \frac { 1 } { \mathcal { L } ( \theta _ { m } ) } } \right) \right) \cdot \frac { \log \frac { 1 } { \mathcal { L } ( \theta _ { m } ) } } { \log \frac { 1 } { \mathcal { L } ( \theta _ { m } ) } + \mathcal { O } ( 1 ) } } \\ & { \quad \quad \quad = 1 - \mathcal { O } \left( \left( \log \frac { 1 } { \mathcal { L } ( \theta _ { m } ) } \right) ^ { - 1 } \right) . } \end{array} +$$ + +Now we specify the value of $C _ { \eta }$ . By (S1) and (S2), we can define $B _ { 0 } , B _ { 1 } , B _ { 2 }$ as follows: + +$$ +\begin{array} { r l } & { B _ { 0 } : = \operatorname* { s u p } \left\{ q _ { n } : \pmb \theta \in S ^ { d - 1 } , n \in [ N ] \right\} , } \\ & { B _ { 1 } : = \operatorname* { s u p } \left\{ \left\| \nabla q _ { n } \right\| _ { 2 } : \pmb \theta \in S ^ { d - 1 } , n \in [ N ] \right\} , } \\ & { B _ { 2 } : = \operatorname* { s u p } \left\{ \left\| \nabla ^ { 2 } q _ { n } \right\| _ { 2 } : \pmb \theta \in S ^ { d - 1 } , n \in [ N ] \right\} . } \end{array} +$$ + +Then we set $\begin{array} { r l } & { C _ { \eta } : = \frac { 1 } { 2 } \left( B _ { 1 } ^ { 2 } + \rho ( t _ { 0 } ) ^ { - L } B _ { 2 } \right) \operatorname* { m i n } \Big \{ \hat { \gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \Big \} . } \end{array}$ + +# E.3 THEOREMS + +Now we state our main theorems for the monotonicity of the normalized margin and the convergence to KKT points. We will prove Theorem E.2 in Appendix E.4, and prove Theorem E.3 and E.4 in Appendix E.5. + +Theorem E.2. Under assumptions (S1), (A2) - (A4), (S5), the following are true for gradient descent: + +1. For all $t \geq t _ { 0 }$ , $\hat { \gamma } ( t + 1 ) \geq \hat { \gamma } ( t )$ ; + +2. For all $t \geq t _ { 0 }$ , either $\hat { \gamma } ( t + 1 ) > \hat { \gamma } ( t )$ or $\hat { \pmb { \theta } } ( t + 1 ) = \hat { \pmb { \theta } } ( t )$ ; + +3. $\mathcal { L } ( t ) 0$ and $\rho ( t ) \infty$ as $t \to + \infty$ ; therefore, $| \bar { \gamma } ( t ) - \tilde { \gamma } ( t ) | 0$ . + +Theorem E.3. Consider gradient flow under assumptions (S1), (A2) - (A4), (S5). For every limit point $\bar { \pmb { \theta } }$ of $\left\{ { \hat { \pmb { \theta } } } ( t ) : t \geq 0 \right\}$ , $\bar { \theta } / q _ { \mathrm { m i n } } ( \bar { \theta } ) ^ { 1 / L }$ is a KKT point of $( P )$ . + +Theorem E.4. Consider gradient descent under assumptions (S1), (A2) - (A4), (S5). For any $\epsilon , \delta >$ 0, there exists $r : = \Theta ( \log { \delta ^ { - 1 } } )$ and $\Delta : = \Theta ( \epsilon ^ { - 2 } )$ such that $\pmb { \theta } / q _ { \mathrm { m i n } } ( \pmb { \theta } ) ^ { 1 / L }$ is an $( \epsilon , \delta )$ -KKT point at some time $t _ { * }$ satisfying $\log \rho ( t _ { * } ) \in ( r , r + \Delta )$ . + +With a refined analysis, we can also derive tight rates for loss convergence and weight growth. We defer the proof to Appendix E.6. + +Theorem E.5. Under assumptions (S1), (A2) - (A4), (S5), we have the following tight rates for training loss and weight norm: + +$$ +\mathcal { L } ( t ) = \Theta \left( \frac { 1 } { T ( \log T ) ^ { 2 - 2 / L } } \right) \quad a n d \quad \rho ( t ) = \Theta ( ( \log T ) ^ { 1 / L } ) , +$$ + +where $\begin{array} { r } { T = \sum _ { \tau = t _ { 0 } } ^ { t - 1 } \eta ( \tau ) } \end{array}$ + +# E.4 PROOF FOR THEOREM E.2 + +Wefor $\begin{array} { r } { \nu ( t ) : = \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( t ) } q _ { n } ( t ) } \end{array}$ as we do for gradient flow. Then we can get a clary B.3. Also, we can get a lower bound for d formusing $\langle \pmb { \theta } ( t ) , - \nabla \mathcal { L } ( t ) \rangle$ $\nu ( t )$ Lemma B.5 for exponential loss directly. + +Corollary E.6. hθ(t), −∇L(t)i = Lν(t). If L(t) < 1, then ν(t) ≥ L(t)λ(L(t)) . + +As we are analyzing gradient descent, the norm of $\nabla \mathcal { L }$ and $\nabla ^ { 2 } \boldsymbol { \mathcal { L } }$ appear naturally in discretization error terms. To bound them, we have the following lemma when $\widetilde \gamma ( \pmb \theta )$ and $\rho ( \pmb \theta )$ have lower bounds: + +Lemma E.7. For any $\pmb \theta , i f \tilde { \gamma } ( \pmb \theta ) \geq \hat { \gamma } ( t _ { 0 } ) , \rho ( \pmb \theta ) \geq \rho ( t _ { 0 } )$ , then + +$$ +\begin{array} { r } { \| \nabla \mathcal { L } ( \pmb { \theta } ) \| _ { 2 } ^ { 2 } \leq 2 C _ { \eta } \kappa ( \mathcal { L } ( \pmb { \theta } ) ) \mathcal { L } ( \pmb { \theta } ) \quad a n d \quad \left\| \nabla ^ { 2 } \mathcal { L } ( \pmb { \theta } ) \right\| _ { 2 } \leq 2 C _ { \eta } \kappa ( \mathcal { L } ( \pmb { \theta } ) ) . } \end{array} +$$ + +Proof. By the chain rule and the definitions of $B _ { 1 } , B _ { 2 }$ , we have + +$$ +\begin{array} { r l } & { \displaystyle \left\| \nabla \mathcal { L } \right\| _ { 2 } = \left\| - \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \nabla q _ { n } \right\| _ { 2 } \leq \mathcal { L } \rho ^ { L - 1 } B _ { 1 } } \\ & { \displaystyle \left\| \nabla ^ { 2 } \mathcal { L } \right\| _ { 2 } = \left\| \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \big ( \nabla q _ { n } \nabla q _ { n } ^ { \top } - \nabla ^ { 2 } q _ { n } \big ) \right\| _ { 2 } } \\ & { \qquad \leq \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \left( B _ { 1 } ^ { 2 } \rho ^ { 2 L - 2 } + B _ { 2 } \rho ^ { L - 2 } \right) \leq \mathcal { L } \rho ^ { 2 L - 2 } \left( B _ { 1 } ^ { 2 } + \rho ^ { - L } B _ { 2 } \right) . } \end{array} +$$ + +Note that $\begin{array} { r } { \hat { \gamma } ( t _ { 0 } ) \rho ^ { L } \leq \tilde { \gamma } \rho ^ { L } \leq \log \frac { 1 } { \mathcal { L } } \leq B _ { 0 } \rho ^ { L } } \end{array}$ . So $\begin{array} { r } { \hat { \gamma } ( t _ { 0 } ) ^ { - 1 } \log \frac { 1 } { \mathcal { L } } \leq \rho ^ { L } \leq B _ { 0 } ^ { - 1 } \log \frac { 1 } { \mathcal { L } } } \end{array}$ . Combining all these formulas together gives + +$$ +\begin{array} { r l } & { \left\| \nabla { \mathcal { L } } \right\| _ { 2 } ^ { 2 } \leq B _ { 1 } ^ { 2 } \operatorname* { m i n } \left\{ \widehat { \gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \right\} { \mathcal { L } } ^ { 2 } \cdot \left( \log \displaystyle \frac { 1 } { \mathcal { L } } \right) ^ { 2 - 2 / L } \leq 2 C _ { \eta } \kappa ( { \mathcal { L } } ) \mathcal { L } } \\ & { \left\| \nabla ^ { 2 } { \mathcal { L } } \right\| _ { 2 } \leq \left( B _ { 1 } ^ { 2 } + \rho ^ { - L } B _ { 2 } \right) \operatorname* { m i n } \left\{ \widehat { \gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \right\} { \mathcal { L } } \cdot \left( \log \displaystyle \frac { 1 } { \mathcal { L } } \right) ^ { 2 - 2 / L } \leq 2 C _ { \eta } \kappa ( { \mathcal { L } } ) , } \end{array} +$$ + +which completes the proof. + +For proving the first two propositions in Theorem E.2, we only need to prove Lemma E.8. (P1) gives a lower bound for $\tilde { \gamma }$ . (P2) gives both lower and upper bounds for the weight growth using $\nu ( t )$ . (P3) gives a lower bound for the decrement of training loss. Finally, (P4) shows the monotonicity of $\hat { \gamma }$ , and it is trivial to deduce the first two propositions in Theorem E.2 from (P4). + +Lemma E.8. For all $t = t _ { 0 } , t _ { 0 } + 1 , \ldots$ , we interpolate between $\pmb \theta ( t )$ and $\pmb \theta ( t + 1 )$ by defining $\pmb \theta ( t + \alpha ) = \pmb \theta ( t ) - \alpha \eta ( t ) \nabla \mathcal L ( t )$ for $\alpha \in ( 0 , 1 )$ . Then for all integer $t \geq t _ { 0 }$ , $\nu ( t ) > 0$ , and the following holds for all $\alpha \in [ 0 , 1 ]$ : + +$$ +\begin{array} { r l } & { ( P I ) . \ \tilde { \gamma } ( t + \alpha ) > \hat { \gamma } ( t _ { 0 } ) . } \\ & { ( P 2 ) . \ 2 L \alpha \eta ( t ) \nu ( t ) \leq \rho ( t + \alpha ) ^ { 2 } - \rho ( t ) ^ { 2 } \leq 2 L \alpha \eta ( t ) \nu ( t ) \left( 1 + \frac { \lambda ( \mathcal { L } ( t ) ) \mu ( \mathcal { L } ( t ) ) } { L } \right) . } \\ & { ( P 3 ) . \ \mathcal { L } ( t + \alpha ) - \mathcal { L } ( t ) \leq - \alpha \eta ( t ) ( 1 - \mu ( \mathcal { L } ( t ) ) ) \left. \nabla \mathcal { L } ( t ) \right. _ { 2 } ^ { 2 } . } \\ & { ( P 4 ) . \ \log \hat { \gamma } ( t + \alpha ) - \log \hat { \gamma } ( t ) \geq \frac { \rho ( t ) ^ { 2 } } { L \nu ( t ) ^ { 2 } } \left. \left( I - \hat { \theta } ( t ) \hat { \theta } ( t ) ^ { \top } \right) \nabla \mathcal { L } ( t ) \right. _ { 2 } ^ { 2 } . \log \frac { \rho ( t + \alpha ) } { \rho ( t ) } . } \end{array} +$$ + +To prove Lemma E.8, we only need to prove the following lemma and then use an induction: + +Lemma E.9. Fix an integer $T \geq t _ { 0 }$ . Suppose that $( P I ) , ( P 2 ) , ( P 3 ) , ( P 4 )$ hold for any $t + \alpha \leq T$ . Then $i f ( P l )$ holds for $( t , \alpha ) \in \{ T \} \times [ 0 , A )$ for some $A \in ( 0 , 1 ] ,$ , then all of $( P I ) , ( P 2 ) , ( P 3 ) , ( P 4 )$ hold for $( t , \alpha ) \in \{ T \} \times [ 0 , A ]$ . + +Proof for Lemma E.8. We prove this lemma by induction. For $t = t _ { 0 } , \alpha = 0$ , $\nu ( t ) > 0$ by (S4) and Corollary E.6. (P2), (P3), (P4) hold trivially since $\log \hat { \gamma } ( t + \alpha ) = \log \hat { \gamma } ( t )$ , $\textstyle { \mathcal { L } } ( t + \alpha ) = { \mathcal { L } } ( t )$ and $\log \hat { \gamma } ( t + \alpha ) = \log \hat { \gamma } ( t )$ . By Lemma E.1, (P1) also holds trivially. + +Now we fix an integer $T \geq t _ { 0 }$ and assume that (P1), (P2), (P3), (P4) hold for any $t + \alpha \leq T$ (where $t \geq t _ { 0 }$ is an integer and $\alpha \in [ 0 , 1 ] ,$ ). By (P3), $\mathcal { L } ( t ) \leq \mathcal { L } ( t _ { 0 } ) < 1$ , so $\nu ( t ) > 0$ . We only need to show that (P1), (P2), (P3), (P4) hold for $t = T$ and $\alpha \in [ 0 , 1 ]$ . + +Let $A : = \operatorname* { i n f } \{ \alpha \in [ 0 , 1 ] : \alpha = 1$ or (P1) does not hold for $( T , \alpha ) \}$ . If $A = 0$ , then (P1) holds for $( T , A )$ since (P1) holds for $( T - 1 , 1 )$ ; if $A > 0$ , we can also know that (P1) holds for $( T , A )$ by Lemma E.9. Suppose that $A < 1$ . Then by the continuity of $\tilde { \gamma } ( T + \alpha )$ (with respect to $\alpha$ ), we know that there exists $A ^ { \prime } > A$ such that $\tilde { \gamma } ( T + \alpha ) > \hat { \gamma } ( t _ { 0 } )$ for all $\alpha \in [ A , A ^ { \prime } ]$ , which contradicts to the definition of $A$ . Therefore, $A = 1$ . Using Lemma E.9 again, we can conclude that (P1), (P2), (P3), (P4) hold for $t = T$ and $\alpha \in [ 0 , 1 ]$ . □ + +Now we turn to prove Lemma E.9. + +Proof for Lemma E.9. Applying (P3) on $( t , \alpha ) \in \{ t _ { 0 } , \ldots , T - 1 \} \times 1$ , we have ${ \mathcal { L } } ( t ) \leq { \mathcal { L } } ( t _ { 0 } ) < 1$ . Then by Corollary E.6, we have $\nu ( t ) > 0$ . Applying (P2) on $( t , \alpha ) \in \{ t _ { 0 } , \ldots , T - 1 \} \times 1$ , we can get $\rho ( t ) \geq \rho ( t _ { 0 } )$ . + +Fix $t = T$ . By (P2) with $\alpha \in [ 0 , A )$ and the continuity of $\tilde { \gamma }$ , we have $\tilde { \gamma } ( t + A ) \geq \hat { \gamma } ( t _ { 0 } )$ . Thus, $\tilde { \gamma } ( t + \alpha ) \geq \hat { \gamma } ( t _ { 0 } )$ for all $\alpha \in [ 0 , A ]$ . + +Proof for (P2). By definition, $\begin{array} { r } { \rho ( t + \alpha ) ^ { 2 } = \| \pmb { \theta } ( t ) - \alpha \eta ( t ) \nabla \mathcal { L } ( t ) \| _ { 2 } ^ { 2 } = \rho ( t ) ^ { 2 } + \alpha ^ { 2 } \eta ( t ) ^ { 2 } \left\| \nabla \mathcal { L } ( t ) \right\| _ { 2 } ^ { 2 } - } \end{array}$ $2 \alpha \eta ( t ) \left. \pmb { \theta } ( t ) , \nabla \mathcal { L } ( t ) \right.$ . By Corollary E.6, we have + +$$ +\rho ( t + \alpha ) ^ { 2 } - \rho ( t ) ^ { 2 } = \alpha ^ { 2 } \eta ( t ) ^ { 2 } \left. \nabla \mathcal { L } ( t ) \right. _ { 2 } ^ { 2 } + 2 L \alpha \eta ( t ) \nu ( t ) . +$$ + +So $\rho ( t + \alpha ) ^ { 2 } - \rho ( t ) ^ { 2 } \geq 2 L \alpha \eta ( t ) \nu ( t )$ . For the other direction, we have the following using Corollary E.6 and Lemma E.7, + +$$ +\begin{array} { r l } & { \rho ( t + \alpha ) ^ { 2 } - \rho ( t ) ^ { 2 } = 2 L \alpha \eta ( t ) \nu ( t ) \left( 1 + \frac { \alpha \eta ( t ) \left\| \nabla \mathcal { L } ( t ) \right\| _ { 2 } ^ { 2 } } { 2 L \nu ( t ) } \right) } \\ & { \qquad \le 2 L \alpha \eta ( t ) \nu ( t ) \left( 1 + \frac { C _ { \eta } ^ { - 1 } \kappa ( \mathcal { L } ( t ) ) ^ { - 1 } \mu ( \mathcal { L } ( t ) ) \cdot 2 C _ { \eta } \kappa ( \mathcal { L } ( t ) ) \mathcal { L } ( t ) } { 2 L \mathcal { L } ( t ) \lambda ( \mathcal { L } ( t ) ) ^ { - 1 } } \right) } \\ & { \qquad = 2 L \alpha \eta ( t ) \nu ( t ) \left( 1 + \lambda ( \mathcal { L } ( t ) ) \mu ( \mathcal { L } ( t ) ) / L \right) . } \end{array} +$$ + +Proof for (P3). (P3) holds trivially for $\alpha = 0$ or $\nabla \mathcal { L } ( t ) = \mathbf { 0 }$ . So now we assume that $\alpha \neq 0$ and $\nabla \mathcal { L } ( t ) \neq \mathbf { 0 }$ . By the update rule (18) and Taylor expansion, there exists $\xi \in ( 0 , \alpha )$ such that + +$$ +\begin{array} { r l } & { \mathrel { \mathop : } ( t + \alpha ) = \mathcal { L } ( t ) + ( \theta ( t + \alpha ) - \theta ( t ) ) ^ { \top } \nabla \mathcal { L } ( t ) + \frac { 1 } { 2 } ( \theta ( t + \alpha ) - \theta ( t ) ) ^ { \top } \nabla ^ { 2 } \mathcal { L } ( t + \xi ) ( \theta ( t + \alpha ) - \theta ( t ) ) } \\ & { \qquad \le \mathcal { L } ( t ) - \alpha \eta ( t ) \| \nabla \mathcal { L } ( t ) \| _ { 2 } ^ { 2 } + \frac { 1 } { 2 } \alpha ^ { 2 } \eta ( t ) ^ { 2 } \| \nabla ^ { 2 } \mathcal { L } ( t + \xi ) \| _ { 2 } \| \nabla \mathcal { L } ( t ) \| _ { 2 } ^ { 2 } } \\ & { \qquad = \mathcal { L } ( t ) - \alpha \eta ( t ) ( 1 - \frac { 1 } { 2 } \alpha \eta ( t ) ) \| \nabla ^ { 2 } \mathcal { L } ( t + \xi ) \| _ { 2 } ) \| \nabla \mathcal { L } ( t ) \| _ { 2 } ^ { 2 } . } \end{array} +$$ + +By Lemma E.7, $\left\| \nabla ^ { 2 } \mathcal { L } ( t + \xi ) \right\| _ { 2 } \leq 2 C _ { \eta } \cdot \kappa ( \mathcal { L } ( t + \xi ) )$ , so we have + +$$ +\mathcal { L } ( t + \alpha ) \leq \mathcal { L } ( t ) - \alpha \eta ( t ) \left( 1 - \alpha C _ { \eta } \eta ( t ) \kappa ( \mathcal { L } ( t + \xi ) ) \right) \left. \nabla \mathcal { L } ( t ) \right. _ { 2 } ^ { 2 } . +$$ + +Now we only need to show that $\mathscr { L } ( t + \alpha ) < \mathscr { L } ( t )$ for all $\alpha \in ( 0 , A ]$ . Assuming this, we can have $\kappa ( \mathcal { L } ( t + \xi ) ) \leq \kappa ( \mathcal { L } ( t ) )$ by the monotonicity of $\kappa$ , and thus + +$$ +\begin{array} { r l } & { \mathcal { L } ( t + \alpha ) \leq \mathcal { L } ( t ) - \alpha \eta ( t ) \left( 1 - \alpha C _ { \eta } \eta ( t ) \kappa ( \mathcal { L } ( t ) ) \right) \left. \nabla \mathcal { L } ( t ) \right. _ { 2 } ^ { 2 } } \\ & { \qquad \leq \mathcal { L } ( t ) - \alpha \eta ( t ) \left( 1 - \mu ( \mathcal { L } ( t ) ) \right) \left. \nabla \mathcal { L } ( t ) \right. _ { 2 } ^ { 2 } . } \end{array} +$$ + +Now we show that $\mathcal { L } ( t + \alpha ) ~ < ~ \mathcal { L } ( t )$ for all $\alpha \in \mathsf { \Gamma } ( 0 , A ]$ . Assume to the contrary that $\alpha _ { 0 } : =$ $\operatorname* { i n f } \{ \alpha ^ { \prime } \in ( 0 , A ] : { \mathcal { L } } ( t + \alpha ^ { \prime } ) \geq { \mathcal { L } } ( t ) \}$ exists. If $\alpha _ { 0 } = 0$ , let $p : [ 0 , 1 ] \to \mathbb { R } , \alpha \mapsto \mathcal { L } ( t + \alpha )$ , then $\begin{array} { r } { p ^ { \prime } ( 0 ) = \operatorname* { l i m } _ { \alpha \downarrow 0 } \frac { \mathcal { L } ( t + \alpha ) - \mathcal { L } ( t ) } { \alpha } \geq 0 } \end{array}$ L(t+α)−L(t) ≥ 0, but it contradicts to p0(0) = −η(t) k∇L(t)k22 < 0. If α0 > 0, $\kappa$ have $\kappa ( \mathcal { L } ( t + \xi ) ) < \kappa ( \dot { \mathcal { L } } ( t ) )$ $\xi \in ( 0 , \alpha _ { 0 } )$ thus + +$$ +\begin{array} { r } { \mathcal { L } ( t + \alpha ) \leq \mathcal { L } ( t ) - \alpha \eta ( t ) \left( 1 - \mu ( \mathcal { L } ( t ) ) \right) \left. \nabla \mathcal { L } ( t ) \right. _ { 2 } ^ { 2 } < \mathcal { L } ( t ) , } \end{array} +$$ + +which leads to a contradiction. + +Proof for $( \mathbf { P 4 } )$ . We define $\pmb { v } ( t ) : = \hat { \pmb { \theta } } ( t ) \hat { \pmb { \theta } } ( t ) ^ { \top } ( - \nabla \mathcal { L } ( t ) )$ and $\pmb { u } ( t ) : = \left( \pmb { I } - \hat { \pmb { \theta } } ( t ) \hat { \pmb { \theta } } ( t ) ^ { \top } \right) ( - \nabla \mathcal { L } ( t ) )$ similarly as in the analysis for gradient flow. For ${ \pmb v } ( t )$ , we have + +$$ +\| \pmb { v } ( t ) \| _ { 2 } = \frac { 1 } { \rho ( t ) } \left. \pmb { \theta } ( t ) , - \pmb { \mathcal { L } } ( t ) \right. = \frac { 1 } { \rho ( t ) } \cdot L \nu ( t ) . +$$ + +Decompose $\left\| \nabla \mathcal { L } ( t ) \right\| _ { 2 } ^ { 2 } = \left\| \pmb { v } ( t ) \right\| _ { 2 } ^ { 2 } + \left\| \pmb { u } ( t ) \right\| _ { 2 } ^ { 2 }$ . Then by (P3), we have + +$$ +\mathcal { L } ( t + \alpha ) - \mathcal { L } ( t ) \leq - \alpha \eta ( t ) ( 1 - \mu ( \mathcal { L } ( t ) ) ) \left( \frac { 1 } { \rho ( t ) ^ { 2 } } \cdot L ^ { 2 } \nu ( t ) ^ { 2 } + \| \pmb { u } \| _ { 2 } ^ { 2 } \right) . +$$ + +Multiplying 1+λ(L(t))µ(L(t))/L on both sides, we have + +$$ +\frac { \iota + \lambda ( \mathcal { L } ( t ) ) \mu ( \mathcal { L } ( t ) ) / L } { ( 1 - \mu ( \mathcal { L } ( t ) ) ) \nu ( t ) } ( \mathcal { L } ( t + \alpha ) - \mathcal { L } ( t ) ) \leq - \alpha \eta ( t ) \left( 1 + \frac { \lambda ( \mathcal { L } ( t ) ) \mu ( \mathcal { L } ( t ) ) } { L } \right) \left( \frac { L ^ { 2 } \nu ( t ) } { \rho ( t ) ^ { 2 } } + \frac { \| u \| _ { 2 } ^ { 2 } } { \nu ( t ) } \right) +$$ + +By Corollary E.6, we can bound $\nu ( t )$ by $\nu ( t ) \geq \mathcal { L } ( t ) / \lambda ( \mathcal { L } ( t ) )$ . By (P2), we have the inequality $\begin{array} { r } { \alpha \eta ( t ) \nu ( t ) \left( 1 + \frac { \lambda ( \mathcal { L } ( t ) ) \mu ( \mathcal { L } ( t ) ) } { L } \right) \ge \frac { \rho ( t + \alpha ) ^ { 2 } - \rho ( t ) ^ { 2 } } { 2 L } } \end{array}$ . So we further have + +$$ +\frac { 1 + \lambda ( \mathcal { L } ( t ) ) \mu ( \mathcal { L } ( t ) ) / L } { 1 - \mu ( \mathcal { L } ( t ) ) ) \mathcal { L } ( t ) / \lambda ( \mathcal { L } ( t ) ) } ( \mathcal { L } ( t + \alpha ) - \mathcal { L } ( t ) ) \leq - \frac { 1 } { \rho ( t ) ^ { 2 } } ( \rho ( t + \alpha ) ^ { 2 } - \rho ( t ) ^ { 2 } ) \left( \frac { L } { 2 } + \frac { \rho ( t ) ^ { 2 } } { 2 L \nu ( t ) ^ { 2 } } \| u \| _ { 2 } ^ { 2 } \right) +$$ + +From the definition φ, it is easy to see that −φ0(L(t)) ≥ 1+λ(L(t))µ(L(t))/L(1−µ(L(t)))L(t)/λ(L(t)) . Let $\psi ( x ) = - \log x .$ , then $\begin{array} { r } { \psi ^ { \prime } ( x ) = - \frac { 1 } { x } } \end{array}$ . Combining these together gives + +$$ +\phi ^ { \prime } ( \mathcal { L } ( t ) ) ( \mathcal { L } ( t + \alpha ) - \mathcal { L } ( t ) ) + \psi ^ { \prime } ( \rho ( t ) ^ { 2 } ) ( \rho ( t + \alpha ) ^ { 2 } - \rho ( t ) ^ { 2 } ) \left( \frac { L } { 2 } + \frac { \rho ( t ) ^ { 2 } } { 2 L \nu ( t ) ^ { 2 } } \left. \boldsymbol { u } \right. _ { 2 } ^ { 2 } \right) \geq 0 +$$ + +Then by convexity of $\phi$ and $\psi$ , we have + +$$ +\left( \phi ( \mathcal { L } ( t + \alpha ) ) - \phi ( \mathcal { L } ( t ) ) \right) + \left( \log \frac { 1 } { \rho ( t + \alpha ) ^ { 2 } } - \log \frac { 1 } { \rho ( t ) ^ { 2 } } \right) \left( \frac { L } { 2 } + \frac { \rho ( t ) ^ { 2 } } { 2 L \nu ( t ) ^ { 2 } } \left\| \boldsymbol { u } \right\| _ { 2 } ^ { 2 } \right) \ge 0 +$$ + +And by definition of $\hat { \gamma }$ , this can be re-written as + +$$ +\begin{array} { l } { \log \widehat { \gamma } ( t + \alpha ) - \log \widehat { \gamma } ( t ) = \left( \phi ( \mathcal { L } ( t + \alpha ) ) - \phi ( \mathcal { L } ( t ) ) \right) + L \left( \log \displaystyle \frac { 1 } { \rho ( t + \alpha ) } - \log \displaystyle \frac { 1 } { \rho ( t ) } \right) } \\ { \qquad \geq - \left( \log \displaystyle \frac { 1 } { \rho ( t + \alpha ) ^ { 2 } } - \log \displaystyle \frac { 1 } { \rho ( t ) ^ { 2 } } \right) \displaystyle \frac { \rho ( t ) ^ { 2 } } { 2 L \nu ( t ) ^ { 2 } } \left. u \right. _ { 2 } ^ { 2 } } \\ { \qquad = \displaystyle \frac { \rho ( t ) ^ { 2 } } { L \nu ( t ) ^ { 2 } } \left. u \right. _ { 2 } ^ { 2 } \cdot \log \displaystyle \frac { \rho ( t + \alpha ) } { \rho ( t ) } . } \end{array} +$$ + +Proof for $( \mathbf { P 1 } )$ . By (P4), $\log \hat { \gamma } ( t + \alpha ) \geq \log \hat { \gamma } ( t ) \geq \log \hat { \gamma } ( t _ { 0 } )$ . Note that $\phi ( x ) \geq \log \log \frac { 1 } { x }$ . So we + +$$ +\tilde { \gamma } ( t + \alpha ) > \hat { \gamma } ( t + \alpha ) \geq \hat { \gamma } ( t _ { 0 } ) , +$$ + +which completes the proof. + +For showing the third proposition in Theorem E.2, we use (P1) to give a lower bound for $\| \nabla \mathcal { L } ( t ) \| _ { 2 }$ , and use (P3) to show the speed of loss decreasing. Then it can be seen that $\mathcal { L } ( t ) 0$ and $\rho ( t ) $ $+ \infty$ . By Lemma E.1, we then have $| \bar { \gamma } - \hat { \gamma } | \to 0$ . + +Lemma E.10. Let $\begin{array} { r } { E _ { 0 } : = \mathcal { L } ( t _ { 0 } ) ^ { 2 } ( \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } ) ^ { 2 - 2 / L } } \end{array}$ . Then for all $t > t _ { 0 }$ + +$$ +E ( \mathcal { L } ( t ) ) \geq \frac { 1 } { 2 } L ^ { 2 } \hat { \gamma } ( t _ { 0 } ) ^ { 2 / L } \sum _ { \tau = t _ { 0 } } ^ { t - 1 } \eta ( \tau ) ~ f o r ~ E ( x ) : = \int _ { x } ^ { \mathcal { L } ( t _ { 0 } ) } \frac { 1 } { \operatorname* { m i n } \{ u ^ { 2 } ( \log \frac { 1 } { u } ) ^ { 2 - 2 / L } , E _ { 0 } \} } d u . +$$ + +Therefore, $\mathcal { L } ( t ) 0$ and $\rho ( t ) \to + \infty$ as $t \to \infty$ . + +Proof. For any integer $t \geq t _ { 0 }$ , $\begin{array} { r } { \mu ( \mathcal { L } ( t ) ) \leq \frac { 1 } { 2 } } \end{array}$ and $\| \nabla \mathcal { L } ( t ) \| _ { 2 } \geq \| \pmb { v } ( t ) \| _ { 2 }$ . Combining these with (P3), we have + +$$ +\mathcal { L } ( t + 1 ) - \mathcal { L } ( t ) \leq - \frac { 1 } { 2 } \eta ( t ) \left. \pmb { v } ( t ) \right. _ { 2 } ^ { 2 } = - \frac { 1 } { 2 } \eta ( t ) \frac { L ^ { 2 } \nu ( t ) ^ { 2 } } { \rho ( t ) ^ { 2 } } . +$$ + +By (P1), $\begin{array} { r } { \rho ( t ) ^ { - 2 } \leq \hat { \gamma } ( t _ { 0 } ) ^ { 2 / L } \left( \log \frac { 1 } { \mathcal { L } ( t ) } \right) ^ { - 2 / L } } \end{array}$ /L. By Corollary E.6, ν(t)2 ≥ L(t)2 log 1L(t) 2. Thus we have + +$$ +\mathcal { L } ( t + 1 ) - \mathcal { L } ( t ) \leq - \frac { 1 } { 2 } \eta ( t ) \cdot L ^ { 2 } \hat { \gamma } ( t _ { 0 } ) ^ { 2 / L } \mathcal { L } ( t ) ^ { 2 } \left( \log \frac { 1 } { \mathcal { L } ( t ) } \right) ^ { 2 - 2 / L } . +$$ + +It is easy to see that 1u2(log 1 )2−2/L is unimodal in (0, 1), so E0(x) is non-decreasing and E(x) is convex. So we have + +$$ +\begin{array} { r l } & { E ( \mathcal { L } ( t + 1 ) ) - E ( \mathcal { L } ( t ) ) \geq E ^ { \prime } ( \mathcal { L } ( t ) ) \left( \mathcal { L } ( t + 1 ) - \mathcal { L } ( t ) \right) } \\ & { \qquad \quad \geq \frac { 1 } { 2 } \eta ( t ) \cdot L ^ { 2 } \widehat { \gamma } ( t _ { 0 } ) ^ { 2 / L } \cdot \frac { \mathcal { L } ( t ) ^ { 2 } \left( \log \frac { 1 } { \mathcal { L } ( t ) } \right) ^ { 2 - 2 / L } } { \operatorname* { m i n } \{ \mathcal { L } ( t ) ^ { 2 } ( \log \frac { 1 } { \mathcal { L } ( t ) } ) ^ { 2 - 2 / L } , E _ { 0 } \} } } \\ & { \qquad \quad \geq \frac { 1 } { 2 } \eta ( t ) \cdot L ^ { 2 } \widehat { \gamma } ( t _ { 0 } ) ^ { 2 / L } , } \end{array} +$$ + +which proves $\begin{array} { r } { E ( \mathcal { L } ( t ) ) \ge \frac { 1 } { 2 } L ^ { 2 } \hat { \gamma } ( t _ { 0 } ) ^ { 2 / L } \sum _ { \tau = t _ { 0 } } ^ { t - 1 } \eta ( \tau ) } \end{array}$ . Note that $\mathcal { L }$ is non-decreasing. If $\mathcal { L }$ does not decreases to 0, then neither does $E ( \mathcal { L } )$ . But the RHS grows to $+ \infty$ , which leads to a contradiction. So $\mathcal { L } \to 0$ . To make $\mathcal { L } \to 0$ , $q _ { \mathrm { m i n } }$ must converge to $+ \infty$ . So $\rho \to + \infty$ . □ + +# E.5 PROOF FOR THEOREM E.3 AND E.4 + +The proofs for Theorem E.3 and E.4 are similar as those for Theorem A.8 and A.9 in Appendix C. + +$\begin{array} { r } { \beta ( t ) : = \frac { 1 } { \| \nabla \mathcal { L } ( t ) \| _ { 2 } } \left. \hat { \pmb { \theta } } , - \nabla \mathcal { L } ( t ) \right. } \end{array}$ as we do in Appendix C. It is easy to see that Lemma C.8 still holds if we replace $\widetilde { \gamma } ( t _ { 0 } )$ with $\hat { \gamma } ( t _ { 0 } )$ . So we only need to show $\mathcal { L } \to 0$ and $\beta 1$ for proving convergence to KKT points. $\mathcal { L } \to 0$ can be followed from Theorem E.2. Similar as the proof for Lemma C.9, it follows from Lemma E.8 and (15) that for all $t _ { 2 } > t _ { 1 } \geq t _ { 0 }$ , + +$$ +\sum _ { t = t _ { 1 } } ^ { t _ { 2 } - 1 } \left( \beta ( t ) ^ { - 2 } - 1 \right) \cdot \log \frac { \rho ( t + 1 ) } { \rho ( t ) } \leq \frac { 1 } { L } \log \frac { \hat { \gamma } ( t _ { 2 } ) } { \hat { \gamma } ( t _ { 1 } ) } . +$$ + +Now we prove Theorem E.4. + +Proof for Theorem E.4. We make the following changes in the proof for Theorem A.9. First, we replace $\tilde { \gamma } ( t _ { 0 } )$ with $\hat { \gamma } ( t _ { 0 } )$ , since $\tilde { \gamma } ( t )$ $\left( t \geq t _ { 0 } \right)$ is lower bounded by $\hat { \gamma } ( t _ { 0 } )$ rather than $\tilde { \gamma } ( t _ { 0 } )$ . Second, when choosing $t _ { 1 }$ and $t _ { 2 }$ , we make $\log \rho ( t _ { 1 } )$ and $\log \rho ( t _ { 2 } )$ equal to the chosen values approximately with an additive error $o ( 1 )$ , rather than make them equal exactly. This is possible because it can be shown from (P2) in Lemma E.8 that the following holds: + +$$ +\begin{array} { r l } & { \rho ( t + 1 ) ^ { 2 } - \rho ( t ) ^ { 2 } = O ( \eta ( t ) \nu ( t ) ) } \\ & { \qquad = o ( \kappa ( \mathcal { L } ( t ) ) ^ { - 1 } \cdot \mathcal { L } ( t ) \rho ( t ) ^ { L } ) = o \left( \frac { \rho ( t ) ^ { L } } { ( \log \frac { 1 } { \mathcal { L } ( t ) } ) ^ { 2 - 2 / L } } \right) = o ( \rho ( t ) ^ { - L + 2 } ) . } \end{array} +$$ + +Dividing $\rho ( t ) ^ { 2 }$ on the leftmost and rightmost sides, we have + +$$ +\rho ( t + 1 ) / \rho ( t ) = 1 + o ( \rho ( t ) ^ { - L } ) = 1 + o ( 1 ) , +$$ + +which implies that $\log \rho ( t + 1 ) - \log \rho ( t ) = o ( 1 )$ . Therefore, for any $R$ , we can always find the minimum time $t$ such that $\log \rho ( t ) \geq R$ , and it holds for sure that $\log \rho ( t ) - R \to 0$ as $R \to + \infty$ . + +For proving Theorem E.3, we also need the following lemma as a variant of Lemma C.11. + +Lemma E.11. For all $t \geq t _ { 0 }$ + +$$ +\left\| \hat { \pmb \theta } ( t + 1 ) - \hat { \pmb \theta } ( t ) \right\| _ { 2 } \leq \left( \frac { B _ { 1 } } { L \tilde { \gamma } ( t ) } \cdot \frac { \rho ( t + 1 ) } { \rho ( t ) } + 1 \right) \log \frac { \rho ( t + 1 ) } { \rho ( t ) } +$$ + +Proof. According to the update rule, we have + +$$ +\begin{array} { r l } & { \left\| \hat { \theta } ( t + 1 ) - \hat { \theta } ( t ) \right\| _ { 2 } = \displaystyle \frac { 1 } { \rho ( t + 1 ) } \left\| \theta ( t + 1 ) - \frac { \rho ( t + 1 ) } { \rho ( t ) } \pmb { \theta } ( t ) \right\| _ { 2 } } \\ & { \qquad \leq \displaystyle \frac { 1 } { \rho ( t + 1 ) } \left( \| \theta ( t + 1 ) - \theta ( t ) \| _ { 2 } + \left\| \left( \frac { \rho ( t + 1 ) } { \rho ( t ) } - 1 \right) \pmb { \theta } ( t ) \right\| _ { 2 } \right) } \\ & { \qquad = \displaystyle \frac { \eta ( t ) } { \rho ( t + 1 ) } \| \nabla \mathcal { L } ( t ) \| _ { 2 } + \left( 1 - \frac { \rho ( t ) } { \rho ( t + 1 ) } \right) . } \end{array} +$$ + +By (16), $\begin{array} { r } { \| \nabla \mathcal { L } ( t ) \| _ { 2 } \le \frac { B _ { 1 } \nu ( t ) } { \widetilde { \gamma } ( t ) \rho ( t ) } } \end{array}$ . So we can bound the first term as + +$$ +\begin{array} { r l } & { \displaystyle \frac { \eta ( t ) } { \rho ( t + 1 ) } \| \nabla \mathcal { L } ( t ) \| _ { 2 } \le \frac { B _ { 1 } } { \widetilde { \gamma } ( t ) } \cdot \frac { \eta ( t ) \nu ( t ) } { \rho ( t + 1 ) \rho ( t ) } \le \frac { B _ { 1 } } { \widetilde { \gamma } ( t ) } \cdot \displaystyle \frac { \rho ( t + 1 ) ^ { 2 } - \rho ( t ) ^ { 2 } } { 2 L \rho ( t + 1 ) \rho ( t ) } } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad = \frac { B _ { 1 } } { 2 L \widetilde { \gamma } ( t ) } \cdot \displaystyle \frac { \rho ( t + 1 ) ^ { 2 } - \rho ( t ) ^ { 2 } } { \rho ( t + 1 ) ^ { 2 } } \cdot \displaystyle \frac { \rho ( t + 1 ) } { \rho ( t ) } } \\ & { \quad \quad \quad \quad \quad \le \frac { B _ { 1 } } { L \widetilde { \gamma } ( t ) } \cdot \displaystyle \frac { \rho ( t + 1 ) } { \rho ( t ) } \log \displaystyle \frac { \rho ( t + 1 ) } { \rho ( t ) } } \end{array} +$$ + +where the last inequality uses the inequality bound the second term by $\begin{array} { r } { \frac { a - b } { a } \leq \log ( a / b ) } \end{array}$ . Using this inequality again, we can + +$$ +\begin{array} { r l } & { \quad 1 - \displaystyle \frac { \rho ( t ) } { \rho ( t + 1 ) } \leq \log \displaystyle \frac { \rho ( t + 1 ) } { \rho ( t ) } . } \\ & { \quad \left\| \hat { \pmb \theta } ( t + 1 ) - \hat { \pmb \theta } ( t ) \right\| _ { 2 } \leq \left( \frac { B _ { 1 } } { L \tilde { \gamma } ( t ) } \cdot \displaystyle \frac { \rho ( t + 1 ) } { \rho ( t ) } + 1 \right) \log \frac { \rho ( t + 1 ) } { \rho ( t ) } . } \end{array} +$$ + +Combining these together gives + +Now we are ready to prove Theorem E.3. + +Proof for Theorem E.3. As discussed above, we only need to show a variant of Lemma C.12 for gradient descent: for every limit point $\bar { \pmb { \theta } }$ of $\left\{ { \hat { \theta } } ( t ) : t \geq 0 \right\}$ , there exists a sequence of $\{ t _ { m } : m \in \mathbb { N } \}$ such that $t _ { m } \uparrow + \infty , \hat { \pmb \theta } ( t _ { m } ) \bar { \pmb \theta }$ , and $\beta ( t _ { m } ) 1$ . + +We only need to change the choices of $s _ { m } , s _ { m } ^ { \prime } , t _ { m }$ in the proof for Lemma C.12. We choose $s _ { m } >$ $t _ { m - 1 }$ to be a time such that + +$$ +\| \hat { \pmb \theta } ( s _ { m } ) - \bar { \pmb \theta } \| _ { 2 } \leq \epsilon _ { m } \quad \mathrm { a n d } \quad \frac { 1 } { L } \log ( \frac { \operatorname* { l i m } _ { t + \infty } \hat { \gamma } ( t ) } { \hat { \gamma } ( s _ { m } ) } ) \leq \epsilon _ { m } ^ { 3 } . +$$ + +Then we let $s _ { m } ^ { \prime } > s _ { m }$ be the minimum time such that $\log { \rho ( s _ { m } ^ { \prime } ) } \ge \log { \rho ( s _ { m } ) } + \epsilon _ { m }$ . According to Theorem E.2, $s _ { m }$ and $s _ { m } ^ { \prime }$ must exist. Finally, we construct $t _ { m } \in \{ s _ { m } , \ldots , s _ { m } ^ { \prime } - 1 \}$ to be a time that $\beta ( t _ { m } ) ^ { - 2 } - 1 \leq \epsilon _ { m } ^ { 2 }$ , where the existence can be shown by (22). + +To see that this construction meets our requirement, note that $\beta ( t _ { m } ) ^ { - 2 } - 1 \le \epsilon _ { m } ^ { 2 } \to 0$ and + +$$ +\Big \| \hat { \pmb \theta } ( t _ { m } ) - \bar { \pmb \theta } \Big \| _ { 2 } \leq \Big \| \hat { \pmb \theta } ( t _ { m } ) - \hat { \pmb \theta } ( s _ { m } ) \Big \| _ { 2 } + \Big \| \hat { \pmb \theta } ( s _ { m } ) - \bar { \pmb \theta } \Big \| _ { 2 } \leq \bigg ( \frac { B _ { 1 } } { L \hat { \gamma } ( t _ { 0 } ) } e ^ { \epsilon _ { m } } + 1 \bigg ) \cdot \epsilon _ { m } + \epsilon _ { m } 0 , +$$ + +where the last inequality is by Lemma E.11. + +# E.6 PROOF FOR THEOREM E.5 + +Proof. By a similar analysis as Lemma D.4, we have + +$$ +\begin{array} { r l r } { { E ( x ) = \int _ { x } ^ { \mathcal { L } ( t _ { 0 } ) } \Theta ( \frac { 1 } { u ^ { 2 } ( \log \frac { 1 } { u } ) ^ { 2 - 2 / L } } ) d u = \int _ { 1 / \mathcal { L } ( t _ { 0 } ) } ^ { 1 / x } \Theta ( \frac { 1 } { ( \log u ) ^ { 2 - 2 / L } } ) d u } } \\ & { } & { = \Theta ( \frac { 1 } { x ( \log \frac { 1 } { x } ) ^ { 2 - 2 / L } } ) . } \end{array} +$$ + +We can also bound the inverse function $E ^ { - 1 } ( y )$ by $\Theta \left( { \frac { 1 } { y ( \log y ) ^ { 2 - 2 / L } } } \right)$ . With these, we can use a similar analysis as Theorem A.10 to prove Theorem E.5. + +First, using a similar proof as for (17), we have $\rho ^ { L } = \Theta ( \log \frac { 1 } { \mathcal { L } } )$ . So we only need to show ${ \mathcal { L } } ( t ) =$ $\Theta \big ( { \frac { 1 } { T ( \log T ) ^ { 2 - 2 / L } } } \big )$ . With a similar analysis as for (P3) in Lemma E.9, we have the following bound for $\mathcal { L } ( \tau + 1 ) - \mathcal { L } ( \tau )$ : + +$$ +\begin{array} { r } { \mathcal { L } ( \tau + 1 ) - \mathcal { L } ( \tau ) \geq - \eta ( \tau ) ( 1 + \mu ( \mathcal { L } ( \tau ) ) ) \left. \nabla \mathcal { L } ( \tau ) \right. _ { 2 } ^ { 2 } . } \end{array} +$$ + +Using the fact that $\mu \leq 1 / 2$ , we have $\begin{array} { r } { \mathcal { L } ( \tau + 1 ) - \mathcal { L } ( \tau ) \geq - \frac { 3 } { 2 } \eta ( \tau ) \left. \nabla \mathcal { L } ( \tau ) \right. _ { 2 } ^ { 2 } } \end{array}$ . By Lemma E.7, $\| \nabla \mathcal { L } ( \tau ) \| _ { 2 } \le 2 C _ { \eta } \kappa ( \mathcal { L } ( \tau ) ) \mathcal { L } ( \tau )$ . Using a similar proof as for Lemma E.10, we can show that $E ( \mathcal { L } ( t ) ) \ \stackrel { } { \leq } \ O ( T )$ . Combining this with Lemma E.10, we have $E ( \mathcal { L } ( t ) ) ~ = ~ \Theta ( T )$ . Therefore, $\begin{array} { r } { \mathcal { L } ( t ) = \Theta \big ( \frac { 1 } { T ( \log T ) ^ { 2 - 2 / L } } \big ) } \end{array}$ . □ + +# F GRADIENT DESCENT: GENERAL LOSS FUNCTIONS + +It is worth to note that the above analysis can be extended to other loss functions. For this, we need to replace (B3) with a strong assumption (S3), which takes into account the second-order derivatives of $f$ . + +(S3). The loss function $\ell ( q )$ can be expressed as $\ell ( q ) = e ^ { - f ( q ) }$ such that + +(S3.1). $f : \mathbb { R } \to \mathbb { R }$ is $\mathcal { C } ^ { 2 }$ -smooth. + +(S3.2). $f ^ { \prime } ( q ) > 0$ for all $q \in \mathbb { R }$ . + +(S3.3). There exists $b _ { f } \geq 0$ such that $f ^ { \prime } ( q ) q$ is non-decreasing for $q \in ( b _ { f } , + \infty )$ , and $f ^ { \prime } ( q ) q \to + \infty$ as $q \to + \infty$ . + +(S3.4). Let $g \ : \ [ f ( b _ { f } ) , + \infty ) \ \ [ b _ { f } , + \infty )$ be the inverse function of $f$ on the domain $[ b _ { f } , + \infty )$ . There exists $p \geq 0$ such that for all $x > f ( b _ { f } ) , y > b _ { f }$ , + +$$ +\left| { \frac { g ^ { \prime \prime } ( x ) } { g ^ { \prime } ( x ) } } \right| \leq { \frac { p } { x } } \quad { \mathrm { a n d } } \quad \left| { \frac { f ^ { \prime \prime } ( y ) } { f ^ { \prime } ( y ) } } \right| \leq { \frac { p } { y } } . +$$ + +It can be verified that (S3) is satisfied by exponential loss and logistic loss. Now we explain each of the assumptions in (S3). (S3.2) and (S3.3) are essentially the same as (B3.2) and (B3.3). (S3.1) and (B3.1) are the same except that (S3.1) assumes $f$ is $\mathcal { C } ^ { 2 }$ -smooth rather than ${ \mathcal { C } } ^ { 1 }$ -smooth. + +(S3.4) can also be written in the following form: + +$$ +\left| { \frac { d } { d x } } \log g ^ { \prime } ( x ) \right| \leq p \cdot { \frac { d } { d x } } \log x \quad { \mathrm { a n d } } \quad \left| { \frac { d } { d y } } \log f ^ { \prime } ( y ) \right| \leq p \cdot { \frac { d } { d y } } \log y . +$$ + +That is, $\log g ^ { \prime } ( x )$ and $\log f ^ { \prime } ( y )$ grow no faster than $O ( \log x )$ and $O ( \log y )$ , respectively. In fact, (B3.4) can be deduced from (S3.4). Recall that (B3.4) ensures that ${ \Theta } \bar { ( g ^ { \prime } ( x ) ) \ } \stackrel { - } { = } \ g ^ { \prime } ( \stackrel { . } { \Theta } ( x ) )$ and $\Theta ( f ^ { \prime } ( y ) ) = f ^ { \prime } ( \Theta ( y ) )$ . Thus, (S3.4) also gives us the interchangeability between $f ^ { \prime } , g ^ { \prime }$ and $\Theta$ . + +Lemma F.1. (S3.4) implies (B3.4) with $b _ { g } = \operatorname* { m a x } \{ 2 f ( b _ { f } ) , f ( 2 b _ { f } ) \}$ and $K = 2 ^ { p }$ . + +Proof. Fix $x \in ( b _ { g } , + \infty ) , y \in ( g ( b _ { g } ) , + \infty )$ and $\theta \in [ 1 / 2 , 1 )$ . Integrating (23) on both sides of the inequalities from $\theta x$ to $x$ and $\theta y$ to $y$ , we have + +$$ +| \log g ^ { \prime } ( x ) - \log g ^ { \prime } ( \theta x ) | \leq p \cdot \log { \frac { 1 } { \theta } } \quad { \mathrm { a n d } } \quad | \log f ^ { \prime } ( y ) - \log f ^ { \prime } ( \theta y ) | \leq p \cdot \log { \frac { 1 } { \theta } } . +$$ + +Therefore, we have $g ^ { \prime } ( x ) \leq \theta ^ { - p } g ^ { \prime } ( \theta x ) \leq K g ^ { \prime } ( \theta x )$ and $f ^ { \prime } ( y ) \leq \theta ^ { - p } f ^ { \prime } ( \theta y ) \leq K f ^ { \prime } ( \theta x ) ,$ . + +To extend our results to general loss functions, we need to redefine $\kappa ( x ) , \lambda ( x ) , \phi ( x ) , C _ { \eta } , H ( x )$ in order. For $\kappa ( x )$ and $\lambda ( x )$ , we redefine them as follows: + +$$ +\kappa ( x ) : = \operatorname* { s u p } \left\{ \frac { w ( \log \frac { 1 } { w } ) ^ { 2 - 2 / L } } { g ^ { \prime } ( \log \frac { 1 } { w } ) ^ { 2 } } : w \in ( 0 , x ] \right\} \qquad \lambda ( x ) : = \frac { g ^ { \prime } ( \log \frac { 1 } { x } ) } { g ( \log \frac { 1 } { x } ) } . +$$ + +By Lemma D.1, $\begin{array} { r } { \frac { w ( \log \frac { 1 } { w } ) ^ { 2 - 2 / L } } { g ^ { \prime } ( \log \frac { 1 } { w } ) ^ { 2 } } = O \left( w ( \log \frac { 1 } { w } ) ^ { 4 - 2 / L } / g ( \log \frac { 1 } { w } ) ^ { 2 } \right) \to 0 } \end{array}$ . So $\kappa ( x )$ is well-defined. +Using $\lambda ( x )$ , we can define $\phi ( x )$ and $\hat { \gamma } ( \pmb \theta )$ as follows. + +$$ +\begin{array} { r l } & { \phi ^ { \prime } ( x ) : = - \operatorname* { s u p } \left\{ \displaystyle \frac { \lambda ( w ) } { w } \left( 1 + 2 ( 1 + \lambda ( w ) / L ) \mu ( w ) \right) : w \in [ x , \mathcal { L } ( t _ { 0 } ) ] \right\} } \\ & { \displaystyle \phi ( x ) : = \log \left( g ( \log \frac { 1 } { x } ) \right) + \int _ { 0 } ^ { x } \left( \phi ^ { \prime } ( w ) + \frac { \lambda ( w ) } { w } \right) d w } \\ & { \displaystyle \hat { \gamma } ( \pmb { \theta } ) : = \frac { e ^ { \phi ( \mathcal { L } ( \pmb { \theta } ) ) } } { \rho ^ { L } } . } \end{array} +$$ + +Using a similar argument as in Lemma E.1, we can show that $\hat { \gamma } ( \pmb \theta )$ is well-defined and $\hat { \gamma } ( \pmb \theta ) ~ <$ $\begin{array} { r } { \tilde { \gamma } ( \pmb { \theta } ) : = g ( \log \frac { 1 } { \mathcal { L } } ) \breve { / } \rho ^ { L } \leq \bar { \gamma } ( \pmb { \theta } ) } \end{array}$ . When ${ \mathcal { L } } ( \pmb { \theta } ) \to 0$ , we also have $\hat { \gamma } ( \pmb { \theta } ) / \bar { \gamma } ( \pmb { \theta } ) 1$ . + +For $C _ { \eta }$ , we define it to be the following. + +$$ +\begin{array} { r } { \gamma _ { \eta } : = \frac { 1 } { 2 } B _ { 1 } \left( \left( \frac { B _ { 0 } } { \hat { \gamma } ( t _ { 0 } ) } \right) ^ { p } B _ { 1 } + \frac { 2 ^ { p + 1 } } { \log \frac { 1 } { C ( t _ { 0 } ) } } ( p B _ { 1 } + B _ { 2 } ) \right) \left( \frac { B _ { 0 } } { \hat { \gamma } ( t _ { 0 } ) } \right) ^ { p } \operatorname* { m i n } \left\{ \hat { \gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \right\} . } \end{array} +$$ + +Finally, the definitions for $\begin{array} { r } { \mu ( x ) : = ( \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } ) / ( 2 \log \frac { 1 } { x } ) } \end{array}$ and $H ( x ) : = \mu ( x ) / ( C _ { \eta } \kappa ( x ) )$ in (S5) remain unchanged except that $\kappa ( x )$ and $C _ { \eta }$ now use the new definitions. + +Similar as gradient flow, we define $\begin{array} { r } { \nu ( t ) : = \sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ( t ) ) } f ^ { \prime } ( q _ { n } ( t ) ) q _ { n } ( t ) } \end{array}$ . The key idea behind the above definitions is that we can prove similar bounds for $\nu ( t ) , \| \nabla \mathcal { L } \| _ { 2 } , \| \nabla ^ { 2 } \mathcal { L } \| _ { 2 }$ as Corollary E.6 and Lemma E.7. + +Lemma F.2. $\langle \pmb \theta ( t ) , - \nabla \mathcal L ( t ) \rangle = L \nu ( t ) .$ . If $\mathcal { L } ( t ) < e ^ { - f ( b _ { f } ) }$ , then $\begin{array} { r } { \nu ( t ) \geq \frac { \mathcal { L } ( t ) } { \lambda ( \mathcal { L } ( t ) ) } } \end{array}$ and $\lambda ( \mathcal { L } ( t ) )$ has the lower bound $\begin{array} { r } { \lambda ( \mathcal { L } ( t ) ) \leq 2 ^ { p + 1 } \left( \log \frac { 1 } { \mathcal { L } ( t ) } \right) ^ { - 1 } } \end{array}$ . + +Proof. It can be easily proved by combining Theorem B.4, Lemma B.5 and Lemma D.1 together. □ + +Lemma F.3. For any $\pmb \theta$ , if $\mathcal { L } ( \pmb \theta ) \leq \mathcal { L } ( t _ { 0 } ) , \tilde { \gamma } ( \pmb \theta ) \geq \hat { \gamma } ( t _ { 0 } )$ , then + +$$ +\begin{array} { r } { \| \nabla \mathcal { L } ( \pmb { \theta } ) \| _ { 2 } ^ { 2 } \leq 2 C _ { \eta } \kappa ( \mathcal { L } ( \pmb { \theta } ) ) \mathcal { L } ( \pmb { \theta } ) \quad a n d \quad \left\| \nabla ^ { 2 } \mathcal { L } ( \pmb { \theta } ) \right\| _ { 2 } \leq 2 C _ { \eta } \kappa ( \mathcal { L } ( \pmb { \theta } ) ) . } \end{array} +$$ + +Proof. Note that a direct corollary of (23) is that $f ^ { \prime } ( x _ { 1 } ) \leq f ^ { \prime } ( x _ { 2 } ) ( x _ { 1 } / x _ { 2 } ) ^ { p }$ for $x _ { 1 } \geq x _ { 2 } > b _ { f }$ . So we have + +$$ +f ^ { \prime } ( q _ { n } ) \leq f ^ { \prime } \left( g ( \log { \frac { 1 } { \mathcal { L } } } ) \right) \left( { \frac { q _ { n } } { g ( \log { \frac { 1 } { \mathcal { L } } } ) } } \right) ^ { p } = { \frac { 1 } { g ^ { \prime } ( \log { \frac { 1 } { \mathcal { L } } } ) } } \left( { \frac { q _ { n } / \rho ^ { L } } { \widetilde { \gamma } } } \right) ^ { p } \leq { \frac { R } { g ^ { \prime } ( \log { \frac { 1 } { \mathcal { L } } } ) } } , +$$ + +where $R : = ( B _ { 0 } / \hat { \gamma } ( t _ { 0 } ) ) ^ { p }$ . Applying (S3.4), we can also deduce that + +$$ +| f ^ { \prime \prime } ( q _ { n } ) | \leq { \frac { p } { q _ { n } } } \cdot f ^ { \prime } ( q _ { n } ) \leq { \frac { p R } { g ( \log { \frac { 1 } { \mathcal { L } } } ) g ^ { \prime } ( \log { \frac { 1 } { \mathcal { L } } } ) } } . +$$ + +Now we bound $\lVert \nabla \mathcal { L } \rVert _ { 2 }$ and $\left\| \nabla ^ { 2 } \mathcal { L } \right\| _ { 2 }$ . By the chain rule, we have + +$$ +\begin{array} { r l } & { \left\| \nabla \mathcal { L } \right\| _ { 2 } = \left\| - \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - f ( \boldsymbol { q } _ { n } ) } f ^ { \prime } ( \boldsymbol { q } _ { n } ) \nabla \boldsymbol { q } _ { n } \right\| _ { 2 } \leq B _ { 1 } R \cdot \frac { \mathcal { L } \rho ^ { L - 1 } } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } } \\ & { \left\| \nabla ^ { 2 } \mathcal { L } \right\| _ { 2 } = \left\| \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - f ( \boldsymbol { q } _ { n } ) } \left( \big ( f ^ { \prime } ( \boldsymbol { q } _ { n } ) ^ { 2 } - f ^ { \prime \prime } ( \boldsymbol { q } _ { n } ) \big ) \nabla q _ { n } \nabla q _ { n } ^ { \top } - f ^ { \prime } ( q _ { n } ) \nabla ^ { 2 } q _ { n } \right) \right\| _ { 2 } } \\ & { \qquad \leq \displaystyle \sum _ { n = 1 } ^ { N } e ^ { - f ( \boldsymbol { q } _ { n } ) } \left( \left( \frac { R ^ { 2 } } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 } } + \frac { p R } { g ( \log \frac { 1 } { \mathcal { L } } ) g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } \right) B _ { 1 } ^ { 2 } \rho ^ { 2 L - 2 } + \frac { R } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } B _ { 2 } \rho ^ { L - 2 } \right) } \\ & { \qquad \leq \frac { \mathcal { L } \rho ^ { 2 L - 2 } } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 } } \left( \left( R + p \lambda ( \mathcal { L } ) \right) B _ { 1 } ^ { 2 } + \frac { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } { \rho ^ { L } } B _ { 2 } \right) R . } \end{array} +$$ + +For $\rho$ we have $\begin{array} { r } { \hat { \gamma } ( t _ { 0 } ) \rho ^ { L } \le \log \frac { 1 } { \mathcal L } \le B _ { 0 } \rho ^ { L } , } \end{array}$ , so we can bound $\rho ^ { 2 L - 2 }$ by $\rho ^ { 2 L - 2 } \leq M ( \log { \frac { 1 } { \mathcal { L } } } ) ^ { 2 L - 2 }$ , where ${ \cal M } : = \operatorname * { m i n } \left\{ \hat { \gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \right\}$ . By Lemma F.2, we have $\begin{array} { r } { \lambda ( \mathcal { L } ) \ \leq \ 2 ^ { p + 1 } \frac { 1 } { \log \frac { 1 } { \mathcal { L } } } \ \leq \ } \end{array}$ $\frac { 2 ^ { p + 1 } } { \log { \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } } }$ , and also $\begin{array} { r } { \frac { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } { \rho ^ { L } } = \lambda ( \mathcal { L } ) \cdot \tilde { \gamma } \leq \frac { 2 ^ { p + 1 } B _ { 1 } } { \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } } } \end{array}$ . Combining all these formulas together gives + +$$ +\begin{array} { r l r } & { \| \nabla \mathcal { L } \| _ { 2 } ^ { 2 } \leq B _ { 1 } ^ { 2 } R ^ { 2 } M \displaystyle \frac { \mathcal { L } ^ { 2 } } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 } } \left( \log \frac { 1 } { \mathcal { L } } \right) ^ { 2 - 2 / L } \leq 2 C _ { \eta ^ { \kappa } } ( \mathcal { L } ) \mathcal { L } } & \\ & { \big | \nabla ^ { 2 } \mathcal { L } \big \| _ { 2 } \leq \left( \left( R + \frac { p 2 ^ { p + 1 } } { \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } } \right) B _ { 1 } ^ { 2 } + \frac { 2 ^ { p + 1 } B _ { 1 } B _ { 2 } } { \log \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } } \right) R M \displaystyle \frac { \mathcal { L } } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) ^ { 2 } } \left( \log \frac { 1 } { \mathcal { L } } \right) ^ { 2 - 2 / L } \leq 2 C _ { \eta ^ { \kappa } } ( \mathcal { L } ) , } & \end{array} +$$ + +which completes the proof. + +With Corollary E.6 and Lemma E.7, we can prove Lemma E.8 with exactly the same argument. Then Theorem E.2, E.3, E.4 can also be proved similarly. For Theorem E.5, we can follow the argument for gradient flow to show that it holds with slightly different tight bounds: + +Theorem F.4. Under assumptions (S1), (A2), (S3), (A4), (S5), we have the following tight rates for training loss and weight norm: + +$$ +\mathcal { L } ( t ) = \Theta \left( \frac { g ( \log T ) ^ { 2 / L } } { T ( \log T ) ^ { 2 } } \right) \quad a n d \quad \rho ( t ) = \Theta ( g ( \log T ) ^ { 1 / L } ) , +$$ + +where $\begin{array} { r } { T = \sum _ { \tau = t _ { 0 } } ^ { t - 1 } \eta ( \tau ) } \end{array}$ + +# G EXTENSION: MULTI-CLASS CLASSIFICATION + +In this section, we generalize our results to multi-class classification with cross-entropy loss. This part of analysis is inspired by Theorem 1 in (Zhang et al., 2019), which gives a lower bound for the gradient in terms of the loss $\mathcal { L }$ . + +Since now a neural network has multiple outputs, we need to redefine our notations. Let $C$ be the number of classes. The output of a neural network $\Phi$ is a vector $\Phi ( \pmb { \theta } ; \pmb { x } ) \in \mathbb { R } ^ { C }$ . We use $\Phi _ { j } ( \pmb { \theta } ; \pmb { x } ) \in \mathbb { R }$ to denote the $j$ -th output of $\Phi$ on the input $\textbf { \em x } \in \mathbb { R } ^ { d _ { \mathrm { { x } } } }$ . A dataset is denoted by $\bar { \mathcal { D } } = \{ \pmb { x } _ { n } , y _ { n } \} _ { n = 1 } ^ { N } = \{ ( \pmb { x } _ { n } , y _ { n } ) : n \in [ N ] \}$ , where on the d $\pmb { x } _ { n } \in \mathbb { R } ^ { d _ { \mathbf { x } } }$ is a data input and defined as $y _ { n } \in [ C ]$ is the $\Phi$ $\mathcal { D }$ + +$$ +\mathcal { L } ( \pmb \theta ) : = \sum _ { n = 1 } ^ { N } - \log \frac { e ^ { \Phi _ { y _ { n } } ( \pmb \theta ; \pmb x _ { n } ) } } { \sum _ { j = 1 } ^ { C } e ^ { \Phi _ { j } ( \pmb \theta ; \pmb x _ { n } ) } } . +$$ + +The margin for a single data point $( { \pmb x } _ { n } , y _ { n } )$ is defined to be $\begin{array} { r c l } { q _ { n } ( \pmb { \theta } ) } & { : = } & { \Phi _ { y _ { n } } ( \pmb { \theta } ; \pmb { x } _ { n } ) - } \end{array}$ $\operatorname* { m a x } _ { j \neq y _ { n } } \bar { \{ \Phi _ { j } ( \pmb { \theta } ; \pmb { x } _ { n } ) \} }$ , and the margin for the entire dataset is defined to be $q _ { \mathrm { m i n } } ( \pmb { \theta } ) =$ $\mathrm { m i n } _ { n \in [ N ] } q _ { n } ( \pmb \theta )$ . We define the normalized margin to be $\bar { \gamma } ( \pmb \theta ) : = q _ { \mathrm { m i n } } ( \pmb \theta ) = q _ { \mathrm { m i n } } ( \pmb \theta ) / \rho ^ { L }$ , where $\rho : = \| \pmb \theta \| _ { 2 }$ and $\hat { \pmb \theta } : = \pmb \theta / \rho \in S ^ { d - 1 }$ as usual. + +Let $\ell ( q ) : = \log ( 1 + e ^ { - q } )$ be the logistic loss. Recall that $\ell ( q )$ satisfies (B3). Let $f ( q ) = - \log \ell ( q ) =$ $- \log \log ( 1 + e ^ { - q } )$ . Let $g$ be the inverse function of $f$ . So $g ( q ) = - \log ( e ^ { e ^ { - q } } - 1 )$ . + +The cross-entropy loss can be rewritten in other ways. Let $s _ { n j } : = \Phi _ { y _ { n } } ( \pmb { \theta } ; \pmb { x } _ { n } ) - \Phi _ { j } ( \pmb { \theta } ; \pmb { x } _ { n } )$ . Let $\begin{array} { r } { \widetilde { q } _ { n } : = - \mathrm { L S E } ( \{ - s _ { n j } : j \neq y _ { n } \} ) = - \log \left( \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } } \right) } \end{array}$ . Then + +$$ +\mathcal { L } ( \pmb { \theta } ) : = \sum _ { n = 1 } ^ { N } \log \left( 1 + \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } } \right) = \sum _ { n = 1 } ^ { N } \log ( 1 + e ^ { - \tilde { q } _ { n } } ) = \sum _ { n = 1 } ^ { N } \ell ( \tilde { q } _ { n } ) = \sum _ { n = 1 } ^ { N } e ^ { - f ( \tilde { q } _ { n } ) } . +$$ + +Gradient Flow. For gradient flow, we assume the following: + +(M1). (Regularity). For any fixed $_ { \textbf { \em x } }$ and $j \in [ C ] , \Phi _ { j } ( \cdot ; x )$ is locally Lipschitz and admits a chain rule; + +(M2). (Homogeneity). There exists $L \ > \ 0$ such that $\forall j ~ \in ~ [ C ] , \forall \alpha ~ > ~ 0 ~ : ~ \Phi _ { j } ( \alpha \pmb \theta ; \pmb x ) ~ =$ $\alpha ^ { L } \Phi _ { j } ( \pmb { \theta } ; \pmb { x } )$ ; + +(M3). (Cross-entropy Loss). $\mathcal { L } ( \pmb { \theta } )$ is defined as the cross-entropy loss on the training set; + +(M4). (Separability). There exists a time $t _ { 0 }$ such that $\mathcal { L } ( t _ { 0 } ) < \log 2$ . + +$\mathcal { L } < \log 2$ , then $\begin{array} { r } { \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } } < 1 } \end{array}$ for all $n \in [ N ]$ , and thus $s _ { n j } > 0$ for all $n \in [ N ] , j \in [ C ]$ . So (M4) ensures the separability of training data. + +Definition G.1. For cross-entropy loss, the smoothed normalized margin $\widetilde \gamma ( \pmb \theta )$ of $\pmb \theta$ is defined as + +$$ +\widetilde { \gamma } ( \pmb \theta ) : = \frac { \ell ^ { - 1 } ( \mathcal { L } ) } { \rho ^ { L } } = \frac { - \log ( e ^ { \mathcal { L } } - 1 ) } { \rho ^ { L } } , +$$ + +where $\ell ^ { - 1 } ( \cdot )$ is the inverse function of the logistic loss $\ell ( \cdot )$ . + +Theorem 4.1 and 4.4 still hold. Here we redefine the optimization problem (P) to be + +$$ +\begin{array} { r l } { \operatorname* { m i n } } & { \frac { 1 } { 2 } \left\| \pmb { \theta } \right\| _ { 2 } ^ { 2 } } \\ { \mathrm { s . t . } } & { s _ { n j } ( \pmb { \theta } ) \geq 1 \qquad \forall n \in [ N ] , \forall j \in [ C ] \setminus \{ y _ { n } \} } \end{array} +$$ + +Most of our proofs are very similar as before. Here we only show the proof for the generalized version of Lemma 5.1. + +Lemma G.2. Lemma 5.1 is also true for the smoothed normalized margin $\tilde { \gamma }$ defined in Definition G.1. + +Proof. Define $\nu ( t )$ by the following formula: + +$$ +\nu ( t ) : = \sum _ { n = 1 } ^ { N } \frac { \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } } s _ { n j } } { 1 + \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } } } . +$$ + +Using a similar argument as in Theorem B.4, it can be proved that $\textstyle { \frac { 1 } { 2 } } { \frac { d \rho ^ { 2 } } { d t } } = L \nu ( t )$ for a.e. $t > 0$ . + +It can be shown that Lemma B.5, which asserts that $\begin{array} { r } { \nu ( t ) \geq \frac { g ( \log \frac { 1 } { \mathcal { L } } ) } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } \mathcal { L } } \end{array}$ , still holds for this new definition of $\nu ( t )$ . By definition, $s _ { n j } \geq q _ { n }$ for $j \neq y _ { n }$ . Also note that $\tilde { e } ^ { - \tilde { q } _ { n } } \geq e ^ { - q _ { n } }$ . So $s _ { n j } \geq q _ { n } \geq$ $\tilde { q } _ { n }$ . Then we have + +$$ +\nu ( t ) \geq \sum _ { n = 1 } ^ { N } \frac { \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } } } { 1 + \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } } } \cdot \tilde { q } _ { n } = \sum _ { n = 1 } ^ { N } \frac { e ^ { - \tilde { q } _ { n } } } { 1 + e ^ { - \tilde { q } _ { n } } } \cdot \tilde { q } _ { n } = \sum _ { n = 1 } ^ { N } e ^ { - f ( \tilde { q } _ { n } ) } f ^ { \prime } ( \tilde { q } _ { n } ) \tilde { q } _ { n } . +$$ + +Note that $\begin{array} { r } { \mathcal { L } = \sum _ { n = 1 } ^ { N } e ^ { - f ( \tilde { q } _ { n } ) } } \end{array}$ . Then using Lemma B.5 for logistic loss can conclude that $\nu ( t ) \geq$ $\begin{array} { r } { \frac { g ( \log \frac { 1 } { \mathcal { L } } ) } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } \mathcal { L } } \end{array}$ + +The rest of the proof for this lemma is exactly the same as that for Lemma 5.1. + +Gradient Descent. For gradient descent, we only need to replace (M1) with (S1) and make the assumption (S5) on the learning rate. + +(S1). (Smoothness). For any fixed $_ { \textbf { \em x } }$ and $j \in [ C ] , \Phi _ { j } ( \cdot ; x )$ is $\mathcal { C } ^ { 2 }$ -smooth; +(S5). (Learing rate condition). $\begin{array} { r } { \sum _ { t \geq 0 } \eta ( t ) = + \infty } \end{array}$ and $\eta ( t ) \leq H ( \mathcal { L } ( \pmb { \theta } ( t ) ) )$ . + +Here $H ( x ) : = \mu ( x ) / ( C _ { \eta } \kappa ( x ) )$ is defined to be the same as in Appendix $\mathrm { F }$ (when $\ell ( \cdot )$ is set to logistic loss) except that we use another $C _ { \eta }$ which will be specified later. + +We only need to show that Lemma F.2 and Lemma F.3 continue to hold. Using the same argument as we do for gradient flow, we can show that $\gamma ( t )$ and $\lambda ( x )$ do satisfy the propositions in Lemma F.2. + +For Lemma F.3, we first note the original definition of $C _ { \eta }$ involves $B _ { 0 } , B _ { 1 } , B _ { 2 }$ , which are undefined in the multi-class setting. So now we redefine them as + +$$ +\begin{array} { r l } & { B _ { 0 } : = \operatorname* { s u p } \left. s _ { n j } : \pmb \theta \in { \cal S } ^ { d - 1 } , n \in [ N ] , j \in [ C ] \right. , } \\ & { B _ { 1 } : = \operatorname* { s u p } \left. \left\| \nabla s _ { n j } \right\| _ { 2 } : \pmb \theta \in { \cal S } ^ { d - 1 } , n \in [ N ] , j \in [ C ] \right. , } \\ & { B _ { 2 } : = \operatorname* { s u p } \left. \left\| \nabla ^ { 2 } s _ { n j } \right\| _ { 2 } : \pmb \theta \in { \cal S } ^ { d - 1 } , n \in [ N ] , j \in [ C ] \right. . } \end{array} +$$ + +By the property of LSE, we can use $B _ { 0 } , B _ { 1 } , B _ { 2 }$ to bound $\tilde { q } _ { n } , \nabla \tilde { q } _ { n } , \nabla ^ { 2 } \tilde { q } _ { n }$ . + +$$ +\begin{array} { c } { \displaystyle \tilde { q } _ { n } \leq q _ { n } \leq B _ { 0 } \rho ^ { L } , } \\ { \| \nabla \tilde { q } _ { n } \| _ { 2 } = \left\| \sum _ { j = 1 } ^ { C } \frac { e ^ { - s _ { n j } } } { \sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } \nabla s _ { n j } \right\| _ { 2 } \leq \sum _ { j = 1 } ^ { C } \frac { e ^ { - s _ { n j } } } { \sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } \| \nabla s _ { n j } \| _ { 2 } \leq B _ { 1 } \rho ^ { L - 1 } , } \\ { \| \nabla ^ { 2 } \tilde { q } _ { n } \| _ { 2 } = \left\| \sum _ { j = 1 } ^ { C } \left( \frac { e ^ { - s _ { n j } } } { \sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } ( \nabla ^ { 2 } s _ { n j } - \nabla s _ { n j } \nabla s _ { n j } ^ { \top } ) + \frac { e ^ { - 2 s _ { n j } } } { \left( \sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } \right) ^ { 2 } } \nabla s _ { n j } \nabla s _ { n j } ^ { \top } \right) \right\| _ { 2 } } \\ { \leq B _ { 2 } \rho ^ { L - 2 } + 2 B _ { 1 } ^ { 2 } \rho ^ { 2 L - 2 } . } \end{array} +$$ + +Using these bounds, we can prove with the constant $C _ { \eta }$ defined as follows: + +$$ +\begin{array} { r l } & { C _ { \eta } : = \frac { 1 } { 2 } \left( \left( \left( \frac { B _ { 0 } } { \hat { \gamma } ( t _ { 0 } ) } \right) ^ { p } + \frac { p 2 ^ { p + 1 } } { \log { \frac { 1 } { \mathcal { L } ( t _ { 0 } ) } } } \right) B _ { 1 } ^ { 2 } + ( 2 \log { 2 } ) \left( \frac { B _ { 2 } } { \rho ( t _ { 0 } ) ^ { L } } + 2 B _ { 1 } ^ { 2 } \right) \right) \left( \frac { B _ { 0 } } { \hat { \gamma } ( t _ { 0 } ) } \right) ^ { p } M , } \\ & { \textsf { r e } M : = \operatorname* { m i n } \Big \{ \hat { \gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \Big \} . } \end{array} +$$ + +# H EXTENSION: MULTI-HOMOGENEOUS MODELS + +In this section, we extend our results to multi-homogeneous models. For this, the main difference from the proof for homogeneous models is that now we have to separate the norm of each homogeneous parts of the parameter, rather than consider them as a whole. So only a small part to proof needs to be changed. We focus on gradient flow, but it is worth to note that following the same argument, it is not hard to extend the results to gradient descent. + +Let smo $\Phi ( { \pmb w } _ { 1 } , \ldots , { \pmb w } _ { m } ; { \pmb x } )$ be arg $\left( k _ { 1 } , \ldots , k _ { m } \right)$ -homogeneous. Let (5) can be rewritten a $\rho _ { i } = \| \pmb { w } _ { i } \| _ { 2 }$ and $\begin{array} { r } { \hat { \pmb { w } } _ { i } = \frac { { \pmb w } _ { i } } { \| { \pmb w } _ { i } \| _ { 2 } } } \end{array}$ . The + +Definition H.1. For a multi-homogeneous model with loss function $\ell ( \cdot )$ satisfying (B3), the smoothed normalized margin $\widetilde \gamma ( \pmb \theta )$ of $\pmb \theta$ is defined as + +$$ +{ \tilde { \gamma } } ( \pmb { \theta } ) : = \frac { g ( \log \frac { 1 } { \mathcal { L } } ) } { \prod _ { i = 1 } ^ { m } \rho _ { i } ^ { k _ { i } } } = \frac { \ell ^ { - 1 } ( \mathcal { L } ) } { \prod _ { i = 1 } ^ { m } \rho _ { i } ^ { k _ { i } } } . +$$ + +We only prove the generalized version of Lemma 5.1 here. The other proofs are almost the same. + +Lemma H.2. For all $t > t _ { 0 }$ , + +$$ +{ \frac { d } { d t } } \log \rho _ { i } > 0 f o r a l l i \in [ m ] \quad a n d \quad { \frac { d } { d t } } \log { \tilde { \gamma } } \geq \sum _ { i = 1 } ^ { m } k _ { i } \left( { \frac { d } { d t } } \log \rho _ { i } \right) ^ { - 1 } \left\| { \frac { d { \hat { w } } _ { i } } { d t } } \right\| _ { 2 } ^ { 2 } . +$$ + +Proof. Note that ddt log ρi = 12ρ2 dρidt $\begin{array} { r } { \frac { d } { d t } \log \rho _ { i } = \frac { 1 } { 2 \rho _ { i } ^ { 2 } } \frac { d \rho _ { i } ^ { 2 } } { d t } = \frac { k _ { i } \nu ( t ) } { \rho _ { i } ^ { 2 } } } \end{array}$ by Theorem B.4. It simply follows from Lemma B.5 that $\begin{array} { r } { \frac { d } { d t } \log \rho > 0 } \end{array}$ for a.e. $t > t _ { 0 }$ . And it is easy to see that $\begin{array} { r } { \log { \tilde { \gamma } } = \log \left( g ( \log \frac { 1 } { \mathcal { L } } ) / \rho ^ { L } \right) } \end{array}$ exists for all + +$t \geq t _ { 0 }$ . By the chain rule and Lemma B.5, we have + +$$ +\begin{array} { r } { \frac { d } { d t } \log \tilde { \gamma } = \frac { d } { d t } \left( \log \left( g ( \log \frac { 1 } { \mathcal { L } } ) \right) - \displaystyle \sum _ { i = 1 } ^ { m } k _ { i } \log \rho _ { i } \right) = \frac { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } { g ( \log \frac { 1 } { \mathcal { L } } ) } \cdot \frac { 1 } { \mathcal { L } } \cdot \left( - \frac { d \mathcal { L } } { d t } \right) - \displaystyle \sum _ { i = 1 } ^ { m } \frac { k _ { i } ^ { 2 } \nu ( t ) } { \rho _ { i } ^ { 2 } } } \\ { \geq \frac { 1 } { \nu ( t ) } \cdot \left( - \frac { d \mathcal { L } } { d t } \right) - \displaystyle \sum _ { i = 1 } ^ { m } \frac { k _ { i } ^ { 2 } \nu ( t ) } { \rho _ { i } ^ { 2 } } } \\ { \geq \frac { 1 } { \nu ( t ) } \cdot \left( - \frac { d \mathcal { L } } { d t } - \displaystyle \sum _ { i = 1 } ^ { m } \frac { k _ { i } ^ { 2 } \nu ( t ) ^ { 2 } } { \rho _ { i } ^ { 2 } } \right) . } \end{array} +$$ + +On the one hand, $\begin{array} { r } { - \frac { d \mathcal { L } } { d t } = \sum _ { i = 1 } ^ { m } \left\| \frac { d \pmb { w } _ { i } } { d t } \right\| _ { 2 } ^ { 2 } } \end{array}$ for a.e. $t > 0$ by Lemma I.3; on the other hand, $k _ { i } \nu ( t ) =$ $\begin{array} { r } { \left. w _ { i } , \frac { d w _ { i } } { d t } \right. } \end{array}$ dt i=1 dt 2 by Theorem B.4. Combining these together yields + +$$ +\frac { d } { d t } \log \tilde { \gamma } \geq \frac { 1 } { \nu ( t ) } \sum _ { i = 1 } ^ { m } \left( \left\| \frac { d w _ { i } } { d t } \right\| _ { 2 } ^ { 2 } - \left. \hat { w } _ { i } , \frac { d w _ { i } } { d t } \right. ^ { 2 } \right) = \frac { 1 } { \nu ( t ) } \left\| ( I - \hat { w } _ { i } \hat { w } _ { i } ^ { \top } ) \frac { d w _ { i } } { d t } \right\| _ { 2 } ^ { 2 } . +$$ + +By the chain rule, $\begin{array} { r } { \frac { d \hat { \pmb { w } } _ { i } } { d t } = \frac { 1 } { \rho _ { i } } ( \pmb { I } - \hat { \pmb { w } _ { i } } \hat { \pmb { w } _ { i } } ^ { \top } ) \frac { d \pmb { w } _ { i } } { d t } } \end{array}$ i>) dwidt for a.e. t > 0. So we have + +$$ +\frac { d } { d t } \log \tilde { \gamma } \geq \sum _ { i = 1 } ^ { m } \frac { \rho _ { i } ^ { 2 } } { \nu ( t ) } \left\| \frac { d \hat { w } _ { i } } { d t } \right\| _ { 2 } ^ { 2 } = \sum _ { i = 1 } ^ { m } k _ { i } \left( \frac { d } { d t } \log \rho _ { i } \right) ^ { - 1 } \left\| \frac { d \hat { w } _ { i } } { d t } \right\| _ { 2 } ^ { 2 } . +$$ + +For cross-entropy loss, we can combine the proofs in Appendix $\mathbf { G }$ to show that Lemma H.2 holds if we use the following definition of the smoothed normalized margin: + +Definition H.3. For a multi-homogeneous model with cross-entropy, the smoothed normalized margin $\widetilde \gamma ( \pmb \theta )$ of $\pmb { \theta }$ is defined as + +$$ +\widetilde { \gamma } = \frac { \ell ^ { - 1 } ( \mathcal { L } ) } { \prod _ { i = 1 } ^ { m } \rho _ { i } ^ { k _ { i } } } = \frac { - \log ( e ^ { \mathcal { L } } - 1 ) } { \prod _ { i = 1 } ^ { m } \rho _ { i } ^ { k _ { i } } } . +$$ + +where $\ell ^ { - 1 } ( \cdot )$ is the inverse function of the logistic loss $\ell ( \cdot )$ . + +The only place we need to change in the proof for Lemma H.2 is that instead of using Lemma B.5, we need to prove $\begin{array} { r } { \nu ( t ) \geq \frac { g ( \log \frac { 1 } { \mathcal { L } } ) } { g ^ { \prime } ( \log \frac { 1 } { \mathcal { L } } ) } \mathcal { L } } \end{array}$ g(log 1L )g0(log 1 ) L in a similar way as in Lemma G.2. The other parts of the proof are exactly the same as before. + +# I CHAIN RULES FOR NON-DIFFERENTIABLE FUNCTIONS + +In this section, we provide some background on the chain rule for non-differentiable functions. The ordinary chain rule for differentiable functions is a very useful formula for computing derivatives in calculus. However, for non-differentiable functions, it is difficult to find a natural definition of subdifferential so that the chain rule equation holds exactly. To solve this issue, Clarke proposed Clarke’s subdifferential (Clarke, 1975; 1990; Clarke et al., 2008) for locally Lipschitz functions, for which the chain rule holds as an inclusion rather than an equation: + +Theorem I.1 (Theorem 2.3.9 and 2.3.10 of (Clarke, 1990)). Let $z _ { 1 } , \dots , z _ { n } : \mathbb { R } ^ { d } \to \mathbb { R }$ and $f :$ $\mathbb { R } ^ { n } \to \mathbb { R }$ be locally Lipschitz functions. Let $( f \circ z ) ( { \pmb x } ) = f ( z _ { 1 } ( { \pmb x } ) , \ldots , z _ { n } ( { \pmb x } ) )$ be the composition of $f$ and $_ z$ . Then, + +$$ +\partial ^ { \circ } ( f \circ z ) ( \pmb { x } ) \subseteq \mathrm { c o n v } \left\{ \sum _ { i = 1 } ^ { n } \alpha _ { i } \pmb { h } _ { i } : \pmb { \alpha } \in \partial ^ { \circ } f ( z _ { 1 } ( \pmb { x } ) , \ldots , z _ { n } ( \pmb { x } ) ) , \pmb { h } _ { i } \in \partial ^ { \circ } z _ { i } ( \pmb { x } ) \right\} . +$$ + +For analyzing gradient flow, the chain rule is crucial. For a differentiable loss function $\mathcal { L } ( \pmb { \theta } )$ , we the chain rule that the function value keeps decreasing along the gradient flow : $\begin{array} { r l r } { \mathrm { ~ } } & { { } } & { \frac { d \pmb \theta ( t ) } { d t } = } \end{array}$ $- \nabla \mathcal { L } ( \pmb \theta ( t ) )$ + +$$ +\frac { d \mathcal { L } ( \pmb { \theta } ( t ) ) } { d t } = \left. \nabla \mathcal { L } ( \pmb { \theta } ( t ) ) , \frac { d \pmb { \theta } ( t ) } { d t } \right. = - \left\| \frac { d \pmb { \theta } ( t ) } { d t } \right\| _ { 2 } ^ { 2 } . +$$ + +But for locally Lipschitz functions which could be non-differentiable, (25) may not hold in general since Theorem I.1 only holds for an inclusion. + +Following (Davis et al., 2020; Drusvyatskiy et al., 2015), we consider the functions that admit a chain rule for any arc. + +Definition I.2 (Chain Rule). A locally Lipschitz function $f : { \mathbb { R } ^ { d } } \to { \mathbb { R } }$ admits a chain rule if for any arc $z : [ 0 , + \infty ) \to \mathbb { R } ^ { d }$ , + +$$ +\forall h \in \partial ^ { \circ } f ( z ( t ) ) : ( f \circ z ) ^ { \prime } ( t ) = \langle h , z ^ { \prime } ( t ) \rangle +$$ + +holds for a.e. $t > 0$ . + +It is shown in (Davis et al., 2020; Drusvyatskiy et al., 2015) that a generalized version of (25) holds for such functions: + +Lemma I.3 (Lemma 5.2 (Davis et al., 2020)). Let $\mathcal { L } : \mathbb { R } ^ { d } \mathbb { R }$ be a locally Lipschitz function that admits a chain rule. Let $\pmb \theta : [ 0 , + \infty ) \mathbb R ^ { d }$ be the gradient flow on $\mathcal { L }$ : + +$$ +\frac { d \pmb { \theta } ( t ) } { d t } \in - \partial ^ { \circ } \mathcal { L } ( \pmb { \theta } ( t ) ) \qquad f o r a . e . \ t \geq 0 , +$$ + +Then + +$$ +{ \frac { d { \mathcal { L } } ( \pmb { \theta } ( t ) ) } { d t } } = - \left\| { \frac { d \pmb { \theta } ( t ) } { d t } } \right\| _ { 2 } ^ { 2 } = - \operatorname* { m i n } \{ \| \pmb { h } \| _ { 2 } ^ { 2 } : \pmb { h } \in \partial ^ { \circ } { \mathcal { L } } ( \pmb { \theta } ( t ) ) \} +$$ + +holds for a.e. $t > 0$ . + +We can see that ${ \mathcal { C } } ^ { 1 }$ -smooth functions admit chain rules. As shown in (Davis et al., 2020), if a locally Lipschitz function is subdifferentiablly regular or Whitney $\mathcal { C } ^ { 1 }$ -stratifiable, then it admits a chain rule. The latter one includes a large family of functions, e.g., semi-algebraic functions, semianalytic functions, and definable functions in an o-minimal structure (Coste, 2002; van den Dries & Miller, 1996). + +It is worth noting that the class of functions that admits chain rules is closed under composition. +This is indeed a simple corollary of Theorem I.1. + +Theorem I.4. Let $z _ { 1 } , \dots , z _ { n } : \mathbb { R } ^ { d } \to \mathbb { R }$ and $f : \mathbb { R } ^ { n } \mathbb { R }$ be locally Lipschitz functions and assume all of them admit chain rules. Let $( f \circ z ) ( \pmb { x } ) = f ( z _ { 1 } ( \pmb { x } ) , \dots , z _ { n } ( \pmb { x } ) )$ be the composition of $f$ and $_ z$ . Then $f \circ z$ also admits a chain rule. + +Proof. We can see that $f \circ z$ is locally Lipschitz. Let $\pmb { x } : [ 0 , + \infty ) \mathbb { R } ^ { d } , t \mapsto \pmb { x } ( t )$ be an arc. First, we show that $z \circ \dot { \pmb x } : [ 0 , + \infty ) \mathbb { R } ^ { d } , t \mapsto z ( { \pmb x } ( t ) )$ is also an arc. For any closed sub-interval $I$ , $z ( \pmb { x } ( I ) )$ must be contained in a compact set $U$ . Then it can be shown that the locally Lipschitz continuous function $_ z$ is (globally) Lipschitz continuous on $U$ . By the fact that the composition of a Lipschitz continuous and an absolutely continuous function is absolutely continuous, $z \circ x$ is absolutely continuous on $I$ , and thus it is an arc. + +Since $f$ and $_ z$ admit chain rules on arcs $_ { z \mathrm { ~ O ~ } \pmb { x } }$ and $_ { \textbf { \em x } }$ respectively, the following holds for a.e. $t > 0$ + +$$ +\begin{array} { r l } { \forall \pmb { \alpha } \in \partial ^ { \circ } f ( \pmb { z } ( \pmb { x } ( t ) ) ) : } & { ( f \circ ( \pmb { z } \circ \pmb { x } ) ) ^ { \prime } ( t ) = \langle \pmb { \alpha } , ( \pmb { z } \circ \pmb { x } ) ^ { \prime } ( t ) \rangle , } \\ { \forall \pmb { h } _ { i } \in \partial ^ { \circ } z _ { i } ( \pmb { x } ( t ) ) : } & { ( z _ { i } \circ \pmb { x } ) ^ { \prime } ( t ) = \langle \pmb { h } _ { i } , \pmb { x } ^ { \prime } ( t ) \rangle . } \end{array} +$$ + +Combining these we obtain that for a.e. $t > 0$ , + +$$ +( f \circ z \circ x ) ^ { \prime } ( t ) = \sum _ { i = 1 } ^ { n } \alpha _ { i } \left. h _ { i } , x ^ { \prime } ( t ) \right. , +$$ + +for all $\alpha ~ \in ~ \partial ^ { \circ } f ( z ( { \pmb x } ( t ) ) )$ and for all $\begin{array} { r } { \pmb { h } _ { i } \in \partial ^ { \circ } z _ { i } ( \pmb { x } ( t ) ) } \end{array}$ . The RHS can be rewritten as $\textstyle \langle \sum _ { i = 1 } ^ { n } \alpha _ { i } \pmb { h } _ { i } , \pmb { x } ^ { \prime } ( t ) \rangle$ . By Theorem I.1, every $k \in \partial ^ { \circ } ( f \circ z ) ( { \pmb x } ( t ) )$ can be written as a convex combination of a finite set of points in the form of $\scriptstyle \sum _ { i = 1 } ^ { n } \alpha _ { i } h _ { i }$ . So $( f \circ z \circ x ) ^ { \prime } ( t ) = \langle k , x ^ { \prime } ( t ) \rangle$ holds for a.e. $t > 0$ . □ + +# J MEXICAN HAT + +In this section, we give an example to illustrate that gradient flow does not necessarily converge in direction, even for $C ^ { \infty }$ -smooth homogeneous models. + +It is known that gradient flow (or gradient descent) may not converge to any point even when optimizing an $C ^ { \infty }$ function (Curry, 1944; Zoutendijk, 1976; Palis & De Melo, 2012; Absil et al., 2005). One famous counterexample is the “Mexican Hat” function described in (Absil et al., 2005): + +$$ +f ( u , v ) : = f ( r \cos \varphi , r \sin \varphi ) : = \left\{ { \begin{array} { l l } { e ^ { - { \frac { 1 } { 1 - r ^ { 2 } } } } \left( 1 - C ( r ) \sin \left( \varphi - { \frac { 1 } { 1 - r ^ { 2 } } } \right) \right) \quad } & { r < 1 } \\ { 0 \quad } & { r \geq 1 } \end{array} } \right. +$$ + +where $\begin{array} { r } { C ( r ) = \frac { 4 r ^ { 4 } } { 4 r ^ { 4 } + ( 1 - r ^ { 2 } ) ^ { 4 } } \in [ 0 , 1 ] } \end{array}$ . It can be shown that $f$ is ${ \mathcal { C } } ^ { \infty }$ -smooth on $\mathbb { R } ^ { 2 }$ but not analytic. +See Figure 2 for a plot for $f ( u , v )$ . + +![](images/fdd8b960a9c32bef3e12b8b8208114e0c2be68507e93e7f18e6e76881a178ab2.jpg) +Figure 2: A plot for the Mexican Hat function $f ( u , v )$ . + +However, the Maxican Hat function is not homogeneous, and Absil et al. (2005) did not consider the directional convergence, either. To make it homogeneous, we introduce an extra variable $z$ , and normalize the parameter before evaluate $f$ . In particular, we fix $L > 0$ and define + +$$ +h ( \theta ) = h ( x , y , z ) = \rho ^ { L } ( 1 - f ( u , v ) ) \qquad \mathrm { w h e r e } \quad u = x / \rho , \quad v = y / \rho , \quad \rho = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } . +$$ + +Theorem J.1. Consider gradient flow on $\begin{array} { r } { { \mathcal L } ( \pmb { \theta } ) = \sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( \pmb { \theta } ) } } \end{array}$ , where $q _ { n } ( \pmb \theta ) \ = \ h ( \pmb \theta )$ for all $n \in [ N ]$ . Suppose the polar representation of $( u , v )$ is $( r \cos \varphi , r \sin \varphi )$ . If $0 ~ < ~ r ~ < ~ 1$ and $\textstyle \varphi = { \frac { 1 } { 1 - r ^ { 2 } } }$ holds at time $t = 0$ , then $\frac { \pmb { \theta } ( t ) } { \| \pmb { \theta } ( t ) \| _ { 2 } }$ does not converge to any point, and the limit points of $\big \{ \frac { \pmb \theta ( t ) } { \lVert \pmb \theta ( t ) \rVert _ { 2 } } : t > 0 \big \}$ form a circle $\left\{ \left( x , y , z \right) \in S ^ { 2 } : x ^ { 2 } + y ^ { 2 } = 1 , z = 0 \right\}$ . + +Proof. Define $\textstyle \psi = \varphi - { \frac { 1 } { 1 - r ^ { 2 } } }$ . Our proof consists of two parts, following from the idea in (Absil et al., 2005). First, we show that $\begin{array} { r } { { \frac { d \psi } { d t } } = 0 } \end{array}$ as long as $\psi = 0$ . Then we can infer that $\psi = 0$ for all $t \geq 0$ . Next, we show that $r 1$ as $t \to + \infty$ . Using $\psi = 0$ , we know that the polar angle $\varphi + \infty$ as $t \to + \infty$ . Therefore, $( u , v )$ circles around $\{ ( u , v ) : u ^ { 2 } + v ^ { 2 } = 1 \}$ , and thus it does not converge. + +Proof for $\begin{array} { r } { \frac { d \psi } { d t } = 0 } \end{array}$ . For convenience, we use $w$ to denote $z / \rho$ . By simple calculation, we have the following formulas for partial derivatives: + +$$ +\left\{ \begin{array} { l l } { \frac { \partial h } { \partial x } = \rho ^ { L - 1 } \left( L u ( 1 - f ) - ( 1 - u ^ { 2 } ) \frac { \partial f } { \partial u } + u v \frac { \partial f } { \partial v } \right) } \\ { \frac { \partial h } { \partial y } = \rho ^ { L - 1 } \left( L v ( 1 - f ) + u v \frac { \partial f } { \partial u } - ( 1 - v ^ { 2 } ) \frac { \partial f } { \partial v } \right) } \\ { \frac { \partial h } { \partial z } = \rho ^ { L - 1 } \left( L w ( 1 - f ) + u w \frac { \partial f } { \partial u } + v w \frac { \partial f } { \partial v } \right) } \end{array} \right. \qquad \left\{ \begin{array} { l l } { \frac { \partial f } { \partial u } = \frac { u } { r } \cdot \frac { \partial f } { \partial r } - \frac { v } { r ^ { 2 } } \cdot \frac { \partial f } { \partial \varphi } } \\ { \frac { \partial f } { \partial v } = \frac { v } { r } \cdot \frac { \partial f } { \partial r } + \frac { u } { r ^ { 2 } } \cdot \frac { \partial f } { \partial \varphi } } \end{array} \right. +$$ + +For gradient flow, we have + +$$ +\begin{array} { r l } & { \frac { d \theta } { d t } = N e ^ { - h } \nabla h } \\ & { \frac { d u } { d t } = \frac { 1 } { \rho } \left( ( 1 - u ^ { 2 } ) \frac { d x } { d t } - u v \frac { d y } { d t } - u w \frac { d z } { d t } \right) = N e ^ { - h } \rho ^ { L - 2 } \left( - ( 1 - u ^ { 2 } ) \frac { \partial f } { \partial u } + u v \frac { \partial f } { \partial v } \right) } \\ & { \frac { d v } { d t } = \frac { 1 } { \rho } \left( ( 1 - v ^ { 2 } ) \frac { d y } { d t } - u v \frac { d x } { d t } - v w \frac { d z } { d t } \right) = N e ^ { - h } \rho ^ { L - 2 } \left( - ( 1 - v ^ { 2 } ) \frac { \partial f } { \partial v } + u v \frac { \partial f } { \partial u } \right) } \end{array} +$$ + +By writing down the movement of $( u , v )$ in the polar coordinate system, we have + +$$ +\begin{array} { r l } & { \frac { d r } { d t } = \frac { u } { r } \cdot \frac { d u } { d t } + \frac { v } { r } \cdot \frac { d v } { d t } } \\ & { \quad = - \frac { N e ^ { - h } \rho ^ { L - 2 } } { r } \left( u ( 1 - r ^ { 2 } ) \frac { \partial f } { \partial u } + v ( 1 - r ^ { 2 } ) \frac { \partial f } { \partial v } \right) } \\ & { \quad = - N e ^ { - h } \rho ^ { L - 2 } ( 1 - r ^ { 2 } ) \frac { \partial f } { \partial r } } \\ & { \frac { d \varphi } { d t } = - \frac { v } { r ^ { 2 } } \cdot \frac { d u } { d t } + \frac { u } { r ^ { 2 } } \cdot \frac { d v } { d t } } \\ & { \quad = \frac { N e ^ { - h } \rho ^ { L - 2 } } { r ^ { 2 } } \left( v \frac { \partial f } { \partial u } - u \frac { \partial f } { \partial v } \right) } \\ & { \quad = - N e ^ { - h } \rho ^ { L - 2 } \frac { 1 } { r ^ { 2 } } \cdot \frac { \partial f } { \partial \varphi } } \end{array} +$$ + +For $\psi = 0$ , the partial derivatives of $f$ with respect to $r$ and $\varphi$ can be evaluated as follows: + +$$ +\begin{array} { l } { \displaystyle \frac { \partial f } { \partial r } = \frac { d } { d r } \left( e ^ { - \frac { 1 } { 1 - r ^ { 2 } } } \right) - e ^ { - \frac { 1 } { 1 - r ^ { 2 } } } C ^ { \prime } ( r ) \sin \psi - e ^ { - \frac { 1 } { 1 - r ^ { 2 } } } C ( r ) \cos \psi \cdot \frac { \partial \psi } { \partial r } } \\ { = \frac { d } { d r } \left( e ^ { - \frac { 1 } { 1 - r ^ { 2 } } } \right) + e ^ { - \frac { 1 } { 1 - r ^ { 2 } } } C ( r ) \frac { d } { d r } \left( \frac { 1 } { 1 - r ^ { 2 } } \right) } \\ { = - \frac { 2 r } { ( 1 - r ^ { 2 } ) ^ { 2 } } ( 1 - C ( r ) ) e ^ { - \frac { 1 } { 1 - r ^ { 2 } } } . } \\ { \displaystyle \frac { \partial f } { \partial \varphi } = - e ^ { - \frac { 1 } { 1 - r ^ { 2 } } } C ( r ) \cos \psi \cdot \frac { \partial \psi } { \partial \varphi } } \\ { = - e ^ { - \frac { 1 } { 1 - r ^ { 2 } } } C ( r ) . } \end{array} +$$ + +So if $\psi = 0$ , then dψdt = 0 by the direct calculation below: + +$$ +{ \begin{array} { r l } & { { \frac { d \psi } { d t } } = { \frac { d \varphi } { d t } } - { \frac { d } { d r } } \left( { \frac { 1 } { 1 - r ^ { 2 } } } \right) \cdot { \frac { d r } { d t } } } \\ & { \qquad = N e ^ { - h } \rho ^ { L - 2 } \left( - { \frac { 1 } { r ^ { 2 } } } { \frac { \partial f } { \partial \varphi } } + { \frac { 2 r } { ( 1 - r ^ { 2 } ) ^ { 2 } } } { \frac { \partial f } { \partial r } } \right) } \\ & { \qquad = N e ^ { - h } \rho ^ { L - 2 } e ^ { - { \frac { 1 } { 1 - r ^ { 2 } } } } \left( \left( { \frac { 1 } { r ^ { 2 } } } + { \frac { 4 r ^ { 2 } } { ( 1 - r ^ { 2 } ) ^ { 4 } } } \right) C ( r ) - { \frac { 4 r ^ { 2 } } { ( 1 - r ^ { 2 } ) ^ { 4 } } } \right) = 0 . } \end{array} } +$$ + +Proof for $r 1$ . Let $( \bar { u } , \bar { v } )$ be a convergent point of $\{ ( u , v ) : t \geq 0 \}$ . Define $\bar { r } = \sqrt { u ^ { 2 } + v ^ { 2 } }$ . It is easy to see that $r \leq 1$ from the normalization of $\pmb { \theta }$ in the definition. According to Theorem 4.4, we know that $( \bar { u } , \bar { v } )$ is a stationary point of $\bar { \gamma } ( u ( t ) , v ( t ) ) \ : = \ : 1 \ : - \ : f ( u ( t ) , v ( t ) )$ . If $\bar { r } \ = \ 0$ , then $f ( \bar { u } , \bar { v } ) > f ( u ( 0 ) , v ( 0 ) )$ , which contradicts to the monotonicity of $\tilde { \gamma } ( t ) = \bar { \gamma } ( t ) = 1 - f ( u ( t ) , v ( t ) )$ . If $\bar { r } < 1 $ , then $\bar { \psi } = 0$ (defined as $\psi$ for $( \bar { u } , \bar { v } ) .$ ). So − 2r(1−r2)2 (1 − C(r))e− 1−r2 6= 0, which again leads to a contradiction. Therefore, $\bar { r } = 1$ , and thus $r 1$ . □ + +# K EXPERIMENTS + +To validate our theoretical results, we conduct several experiments. We mainly focus on MNIST dataset. We trained two models with Tensorflow. The first one (called the CNN with bias) is a standard 4-layer CNN with exactly the same architecture as that used in MNIST Adversarial Examples + +Challenge5. The layers of this model can be described as conv-32 with filter size $5 \times 5$ , max-pool, conv-64 with filter size $3 \times 3$ , max-pool, fc-1024, fc-10 in order. Notice that this model has bias terms in each layer, and thus does not satisfy homogeneity. To make its outputs homogeneous to its parameters, we also trained this model after removing all the bias terms except those in the first layer (the modified model is called the CNN without bias). Note that keeping the bias terms in the first layer prevents the model to be homogeneous in the input data while retains the homogeneity in parameters. We initialize all layer weights by He normal initializer (He et al., 2015) and all bias terms by zero. In training the models, we use SGD with batch size 100 without momentum. We normalize all the images to $[ 0 , 1 ] ^ { 3 2 \times 3 2 }$ by dividing 255 for each pixel. + +# K.1 EVALUATION FOR NORMALIZED MARGIN + +In the first part of our experiments, we evaluate the normalized margin every few epochs to see how it changes over time. From now on, we view the bias term in the first layer as a part of the weight in the first layer for convenience. Observe that the CNN without bias is multi-homogeneous in layer weights (see (4) in Section 4.4). So for the CNN without bias, we define the normalized margin $\bar { \gamma }$ as the margin divided by the product of the $L ^ { 2 }$ -norm of all layer weights. Here we compute the $L ^ { 2 }$ -norm of a layer weight parameter after flattening it into a one-dimensional vector. For the CNN with bias, we still compute the smoothed normalized margin in this way. When computing the $L ^ { 2 }$ - norm of every layer weight, we simply ignore the bias terms if they are not in the first layer. For completeness, we include the plots for the normalized margin using the original definition (2) in Figure 3 and 4. + +![](images/537667231ed5f10711f6cfa0e044280666fd050879ce6aa6d81b43708667e02e.jpg) +Figure 3: Training CNNs with and without bias on MNIST, using SGD with learning rate 0.01. The training accuracy (left) increases to $1 0 0 \%$ after about 100 epochs, and the normalized margin with the original definition (right) keeps increasing after the model is fitted. + +![](images/6a977df3714686b1cdc63609ca27ce2478cbf361e5fd64ab3db15c2a4a1067fc.jpg) +Figure 4: Training CNNs with and without bias on MNIST, using SGD with the loss-based learning rate scheduler. The training accuracy (left) increases to $1 0 0 \%$ after about 20 epochs, and the normalized margin with the original definition (middle) increases rapidly after the model is fitted. The right figure shows the change of the relative learning rate $\alpha ( t )$ (see (27) for its definition) during training. + +SGD with Constant Learning Rate. We first train the CNNs using SGD with constant learning rate 0.01. After about 100 epochs, both CNNs have fitted the training set. After that, we can see that the normalized margins of both CNNs increase. However, the growth rate of the normalized margin is rather slow. The results are shown in Figure 1 in Section 1. We also tried other learning rates other than 0.01, and similar phenomena can be observed. + +SGD with Loss-based Learning Rate. Indeed, we can speed up the training by using a proper scheduling of learning rates for SGD. We propose a heuristic learning rate scheduling method, called the loss-based learning rate scheduling. The basic idea is to find the maximum possible learning rate at each epoch based on the current training loss (in a similar way as the line search method). See Appendix L.1 for the details. As shown in Figure 1, SGD with loss-based learning rate scheduling decreases the training loss exponentially faster than SGD with constant learning rate. Also, a rapid growth of normalized margin is observed for both CNNs. Note that with this scheduling the training loss can be as small as $\overline { { 1 0 ^ { - 8 0 0 } } }$ , which may lead to numerical issues. To address such issues, we applied some re-parameterization tricks and numerical tricks in our implementation. See Appendix L.2 for the details. + +Experiments on CIFAR-10. To verify whether the normalized margin is increasing in practice, we also conduct experiments on CIFAR-10. We use a modified version of VGGNet-16. The layers of this model can be described as conv- $6 4 \times 2$ , max-pool, conv- $1 2 8 \times 2$ , max-pool, conv-256 $\times 3$ , max-pool, conv- $. 5 1 2 \times 3$ , $\mathtt { m a x - p o o l }$ , $\mathsf { c o n v } { - } 5 1 2 \times 3$ , $\mathtt { m a x - p o o l }$ , $\pm _ { \mathrm { C - 1 0 } }$ in order, where each conv has filter size $3 \times 3$ . We train two networks: one is exactly the same as the VGGNet we described, and the other one is the VGGNet without any bias terms except those in the first layer (similar as in the experiments on MNIST). The experiment results are shown in Figure 5 and 6. We can see that the normalize margin is increasing over time. + +![](images/58ff1562f78c324281e18d2aa99b831e549164308703f02119c458659d4d2e3c.jpg) +Figure 5: Training VGGNet with and without bias on CIFAR-10, using SGD with learning rate 0.1. + +![](images/495a9fd0459b12209dbeccc75dc0894e2a64a5daed8bfc4e279791f66c40feb6.jpg) +Figure 6: Training VGGNet with and without bias on CIFAR-10, using SGD with the loss-based learning rate scheduler. + +Test Accuracy. Previous works on margin-based generalization bounds (Neyshabur et al., 2018; Bartlett et al., 2017; Golowich et al., 2018; Li et al., 2018a; Wei et al., 2019; Banburski et al., + +2019) usually suggest that a larger margin implies a better generalization bound. To see whether the generalization error also gets smaller in practice, we plot train and test accuracy for both MNIST and CIFAR-10. As shown in Figure 7, the test accuracy changes only slightly after training with lossbased learning rate scheduling for 10000 epochs, although the normalized margin does increase a lot. We leave it as a future work to study this interesting gap between margin-based generalization bound and generalization error. Concurrent to this work, Wei & Ma (2020) proposed a generalization bound based on a new notion of margin called all-layer margin, and showed via experiments that enlarging all-layer margin can indeed improve generalization. It would be an interesting research direction to study how different definitions of margin may lead to different generalization abilities. + +![](images/30c454b39b58b3eb361944408c107dca51e6a4f80df7e6e0667b0ce844855b9b.jpg) +Figure 7: (Left). Training and test accuracy during training CNNs without bias on MNIST, using SGD with the loss-based learning rate scheduler. Every number is averaged over 10 runs. (Right). Training and test accuracy during training VGGNet without bias on CIFAR-10, using SGD with the loss-based learning rate scheduler. Every number is averaged over 3 runs. + +# K.2 EVALUATION FOR ROBUSTNESS + +Recently, robustness of deep learning has received considerable attention (Szegedy et al., 2013; Biggio et al., 2013; Athalye et al., 2018), since most state-of-the-arts deep neural networks are found to be very vulnerable against small but adversarial perturbations of the input points. In our experiments, we found that enlarging the normalized margin can improve the robustness. In particular, by simply training the neural network for a longer time with our loss-based learning rate, we observe noticeable improvements of $L ^ { 2 }$ -robustness on both the training set and test set. + +We first elaborate the relationship between the normalized margin and the robustness from a theoretical perspective. For a data point $\boldsymbol { z } = ( \boldsymbol { x } , y )$ , we can define the robustness (with respect to some norm $\| \cdot \| )$ of a neural network $\Phi$ for $_ z$ to be + +$$ +R _ { \theta } ( z ) : = \operatorname* { i n f } _ { x ^ { \prime } \in X } \left\{ \| x - x ^ { \prime } \| : ( x ^ { \prime } , y ) { \mathrm { ~ i s ~ m i s c l a s s i f i e d } } \right\} . +$$ + +where $X$ is the data domain (which is $[ 0 , 1 ] ^ { 3 2 \times 3 2 }$ for MNIST). It is well-known that the normalized margin is a lower bound of $L ^ { 2 }$ -robustness for fully-connected networks (See, e.g., Theorem 4 in (Sokolic et al., 2017)). Indeed, a general relationship between those two quantities can be easily shown. Note that a data point $_ { z }$ is correctly classified iff the margin for $_ z$ , denoted as $q _ { \pmb { \theta } } ( z )$ , is larger than 0. For homogeneous models, the margin $q _ { \pmb { \theta } } ( z )$ and the normalized margin $q _ { \hat { \theta } } ( z )$ for $_ { \textbf { \em x } }$ have the same sign. If $q _ { \hat { \theta } } ( \cdot ) : \mathbb { R } ^ { d _ { \mathrm { x } } } \mathbb { R }$ is $\beta$ -Lipschitz (with respect to some norm $\| \cdot \| )$ , then it is easy to see that $R _ { \pmb \theta } ( z ) \ge q _ { \hat { \pmb \theta } } ( z ) / \beta$ . This suggests that improving the normalize margin on the training set can improve the robustness on the training set. Therefore, our theoretical analysis suggests that training longer can improve the robustness of the model on the training set. + +This observation does match with our experiment results. In the experiments, we measure the $L ^ { 2 }$ - robustness of the CNN without bias for the first time its loss decreases below $1 0 ^ { - 1 0 }$ , $1 0 ^ { - 1 5 }$ , $1 0 ^ { - 2 0 }$ , $1 0 ^ { - 1 2 0 }$ (labelled as $\mathtt { m o d e l - 1 }$ to $\mathtt { m o d e l - 4 }$ respectively). We also measure the $L ^ { 2 }$ -robustness for the final model after training for 10000 epochs (labelled as mode $\boldsymbol { \perp } - \boldsymbol { 5 }$ ), whose training loss is about $1 0 ^ { - 8 8 2 }$ . The normalized margin of each model is monotone increasing with respect to the number of epochs, as shown in Table 1. + +![](images/a031802da08e0e1c7aacc79fcc3841f0f08566e90e8c918e7dfc0c538827da67.jpg) +Figure 8: $L ^ { 2 }$ -robustness of the models of CNNs without bias trained for different number of epochs (see Table 1 for the statistics of each model). Figures on the first row show the robust accuracy on the training set, and figures on the second row show that on the test set. On every row, the left figure and the right figure plot the same curves but they are in different scales. From model-1 to model-4, noticeable robust accuracy improvements can be observed. The improvement of model-5 upon model-4 is marginal or nonexistent for some , but the improvement upon model-1 is always significant. + +Table 1: Statistics of the CNN without bias after training for different number of epochs. + +
model namenumber of epochstrain lossnormalized margintrain acctest acc
model-13810-10.045.65× 10-5100%99.3%
model-27510-15.129.50 ×10-5100%99.3%
model-310710-20.071.30 × 10-4100%99.3%
model-493510- -120.014.61 × 10-4100%99.2%
model-51000010 -881.511.18 × 10-3100%99.1%
+ +We use the standard method for evaluating $L ^ { 2 }$ -robustness in (Carlini & Wagner, 2017) and the source code from the authors with default hyperparameters6. We plot the robust accuracy (the percentage of data with robustness $> \epsilon \bar { }$ ) for the training set in the figures on the first row of Figure 8. It can be seen from the figures that for small $\epsilon$ (e.g., $\epsilon < 0 . 3 )$ ), the relative order of robust accuracy is just the order of model-1 to model-5. For relatively large $\epsilon$ (e.g., $\epsilon > 0 . 3$ ), the improvement of model-5 upon model-2 to model-4 becomes marginal or nonexistent in certain intervals of $\epsilon$ , but model-1 to model-4 still have an increasing order of robust accuracy and the improvement of model-5 upon model-1 is always significant. This shows that training longer can help to improve the $L ^ { 2 }$ -robust accuracy on the training set. + +We also evaluate the robustness on the test set, in which a misclassified test sample is considered to have robustness 0, and plot the robust accuracy in the figures on the second row of Figure 8. It can be seen from the figures that for small $\epsilon$ (e.g., $\epsilon < 0 . 2 $ ), the curves of the robust accuracy of model-1 to model-5 are almost indistinguishable. However, for relatively large $\epsilon$ (e.g., $\epsilon > 0 . 2$ ), again, model-1 to model-4 have an increasing order of robust accuracy and the improvement of model-5 upon model-1 is always significant. This shows that training longer can also help to improve the $\bar { L } ^ { 2 }$ -robust accuracy on the test set. + +We tried various different settings of hyperparameters for the evaluation method (including different learning rates, different binary search steps, etc.) and we observed that the shapes and relative positions of the curves in Figure 8 are stable across different hyperparameter settings. + +It is worth to note that the normalized margin and robustness do not grow in the same speed in our experiments, although the theory suggests $R _ { \pmb \theta } ( \pmb z ) \geq q _ { \hat { \pmb \theta } } ( \pmb z ) / \beta$ . This may be because the Lipschitz constant $\beta$ (if defined locally) is also changing during training. Combining training longer with existing techniques for constraining Lipschitz number (Anil et al., 2019; Cisse et al., 2017) could potentially alleviate this issue, and we leave it as a future work. + +# L ADDITIONAL EXPERIMENTAL DETAILS + +In this section, we provide additional details of our experiments. + +# L.1 LOSS-BASED LEARNING RATE SCHEDULING + +The intuition of the loss-based learning rate scheduling is as follows. If the training loss is $\alpha$ - smooth, then optimization theory suggests that we should set the learning rate to roughly $1 / \alpha$ . For a homogeneous model with cross-entropy loss, if the training accuracy is $1 0 0 \%$ at $\pmb \theta$ , then a simple calculation can show that the smoothness (the $L ^ { 2 }$ -norm of the Hessian matrix) at $\pmb \theta$ is $\mathcal { \dot { O } } ( \bar { \mathcal { L } } { \cdot } \mathrm { p o l y } ( \bar { \rho } ) )$ , where $\bar { \mathcal { L } }$ is the average training loss and $\operatorname { p o l y } ( \rho )$ is some polynomial. Motivated by this fact, we parameterize the learning rate $\eta ( t )$ at epoch $t$ as + +$$ +\eta ( t ) : = \frac { \alpha ( t ) } { \bar { \mathcal { L } } ( t - 1 ) } , +$$ + +where $\bar { \mathcal { L } } ( t - 1 )$ is the average training loss at epoch $t - 1$ , and $\alpha ( t )$ is a relative learning rate to be tuned (Similar parameterization has been considiered in (Nacson et al., 2019b) for linear model). The loss-based learning rate scheduling is indeed a variant of line search. In particular, we initialize $\alpha ( 0 )$ by some value, and do the following at each epoch $t$ : + +Step 1. Initially $\alpha ( t ) \alpha ( t - 1 )$ ; Let $\bar { \mathcal { L } } ( t - 1 )$ be the training loss at the end of the last epoch; +Step 2. Run SGD through the whole training set with learning rate $\eta ( t ) : = \alpha ( t ) / \mathcal { L } ( t - 1 )$ ; +Step 3. Evaluate the training loss $\bar { \mathcal { L } } ( t )$ on the whole training set; +Step 4. If $\bar { \mathcal { L } } ( t ) < \bar { \mathcal { L } } ( t - 1 )$ , $\alpha ( t ) \gets \alpha ( t ) \cdot r _ { u }$ and end this epoch; otherwise, $\alpha ( t ) \gets \alpha ( t ) / r _ { d }$ and go to Step 2. + +In all our experiments, we set $\alpha ( 0 ) : = 0 . 1 , r _ { u } : = 2 ^ { 1 / 5 } \approx 1 . 1 4 9 , r _ { d } : = 2 ^ { 1 / 1 0 } \approx 1 . 0 7 2 .$ This specific choice of those hyperparameters is not important; other choices can only affact the computational efficiency, but not the overall tendency of normalized margin. + +# L.2 ADDRESSING NUMERICAL ISSUES + +Since we are dealing with extremely small loss (as small as $1 0 ^ { - 8 0 0 }$ ), the current Tensorflow implementation would run into numerical issues. To address the issues, we work as follows. Let $\bar { \mathcal { L } } _ { B } ( \pmb { \theta } )$ be the (average) training loss within a batch $B \subseteq [ N ]$ . We use the notations $C , s _ { n j } , \tilde { q } _ { n } , q _ { n }$ from Appendix G. We only need to show how to perform forward and backward passes for $\bar { \mathcal { L } } _ { B } ( \pmb { \theta } )$ . + +Forward Pass. Suppose we have a good estimate $\widetilde { \mathcal F }$ for $\log \bar { \mathcal { L } } _ { B } ( \pmb { \theta } )$ in the sense that + +$$ +\mathcal { R } _ { B } ( \pmb { \theta } ) : = \bar { \mathcal { L } } _ { B } ( \pmb { \theta } ) e ^ { - \widetilde { \mathcal { F } } } = \frac { 1 } { B } \sum _ { n \in B } \log \left( 1 + \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } ( \pmb { \theta } ) } \right) e ^ { - \widetilde { \mathcal { F } } } +$$ + +is in the range of float64. ${ \mathcal { R } } _ { B } ( \theta )$ can be thought of a relative training loss with respect to $\widetilde { \mathcal F }$ . Instead of evaluating the training loss $\bar { \mathcal { L } } _ { B } ( \pmb { \theta } )$ directly, we turn to evaluate this relative training loss in a numerically stable way: + +Step 1. Perform forward pass to compute the values of $s _ { n j }$ with float32, and convert them into float64; + +Step 2. Let $Q : = 3 0$ . If $q _ { n } ( \pmb { \theta } ) > Q$ for all $n \in B$ , then we compute + +$$ +\mathcal { R } _ { B } ( \pmb { \theta } ) = \frac { 1 } { B } \sum _ { n \in B } e ^ { - ( \tilde { q } _ { n } ( \pmb { \theta } ) + \widetilde { \mathcal { F } } ) } , +$$ + +where $\begin{array} { r } { \widetilde { q } _ { n } ( \pmb { \theta } ) : = - \mathrm { L S E } ( \{ - s _ { n j } : j \neq y _ { n } \} ) = - \log \left( \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } } \right) } \end{array}$ is evaluated in a numerically stable way; otherwise, we compute + +$$ +\mathcal { R } _ { B } ( \pmb { \theta } ) = \frac { 1 } { B } \sum _ { n \in B } \log 1 \mathrm { p } \left( \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } ( \pmb { \theta } ) } \right) e ^ { - \widetilde { \mathcal { F } } } , +$$ + +where $\log 1 \mathrm { p } ( x )$ is a numerical stable implementation of $\log ( 1 + x )$ . + +This algorithm can be explained as follows. Step 1 is numerically stable because we observe from the experiments that the layer weights and layer outputs grow slowly. Now we consider Step 2. If $q _ { n } ( \pmb \theta ) \leq Q$ for some $n \in [ B ]$ , then $\bar { \mathcal { L } } _ { B } ( \pmb { \theta } ) \dot { = } \Omega ( \dot { e } ^ { - Q } )$ is in the range of float64, so we can compute ${ \mathcal { R } } _ { B } ( \theta )$ by (28) directly except that we need to use a numerical stable implementation of $\log ( 1 + x )$ . For $q _ { n } ( \pmb { \theta } ) > Q$ , arithmetic underflow can occur. By Taylor expansion of $\log ( 1 + x )$ , we know that when $x$ is small enough $\log ( 1 + x ) \approx x$ in the sense that the relative error $\textstyle { \frac { | \log ( 1 + x ) - x | } { \log ( 1 + x ) } } =$ $O ( x )$ . Thus, we can do the following approximation + +$$ +\log \left( 1 + \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } ( \theta ) } \right) e ^ { - \widetilde { \mathcal { F } } } \approx \sum _ { j \neq y _ { n } } e ^ { - s _ { n j } ( \theta ) } \cdot e ^ { - \widetilde { \mathcal { F } } } +$$ + +for $q _ { n } ( \pmb { \theta } ) > Q$ , and only introduce a relative error of $O ( C e ^ { - Q } )$ (recall that $C$ is the number of classes). Using a numerical stable implementation of LSE, we can compute $\tilde { q } _ { n }$ easily. Then the RHS of (29) can be rewritten as $e ^ { - ( \tilde { q } _ { n } ( \pmb { \theta } ) + \widetilde { \mathcal { F } } ) }$ . Note that computing $e ^ { - ( \tilde { q } _ { n } ( \pmb { \theta } ) + \widetilde { \mathcal { F } } ) }$ does not have underflow or overflow problems if $\widetilde { \mathcal F }$ is a good approximation for $\log \bar { \mathcal { L } } _ { B } ( \pmb { \theta } )$ . + +Backward Pass. To perform backward pass, we build a computation graph in Tensorflow for the above forward pass for the relative training loss and use the automatic differentiation. We parameterize the learning rate as $\boldsymbol \eta = \boldsymbol \hat { \eta } \cdot \boldsymbol e ^ { \widetilde { \mathcal F } }$ . Then it is easy to see that taking a step of gradient descent for $\mathcal { L } _ { B } ( \pmb { \theta } )$ with learning rate $\eta$ is equivalent to taking a step for ${ \mathcal { R } } _ { B } ( \theta )$ with $\hat { \eta }$ . Thus, as long as $\hat { \eta }$ can fit into float64, we can perform gradient descent on ${ \mathcal { R } } _ { B } ( \theta )$ to ensure numerical stability. + +The Choice of $\widetilde { \mathcal F }$ . The only question remains is how to choose $\widetilde { \mathcal F }$ . In our experiments, we set $\widetilde { \mathcal { F } } ( t ) : = \log \bar { \mathcal { L } } ( t - 1 )$ to be the training loss at the end of the last epoch, since the training loss cannot change a lot within one single epoch. For this, we need to maintain $\log \bar { \mathcal { L } } ( t )$ during training. This can be done as follows: after evaluating the relative training loss $\mathcal { R } ( t )$ on the whole training set, we can obtain $\log \bar { \mathcal { L } } ( t )$ by adding $\widetilde { \mathcal { F } } ( t )$ and $\log \mathcal { R } ( t )$ together. + +It is worth noting that with this choice of $\widetilde { \mathcal F }$ , $\hat { \eta } ( t ) = \alpha ( t )$ in the loss-based learning rate scheduling. As shown in the right figure of Figure 4, $\alpha ( t )$ is always between $1 0 ^ { - 9 }$ and $1 0 ^ { 0 }$ , which ensures the numerical stability of backward pass. \ No newline at end of file diff --git a/parse/train/SJeLIgBKPS/SJeLIgBKPS_content_list.json b/parse/train/SJeLIgBKPS/SJeLIgBKPS_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b3308f5e1a95c90cbbe305ae376d6ad18af2a3b4 --- /dev/null +++ b/parse/train/SJeLIgBKPS/SJeLIgBKPS_content_list.json @@ -0,0 +1,9396 @@ +[ + { + "type": "text", + "text": "GRADIENT DESCENT MAXIMIZES THE MARGIN OF HOMOGENEOUS NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Kaifeng Lyu & Jian Li \nInstitute for Interdisciplinary Information Sciences \nTsinghua University \nBeijing, China \nvfleaking@gmail.com,lijian83@mail.tsinghua.edu.cn ", + "bbox": [ + 184, + 170, + 665, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 276, + 544, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we study the implicit regularization of the gradient descent algorithm in homogeneous neural networks, including fully-connected and convolutional neural networks with ReLU or LeakyReLU activations. In particular, we study the gradient descent or gradient flow (i.e., gradient descent with infinitesimal step size) optimizing the logistic loss or cross-entropy loss of any homogeneous model (possibly non-smooth), and show that if the training loss decreases below a certain threshold, then we can define a smoothed version of the normalized margin which increases over time. We also formulate a natural constrained optimization problem related to margin maximization, and prove that both the normalized margin and its smoothed version converge to the objective value at a KKT point of the optimization problem. Our results generalize the previous results for logistic regression with one-layer or multi-layer linear networks, and provide more quantitative convergence results with weaker assumptions than previous results for homogeneous smooth neural networks. We conduct several experiments to justify our theoretical finding on MNIST and CIFAR-10 datasets. Finally, as margin is closely related to robustness, we discuss potential benefits of training longer for improving the robustness of the model. ", + "bbox": [ + 233, + 306, + 764, + 541 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 566, + 336, + 582 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A major open question in deep learning is why gradient descent or its variants, are biased towards solutions with good generalization performance on the test set. To achieve a better understanding, previous works have studied the implicit bias of gradient descent in different settings. One simple but insightful setting is linear logistic regression on linearly separable data. In this setting, the model is parameterized by a weight vector $\\textbf { \\em w }$ , and the class prediction for any data point $_ { \\textbf { \\em x } }$ is determined by the sign of ${ \\pmb w } ^ { \\top } { \\pmb x }$ . Therefore, only the direction ${ w / \\| \\boldsymbol { w } \\| _ { 2 } }$ is important for making prediction. Soudry et al. (2018a;b); Ji & Telgarsky (2018; 2019c); Nacson et al. (2019c) investigated this problem and proved that the direction of $\\pmb { w }$ converges to the direction that maximizes the $L ^ { 2 }$ -margin while the norm of $\\pmb { w }$ diverges to $+ \\infty$ , if we train $\\pmb { w }$ with (stochastic) gradient descent on logistic loss. Interestingly, this convergent direction is the same as that of any regularization path: any sequence of weight vectors $\\{ { \\pmb w } _ { t } \\}$ such that every ${ \\pmb w } _ { t }$ is a global minimum of the $L ^ { 2 }$ -regularized loss $\\begin{array} { r } { \\mathcal { L } ( \\pmb { w } ) + \\frac { \\lambda _ { t } } { 2 } \\| \\pmb { w } \\| _ { 2 } ^ { 2 } } \\end{array}$ with $\\lambda _ { t } \\to 0$ (Rosset et al., 2004). Indeed, the trajectory of gradient descent is also pointwise close to a regularization path (Suggala et al., 2018). ", + "bbox": [ + 174, + 597, + 825, + 780 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The aforementioned linear logistic regression can be viewed as a single-layer neural network. A natural and important question is to what extent gradient descent has similiar implicit bias for modern deep neural networks. For theoretical analysis, a natural candidate is to consider homogeneous neural networks. Here a neural network $\\Phi$ is said to be (positively) homogeneous if there is a number $L > 0$ (called the order) such that the network output $\\Phi ( \\pmb \\theta ; \\pmb x )$ , where $\\pmb { \\theta }$ stands for the parameter and $_ { \\textbf { \\em x } }$ stands for the input, satisfies the following: ", + "bbox": [ + 174, + 786, + 823, + 871 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/e3ce4a0c9a7c979bfaa42d4d19d9cc3e042356de9fed3b5d78284d8b31c44bbd.jpg", + "text": "$$\n\\forall c > 0 : \\Phi ( c \\pmb \\theta ; \\pmb x ) = c ^ { L } \\Phi ( \\pmb \\theta ; \\pmb x ) \\mathrm { f o r ~ a l l ~ } \\pmb \\theta \\mathrm { ~ a n d ~ } \\pmb x .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 875, + 658, + 892 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "It is important to note that many neural networks are homogeneous (Neyshabur et al., 2015a; Du et al., 2018). For example, deep fully-connected neural networks or deep CNNs with ReLU or ", + "bbox": [ + 173, + 895, + 828, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "LeakyReLU activations can be made homogeneous if we remove all the bias terms, and the order $L$ is exactly equal to the number of layers. ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In (Wei et al., 2019), it is shown that the regularization path does converge to the max-margin direction for homogeneous neural networks with cross-entropy or logistic loss. This result suggests that gradient descent or gradient flow may also converges to the max-margin direction by assuming homogeneity, and this is indeed true for some sub-classes of homogeneous neural networks. For gradient flow, this convergent direction is proven for linear fully-connected networks (Ji & Telgarsky, 2019a). For gradient descent on linear fully-connected and convolutional networks, (Gunasekar et al., 2018b) formulate a constrained optimization problem related to margin maximization and prove that gradient descent converges to the direction of a KKT point or even the max-margin direction, under various assumptions including the convergence of loss and gradient directions. In an independent work, (Nacson et al., 2019a) generalize the result in (Gunasekar et al., 2018b) to smooth homogeneous models (we will discuss this work in more details in Section 2). ", + "bbox": [ + 174, + 138, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 MAIN RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 309, + 323, + 324 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we identify a minimal set of assumptions for proving our theoretical results for homogeneous neural networks on classification tasks. Besides homogeneity, we make two additional assumptions: ", + "bbox": [ + 176, + 337, + 823, + 378 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. Exponential-type Loss Function. We require the loss function to have certain exponential tail (see Appendix A for the details). This assumption is not restrictive as it includes the most popular classfication losses: exponential loss, logistic loss and cross-entropy loss. \n2. Separability. The neural network can separate the training data during training (i.e., the neural network can achieve $1 0 0 \\%$ training accuracy)1. ", + "bbox": [ + 209, + 390, + 825, + 467 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While the first assumption is natural, the second requires some explanation. In fact, we assume that at some time $t _ { 0 }$ , the training loss is smaller than a threshold, and the threshold here is chosen to be so small that the training accuracy is guaranteed to be $1 0 0 \\%$ (e.g., for the logistic loss and cross-entropy loss, the threshold can be set to $\\mathrm { l n } 2$ ). Empirically, state-of-the-art CNNs for image classification can even fit randomly labeled data easily (Zhang et al., 2017). Recent theoretical work on over-parameterized neural networks (Allen-Zhu et al., 2019; Zou et al., 2018) show that gradient descent can fit the training data if the width is large enough. Furthermore, in order to study the margin, ensuring the training data can be separated is inevitable; otherwise, there is no positive margin between the data and decision boundary. ", + "bbox": [ + 174, + 479, + 825, + 604 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our Contribution. Similar to linear models, for homogeneous models, only the direction of parameter $\\pmb \\theta$ is important for making predictions, and one can see that the margin $\\gamma ( \\pmb \\theta )$ scales linearly with $\\lVert \\pmb { \\theta } \\rVert _ { 2 } ^ { L }$ , when fixing the direction of $\\pmb \\theta$ . To compare margins among $\\pmb { \\theta }$ in different directions, it makes sense to study the normalized margin, $\\bar { \\gamma } ( \\pmb \\theta ) : = \\gamma ( \\pmb \\theta ) / \\| \\pmb \\theta \\| _ { 2 } ^ { L }$ . ", + "bbox": [ + 176, + 611, + 825, + 669 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we focus on the training dynamics of the network after $t _ { 0 }$ (recall that $t _ { 0 }$ is a time that the training loss is less than the threshold). Our theoretical results can answer the following questions regarding the normalized margin. ", + "bbox": [ + 174, + 674, + 825, + 717 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "First, how does the normalized margin change during training? The answer may seem complicated since one can easily come up with examples in which $\\bar { \\gamma }$ increases or decreases in a short time interval. However, we can show that the overall trend of the normalized margin is to increase in the following sense: there exists a smoothed version of the normalized margin, denoted as $\\tilde { \\gamma }$ , such that (1) $| \\tilde { \\gamma } - \\bar { \\tilde { \\gamma } } | \\to 0$ as $t \\to \\infty$ ; and $( 2 ) \\tilde { \\gamma }$ is non-decreasing for $t > t _ { 0 }$ . ", + "bbox": [ + 174, + 723, + 825, + 794 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Second, how large is the normalized margin at convergence? To answer this question, we formulate a natural constrained optimization problem which aims to directly maximize the margin. We show that every limit point of $\\{ \\pmb \\theta ( t ) / \\| \\pmb \\theta ( t ) \\| _ { 2 } : t > 0 \\}$ is along the direction of a KKT point of the maxmargin problem. This indicates that gradient descent/gradient flow performs margin maximization implicitly in deep homogeneous networks. This result can be seen as a significant generalization of previous works (Soudry et al., 2018a;b; Ji & Telgarsky, $2 0 1 9 \\mathrm { a }$ ; Gunasekar et al., 2018b) from linear classifiers to homogeneous classifiers. ", + "bbox": [ + 174, + 800, + 825, + 897 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/2e818eb552978f56e83f486e74331e74a063c35463a07ca1e458303572110c47.jpg", + "image_caption": [ + "Figure 1: (a) Training CNNs with and without bias on MNIST, using SGD with learning rate 0.01. The training loss (left) decreases over time, and the normalized margin (right) keeps increasing after the model is fitted, but the growth rate is slow $( \\approx 1 . 8 \\times 1 0 ^ { - 4 }$ after 10000 epochs). (b) Training CNNs with and without bias on MNIST, using SGD with the loss-based learning rate scheduler. The training loss (left) decreases exponentially over time $( < 1 0 ^ { - 8 0 0 }$ after 9000 epochs), and the normalized margin (right) increases rapidly after the model is fitted $( \\approx 1 . 2 \\times 1 0 ^ { - 3 }$ after 10000 epochs, $1 0 \\times$ larger than that of SGD with learning rate 0.01). Experimental details are in Appendix $\\mathbf { K }$ . " + ], + "image_footnote": [], + "bbox": [ + 184, + 108, + 813, + 237 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As by-products of the above results, we derive tight asymptotic convergence/growth rates of the loss and weights. It is shown in (Soudry et al., 2018a;b; Ji & Telgarsky, 2018; 2019c) that the loss decreases at the rate of $O ( 1 / t )$ , the weight norm grows as $O ( \\log t )$ for linear logistic regression. In this work, we generalize the result by showing that the loss decreases at the rate of $O ( 1 / ( t ( \\log t ) ^ { 2 - 2 / L } ) )$ and the weight norm grows as $O ( ( \\log t ) ^ { 1 / L } )$ for homogeneous neural networks with exponential loss, logistic loss, or cross-entropy loss. ", + "bbox": [ + 174, + 376, + 825, + 463 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Experiments.2 The main practical implication of our theoretical result is that training longer can enlarge the normalized margin. To justify this claim empiricaly, we train CNNs on MNIST and CIFAR-10 with SGD (see Section K.1). Results on MNIST are presented in Figure 1. For constant step size, we can see that the normalized margin keeps increasing, but the growth rate is rather slow (because the gradient gets smaller and smaller). Inspired by our convergence results for gradient descent, we use a learning rate scheduling method which enlarges the learning rate according to the current training loss, then the training loss decreases exponentially faster and the normalized margin increases significantly faster as well. ", + "bbox": [ + 174, + 469, + 825, + 582 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For feedforward neural networks with ReLU activation, the normalized margin on a training sample is closely related to the $L ^ { 2 }$ -robustness (the $L ^ { 2 }$ -distance from the training sample to the decision boundary). Indeed, the former divided by a Lipschitz constant is a lower bound for the latter. For example, the normalized margin is a lower bound for the $L ^ { 2 }$ -robustness on fully-connected networks with ReLU activation (see, e.g., Theorem 4 in (Sokolic et al., 2017)). This fact suggests that training longer may have potential benefits on improving the robustness of the model. In our experiments, we observe noticeable improvements of $L ^ { 2 }$ -robustness on both training and test sets (see Section K.2). ", + "bbox": [ + 174, + 589, + 825, + 686 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 707, + 344, + 723 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Implicit Bias in Training Linear Classifiers. For linear logistic regression on linearly separable data, Soudry et al. (2018a;b) showed that full-batch gradient descent converges in the direction of the max $L ^ { 2 }$ -margin solution of the corresponding hard-margin Support Vector Machine (SVM). Subsequent works extended this result in several ways: Nacson et al. (2019c) extended the results to the case of stochastic gradient descent; Gunasekar et al. (2018a) considered other optimization methods; Nacson et al. (2019b) considered other loss functions including those with poly-exponential tails; Ji & Telgarsky (2018; 2019c) characterized the convergence of weight direction without assuming separability; Ji & Telgarsky (2019b) proved a tighter convergence rate for the weight direction. ", + "bbox": [ + 173, + 738, + 825, + 851 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Those results on linear logistic regression have been generalized to deep linear networks. Ji & Telgarsky (2019a) showed that the product of weights in a deep linear network with strictly decreasing loss converges in the direction of the max $L ^ { 2 }$ -margin solution. Gunasekar et al. (2018b) showed more general results for gradient descent on linear fully-connected and convolutional networks with exponential loss, under various assumptions on the convergence of the loss and gradient direction. ", + "bbox": [ + 176, + 857, + 825, + 900 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Margin maximization phenomenon is also studied for boosting methods (Schapire et al., 1998; Schapire & Freund, 2012; Shalev-Shwartz & Singer, 2010; Telgarsky, 2013) and Normalized Perceptron (Ramdas & Pena, 2016). ", + "bbox": [ + 178, + 138, + 821, + 181 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Implicit Bias in Training Nonlinear Classifiers. Soudry et al. (2018a) analyzed the case where there is only one trainable layer of a ReLU network. Xu et al. (2018) characterized the implicit bias for the model consisting of one single ReLU unit. Our work is closely related to a recent independent work by (Nacson et al., 2019a) which we discuss in details below. ", + "bbox": [ + 174, + 188, + 825, + 243 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Comparison with (Nacson et al., 2019a). Very recently, (Nacson et al., 2019a) analyzed gradient descent for smooth homogeneous models and proved the convergence of parameter direction to a KKT point of the aforementioned max-margin problem. Compared with their work, our work adopt much weaker assumptions: (1) They assume the training loss converges to 0, but in our work we only require that the training loss is lower than a small threshold value at some time $t _ { 0 }$ (and we prove the exact convergence rate of the loss after $t _ { 0 }$ ); (2) They assume the convergence of parameter direction3, while we prove that KKT conditions hold for all limit points of $\\{ \\pmb \\theta ( t ) \\bar { / } \\lVert \\pmb \\theta ( t ) \\rVert _ { 2 } : t > 0 \\}$ , without requiring any convergence assumption; (3) They assume the convergence of the direction of losses (the direction of the vector whose entries are loss values on every data point) and Linear Independence Constraint Qualification (LICQ) for the max-margin problem, while we do not need such assumptions. Besides the above differences in assumptions, we also prove the monotonicity of the normalized margin and provide tight convergence rate for training loss. We believe both results are interesting in their own right. ", + "bbox": [ + 174, + 251, + 825, + 431 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Another technical difference is that their work analyzes discrete gradient descent on smooth homogeneous models (which fails to capture ReLU networks). In our work, we analyze both gradient descent on smooth homogeneous models and also gradient flow on homogeneous models which could be non-smooth. ", + "bbox": [ + 174, + 438, + 825, + 494 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Other Works on Implicit Bias. Banburski et al. (2019) also studied the dynamics of gradient flow and among other things, provided mathematical insights to the implicit bias towards max margin solution for homogeneous networks. We note that their analysis of gradient flow decomposes the dynamics to the tangent component and radial component, which is similar to our proof of Theorem 4.1 in spirit. Wilson et al. (2017); Ali et al. (2019); Gunasekar et al. (2018a) showed that for the linear least-square problem gradient-based methods converge to the unique global minimum that is closest to the initialization in $L ^ { 2 }$ distance. Du et al. (2019); Jacot et al. (2018); Lee et al. (2019); Arora et al. (2019b) showed that over-parameterized neural networks of sufficient width (or infinite width) behave as linear models with Neural Tangent Kernel (NTK) with proper initialization and gradient descent converges linearly to a global minimum near the initial point. Other related works include (Ma et al., 2019; Gidel et al., 2019; Arora et al., $2 0 1 9 \\mathrm { a }$ ; Suggala et al., 2018; Blanc et al., 2019; Neyshabur et al., 2015b;a). ", + "bbox": [ + 173, + 501, + 825, + 667 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 688, + 339, + 704 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Basic Notations. For any $N \\in \\mathbb N$ , let $[ N ] = \\{ 1 , \\dots , N \\}$ . $\\lVert \\pmb { v } \\rVert _ { 2 }$ denotes the $L ^ { 2 }$ -norm of a vector $\\textbf { { v } }$ . The default base of log is $e$ . For a function $f : { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }$ , $\\nabla f ( { \\pmb x } )$ stands for the gradient at $_ { \\textbf { \\em x } }$ if it exists. A function $f : X \\to \\mathbb { R } ^ { d }$ is $\\mathcal { C } ^ { k }$ -smooth if $f$ is $k$ times continuously differentiable. A function $f : X \\to \\mathbb { R }$ is locally Lipschitz if for every $\\pmb { x } \\in X$ there exists a neighborhood $U$ of $_ { \\textbf { \\em x } }$ such that the restriction of $f$ on $U$ is Lipschitz continuous. ", + "bbox": [ + 173, + 718, + 825, + 790 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Non-smooth Analysis. For a locally Lipschitz function $f : X \\to \\mathbb { R }$ , the Clarke’s subdifferential (Clarke, 1975; Clarke et al., 2008; Davis et al., 2020) at $\\pmb { x } \\in X$ is the convex set ", + "bbox": [ + 173, + 795, + 823, + 824 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9688ef80934f5a16d082b5a90c9a8931e9280e16d81f8bfa00cab49e6857c185.jpg", + "text": "$$\n\\partial ^ { \\circ } f ( { \\pmb x } ) : = \\operatorname { c o n v } \\left. \\operatorname* { l i m } _ { k \\to \\infty } \\nabla f ( { \\pmb x } _ { k } ) : { \\pmb x } _ { k } \\to { \\pmb x } , f \\mathrm { ~ i s ~ d i f f e r e n t i a b l e ~ a t } { \\pmb x } _ { k } \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 829, + 732, + 864 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For brevity, we say that a function $z : I \\mathbb { R } ^ { d }$ on the interval $I$ is an arc if $_ z$ is absolutely continuous for any compact sub-interval of exists. Following the terminolo $I$ . For an arc in (Davis e $_ z$ , a $z ^ { \\prime } ( t )$ (or 20), $\\textstyle { \\frac { d z } { d t } } ( t ) )$ stands for the derivative at y that a locally Lipschitz fu $t$ if ittion $f : { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }$ admits a chain rule if for any arc $z : [ 0 , + \\infty ) \\to \\mathbb { R } ^ { \\bar { d } }$ , $\\forall h \\in \\partial ^ { \\circ } f ( \\bar { z } ( t ) ) \\bar { : } ( f \\circ z ) ^ { \\prime } ( t ) =$ $\\langle h , z ^ { \\prime } ( t ) \\rangle$ holds for a.e. $t > 0$ (see also Appendix I). ", + "bbox": [ + 173, + 102, + 825, + 174 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Binary Classification. Let $\\Phi$ be a neural network, assumed to be parameterized by $\\pmb { \\theta }$ . The output of $\\Phi$ on an input $\\pmb { x } \\in \\mathbb { R } ^ { d _ { \\mathbf { x } } }$ is a real number $\\Phi ( \\pmb \\theta ; \\pmb x )$ , and the sign of $\\Phi ( \\pmb \\theta ; \\pmb x )$ stands for the classification result. A dataset is denoted by $\\mathcal { D } = \\{ ( x _ { n } , y _ { n } ) : n \\in [ N ] \\}$ , where $\\pmb { x } _ { n } \\in \\mathbb { R } ^ { d _ { \\mathrm { { x } } } }$ stands for a data input and $y _ { n } \\in \\{ \\pm 1 \\}$ stands for the corresponding label. For a loss function $\\ell : \\mathbb { R } \\to \\mathbb { R }$ , we define the training loss of $\\Phi$ on the dataset $\\mathcal { D }$ to be $\\begin{array} { r } { \\mathcal { L } ( \\pmb { \\theta } ) : = \\sum _ { n = 1 } ^ { N } \\ell ( y _ { n } \\Phi ( \\pmb { \\theta } ; \\pmb { x } _ { n } ) ) } \\end{array}$ . ", + "bbox": [ + 174, + 180, + 825, + 255 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Gradient Descent. We consider the process of training this neural network $\\Phi$ with either gradient descent or gradient flow. For gradient descent, we assume the training loss $\\mathcal { L } ( \\pmb \\theta )$ is $\\mathcal { C } ^ { 2 }$ -smooth and describe the gradient descnet process as $\\pmb { \\theta } ( t + 1 ) = \\pmb { \\theta } ( t ) - \\eta ( t ) \\nabla \\mathcal { L } ( \\pmb { \\theta } ( \\bar { t } ) )$ , where $\\dot { \\eta } ( t )$ is the learning rate at time $t$ and $\\nabla \\mathcal { L } ( \\pmb \\theta ( t ) )$ is the gradient of $\\mathcal { L }$ at $\\pmb \\theta ( t )$ . ", + "bbox": [ + 174, + 260, + 825, + 316 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Gradient Flow. For gradient flow, we do not assume the differentibility but only some regularity assumptions including locally Lipschitz. Gradient flow can be seen as gradient descent with infinitesimal step size. In this model, $\\pmb \\theta$ changes continuously with time, and the trajectory of parameter $\\pmb { \\theta }$ during training is an arc $\\pmb \\theta : [ 0 , + \\infty ) \\dot { } \\mathbb R ^ { d } , t \\mapsto \\pmb \\theta ( \\dot { t } )$ that satisfies the differential inclusion $\\begin{array} { r } { \\frac { d { \\pmb \\theta } ( t ) } { d t } \\in - \\partial ^ { \\circ } \\mathcal { L } ( { \\pmb \\theta } ( t ) ) } \\end{array}$ for a.e. $t \\geq 0$ . The Clarke’s subdifferential $\\partial ^ { \\circ } { \\mathcal { L } }$ is a natural generalization of the usual differential to non-differentiable functions. If $\\mathcal { L } ( \\pmb \\theta )$ is actually a ${ \\mathcal { C } } ^ { 1 }$ -smooth function, the above differential inclusion reduces to $\\begin{array} { r } { \\frac { d \\pmb { \\theta } ( t ) } { d t } = - \\nabla \\mathcal { L } ( \\pmb { \\theta } ( t ) ) } \\end{array}$ for all $t \\geq 0$ , which corresponds to the gradient flow with differential in the usual sense. ", + "bbox": [ + 173, + 321, + 825, + 443 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 GRADIENT DESCENT / GRADIENT FLOW ON HOMOGENEOUS MODEL ", + "text_level": 1, + "bbox": [ + 171, + 463, + 777, + 481 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we first state our results for gradient flow and gradient descent on homogeneous models with exponential loss $\\ell ( q ) : = e ^ { - q }$ for simplicity of presentation. Due to space limit, we defer the more general results which hold for a large family of loss functions (including logistic loss and cross-entropy loss) to Appendix A, F and $\\mathbf { G }$ . ", + "bbox": [ + 174, + 494, + 825, + 553 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 ASSUMPTIONS ", + "text_level": 1, + "bbox": [ + 174, + 569, + 316, + 583 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Gradient Flow. For gradient flow, we assume the following: ", + "bbox": [ + 173, + 594, + 580, + 611 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(A1). (Regularity). For any fixed $_ { \\textbf { \\em x } }$ , $\\Phi ( \\cdot ; x )$ is locally Lipschitz and admits a chain rule; \n(A2). (Homogeneity). There exists $L > 0$ such that $\\forall \\alpha > 0 : \\Phi ( \\alpha \\pmb { \\theta } ; \\pmb { x } ) = \\alpha ^ { L } \\Phi ( \\pmb { \\theta } ; \\pmb { x } )$ ; \n(A3). (Exponential Loss). $\\ell ( q ) = e ^ { - q }$ ; \n(A4). (Separability). There exists a time $t _ { 0 }$ such that $\\mathcal { L } ( \\pmb { \\theta } ( t _ { 0 } ) ) < 1$ . ", + "bbox": [ + 181, + 621, + 772, + 695 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(A1) is a technical assumption about the regularity of the network output. As shown in (Davis et al., 2020), the output of almost every neural network admits a chain rule (as long as the neural network is composed by definable pieces in an o-minimal structure, e.g., ReLU, sigmoid, LeakyReLU). ", + "bbox": [ + 176, + 707, + 823, + 750 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(A2) assumes the homogeneity, the main property we assume in this work. (A3), (A4) correspond to the two conditions introduced in Section 1. The exponential loss in (A3) is main focus of this section. (A4) is a separability assumption: the condition $\\mathcal { L } ( \\pmb \\theta ( t _ { 0 } ) ) < 1$ ensures that $\\ell ( y _ { n } \\Phi ( \\pmb \\theta ( t _ { 0 } ) ; \\pmb x _ { n } ) ) < 1$ for all $n \\in [ N ]$ , and thus $y _ { n } \\bar { \\Phi } ( \\pmb \\theta ( t _ { 0 } ) ; \\pmb x _ { n } ) > 0$ , meaning that $\\Phi$ classifies every ${ \\mathbf { \\mathcal { x } } } _ { n }$ correctly. ", + "bbox": [ + 174, + 756, + 825, + 813 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Gradient Descent. For gradient descent, we assume (A2), (A3), (A4) similarly as for gradient flow, and the following two assumptions (S1) and (S5). ", + "bbox": [ + 174, + 819, + 823, + 848 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(S1). (Smoothness). For any fixed $_ { \\textbf { \\em x } }$ , $\\Phi ( \\cdot ; x )$ is $\\mathcal { C } ^ { 2 }$ -smooth on $\\mathbb { R } ^ { d } \\backslash \\{ \\mathbf { 0 } \\}$ . \n(S5). (Learning rate condition, Informal). $\\eta ( t ) = \\eta _ { 0 }$ for a sufficiently small constant $\\eta _ { 0 }$ . In fact, $\\eta ( t )$ is even allowed to be as large as $\\begin{array} { r } { \\dot { O } ( \\mathcal { L } ( \\dot { t } ) ^ { - 1 } \\mathrm { p o l y l o g } \\frac { 1 } { \\mathcal { L } ( t ) } ) } \\end{array}$ ). See Appendix E.1 for the details. ", + "bbox": [ + 189, + 858, + 825, + 922 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(S5) is natural since deep neural networks are usually trained with constant learning rates. (S1) ensures the smoothness of $\\Phi$ , which is often assumed in the optimization literature in order to analyze gradient descent. While (S1) does not hold for neural networks with ReLU, it does hold for neural networks with smooth homogeneous activation such as the quadratic activation $\\phi ( x ) : = x ^ { 2 }$ (Li et al., 2018b; Du & Lee, 2018) or powers of ReLU $\\phi ( x ) : = \\mathrm { R e L U } ( x ) ^ { \\alpha }$ for $\\alpha > 2$ (Zhong et al., 2017; Klusowski & Barron, 2018; Li et al., 2019). ", + "bbox": [ + 173, + 103, + 825, + 188 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 MAIN THEOREM: MONOTONICITY OF NORMALIZED MARGINS ", + "text_level": 1, + "bbox": [ + 174, + 202, + 650, + 218 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The margin for a single data point $( { \\pmb x } _ { n } , y _ { n } )$ is defined to be $q _ { n } ( \\pmb \\theta ) : = y _ { n } \\Phi ( \\pmb \\theta ; \\pmb x _ { n } )$ , and the margin for the entire dataset is defined to be $q _ { \\operatorname* { m i n } } ( \\pmb { \\theta } ) : = \\operatorname* { m i n } _ { n \\in [ N ] } q _ { n } ( \\pmb { \\theta } )$ . By homogenity, the margin $q _ { \\mathrm { m i n } } ( \\pmb { \\theta } )$ scales linearly with $\\lVert \\pmb { \\theta } \\rVert _ { 2 } ^ { L }$ for any fixed direction since $q _ { \\mathrm { m i n } } ( c \\pmb \\theta ) = c ^ { L } q _ { \\mathrm { m i n } } ( \\pmb \\theta )$ . So we consider the normalized margin defined as below: ", + "bbox": [ + 173, + 228, + 825, + 287 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d2e15873428b21f2f19058c3a10617e16b49f92c60197383dac6ac706aebf063.jpg", + "text": "$$\n\\bar { \\gamma } ( \\pmb \\theta ) : = q _ { \\mathrm { m i n } } \\left( \\frac { \\pmb \\theta } { \\lVert \\pmb \\theta \\rVert _ { 2 } } \\right) = \\frac { q _ { \\mathrm { m i n } } ( \\pmb \\theta ) } { \\lVert \\pmb \\theta \\rVert _ { 2 } ^ { L } } .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 287, + 617, + 323 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We say $f$ is an $\\epsilon$ -additive approximation for the normalized margin if $\\bar { \\gamma } \\mathrm { ~ - ~ } \\epsilon \\le f \\le \\bar { \\gamma }$ , and $c$ - multiplicative approximation if $c \\bar { \\gamma } \\leq f \\leq \\bar { \\gamma }$ . ", + "bbox": [ + 174, + 324, + 826, + 353 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Gradient Flow. Our first result is on the overall trend of the normalized margin $\\bar { \\gamma } ( \\pmb \\theta ( t ) )$ . For both gradient flow and gradient descent, we identify a smoothed version of the normalized margin, and show that it is non-decreasing during training. More specifically, we have the following theorem for gradient flow. ", + "bbox": [ + 174, + 358, + 825, + 414 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 4.1 (Corollary of Theorem A.7). Under assumptions $( A I ) ~ - ~ ( A 4 )$ , there exists an $O ( \\| \\pmb { \\theta } \\| _ { 2 } ^ { - L } )$ -additive approximation function $\\widetilde \\gamma ( \\pmb \\theta )$ for the normalized margin such that the following statements are true for gradient flow: ", + "bbox": [ + 174, + 416, + 825, + 460 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1. For a.e. $t > t _ { 0 }$ , $\\begin{array} { r } { \\frac { d } { d t } \\tilde { \\gamma } ( \\pmb { \\theta } ( t ) ) \\geq 0 } \\end{array}$ ; ", + "bbox": [ + 214, + 468, + 441, + 486 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$t > t _ { 0 }$ $\\begin{array} { r } { \\frac { d } { d t } \\tilde { \\gamma } ( \\pmb { \\theta } ( t ) ) > 0 } \\end{array}$ ${ \\frac { d } { d t } } { \\frac { \\pmb { \\theta } ( t ) } { \\| \\pmb { \\theta } ( t ) \\| _ { 2 } } } = 0$ ", + "bbox": [ + 212, + 493, + 598, + 516 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3. $\\mathcal { L } ( \\pmb \\theta ( t ) ) 0$ and $\\| \\pmb \\theta ( t ) \\| _ { 2 } \\to \\infty$ as $t \\to + \\infty$ ; therefore, $| \\bar { \\gamma } ( \\pmb \\theta ( t ) ) - \\tilde { \\gamma } ( \\pmb \\theta ( t ) ) | 0$ . ", + "bbox": [ + 207, + 522, + 771, + 540 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "More concretely, the function $\\tilde { \\gamma } ( \\pmb \\theta )$ in Theorem 4.1 is defined as ", + "bbox": [ + 174, + 546, + 594, + 563 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/951e8ac53f1ca499cb06974b9c9a00d287116613be412b3e7d5ef1bf5a2a049f.jpg", + "text": "$$\n\\widetilde { \\gamma } ( \\pmb \\theta ) : = \\frac { \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } } { \\| \\pmb \\theta \\| _ { 2 } ^ { L } } = \\frac { - \\log \\left( \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( \\pmb \\theta ) } \\right) } { \\| \\pmb \\theta \\| _ { 2 } ^ { L } } .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 564, + 650, + 607 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note that the only difference between $\\bar { \\gamma } ( \\pmb \\theta )$ and $\\tilde { \\gamma } ( \\pmb \\theta )$ is that the margin $q _ { \\mathrm { m i n } } ( \\pmb { \\theta } )$ in $\\bar { \\gamma } ( \\pmb \\theta )$ is replaced by $\\begin{array} { r } { \\log \\frac { 1 } { \\mathcal { L } ( \\pmb { \\theta } ) } = - \\mathrm { L S E } ( - q _ { 1 } ( \\pmb { \\theta } ) , \\dots , - q _ { N } ( \\pmb { \\theta } ) ) } \\end{array}$ , where $\\operatorname { L S E } ( a _ { 1 } , \\dots , a _ { N } ) = \\log ( \\exp ( a _ { 1 } ) +$ $\\cdot \\cdot \\cdot + \\exp ( a _ { N } ) )$ is the LogSumExp function. This is indeed a very natural idea, and previous works on linear models (e.g., (Telgarsky, 2013; Nacson et al., 2019b)) also approximate $q _ { \\mathrm { m i n } }$ with LogSumExp in the analysis of margin. It is easy to see why $\\widetilde \\gamma ( \\pmb \\theta )$ is an $O ( \\| \\pmb { \\theta } \\| _ { 2 } ^ { - L } )$ -additive approximation for $\\bar { \\gamma } ( \\pmb \\theta )$ : $\\begin{array} { r } { \\mathrm { ~ : ~ } e ^ { a _ { \\mathrm { m a x } } } \\leq \\sum _ { n = 1 } ^ { N } e ^ { a _ { n } } \\leq N e ^ { a _ { \\mathrm { m a x } } } } \\end{array}$ holds for $a _ { \\mathrm { { m a x } } } = \\operatorname* { m a x } \\{ a _ { 1 } , \\dots , a _ { N } \\}$ , so $a _ { \\mathrm { m a x } } \\leq \\mathrm { L S E } ( a _ { 1 } , \\ldots , a _ { N } ) \\leq a _ { \\mathrm { m a x } } \\bar { + } \\log N$ ; combining this with the definition of $\\widetilde \\gamma ( \\pmb \\theta )$ gives $\\bar { \\gamma } ( \\pmb \\theta ) - \\lVert \\pmb \\theta \\rVert _ { 2 } ^ { - L } \\log N \\leq \\tilde { \\gamma } ( \\pmb \\theta ) \\leq \\bar { \\gamma } ( \\pmb \\theta )$ . ", + "bbox": [ + 173, + 608, + 825, + 731 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Gradient Descent. For gradient descent, Theorem 4.1 holds similarly with a slightly different function $\\hat { \\gamma } ( \\pmb \\theta )$ that approximates $\\bar { \\gamma } ( \\pmb \\theta )$ multiplicatively rather than additively. ", + "bbox": [ + 174, + 736, + 821, + 765 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 4.2 (Corollary of Theorem E.2). Under assumptions $( S I )$ , (A2) - (A4), (S5), there exists an $( 1 - O ( 1 / ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ) )$ -multiplicative approximation function $\\hat { \\gamma } ( \\pmb \\theta )$ for the normalized margin such that the following statements are true for gradient descent: ", + "bbox": [ + 173, + 766, + 825, + 809 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Due to the discreteness of gradient descent, the explicit formula for $\\hat { \\gamma } ( \\pmb \\theta )$ is somewhat technical, and we refer the readers to Appendix E for full details. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Convergence Rates. It is shown in Theorem 4.1, 4.2 that $\\mathcal { L } ( \\pmb \\theta ( t ) ) 0$ and $\\| \\pmb \\theta ( t ) \\| _ { 2 } \\to \\infty$ . In fact, with a more refined analysis, we can prove tight loss convergence and weight growth rates using the monotonicity of normalized margins. ", + "bbox": [ + 176, + 102, + 823, + 146 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 4.3 (Corollary of Theorem A.10 and E.5). For gradient flow under assumptions (A1) - (A4) or gradient descent under assumptions (S1), (A2) - (A4), (S5), we have the following tight bounds for training loss and weight norm: ", + "bbox": [ + 173, + 150, + 826, + 193 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/bf296f66e9bbf0a5e8d9ae747033439ecb368de26010005c8aa49638bcdecef3.jpg", + "text": "$$\n\\mathcal { L } ( \\pmb \\theta ( t ) ) = \\Theta \\left( \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } \\right) \\qquad a n d \\qquad \\| \\pmb \\theta ( t ) \\| _ { 2 } = \\Theta \\left( ( \\log T ) ^ { 1 / L } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 248, + 199, + 746, + 234 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $T = t$ for gradient flow and $\\begin{array} { r } { T = \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}$ for gradient descent. ", + "bbox": [ + 173, + 241, + 655, + 261 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 MAIN THEOREM: CONVERGENCE TO KKT POINTS ", + "text_level": 1, + "bbox": [ + 173, + 273, + 570, + 290 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For gradient flow, $\\tilde { \\gamma }$ is upper-bounded by $\\tilde { \\gamma } \\le \\bar { \\gamma } \\le \\operatorname* { s u p } \\{ q _ { n } ( \\pmb \\theta ) : \\lVert \\pmb \\theta \\rVert _ { 2 } = 1 \\}$ . Combining this with Theorem 4.1 and the monotone convergence theorem, it is not hard to see that $\\scriptstyle \\operatorname* { l i m } _ { t \\to + \\infty } { \\bar { \\gamma } } ( \\pmb { \\theta } ( t ) )$ and $\\scriptstyle \\operatorname* { l i m } _ { t \\to + \\infty } { \\tilde { \\gamma } } ( \\pmb { \\theta } ( t ) )$ exist and equal to the same value. Using a similar argument, we can draw the same conclusion for gradient descent. ", + "bbox": [ + 173, + 299, + 825, + 357 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To understand the implicit regularization effect, a natural question arises: what optimality property does the limit of normalized margin have? To this end, we identify a natural constrained optimization problem related to margin maximization, and prove that $\\pmb \\theta ( t )$ directionally converges to its KKT points, as shown below. We note that we can extend this result to the finite time case, and show that gradient flow or gradient descent passes through an approximate KKT point after a certain amount of time. See Theorem A.9 in Appendix A and Theorem E.4 in Appendix E for the details. We will briefly review the definition of KKT points and approximate KKT points for a constraint optimization problem in Appendix C.1. ", + "bbox": [ + 173, + 363, + 825, + 476 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 4.4 (Corollary of Theorem A.8 and E.3). For gradient flow under assumptions (A1) - (A4) or gradient descent under assumptions (S1), (A2) - (A4), (S5), any limit point $\\bar { \\pmb { \\theta } }$ of $\\left\\{ \\frac { \\pmb \\theta ( t ) } { \\| \\pmb \\theta ( t ) \\| _ { 2 } } : t \\geq 0 \\right\\}$ is along the direction of a KKT point of the following constrained optimization problem $( P )$ : ", + "bbox": [ + 173, + 478, + 825, + 531 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/b658d9b248bc89608d0277c369d86ef3ba9b1253168693340ce7ae371032716e.jpg", + "text": "$$\n\\operatorname* { m i n } \\quad \\frac { 1 } { 2 } \\| \\pmb { \\theta } \\| _ { 2 } ^ { 2 } \\qquad \\quad s . t . \\quad q _ { n } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ]\n$$", + "text_format": "latex", + "bbox": [ + 325, + 536, + 669, + 568 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "That is, for any limit point $\\bar { \\theta }$ , there exists a scaling factor $\\alpha > 0$ such that $\\alpha \\bar { \\pmb { \\theta } }$ satisfies Karush-KuhnTucker (KKT) conditions of $( P )$ . ", + "bbox": [ + 174, + 574, + 823, + 603 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Minimizing (P) over its feasible region is equivalent to maximizing the normalized margin over all possible directions. The proof is as follows. Note that we only need to consider all feasible points $\\pmb { \\theta }$ with $q _ { \\mathrm { m i n } } ( \\pmb \\theta ) > 0$ . For a fixed $\\pmb { \\theta }$ , $\\alpha \\pmb { \\theta }$ is a feasible point of (P) iff $\\alpha \\ge q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { - 1 / L }$ . Thus, the minimum objective value over all feasible points of (P) in the direction of $\\pmb \\theta$ is $\\begin{array} { r l } { \\frac { 1 } { 2 } \\| \\pmb { \\theta } / q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { 1 / L } \\| _ { 2 } ^ { 2 } = } \\end{array}$ $\\frac { 1 } { 2 } \\bar { \\gamma } ( \\pmb { \\theta } ) ^ { - 2 / L }$ . Taking minimum over all possible directions, we can conclude that if the maximum normalized margin is $\\bar { \\gamma } _ { * }$ , then the minimum objective of (P) is $\\frac { 1 } { 2 } \\bar { \\gamma } _ { * } ^ { - 2 / L }$ . ", + "bbox": [ + 173, + 614, + 825, + 712 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "It can be proved that (P) satisfies the Mangasarian-Fromovitz Constraint Qualification (MFCQ) (See Lemma C.7). Thus, KKT conditions are first-order necessary conditions for global optimality. For linear models, KKT conditions are also sufficient for ensuring global optimality; however, for deep homogeneous networks, $q _ { n } ( \\pmb \\theta )$ can be highly non-convex. Indeed, as gradient descent is a first-order optimization method, if we do not make further assumptions on $q _ { n } ( \\pmb \\theta )$ , then it is easy to construct examples that gradient descent does not lead to a normalized margin that is globally optimal. Thus, proving the convergence to KKT points is perhaps the best we can hope for in our setting, and it is an interesting future work to prove stronger convergence results with further natural assumptions. ", + "bbox": [ + 173, + 715, + 825, + 829 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Moreover, we can prove the following corollary, which characterizes the optimality of the normalized margin using SVM with Neural Tangent Kernel (NTK, introduced in (Jacot et al., 2018)) defined at limit points. The proof is deferred to Appendix C.6. ", + "bbox": [ + 174, + 834, + 825, + 877 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Corollary 4.5 (Corollary of Theorem 4.4). Assume (S1). Then for gradient flow under assumptions $( A 2 ) \\textrm { - } ( A 4 )$ or gradient descent under assumptions (A2) - (A4), (S5), any limit point $\\bar { \\pmb { \\theta } }$ of $\\{ \\pmb \\theta ( t ) / \\| \\pmb \\theta ( t ) \\| _ { 2 } : t \\geq 0 \\}$ is along the max-margin direction for the hard-margin SVM with kernel ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "$K _ { \\bar { \\theta } } ( x , x ^ { \\prime } ) = \\left. \\nabla \\Phi _ { x } ( \\bar { \\theta } ) , \\nabla \\Phi _ { x ^ { \\prime } } ( \\bar { \\theta } ) \\right.$ , where $\\Phi _ { x } ( \\pmb { \\theta } ) : = \\Phi ( \\pmb { \\theta } ; \\pmb { x } )$ . That is, for some $\\alpha > 0$ , $\\alpha \\bar { \\pmb { \\theta } }$ is the optimal solution for the following constrained optimization problem: ", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/4c459abaf9318ba5d7a7ad87e0c53f28e8a58b443c42245c1d4e9e6b3243bd74.jpg", + "text": "$$\n\\operatorname* { m i n } \\quad \\frac { 1 } { 2 } \\| \\pmb { \\theta } \\| _ { 2 } ^ { 2 } \\qquad \\quad s . t . \\quad y _ { n } \\left. \\pmb { \\theta } , \\nabla \\Phi _ { \\pmb { x } _ { n } } ( \\bar { \\pmb { \\theta } } ) \\right. \\geq 1 \\qquad \\forall n \\in [ N ]\n$$", + "text_format": "latex", + "bbox": [ + 285, + 136, + 710, + 165 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "If we assume $( A I )$ instead of $( S I )$ for gradient flow, then there exists a mapping $h ( x ) \\in \\partial ^ { \\circ } \\Phi _ { x } ( \\bar { \\theta } )$ such that the same conclusion holds for $K _ { \\bar { \\theta } } ( x , x ^ { \\prime } ) = \\langle h ( x ) , h ( x ^ { \\prime } ) \\rangle$ . ", + "bbox": [ + 171, + 170, + 821, + 200 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 OTHER MAIN RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 215, + 377, + 229 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The above results can be extended to other settings as shown below. ", + "bbox": [ + 174, + 241, + 617, + 256 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Other Binary Classification Loss. The results on exponential loss can be generalized to a much broader class of binary classification loss. The class includes the logistic loss which is one of the most popular loss functions, $\\ell ( q ) = \\log ( 1 + e ^ { - q } )$ . The function class also includes other losses with exponential tail, e.g., $\\ell ( q ) = e ^ { - q ^ { 3 } } , \\ell ( q ) = \\log ( 1 + e ^ { - q ^ { 3 } } )$ . For all those loss functions, we can use its inverse function $\\ell ^ { - 1 }$ to define the smoothed normalized margin as follows ", + "bbox": [ + 173, + 262, + 825, + 335 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/21749494740c2b50e962f8915088cb2872acdb2696ccd7a5a5173548dddf35db.jpg", + "text": "$$\n\\tilde { \\gamma } ( \\pmb { \\theta } ) = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ( \\pmb { \\theta } ) ) } { \\| \\pmb { \\theta } \\| _ { 2 } ^ { L } } .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 338, + 566, + 375 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Then all our results for gradient flow continue to hold (Appendix A). Using a similar modification, we can also extend it to gradient descent (Appendix F). ", + "bbox": [ + 173, + 377, + 823, + 407 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Cross-entropy Loss. In multi-class classification, we can define $q _ { n }$ to be the difference between the classification score for the true label and the maximum score for the other labels, then the margin qmin := minn∈[N] qn and the normalized margin γ¯(θ) := qmin(θ)kθkL can be similarly defined as before. In Appendix G, we define the smoothed normalized margin for cross-entropy loss to be the same as that for logistic loss (See Remark A.4). Then we show that Theorem 4.1 and Theorem 4.4 still hold (but with a slightly different definition of (P)) for gradient flow, and we also extend the results to gradient descent. ", + "bbox": [ + 173, + 412, + 825, + 517 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Multi-homogeneous Models. Some neural networks indeed possess a stronger property than homogeneity, which we call multi-homogeneity. For example, the output of a CNN (without bias terms) is 1-homogeneous with respect to the weights of each layer. In general, we say that a neural network $\\Phi ( \\pmb \\theta ; \\pmb x )$ with $\\pmb \\theta = ( \\pmb w _ { 1 } , \\dots , \\pmb w _ { m } )$ is $\\left( k _ { 1 } , \\ldots , k _ { m } \\right)$ -homogeneous if for any and any $c _ { 1 } , \\ldots , c _ { m } > 0$ , we have $\\begin{array} { r } { \\Phi \\big ( c _ { 1 } \\pmb { w } _ { 1 } , \\ldots , c _ { m } \\pmb { w } _ { m } ; \\pmb { x } \\big ) = \\prod _ { i = 1 } ^ { m } c _ { i } ^ { k _ { i } } \\cdot \\Phi \\big ( \\pmb { w } _ { 1 } , \\ldots , \\pmb { w } _ { m } ; \\pmb { x } \\big ) } \\end{array}$ . In the previous example, an $L$ -layer CNN with layer weights $\\pmb \\theta = ( \\pmb w _ { 1 } , \\dots , \\pmb w _ { L } )$ is $( 1 , \\ldots , 1 )$ -homogeneous. ", + "bbox": [ + 173, + 523, + 825, + 612 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "easily see that that , so our previous analys $\\left( k _ { 1 } , \\ldots , k _ { m } \\right)$ -homogeneity implies neous models still appli $L$ -homogeneity, where to multi-homogeneous $L \\ =$ $\\textstyle \\sum _ { i = 1 } ^ { m } k _ { i }$ ", + "bbox": [ + 174, + 616, + 825, + 659 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/c983cfcda7cfeee0cf942cc87cccfcaa64bbf4a2f49dbbd9b79d956287bca0e2.jpg", + "text": "$$\n\\bar { \\gamma } ( \\pmb { w } _ { 1 } , \\ldots , \\pmb { w } _ { m } ) : = q _ { \\operatorname* { m i n } } ( \\frac { \\pmb { w } _ { 1 } } { \\lVert \\pmb { w } _ { 1 } \\rVert _ { 2 } } , \\allowbreak \\ldots , \\frac { \\pmb { w } _ { m } } { \\lVert \\pmb { w } _ { m } \\rVert _ { 2 } } ) = \\frac { q _ { \\operatorname* { m i n } } } { \\prod _ { i = 1 } ^ { m } \\lVert \\pmb { w } _ { i } \\rVert _ { 2 } ^ { k _ { i } } } .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 662, + 715, + 695 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this case, the smoothed approximation of $\\bar { \\gamma }$ for general binary classification loss (under some conditions) can be similarly defined for gradient flow: ", + "bbox": [ + 174, + 699, + 825, + 727 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/6f61b68c966b00aa003d66d58ce28f342b2d5d2a6005bf234a614e207c517d7f.jpg", + "text": "$$\n\\widetilde { \\gamma } ( \\pmb { w } _ { 1 } , \\dots , \\pmb { w } _ { m } ) : = \\frac { \\ell ^ { - 1 } ( \\pmb { \\mathscr { L } } ) } { \\prod _ { i = 1 } ^ { m } \\| \\pmb { w } _ { i } \\| _ { 2 } ^ { k _ { i } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 729, + 614, + 768 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "It can be shown that $\\tilde { \\gamma }$ is also non-decreasing during training when the loss is small enough (Appendix $_ \\mathrm { H }$ ). In the case of cross-entropy loss, we can still define $\\tilde { \\gamma }$ by (5) while $\\ell ( \\cdot )$ is set to the logistic loss in the formula. ", + "bbox": [ + 173, + 771, + 825, + 814 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 PROOF SKETCH: GRADIENT FLOW ON HOMOGENEOUS MODEL WITH EXPONENTIAL LOSS ", + "text_level": 1, + "bbox": [ + 176, + 832, + 781, + 867 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we present a proof sketch in the case of gradient flow on homogeneous model with exponential loss to illustrate our proof ideas. Due to space limit, the proof for the main theorems on gradient flow and gradient descent in Section 4 are deferred to Appendix A and E respectively. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For convenience, we introduce a few more notations for a $L$ -homogeneous neural network $\\Phi ( \\pmb \\theta ; \\pmb x )$ . Let $S ^ { d - 1 } = \\{ \\pmb \\theta \\in \\mathbb { R } ^ { d } : \\| \\pmb \\theta \\| _ { 2 } = 1 \\}$ be the set of $L ^ { 2 }$ -normalized parameters. Define $\\rho : = \\| \\pmb \\theta \\| _ { 2 }$ and $\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\Vert \\pmb \\theta \\Vert _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}$ to be the length and direction of $\\pmb \\theta$ . For both gradient descent and gradient flow, $\\pmb \\theta$ is a function of time $t$ . For convenience, we also view the functions of $\\pmb \\theta$ , including $\\mathcal { L } ( \\pmb { \\theta } ) , q _ { n } ( \\pmb { \\theta } ) , q _ { \\mathrm { m i n } } ( \\pmb { \\theta } )$ , as functions of $t$ . So we can write $\\mathscr { L } ( t ) : = \\mathscr { L } ( \\pmb \\theta ( t ) ) , q _ { n } ( t ) : =$ $q _ { n } ( \\pmb \\theta ( t ) ) , q _ { \\mathrm { m i n } } ( t ) : = q _ { \\mathrm { m i n } } ( \\pmb \\theta ( t ) )$ . ", + "bbox": [ + 173, + 102, + 825, + 195 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Lemma 5.1 below is the key lemma in our proof. It decomposes the growth of the smoothed normalized margin into the ratio of two quantities related to the radial and tangential velocity components of $\\pmb { \\theta }$ respectively. We will give a proof sketch for this later in this section. We believe that this lemma is of independent interest. ", + "bbox": [ + 173, + 199, + 825, + 256 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Lemma 5.1 (Corollary of Lemma B.1). For a.e. $t > t _ { 0 }$ , ", + "bbox": [ + 174, + 257, + 542, + 273 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/965649f468c15c8e964a4520f5c3cbf7c57a6c5b2f4a7a2a6b09815e4dc23ac0.jpg", + "text": "$$\n\\frac { d } { d t } \\log \\rho > 0 \\quad a n d \\quad \\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 275, + 692, + 319 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Using Lemma 5.1, the first two claims in Theorem 4.1 can be directly proved. For the third claim, we make use of the monotonicity of the margin to lower bound the gradient, and then show $\\mathcal { L } \\to 0$ and $\\rho + \\infty$ . Recall that $\\tilde { \\gamma }$ is an $O ( \\rho ^ { - L } )$ -additive approximation for $\\bar { \\gamma }$ . So this proves the third claim. We defer the detailed proof to Appendix B. ", + "bbox": [ + 173, + 325, + 826, + 383 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To show Theorem 4.4, we first change the time measure to $\\log \\rho$ , i.e., now we see $t$ as a function of $\\log \\rho$ . So the second inequality in Lemma 5.1 can be rewritten as $\\begin{array} { r } { \\frac { d \\log \\tilde { \\gamma } } { d \\log \\rho } \\geq L \\| \\frac { d \\hat { \\pmb \\theta } } { d \\log \\rho } \\| _ { 2 } ^ { 2 } } \\end{array}$ . Integrating on both sides and noting that $\\tilde { \\gamma }$ is upper-bounded, we know that there must be many instant $\\log \\rho$ such that $\\big \\| \\frac { d \\hat { \\pmb { \\theta } } } { d \\log \\rho } \\big \\| _ { 2 }$ is small. By analyzing the landscape of training loss, we show that these points are “approximate” KKT points. Then we show that every convergent sub-sequence of $\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\}$ can be modified to be a sequence of “approximate” KKT points which converges to the same limit. Then we conclude the proof by applying a theorem from (Dutta et al., 2013) to show that the limit of this convergent sequence of “approximate” KKT points is a KKT point. We defer the detailed proof to Appendix C. ", + "bbox": [ + 173, + 388, + 826, + 530 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Now we give a proof sketch for Lemma 5.1, in which we derive the formula of $\\tilde { \\gamma }$ step by step. In the proof, we obtain several clean close form formulas for several relevant quantities, by using the chain rule and Euler’s theorem for homogeneous functions extensively. ", + "bbox": [ + 173, + 536, + 825, + 579 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Proof Sketch of Lemma 5.1. For ease of presentation, we ignore the regularity issues of taking derivatives in this proof sketch. We start from the equation $\\begin{array} { r } { \\frac { d \\mathcal { L } } { d t } = - \\left. \\partial ^ { \\circ } \\mathcal { L } ( \\pmb { \\theta } ( t ) ) , \\frac { d \\pmb { \\theta } } { d t } \\right. = - \\left\\| \\frac { d \\pmb { \\theta } } { d t } \\right\\| _ { 2 } ^ { 2 } } \\end{array}$ which follows from the chain rule (see also Lemma I.3). Then we note that $\\textstyle { \\frac { d \\theta } { d t } }$ can be decomposed into two parts: the radial component $\\begin{array} { r } { \\pmb { v } : = \\hat { \\pmb { \\theta } } \\hat { \\pmb { \\theta } } ^ { \\top } \\frac { d \\pmb { \\theta } } { d t } } \\end{array}$ and the tangent component $\\begin{array} { r } { \\mathbf { \\hat { \\boldsymbol { u } } } : = ( \\mathbf { \\boldsymbol { I } } - \\hat { \\theta } \\hat { \\theta } ^ { \\top } ) \\frac { d \\theta } { d t } } \\end{array}$ . ", + "bbox": [ + 173, + 593, + 825, + 664 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The radial component is easier to analyze. By the chain rule, $\\begin{array} { r } { \\| \\pmb { v } \\| _ { 2 } = \\hat { \\pmb { \\theta } } ^ { \\top } \\frac { d \\pmb { \\theta } } { d t } = \\frac { 1 } { \\rho } \\left. \\pmb { \\theta } , \\frac { d \\pmb { \\theta } } { d t } \\right. = \\frac { 1 } { \\rho } \\cdot \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } } \\end{array}$ . Fo r 12 dρ2dt , we have an exact formula: ", + "bbox": [ + 173, + 670, + 823, + 709 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/35a1eb0d3944fb2de6f893f0d95c9248a8a4b75bbc7d53bb93d1bb4fc0121251.jpg", + "text": "$$\n{ \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } = \\left. \\theta , { \\frac { d \\theta } { d t } } \\right. = \\left. \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\partial ^ { \\circ } q _ { n } , \\theta \\right. = L \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 710, + 700, + 755 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The last equality is due to $\\langle \\partial ^ { \\circ } q _ { n } , \\theta \\rangle = L q _ { n }$ by homogeneity of $q _ { n }$ . This is sometimes called Euler’s theorem for homogeneous functions (see Theorem B.2). For differentiable $q _ { n }$ , it can be easily proved by taking the derivative over $c$ on both sides of $q _ { n } ( c \\pmb { \\theta } ) = c ^ { L } q _ { n } ( \\pmb { \\theta } )$ and letting $c = 1$ . ", + "bbox": [ + 174, + 756, + 825, + 799 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "With (6), we can lower bound $\\textstyle { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } }$ by ", + "bbox": [ + 176, + 806, + 429, + 824 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/aa90c976b89dda2f0cccb3a4dbadb19b4be6f7b1e3486c23865443d0cea35669.jpg", + "text": "$$\n\\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { n } \\ge L \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { \\operatorname* { m i n } } \\ge L \\cdot \\mathcal { L } \\log \\frac { 1 } { \\mathcal { L } } ,\n$$", + "text_format": "latex", + "bbox": [ + 307, + 825, + 691, + 869 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where the last inequality uses the fact that $e ^ { - q _ { \\mathrm { m i n } } } \\leq { \\mathcal { L } }$ . (7) also implies that $\\textstyle { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } > 0$ for $t > t _ { 0 }$ since $\\mathcal { L } ( t _ { 0 } ) < 1$ and $\\mathcal { L }$ is non-increasing. As $\\begin{array} { r } { \\frac { d } { d t } \\log \\rho = \\frac { 1 } { 2 \\rho ^ { 2 } } \\frac { d \\rho ^ { 2 } } { d t } } \\end{array}$ , this also proves the first inequality of Lemma 5.1. ", + "bbox": [ + 174, + 872, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Now, we have $\\begin{array} { r } { \\left\\| \\pmb { v } \\right\\| _ { 2 } ^ { 2 } = \\frac { 1 } { \\rho ^ { 2 } } \\left( \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } \\right) ^ { 2 } = \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } \\cdot \\frac { d } { d t } \\log \\rho } \\end{array}$ on the one hand; on the other hand, by the chain rule we have $\\begin{array} { r } { \\frac { d \\hat { \\pmb { \\theta } } } { d t } = \\frac { 1 } { \\rho ^ { 2 } } ( \\rho \\frac { d \\pmb { \\theta } } { d t } - \\frac { d \\rho } { d t } \\pmb { \\theta } ) = \\frac { 1 } { \\rho ^ { 2 } } ( \\rho \\frac { d \\pmb { \\theta } } { d t } - ( \\hat { \\pmb { \\theta } } ^ { \\top } \\frac { d \\pmb { \\theta } } { d t } ) \\pmb { \\theta } ) = \\frac { \\pmb { u } } { \\rho } } \\end{array}$ . So we have ", + "bbox": [ + 173, + 101, + 825, + 150 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/ca355ccf7e1a696811895fc82cec9e80f2776d9d6e70d92ff473a3d270ec5f78.jpg", + "text": "$$\n- { \\frac { d { \\mathcal { L } } } { d t } } = \\left\\| { \\frac { d \\theta } { d t } } \\right\\| _ { 2 } ^ { 2 } = \\left\\| v \\right\\| _ { 2 } ^ { 2 } + \\left\\| u \\right\\| _ { 2 } ^ { 2 } = { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } \\cdot { \\frac { d } { d t } } \\log \\rho + \\rho ^ { 2 } \\left\\| { \\frac { d \\hat { \\theta } } { d t } } \\right\\| _ { 2 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 279, + 156, + 717, + 203 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Dividing 12 dρ2dt on the leftmost and rightmost sides, we have ", + "bbox": [ + 173, + 212, + 566, + 231 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/d4f906716e4572d65d93729f4d450c357f426bc223e5dc8c9ba6ad398322e017.jpg", + "text": "$$\n- { \\frac { d { \\mathcal { L } } } { d t } } \\cdot \\left( { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } \\right) ^ { - 1 } = { \\frac { d } { d t } } \\log \\rho + \\left( { \\frac { d } { d t } } \\log \\rho \\right) ^ { - 1 } \\left\\| { \\frac { d { \\hat { \\pmb { \\theta } } } } { d t } } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 239, + 689, + 285 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "By $\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } \\ge 0 } \\end{array}$ and (7), the LHS is no greater than $\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } \\cdot \\left( L \\cdot \\mathcal { L } \\log \\frac { 1 } { \\mathcal { L } } \\right) ^ { - 1 } = \\frac { 1 } { L } \\log \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}$ . Thus we dt 2 have $\\begin{array} { r } { \\frac { d } { d t } \\log \\log \\frac { 1 } { \\mathcal { L } } - L \\frac { d } { d t } \\log \\rho \\geq L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left. \\frac { d \\hat { \\theta } } { d t } \\right. _ { 2 } ^ { 2 } } \\end{array}$ , where the LHS is exactly $\\begin{array} { r } { \\frac { d } { d t } \\log \\tilde { \\gamma } } \\end{array}$ . ", + "bbox": [ + 173, + 294, + 825, + 340 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 DISCUSSION AND FUTURE DIRECTIONS ", + "text_level": 1, + "bbox": [ + 174, + 358, + 537, + 376 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we analyze the dynamics of gradient flow/descent of homogeneous neural networks under a minimal set of assumptions. The main technical contribution of our work is to prove rigorously that for gradient flow/descent, the normalized margin is increasing and converges to a KKT point of a natural max-margin problem. Our results leads to some natural further questions: ", + "bbox": [ + 174, + 391, + 825, + 448 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "• Can we generalize our results for gradient descent on smooth neural networks to nonsmooth ones? In the smooth case, we can lower bound the decrement of training loss by the gradient norm squared, multiplied by a factor related to learning rate. However, in the non-smooth case, no such inequality is known in the optimization literature. \n• Can we make more structural assumptions on the neural network to prove stronger results? In this work, we use a minimal set of assumptions to show that the convergent direction of parameters is a KKT point. A potential research direction is to identify more key properties of modern neural networks and show that the normalized margin at convergence is locally or globally optimal (in terms of optimizing (P)). Can we extend our results to neural networks with bias terms? In our experiments, the normalized margin of the CNN with bias also increases during training despite that its output is non-homogeneous. It is very interesting (and technically challenging) to provide a rigorous proof for this fact. ", + "bbox": [ + 217, + 460, + 825, + 654 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 678, + 356, + 693 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The research is supported in part by the National Natural Science Foundation of China Grant 61822203, 61772297, 61632016, 61761146003and the Zhongguancun Haihua Institute for Frontier Information Technology and Turing AI Institute of Nanjing. We thank Liwei Wang for helpful suggestions on the connection between margin and robustness. We thank Sanjeev Arora, Tianle Cai, Simon Du, Jason D. Lee, Zhiyuan Li, Tengyu Ma, Ruosong Wang for helpful discussions. ", + "bbox": [ + 174, + 709, + 825, + 780 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 803, + 285, + 818 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Pierre-Antoine Absil, Robert Mahony, and Benjamin Andrews. Convergence of the iterates of descent methods for analytic cost functions. SIAM Journal on Optimization, 16(2):531–547, 2005. ", + "bbox": [ + 174, + 827, + 823, + 854 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Alnur Ali, J. Zico Kolter, and Ryan J. Tibshirani. A continuous-time view of early stopping for least squares regression. 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", + "bbox": [ + 173, + 478, + 584, + 493 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A RESULTS FOR GENERAL LOSS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 460, + 118 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In this section, we state our results for a broad class of binary classification loss. A major consequence of this generalization is that the logistic loss, one of the most popular loss functions, $\\ell ( \\bar { q } ) = \\log ( 1 + e ^ { - \\bar { q } } )$ is included. The function class also includes other losses with exponential tail, e.g., $\\ell ( q ) = e ^ { - q ^ { 3 } } , \\ell ( q ) = \\log ( 1 + e ^ { - q ^ { 3 } } )$ . ", + "bbox": [ + 173, + 132, + 825, + 193 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.1 ASSUMPTIONS ", + "text_level": 1, + "bbox": [ + 174, + 208, + 321, + 223 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We first focus on gradient flow. We assume (A1), (A2) as we do for exponential loss. For (A3), (A4), we replace them with two weaker assumptions (B3), (B4). All the assumptions are listed below: ", + "bbox": [ + 178, + 234, + 825, + 263 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(A1). (Regularity). For any fixed $_ { \\textbf { \\em x } }$ , $\\Phi ( \\cdot ; x )$ is locally Lipschitz and admits a chain rule; \n(A2). (Homogeneity). There exists $L > 0$ such that $\\forall \\alpha > 0 : \\Phi ( \\alpha \\pmb { \\theta } ; \\pmb { x } ) = \\alpha ^ { L } \\Phi ( \\pmb { \\theta } ; \\pmb { x } )$ ; \n(B3). The loss function $\\ell ( q )$ can be expressed as $\\ell ( q ) = e ^ { - f ( q ) }$ such that (B3.1). $f : \\mathbb { R } \\to \\mathbb { R }$ is ${ \\mathcal { C } } ^ { 1 }$ -smooth. (B3.2). $f ^ { \\prime } ( q ) > 0$ for all $q \\in \\mathbb { R }$ . (B3.3). There exists $b _ { f } \\ \\geq \\ 0$ such that $f ^ { \\prime } ( q ) q$ is non-decreasing for $q \\in ( b _ { f } , + \\infty )$ , and $f ^ { \\prime } ( q ) q \\to + \\infty$ as $q \\to + \\infty$ . (B3.4). Let $g \\ : \\ [ f ( b _ { f } ) , + \\infty ) \\ \\ [ b _ { f } , + \\infty )$ be the inverse function of $f$ on the domain $[ b _ { f } , + \\infty )$ . There exists $b _ { g } ~ \\overset { \\cdot } { = } ~ \\operatorname* { m a x } \\{ 2 f ( b _ { f } ) , f ( 2 b _ { f } ) \\} , K \\geq 1$ such that $g ^ { \\prime } ( x ) \\leq$ $K g ^ { \\prime } ( \\theta x )$ and $f ^ { \\prime } ( y ) ~ \\leq ~ \\breve { K } f ^ { \\prime } ( \\theta y )$ for all $\\dot { x } \\in ( \\bar { b _ { g } } , + \\infty ) , y \\in ( g ( b _ { g } ) , + \\infty )$ and $\\theta \\in [ 1 / 2 , 1 )$ . ", + "bbox": [ + 187, + 272, + 828, + 450 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(B4). (Separability). There exists a time $t _ { 0 }$ such that $\\mathcal { L } ( t _ { 0 } ) < e ^ { - f ( b _ { f } ) } = \\ell ( b _ { f } )$ . ", + "bbox": [ + 189, + 453, + 709, + 469 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(A1) and (A2) remain unchanged. (B3) is satisfied by exponential loss $\\ell ( q ) = e ^ { - q }$ (with $f ( q ) = q )$ and logistic loss $\\ell ( q ) = \\log ( \\bar { 1 } + e ^ { - q } )$ (with $f ( q ) \\stackrel { \\cdot } { = } - \\log \\log ( 1 + e ^ { - q } ) \\stackrel { \\cdot } { ) }$ . (B4) are essentially the same as (A4) but (B4) uses a threshold value that depends on the loss function. Assuming (B3), it is easy to see that (B4) ensures the separability of data since $\\ell ( q _ { n } ) < e ^ { - f ( b _ { f } ) }$ implies $q _ { n } > b _ { f } \\geq 0$ . For logistic loss, we can set $b _ { f } = 0$ (see Remark A.2). So the corresponding threshold value in (B4) is $\\ell ( 0 ) = \\log 2$ . ", + "bbox": [ + 173, + 479, + 825, + 565 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Now we discuss each of the assumptions in (B3). (B3.1) is a natural assumption on smoothness. (B3.2) requires $\\ell ( \\cdot )$ to be monotone decreasing, which is also natural since $\\ell ( \\cdot )$ is used for binary classification. The rest of two assumptions in (B3) characterize the properties of $\\ell ^ { \\prime } ( q )$ when $q$ is large enough. (B3.3) is an assumption that appears naturally from the proof. For exponential loss, $f ^ { \\prime } ( q ) q = q$ is always non-decreasing, so we can set $b _ { f } = 0$ . In (B3.4), the inverse function $g$ is defined. It is guaranteed by (B3.1) and (B3.2) that $g$ always exists and $g$ is also ${ \\mathcal { C } } ^ { 1 }$ -smooth. Though (B3.4) looks very complicated, it essentially says that $f ^ { \\prime } ( \\Theta ( q ) ) = \\Theta ( \\bar { f } ^ { \\prime } ( q ) ) , g ^ { \\prime } ( \\Theta ( q ) ) = \\Theta ( g ^ { \\prime } ( \\bar { q } ) )$ as $q \\infty$ . (B3.4) is indeed a technical assumption that enables us to asymptotically compare the loss or the length of gradient at different data points. It is possible to base our results on weaker assumptions than (B3.4), but we use (B3.4) for simplicity since it has already been satisfied by many loss functions such as the aforementioned examples. ", + "bbox": [ + 173, + 570, + 825, + 727 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We summarize the corresponding $f , g$ and $b _ { f }$ for exponential loss and logistic loss below: ", + "bbox": [ + 173, + 732, + 759, + 748 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Remark A.1. Exponential loss $\\ell ( q ) = e ^ { - q }$ satisfies (B3) with ", + "bbox": [ + 174, + 751, + 586, + 766 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/a25fb0e480632ea559ad84b365b194aa5460fcbd8ffffe15715d67bf92d7e7d7.jpg", + "text": "$$\nf ( q ) = q \\qquad f ^ { \\prime } ( q ) = 1 \\qquad g ( q ) = q \\qquad g ^ { \\prime } ( q ) = 1 \\qquad b _ { f } = 0 \\qquad \\ell ( b _ { f } ) = 1 .\n$$", + "text_format": "latex", + "bbox": [ + 235, + 768, + 763, + 787 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Remark A.2. Logistic loss $\\ell ( q ) = \\log ( 1 + e ^ { - q } )$ satisfies $( B 3 )$ with ", + "bbox": [ + 173, + 791, + 617, + 808 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/9aeacebecb22545acf7006232f10f6e95a6f872953d3e0452bcf59b04b4d5f1c.jpg", + "text": "$$\n\\begin{array} { l l } { { f ( q ) = - \\log \\log ( 1 + e ^ { - q } ) = \\Theta ( q ) ~ } } & { { ~ f ^ { \\prime } ( q ) = \\displaystyle \\frac { e ^ { - q } } { ( 1 + e ^ { - q } ) \\log ( 1 + e ^ { - q } ) } = \\Theta ( 1 ) } } \\\\ { { { } } } & { { { } } } \\\\ { { g ( q ) = - \\log ( e ^ { e ^ { - q } } - 1 ) = \\Theta ( q ) ~ } } & { { ~ g ^ { \\prime } ( q ) = \\displaystyle \\frac { e ^ { - q } } { e ^ { e ^ { - q } } - 1 } = \\Theta ( 1 ) } } \\\\ { { { } } } & { { { } } } \\\\ { { b _ { f } = 0 ~ } } & { { ~ \\ell ( b _ { f } ) = \\log 2 . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 207, + 813, + 789, + 900 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The proof for Remark A.1 is trivial. For Remark A.2, we give a proof below. ", + "bbox": [ + 173, + 909, + 676, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof for Remark A.2. By simple calculations, the formulas for $f ( q ) , f ^ { \\prime } ( q ) , g ( q ) , g ^ { \\prime } ( q )$ are correct. (B3.1) is trivial. $f ^ { \\prime } ( q ) ~ = ~ \\frac { e ^ { - q } } { ( 1 + e ^ { - q } ) \\log ( 1 + e ^ { - q } ) } ~ > ~ 0$ , so (B3.2) is satisfied. For (B3.3), note that $\\begin{array} { r } { f ^ { \\prime } ( q ) q = \\frac { q } { ( 1 + e ^ { q } ) \\log ( 1 + e ^ { - q } ) } } \\end{array}$ . The denominator is a decreasing function since ", + "bbox": [ + 173, + 102, + 825, + 156 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/ba9ead046cd453e239d2c935dfde05a2254e3890d255d93b27a1cbeb40f1e0e0.jpg", + "text": "$$\n{ \\frac { d } { d q } } \\left( ( 1 + e ^ { q } ) \\log ( 1 + e ^ { - q } ) \\right) = e ^ { q } \\log ( 1 + e ^ { - q } ) - 1 < e ^ { q } \\cdot e ^ { - q } - 1 = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 261, + 160, + 736, + 193 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Thus, $f ^ { \\prime } ( q ) q$ is a strictly increasing function on $\\mathbb { R }$ . As $b _ { f }$ is required to be non-negative, we set $b _ { f } = 0$ . For proving that $f ^ { \\prime } ( q ) q \\to + \\infty$ and (B4), we only need to notice that $\\begin{array} { r } { f ^ { \\prime } ( q ) \\sim \\frac { e ^ { - q } } { 1 \\cdot e ^ { - q } } = 1 } \\end{array}$ and $g ^ { \\prime } ( x ) = \\bar { 1 } / f ^ { \\prime } ( \\bar { g ( x ) } ) \\sim 1$ . □ ", + "bbox": [ + 173, + 196, + 826, + 244 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.2 SMOOTHED NORMALIZED MARGIN ", + "text_level": 1, + "bbox": [ + 176, + 258, + 464, + 273 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For a loss function $\\ell ( \\cdot )$ satisfying (B3), it is easy to see from (B3.2) that its inverse function $\\ell ^ { - 1 } ( \\cdot )$ must exist. For this kind of loss functions, we define the smoothed normalized margin as follows: ", + "bbox": [ + 173, + 284, + 823, + 314 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Definition A.3. For a loss function $\\ell ( \\cdot )$ satisfying (B3), the smoothed normalized margin $\\widetilde \\gamma ( \\pmb \\theta )$ of $\\pmb \\theta$ is defined as ", + "bbox": [ + 171, + 315, + 825, + 344 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/8b913b60edcebfcd8526eedfbe5f446b272c22f815e963b107ad97eb2188564a.jpg", + "text": "$$\n\\widetilde { \\gamma } ( \\pmb { \\theta } ) : = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\rho ^ { L } } = \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\rho ^ { L } } = \\frac { g \\left( - \\log \\left( \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ( \\pmb { \\theta } ) ) } \\right) \\right) } { \\rho ^ { L } } ,\n$$", + "text_format": "latex", + "bbox": [ + 282, + 345, + 714, + 390 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\ell ^ { - 1 } ( \\cdot )$ is the inverse function of $\\ell ( \\cdot )$ and $\\rho : = \\| \\pmb \\theta \\| _ { 2 }$ ", + "bbox": [ + 174, + 393, + 558, + 409 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Remark A.4. For logistic loss $\\ell ( q ) = \\log ( 1 + e ^ { - q } )$ , $\\begin{array} { r } { \\widetilde \\gamma ( \\pmb \\theta ) = \\rho ^ { - L } \\log \\frac { 1 } { \\exp ( \\mathcal L ) - 1 } } \\end{array}$ ; for exponential loss $\\ell ( q ) = e ^ { - q }$ , $\\begin{array} { r } { \\tilde { \\gamma } ( \\pmb { \\theta } ) = \\rho ^ { - L } \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}$ , which is the same as (3). ", + "bbox": [ + 174, + 410, + 821, + 446 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Now we give some insights on how well $\\tilde { \\gamma } ( \\pmb \\theta )$ approximates $\\bar { \\gamma } ( \\pmb \\theta )$ using a similar argument as in Section 4.2. Using the LogSumExp function, the smoothed normalized margin $\\widetilde \\gamma ( \\pmb \\theta )$ can also be written as ", + "bbox": [ + 174, + 454, + 820, + 494 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/6da21a0eaa2407d08cd6f7318527e50b01a85acc2a2d07edbc7204f22ecf0aee.jpg", + "text": "$$\n\\tilde { \\gamma } ( \\pmb \\theta ) = \\frac { g ( - \\mathrm { L S E } ( - f ( q _ { 1 } ) , \\dots , - f ( q _ { N } ) ) ) } { \\rho ^ { L } } .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 492, + 640, + 526 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "LSE is a $( \\log N )$ -additive approximation for max. So we can roughly approximate $\\widetilde \\gamma ( \\pmb \\theta )$ by ", + "bbox": [ + 169, + 527, + 769, + 542 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/cb28978f2607d2b1cc9719bfbe8f62a00a8937b9042e17c82b0c0a2147c13a22.jpg", + "text": "$$\n\\tilde { \\gamma } ( \\pmb \\theta ) \\approx \\frac { g ( - \\operatorname* { m a x } \\{ - f ( q _ { 1 } ) , \\dots , - f ( q _ { N } ) \\} ) } { \\rho ^ { L } } = \\frac { g ( f ( q _ { \\operatorname* { m i n } } ) ) } { \\rho ^ { L } } = \\bar { \\gamma } ( \\pmb \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 545, + 720, + 580 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Note that (B3.3) is crucial to make the above approximation reasonable. Similar to exponential loss, we can show the following lemma asserting that $\\tilde { \\gamma }$ is a good approximation of $\\bar { \\gamma }$ . ", + "bbox": [ + 173, + 582, + 823, + 611 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma A.5. Assuming $( B 3 ) ^ { 4 }$ , we have the following properties about the margin: ", + "bbox": [ + 176, + 613, + 718, + 628 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(a) $\\begin{array} { r } { f ( q _ { \\mathrm { m i n } } ) - \\log N \\leq \\log \\frac { 1 } { \\mathcal { L } } \\leq f ( q _ { \\mathrm { m i n } } ) . } \\end{array}$ ", + "bbox": [ + 176, + 637, + 449, + 656 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(b) $\\begin{array} { r } { I f \\log { \\frac { 1 } { \\mathcal { L } } } > f ( b _ { f } ) } \\end{array}$ , then there exists $\\xi \\in \\left( f ( q _ { \\operatorname* { m i n } } ) - \\log N , f ( q _ { \\operatorname* { m i n } } ) \\right) \\cap \\left( b _ { f } , + \\infty \\right)$ such that ", + "bbox": [ + 171, + 662, + 787, + 681 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/47307422b79dc5f6ccec5c3ed6587478b29e621b522846f60e153ef29ed620f4.jpg", + "text": "$$\n\\bar { \\gamma } - \\frac { g ^ { \\prime } ( \\xi ) \\log N } { \\rho ^ { L } } \\leq \\tilde { \\gamma } \\leq \\bar { \\gamma } .\n$$", + "text_format": "latex", + "bbox": [ + 421, + 684, + 599, + 718 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(c) For a seuqnce of parameters $\\{ \\pmb \\theta _ { m } \\in \\mathbb { R } ^ { d } : m \\in \\mathbb { N } \\}$ , if $\\mathcal { L } ( \\pmb { \\theta } _ { m } ) 0$ , then $\\lvert \\tilde { \\gamma } ( \\pmb { \\theta } _ { m } ) - \\bar { \\gamma } ( \\pmb { \\theta } _ { m } ) \\rvert 0$ . ", + "bbox": [ + 171, + 728, + 823, + 747 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. (a) can be easily deduced from $e ^ { - f ( q _ { \\mathrm { m i n } } ) } \\leq \\mathcal { L } \\leq N e ^ { - f ( q _ { \\mathrm { m i n } } ) }$ . Combining $( a )$ and the monotonicity of $g ( \\cdot )$ , we further have $g ( s ) \\leq g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\leq q _ { \\operatorname* { m i n } }$ for $s : = \\operatorname* { m a x } \\{ f ( b _ { f } ) , f ( q _ { \\operatorname* { m i n } } ) \\ : -$ $\\log N \\}$ . By the mean value Theorem, there exists $\\xi \\in \\widetilde { ( s , f ( q _ { \\operatorname* { m i n } } ) ) } )$ such that $g ( s ) = { \\dot { g } } ( f ( q _ { \\mathrm { m i n } } ) ) -$ $g ^ { \\prime } ( \\xi ) ( f ( q _ { \\mathrm { m i n } } ) - s ) \\geq q _ { \\mathrm { m i n } } - g ^ { \\prime } ( \\xi ) \\log N$ . Dividing $\\rho ^ { L }$ on each side of $q _ { \\operatorname* { m i n } } - g ^ { \\prime } ( \\xi ) \\log N \\le$ $g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) \\leq q _ { \\operatorname* { m i n } }$ proves $( b )$ . ", + "bbox": [ + 173, + 758, + 825, + 833 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Now we prove $( c )$ . Without loss of generality, we assume log $\\frac { 1 } { \\mathcal { L } ( \\pmb { \\theta } _ { m } ) } > f ( b _ { f } )$ for all $\\pmb { \\theta } _ { m }$ . It follows from $( b )$ that for every $\\pmb { \\theta } _ { m }$ there exists $\\xi _ { m } \\in ( f ( q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) ) - \\log N , f ( q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) ) ) \\cap ( b _ { f } , + \\infty )$ such that ", + "bbox": [ + 173, + 840, + 825, + 885 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/5bd289c0893766d5b9254ca13097eee09973bf064d805c5311650e27a2430b9b.jpg", + "text": "$$\n\\bar { \\gamma } ( \\pmb \\theta _ { m } ) - ( g ^ { \\prime } ( \\xi _ { m } ) \\log N ) / \\rho ( \\pmb \\theta _ { m } ) ^ { L } \\leq \\widetilde \\gamma ( \\pmb \\theta _ { m } ) \\leq \\bar { \\gamma } ( \\pmb \\theta _ { m } ) .\n$$", + "text_format": "latex", + "bbox": [ + 318, + 883, + 679, + 902 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Note that $\\begin{array} { r c l } { \\xi _ { m } } & { \\ge } & { f \\big ( q _ { \\mathrm { m i n } } ( \\pmb \\theta _ { m } ) \\big ) - \\log N \\ge \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta _ { m } ) } - \\log N + \\infty } \\end{array}$ . So $\\begin{array} { r l } { \\frac { g ( \\xi _ { m } ) } { g ^ { \\prime } ( \\xi _ { m } ) } } & { { } = } \\end{array}$ $f ^ { \\prime } ( g ( \\xi _ { m } ) ) g ( \\xi _ { m } ) \\to + \\infty$ by (B3.3). Also note that there exists a constant $B _ { 0 }$ such that $\\bar { \\gamma } ( \\pmb { \\theta } _ { m } ) \\leq B _ { 0 }$ for all $m$ since $\\bar { \\gamma }$ is continuous on the unit sphere $S ^ { d - 1 }$ . So we have ", + "bbox": [ + 173, + 101, + 826, + 150 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/12d6697634102808eb7462e498cc1d2cbbc1209f368febcaea930ba6d9f02bdd.jpg", + "text": "$$\n\\frac { g ^ { \\prime } ( \\xi _ { m } ) } { \\rho ( \\theta _ { m } ) ^ { L } } = \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\frac { g ( \\xi _ { m } ) } { \\rho ( \\theta _ { m } ) ^ { L } } \\le \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\frac { q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) } { \\rho ( \\theta _ { m } ) ^ { L } } = \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\cdot \\bar { \\gamma } ( \\theta _ { m } ) \\le \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\cdot B \\to 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 205, + 152, + 789, + 188 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where the first inequality follows since $\\xi _ { m } \\leq f ( q _ { \\operatorname* { m i n } } ( \\pmb \\theta _ { m } ) )$ . Together with (8), we have $\\tilde { \\gamma } ( \\pmb { \\theta } _ { m } ) -$ $\\bar { \\gamma } ( \\pmb \\theta _ { m } ) | ~ 0$ . ", + "bbox": [ + 171, + 190, + 825, + 220 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Remark A.6. For exponential loss, we have already shown in Section 4.2 that $\\tilde { \\gamma } ( \\pmb \\theta )$ is an $O ( \\rho ^ { - L } )$ - additive approximation for $\\bar { \\gamma } ( \\pmb \\theta )$ . For logistic loss, it follows easily from $g ^ { \\prime } ( q ) = \\Theta ( 1 )$ and $( b )$ of Lemma A.5 that $\\tilde { \\gamma } ( \\pmb \\theta )$ is an $O ( \\rho ^ { - L } )$ -additive approximation for $\\bar { \\gamma } ( \\pmb \\theta )$ if $\\mathcal { L }$ is sufficiently small. ", + "bbox": [ + 173, + 227, + 826, + 271 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.3 THEOREMS ", + "text_level": 1, + "bbox": [ + 174, + 286, + 299, + 300 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Now we state our main theorems. For the monotonicity of the normalized margin, we have the following theorem. The proof is provided in Appendix B. ", + "bbox": [ + 173, + 311, + 823, + 340 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Theorem A.7. Under assumptions (A1), (A2), $( B 3 ) ^ { 4 }$ , $( B 4 )$ , the following statements are true for gradient flow: ", + "bbox": [ + 173, + 343, + 821, + 372 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "1. For a.e. $t > t _ { 0 }$ , $\\begin{array} { r } { \\frac { d } { d t } \\tilde { \\gamma } ( \\pmb { \\theta } ( t ) ) \\geq 0 } \\end{array}$ ; \n2. For a.e. t > t0, either ddt γ˜(θ(t)) > 0 or ddt θ(t)kθ(t)k2 ; \n3. $\\mathcal { L } ( \\pmb \\theta ( t ) ) 0$ and $\\| \\pmb \\theta ( t ) \\| _ { 2 } \\to \\infty$ as $t \\to + \\infty$ ; therefore, $| \\bar { \\gamma } ( \\pmb \\theta ( t ) ) - \\tilde { \\gamma } ( \\pmb \\theta ( t ) ) | 0 .$ . ", + "bbox": [ + 210, + 381, + 774, + 455 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For the normalized margin at convergence, we have two theorems, one for infinite-time limiting case, and the other being a finite-time quantitative result. Their proofs can be found in Appendix $\\textrm { C }$ As in the exponential loss case, we define the constrained optimization problem (P) as follows: ", + "bbox": [ + 176, + 463, + 821, + 506 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/c75aa4259ad04a173fdd66e7e2e38ab68f8bcd8c2839aca4e7d52cbd97febe84.jpg", + "text": "$$\n\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\mathrm { s . t . } } & { q _ { n } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 392, + 508, + 604, + 558 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "First, we show the directional convergence of $\\pmb \\theta ( t )$ to a KKT point of (P). ", + "bbox": [ + 176, + 560, + 653, + 577 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Theorem A.8. Consider gradient flow under assumptions (A1), (A2), $( B 3 )$ , $( B 4 )$ . For every limit point $\\bar { \\pmb { \\theta } }$ of $\\left\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\right\\}$ , $\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }$ is a KKT point of $( P )$ . ", + "bbox": [ + 173, + 578, + 820, + 616 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Second, we show that after finite time, gradient flow can pass through an approximate KKT point. ", + "bbox": [ + 173, + 626, + 813, + 641 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Theorem A.9. Consider gradient flow under assumptions (A1), (A2), $( B 3 )$ , $( B 4 )$ . For any $\\epsilon , \\delta > 0$ , there exists $r : = \\Theta ( \\log \\delta ^ { \\bar { - } 1 } )$ and $\\bar { \\Delta } : = \\Theta ( \\epsilon ^ { - 2 } )$ such that $\\theta / \\dot { q } _ { \\mathrm { m i n } } ( \\theta ) ^ { 1 / L }$ is an $( \\epsilon , \\delta )$ -KKT point at some time $t _ { * }$ satisfying $\\log \\| \\pmb { \\theta } ( t _ { * } ) \\| _ { 2 } \\in ( r , r + \\Delta )$ . ", + "bbox": [ + 174, + 643, + 825, + 688 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For the definitions for KKT points and approximate KKT points, we refer the readers to Appendix C.1 for more details. ", + "bbox": [ + 174, + 695, + 823, + 726 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "With a refined analysis, we can also provide tight rates for loss convergence and weight growth. The proof is given in Appendix D. ", + "bbox": [ + 174, + 731, + 823, + 761 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Theorem A.10. Under assumptions (A1), (A2), $( B 3 )$ , $( B 4 )$ , we have the following tight rates for loss convergence and weight growth: ", + "bbox": [ + 173, + 763, + 823, + 792 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/ef93c471a7a9aa2b5d7129d9b54e552aa5cdb46846903a7a95270900ab83f5cd.jpg", + "text": "$$\n\\mathcal { L } ( \\pmb \\theta ( t ) ) = \\Theta \\left( \\frac { g ( \\log t ) ^ { 2 / L } } { t ( \\log t ) ^ { 2 } } \\right) \\quad a n d \\quad \\| \\pmb \\theta ( t ) \\| _ { 2 } = \\Theta ( g ( \\log t ) ^ { 1 / L } ) .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 795, + 715, + 832 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Applying Theorem A.10 to exponential loss and logistic loss, in which $g ( x ) = \\Theta ( x )$ , we have the following corollary: ", + "bbox": [ + 173, + 842, + 823, + 872 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Corollary A.11. If $\\ell ( \\cdot )$ is the exponential or logistic loss, then, ", + "bbox": [ + 173, + 875, + 589, + 890 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/225daa3015c64476ab96ce5ea5eb64367ea28e66eb24ea655f9955ca65643b41.jpg", + "text": "$$\n\\mathcal { L } ( \\pmb \\theta ( t ) ) = \\Theta \\left( \\frac { 1 } { t ( \\log t ) ^ { 2 - 2 / L } } \\right) \\quad a n d \\quad \\| \\pmb \\theta ( t ) \\| _ { 2 } = \\Theta ( ( \\log t ) ^ { 1 / L } ) .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 893, + 717, + 929 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B MARGIN MONOTONICITY FOR GENERAL LOSS ", + "text_level": 1, + "bbox": [ + 173, + 102, + 599, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this section, we consider gradient flow and prove Theorem A.7. We assume (A1), (A2), (B3), (B4) as mentioned in Appendix A. ", + "bbox": [ + 173, + 132, + 823, + 161 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We follow the notations in Section 5 to define $\\rho : = \\| \\pmb \\theta \\| _ { 2 }$ and $\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\| \\pmb \\theta \\| _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}$ , and sometimes we view the functions of $\\pmb { \\theta }$ as functions of $t$ . ", + "bbox": [ + 176, + 167, + 823, + 202 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.1 PROOF FOR PROPOSITION 1 AND 2 ", + "text_level": 1, + "bbox": [ + 174, + 217, + 459, + 233 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "To prove the first two propositions, we generalize our key lemma (Lemma 5.1) to general loss. ", + "bbox": [ + 166, + 243, + 787, + 258 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Lemma B.1. For $\\tilde { \\gamma }$ defined in Definition A.3, the following holds for all $t > t _ { 0 }$ , ", + "bbox": [ + 176, + 262, + 700, + 277 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/e8a1cd862b6a3eb2c0ce049b244bfce75b01b358d707f0e8d5e497fa60007b6f.jpg", + "text": "$$\n\\frac { d } { d t } \\log \\rho > 0 \\quad a n d \\quad \\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 282, + 692, + 327 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Before proving Lemma B.1, we review two important properties of homogeneous functions. Note that these two properties are usually shown for smooth functions. By considering Clarke’s subdifferential, we can generalize it to locally Lipschitz functions that admit chain rules: ", + "bbox": [ + 173, + 337, + 825, + 381 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Theorem B.2. Let $F : \\mathbb { R } ^ { d } \\mathbb { R }$ be a locally Lipschitz function that admits a chain rule. If $F$ is $k$ -homogeneous, then ", + "bbox": [ + 173, + 382, + 823, + 412 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(a) For all $\\pmb { x } \\in \\mathbb { R } ^ { d }$ and $\\alpha > 0$ , $\\partial ^ { \\circ } F ( \\alpha x ) = \\alpha ^ { k - 1 } \\partial ^ { \\circ } F ( x )$ That is, $\\partial ^ { \\circ } F ( \\alpha x ) = \\{ \\alpha ^ { k - 1 } h : h \\in \\partial ^ { \\circ } F ( x ) \\} .$ . ", + "bbox": [ + 174, + 421, + 599, + 470 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(b) (Euler’s Theorem for Homogeneous Functions). For all $\\pmb { x } \\in \\mathbb { R } ^ { d }$ , ", + "bbox": [ + 171, + 477, + 624, + 493 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "$\\langle { \\pmb x } , \\partial ^ { \\circ } F ( { \\pmb x } ) \\rangle = k \\cdot F ( { \\pmb x } )$ That is, $\\langle { \\pmb x } , { \\pmb h } \\rangle = { \\boldsymbol k } \\cdot { \\boldsymbol F } ( { \\pmb x } )$ for all $\\pmb { h } \\in \\partial ^ { \\circ } F ( \\pmb { x } )$ . ", + "bbox": [ + 197, + 497, + 596, + 535 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. Let $D$ be the set of points $_ { \\textbf { \\em x } }$ such that $F$ is differentiable at $_ { \\textbf { \\em x } }$ . According to the definition of Clarke’s subdifferential, for proving $( a )$ , it is sufficient to show that ", + "bbox": [ + 173, + 549, + 823, + 578 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/96a2d197e1cf0d214a8ef0139b5d3ed622e9eddfaf3582612ee5391132ab48d8.jpg", + "text": "$$\n\\{ \\operatorname* { l i m } _ { k \\to \\infty } \\nabla F ( \\alpha { \\pmb x } _ { k } ) : { \\pmb x } _ { k } \\to { \\pmb x } , \\alpha { \\pmb x } _ { k } \\in D \\} = \\{ \\alpha ^ { k - 1 } \\operatorname* { l i m } _ { k \\to \\infty } \\nabla F ( { \\pmb x } _ { k } ) : { \\pmb x } _ { k } \\to { \\pmb x } , { \\pmb x } _ { k } \\in D \\}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 582, + 767, + 607 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Fix $\\pmb { x } _ { k } \\in D$ . Let $U$ be a neighborhood of $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }$ . By definition of homogeneity, for any $\\pmb { h } \\in \\mathbb { R } ^ { d }$ and any $\\pmb { y } \\in U \\setminus \\{ \\pmb { x } _ { k } \\}$ , ", + "bbox": [ + 173, + 612, + 826, + 642 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/a37216eff39b5cd3893270c9b46d87079cc3b27146ca426e8e90c9864696f658.jpg", + "text": "$$\n\\frac { F ( \\alpha y ) - F ( \\alpha x _ { k } ) - \\left. \\alpha y - \\alpha x _ { k } , \\alpha ^ { k - 1 } h \\right. } { \\| \\alpha y - \\alpha x _ { k } \\| _ { 2 } } = \\alpha ^ { k - 1 } \\cdot \\frac { F ( y ) - F ( { \\pmb x } _ { k } ) - \\left. y - { \\pmb x } _ { k } , h \\right. } { \\| { \\pmb y } - { \\pmb x } _ { k } \\| _ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 215, + 645, + 781, + 684 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Taking limits ${ \\pmb y } { \\pmb x } _ { k }$ on both sides, we know that the LHS converges to 0 iff the RHS converges to 0. Then by definition of differetiability and gradient, $F$ is differentiable at $\\alpha { \\pmb x } _ { k }$ iff it is differentiable at $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }$ , and $\\nabla F ( \\alpha \\mathbf { x } _ { k } ) = \\alpha ^ { k - 1 } h$ iff $\\nabla \\tilde { F } ( \\mathbf { \\boldsymbol { x } } _ { k } ) = h$ . This proves (10). ", + "bbox": [ + 173, + 686, + 825, + 729 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "To prove $( b )$ , we fix $\\textbf { \\em x } \\in \\mathbb { R } ^ { d }$ . Let $z \\ : \\ [ 0 , + \\infty ) \\ \\to \\ \\mathbb { R } ^ { d } , \\alpha \\ \\mapsto \\ \\alpha x$ be an arc. By definition of homogeneity, $( F \\circ z ) ( \\alpha ) = \\alpha ^ { k } F ( \\pmb { x } )$ for $\\alpha > 0$ . Taking derivative with respect to $\\alpha$ on both sides (for differentiable points), we have ", + "bbox": [ + 173, + 734, + 825, + 779 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/45b9ba5648765fdca135ac61a4e11c907a17a12adf0d20570bc92677dfa0299c.jpg", + "text": "$$\n\\forall \\pmb { h } \\in \\partial ^ { \\circ } F ( \\alpha \\pmb { x } ) : \\langle \\pmb { x } , \\pmb { h } \\rangle = k \\alpha ^ { k - 1 } F ( \\pmb { x } )\n$$", + "text_format": "latex", + "bbox": [ + 364, + 782, + 632, + 801 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "holds for a.e. $\\alpha > 0$ . Pick an arbitrary $\\alpha > 0$ making (11) hold. Then by $( a )$ , (11) is equivalent to $\\forall { \\pmb { h } } \\in \\partial ^ { \\circ } F ( { \\pmb x } ) : \\langle { \\pmb x } , \\alpha ^ { k - 1 } { \\pmb h } \\rangle = k \\alpha ^ { k - 1 } F ( { \\pmb x } )$ , which proves $( b )$ . □ ", + "bbox": [ + 173, + 806, + 823, + 835 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Applying Theorem B.2 to homogeneous neural networks, we have the following corollary: ", + "bbox": [ + 169, + 849, + 767, + 864 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Corollary B.3. Under the assumptions (A1) and (A2), for any $\\pmb \\theta \\in \\mathbb { R } ^ { d }$ and $\\pmb { x } \\in \\mathbb { R } ^ { d _ { \\mathbf { x } } }$ , ", + "bbox": [ + 176, + 867, + 730, + 883 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/1b68bb55bdd66175a9f968c7093368496ce7d564841a625caa7b53c16b5eb5d8.jpg", + "text": "$$\n\\langle \\pmb { \\theta } , \\partial ^ { \\circ } \\Phi _ { \\pmb { x } } ( \\pmb { \\theta } ) \\rangle = L \\cdot \\Phi _ { \\pmb { x } } ( \\pmb { \\theta } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 405, + 887, + 589, + 905 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $\\Phi _ { x } ( \\pmb { \\theta } ) = \\Phi ( \\pmb { \\theta } ; \\pmb { x } )$ is the network output for a fixed input $_ { \\textbf { \\em x } }$ . ", + "bbox": [ + 173, + 909, + 604, + 925 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Corollary B.3 can be used to derive an exact formula for the weight growth during training. ", + "bbox": [ + 169, + 102, + 771, + 119 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Theorem B.4. For a.e. $t \\geq 0$ , ", + "bbox": [ + 174, + 122, + 372, + 137 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/ac3c6044b8852109467d58ccdc24c96868de854235173601b5edb1f564158ec7.jpg", + "text": "$$\n\\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 141, + 609, + 185 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. The proof idea is to use Corollary B.3 and chain rules (See Appendix I for chain rules in Clarke’s sense). Applying the chain rule on $t \\mapsto \\rho ^ { 2 } = \\| \\pmb \\theta \\| _ { 2 } ^ { 2 }$ yields $\\begin{array} { r } { \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } \\ = \\ - \\langle \\theta , h \\rangle } \\end{array}$ for all $h \\in \\partial ^ { \\circ } { \\mathcal { L } }$ and a.e. $t > 0$ . Then applying the chain rule on $\\pmb \\theta \\mapsto \\mathcal L$ , we have ", + "bbox": [ + 173, + 196, + 826, + 244 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/9109f516043faf884f5cb79eabea68297191ae253402f05e16f9eaa28c025c6d.jpg", + "text": "$$\n- \\partial ^ { \\circ } { \\mathcal { L } } \\subseteq \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) \\partial ^ { \\circ } q _ { n } = \\left\\{ \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) { h _ { n } } : { h _ { n } } \\in \\partial ^ { \\circ } q _ { n } \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 245, + 250, + 750, + 294 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "By Corollary B.3, $\\langle \\pmb \\theta , \\pmb h _ { n } \\rangle = L q _ { n }$ , and thus $\\begin{array} { r } { \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } } \\end{array}$ . ", + "bbox": [ + 173, + 299, + 691, + 320 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For convenience, we define $\\begin{array} { r } { \\nu ( t ) : = \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } } \\end{array}$ for all $t \\geq 0$ . Then Theorem B.4 can be rephrased as $\\begin{array} { r } { \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\nu ( t ) } \\end{array}$ for a.e. $t \\geq 0$ . ", + "bbox": [ + 176, + 333, + 825, + 371 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma B.5. For all $t > t _ { 0 }$ ", + "bbox": [ + 173, + 375, + 361, + 388 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/3e0297bdd5cee400141a2f7def5edba66393efe1503e7488e9133cb6347f0a36.jpg", + "text": "$$\n\\nu ( t ) \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 385, + 566, + 424 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. By Lemma A.5, $q _ { n } \\geq g ( \\log { \\frac { 1 } { \\mathcal { L } } } )$ for all $n \\in [ N ]$ . Then by Assumption (B3), ", + "bbox": [ + 173, + 436, + 715, + 455 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/ecaf2c6b9571079c80d2249dbf4350e91c9ad62abf14b593e8e7b88e63685840.jpg", + "text": "$$\nf ^ { \\prime } ( q _ { n } ) q _ { n } \\geq f ^ { \\prime } ( g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ) \\cdot g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) = { \\frac { g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } { g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 330, + 460, + 666, + 500 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Combining this with the definitions of $\\nu ( t )$ and $\\mathcal { L }$ gives ", + "bbox": [ + 173, + 503, + 539, + 518 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/21914080f13a4faa6dd5c62496274d03818eccbde3f4225e09449f6f978b16f0.jpg", + "text": "$$\n\\begin{array} { r } { \\nu ( t ) = \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } \\geq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } \\displaystyle \\frac { g ( \\log \\frac { 1 } { \\mathscr { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathscr { L } } ) } = \\displaystyle \\frac { g ( \\log \\frac { 1 } { \\mathscr { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathscr { L } } ) } \\mathscr { L } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 525, + 728, + 568 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof for Lemma B.1. Note that $\\begin{array} { r } { \\frac { d } { d t } \\log \\rho = \\frac { 1 } { 2 \\rho ^ { 2 } } \\frac { d \\rho ^ { 2 } } { d t } = L \\frac { \\nu ( t ) } { \\rho ^ { 2 } } } \\end{array}$ by Theorem B.4. Then it simply follows from Lemma B.5 that $\\begin{array} { r } { \\frac { d } { d t } \\log \\rho > 0 } \\end{array}$ for a.e. $t > t _ { 0 }$ . For the second inequality, we first prove that $\\begin{array} { r } { \\log \\tilde { \\gamma } = \\log \\left( \\ell ^ { - 1 } ( \\mathcal { L } ) / \\rho ^ { L } \\right) = \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\rho ^ { L } \\right) } \\end{array}$ exists for all $t \\geq t _ { 0 }$ . $\\mathcal { L } ( t )$ is non-increasing with $t$ . So $\\mathcal { L } ( t ) < e ^ { - f ( b _ { f } ) }$ for all $t \\geq t _ { 0 }$ . This implies that (1) $\\log { \\frac { 1 } { \\mathcal { L } } }$ is always in the domain of $g$ ; (2) $\\rho > 0$ (otherwise $\\mathcal { L } \\geq N e ^ { - f ( 0 ) } > e ^ { - f ( b _ { f } ) }$ , contradicting (B4)). Therefore, $\\begin{array} { r } { \\widetilde { \\gamma } : = g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\rho ^ { L } } \\end{array}$ exists and is always positive for all $t \\geq t _ { 0 }$ , which proves the existence of $\\log { \\tilde { \\gamma } }$ . ", + "bbox": [ + 173, + 599, + 826, + 702 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "By the chain rule and Lemma B.5, we have ", + "bbox": [ + 173, + 707, + 459, + 722 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/63b24edc3f36a85ec0a4c3e33860a27487032315f67b9dc1305ab9c184118bf3.jpg", + "text": "$$\n\\begin{array} { r l r } & { \\frac { d } { d t } \\log \\tilde { \\gamma } = \\frac { d } { d t } \\left( \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\right) - L \\log \\rho \\right) = \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\cdot \\frac { 1 } { \\mathcal { L } } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - L ^ { 2 } \\cdot \\frac { \\nu ( t ) } { \\rho ^ { 2 } } } & \\\\ & { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - L ^ { 2 } \\cdot \\frac { \\nu ( t ) } { \\rho ^ { 2 } } } & \\\\ & { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } - \\frac { L ^ { 2 } \\nu ( t ) ^ { 2 } } { \\rho ^ { 2 } } \\right) . } & \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 727, + 784, + 838 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "On the one hand, by Theorem B.4. $\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } = \\left. \\frac { d \\pmb { \\theta } } { d t } \\right. _ { 2 } ^ { 2 } } \\\\ { \\infty } \\end{array}$ for a.e. se toget $t > 0$ by Lemma I.3; on the other hand, lds $\\begin{array} { r } { L \\nu ( t ) = \\left. \\theta , \\frac { d \\theta } { d t } \\right. } \\end{array}$ ", + "bbox": [ + 174, + 842, + 823, + 875 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/f88bde66b96035a463935ca77b3e84afdbaba801adc29f134a065671b81f43ec.jpg", + "text": "$$\n{ \\frac { d } { d t } } \\log { \\tilde { \\gamma } } \\geq { \\frac { 1 } { \\nu ( t ) } } \\left( \\left\\| { \\frac { d \\theta } { d t } } \\right\\| _ { 2 } ^ { 2 } - \\left. { \\hat { \\theta } } , { \\frac { d \\theta } { d t } } \\right. ^ { 2 } \\right) = { \\frac { 1 } { \\nu ( t ) } } \\left\\| ( I - { \\hat { \\theta } } { \\hat { \\theta } } ^ { \\top } ) { \\frac { d \\theta } { d t } } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 880, + 730, + 922 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "By the chain rule, $\\begin{array} { r } { \\frac { d \\hat { \\pmb { \\theta } } } { d t } = \\frac { 1 } { \\rho } ( \\pmb { I } - \\hat { \\pmb { \\theta } } \\hat { \\pmb { \\theta } } ^ { \\top } ) \\frac { d \\pmb { \\theta } } { d t } } \\end{array}$ for a.e. $t > 0$ . So we have ", + "bbox": [ + 173, + 99, + 612, + 122 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/ebf30250bc461ae48bbd6db054e790d690872f7ce0600982803024f156831d8d.jpg", + "text": "$$\n\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq \\frac { \\rho ^ { 2 } } { \\nu ( t ) } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } = L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 125, + 673, + 171 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "B.2 PROOF FOR PROPOSITION 3 ", + "text_level": 1, + "bbox": [ + 174, + 203, + 410, + 218 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "To prove the third proposition, we prove the following lemma to show that $\\mathcal { L } 0$ by giving an upper bound for $\\mathcal { L }$ . Since $\\mathcal { L }$ can never be 0 for bounded $\\rho$ , $\\mathcal { L } \\to 0$ directly implies $\\rho + \\infty$ . For showing $| \\bar { \\gamma } - \\tilde { \\gamma } | \\to 0$ , we only need to apply $( c )$ in Lemma A.5, which shows this when $\\mathcal { L } \\to 0$ . ", + "bbox": [ + 174, + 229, + 826, + 272 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Lemma B.6. For all $t > t _ { 0 }$ ", + "bbox": [ + 173, + 275, + 361, + 289 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/28b25dce057457ac09c7824cc0ec94de27dcfe36225c937c7724662c01a68540.jpg", + "text": "$$\nG ( 1 / \\mathcal { L } ( t ) ) \\geq L ^ { 2 } \\widetilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } ( t - t _ { 0 } ) \\qquad f o r G ( x ) : = \\int _ { 1 / \\mathcal { L } ( t _ { 0 } ) } ^ { x } \\frac { g ^ { \\prime } ( \\log u ) ^ { 2 } } { g ( \\log u ) ^ { 2 - 2 / L } } d u .\n$$", + "text_format": "latex", + "bbox": [ + 240, + 290, + 758, + 328 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Therefore, $\\mathcal { L } ( t ) 0$ and $\\rho ( t ) \\to + \\infty$ as $t \\to \\infty$ . ", + "bbox": [ + 174, + 330, + 500, + 347 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof for Lemma B.6. By Lemma I.3 and Theorem B.4, ", + "bbox": [ + 173, + 359, + 544, + 375 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/49aded740ec06573e6614b08c4628bebfd2d5809f147ca242767773616a537af.jpg", + "text": "$$\n- { \\frac { d { \\mathcal { L } } } { d t } } = \\left\\| { \\frac { d \\pmb { \\theta } } { d t } } \\right\\| _ { 2 } ^ { 2 } \\geq \\left. { \\hat { \\pmb { \\theta } } } , { \\frac { d \\pmb { \\theta } } { d t } } \\right. ^ { 2 } = L ^ { 2 } \\cdot { \\frac { \\nu ( t ) ^ { 2 } } { \\rho ^ { 2 } } }\n$$", + "text_format": "latex", + "bbox": [ + 349, + 378, + 645, + 416 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Using Lemma B.5 to lower bound $\\nu$ and replacing $\\rho$ with $\\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\tilde { \\gamma } \\right) ^ { 1 / L }$ by the definition of $\\tilde { \\gamma }$ we have ", + "bbox": [ + 173, + 421, + 823, + 452 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/86ca346aa77772599b25aa77d54282aa3bf8cea82f6e930d3a4bc68b7efff397.jpg", + "text": "$$\n- \\frac { d \\mathcal { L } } { d t } \\geq L ^ { 2 } \\cdot \\bigg ( \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } \\bigg ) ^ { 2 } \\cdot \\bigg ( \\frac { \\widetilde { \\gamma } ( t ) } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\bigg ) ^ { 2 / L } \\geq L ^ { 2 } \\widetilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\cdot \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 - 2 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\mathcal { L } ,\n$$", + "text_format": "latex", + "bbox": [ + 222, + 452, + 772, + 493 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where the last inequality uses the monotonicity of $\\tilde { \\gamma }$ . So the following holds for a.e. $t \\geq t _ { 0 }$ ", + "bbox": [ + 173, + 494, + 769, + 510 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/aef9ca675b70538a2a335a8f196ae2fcc98b67ee53e099398a97e520273bbca5.jpg", + "text": "$$\n\\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 - 2 / L } } \\cdot \\frac { d } { d t } \\frac { 1 } { \\mathcal { L } } \\ge L ^ { 2 } \\tilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 512, + 625, + 550 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Integrating on both sides from $t _ { 0 }$ to $t$ , we can conclude that ", + "bbox": [ + 173, + 553, + 565, + 568 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/d72e2b18dad626c7ea80f7b77b588e5ee43c630c2afaca93c9909af530bf576a.jpg", + "text": "$$\nG ( 1 / \\mathcal { L } ) \\geq L ^ { 2 } \\tilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } ( t - t _ { 0 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 393, + 569, + 604, + 589 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Note that $1 / \\mathcal { L }$ is non-decreasing. If $1 / \\mathcal { L }$ does not grow to $+ \\infty$ , then neither does $G ( 1 / \\mathcal { L } )$ . But the RHS grows to $+ \\infty$ , which leads to a contradiction. So $\\mathcal { L } \\to 0$ . ", + "bbox": [ + 171, + 592, + 825, + 621 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "To make $\\mathcal { L } \\to 0$ , $q _ { \\mathrm { m i n } }$ must converge to $+ \\infty$ . So $\\rho \\to + \\infty$ . ", + "bbox": [ + 173, + 627, + 565, + 642 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C CONVERGENCE TO THE MAX-MARGIN SOLUTION ", + "text_level": 1, + "bbox": [ + 174, + 661, + 622, + 679 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In this section, we analyze the convergent direction of $\\pmb { \\theta }$ and prove Theorem A.8 and A.9, assuming (A1), (A2), (B3), (B4) as mentioned in Section A. ", + "bbox": [ + 171, + 691, + 825, + 720 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We follow the notations in Section 5 to define $\\rho : = \\| \\pmb \\theta \\| _ { 2 }$ and $\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\| \\pmb \\theta \\| _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}$ , and sometimes we view the functions of $\\pmb { \\theta }$ as functions of $t$ . ", + "bbox": [ + 176, + 727, + 823, + 761 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C.1 PRELIMINARIES FOR KKT CONDITIONS ", + "text_level": 1, + "bbox": [ + 173, + 776, + 496, + 791 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We first review the definition of Karush-Kuhn-Tucker (KKT) conditions for non-smooth optimization problems following from (Dutta et al., 2013). ", + "bbox": [ + 174, + 801, + 823, + 832 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Consider the following optimization problem (P) for $\\pmb { x } \\in \\mathbb { R } ^ { d }$ : ", + "bbox": [ + 173, + 837, + 578, + 853 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/35be983d7db429f620f565512bad71d74b7d66612203f761ae8e1879acade459.jpg", + "text": "$$\n\\begin{array} { r l } { \\operatorname* { m i n } } & { { } f ( { \\pmb x } ) } \\\\ { \\mathrm { s . t . } } & { { } g _ { n } ( { \\pmb x } ) \\le 0 \\qquad \\forall n \\in [ N ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 392, + 856, + 604, + 891 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $f , g _ { 1 } , \\ldots , g _ { n } : \\mathbb { R } ^ { d } \\to \\mathbb { R }$ are locally Lipschitz functions. We say that $\\pmb { x } \\in \\mathbb { R } ^ { d }$ is a feasible point of (P) if $_ { \\textbf { \\em x } }$ satisfies $g _ { n } ( { \\pmb x } ) \\le 0$ for all $n \\in [ N ]$ . ", + "bbox": [ + 174, + 895, + 820, + 925 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Definition C.1 (KKT Point). A feasible point $_ { \\textbf { \\em x } }$ of (P) is a KKT point if $_ { \\textbf { \\em x } }$ satisfies KKT conditions: there exists $\\lambda _ { 1 } , \\ldots , \\lambda _ { N } \\geq 0$ such that ", + "bbox": [ + 168, + 103, + 823, + 132 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/74aedca2a3f5a1b8a3694e9f58b86a2e1cbae12024ea5703aa8b45f248601048.jpg", + "text": "$$\n2 . \\ \\forall n \\in [ N ] : \\lambda _ { n } g _ { n } ( { \\pmb x } ) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 215, + 169, + 400, + 185 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "It is important to note that a global minimum of (P) may not be a KKT point, but under some regularity assumptions, the KKT conditions become a necessary condition for global optimality. The regularity condition we shall use in this paper is the non-smooth version of Mangasarian-Fromovitz Constraint Qualification (MFCQ) (see, e.g., the constraint qualification (C.Q.5) in (Giorgi et al., 2004)): ", + "bbox": [ + 173, + 195, + 825, + 266 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Definition C.2 (MFCQ). For a feasible point $_ { \\textbf { \\em x } }$ of (P), (P) is said to satisfy MFCQ at $_ { \\textbf { \\em x } }$ if there exists $\\pmb { v } \\in \\mathbb { R } ^ { d }$ such that for all $n \\in [ N ]$ with $g _ { n } ( { \\pmb x } ) = 0$ , ", + "bbox": [ + 173, + 268, + 823, + 299 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/deaf377e4b6acae33659aab9cfeaa4d726fedbd90548f9a387f6e3833667c928.jpg", + "text": "$$\n\\forall h \\in \\partial ^ { \\circ } g _ { n } ( { \\pmb x } ) : \\langle h , { \\pmb v } \\rangle > 0 .\n$$", + "text_format": "latex", + "bbox": [ + 403, + 304, + 593, + 320 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Following from (Dutta et al., 2013), we define an approximate version of KKT point, as shown below. Note that this definition is essentially the modified $\\epsilon$ -KKT point defined in their paper, but these two definitions differ in the following two ways: (1) First, in their paper, the subdifferential is allowed to be evaluated in a neighborhood of $_ { \\textbf { \\em x } }$ , so our definition is slightly stronger; (2) Second, their paper fixes $\\delta = \\epsilon ^ { 2 }$ , but in our definition we make them independent. ", + "bbox": [ + 173, + 333, + 825, + 404 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Definition C.3 (Approximate KKT Point). For $\\epsilon , \\delta > 0$ , a feasible point $_ { \\textbf { \\em x } }$ of (P) is an $( \\epsilon , \\delta )$ -KKT point if there exists $\\lambda _ { n } \\geq 0 , \\pmb { k } \\in \\partial ^ { \\circ } f ( \\pmb { x } ) , \\pmb { h } _ { n } \\in \\partial ^ { \\circ } g _ { n } ( \\pmb { x } )$ for all $n \\in [ N ]$ such that ", + "bbox": [ + 171, + 406, + 820, + 438 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/51e96b6f10fffba71794ea436b968fb2994e9a0dba5a53fc1931e5073c9efe1e.jpg", + "text": "$$\n\\begin{array} { r } { 1 . \\ \\left\\| k + \\sum _ { n \\in [ N ] } \\lambda _ { n } \\pmb { h } _ { n } ( \\pmb { x } ) \\right\\| _ { 2 } \\leq \\epsilon ; } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 445, + 441, + 474 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/7ea9732df11d18814c1b4d7f5de20ba948f8a9480b5d152a38bf9fbec31459b5.jpg", + "text": "$$\n2 . \\forall n \\in [ N ] : \\lambda _ { n } g _ { n } ( { \\pmb x } ) \\geq - \\delta .\n$$", + "text_format": "latex", + "bbox": [ + 210, + 479, + 415, + 497 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "As shown in (Dutta et al., 2013), $( \\epsilon , \\delta )$ -KKT point is an approximate version of KKT point in the sense that a series of $( \\epsilon , \\delta )$ -KKT points can converge to a KKT point. We restate their theorem in our setting: ", + "bbox": [ + 174, + 506, + 825, + 550 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Theorem C.4 (Corollary of Theorem 3.6 in (Dutta et al., 2013)). Let $\\{ \\pmb { x } _ { k } \\in \\mathbb { R } ^ { d } : k \\in \\mathbb { N } \\}$ be a sequence of feasible points of $( P )$ , $\\{ \\epsilon _ { k } > 0 : k \\in \\mathbb { N } \\}$ and $\\{ \\delta _ { k } > 0 : k \\in \\tilde { \\mathbb { N } } \\}$ be two sequences. $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }$ is an $( \\epsilon _ { k } , \\delta _ { k } )$ -KKT point for every $k$ , and $\\epsilon _ { k } 0 , \\delta _ { k } 0$ . If $\\mathbfit { x } _ { k } \\to \\mathbfit { x }$ as $k \\to + \\infty$ and MFCQ holds at $_ { \\textbf { \\em x } }$ , then $_ { \\textbf { \\em x } }$ is a KKT point of $( P )$ . ", + "bbox": [ + 173, + 553, + 825, + 611 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "C.2 KKT CONDITIONS FOR (P) ", + "text_level": 1, + "bbox": [ + 174, + 626, + 406, + 641 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Recall that for a homogeneous neural network, the optimization problem (P) is defined as follows: ", + "bbox": [ + 171, + 651, + 815, + 667 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/6fb4f7835d0f9d076ad7003b4eb71d439d4eaceac823d4a752c73979f1032d90.jpg", + "text": "$$\n\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\mathrm { s . t . } } & { q _ { n } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 392, + 671, + 604, + 722 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Using the terminologies and notations in Appendix C.1, the objective and constraints are $f ( { \\pmb x } ) =$ $\\begin{array} { r l } { { { \\frac { 1 } { 2 } } \\| \\pmb { x } \\| _ { 2 } ^ { 2 } } } \\end{array}$ and $g _ { n } ( \\pmb { x } ) = 1 - q _ { n } ( \\pmb { x } )$ . The KKT points and approximate KKT points for (P) are defined as follows: ", + "bbox": [ + 174, + 732, + 826, + 775 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Definition C.5 (KKT Point of (P)). A feasible point $\\pmb \\theta$ of (P) is a KKT point if there exist $\\lambda _ { 1 } , \\dots , \\lambda _ { N } \\ge 0$ such that ", + "bbox": [ + 169, + 777, + 823, + 808 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "1. $\\begin{array} { r } { \\pmb { \\theta } - \\sum _ { n = 1 } ^ { N } \\lambda _ { n } \\pmb { h } _ { n } = \\mathbf { 0 } } \\end{array}$ for some $h _ { 1 } , \\ldots , h _ { N }$ satisfying $h _ { n } \\in \\partial ^ { \\circ } q _ { n } ( \\theta )$ \n2. $\\forall n \\in [ N ] : \\lambda _ { n } ( q _ { n } ( \\pmb \\theta ) - 1 ) = 0 .$ ", + "bbox": [ + 210, + 818, + 696, + 859 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Definition C.6 (Approximate KKT Point of (P)). A feasible point $\\pmb { \\theta }$ of $( \\mathrm { P } )$ is an $( \\epsilon , \\delta )$ -KKT point of (P) if there exists $\\lambda _ { 1 } , \\ldots , \\lambda _ { N } \\geq 0$ such that ", + "bbox": [ + 171, + 862, + 825, + 892 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/1066bfc89940fed928fe523afbc46b192c39c4911be75af62e8362785f9119b1.jpg", + "text": "$$\n2 . \\forall n \\in [ N ] : \\lambda _ { n } ( q _ { n } ( \\pmb \\theta ) - 1 ) \\leq \\delta .\n$$", + "text_format": "latex", + "bbox": [ + 212, + 102, + 442, + 119 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "By the homogeneity of $q _ { n }$ , it is easy to see that (P) satisfies MFCQ, and thus KKT conditions are first-order necessary condition for global optimality. ", + "bbox": [ + 171, + 128, + 823, + 160 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Lemma C.7. $( P )$ satisfies MFCQ at every feasible point $\\pmb { \\theta }$ . ", + "bbox": [ + 173, + 162, + 562, + 178 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. Take $\\boldsymbol { v } : = \\boldsymbol { \\theta }$ . For all $n \\in [ N ]$ satisfying $q _ { n } = 1$ , by homogeneity of $q _ { n }$ , ", + "bbox": [ + 174, + 191, + 689, + 208 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/6149a08561a4ab5601c13b2a5e89ab042772d7c69ba9de6408f12dea2b7fccd8.jpg", + "text": "$$\n\\langle v , h \\rangle = L q _ { n } ( \\pmb { \\theta } ) = L > 0\n$$", + "text_format": "latex", + "bbox": [ + 408, + 214, + 588, + 231 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "holds for any $\\pmb { h } \\in - \\partial ^ { \\circ } q _ { n } ( \\pmb { \\theta } ) = \\partial ^ { \\circ } \\big ( 1 - q _ { n } ( \\pmb { \\theta } ) \\big )$ . ", + "bbox": [ + 173, + 237, + 486, + 253 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C.3 KEY LEMMAS ", + "text_level": 1, + "bbox": [ + 174, + 268, + 316, + 284 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Define β(t) := $\\begin{array} { r } { \\beta ( t ) : = \\frac { 1 } { \\left\\| \\frac { d \\theta } { d t } \\right\\| _ { 2 } } \\left. \\hat { \\pmb { \\theta } } , \\frac { d \\pmb { \\theta } } { d t } \\right. } \\end{array}$ to be the cosine of the angle between $\\pmb \\theta$ and $\\textstyle { \\frac { d \\theta } { d t } }$ . Here $\\beta ( t )$ is only defined for a.e. $t > 0$ . Since $q _ { n }$ is locally Lipschitz, it can be shown that $q _ { n }$ is (globally) Lipschitz on the compact set $S ^ { d - 1 }$ , which is the unit sphere in $\\mathbb { R } ^ { d }$ . Define ", + "bbox": [ + 174, + 292, + 826, + 347 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/ea5c88a2428113a373c5ff93faae948c1cec85843257137e416126bfacc8b677.jpg", + "text": "$$\n\\begin{array} { r l } & { B _ { 0 } : = \\operatorname* { s u p } \\{ \\cfrac { q _ { n } } { \\rho ^ { L } } : \\theta \\in \\mathbb { R } ^ { d } \\setminus \\{ \\mathbf { 0 } \\} \\} } \\\\ & { \\quad = \\operatorname* { s u p } \\{ q _ { n } : \\theta \\in \\mathcal { S } ^ { d - 1 } \\} < \\infty . } \\\\ & { B _ { 1 } : = \\operatorname* { s u p } \\{ \\cfrac { \\| h \\| _ { 2 } } { \\rho ^ { L - 1 } } : \\theta \\in \\mathbb { R } ^ { d } \\setminus \\{ \\mathbf { 0 } \\} , h \\in \\partial ^ { \\circ } q _ { n } , n \\in [ N ] \\} } \\\\ & { \\quad = \\operatorname* { s u p } \\{ \\| h \\| _ { 2 } : \\theta \\in \\mathcal { S } ^ { d - 1 } , \\boldsymbol { h } \\in \\partial ^ { \\circ } q _ { n } , n \\in [ N ] \\} < \\infty . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 307, + 353, + 689, + 467 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "For showing Theorem A.8 and Theorem A.9, we first prove Lemma C.8. In light of this lemma, if we aim to show that $\\pmb \\theta$ is along the direction of an approximate KKT point, we only need to show $\\beta 1$ (which makes $\\epsilon \\to 0$ ) and $\\mathcal { L } \\to 0$ (which makes $\\delta 0$ ). ", + "bbox": [ + 173, + 477, + 828, + 520 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Lemma C.8. Let $C _ { 1 } , C _ { 2 }$ be two constants defined as ", + "bbox": [ + 174, + 523, + 524, + 539 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/1d690195c3487763fde4df2134558983f5a2e2d36a8d53fccef60d9395966b76.jpg", + "text": "$$\nC _ { 1 } : = \\frac { \\sqrt { 2 } } { \\tilde { \\gamma } ( t _ { 0 } ) ^ { 1 / L } } , \\qquad C _ { 2 } : = \\frac { 2 e N K ^ { 2 } } { L \\tilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } } \\left( \\frac { B _ { 1 } } { \\tilde { \\gamma } ( t _ { 0 } ) } \\right) ^ { \\log _ { 2 } K } ,\n$$", + "text_format": "latex", + "bbox": [ + 308, + 545, + 687, + 583 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where $K , b _ { g }$ are constants specified in (B3.4). $I f \\log { \\frac { 1 } { \\mathcal { L } } } \\geq b _ { g }$ at time $t > t _ { 0 }$ , then $\\tilde { \\pmb { \\theta } } : = \\pmb { \\theta } / q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { 1 / L }$ is an $( \\epsilon , \\delta )$ -KKT point of $( P )$ , where $\\epsilon : = C _ { 1 } \\sqrt { 1 - \\beta }$ and $\\delta : = C _ { 2 } / ( \\log \\frac { 1 } { \\mathcal { L } } )$ . ", + "bbox": [ + 173, + 590, + 821, + 625 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. Let $\\begin{array} { r } \\begin{array} { r } { \\begin{array} { r } { \\boldsymbol { h } ( t ) : = \\frac { d \\theta } { d t } ( t ) } \\end{array} } \\end{array} \\end{array}$ for a.e. $t > 0$ . By the chain rule, there exist $\\pmb { h } _ { 1 } , \\dots , \\pmb { h } _ { N }$ such that $h _ { n } \\in \\partial ^ { \\circ } q _ { n }$ and $\\begin{array} { r } { \\pmb { h } = \\sum _ { \\pmb { n } \\in [ N ] } e ^ { - \\pmb { f } ( \\ v { q } _ { n } ) } f ^ { \\prime } ( \\ v { q } _ { n } ) \\pmb { h } _ { n } } \\end{array}$ . Let $\\tilde { \\pmb { h } } _ { n } : = \\pmb { h } _ { n } / q _ { \\mathrm { m i n } } ^ { 1 - 1 / L } \\in \\partial ^ { \\circ } q _ { n } ( \\tilde { \\pmb { \\theta } } )$ q1−1/Lmin ∈ ∂◦qn(θ˜) (Recall that θ˜ := θ/qmin(θ)1/L). Construct λn := q1−2/min $\\lambda _ { n } : = q _ { \\mathrm { m i n } } ^ { 1 - 2 / L } \\rho \\cdot e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) / \\left\\| h \\right\\| _ { 2 }$ . Now we only need to show ", + "bbox": [ + 173, + 637, + 825, + 696 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/419bad4bb3a0475390aabacd1d37e592e7f903894afed489d1a7f03ffe6932ec.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left\\| \\tilde { \\theta } - \\displaystyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } \\tilde { h } _ { n } \\right\\| _ { 2 } ^ { 2 } \\leq \\frac { 2 } { \\tilde { \\gamma } ^ { 2 / L } } ( 1 - \\beta ) } \\\\ & { \\displaystyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) \\leq \\left( \\frac { 2 e N K ^ { 2 } } { L \\tilde { \\gamma } ^ { 2 / L } } \\left( \\frac { B _ { 1 } } { \\tilde { \\gamma } } \\right) ^ { \\log _ { 2 } K } \\right) \\frac { 1 } { f ( \\tilde { \\gamma } \\rho ^ { L } ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 702, + 692, + 794 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Then $\\tilde { \\theta }$ can be shown to be an $( \\epsilon , \\delta )$ -KKT point by the monotonicity $\\tilde { \\gamma } ( t ) \\geq \\tilde { \\gamma } ( t _ { 0 } )$ for $t > t _ { 0 }$ ", + "bbox": [ + 171, + 800, + 782, + 818 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof of (12). From our construction, $\\begin{array} { r } { \\sum _ { n = 1 } ^ { N } \\lambda _ { n } \\tilde { \\pmb { h } } _ { n } = q _ { \\mathrm { m i n } } ^ { - 1 / L } \\rho \\pmb { h } / \\left\\| \\pmb { h } \\right\\| _ { 2 } } \\end{array}$ . So ", + "bbox": [ + 173, + 830, + 678, + 852 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/19035e35b79d57b0e90e8ef3adc8c5c9923b6bcf01a857920e8c1dec3f761136.jpg", + "text": "$$\n\\left\\| \\tilde { \\theta } - \\sum _ { n = 1 } ^ { N } \\lambda _ { n } \\tilde { h } _ { n } \\right\\| _ { 2 } ^ { 2 } = q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } \\left\\| \\hat { \\theta } - \\frac { h } { \\| h \\| _ { 2 } } \\right\\| _ { 2 } ^ { 2 } = q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } ( 2 - 2 \\beta ) \\leq \\frac { 2 } { \\tilde { \\gamma } ^ { 2 / L } } ( 1 - \\beta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 222, + 856, + 772, + 904 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where the last equality is by Lemma A.5. ", + "bbox": [ + 174, + 909, + 444, + 924 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof for (13). According to our construction, ", + "bbox": [ + 174, + 103, + 490, + 118 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/a57c568281e51a23a2ff29bba9b5b1f407d0bd6541d8850dfe0f80d7b78f18a2.jpg", + "text": "$$\n\\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) = \\frac { q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho } { \\| h \\| _ { 2 } } \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 123, + 704, + 167 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Note that $\\| h \\| _ { 2 } \\geq \\left. h , \\hat { \\theta } \\right. = L \\nu / \\rho$ . By Lemma B.5 and Lemma D.1, we have ", + "bbox": [ + 171, + 171, + 684, + 195 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/affb1be95e3ec6eca81bc912efabcc91b8769aec9401d359158fed6344f9955d.jpg", + "text": "$$\n\\nu \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } \\geq \\frac { 1 } { 2 K } \\log \\frac { 1 } { \\mathcal { L } } \\cdot \\mathcal { L } \\geq \\frac { 1 } { 2 K } f ( \\widetilde { \\gamma } \\rho ^ { L } ) e ^ { - f ( q _ { \\operatorname* { m i n } } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 202, + 692, + 241 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where the last inequality uses $\\begin{array} { r } { f ( \\tilde { \\gamma } \\rho ^ { L } ) = \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}$ and $\\mathcal { L } \\geq e ^ { - f ( q _ { \\mathrm { m i n } } ) }$ . Combining these gives ", + "bbox": [ + 168, + 244, + 764, + 263 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/15cb0d557a767c7431bba1921d27bb346a1ecdf6ff2f0721ac08ec3a1923f999.jpg", + "text": "$$\n\\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) \\leq \\frac { 2 K q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } } { L f ( \\tilde { \\gamma } \\rho ^ { L } ) } \\sum _ { n = 1 } ^ { N } e ^ { f ( q _ { \\operatorname* { m i n } } ) - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 254, + 268, + 740, + 311 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "If $q _ { n } > q _ { \\mathrm { m i n } }$ , then by the mean value theorem there exists $\\xi _ { n } ~ \\in ~ ( q _ { \\operatorname* { m i n } } , q _ { n } )$ such that $f ( q _ { n } ) \\mathrm { ~ - ~ }$ $f ( q _ { \\mathrm { m i n } } ) = f ^ { \\prime } ( \\xi _ { n } ) ( q _ { n } - q _ { \\mathrm { m i n } } )$ . By Assumption (B3.4), we know that $f ^ { \\prime } ( q _ { n } ) \\leq K ^ { \\lceil \\log _ { 2 } ( q _ { n } / \\xi _ { n } ) \\rceil } f ^ { \\prime } ( \\xi _ { n } )$ . Note that $\\begin{array} { r } { \\lceil \\log _ { 2 } ( q _ { n } / \\xi _ { n } ) \\rceil \\leq \\log _ { 2 } ( 2 B _ { 1 } \\rho ^ { L } / q _ { \\mathrm { m i n } } ) \\leq \\log _ { 2 } ( 2 B _ { 1 } / \\tilde { \\gamma } ) . } \\end{array}$ . Then we have ", + "bbox": [ + 174, + 316, + 823, + 363 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/440dc95ad072a4d575fa597b8aa69c00b5ac0cde52c186d30ecd36b3b4fa4b86.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) \\le \\frac { 2 K q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } } { L f ( \\tilde { \\gamma } \\rho ^ { L } ) } K ^ { \\log _ { 2 } ( 2 B _ { 1 } / \\tilde { \\gamma } ) } \\displaystyle \\sum _ { n \\colon { q _ { n } \\ne q _ { \\operatorname* { m i n } } } } e ^ { - f ^ { \\prime } ( \\xi _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) } f ^ { \\prime } ( \\xi _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) } \\\\ & { \\displaystyle \\le \\frac { 2 K \\tilde { \\gamma } ^ { - 2 / L } \\rho ^ { 2 } } { L f ( \\tilde { \\gamma } \\rho ^ { L } ) } K ^ { \\log _ { 2 } ( 2 B _ { 1 } / \\tilde { \\gamma } ) } \\cdot N e } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 183, + 368, + 810, + 449 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "sec uality uses $q _ { \\mathrm { m i n } } ^ { - 2 / L } \\rho ^ { 2 } \\le \\tilde { \\gamma } ^ { - 2 / L }$ b ma A.5 and the fact that the function $x \\mapsto e ^ { - x } x$ $( 0 , + \\infty )$ attains the maximum value $e$ $x = 1$ ", + "bbox": [ + 173, + 454, + 821, + 487 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "By Theorem A.7, we have already known that $\\mathcal { L } 0$ . So it remains to bound $\\beta ( t )$ . For this, we first prove the following lemma to bound the integral of $\\beta ( t )$ . ", + "bbox": [ + 173, + 501, + 823, + 531 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Lemma C.9. For all $t _ { 2 } > t _ { 1 } \\geq t _ { 0 }$ , ", + "bbox": [ + 173, + 534, + 405, + 549 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/dac78b74f96b4e463429a84c11ed07ee6980b2a764b99211ea13d8234d00f21a.jpg", + "text": "$$\n\\int _ { t _ { 1 } } ^ { t _ { 2 } } \\left( \\beta ( \\tau ) ^ { - 2 } - 1 \\right) \\cdot \\frac { d } { d \\tau } \\log \\rho ( \\tau ) \\cdot d \\tau \\leq \\frac { 1 } { L } \\log \\frac { \\tilde { \\gamma } ( t _ { 2 } ) } { \\tilde { \\gamma } ( t _ { 1 } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 554, + 674, + 589 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proof. By Lemma B.4, $\\begin{array} { r } { \\frac { d } { d t } \\log \\rho = \\frac 1 { 2 \\rho ^ { 2 } } \\frac { d \\rho ^ { 2 } } { d t } = \\frac { L \\nu } { \\rho ^ { 2 } } } \\end{array}$ for a.e. $t > 0$ . By Lemma B.1, for a.e. $t \\in ( t _ { 1 } , t _ { 2 } )$ ", + "bbox": [ + 173, + 604, + 823, + 626 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/6508e6afe1e2ffd15241a4c742d42e547fb849be66ea1cf6e12ad431825d7fdd.jpg", + "text": "$$\n\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\cdot \\left. \\frac { \\rho ^ { 2 } } { L \\nu } \\frac { d \\hat { \\pmb \\theta } } { d t } \\right. _ { 2 } ^ { 2 } \\cdot \\frac { d } { d t } \\log \\rho .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 630, + 620, + 676 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "By the chain rule, $\\begin{array} { r } { \\frac { d \\hat { \\pmb { \\theta } } } { d t } = \\frac { 1 } { \\rho } ( \\pmb { I } - \\hat { \\pmb { \\theta } } \\hat { \\pmb { \\theta } } ^ { \\top } ) \\frac { d \\pmb { \\theta } } { d t } } \\end{array}$ . So we have ", + "bbox": [ + 173, + 681, + 524, + 703 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/210dfa36e4abcc2585e666992efad29ac29c5f3292a0db70948a091544ed1133.jpg", + "text": "$$\n\\left\\| \\frac { \\rho ^ { 2 } } { L \\nu } \\frac { d \\hat { \\theta } } { d t } \\right\\| _ { 2 } ^ { 2 } = \\left\\| \\frac { \\rho } { L \\nu ( t ) } ( I - \\hat { \\theta } \\hat { \\theta } ^ { \\top } ) \\frac { d \\theta } { d t } \\right\\| _ { 2 } ^ { 2 } = \\frac { \\left\\| \\frac { d \\theta } { d t } \\right\\| _ { 2 } ^ { 2 } - \\left. \\frac { d \\theta } { d t } , \\hat { \\theta } \\right. ^ { 2 } } { \\left. \\frac { d \\theta } { d t } , \\hat { \\theta } \\right. ^ { 2 } } = \\beta ^ { - 2 } - 1 .\n$$", + "text_format": "latex", + "bbox": [ + 251, + 708, + 746, + 767 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where the last equality follows from the definition of $\\beta$ . Combining 14 and 15, we have ", + "bbox": [ + 173, + 771, + 746, + 786 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/bfd5cbfa6b4b2fa170dd08b40dc378686678d761b74ae96f5f573362ffca551e.jpg", + "text": "$$\n\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\left( \\beta ^ { - 2 } - 1 \\right) \\cdot \\frac { d } { d t } \\log \\rho .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 791, + 616, + 821 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Integrating on both sides from $t _ { 1 }$ to $t _ { 2 }$ proves the lemma. ", + "bbox": [ + 173, + 825, + 547, + 842 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "A direct corollary of Lemma C.9 is the upper bound for the minimum $\\beta ^ { 2 } - 1$ within a time interval: Corollary C.10. For all $t _ { 2 } > t _ { 1 } \\ge t _ { 0 }$ , then there exists $t _ { * } \\in ( t _ { 1 } , t _ { 2 } )$ such that ", + "bbox": [ + 173, + 854, + 830, + 886 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/8316fe2ff9b5754e266daa69050f0d52f561584e2ad688a3dfc2fd823952a4e6.jpg", + "text": "$$\n\\beta ( t _ { * } ) ^ { - 2 } - 1 \\le \\frac { 1 } { L } \\cdot \\frac { \\log \\tilde { \\gamma } ( t _ { 2 } ) - \\log \\tilde { \\gamma } ( t _ { 1 } ) } { \\log \\rho ( t _ { 2 } ) - \\log \\rho ( t _ { 1 } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 895, + 638, + 929 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proof. Denote the RHS as $C$ . Assume to the contrary that $\\beta ( \\tau ) ^ { - 2 } - 1 > C$ for a.e. $\\tau \\in ( t _ { 1 } , t _ { 2 } )$ . By Lemma B.1, $\\log \\rho ( \\tau ) > 0$ for a.e. $\\tau \\in ( t _ { 1 } , t _ { 2 } )$ . Then by Lemma C.9, we have ", + "bbox": [ + 169, + 102, + 825, + 133 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/70b4604217f9bf52fea8326ade97d9ae6b35ab4a1782b21ccbdc1610c54ae7f2.jpg", + "text": "$$\n\\frac { 1 } { L } \\log \\frac { \\tilde { \\gamma } ( t _ { 2 } ) } { \\tilde { \\gamma } ( t _ { 1 } ) } > \\int _ { t _ { 1 } } ^ { t _ { 2 } } { C \\cdot \\frac { d } { d \\tau } \\log \\rho ( \\tau ) \\cdot d \\tau } = C \\cdot ( \\log \\rho ( t _ { 2 } ) - \\log \\rho ( t _ { 1 } ) ) = \\frac { 1 } { L } \\log \\frac { \\tilde { \\gamma } ( t _ { 2 } ) } { \\tilde { \\gamma } ( t _ { 1 } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 214, + 138, + 784, + 176 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "which leads to a contradiction. ", + "bbox": [ + 174, + 183, + 375, + 198 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In the rest of this section, we present both asymptotic and non-asymptotic analyses for the directional convergence by using Corollary C.10 to bound $\\beta ( t )$ . ", + "bbox": [ + 171, + 214, + 825, + 244 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "C.4 ASYMPTOTIC ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 260, + 387, + 275 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We first prove an auxiliary lemma which gives an upper bound for the change of $\\hat { \\pmb { \\theta } }$ . ", + "bbox": [ + 173, + 285, + 717, + 301 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Lemma C.11. For a.e. $t > t _ { 0 }$ , ", + "bbox": [ + 173, + 305, + 377, + 320 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/800af00bec61077132a4a2cdac85a1e820897f5661056be725fa7507524b6f78.jpg", + "text": "$$\n\\left\\| \\frac { d \\hat { \\pmb \\theta } } { d t } \\right\\| _ { 2 } \\le \\frac { B _ { 1 } } { L \\tilde { \\gamma } } \\cdot \\frac { d } { d t } \\log \\rho .\n$$", + "text_format": "latex", + "bbox": [ + 411, + 318, + 581, + 361 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. Observe that $\\begin{array} { r } { \\left\\| \\frac { d \\hat { \\pmb \\theta } } { d t } \\right\\| _ { 2 } = \\frac { 1 } { \\rho } \\left\\| ( \\pmb I - \\hat { \\pmb \\theta } \\hat { \\pmb \\theta } ^ { \\top } ) \\frac { d \\pmb \\theta } { d t } \\right\\| _ { 2 } \\leq \\frac { 1 } { \\rho } \\left\\| \\frac { d \\pmb \\theta } { d t } \\right\\| _ { 2 } . } \\end{array}$ . It is sufficient to bound $\\left\\| { \\frac { d \\theta } { d t } } \\right\\| _ { 2 }$ . By and the chain rule, there exists $\\begin{array} { r } { \\frac { d \\theta } { d t } = \\sum _ { n \\in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) { h _ { n } ( t ) } } \\end{array}$ $h _ { 1 } , \\ldots , h _ { N } : [ 0 , + \\infty ) \\to \\mathbb { R } ^ { d }$ . By definition of satisfying that for a.e. $B _ { 1 }$ , $\\| h _ { n } \\| _ { 2 } \\leq B _ { 1 } \\rho ^ { L - 1 }$ $t > 0$ for a.e. , $h _ { n } ( t ) \\in \\partial ^ { \\circ } q _ { n }$ $t > 0$ . So ", + "bbox": [ + 171, + 376, + 826, + 448 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/952e58f4d575390690d4caad3e44e31d48b480a262f8961ef774e48e9cfa3e69.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\left\\| \\frac { d \\pmb { \\theta } } { d t } \\right\\| _ { 2 } \\leq \\sum _ { n \\in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) \\left\\| \\pmb { h } _ { n } \\right\\| _ { 2 } } \\\\ { \\displaystyle \\qquad \\leq \\sum _ { n \\in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } \\cdot \\frac { 1 } { q _ { n } } \\cdot B _ { 1 } \\rho ^ { L - 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 453, + 661, + 536 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Note that every summand is positive. By Lemma A.5, $q _ { n }$ is lower-bounded by $q _ { n } \\geq q _ { \\mathrm { m i n } } \\geq$ $g ( \\log { \\frac { 1 } { \\mathcal { L } } } )$ , so we can replace $q _ { n }$ with $g ( \\log { \\frac { 1 } { \\mathcal { L } } } )$ in the above inequality. Combining with the fact that $\\begin{array} { r } { \\sum _ { n \\in [ N ] } { \\stackrel { \\sim } { e } } ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } \\quad } \\end{array}$ is just $\\nu$ , we have ", + "bbox": [ + 176, + 541, + 820, + 588 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/4bd63c906f7dc11a45aa59373ba0d8180344b4ec650568da529f5954ad5d2bfa.jpg", + "text": "$$\n\\left\\| \\frac { d \\pmb { \\theta } } { d t } \\right\\| _ { 2 } \\leq \\frac { \\nu } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\cdot B _ { 1 } \\rho ^ { L - 1 } = \\frac { B _ { 1 } \\nu } { \\tilde { \\gamma } \\rho } .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 597, + 630, + 633 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "So we have $\\begin{array} { r } { \\left\\| \\frac { d \\hat { \\pmb \\theta } } { d t } \\right\\| _ { 2 } \\le \\frac { 1 } { \\rho } \\left\\| \\frac { d \\pmb \\theta } { d t } \\right\\| _ { 2 } \\le \\frac { B _ { 1 } } { L \\tilde { \\gamma } } \\cdot \\frac { L \\nu } { \\rho ^ { 2 } } = \\frac { B _ { 1 } } { L \\tilde { \\gamma } } \\cdot \\frac { d } { d t } \\log \\rho . } \\end{array}$ ", + "bbox": [ + 173, + 640, + 560, + 667 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "To prove Theorem A.8, we consider each limit point $\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }$ , and construct a series of approximate KKT points converging to it. Then $\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }$ can be shown to be a KKT point by Theorem C.4. The following lemma ensures that such construction exists. ", + "bbox": [ + 173, + 683, + 825, + 729 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Lemma C.12. For every limit point $\\bar { \\pmb { \\theta } }$ of $\\left\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\right\\}$ , there exists a sequence of $\\{ t _ { m } : m \\in \\mathbb { N } \\}$ such that $t _ { m }$ $\\cdot + \\infty , \\hat { \\pmb { \\theta } } ( t _ { m } ) \\bar { \\pmb { \\theta } }$ , and $\\beta ( t _ { m } ) 1$ . ", + "bbox": [ + 173, + 733, + 825, + 776 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. Let $\\{ \\epsilon _ { m } > 0 : m \\in \\mathbb { N } \\}$ be an arbitrary sequence with $\\epsilon _ { m } \\to 0$ . Now we construct $\\left\\{ { { t } _ { m } } \\right\\}$ by induction. Suppose $t _ { 1 } < t _ { 2 } < \\dots < t _ { m - 1 }$ have already been constructed. Since $\\bar { \\pmb { \\theta } }$ is a limit point and $\\tilde { \\gamma } ( t ) \\uparrow \\tilde { \\gamma } _ { \\infty }$ (recall that $\\tilde { \\gamma } _ { \\infty } : = \\mathrm { l i m } _ { t + \\infty } \\tilde { \\gamma } ( t ) )$ , there exists $s _ { m } > t _ { m - 1 }$ such that ", + "bbox": [ + 173, + 790, + 825, + 835 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/bcb8951d33d8c5dbf3c118211ee2e04e3fe19fe8f13713f580a9a223007cb6f8.jpg", + "text": "$$\n\\left\\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\right\\| _ { 2 } \\leq \\epsilon _ { m } \\quad \\mathrm { a n d } \\quad \\frac { 1 } { L } \\log \\frac { \\widetilde { \\gamma } _ { \\infty } } { \\widetilde { \\gamma } ( s _ { m } ) } \\leq \\epsilon _ { m } ^ { 3 } .\n$$", + "text_format": "latex", + "bbox": [ + 330, + 840, + 663, + 875 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Let $s _ { m } ^ { \\prime } > s _ { m }$ be a time such that $\\log { \\rho ( s _ { m } ^ { \\prime } ) } = \\log { \\rho ( s _ { m } ) } + \\epsilon _ { m }$ . According to Theorem A.7, $\\log \\rho \\to + \\infty$ , so $s _ { m } ^ { \\prime }$ must exist. We construct $t _ { m } \\in ( s _ { m } , s _ { m } ^ { \\prime } )$ to be a time that $\\bar { \\beta } ( t _ { m } ) ^ { - 2 } - 1 \\leq \\epsilon _ { m } ^ { 2 }$ , where the existence can be shown by Corollary C.10. ", + "bbox": [ + 173, + 881, + 825, + 924 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Now we show that this construction meets our requirement. It follows from $\\beta ( t _ { m } ) ^ { - 2 } - 1 \\leq \\epsilon _ { m } ^ { 2 }$ that $\\beta ( t _ { m } ) \\geq 1 / \\sqrt { 1 + \\epsilon _ { m } ^ { 2 } } \\to 1$ . By Lemma C.11, we also know that ", + "bbox": [ + 168, + 102, + 825, + 135 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/e1a5de3dc6b09edbe64e933fc2b31000581c0b9e36d779af1d2de2ae88d2a1c8.jpg", + "text": "$$\n\\biggl \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\bar { \\pmb \\theta } \\biggl \\| _ { 2 } \\leq \\biggl \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\hat { \\pmb \\theta } ( s _ { m } ) \\biggr \\| _ { 2 } + \\biggl \\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\biggl \\| _ { 2 } \\leq \\frac { B _ { 1 } } { L \\tilde { \\gamma } ( t _ { 0 } ) } \\cdot \\epsilon _ { m } + \\epsilon _ { m } 0 .\n$$", + "text_format": "latex", + "bbox": [ + 223, + 140, + 772, + 174 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "This completes the proof. ", + "bbox": [ + 174, + 178, + 343, + 193 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof of Theorem A.8. Let $\\tilde { \\pmb { \\theta } } : = \\bar { \\pmb { \\theta } } / q _ { \\mathrm { m i n } } ( \\bar { \\pmb { \\theta } } ) ^ { 1 / L }$ for short. Let $\\{ t _ { m } : m \\in \\mathbb { N } \\}$ be the sequence constructed as in Lemma C.12. For each $t _ { m }$ , define $\\epsilon ( t _ { m } )$ and $\\delta ( t _ { m } )$ as in Lemma C.8. Then we know that $\\pmb { \\theta } ( t _ { m } ) / q _ { \\mathrm { m i n } } ( t _ { m } ) ^ { 1 / L }$ is an $( \\epsilon ( t _ { m } ) , \\delta ( t _ { m } ) )$ -KKT point and $\\pmb { \\theta } ( t _ { m } ) / q _ { \\mathrm { m i n } } ( t _ { m } ) ^ { 1 / L } \\tilde { \\pmb { \\theta } } , \\epsilon ( t _ { m } ) $ $0 , \\delta ( t _ { m } ) \\to 0$ . By Lemma C.7, (P) satisfies MFCQ. Applying Theorem C.4 proves the theorem. ", + "bbox": [ + 173, + 208, + 825, + 270 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "C.5 NON-ASYMPTOTIC ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 284, + 423, + 299 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof of Theorem A.9. Let C0 := 1L log γ˜∞γ˜(t0) . Without loss of generality, we assume $\\begin{array} { r } { \\epsilon < \\frac { \\sqrt { 6 } } { 2 } C _ { 1 } } \\end{array}$ and $\\delta < C _ { 2 } / f ( b _ { f } )$ . ", + "bbox": [ + 174, + 308, + 821, + 344 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Let $t _ { 1 }$ be the time such that $\\begin{array} { r } { \\log \\rho ( t _ { 1 } ) = \\frac { 1 } { L } \\log \\frac { g ( C _ { 2 } \\delta ^ { - 1 } ) } { \\tilde { \\gamma } ( t _ { 0 } ) } = \\Theta ( \\log \\delta ^ { - 1 } ) } \\end{array}$ g g(C2δ−1) = Θ(log δ−1) and t2 be the time such that $\\begin{array} { r } { \\log \\rho ( t _ { 2 } ) - \\log \\rho ( t _ { 1 } ) = \\frac { 1 } { 2 } C _ { 0 } C _ { 1 } ^ { 2 } \\epsilon ^ { - 2 } = \\Theta ( \\epsilon ^ { - 2 } ) } \\end{array}$ . By Corollary C.10, there exists $t _ { * } \\in ( t _ { 1 } , t _ { 2 } )$ such that $\\beta ( t _ { * } ) ^ { - 2 } - 1 \\le 2 \\epsilon ^ { 2 } C _ { 1 } ^ { - 2 }$ . ", + "bbox": [ + 173, + 349, + 825, + 405 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Now we argue that $\\tilde { \\pmb { \\theta } } ( t _ { * } )$ is an $( \\epsilon , \\delta )$ -KKT point. By Lemma C.8, we only need to show $C _ { 1 } \\sqrt { 1 - \\beta ( t _ { * } ) } \\le \\epsilon$ and $C _ { 2 } / f ( \\tilde { \\gamma } ( t _ { * } ) \\rho ( t _ { * } ) ^ { L } ) \\leq \\delta$ . For the first inequality, by assumption $\\epsilon < \\frac { \\sqrt { 6 } } { 2 } C _ { 1 }$ , we know that $\\beta ( t _ { * } ) ^ { - 2 } - 1 < 3$ , which implies $| \\beta ( t _ { * } ) | ~ < ~ \\frac { 1 } { 2 }$ . Then we have $1 - \\beta ( t _ { * } ) \\ \\leq$ 2\u000f2C −2 · β(t∗)2 $\\frac { \\beta ( t _ { * } ) ^ { 2 } } { 1 + \\beta ( t _ { * } ) } ~ \\leq ~ \\epsilon ^ { 2 } C _ { 1 } ^ { - 2 }$ . Therefore, $C _ { 1 } \\sqrt { 1 - \\beta ( t _ { * } ) } \\le \\epsilon$ holds. For the second inequality, $\\begin{array} { r } { \\tilde { \\gamma } ( t _ { * } ) \\rho ( t _ { * } ) ^ { L } \\geq \\tilde { \\gamma } ( t _ { * } ) \\cdot \\frac { g ( C _ { 2 } \\delta ^ { - 1 } ) } { \\tilde { \\gamma } ( t _ { 0 } ) } \\geq g ( C _ { 2 } \\delta ^ { - 1 } ) } \\end{array}$ . Therefore, $C _ { 2 } / f ( \\tilde { \\gamma } ( t _ { * } ) \\rho ( t _ { * } ) ^ { L } ) \\leq \\delta$ holds. □ ", + "bbox": [ + 173, + 411, + 826, + 507 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "C.6 PROOF FOR COROLLARY 4.5 ", + "text_level": 1, + "bbox": [ + 176, + 521, + 416, + 536 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "By the homogeneity of $q _ { n }$ , we can characterize KKT points using kernel SVM. ", + "bbox": [ + 173, + 547, + 692, + 563 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Lemma C.13. If $\\pmb { \\theta } _ { \\ast }$ is KKT point of $( P )$ , then there exists $h _ { n } \\in \\partial ^ { \\circ } \\Phi _ { x _ { n } } ( \\theta _ { * } )$ for $n \\in [ N ]$ such that $\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }$ is an optimal solution for the following constrained optimization problem $( Q )$ : ", + "bbox": [ + 171, + 565, + 821, + 595 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/0aada129d097b964a4f06db44f35cd70871add69082af91fec96a1d54ad6e955.jpg", + "text": "$$\n\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { s . t . } & { y _ { n } \\left. \\pmb { \\theta } , \\pmb { h } _ { n } \\right. \\geq 1 \\qquad \\forall n \\in [ N ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 375, + 601, + 619, + 650 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. It is easy to see that (Q) is a convex optimization problem. For $\\begin{array} { r } { \\pmb { \\theta } = \\frac { 2 } { L } \\pmb { \\theta } _ { * } } \\end{array}$ , from Theorem B.2, we can see that $y _ { n _ { \\cdot } } \\langle \\theta , h _ { n } \\rangle = 2 q _ { n } ( \\theta _ { \\ast } ) \\geq 2 > 1$ , which implies Slater’s condition. Thus, we only need to show that $\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }$ satisfies KKT conditions for (Q). ", + "bbox": [ + 173, + 665, + 825, + 708 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "By the KKT conditions for $( \\mathrm { P } )$ , we can construct $h _ { n } \\in \\partial ^ { \\circ } q _ { n } ( \\theta _ { * } )$ for $n \\in [ N ]$ such that $\\theta _ { * } \\mathrm { ~ - ~ }$ $\\textstyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } y _ { n } h _ { n } = \\mathbf { 0 }$ for some $\\lambda _ { 1 } , \\dots , \\lambda _ { N } \\ge 0$ and $\\lambda _ { n } ( q _ { n } ( \\pmb { \\theta } _ { * } ) - 1 ) = 0$ . Thus, $\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }$ satisfies ", + "bbox": [ + 169, + 713, + 825, + 747 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/4d818b9aa35a65ee65b2d1dee1ff8fe779e64efd59d0d4a14cd445774a1ba0c2.jpg", + "text": "$$\n\\begin{array} { r } { \\frac { 1 } { L } \\lambda _ { n } \\left( y _ { n } \\left. \\frac { 1 } { L } \\pmb { \\theta } _ { * } , \\pmb { h } _ { n } \\right. - 1 \\right) = \\frac { 1 } { L } \\lambda _ { n } \\left( q _ { n } ( \\pmb { \\theta } ) - 1 \\right) \\geq 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 787, + 573, + 808 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "So $\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }$ satisfies KKT conditions for (Q). ", + "bbox": [ + 174, + 820, + 446, + 837 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Now we prove Corollary 4.5 in Section 4.3. ", + "bbox": [ + 176, + 851, + 459, + 866 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. By Theorem A.8, every limit point $\\bar { \\pmb { \\theta } }$ is along the direction of a KKT point of (P). Combining this with Lemma C.13, we know that every limit point $\\bar { \\pmb { \\theta } }$ is also along the max-margin direction of (Q). ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "For smooth models, $h _ { n }$ in $( \\mathrm { Q } )$ is exactly the gradient $\\nabla \\Phi _ { x _ { n } } ( { \\bar { \\theta } } )$ . So, (Q) is the optimization problem for SVM with kernel $K _ { \\bar { \\theta } } ( x , x ^ { \\prime } ) = \\left. \\nabla \\Phi _ { x } ( \\bar { \\theta } ) , \\nabla \\Phi _ { x ^ { \\prime } } ( \\bar { \\theta } ) \\right.$ . For non-smooth models, we can construct an arbitrary function $h ( x ) \\in \\partial ^ { \\circ } \\Phi _ { x } ( \\bar { \\theta } )$ that ensures $\\pmb { h } ( \\pmb { x } _ { n } ) = \\pmb { h } _ { n }$ . Then, (Q) is the optimization problem for SVM with kernel $K _ { \\bar { \\theta } } ( x , x ^ { \\prime } ) = \\langle h ( x ) , h ( x ^ { \\prime } ) \\rangle$ . □ ", + "bbox": [ + 173, + 102, + 825, + 165 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "D TIGHT BOUNDS FOR LOSS CONVERGENCE AND WEIGHT GROWTH", + "text_level": 1, + "bbox": [ + 171, + 184, + 763, + 202 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "In this section, we give proof for Theorem A.10, which gives tight bounds for loss convergence and weight growth under Assumption (A1), (A2), (B3), (B4). ", + "bbox": [ + 173, + 215, + 825, + 246 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "D.1 CONSEQUENCES OF (B3.4) ", + "text_level": 1, + "bbox": [ + 176, + 262, + 406, + 277 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Before proving Theorem A.10, we show some consequences of (B3.4). ", + "bbox": [ + 173, + 289, + 638, + 304 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Lemma D.1. For $f ( \\cdot )$ and $g ( \\cdot )$ , we have ", + "bbox": [ + 174, + 308, + 444, + 324 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "1. For all $x \\in [ b _ { g } , + \\infty )$ , $\\begin{array} { r } { \\frac { g ( x ) } { g ^ { \\prime } ( x ) } \\in [ \\frac { 1 } { 2 K } x , 2 K x ] , } \\end{array}$ ; ", + "bbox": [ + 210, + 335, + 527, + 358 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "2. For all $y \\in [ g ( b _ { g } ) , + \\infty )$ , $\\begin{array} { r } { \\frac { f ( y ) } { f ^ { \\prime } ( y ) } \\in [ \\frac { 1 } { 2 K } y , 2 K y ] . } \\end{array}$ . ", + "bbox": [ + 210, + 367, + 539, + 388 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Thus, $g ( x ) = \\Theta ( x g ^ { \\prime } ( x ) ) , f ( y ) = \\Theta ( y f ^ { \\prime } ( y ) ) .$ ", + "bbox": [ + 174, + 398, + 470, + 416 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof. To prove Item 1, it is sufficient to show that ", + "bbox": [ + 173, + 433, + 509, + 449 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/4d35739734869fa07a2729eef49c68ae1cf0b3fe1c654e169b8455314239cf2c.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { g ( x ) = b _ { f } + \\int _ { f ( b _ { f } ) } ^ { x } g ^ { \\prime } ( u ) d u \\geq \\int _ { x / 2 } ^ { x } g ^ { \\prime } ( u ) d u } } \\\\ { \\displaystyle { \\qquad \\geq ( x / 2 ) \\cdot \\frac { g ^ { \\prime } ( x ) } { K } = \\frac { 1 } { 2 K } \\cdot x g ^ { \\prime } ( x ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 454, + 700, + 526 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/ba03a73a05e6f318e113dd615af355c491153fb124c3f086da985ca5877875c8.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle x = f ( b _ { f } ) + \\int _ { f ( b _ { f } ) } ^ { x } g ^ { \\prime } ( u ) f ^ { \\prime } ( g ( u ) ) d u \\geq \\frac { f ^ { \\prime } ( g ( x ) ) } { K } \\int _ { f ( g ( x ) / 2 ) } ^ { x } g ^ { \\prime } ( u ) d u } } \\\\ { { \\displaystyle \\quad = ( g ( x ) / 2 ) \\cdot \\frac { f ^ { \\prime } ( g ( x ) ) } { K } = \\frac { 1 } { 2 K } \\cdot \\frac { g ( x ) } { g ^ { \\prime } ( x ) } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 539, + 751, + 611 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "To prove Item 2, we only need to notice that Item 1 implies yf 0(y) = g(f(y))g0(f(y)) $\\begin{array} { r } { y f ^ { \\prime } ( y ) = \\frac { g ( f ( y ) ) } { g ^ { \\prime } ( f ( y ) ) } \\in [ \\frac { 1 } { 2 K } f ( y ) , 2 K f ( y ) ] } \\end{array}$ for all $y \\in [ g ( b _ { g } ) , + \\infty )$ . □ ", + "bbox": [ + 174, + 619, + 823, + 655 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Recall that (B3.4) directly implies that $f ^ { \\prime } ( \\Theta ( x ) ) = \\Theta ( f ^ { \\prime } ( x ) )$ and $g ^ { \\prime } ( \\Theta ( x ) ) = \\Theta ( g ^ { \\prime } ( x ) )$ . Combining this with Lemma D.1, we have the following corollary: ", + "bbox": [ + 174, + 671, + 823, + 700 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Corollary D.2. $f ( \\Theta ( x ) ) = \\Theta ( f ( x ) )$ and $g ( \\Theta ( x ) ) = \\Theta ( g ( x ) ) .$ . ", + "bbox": [ + 174, + 704, + 589, + 722 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Also, note that Lemma D.1 essentially shows that $( \\log f ( x ) ) ^ { \\prime } = \\Theta ( 1 / x )$ and $( \\log g ( x ) ) ^ { \\prime } = \\Theta ( 1 / x )$ . S $) \\log f ( x ) = \\Theta ( \\log x )$ and $\\log g ( x ) = \\Theta ( \\log x )$ , which means that $f$ and $g$ grow at most polynomially. ", + "bbox": [ + 174, + 731, + 825, + 775 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Corollary D.3. $f ( x ) = x ^ { \\Theta ( 1 ) }$ and $g ( x ) = x ^ { \\Theta ( 1 ) }$ . ", + "bbox": [ + 173, + 779, + 500, + 796 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "D.2 PROOF FOR THEOREM A.10 ", + "text_level": 1, + "bbox": [ + 176, + 813, + 413, + 828 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "We follow the notations in Section 5 to define $\\rho : = \\| \\pmb \\theta \\| _ { 2 }$ and $\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\| \\pmb \\theta \\| _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}$ , and sometimes we view the functions of as functions of $t$ . And we use the notations $B _ { 0 } , B _ { 1 }$ from Appendix C.3. ", + "bbox": [ + 173, + 837, + 825, + 871 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "The key idea to prove Theorem A.10 is to utilize Lemma B.6, in which $\\mathcal { L } ( t )$ is bounded from above by $\\frac { 1 } { G ^ { - 1 } ( \\Omega ( t ) ) }$ . So upper bounding $\\mathcal { L } ( t )$ reduces to lower bounding $G ^ { - 1 }$ . In the following lemma, we obtain tight asymptotic bounds for $G ( \\cdot )$ and $G ^ { - 1 } ( \\cdot )$ : ", + "bbox": [ + 174, + 876, + 825, + 925 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Lemma D.4. For function $G ( \\cdot )$ defined in Lemma B.6 and its inverse function $G ^ { - 1 } ( \\cdot )$ , we have the following bounds: ", + "bbox": [ + 166, + 102, + 825, + 132 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/873b38301550e74e24092b0508febd59e7b0e41111c53a61297064017b84fe92.jpg", + "text": "$$\nG ( x ) = \\Theta \\left( { \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } } x \\right) \\quad a n d \\quad G ^ { - 1 } ( y ) = \\Theta \\left( { \\frac { ( \\log y ) ^ { 2 } } { g ( \\log y ) ^ { 2 / L } } } y \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 138, + 728, + 176 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof. We first prove the bounds for $G ( x )$ , and then prove the bounds for $G ^ { - 1 } ( y )$ . ", + "bbox": [ + 173, + 194, + 715, + 210 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Bounding for $G ( x )$ . Let $\\begin{array} { r } { C _ { G } = \\int _ { 1 / \\mathcal { L } ( t _ { 0 } ) } ^ { \\exp ( b _ { g } ) } \\frac { g ^ { \\prime } ( \\log u ) ^ { 2 } } { g ( \\log u ) ^ { 2 - 2 / L } } d u } \\end{array}$ . For x ≥ exp(bg), ", + "bbox": [ + 173, + 224, + 679, + 250 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/b2882dceff5a604fa691927c857b52c892df903f7f7c1f9d0bcd280c7430f693.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle G ( x ) = C _ { G } + \\int _ { \\exp ( b _ { g } ) } ^ { x } \\frac { g ^ { \\prime } ( \\log u ) ^ { 2 } } { g ( \\log u ) ^ { 2 - 2 / L } } d u } } \\\\ { { \\displaystyle \\ = C _ { G } + \\int _ { \\exp ( b _ { g } ) } ^ { x } \\left( \\frac { g ^ { \\prime } ( \\log u ) \\log u } { g ( \\log u ) } \\right) ^ { 2 } \\frac { g ( \\log u ) ^ { 2 / L } } { ( \\log u ) ^ { 2 } } d u , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 300, + 256, + 696, + 335 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "By Lemma D.1, ", + "bbox": [ + 173, + 340, + 281, + 356 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/a4c8c22821d169edde56026779bc3a844badf68a7be988c5f59c33afe1aa9e6d.jpg", + "text": "$$\nG ( x ) \\leq C _ { G } + 4 K ^ { 2 } g ( \\log x ) ^ { 2 / L } \\int _ { \\exp ( b _ { g } ) } ^ { x } \\frac { 1 } { ( \\log u ) ^ { 2 } } d u = O \\left( \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } x \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 246, + 363, + 748, + 401 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "On the other hand, for $x \\ge \\exp ( 2 b _ { g } )$ , we have ", + "bbox": [ + 173, + 409, + 477, + 424 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/c364f8bc4059ba52e67ee43cb4cf3b48d3b83aa6c8a7800e421a994a9223c02a.jpg", + "text": "$$\nG ( x ) \\geq \\frac { 1 } { 4 K ^ { 2 } } \\int _ { \\sqrt x } ^ { x } \\frac { g ( \\log u ) ^ { 2 / L } } { ( \\log u ) ^ { 2 } } d u \\geq \\frac { g ( ( \\log x ) / 2 ) ^ { 2 / L } } { 4 K ^ { 2 } } \\int _ { \\sqrt x } ^ { x } \\frac { 1 } { ( \\log u ) ^ { 2 } } d u = \\Omega \\left( \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } x \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 186, + 431, + 810, + 469 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Bounding for $G ^ { - 1 } ( y )$ . Let $x = G ^ { - 1 } ( y )$ for $y \\geq 0$ . $G ( x )$ always has a finite value whenever $x$ is finite. So $x \\to + \\infty$ when $y + \\infty$ . According to the first part of the proof, we know that $\\begin{array} { r } { y \\ = \\ \\Theta \\left( { \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } } x \\right) } \\end{array}$ . Taking logarithm on both sides and using Corollary D.3, we have $\\log y = \\Theta ( \\log x )$ . By Corollary D.2, $g ( \\log y ) = g ( \\Theta ( \\log x ) ) = \\Theta ( g ( \\log x ) )$ . Therefore, ", + "bbox": [ + 173, + 484, + 825, + 554 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/d2a2e49e397f77c07e6594ee7b1c4475ae2912cf9503dba747bc1f13e391440c.jpg", + "text": "$$\n{ \\frac { ( \\log y ) ^ { 2 } } { g ( \\log y ) ^ { 2 / L } } } y = \\Theta \\left( { \\frac { ( \\log x ) ^ { 2 } } { g ( \\log x ) ^ { 2 / L } } } y \\right) = \\Theta ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 559, + 650, + 595 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "This implies that $\\begin{array} { r } { x = \\Theta \\left( \\frac { ( \\log y ) ^ { 2 } } { g ( \\log y ) ^ { 2 / L } } y \\right) } \\end{array}$ ", + "bbox": [ + 176, + 603, + 434, + 628 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "For other bounds, we derive them as follows. We first show that $\\begin{array} { r } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) = \\Theta ( \\rho ^ { L } ) } \\end{array}$ . With this equivalence, we derive an upper bound for the gradient at each time $t$ in terms of $\\mathcal { L }$ , and take an integration to bound $\\mathcal { L } ( t )$ from below. Now we have both lower and upper bounds for $\\mathcal { L } ( t )$ . Plugging these two bounds to $\\begin{array} { r } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) = \\Theta ( \\rho ^ { L } ) } \\end{array}$ gives the lower and upper bounds for $\\rho ( t )$ . ", + "bbox": [ + 173, + 646, + 825, + 707 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof for Theorem A.10. We first prove the upper bound for $\\mathcal { L }$ . Then we derive lower and upper bounds for $\\rho$ in terms of $\\mathcal { L }$ , and use these bounds to give a lower bound for $\\mathcal { L }$ . Finally, we plug in the tight bounds for $\\mathcal { L }$ to obtain the lower and upper bounds for $\\rho$ in terms of $t$ . ", + "bbox": [ + 173, + 723, + 825, + 767 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Upper Bounding $\\mathcal { L }$ . By Lemma B.6, we have $\\begin{array} { r } { \\frac { 1 } { \\mathcal { L } } \\geq G ^ { - 1 } ( \\Omega ( t ) ) } \\end{array}$ . Using Lemma D.4, we have \u0010 (log t)2g(log t)2/L t\u0011, which completes the proof. ", + "bbox": [ + 174, + 780, + 825, + 823 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Bounding $\\rho$ in Terms of $\\mathcal { L }$ . $\\tilde { \\gamma } ( t ) \\ \\geq \\ \\tilde { \\gamma } ( t _ { 0 } )$ , so $\\begin{array} { r } { \\rho ^ { L } \\ \\leq \\ \\frac { 1 } { \\tilde { \\gamma } ( t _ { 0 } ) } g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\ = \\ O ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) } \\end{array}$ . On the other hand, $g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) \\leq q _ { \\operatorname* { m i n } } \\leq B _ { 0 } \\rho ^ { L }$ . So $\\begin{array} { r } { \\rho ^ { L } = \\Omega ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) } \\end{array}$ . Therefore, we have the following relationship between $\\rho ^ { L }$ and $g ( \\log { \\frac { 1 } { \\mathcal { L } } } )$ : ", + "bbox": [ + 173, + 837, + 825, + 891 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/fef86a533fa4ff25f7e96e530b5cb9179fd506dfc2e2a0c22279830c787169ef.jpg", + "text": "$$\n\\rho ^ { L } = \\Theta ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 898, + 566, + 929 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Lower Bounding $\\mathcal { L }$ . Let $\\pmb { h } _ { 1 } , \\dots , \\pmb { h } _ { N }$ be a set of vectors such that $\\begin{array} { r } { h _ { n } \\in { \\frac { \\partial q _ { n } } { \\partial \\theta } } } \\end{array}$ and ", + "bbox": [ + 169, + 102, + 714, + 119 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/2046a4f56e514d4773a2bb499f195f078901c66a9903fe6b96090cff56a524e5.jpg", + "text": "$$\n\\frac { d \\pmb { \\theta } } { d t } = \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) \\pmb { h } _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 401, + 128, + 594, + 172 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "By (17) and Corollary D.2, $f ^ { \\prime } ( q _ { n } ) \\ = \\ f ^ { \\prime } ( { \\cal O } ( \\rho ^ { L } ) ) \\ = \\ f ^ { \\prime } ( { \\cal O } ( g ( \\log \\frac { 1 } { \\cal L } ) ) ) \\ = \\ { \\cal O } ( f ^ { \\prime } ( g ( \\log \\frac { 1 } { \\cal L } ) ) ) \\ =$ $\\begin{array} { r } { O ( 1 / g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ) } \\end{array}$ . Again by (17), we have $\\| h _ { n } \\| _ { 2 } \\le B _ { 1 } \\rho ^ { L - 1 } = O ( g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ^ { 1 - 1 / L } )$ . Combining these two bounds together, it follows from Corollary D.2 that ", + "bbox": [ + 173, + 180, + 825, + 228 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/e681658330e304224bcd2459bc8a4ed0e20d67e8f32100d55977931866412fa7.jpg", + "text": "$$\nf ^ { \\prime } ( q _ { n } ) \\left\\| h _ { n } \\right\\| _ { 2 } = O \\left( \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 1 - 1 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\right) = O \\left( \\frac { \\log \\frac { 1 } { \\mathcal { L } } } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 1 / L } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 236, + 702, + 279 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Thus, ", + "bbox": [ + 173, + 286, + 214, + 301 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/bbd5fc34dcd9fca31f99dd860b1c191455438422e5ee18f8f4a82a98bc029080.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle - \\frac { d \\mathcal { L } } { d t } = \\left\\| \\frac { d \\theta } { d t } \\right\\| _ { 2 } ^ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) h _ { n } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\displaystyle \\qquad \\leq \\left( \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } \\cdot \\displaystyle \\operatorname* { m a x } _ { n \\in [ N ] } \\left\\{ f ^ { \\prime } ( q _ { n } ) \\left\\| h _ { n } \\right\\| _ { 2 } \\right\\} \\right) ^ { 2 } \\leq \\mathcal { L } ^ { 2 } \\cdot O \\left( \\frac { ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 / L } } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 309, + 789, + 404 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "By definition of $G ( \\cdot )$ , this implies that there exists a constant $c$ such that $\\begin{array} { r } { \\frac { d } { d t } G ( \\frac { 1 } { \\mathcal { L } } ) \\leq c } \\end{array}$ for any $\\mathcal { L }$ that is small enough. We can complete our proof by applying Lemma D.4. ", + "bbox": [ + 173, + 411, + 823, + 443 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Bounding $\\rho$ in Terms of $t$ . By (17) and the tight bound for $\\mathcal { L } ( t )$ $, ~ \\rho ^ { L } ~ = ~ \\Theta ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) ~ =$ $\\Theta ( g ( \\Theta ( \\log t ) ) )$ ). Using Corollary D.2, we can conclude that $\\rho ^ { L } = \\Theta ( g ( \\log t ) )$ . □ ", + "bbox": [ + 173, + 458, + 823, + 491 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "E GRADIENT DESCENT: SMOOTH HOMOGENEOUS MODELS WITH EXPONENTIAL LOSS ", + "text_level": 1, + "bbox": [ + 176, + 512, + 736, + 546 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "In this section, we discretize our proof to prove similar results for gradient descent on smooth homogeneous models. As usual, the update rule of gradient descent is defined as ", + "bbox": [ + 174, + 564, + 821, + 593 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/6fb50094d1e04ee2654af4670b3920e27bf088cd4cd48b37572f80637967a50e.jpg", + "text": "$$\n\\pmb \\theta ( t + 1 ) = \\pmb \\theta ( t ) - \\eta ( t ) \\nabla \\mathcal L ( t )\n$$", + "text_format": "latex", + "bbox": [ + 398, + 601, + 599, + 619 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Here $\\eta ( t )$ is the learning rate, and $\\nabla \\mathcal { L } ( t ) : = \\nabla \\mathcal { L } ( \\pmb { \\theta } ( t ) )$ is the gradient of $\\mathcal { L }$ at $\\pmb \\theta ( t )$ . ", + "bbox": [ + 174, + 626, + 715, + 643 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "We first focus on the exponential loss. At the end of this section (Appendix F), we discuss how to extend the proof to general loss functions with a similar assumption as (B3). ", + "bbox": [ + 174, + 648, + 825, + 678 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "The main difficulty for discretizing our previous analysis comes from the fact that the original version of the smoothed normalized margin $\\begin{array} { r } { \\tilde { \\gamma } ( \\pmb { \\theta } ) : = \\rho ^ { - \\tilde { L } } \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}$ becomes less smooth when $\\rho \\to + \\infty$ . Thus, if we take a Taylor expansion for $\\tilde { \\gamma } ( \\pmb { \\theta } ( t + 1 ) )$ from the point $\\pmb \\theta ( t )$ , although one can show that the first-order term is positive as in the gradient flow analysis, the second-order term is unlikely to be bounded during gradient descent with a constant step size. To get a smoothed version of the normalized margin that is monotone increasing, we need to define another one that is even smoother than $\\tilde { \\gamma }$ . ", + "bbox": [ + 173, + 684, + 825, + 782 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Technically, recall that $\\begin{array} { r } { \\frac { d \\mathcal { L } } { d t } = - \\| \\nabla \\mathcal { L } \\| _ { 2 } ^ { 2 } } \\end{array}$ does not hold exactly for gradient descent. However, if the smoothness can be bounded by $s ( t )$ , then it is well-known that ", + "bbox": [ + 174, + 786, + 823, + 818 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/aaacdcd760519b27b1e1ddb258d530f907b63adb2ed34eb23e35cdba79a065d1.jpg", + "text": "$$\n\\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\leq - \\eta ( t ) ( 1 - s ( t ) \\eta ( t ) ) \\| \\nabla \\mathcal { L } \\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 827, + 658, + 845 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "By analyzing the landscape of $\\mathcal { L }$ , one can easily find that the smoothness is bounded locally by $\\dot { O ( } \\mathcal { L } \\cdot \\mathrm { p o l y l o g } ( \\frac { 1 } { \\mathcal { L } } ) \\big )$ . Thus, if we set $\\eta ( t )$ to a constant or set it appropriately according to the loss, then this discretization error becomes negligible. Using this insight, we define the new smoothed normalized margin $\\hat { \\gamma }$ in a way that it increases slightly slower than $\\tilde { \\gamma }$ during training to cancel the effect of discretization error. ", + "bbox": [ + 173, + 853, + 825, + 924 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "E.1 ASSUMPTIONS ", + "text_level": 1, + "bbox": [ + 174, + 103, + 320, + 118 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "As stated in Section 4.1, we assume (A2), (A3), (A4) similarly as for gradient flow, and two additional assumptions (S1) and (S5). ", + "bbox": [ + 173, + 128, + 825, + 159 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "(S1). (Smoothness). For any fixed $_ { \\textbf { \\em x } }$ , $\\Phi ( \\cdot ; x )$ is $\\mathcal { C } ^ { 2 }$ -smooth on $\\mathbb { R } ^ { d } \\backslash \\{ \\mathbf { 0 } \\}$ . \n(A2). (Homogeneity). There exists $L > 0$ such that $\\forall \\alpha > 0 : \\Phi ( \\alpha \\pmb { \\theta } ; \\pmb { x } ) = \\alpha ^ { L } \\Phi ( \\pmb { \\theta } ; \\pmb { x } ) ;$ ; \n(A3). (Exponential Loss). $\\ell ( q ) = e ^ { - q }$ ; \n(A4). (Separability). There exists a time $t _ { 0 }$ such that $\\mathcal { L } ( \\pmb { \\theta } ( t _ { 0 } ) ) < 1$ . \n(S5). (Learing rate condition). $\\begin{array} { r } { \\sum _ { t \\geq 0 } \\eta ( t ) = + \\infty } \\end{array}$ and $\\eta ( t ) \\leq H ( \\mathcal { L } ( \\pmb { \\theta } ( t ) ) )$ . ", + "bbox": [ + 183, + 167, + 761, + 261 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Here $H ( \\mathcal { L } )$ is a function of the current training loss. The explicit formula of $H ( \\mathcal { L } )$ is given below: ", + "bbox": [ + 173, + 271, + 816, + 286 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/f5fae6732bf21d556254fcdd78830af65d7096820d74ad81d547f0c3d3db5fc9.jpg", + "text": "$$\nH ( x ) : = \\frac { \\mu ( x ) } { C _ { \\eta } \\kappa ( x ) } = \\Theta \\left( \\frac { 1 } { x ( \\log \\frac { 1 } { x } ) ^ { 3 - 2 / L } } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 291, + 643, + 329 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "where $C _ { \\eta }$ is a constant, and $\\kappa ( \\boldsymbol { x } ) , \\mu ( \\boldsymbol { x } )$ are two non-decreasing functions. For constant learning rate $\\eta ( t ) = \\eta _ { 0 }$ , (S5) is satisfied when $\\eta _ { 0 }$ if sufficiently small. ", + "bbox": [ + 173, + 333, + 820, + 362 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Roughly speaking, $C _ { \\eta } \\kappa ( \\boldsymbol { x } )$ is an upper bound for the smoothness of $\\mathcal { L }$ in a neighborhood of $\\pmb \\theta$ when $x = { \\mathcal { L } } ( \\theta )$ . And we set the learning rate $\\eta ( t )$ to be the inverse of the smoothness multiplied by a factor $\\dot { \\mu ( x ) } = o ( 1 )$ . In our analysis, $\\mu ( x )$ can be any non-decreasing function that maps $\\left[ 0 , \\mathcal { L } ( \\dot { t } _ { 0 } ) \\right]$ to $( 0 , 1 / 2 ]$ and makes the integral $\\begin{array} { r } { \\int _ { 0 } ^ { 1 / 2 } \\mu ( x ) d x } \\end{array}$ exist. But for simplicity, we define $\\mu ( x )$ as ", + "bbox": [ + 173, + 368, + 825, + 431 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/2352d95e512440623c62eeb237c35bfa4b09464e58f245be1fc50fee4220b3bf.jpg", + "text": "$$\n\\mu ( x ) : = \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } { 2 \\log \\frac { 1 } { x } } = \\Theta \\left( \\frac { 1 } { \\log \\frac { 1 } { x } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 435, + 614, + 477 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "The value of $C _ { \\eta }$ will be specified later. The definition of $\\kappa ( x )$ depends on $L$ . For $0 < L \\leq 1$ , we define $\\kappa ( x )$ as ", + "bbox": [ + 173, + 481, + 823, + 508 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/bf4b32e6d1a5661fb035b88de86e935a7db858c07294dcfff5039371f7d6ea82.jpg", + "text": "$$\n\\kappa ( x ) : = x \\left( \\log \\frac { 1 } { x } \\right) ^ { 2 - 2 / L }\n$$", + "text_format": "latex", + "bbox": [ + 403, + 507, + 578, + 546 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "For $L > 1$ , we define $\\kappa ( x )$ as ", + "bbox": [ + 173, + 547, + 369, + 563 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/ebeea6dd0c80e8502a07ec59359f13ffe4be2a3da1db296464f62e535b11d889.jpg", + "text": "$$\n\\kappa ( x ) : = \\left\\{ \\begin{array} { l l } { x \\left( \\log \\frac { 1 } { x } \\right) ^ { 2 - 2 / L } \\quad } & { x \\in ( 0 , e ^ { 2 / L - 2 } ] } \\\\ { \\kappa _ { \\operatorname* { m a x } } } & { x \\in ( e ^ { 2 / L - 2 } , 1 ) } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 341, + 568, + 655, + 611 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "where $\\kappa _ { \\mathrm { m a x } } : = e ^ { ( 2 - 2 / L ) ( \\ln ( 2 - 2 / L ) - 1 ) }$ . The specific meaning of $C _ { \\eta } , \\kappa ( x )$ and $\\mu ( x )$ will become clear in our analysis. ", + "bbox": [ + 171, + 616, + 825, + 647 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "E.2 SMOOTHED NORMALIZED MARGIN ", + "text_level": 1, + "bbox": [ + 176, + 662, + 462, + 678 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Now we define the smoothed normalized margins. As usual, we define $\\begin{array} { r } { \\widetilde \\gamma ( \\pmb { \\theta } ) : = \\frac { \\log \\frac { 1 } { \\mathcal { L } } } { \\rho ^ { L } } } \\end{array}$ . At the same time, we also define ", + "bbox": [ + 173, + 685, + 823, + 720 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/f82225d70a6abd7020117828b7ba97bca7b7b4db0cd4743292e0c0c12263edff.jpg", + "text": "$$\n\\hat { \\gamma } ( \\pmb { \\theta } ) : = \\frac { e ^ { \\phi ( \\mathcal { L } ) } } { \\rho ^ { L } } .\n$$", + "text_format": "latex", + "bbox": [ + 446, + 726, + 552, + 761 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Here $\\phi : ( 0 , \\mathcal { L } ( t _ { 0 } ) ] \\to ( 0 , + \\infty )$ is constructed as follows. Construct the first-order derivative of $\\phi ( x )$ as ", + "bbox": [ + 169, + 765, + 825, + 794 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/09dc19c6d9db634bb6bf8da741c90cac20dbd1c78a345832ac82cd9ce3ef3a27.jpg", + "text": "$$\n\\phi ^ { \\prime } ( x ) = - \\operatorname* { s u p } \\left\\{ \\frac { 1 + 2 ( 1 + \\lambda ( w ) / L ) \\mu ( w ) } { w \\log \\frac { 1 } { w } } : w \\in \\left[ x , \\mathcal { L } ( t _ { 0 } ) \\right] \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 290, + 792, + 704, + 829 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\lambda ( x ) : = ( \\log \\frac { 1 } { x } ) ^ { - 1 } } \\end{array}$ . And then we set $\\phi ( x )$ to be ", + "bbox": [ + 176, + 830, + 531, + 848 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/d18ef29e6d3088fac2ea801684d9cc811c727b2d319e2aa63dae973925cd0ed2.jpg", + "text": "$$\n\\phi ( x ) = \\log \\log { \\frac { 1 } { x } } + \\int _ { 0 } ^ { x } \\left( \\phi ^ { \\prime } ( w ) + { \\frac { 1 } { w \\log { \\frac { 1 } { w } } } } \\right) d w .\n$$", + "text_format": "latex", + "bbox": [ + 333, + 853, + 665, + 890 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "It can be verified that $\\phi ( x )$ is well-defined and $\\phi ^ { \\prime } ( x )$ is indeed the first-order derivative of $\\phi ( x )$ . \nMoreover, we have the following relationship among $\\hat { \\gamma } , \\tilde { \\gamma }$ and $\\bar { \\gamma }$ . ", + "bbox": [ + 173, + 895, + 823, + 925 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Lemma E.1. $\\hat { \\gamma } ( \\pmb \\theta )$ is well-defined for $\\mathcal { L } ( \\pmb { \\theta } ) \\leq \\mathcal { L } ( t _ { 0 } )$ and has the following properties: ", + "bbox": [ + 171, + 102, + 738, + 119 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "(a) If $\\mathcal { L } ( \\pmb { \\theta } ) \\leq \\mathcal { L } ( t _ { 0 } )$ , then $\\hat { \\gamma } ( \\pmb \\theta ) < \\tilde { \\gamma } ( \\pmb \\theta ) \\leq \\bar { \\gamma } ( \\pmb \\theta ) .$ . ", + "bbox": [ + 205, + 130, + 524, + 147 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "$( b )$ Let $\\{ \\pmb \\theta _ { m } \\in \\mathbb { R } ^ { d } : m \\in \\mathbb { N } \\}$ be a sequence of parameters. If $\\mathcal { L } ( \\pmb { \\theta } _ { m } ) 0$ , then ", + "bbox": [ + 205, + 154, + 725, + 171 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/efbe5835d504517b82c20b63fc0db86b460a672e6b397378b2e09133e22844d2.jpg", + "text": "$$\n\\frac { \\hat { \\gamma } ( \\pmb \\theta _ { m } ) } { \\bar { \\gamma } ( \\pmb \\theta _ { m } ) } = 1 - O ( ( \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta _ { m } ) } ) ^ { - 1 } ) 1 .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 179, + 676, + 222 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Proof. First we verify that $\\hat { \\gamma }$ is well-defined. To see this, we only need to verify that ", + "bbox": [ + 171, + 239, + 725, + 257 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/51700aaf316ed1b0d60ffdefe3113f1f9e3896b13706f525123e6c3dda575d9f.jpg", + "text": "$$\nI ( x ) : = \\int _ { 0 } ^ { x } \\left( \\phi ^ { \\prime } ( w ) + { \\frac { 1 } { w \\log { \\frac { 1 } { w } } } } \\right) d w\n$$", + "text_format": "latex", + "bbox": [ + 372, + 263, + 625, + 300 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "exists for all $x \\in ( 0 , \\mathcal { L } ( t _ { 0 } ) ]$ , then it is trivial to see that $\\phi ^ { \\prime } ( w )$ is indeed the derivative of $\\phi ( w )$ by I 0 ( x ) = φ 0 ( x ) + 1x log 1 . ", + "bbox": [ + 173, + 308, + 825, + 343 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Note that $I ( x )$ exists for all $x \\in ( 0 , \\mathcal { L } ( t _ { 0 } ) ]$ as long as $I ( x )$ exists for a small enough $x > 0$ . By definition, it is easy to verify that $\\begin{array} { r } { r ( w ) : = \\frac { 1 + 2 ( 1 + \\lambda ( w ) / L ) \\mu ( w ) } { w \\log \\frac { 1 } { w } } } \\end{array}$ is decreasing when $w$ is small enough. Thus, for a small enough $w > 0$ , we have ", + "bbox": [ + 173, + 348, + 825, + 400 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/16a00bf4775b9b516dda8090436b581751f73528e435f67849132bc1958c1cca.jpg", + "text": "$$\n- \\phi ^ { \\prime } ( w ) = r ( w ) = \\frac { 1 + 2 ( 1 + \\frac { 1 } { L } ( \\log \\frac { 1 } { w } ) ^ { - 1 } ) \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } { 2 \\log \\frac { 1 } { w } } } { w \\log \\frac { 1 } { w } } = \\frac { 1 } { w \\log \\frac { 1 } { w } } + \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( 1 + \\frac { 1 } { L } ( \\log \\frac { 1 } { w } ) ^ { - 1 } ) } { w ( \\log \\frac { 1 } { w } ) ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 179, + 407, + 813, + 455 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "So we have the following for small enough $x$ : ", + "bbox": [ + 174, + 462, + 473, + 477 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/0d76a71d36a3071f2c2b07b6ffdce8bad5b562c86abaf274d8e5a8fe04497cc2.jpg", + "text": "$$\n\\begin{array} { r } { I ( x ) = \\displaystyle \\int _ { 0 } ^ { x } - \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( 1 + \\frac { 1 } { L } ( \\log \\frac { 1 } { w } ) ^ { - 1 } ) } { w ( \\log \\frac { 1 } { w } ) ^ { 2 } } d w = - \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( \\frac { 1 } { \\log \\frac { 1 } { w } } + \\frac { 1 } { 2 L ( \\log \\frac { 1 } { w } ) ^ { 2 } } ) _ { 0 } ^ { x } } \\\\ { \\displaystyle = - \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( \\frac { 1 } { \\log \\frac { 1 } { x } } + \\frac { 1 } { 2 L ( \\log \\frac { 1 } { x } ) ^ { 2 } } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 484, + 784, + 565 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "This proves the existence of $I ( x )$ . ", + "bbox": [ + 174, + 571, + 397, + 587 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Now we prove (a). By Lemma A.5, $\\tilde { \\gamma } ( \\pmb \\theta ) \\leq \\tilde { \\gamma } ( \\pmb \\theta )$ , so we only need to prove that $\\hat { \\gamma } _ { \\perp } ( \\pmb { \\theta } ) < \\tilde { \\gamma } ( \\pmb { \\theta } )$ . To see this, note that for all $w \\leq \\mathcal { L } ( t _ { 0 } ) , r ( w ) > \\frac { 1 } { w \\log \\frac { 1 } { w } }$ . So we have $- \\bar { \\phi } ^ { \\prime } ( w ) > \\frac { 1 } { w \\log \\frac { 1 } { w } }$ 1g 1w , and this implies $I ( x ) < 0$ for all $x \\le \\mathcal { L } ( t _ { 0 } )$ . Then it holds for all $\\pmb \\theta$ with $\\mathcal { L } ( \\pmb { \\theta } ) \\leq \\mathcal { L } ( t _ { 0 } )$ that ", + "bbox": [ + 174, + 592, + 825, + 642 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/550b1be6b6722356e10c92dc0dcbf0ab17c7d6770f47bcbb052da087eff9c243.jpg", + "text": "$$\n\\frac { \\widehat { \\gamma } ( \\pmb \\theta ) } { \\widetilde { \\gamma } ( \\pmb \\theta ) } = \\frac { e ^ { \\phi ( \\mathcal { L } ( \\pmb \\theta ) ) } } { \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } } = \\frac { e ^ { \\log \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } + I ( \\mathcal { L } ( \\pmb \\theta ) ) } } { e ^ { \\log \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } } } = e ^ { I ( \\mathcal { L } ( \\pmb \\theta ) ) } < 0 .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 648, + 689, + 691 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "To prove (b), we combine (19) and (20), then for small enough $\\mathcal { L } ( \\pmb { \\theta } _ { m } )$ , we have ", + "bbox": [ + 171, + 707, + 696, + 723 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/31d99d28e06520ea4f5c5dece7d2b24466be3c45b9daa6030f2bb33d54d290e1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { \\hat { \\gamma } ( \\theta ) } { \\hat { \\gamma } ( \\theta ) } = \\frac { \\hat { \\gamma } ( \\theta ) } { \\hat { \\gamma } ( \\theta ) } \\cdot \\frac { \\hat { \\gamma } ( \\theta ) } { \\hat { \\gamma } ( \\theta ) } } \\\\ & { \\quad \\quad \\quad = \\exp \\left( - \\log \\frac { 1 } { \\mathcal { L } ( { t _ { 0 } } ) } \\left( \\frac { 1 } { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } + \\frac { 1 } { 2 L ( \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } ) ^ { 2 } } \\right) \\right) \\cdot \\frac { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } { q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) } } \\\\ & { \\quad \\quad \\quad = \\exp \\left( - \\mathcal { O } \\left( \\frac { 1 } { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } \\right) \\right) \\cdot \\frac { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } + \\mathcal { O } ( 1 ) } } \\\\ & { \\quad \\quad \\quad = 1 - \\mathcal { O } \\left( \\left( \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } \\right) ^ { - 1 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 248, + 731, + 750, + 895 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Now we specify the value of $C _ { \\eta }$ . By (S1) and (S2), we can define $B _ { 0 } , B _ { 1 } , B _ { 2 }$ as follows: ", + "bbox": [ + 173, + 102, + 756, + 119 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/f7c7b94b2919472564c0f5c3e77b005b1ea3ea6243bb887a17bc8a4676f3f94b.jpg", + "text": "$$\n\\begin{array} { r l } & { B _ { 0 } : = \\operatorname* { s u p } \\left\\{ q _ { n } : \\pmb \\theta \\in S ^ { d - 1 } , n \\in [ N ] \\right\\} , } \\\\ & { B _ { 1 } : = \\operatorname* { s u p } \\left\\{ \\left\\| \\nabla q _ { n } \\right\\| _ { 2 } : \\pmb \\theta \\in S ^ { d - 1 } , n \\in [ N ] \\right\\} , } \\\\ & { B _ { 2 } : = \\operatorname* { s u p } \\left\\{ \\left\\| \\nabla ^ { 2 } q _ { n } \\right\\| _ { 2 } : \\pmb \\theta \\in S ^ { d - 1 } , n \\in [ N ] \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 344, + 131, + 651, + 196 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Then we set $\\begin{array} { r l } & { C _ { \\eta } : = \\frac { 1 } { 2 } \\left( B _ { 1 } ^ { 2 } + \\rho ( t _ { 0 } ) ^ { - L } B _ { 2 } \\right) \\operatorname* { m i n } \\Big \\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\Big \\} . } \\end{array}$ ", + "bbox": [ + 174, + 207, + 661, + 234 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "E.3 THEOREMS ", + "text_level": 1, + "bbox": [ + 173, + 255, + 295, + 270 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Now we state our main theorems for the monotonicity of the normalized margin and the convergence to KKT points. We will prove Theorem E.2 in Appendix E.4, and prove Theorem E.3 and E.4 in Appendix E.5. ", + "bbox": [ + 173, + 284, + 826, + 327 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Theorem E.2. Under assumptions (S1), (A2) - (A4), (S5), the following are true for gradient descent: ", + "bbox": [ + 171, + 334, + 821, + 363 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "1. For all $t \\geq t _ { 0 }$ , $\\hat { \\gamma } ( t + 1 ) \\geq \\hat { \\gamma } ( t )$ ; ", + "bbox": [ + 214, + 377, + 444, + 395 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "2. For all $t \\geq t _ { 0 }$ , either $\\hat { \\gamma } ( t + 1 ) > \\hat { \\gamma } ( t )$ or $\\hat { \\pmb { \\theta } } ( t + 1 ) = \\hat { \\pmb { \\theta } } ( t )$ ; ", + "bbox": [ + 210, + 406, + 616, + 425 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "3. $\\mathcal { L } ( t ) 0$ and $\\rho ( t ) \\infty$ as $t \\to + \\infty$ ; therefore, $| \\bar { \\gamma } ( t ) - \\tilde { \\gamma } ( t ) | 0$ . ", + "bbox": [ + 210, + 434, + 679, + 452 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Theorem E.3. Consider gradient flow under assumptions (S1), (A2) - (A4), (S5). For every limit point $\\bar { \\pmb { \\theta } }$ of $\\left\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\right\\}$ , $\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }$ is a KKT point of $( P )$ . ", + "bbox": [ + 174, + 457, + 823, + 494 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Theorem E.4. Consider gradient descent under assumptions (S1), (A2) - (A4), (S5). For any $\\epsilon , \\delta >$ 0, there exists $r : = \\Theta ( \\log { \\delta ^ { - 1 } } )$ and $\\Delta : = \\Theta ( \\epsilon ^ { - 2 } )$ such that $\\pmb { \\theta } / q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { 1 / L }$ is an $( \\epsilon , \\delta )$ -KKT point at some time $t _ { * }$ satisfying $\\log \\rho ( t _ { * } ) \\in ( r , r + \\Delta )$ . ", + "bbox": [ + 176, + 502, + 823, + 546 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "With a refined analysis, we can also derive tight rates for loss convergence and weight growth. We defer the proof to Appendix E.6. ", + "bbox": [ + 173, + 559, + 823, + 588 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Theorem E.5. Under assumptions (S1), (A2) - (A4), (S5), we have the following tight rates for training loss and weight norm: ", + "bbox": [ + 173, + 594, + 823, + 625 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/4b2e0c5ff6fe08becd78235a33e1ba14845baf8dc9f226e051b65cbfefdd8b65.jpg", + "text": "$$\n\\mathcal { L } ( t ) = \\Theta \\left( \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } \\right) \\quad a n d \\quad \\rho ( t ) = \\Theta ( ( \\log T ) ^ { 1 / L } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 294, + 637, + 700, + 672 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { T = \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}$ ", + "bbox": [ + 174, + 685, + 336, + 705 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "E.4 PROOF FOR THEOREM E.2 ", + "text_level": 1, + "bbox": [ + 174, + 726, + 400, + 741 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Wefor $\\begin{array} { r } { \\nu ( t ) : = \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( t ) } q _ { n } ( t ) } \\end{array}$ as we do for gradient flow. Then we can get a clary B.3. Also, we can get a lower bound for d formusing $\\langle \\pmb { \\theta } ( t ) , - \\nabla \\mathcal { L } ( t ) \\rangle$ $\\nu ( t )$ Lemma B.5 for exponential loss directly. ", + "bbox": [ + 173, + 752, + 826, + 799 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Corollary E.6. hθ(t), −∇L(t)i = Lν(t). If L(t) < 1, then ν(t) ≥ L(t)λ(L(t)) . ", + "bbox": [ + 173, + 804, + 669, + 828 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "As we are analyzing gradient descent, the norm of $\\nabla \\mathcal { L }$ and $\\nabla ^ { 2 } \\boldsymbol { \\mathcal { L } }$ appear naturally in discretization error terms. To bound them, we have the following lemma when $\\widetilde \\gamma ( \\pmb \\theta )$ and $\\rho ( \\pmb \\theta )$ have lower bounds: ", + "bbox": [ + 173, + 842, + 825, + 872 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Lemma E.7. For any $\\pmb \\theta , i f \\tilde { \\gamma } ( \\pmb \\theta ) \\geq \\hat { \\gamma } ( t _ { 0 } ) , \\rho ( \\pmb \\theta ) \\geq \\rho ( t _ { 0 } )$ , then ", + "bbox": [ + 171, + 877, + 575, + 895 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/6b9909f7f2e24d69959da3441023ee1ddf18b5d8bfa804e9ac3b3cb0565ddf4c.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\nabla \\mathcal { L } ( \\pmb { \\theta } ) \\| _ { 2 } ^ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) \\mathcal { L } ( \\pmb { \\theta } ) \\quad a n d \\quad \\left\\| \\nabla ^ { 2 } \\mathcal { L } ( \\pmb { \\theta } ) \\right\\| _ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 906, + 730, + 926 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof. By the chain rule and the definitions of $B _ { 1 } , B _ { 2 }$ , we have ", + "bbox": [ + 173, + 102, + 593, + 119 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/3b2d6ecf8165136c6bb28b50d62034c9846f50e4ff3e1d5fe8b793823622b626.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\left\\| \\nabla \\mathcal { L } \\right\\| _ { 2 } = \\left\\| - \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\nabla q _ { n } \\right\\| _ { 2 } \\leq \\mathcal { L } \\rho ^ { L - 1 } B _ { 1 } } \\\\ & { \\displaystyle \\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\big ( \\nabla q _ { n } \\nabla q _ { n } ^ { \\top } - \\nabla ^ { 2 } q _ { n } \\big ) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\left( B _ { 1 } ^ { 2 } \\rho ^ { 2 L - 2 } + B _ { 2 } \\rho ^ { L - 2 } \\right) \\leq \\mathcal { L } \\rho ^ { 2 L - 2 } \\left( B _ { 1 } ^ { 2 } + \\rho ^ { - L } B _ { 2 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 125, + 741, + 260 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Note that $\\begin{array} { r } { \\hat { \\gamma } ( t _ { 0 } ) \\rho ^ { L } \\leq \\tilde { \\gamma } \\rho ^ { L } \\leq \\log \\frac { 1 } { \\mathcal { L } } \\leq B _ { 0 } \\rho ^ { L } } \\end{array}$ . So $\\begin{array} { r } { \\hat { \\gamma } ( t _ { 0 } ) ^ { - 1 } \\log \\frac { 1 } { \\mathcal { L } } \\leq \\rho ^ { L } \\leq B _ { 0 } ^ { - 1 } \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}$ . Combining all these formulas together gives ", + "bbox": [ + 173, + 265, + 825, + 297 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/39183ed1cf78814025b802e05f6085544b47d48f0fea52af899f6bea1e7ddb56.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left\\| \\nabla { \\mathcal { L } } \\right\\| _ { 2 } ^ { 2 } \\leq B _ { 1 } ^ { 2 } \\operatorname* { m i n } \\left\\{ \\widehat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\} { \\mathcal { L } } ^ { 2 } \\cdot \\left( \\log \\displaystyle \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta } \\kappa ( { \\mathcal { L } } ) \\mathcal { L } } \\\\ & { \\left\\| \\nabla ^ { 2 } { \\mathcal { L } } \\right\\| _ { 2 } \\leq \\left( B _ { 1 } ^ { 2 } + \\rho ^ { - L } B _ { 2 } \\right) \\operatorname* { m i n } \\left\\{ \\widehat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\} { \\mathcal { L } } \\cdot \\left( \\log \\displaystyle \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta } \\kappa ( { \\mathcal { L } } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 304, + 805, + 382 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "which completes the proof. ", + "bbox": [ + 174, + 387, + 354, + 402 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "For proving the first two propositions in Theorem E.2, we only need to prove Lemma E.8. (P1) gives a lower bound for $\\tilde { \\gamma }$ . (P2) gives both lower and upper bounds for the weight growth using $\\nu ( t )$ . (P3) gives a lower bound for the decrement of training loss. Finally, (P4) shows the monotonicity of $\\hat { \\gamma }$ , and it is trivial to deduce the first two propositions in Theorem E.2 from (P4). ", + "bbox": [ + 173, + 420, + 825, + 477 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Lemma E.8. For all $t = t _ { 0 } , t _ { 0 } + 1 , \\ldots$ , we interpolate between $\\pmb \\theta ( t )$ and $\\pmb \\theta ( t + 1 )$ by defining $\\pmb \\theta ( t + \\alpha ) = \\pmb \\theta ( t ) - \\alpha \\eta ( t ) \\nabla \\mathcal L ( t )$ for $\\alpha \\in ( 0 , 1 )$ . Then for all integer $t \\geq t _ { 0 }$ , $\\nu ( t ) > 0$ , and the following holds for all $\\alpha \\in [ 0 , 1 ]$ : ", + "bbox": [ + 173, + 481, + 823, + 525 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/316658965977837af2fb156a3fb6bcbdbde6a08040c00be4cb1e023d533b0f84.jpg", + "text": "$$\n\\begin{array} { r l } & { ( P I ) . \\ \\tilde { \\gamma } ( t + \\alpha ) > \\hat { \\gamma } ( t _ { 0 } ) . } \\\\ & { ( P 2 ) . \\ 2 L \\alpha \\eta ( t ) \\nu ( t ) \\leq \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } \\leq 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) } { L } \\right) . } \\\\ & { ( P 3 ) . \\ \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) \\leq - \\alpha \\eta ( t ) ( 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } . } \\\\ & { ( P 4 ) . \\ \\log \\hat { \\gamma } ( t + \\alpha ) - \\log \\hat { \\gamma } ( t ) \\geq \\frac { \\rho ( t ) ^ { 2 } } { L \\nu ( t ) ^ { 2 } } \\left. \\left( I - \\hat { \\theta } ( t ) \\hat { \\theta } ( t ) ^ { \\top } \\right) \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } . \\log \\frac { \\rho ( t + \\alpha ) } { \\rho ( t ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 531, + 727, + 652 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "To prove Lemma E.8, we only need to prove the following lemma and then use an induction: ", + "bbox": [ + 173, + 661, + 781, + 676 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Lemma E.9. Fix an integer $T \\geq t _ { 0 }$ . Suppose that $( P I ) , ( P 2 ) , ( P 3 ) , ( P 4 )$ hold for any $t + \\alpha \\leq T$ . Then $i f ( P l )$ holds for $( t , \\alpha ) \\in \\{ T \\} \\times [ 0 , A )$ for some $A \\in ( 0 , 1 ] ,$ , then all of $( P I ) , ( P 2 ) , ( P 3 ) , ( P 4 )$ hold for $( t , \\alpha ) \\in \\{ T \\} \\times [ 0 , A ]$ . ", + "bbox": [ + 173, + 680, + 825, + 724 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof for Lemma E.8. We prove this lemma by induction. For $t = t _ { 0 } , \\alpha = 0$ , $\\nu ( t ) > 0$ by (S4) and Corollary E.6. (P2), (P3), (P4) hold trivially since $\\log \\hat { \\gamma } ( t + \\alpha ) = \\log \\hat { \\gamma } ( t )$ , $\\textstyle { \\mathcal { L } } ( t + \\alpha ) = { \\mathcal { L } } ( t )$ and $\\log \\hat { \\gamma } ( t + \\alpha ) = \\log \\hat { \\gamma } ( t )$ . By Lemma E.1, (P1) also holds trivially. ", + "bbox": [ + 174, + 741, + 825, + 785 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Now we fix an integer $T \\geq t _ { 0 }$ and assume that (P1), (P2), (P3), (P4) hold for any $t + \\alpha \\leq T$ (where $t \\geq t _ { 0 }$ is an integer and $\\alpha \\in [ 0 , 1 ] ,$ ). By (P3), $\\mathcal { L } ( t ) \\leq \\mathcal { L } ( t _ { 0 } ) < 1$ , so $\\nu ( t ) > 0$ . We only need to show that (P1), (P2), (P3), (P4) hold for $t = T$ and $\\alpha \\in [ 0 , 1 ]$ . ", + "bbox": [ + 174, + 790, + 823, + 834 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Let $A : = \\operatorname* { i n f } \\{ \\alpha \\in [ 0 , 1 ] : \\alpha = 1$ or (P1) does not hold for $( T , \\alpha ) \\}$ . If $A = 0$ , then (P1) holds for $( T , A )$ since (P1) holds for $( T - 1 , 1 )$ ; if $A > 0$ , we can also know that (P1) holds for $( T , A )$ by Lemma E.9. Suppose that $A < 1$ . Then by the continuity of $\\tilde { \\gamma } ( T + \\alpha )$ (with respect to $\\alpha$ ), we know that there exists $A ^ { \\prime } > A$ such that $\\tilde { \\gamma } ( T + \\alpha ) > \\hat { \\gamma } ( t _ { 0 } )$ for all $\\alpha \\in [ A , A ^ { \\prime } ]$ , which contradicts to the definition of $A$ . Therefore, $A = 1$ . Using Lemma E.9 again, we can conclude that (P1), (P2), (P3), (P4) hold for $t = T$ and $\\alpha \\in [ 0 , 1 ]$ . □ ", + "bbox": [ + 173, + 838, + 825, + 925 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Now we turn to prove Lemma E.9. ", + "bbox": [ + 176, + 103, + 401, + 118 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof for Lemma E.9. Applying (P3) on $( t , \\alpha ) \\in \\{ t _ { 0 } , \\ldots , T - 1 \\} \\times 1$ , we have ${ \\mathcal { L } } ( t ) \\leq { \\mathcal { L } } ( t _ { 0 } ) < 1$ . Then by Corollary E.6, we have $\\nu ( t ) > 0$ . Applying (P2) on $( t , \\alpha ) \\in \\{ t _ { 0 } , \\ldots , T - 1 \\} \\times 1$ , we can get $\\rho ( t ) \\geq \\rho ( t _ { 0 } )$ . ", + "bbox": [ + 173, + 132, + 825, + 178 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Fix $t = T$ . By (P2) with $\\alpha \\in [ 0 , A )$ and the continuity of $\\tilde { \\gamma }$ , we have $\\tilde { \\gamma } ( t + A ) \\geq \\hat { \\gamma } ( t _ { 0 } )$ . Thus, $\\tilde { \\gamma } ( t + \\alpha ) \\geq \\hat { \\gamma } ( t _ { 0 } )$ for all $\\alpha \\in [ 0 , A ]$ . ", + "bbox": [ + 173, + 181, + 823, + 213 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof for (P2). By definition, $\\begin{array} { r } { \\rho ( t + \\alpha ) ^ { 2 } = \\| \\pmb { \\theta } ( t ) - \\alpha \\eta ( t ) \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } = \\rho ( t ) ^ { 2 } + \\alpha ^ { 2 } \\eta ( t ) ^ { 2 } \\left\\| \\nabla \\mathcal { L } ( t ) \\right\\| _ { 2 } ^ { 2 } - } \\end{array}$ $2 \\alpha \\eta ( t ) \\left. \\pmb { \\theta } ( t ) , \\nabla \\mathcal { L } ( t ) \\right.$ . By Corollary E.6, we have ", + "bbox": [ + 171, + 226, + 823, + 258 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/8f168583b603c09c8053f0866cd04c928047b67272355679e44287b3fe622000.jpg", + "text": "$$\n\\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } = \\alpha ^ { 2 } \\eta ( t ) ^ { 2 } \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } + 2 L \\alpha \\eta ( t ) \\nu ( t ) .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 263, + 686, + 285 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "So $\\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } \\geq 2 L \\alpha \\eta ( t ) \\nu ( t )$ . For the other direction, we have the following using Corollary E.6 and Lemma E.7, ", + "bbox": [ + 171, + 291, + 823, + 321 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/532123aadf62da03b786440165d395dfb0505ea4db3bcf295f923b1b5c1b0346.jpg", + "text": "$$\n\\begin{array} { r l } & { \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } = 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { \\alpha \\eta ( t ) \\left\\| \\nabla \\mathcal { L } ( t ) \\right\\| _ { 2 } ^ { 2 } } { 2 L \\nu ( t ) } \\right) } \\\\ & { \\qquad \\le 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { C _ { \\eta } ^ { - 1 } \\kappa ( \\mathcal { L } ( t ) ) ^ { - 1 } \\mu ( \\mathcal { L } ( t ) ) \\cdot 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( t ) ) \\mathcal { L } ( t ) } { 2 L \\mathcal { L } ( t ) \\lambda ( \\mathcal { L } ( t ) ) ^ { - 1 } } \\right) } \\\\ & { \\qquad = 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) / L \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 327, + 781, + 433 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof for (P3). (P3) holds trivially for $\\alpha = 0$ or $\\nabla \\mathcal { L } ( t ) = \\mathbf { 0 }$ . So now we assume that $\\alpha \\neq 0$ and $\\nabla \\mathcal { L } ( t ) \\neq \\mathbf { 0 }$ . By the update rule (18) and Taylor expansion, there exists $\\xi \\in ( 0 , \\alpha )$ such that ", + "bbox": [ + 173, + 443, + 828, + 473 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/e61b2ec6f0513fe0fb54efdea65daa4166a3ac97997812cbba20047c9df69747.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrel { \\mathop : } ( t + \\alpha ) = \\mathcal { L } ( t ) + ( \\theta ( t + \\alpha ) - \\theta ( t ) ) ^ { \\top } \\nabla \\mathcal { L } ( t ) + \\frac { 1 } { 2 } ( \\theta ( t + \\alpha ) - \\theta ( t ) ) ^ { \\top } \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) ( \\theta ( t + \\alpha ) - \\theta ( t ) ) } \\\\ & { \\qquad \\le \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } + \\frac { 1 } { 2 } \\alpha ^ { 2 } \\eta ( t ) ^ { 2 } \\| \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) \\| _ { 2 } \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } } \\\\ & { \\qquad = \\mathcal { L } ( t ) - \\alpha \\eta ( t ) ( 1 - \\frac { 1 } { 2 } \\alpha \\eta ( t ) ) \\| \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) \\| _ { 2 } ) \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 478, + 839, + 575 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "By Lemma E.7, $\\left\\| \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) \\right\\| _ { 2 } \\leq 2 C _ { \\eta } \\cdot \\kappa ( \\mathcal { L } ( t + \\xi ) )$ , so we have ", + "bbox": [ + 173, + 580, + 609, + 599 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/83f4dc1988319f0e135ea298c68c1db54722cebfd2ccc210bc1930640b4d19ac.jpg", + "text": "$$\n\\mathcal { L } ( t + \\alpha ) \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\alpha C _ { \\eta } \\eta ( t ) \\kappa ( \\mathcal { L } ( t + \\xi ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 606, + 715, + 627 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Now we only need to show that $\\mathscr { L } ( t + \\alpha ) < \\mathscr { L } ( t )$ for all $\\alpha \\in ( 0 , A ]$ . Assuming this, we can have $\\kappa ( \\mathcal { L } ( t + \\xi ) ) \\leq \\kappa ( \\mathcal { L } ( t ) )$ by the monotonicity of $\\kappa$ , and thus ", + "bbox": [ + 171, + 638, + 823, + 670 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/90fb9d163295e531359ad0271e4b8aee6b96a0f7b1345e45296eb29ca8d4eaa5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } ( t + \\alpha ) \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\alpha C _ { \\eta } \\eta ( t ) \\kappa ( \\mathcal { L } ( t ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } } \\\\ & { \\qquad \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\mu ( \\mathcal { L } ( t ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 675, + 699, + 719 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Now we show that $\\mathcal { L } ( t + \\alpha ) ~ < ~ \\mathcal { L } ( t )$ for all $\\alpha \\in \\mathsf { \\Gamma } ( 0 , A ]$ . Assume to the contrary that $\\alpha _ { 0 } : =$ $\\operatorname* { i n f } \\{ \\alpha ^ { \\prime } \\in ( 0 , A ] : { \\mathcal { L } } ( t + \\alpha ^ { \\prime } ) \\geq { \\mathcal { L } } ( t ) \\}$ exists. If $\\alpha _ { 0 } = 0$ , let $p : [ 0 , 1 ] \\to \\mathbb { R } , \\alpha \\mapsto \\mathcal { L } ( t + \\alpha )$ , then $\\begin{array} { r } { p ^ { \\prime } ( 0 ) = \\operatorname* { l i m } _ { \\alpha \\downarrow 0 } \\frac { \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) } { \\alpha } \\geq 0 } \\end{array}$ L(t+α)−L(t) ≥ 0, but it contradicts to p0(0) = −η(t) k∇L(t)k22 < 0. If α0 > 0, $\\kappa$ have $\\kappa ( \\mathcal { L } ( t + \\xi ) ) < \\kappa ( \\dot { \\mathcal { L } } ( t ) )$ $\\xi \\in ( 0 , \\alpha _ { 0 } )$ thus ", + "bbox": [ + 173, + 729, + 826, + 791 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/78a72fa74328b662e81fda8ae4cfee9030f3e404a05693a1d60124b6373fcde4.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( t + \\alpha ) \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\mu ( \\mathcal { L } ( t ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } < \\mathcal { L } ( t ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 796, + 694, + 818 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "which leads to a contradiction. ", + "bbox": [ + 174, + 823, + 377, + 838 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof for $( \\mathbf { P 4 } )$ . We define $\\pmb { v } ( t ) : = \\hat { \\pmb { \\theta } } ( t ) \\hat { \\pmb { \\theta } } ( t ) ^ { \\top } ( - \\nabla \\mathcal { L } ( t ) )$ and $\\pmb { u } ( t ) : = \\left( \\pmb { I } - \\hat { \\pmb { \\theta } } ( t ) \\hat { \\pmb { \\theta } } ( t ) ^ { \\top } \\right) ( - \\nabla \\mathcal { L } ( t ) )$ similarly as in the analysis for gradient flow. For ${ \\pmb v } ( t )$ , we have ", + "bbox": [ + 173, + 852, + 825, + 890 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/b92fc95da92cc467c3f2cd46a13526f3347b1c4bf9e75014ac3007e2b217ccea.jpg", + "text": "$$\n\\| \\pmb { v } ( t ) \\| _ { 2 } = \\frac { 1 } { \\rho ( t ) } \\left. \\pmb { \\theta } ( t ) , - \\pmb { \\mathcal { L } } ( t ) \\right. = \\frac { 1 } { \\rho ( t ) } \\cdot L \\nu ( t ) .\n$$", + "text_format": "latex", + "bbox": [ + 343, + 895, + 651, + 929 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Decompose $\\left\\| \\nabla \\mathcal { L } ( t ) \\right\\| _ { 2 } ^ { 2 } = \\left\\| \\pmb { v } ( t ) \\right\\| _ { 2 } ^ { 2 } + \\left\\| \\pmb { u } ( t ) \\right\\| _ { 2 } ^ { 2 }$ . Then by (P3), we have ", + "bbox": [ + 173, + 99, + 629, + 121 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/7c9aaf127b18144c4b24a4eec566428ff4e5f9a23c1464022b4b997b660a7913.jpg", + "text": "$$\n\\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) \\leq - \\alpha \\eta ( t ) ( 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\left( \\frac { 1 } { \\rho ( t ) ^ { 2 } } \\cdot L ^ { 2 } \\nu ( t ) ^ { 2 } + \\| \\pmb { u } \\| _ { 2 } ^ { 2 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 261, + 125, + 735, + 161 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Multiplying 1+λ(L(t))µ(L(t))/L on both sides, we have ", + "bbox": [ + 173, + 172, + 531, + 196 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/73a889ce67684e9f3bf0117eb1719cac86d2189e2b6f7bdaa87052072288c303.jpg", + "text": "$$\n\\frac { \\iota + \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) / L } { ( 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\nu ( t ) } ( \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) ) \\leq - \\alpha \\eta ( t ) \\left( 1 + \\frac { \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) } { L } \\right) \\left( \\frac { L ^ { 2 } \\nu ( t ) } { \\rho ( t ) ^ { 2 } } + \\frac { \\| u \\| _ { 2 } ^ { 2 } } { \\nu ( t ) } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 181, + 202, + 823, + 243 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "By Corollary E.6, we can bound $\\nu ( t )$ by $\\nu ( t ) \\geq \\mathcal { L } ( t ) / \\lambda ( \\mathcal { L } ( t ) )$ . By (P2), we have the inequality $\\begin{array} { r } { \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) } { L } \\right) \\ge \\frac { \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } } { 2 L } } \\end{array}$ . So we further have ", + "bbox": [ + 173, + 250, + 825, + 287 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/e87b8b099422fa54774c2f4f7bd6994ec44dd74a88bb7dd95d2c9cc7e7a57135.jpg", + "text": "$$\n\\frac { 1 + \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) / L } { 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\mathcal { L } ( t ) / \\lambda ( \\mathcal { L } ( t ) ) } ( \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) ) \\leq - \\frac { 1 } { \\rho ( t ) ^ { 2 } } ( \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } ) \\left( \\frac { L } { 2 } + \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\| u \\| _ { 2 } ^ { 2 } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 181, + 295, + 826, + 330 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "From the definition φ, it is easy to see that −φ0(L(t)) ≥ 1+λ(L(t))µ(L(t))/L(1−µ(L(t)))L(t)/λ(L(t)) . Let $\\psi ( x ) = - \\log x .$ , then $\\begin{array} { r } { \\psi ^ { \\prime } ( x ) = - \\frac { 1 } { x } } \\end{array}$ . Combining these together gives ", + "bbox": [ + 173, + 337, + 825, + 375 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/a22eef9844716928d49ae2e19d2d41a45d1003cbd5cd08d7e4ebf219c1f34d4c.jpg", + "text": "$$\n\\phi ^ { \\prime } ( \\mathcal { L } ( t ) ) ( \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) ) + \\psi ^ { \\prime } ( \\rho ( t ) ^ { 2 } ) ( \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } ) \\left( \\frac { L } { 2 } + \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\left. \\boldsymbol { u } \\right. _ { 2 } ^ { 2 } \\right) \\geq 0\n$$", + "text_format": "latex", + "bbox": [ + 205, + 381, + 792, + 416 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Then by convexity of $\\phi$ and $\\psi$ , we have ", + "bbox": [ + 174, + 421, + 433, + 436 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/7b9c33f7365b54d05d135e8e630b77fe5470f1fc6f203b3993633338da462ee0.jpg", + "text": "$$\n\\left( \\phi ( \\mathcal { L } ( t + \\alpha ) ) - \\phi ( \\mathcal { L } ( t ) ) \\right) + \\left( \\log \\frac { 1 } { \\rho ( t + \\alpha ) ^ { 2 } } - \\log \\frac { 1 } { \\rho ( t ) ^ { 2 } } \\right) \\left( \\frac { L } { 2 } + \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\left\\| \\boldsymbol { u } \\right\\| _ { 2 } ^ { 2 } \\right) \\ge 0\n$$", + "text_format": "latex", + "bbox": [ + 210, + 443, + 787, + 478 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "And by definition of $\\hat { \\gamma }$ , this can be re-written as ", + "bbox": [ + 173, + 483, + 486, + 498 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/5d4ab5a58ba9a8fa3b99b4af7a49c29d8d7e017e4cbff4001677d3a3bdab56dd.jpg", + "text": "$$\n\\begin{array} { l } { \\log \\widehat { \\gamma } ( t + \\alpha ) - \\log \\widehat { \\gamma } ( t ) = \\left( \\phi ( \\mathcal { L } ( t + \\alpha ) ) - \\phi ( \\mathcal { L } ( t ) ) \\right) + L \\left( \\log \\displaystyle \\frac { 1 } { \\rho ( t + \\alpha ) } - \\log \\displaystyle \\frac { 1 } { \\rho ( t ) } \\right) } \\\\ { \\qquad \\geq - \\left( \\log \\displaystyle \\frac { 1 } { \\rho ( t + \\alpha ) ^ { 2 } } - \\log \\displaystyle \\frac { 1 } { \\rho ( t ) ^ { 2 } } \\right) \\displaystyle \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\left. u \\right. _ { 2 } ^ { 2 } } \\\\ { \\qquad = \\displaystyle \\frac { \\rho ( t ) ^ { 2 } } { L \\nu ( t ) ^ { 2 } } \\left. u \\right. _ { 2 } ^ { 2 } \\cdot \\log \\displaystyle \\frac { \\rho ( t + \\alpha ) } { \\rho ( t ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 503, + 782, + 613 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof for $( \\mathbf { P 1 } )$ . By (P4), $\\log \\hat { \\gamma } ( t + \\alpha ) \\geq \\log \\hat { \\gamma } ( t ) \\geq \\log \\hat { \\gamma } ( t _ { 0 } )$ . Note that $\\phi ( x ) \\geq \\log \\log \\frac { 1 } { x }$ . So we ", + "bbox": [ + 171, + 625, + 823, + 642 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/0e4a0d183ce8236e8e7a64f10fbcc37b1193e05ea26e02f7e53e2a0c859d3311.jpg", + "text": "$$\n\\tilde { \\gamma } ( t + \\alpha ) > \\hat { \\gamma } ( t + \\alpha ) \\geq \\hat { \\gamma } ( t _ { 0 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 397, + 651, + 599, + 669 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "which completes the proof. ", + "bbox": [ + 174, + 671, + 352, + 686 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "For showing the third proposition in Theorem E.2, we use (P1) to give a lower bound for $\\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 }$ , and use (P3) to show the speed of loss decreasing. Then it can be seen that $\\mathcal { L } ( t ) 0$ and $\\rho ( t ) $ $+ \\infty$ . By Lemma E.1, we then have $| \\bar { \\gamma } - \\hat { \\gamma } | \\to 0$ . ", + "bbox": [ + 173, + 700, + 825, + 744 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Lemma E.10. Let $\\begin{array} { r } { E _ { 0 } : = \\mathcal { L } ( t _ { 0 } ) ^ { 2 } ( \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ) ^ { 2 - 2 / L } } \\end{array}$ . Then for all $t > t _ { 0 }$ ", + "bbox": [ + 173, + 747, + 637, + 768 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/f645396e0220c7fed40bcf10891ebe26c16f7ea1e08868316f7f4d3af3b0e824.jpg", + "text": "$$\nE ( \\mathcal { L } ( t ) ) \\geq \\frac { 1 } { 2 } L ^ { 2 } \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) ~ f o r ~ E ( x ) : = \\int _ { x } ^ { \\mathcal { L } ( t _ { 0 } ) } \\frac { 1 } { \\operatorname* { m i n } \\{ u ^ { 2 } ( \\log \\frac { 1 } { u } ) ^ { 2 - 2 / L } , E _ { 0 } \\} } d u .\n$$", + "text_format": "latex", + "bbox": [ + 196, + 775, + 800, + 818 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Therefore, $\\mathcal { L } ( t ) 0$ and $\\rho ( t ) \\to + \\infty$ as $t \\to \\infty$ . ", + "bbox": [ + 174, + 824, + 498, + 840 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof. For any integer $t \\geq t _ { 0 }$ , $\\begin{array} { r } { \\mu ( \\mathcal { L } ( t ) ) \\leq \\frac { 1 } { 2 } } \\end{array}$ and $\\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } \\geq \\| \\pmb { v } ( t ) \\| _ { 2 }$ . Combining these with (P3), we have ", + "bbox": [ + 173, + 853, + 825, + 886 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/9eb663b924ad5ab9003e51e9429420374a61fc6e3b2d8f00ae0a6717681e52c0.jpg", + "text": "$$\n\\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\leq - \\frac { 1 } { 2 } \\eta ( t ) \\left. \\pmb { v } ( t ) \\right. _ { 2 } ^ { 2 } = - \\frac { 1 } { 2 } \\eta ( t ) \\frac { L ^ { 2 } \\nu ( t ) ^ { 2 } } { \\rho ( t ) ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 887, + 686, + 921 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "By (P1), $\\begin{array} { r } { \\rho ( t ) ^ { - 2 } \\leq \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { - 2 / L } } \\end{array}$ /L. By Corollary E.6, ν(t)2 ≥ L(t)2 \u0010log 1L(t) \u00112. Thus we have ", + "bbox": [ + 173, + 101, + 826, + 141 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/80583424360afeb17edd7cddcc971c3dac68aaa4ad5b9e81271adce99d4081d5.jpg", + "text": "$$\n\\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\leq - \\frac { 1 } { 2 } \\eta ( t ) \\cdot L ^ { 2 } \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\mathcal { L } ( t ) ^ { 2 } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { 2 - 2 / L } .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 138, + 718, + 176 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "It is easy to see that 1u2(log 1 )2−2/L is unimodal in (0, 1), so E0(x) is non-decreasing and E(x) is convex. So we have ", + "bbox": [ + 173, + 181, + 823, + 215 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/2f2970618c2bfe5c8bfd0294796f29cb3f4cf9996088db9691cd6a187a70588a.jpg", + "text": "$$\n\\begin{array} { r l } & { E ( \\mathcal { L } ( t + 1 ) ) - E ( \\mathcal { L } ( t ) ) \\geq E ^ { \\prime } ( \\mathcal { L } ( t ) ) \\left( \\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\right) } \\\\ & { \\qquad \\quad \\geq \\frac { 1 } { 2 } \\eta ( t ) \\cdot L ^ { 2 } \\widehat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\cdot \\frac { \\mathcal { L } ( t ) ^ { 2 } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { 2 - 2 / L } } { \\operatorname* { m i n } \\{ \\mathcal { L } ( t ) ^ { 2 } ( \\log \\frac { 1 } { \\mathcal { L } ( t ) } ) ^ { 2 - 2 / L } , E _ { 0 } \\} } } \\\\ & { \\qquad \\quad \\geq \\frac { 1 } { 2 } \\eta ( t ) \\cdot L ^ { 2 } \\widehat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 227, + 219, + 769, + 324 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "which proves $\\begin{array} { r } { E ( \\mathcal { L } ( t ) ) \\ge \\frac { 1 } { 2 } L ^ { 2 } \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}$ . Note that $\\mathcal { L }$ is non-decreasing. If $\\mathcal { L }$ does not decreases to 0, then neither does $E ( \\mathcal { L } )$ . But the RHS grows to $+ \\infty$ , which leads to a contradiction. So $\\mathcal { L } \\to 0$ . To make $\\mathcal { L } \\to 0$ , $q _ { \\mathrm { m i n } }$ must converge to $+ \\infty$ . So $\\rho \\to + \\infty$ . □ ", + "bbox": [ + 174, + 330, + 823, + 378 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "E.5 PROOF FOR THEOREM E.3 AND E.4 ", + "text_level": 1, + "bbox": [ + 176, + 395, + 464, + 410 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "The proofs for Theorem E.3 and E.4 are similar as those for Theorem A.8 and A.9 in Appendix C. ", + "bbox": [ + 169, + 421, + 816, + 438 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\beta ( t ) : = \\frac { 1 } { \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } } \\left. \\hat { \\pmb { \\theta } } , - \\nabla \\mathcal { L } ( t ) \\right. } \\end{array}$ as we do in Appendix C. It is easy to see that Lemma C.8 still holds if we replace $\\widetilde { \\gamma } ( t _ { 0 } )$ with $\\hat { \\gamma } ( t _ { 0 } )$ . So we only need to show $\\mathcal { L } \\to 0$ and $\\beta 1$ for proving convergence to KKT points. $\\mathcal { L } \\to 0$ can be followed from Theorem E.2. Similar as the proof for Lemma C.9, it follows from Lemma E.8 and (15) that for all $t _ { 2 } > t _ { 1 } \\geq t _ { 0 }$ , ", + "bbox": [ + 173, + 444, + 826, + 510 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/987d22e74afb6b535b7b0cb33eeb316a1b20c2bc66a8045bcfa0d45f18e51b63.jpg", + "text": "$$\n\\sum _ { t = t _ { 1 } } ^ { t _ { 2 } - 1 } \\left( \\beta ( t ) ^ { - 2 } - 1 \\right) \\cdot \\log \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } \\leq \\frac { 1 } { L } \\log \\frac { \\hat { \\gamma } ( t _ { 2 } ) } { \\hat { \\gamma } ( t _ { 1 } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 516, + 663, + 560 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Now we prove Theorem E.4. ", + "bbox": [ + 174, + 574, + 364, + 589 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Proof for Theorem E.4. We make the following changes in the proof for Theorem A.9. First, we replace $\\tilde { \\gamma } ( t _ { 0 } )$ with $\\hat { \\gamma } ( t _ { 0 } )$ , since $\\tilde { \\gamma } ( t )$ $\\left( t \\geq t _ { 0 } \\right)$ is lower bounded by $\\hat { \\gamma } ( t _ { 0 } )$ rather than $\\tilde { \\gamma } ( t _ { 0 } )$ . Second, when choosing $t _ { 1 }$ and $t _ { 2 }$ , we make $\\log \\rho ( t _ { 1 } )$ and $\\log \\rho ( t _ { 2 } )$ equal to the chosen values approximately with an additive error $o ( 1 )$ , rather than make them equal exactly. This is possible because it can be shown from (P2) in Lemma E.8 that the following holds: ", + "bbox": [ + 174, + 607, + 825, + 679 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/19d084ef2bf0ce91197061c49b69fba8700147822442605eaa552017e226493a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\rho ( t + 1 ) ^ { 2 } - \\rho ( t ) ^ { 2 } = O ( \\eta ( t ) \\nu ( t ) ) } \\\\ & { \\qquad = o ( \\kappa ( \\mathcal { L } ( t ) ) ^ { - 1 } \\cdot \\mathcal { L } ( t ) \\rho ( t ) ^ { L } ) = o \\left( \\frac { \\rho ( t ) ^ { L } } { ( \\log \\frac { 1 } { \\mathcal { L } ( t ) } ) ^ { 2 - 2 / L } } \\right) = o ( \\rho ( t ) ^ { - L + 2 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 684, + 792, + 748 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Dividing $\\rho ( t ) ^ { 2 }$ on the leftmost and rightmost sides, we have ", + "bbox": [ + 173, + 756, + 568, + 772 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/ef385e89d8ac940acf65c1c02a95badb997b78c3eb502f771e11f641d3d670ef.jpg", + "text": "$$\n\\rho ( t + 1 ) / \\rho ( t ) = 1 + o ( \\rho ( t ) ^ { - L } ) = 1 + o ( 1 ) ,\n$$", + "text_format": "latex", + "bbox": [ + 351, + 779, + 643, + 797 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "which implies that $\\log \\rho ( t + 1 ) - \\log \\rho ( t ) = o ( 1 )$ . Therefore, for any $R$ , we can always find the minimum time $t$ such that $\\log \\rho ( t ) \\geq R$ , and it holds for sure that $\\log \\rho ( t ) - R \\to 0$ as $R \\to + \\infty$ . ", + "bbox": [ + 171, + 804, + 825, + 835 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "For proving Theorem E.3, we also need the following lemma as a variant of Lemma C.11. ", + "bbox": [ + 171, + 852, + 761, + 868 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma E.11. For all $t \\geq t _ { 0 }$ ", + "bbox": [ + 173, + 872, + 369, + 887 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/04e07b017ded711eec7404460b58453680292024ccf2b986bd1ee2f80cf13b60.jpg", + "text": "$$\n\\left\\| \\hat { \\pmb \\theta } ( t + 1 ) - \\hat { \\pmb \\theta } ( t ) \\right\\| _ { 2 } \\leq \\left( \\frac { B _ { 1 } } { L \\tilde { \\gamma } ( t ) } \\cdot \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } + 1 \\right) \\log \\frac { \\rho ( t + 1 ) } { \\rho ( t ) }\n$$", + "text_format": "latex", + "bbox": [ + 295, + 893, + 700, + 929 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Proof. According to the update rule, we have ", + "bbox": [ + 174, + 103, + 473, + 118 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/1e83536a68da147fbf43e0db7553f754c84329ecf0f6347db38b29ccecdce645.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left\\| \\hat { \\theta } ( t + 1 ) - \\hat { \\theta } ( t ) \\right\\| _ { 2 } = \\displaystyle \\frac { 1 } { \\rho ( t + 1 ) } \\left\\| \\theta ( t + 1 ) - \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } \\pmb { \\theta } ( t ) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { 1 } { \\rho ( t + 1 ) } \\left( \\| \\theta ( t + 1 ) - \\theta ( t ) \\| _ { 2 } + \\left\\| \\left( \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } - 1 \\right) \\pmb { \\theta } ( t ) \\right\\| _ { 2 } \\right) } \\\\ & { \\qquad = \\displaystyle \\frac { \\eta ( t ) } { \\rho ( t + 1 ) } \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } + \\left( 1 - \\frac { \\rho ( t ) } { \\rho ( t + 1 ) } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 217, + 123, + 779, + 233 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "By (16), $\\begin{array} { r } { \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } \\le \\frac { B _ { 1 } \\nu ( t ) } { \\widetilde { \\gamma } ( t ) \\rho ( t ) } } \\end{array}$ . So we can bound the first term as ", + "bbox": [ + 171, + 236, + 599, + 258 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/b22a2de5f367e3d9f03863120c7ac46c7b27eeba4eb65e33116274a3f759edf5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\frac { \\eta ( t ) } { \\rho ( t + 1 ) } \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } \\le \\frac { B _ { 1 } } { \\widetilde { \\gamma } ( t ) } \\cdot \\frac { \\eta ( t ) \\nu ( t ) } { \\rho ( t + 1 ) \\rho ( t ) } \\le \\frac { B _ { 1 } } { \\widetilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) ^ { 2 } - \\rho ( t ) ^ { 2 } } { 2 L \\rho ( t + 1 ) \\rho ( t ) } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad = \\frac { B _ { 1 } } { 2 L \\widetilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) ^ { 2 } - \\rho ( t ) ^ { 2 } } { \\rho ( t + 1 ) ^ { 2 } } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\le \\frac { B _ { 1 } } { L \\widetilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } \\log \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 217, + 262, + 781, + 371 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "where the last inequality uses the inequality bound the second term by $\\begin{array} { r } { \\frac { a - b } { a } \\leq \\log ( a / b ) } \\end{array}$ . Using this inequality again, we can ", + "bbox": [ + 174, + 375, + 825, + 404 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/810d60710ce103b6ac97f702d5f73a7fd659607aed9bc0686c52e8f17301147f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad 1 - \\displaystyle \\frac { \\rho ( t ) } { \\rho ( t + 1 ) } \\leq \\log \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } . } \\\\ & { \\quad \\left\\| \\hat { \\pmb \\theta } ( t + 1 ) - \\hat { \\pmb \\theta } ( t ) \\right\\| _ { 2 } \\leq \\left( \\frac { B _ { 1 } } { L \\tilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } + 1 \\right) \\log \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 387, + 400, + 750, + 467 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Combining these together gives ", + "bbox": [ + 176, + 438, + 748, + 465 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Now we are ready to prove Theorem E.3. ", + "bbox": [ + 176, + 478, + 444, + 493 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Proof for Theorem E.3. As discussed above, we only need to show a variant of Lemma C.12 for gradient descent: for every limit point $\\bar { \\pmb { \\theta } }$ of $\\left\\{ { \\hat { \\theta } } ( t ) : t \\geq 0 \\right\\}$ , there exists a sequence of $\\{ t _ { m } : m \\in \\mathbb { N } \\}$ such that $t _ { m } \\uparrow + \\infty , \\hat { \\pmb \\theta } ( t _ { m } ) \\bar { \\pmb \\theta }$ , and $\\beta ( t _ { m } ) 1$ . ", + "bbox": [ + 173, + 507, + 825, + 564 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "We only need to change the choices of $s _ { m } , s _ { m } ^ { \\prime } , t _ { m }$ in the proof for Lemma C.12. We choose $s _ { m } >$ $t _ { m - 1 }$ to be a time such that ", + "bbox": [ + 174, + 569, + 821, + 598 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/629597a695ebe3b0c26369724e6c0c4851c5e766e141044aa84c43276bcf56a7.jpg", + "text": "$$\n\\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\| _ { 2 } \\leq \\epsilon _ { m } \\quad \\mathrm { a n d } \\quad \\frac { 1 } { L } \\log ( \\frac { \\operatorname* { l i m } _ { t + \\infty } \\hat { \\gamma } ( t ) } { \\hat { \\gamma } ( s _ { m } ) } ) \\leq \\epsilon _ { m } ^ { 3 } .\n$$", + "text_format": "latex", + "bbox": [ + 302, + 603, + 694, + 655 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Then we let $s _ { m } ^ { \\prime } > s _ { m }$ be the minimum time such that $\\log { \\rho ( s _ { m } ^ { \\prime } ) } \\ge \\log { \\rho ( s _ { m } ) } + \\epsilon _ { m }$ . According to Theorem E.2, $s _ { m }$ and $s _ { m } ^ { \\prime }$ must exist. Finally, we construct $t _ { m } \\in \\{ s _ { m } , \\ldots , s _ { m } ^ { \\prime } - 1 \\}$ to be a time that $\\beta ( t _ { m } ) ^ { - 2 } - 1 \\leq \\epsilon _ { m } ^ { 2 }$ , where the existence can be shown by (22). ", + "bbox": [ + 174, + 659, + 825, + 703 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "To see that this construction meets our requirement, note that $\\beta ( t _ { m } ) ^ { - 2 } - 1 \\le \\epsilon _ { m } ^ { 2 } \\to 0$ and ", + "bbox": [ + 173, + 708, + 767, + 724 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/3f93d9b25c0dde3e6b9639bf089f54f4edc0df6cc8f9aaaa925fb3b9b1bf4139.jpg", + "text": "$$\n\\Big \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\bar { \\pmb \\theta } \\Big \\| _ { 2 } \\leq \\Big \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\hat { \\pmb \\theta } ( s _ { m } ) \\Big \\| _ { 2 } + \\Big \\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\Big \\| _ { 2 } \\leq \\bigg ( \\frac { B _ { 1 } } { L \\hat { \\gamma } ( t _ { 0 } ) } e ^ { \\epsilon _ { m } } + 1 \\bigg ) \\cdot \\epsilon _ { m } + \\epsilon _ { m } 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 186, + 728, + 808, + 763 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "where the last inequality is by Lemma E.11. ", + "bbox": [ + 176, + 768, + 462, + 784 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "E.6 PROOF FOR THEOREM E.5 ", + "text_level": 1, + "bbox": [ + 174, + 799, + 400, + 815 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Proof. By a similar analysis as Lemma D.4, we have ", + "bbox": [ + 174, + 825, + 524, + 842 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/6bac32ce62dd4860ece8d882769d3d52b411a87de66fc87b777fcf0ff1975e0c.jpg", + "text": "$$\n\\begin{array} { r l r } { { E ( x ) = \\int _ { x } ^ { \\mathcal { L } ( t _ { 0 } ) } \\Theta ( \\frac { 1 } { u ^ { 2 } ( \\log \\frac { 1 } { u } ) ^ { 2 - 2 / L } } ) d u = \\int _ { 1 / \\mathcal { L } ( t _ { 0 } ) } ^ { 1 / x } \\Theta ( \\frac { 1 } { ( \\log u ) ^ { 2 - 2 / L } } ) d u } } \\\\ & { } & { = \\Theta ( \\frac { 1 } { x ( \\log \\frac { 1 } { x } ) ^ { 2 - 2 / L } } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 845, + 753, + 924 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "We can also bound the inverse function $E ^ { - 1 } ( y )$ by $\\Theta \\left( { \\frac { 1 } { y ( \\log y ) ^ { 2 - 2 / L } } } \\right)$ . With these, we can use a similar analysis as Theorem A.10 to prove Theorem E.5. ", + "bbox": [ + 174, + 101, + 825, + 138 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "First, using a similar proof as for (17), we have $\\rho ^ { L } = \\Theta ( \\log \\frac { 1 } { \\mathcal { L } } )$ . So we only need to show ${ \\mathcal { L } } ( t ) =$ $\\Theta \\big ( { \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } } \\big )$ . With a similar analysis as for (P3) in Lemma E.9, we have the following bound for $\\mathcal { L } ( \\tau + 1 ) - \\mathcal { L } ( \\tau )$ : ", + "bbox": [ + 173, + 143, + 825, + 194 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/4833b3bb9de80ed41571b6663bedda4d9a852b5d2d7e435f5dedb5905c0e7093.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\tau + 1 ) - \\mathcal { L } ( \\tau ) \\geq - \\eta ( \\tau ) ( 1 + \\mu ( \\mathcal { L } ( \\tau ) ) ) \\left. \\nabla \\mathcal { L } ( \\tau ) \\right. _ { 2 } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 202, + 678, + 220 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Using the fact that $\\mu \\leq 1 / 2$ , we have $\\begin{array} { r } { \\mathcal { L } ( \\tau + 1 ) - \\mathcal { L } ( \\tau ) \\geq - \\frac { 3 } { 2 } \\eta ( \\tau ) \\left. \\nabla \\mathcal { L } ( \\tau ) \\right. _ { 2 } ^ { 2 } } \\end{array}$ . By Lemma E.7, $\\| \\nabla \\mathcal { L } ( \\tau ) \\| _ { 2 } \\le 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\tau ) ) \\mathcal { L } ( \\tau )$ . Using a similar proof as for Lemma E.10, we can show that $E ( \\mathcal { L } ( t ) ) \\ \\stackrel { } { \\leq } \\ O ( T )$ . Combining this with Lemma E.10, we have $E ( \\mathcal { L } ( t ) ) ~ = ~ \\Theta ( T )$ . Therefore, $\\begin{array} { r } { \\mathcal { L } ( t ) = \\Theta \\big ( \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } \\big ) } \\end{array}$ . □ ", + "bbox": [ + 173, + 228, + 826, + 291 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "F GRADIENT DESCENT: GENERAL LOSS FUNCTIONS ", + "text_level": 1, + "bbox": [ + 174, + 306, + 630, + 324 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "It is worth to note that the above analysis can be extended to other loss functions. For this, we need to replace (B3) with a strong assumption (S3), which takes into account the second-order derivatives of $f$ . ", + "bbox": [ + 173, + 338, + 826, + 381 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "(S3). The loss function $\\ell ( q )$ can be expressed as $\\ell ( q ) = e ^ { - f ( q ) }$ such that ", + "bbox": [ + 191, + 392, + 671, + 409 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "(S3.1). $f : \\mathbb { R } \\to \\mathbb { R }$ is $\\mathcal { C } ^ { 2 }$ -smooth. ", + "bbox": [ + 235, + 411, + 452, + 426 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "(S3.2). $f ^ { \\prime } ( q ) > 0$ for all $q \\in \\mathbb { R }$ . ", + "bbox": [ + 233, + 429, + 444, + 444 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "(S3.3). There exists $b _ { f } \\geq 0$ such that $f ^ { \\prime } ( q ) q$ is non-decreasing for $q \\in ( b _ { f } , + \\infty )$ , and $f ^ { \\prime } ( q ) q \\to + \\infty$ as $q \\to + \\infty$ . ", + "bbox": [ + 228, + 445, + 823, + 474 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "(S3.4). Let $g \\ : \\ [ f ( b _ { f } ) , + \\infty ) \\ \\ [ b _ { f } , + \\infty )$ be the inverse function of $f$ on the domain $[ b _ { f } , + \\infty )$ . There exists $p \\geq 0$ such that for all $x > f ( b _ { f } ) , y > b _ { f }$ , ", + "bbox": [ + 240, + 477, + 823, + 506 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/19221d8c5f4383da742c9294e37b08def7e22fe213443ac5a840b50ba551ac1d.jpg", + "text": "$$\n\\left| { \\frac { g ^ { \\prime \\prime } ( x ) } { g ^ { \\prime } ( x ) } } \\right| \\leq { \\frac { p } { x } } \\quad { \\mathrm { a n d } } \\quad \\left| { \\frac { f ^ { \\prime \\prime } ( y ) } { f ^ { \\prime } ( y ) } } \\right| \\leq { \\frac { p } { y } } .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 511, + 676, + 547 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "It can be verified that (S3) is satisfied by exponential loss and logistic loss. Now we explain each of the assumptions in (S3). (S3.2) and (S3.3) are essentially the same as (B3.2) and (B3.3). (S3.1) and (B3.1) are the same except that (S3.1) assumes $f$ is $\\mathcal { C } ^ { 2 }$ -smooth rather than ${ \\mathcal { C } } ^ { 1 }$ -smooth. ", + "bbox": [ + 173, + 556, + 826, + 599 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "(S3.4) can also be written in the following form: ", + "bbox": [ + 174, + 604, + 491, + 621 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/9c68847be24d355830261446fe2dd4337cfe6393449001bb700e8de41317834c.jpg", + "text": "$$\n\\left| { \\frac { d } { d x } } \\log g ^ { \\prime } ( x ) \\right| \\leq p \\cdot { \\frac { d } { d x } } \\log x \\quad { \\mathrm { a n d } } \\quad \\left| { \\frac { d } { d y } } \\log f ^ { \\prime } ( y ) \\right| \\leq p \\cdot { \\frac { d } { d y } } \\log y .\n$$", + "text_format": "latex", + "bbox": [ + 272, + 626, + 727, + 662 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "That is, $\\log g ^ { \\prime } ( x )$ and $\\log f ^ { \\prime } ( y )$ grow no faster than $O ( \\log x )$ and $O ( \\log y )$ , respectively. In fact, (B3.4) can be deduced from (S3.4). Recall that (B3.4) ensures that ${ \\Theta } \\bar { ( g ^ { \\prime } ( x ) ) \\ } \\stackrel { - } { = } \\ g ^ { \\prime } ( \\stackrel { . } { \\Theta } ( x ) )$ and $\\Theta ( f ^ { \\prime } ( y ) ) = f ^ { \\prime } ( \\Theta ( y ) )$ . Thus, (S3.4) also gives us the interchangeability between $f ^ { \\prime } , g ^ { \\prime }$ and $\\Theta$ . ", + "bbox": [ + 173, + 667, + 826, + 712 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma F.1. (S3.4) implies (B3.4) with $b _ { g } = \\operatorname* { m a x } \\{ 2 f ( b _ { f } ) , f ( 2 b _ { f } ) \\}$ and $K = 2 ^ { p }$ . ", + "bbox": [ + 174, + 714, + 710, + 732 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. Fix $x \\in ( b _ { g } , + \\infty ) , y \\in ( g ( b _ { g } ) , + \\infty )$ and $\\theta \\in [ 1 / 2 , 1 )$ . Integrating (23) on both sides of the inequalities from $\\theta x$ to $x$ and $\\theta y$ to $y$ , we have ", + "bbox": [ + 173, + 746, + 823, + 775 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/60f4ae8b4640b186ededf3922e9f8301286e8b556d8c88cf7798cd093936b682.jpg", + "text": "$$\n| \\log g ^ { \\prime } ( x ) - \\log g ^ { \\prime } ( \\theta x ) | \\leq p \\cdot \\log { \\frac { 1 } { \\theta } } \\quad { \\mathrm { a n d } } \\quad | \\log f ^ { \\prime } ( y ) - \\log f ^ { \\prime } ( \\theta y ) | \\leq p \\cdot \\log { \\frac { 1 } { \\theta } } .\n$$", + "text_format": "latex", + "bbox": [ + 233, + 781, + 766, + 811 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Therefore, we have $g ^ { \\prime } ( x ) \\leq \\theta ^ { - p } g ^ { \\prime } ( \\theta x ) \\leq K g ^ { \\prime } ( \\theta x )$ and $f ^ { \\prime } ( y ) \\leq \\theta ^ { - p } f ^ { \\prime } ( \\theta y ) \\leq K f ^ { \\prime } ( \\theta x ) ,$ . ", + "bbox": [ + 171, + 818, + 754, + 835 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "To extend our results to general loss functions, we need to redefine $\\kappa ( x ) , \\lambda ( x ) , \\phi ( x ) , C _ { \\eta } , H ( x )$ in order. For $\\kappa ( x )$ and $\\lambda ( x )$ , we redefine them as follows: ", + "bbox": [ + 173, + 849, + 830, + 878 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/b21d2113bed7f553675a58b92a5cc488a5ff98be53adf0938cf3b05f219cf04d.jpg", + "text": "$$\n\\kappa ( x ) : = \\operatorname* { s u p } \\left\\{ \\frac { w ( \\log \\frac { 1 } { w } ) ^ { 2 - 2 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { w } ) ^ { 2 } } : w \\in ( 0 , x ] \\right\\} \\qquad \\lambda ( x ) : = \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { x } ) } { g ( \\log \\frac { 1 } { x } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 258, + 886, + 738, + 929 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "By Lemma D.1, $\\begin{array} { r } { \\frac { w ( \\log \\frac { 1 } { w } ) ^ { 2 - 2 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { w } ) ^ { 2 } } = O \\left( w ( \\log \\frac { 1 } { w } ) ^ { 4 - 2 / L } / g ( \\log \\frac { 1 } { w } ) ^ { 2 } \\right) \\to 0 } \\end{array}$ . So $\\kappa ( x )$ is well-defined. \nUsing $\\lambda ( x )$ , we can define $\\phi ( x )$ and $\\hat { \\gamma } ( \\pmb \\theta )$ as follows. ", + "bbox": [ + 174, + 99, + 825, + 140 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/b2233ecccbab3a94fb199d3489c179e9d15a45110c22cf1582d9e7db4f46f7cc.jpg", + "text": "$$\n\\begin{array} { r l } & { \\phi ^ { \\prime } ( x ) : = - \\operatorname* { s u p } \\left\\{ \\displaystyle \\frac { \\lambda ( w ) } { w } \\left( 1 + 2 ( 1 + \\lambda ( w ) / L ) \\mu ( w ) \\right) : w \\in [ x , \\mathcal { L } ( t _ { 0 } ) ] \\right\\} } \\\\ & { \\displaystyle \\phi ( x ) : = \\log \\left( g ( \\log \\frac { 1 } { x } ) \\right) + \\int _ { 0 } ^ { x } \\left( \\phi ^ { \\prime } ( w ) + \\frac { \\lambda ( w ) } { w } \\right) d w } \\\\ & { \\displaystyle \\hat { \\gamma } ( \\pmb { \\theta } ) : = \\frac { e ^ { \\phi ( \\mathcal { L } ( \\pmb { \\theta } ) ) } } { \\rho ^ { L } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 145, + 730, + 252 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Using a similar argument as in Lemma E.1, we can show that $\\hat { \\gamma } ( \\pmb \\theta )$ is well-defined and $\\hat { \\gamma } ( \\pmb \\theta ) ~ <$ $\\begin{array} { r } { \\tilde { \\gamma } ( \\pmb { \\theta } ) : = g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\breve { / } \\rho ^ { L } \\leq \\bar { \\gamma } ( \\pmb { \\theta } ) } \\end{array}$ . When ${ \\mathcal { L } } ( \\pmb { \\theta } ) \\to 0$ , we also have $\\hat { \\gamma } ( \\pmb { \\theta } ) / \\bar { \\gamma } ( \\pmb { \\theta } ) 1$ . ", + "bbox": [ + 171, + 253, + 823, + 285 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "For $C _ { \\eta }$ , we define it to be the following. ", + "bbox": [ + 173, + 290, + 437, + 305 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/ec4bb9e70cec3464c55ef0cd792ecb8d9a5a2e02759e0e0dfed7ae19882259bc.jpg", + "text": "$$\n\\begin{array} { r } { \\gamma _ { \\eta } : = \\frac { 1 } { 2 } B _ { 1 } \\left( \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } B _ { 1 } + \\frac { 2 ^ { p + 1 } } { \\log \\frac { 1 } { C ( t _ { 0 } ) } } ( p B _ { 1 } + B _ { 2 } ) \\right) \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } \\operatorname* { m i n } \\left\\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 309, + 828, + 352 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Finally, the definitions for $\\begin{array} { r } { \\mu ( x ) : = ( \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ) / ( 2 \\log \\frac { 1 } { x } ) } \\end{array}$ and $H ( x ) : = \\mu ( x ) / ( C _ { \\eta } \\kappa ( x ) )$ in (S5) remain unchanged except that $\\kappa ( x )$ and $C _ { \\eta }$ now use the new definitions. ", + "bbox": [ + 174, + 356, + 823, + 391 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Similar as gradient flow, we define $\\begin{array} { r } { \\nu ( t ) : = \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ( t ) ) } f ^ { \\prime } ( q _ { n } ( t ) ) q _ { n } ( t ) } \\end{array}$ . The key idea behind the above definitions is that we can prove similar bounds for $\\nu ( t ) , \\| \\nabla \\mathcal { L } \\| _ { 2 } , \\| \\nabla ^ { 2 } \\mathcal { L } \\| _ { 2 }$ as Corollary E.6 and Lemma E.7. ", + "bbox": [ + 173, + 396, + 826, + 444 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Lemma F.2. $\\langle \\pmb \\theta ( t ) , - \\nabla \\mathcal L ( t ) \\rangle = L \\nu ( t ) .$ . If $\\mathcal { L } ( t ) < e ^ { - f ( b _ { f } ) }$ , then $\\begin{array} { r } { \\nu ( t ) \\geq \\frac { \\mathcal { L } ( t ) } { \\lambda ( \\mathcal { L } ( t ) ) } } \\end{array}$ and $\\lambda ( \\mathcal { L } ( t ) )$ has the lower bound $\\begin{array} { r } { \\lambda ( \\mathcal { L } ( t ) ) \\leq 2 ^ { p + 1 } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { - 1 } } \\end{array}$ . ", + "bbox": [ + 173, + 445, + 825, + 493 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Proof. It can be easily proved by combining Theorem B.4, Lemma B.5 and Lemma D.1 together. □ ", + "bbox": [ + 173, + 505, + 823, + 535 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Lemma F.3. For any $\\pmb \\theta$ , if $\\mathcal { L } ( \\pmb \\theta ) \\leq \\mathcal { L } ( t _ { 0 } ) , \\tilde { \\gamma } ( \\pmb \\theta ) \\geq \\hat { \\gamma } ( t _ { 0 } )$ , then ", + "bbox": [ + 171, + 541, + 580, + 559 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/5e3aa293e04ba12492267f2503a2228a2c9280b89b0018630b475b8731c12dca.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\nabla \\mathcal { L } ( \\pmb { \\theta } ) \\| _ { 2 } ^ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) \\mathcal { L } ( \\pmb { \\theta } ) \\quad a n d \\quad \\left\\| \\nabla ^ { 2 } \\mathcal { L } ( \\pmb { \\theta } ) \\right\\| _ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 561, + 730, + 582 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Proof. Note that a direct corollary of (23) is that $f ^ { \\prime } ( x _ { 1 } ) \\leq f ^ { \\prime } ( x _ { 2 } ) ( x _ { 1 } / x _ { 2 } ) ^ { p }$ for $x _ { 1 } \\geq x _ { 2 } > b _ { f }$ . So we have ", + "bbox": [ + 174, + 594, + 825, + 625 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/73613c208a21c367087550ae5413da2b7d66550b4522a0baeea4c6ac3e6c12b2.jpg", + "text": "$$\nf ^ { \\prime } ( q _ { n } ) \\leq f ^ { \\prime } \\left( g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) \\right) \\left( { \\frac { q _ { n } } { g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } \\right) ^ { p } = { \\frac { 1 } { g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } \\left( { \\frac { q _ { n } / \\rho ^ { L } } { \\widetilde { \\gamma } } } \\right) ^ { p } \\leq { \\frac { R } { g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } ,\n$$", + "text_format": "latex", + "bbox": [ + 235, + 626, + 761, + 665 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "where $R : = ( B _ { 0 } / \\hat { \\gamma } ( t _ { 0 } ) ) ^ { p }$ . Applying (S3.4), we can also deduce that ", + "bbox": [ + 174, + 667, + 622, + 685 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/25293948d1d5ce46b2034a1543169fe865fb0827e643f8636917f4c19324091e.jpg", + "text": "$$\n| f ^ { \\prime \\prime } ( q _ { n } ) | \\leq { \\frac { p } { q _ { n } } } \\cdot f ^ { \\prime } ( q _ { n } ) \\leq { \\frac { p R } { g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 688, + 651, + 724 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Now we bound $\\lVert \\nabla \\mathcal { L } \\rVert _ { 2 }$ and $\\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 }$ . By the chain rule, we have ", + "bbox": [ + 173, + 728, + 602, + 746 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/63f9e0fdb6110c12727f7233454a668cd4590f7bc7d3ef7a807c85fdcca98eb4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left\\| \\nabla \\mathcal { L } \\right\\| _ { 2 } = \\left\\| - \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } f ^ { \\prime } ( \\boldsymbol { q } _ { n } ) \\nabla \\boldsymbol { q } _ { n } \\right\\| _ { 2 } \\leq B _ { 1 } R \\cdot \\frac { \\mathcal { L } \\rho ^ { L - 1 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } } \\\\ & { \\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } \\left( \\big ( f ^ { \\prime } ( \\boldsymbol { q } _ { n } ) ^ { 2 } - f ^ { \\prime \\prime } ( \\boldsymbol { q } _ { n } ) \\big ) \\nabla q _ { n } \\nabla q _ { n } ^ { \\top } - f ^ { \\prime } ( q _ { n } ) \\nabla ^ { 2 } q _ { n } \\right) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } \\left( \\left( \\frac { R ^ { 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } + \\frac { p R } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\right) B _ { 1 } ^ { 2 } \\rho ^ { 2 L - 2 } + \\frac { R } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } B _ { 2 } \\rho ^ { L - 2 } \\right) } \\\\ & { \\qquad \\leq \\frac { \\mathcal { L } \\rho ^ { 2 L - 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\left( \\left( R + p \\lambda ( \\mathcal { L } ) \\right) B _ { 1 } ^ { 2 } + \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\rho ^ { L } } B _ { 2 } \\right) R . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 183, + 751, + 816, + 925 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "For $\\rho$ we have $\\begin{array} { r } { \\hat { \\gamma } ( t _ { 0 } ) \\rho ^ { L } \\le \\log \\frac { 1 } { \\mathcal L } \\le B _ { 0 } \\rho ^ { L } , } \\end{array}$ , so we can bound $\\rho ^ { 2 L - 2 }$ by $\\rho ^ { 2 L - 2 } \\leq M ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ^ { 2 L - 2 }$ , where ${ \\cal M } : = \\operatorname * { m i n } \\left\\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\}$ . By Lemma F.2, we have $\\begin{array} { r } { \\lambda ( \\mathcal { L } ) \\ \\leq \\ 2 ^ { p + 1 } \\frac { 1 } { \\log \\frac { 1 } { \\mathcal { L } } } \\ \\leq \\ } \\end{array}$ $\\frac { 2 ^ { p + 1 } } { \\log { \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } }$ , and also $\\begin{array} { r } { \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\rho ^ { L } } = \\lambda ( \\mathcal { L } ) \\cdot \\tilde { \\gamma } \\leq \\frac { 2 ^ { p + 1 } B _ { 1 } } { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } } \\end{array}$ . Combining all these formulas together gives ", + "bbox": [ + 173, + 101, + 826, + 170 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/94c47cfd0cd7aac7ab543d51c14820eb758303a09fcd837db2e3d5684acfa512.jpg", + "text": "$$\n\\begin{array} { r l r } & { \\| \\nabla \\mathcal { L } \\| _ { 2 } ^ { 2 } \\leq B _ { 1 } ^ { 2 } R ^ { 2 } M \\displaystyle \\frac { \\mathcal { L } ^ { 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\left( \\log \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta ^ { \\kappa } } ( \\mathcal { L } ) \\mathcal { L } } & \\\\ & { \\big | \\nabla ^ { 2 } \\mathcal { L } \\big \\| _ { 2 } \\leq \\left( \\left( R + \\frac { p 2 ^ { p + 1 } } { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } \\right) B _ { 1 } ^ { 2 } + \\frac { 2 ^ { p + 1 } B _ { 1 } B _ { 2 } } { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } \\right) R M \\displaystyle \\frac { \\mathcal { L } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\left( \\log \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta ^ { \\kappa } } ( \\mathcal { L } ) , } & \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 174, + 825, + 260 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "which completes the proof. ", + "bbox": [ + 174, + 261, + 354, + 276 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "With Corollary E.6 and Lemma E.7, we can prove Lemma E.8 with exactly the same argument. Then Theorem E.2, E.3, E.4 can also be proved similarly. For Theorem E.5, we can follow the argument for gradient flow to show that it holds with slightly different tight bounds: ", + "bbox": [ + 174, + 290, + 825, + 334 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Theorem F.4. Under assumptions (S1), (A2), (S3), (A4), (S5), we have the following tight rates for training loss and weight norm: ", + "bbox": [ + 173, + 337, + 823, + 366 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/19d3c481f8f5f29d6dbf2aaa60834dec89dc6a726bf5949e087976f458eabef6.jpg", + "text": "$$\n\\mathcal { L } ( t ) = \\Theta \\left( \\frac { g ( \\log T ) ^ { 2 / L } } { T ( \\log T ) ^ { 2 } } \\right) \\quad a n d \\quad \\rho ( t ) = \\Theta ( g ( \\log T ) ^ { 1 / L } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 300, + 371, + 696, + 406 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { T = \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}$ ", + "bbox": [ + 174, + 411, + 336, + 431 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "G EXTENSION: MULTI-CLASS CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 173, + 449, + 584, + 465 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "In this section, we generalize our results to multi-class classification with cross-entropy loss. This part of analysis is inspired by Theorem 1 in (Zhang et al., 2019), which gives a lower bound for the gradient in terms of the loss $\\mathcal { L }$ . ", + "bbox": [ + 173, + 479, + 825, + 522 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Since now a neural network has multiple outputs, we need to redefine our notations. Let $C$ be the number of classes. The output of a neural network $\\Phi$ is a vector $\\Phi ( \\pmb { \\theta } ; \\pmb { x } ) \\in \\mathbb { R } ^ { C }$ . We use $\\Phi _ { j } ( \\pmb { \\theta } ; \\pmb { x } ) \\in \\mathbb { R }$ to denote the $j$ -th output of $\\Phi$ on the input $\\textbf { \\em x } \\in \\mathbb { R } ^ { d _ { \\mathrm { { x } } } }$ . A dataset is denoted by $\\bar { \\mathcal { D } } = \\{ \\pmb { x } _ { n } , y _ { n } \\} _ { n = 1 } ^ { N } = \\{ ( \\pmb { x } _ { n } , y _ { n } ) : n \\in [ N ] \\}$ , where on the d $\\pmb { x } _ { n } \\in \\mathbb { R } ^ { d _ { \\mathbf { x } } }$ is a data input and defined as $y _ { n } \\in [ C ]$ is the $\\Phi$ $\\mathcal { D }$ ", + "bbox": [ + 173, + 529, + 825, + 602 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/767e11d63b44c6dae3d317aeda636a07489924812f02e176285a4a6d6a4b60d3.jpg", + "text": "$$\n\\mathcal { L } ( \\pmb \\theta ) : = \\sum _ { n = 1 } ^ { N } - \\log \\frac { e ^ { \\Phi _ { y _ { n } } ( \\pmb \\theta ; \\pmb x _ { n } ) } } { \\sum _ { j = 1 } ^ { C } e ^ { \\Phi _ { j } ( \\pmb \\theta ; \\pmb x _ { n } ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 606, + 620, + 650 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "The margin for a single data point $( { \\pmb x } _ { n } , y _ { n } )$ is defined to be $\\begin{array} { r c l } { q _ { n } ( \\pmb { \\theta } ) } & { : = } & { \\Phi _ { y _ { n } } ( \\pmb { \\theta } ; \\pmb { x } _ { n } ) - } \\end{array}$ $\\operatorname* { m a x } _ { j \\neq y _ { n } } \\bar { \\{ \\Phi _ { j } ( \\pmb { \\theta } ; \\pmb { x } _ { n } ) \\} }$ , and the margin for the entire dataset is defined to be $q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) =$ $\\mathrm { m i n } _ { n \\in [ N ] } q _ { n } ( \\pmb \\theta )$ . We define the normalized margin to be $\\bar { \\gamma } ( \\pmb \\theta ) : = q _ { \\mathrm { m i n } } ( \\pmb \\theta ) = q _ { \\mathrm { m i n } } ( \\pmb \\theta ) / \\rho ^ { L }$ , where $\\rho : = \\| \\pmb \\theta \\| _ { 2 }$ and $\\hat { \\pmb \\theta } : = \\pmb \\theta / \\rho \\in S ^ { d - 1 }$ as usual. ", + "bbox": [ + 173, + 655, + 825, + 718 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Let $\\ell ( q ) : = \\log ( 1 + e ^ { - q } )$ be the logistic loss. Recall that $\\ell ( q )$ satisfies (B3). Let $f ( q ) = - \\log \\ell ( q ) =$ $- \\log \\log ( 1 + e ^ { - q } )$ . Let $g$ be the inverse function of $f$ . So $g ( q ) = - \\log ( e ^ { e ^ { - q } } - 1 )$ . ", + "bbox": [ + 173, + 724, + 823, + 758 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "The cross-entropy loss can be rewritten in other ways. Let $s _ { n j } : = \\Phi _ { y _ { n } } ( \\pmb { \\theta } ; \\pmb { x } _ { n } ) - \\Phi _ { j } ( \\pmb { \\theta } ; \\pmb { x } _ { n } )$ . Let $\\begin{array} { r } { \\widetilde { q } _ { n } : = - \\mathrm { L S E } ( \\{ - s _ { n j } : j \\neq y _ { n } \\} ) = - \\log \\left( \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } \\right) } \\end{array}$ . Then ", + "bbox": [ + 174, + 762, + 825, + 803 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/526f356b171e1d20d297ecb616559c62685f08fbf0c7f5dfdd205411ac71340e.jpg", + "text": "$$\n\\mathcal { L } ( \\pmb { \\theta } ) : = \\sum _ { n = 1 } ^ { N } \\log \\left( 1 + \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } \\right) = \\sum _ { n = 1 } ^ { N } \\log ( 1 + e ^ { - \\tilde { q } _ { n } } ) = \\sum _ { n = 1 } ^ { N } \\ell ( \\tilde { q } _ { n } ) = \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\tilde { q } _ { n } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 209, + 808, + 784, + 858 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Gradient Flow. For gradient flow, we assume the following: ", + "bbox": [ + 173, + 869, + 583, + 885 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "(M1). (Regularity). For any fixed $_ { \\textbf { \\em x } }$ and $j \\in [ C ] , \\Phi _ { j } ( \\cdot ; x )$ is locally Lipschitz and admits a chain rule; ", + "bbox": [ + 181, + 893, + 826, + 924 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "(M2). (Homogeneity). There exists $L \\ > \\ 0$ such that $\\forall j ~ \\in ~ [ C ] , \\forall \\alpha ~ > ~ 0 ~ : ~ \\Phi _ { j } ( \\alpha \\pmb \\theta ; \\pmb x ) ~ =$ $\\alpha ^ { L } \\Phi _ { j } ( \\pmb { \\theta } ; \\pmb { x } )$ ; ", + "bbox": [ + 179, + 102, + 825, + 135 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "(M3). (Cross-entropy Loss). $\\mathcal { L } ( \\pmb { \\theta } )$ is defined as the cross-entropy loss on the training set; ", + "bbox": [ + 186, + 138, + 774, + 155 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "(M4). (Separability). There exists a time $t _ { 0 }$ such that $\\mathcal { L } ( t _ { 0 } ) < \\log 2$ . ", + "bbox": [ + 186, + 159, + 637, + 175 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "$\\mathcal { L } < \\log 2$ , then $\\begin{array} { r } { \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } < 1 } \\end{array}$ for all $n \\in [ N ]$ , and thus $s _ { n j } > 0$ for all $n \\in [ N ] , j \\in [ C ]$ . So (M4) ensures the separability of training data. ", + "bbox": [ + 178, + 184, + 823, + 217 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Definition G.1. For cross-entropy loss, the smoothed normalized margin $\\widetilde \\gamma ( \\pmb \\theta )$ of $\\pmb \\theta$ is defined as ", + "bbox": [ + 173, + 220, + 810, + 237 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/c4580bf722dd3b82530e20221db120f440274e72452dae9c0e1287f8c45c5c55.jpg", + "text": "$$\n\\widetilde { \\gamma } ( \\pmb \\theta ) : = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\rho ^ { L } } = \\frac { - \\log ( e ^ { \\mathcal { L } } - 1 ) } { \\rho ^ { L } } ,\n$$", + "text_format": "latex", + "bbox": [ + 379, + 243, + 617, + 280 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "where $\\ell ^ { - 1 } ( \\cdot )$ is the inverse function of the logistic loss $\\ell ( \\cdot )$ . ", + "bbox": [ + 173, + 286, + 565, + 303 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Theorem 4.1 and 4.4 still hold. Here we redefine the optimization problem (P) to be ", + "bbox": [ + 173, + 314, + 723, + 330 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/4eea707e36a5a57fd8bc00c5d254c4a50c40eedf8a156a0885388a87eb0183a0.jpg", + "text": "$$\n\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\mathrm { s . t . } } & { s _ { n j } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ] , \\forall j \\in [ C ] \\setminus \\{ y _ { n } \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 335, + 665, + 387 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Most of our proofs are very similar as before. Here we only show the proof for the generalized version of Lemma 5.1. ", + "bbox": [ + 173, + 392, + 823, + 421 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Lemma G.2. Lemma 5.1 is also true for the smoothed normalized margin $\\tilde { \\gamma }$ defined in Definition G.1. ", + "bbox": [ + 169, + 425, + 825, + 455 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Proof. Define $\\nu ( t )$ by the following formula: ", + "bbox": [ + 173, + 473, + 472, + 489 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/a3a6ef5aea6f1853718c2f6cae40809c10d680847a84088d87e6016c2c48144c.jpg", + "text": "$$\n\\nu ( t ) : = \\sum _ { n = 1 } ^ { N } \\frac { \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } s _ { n j } } { 1 + \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } } .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 497, + 604, + 541 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Using a similar argument as in Theorem B.4, it can be proved that $\\textstyle { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } = L \\nu ( t )$ for a.e. $t > 0$ . ", + "bbox": [ + 171, + 549, + 799, + 570 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "It can be shown that Lemma B.5, which asserts that $\\begin{array} { r } { \\nu ( t ) \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } } \\end{array}$ , still holds for this new definition of $\\nu ( t )$ . By definition, $s _ { n j } \\geq q _ { n }$ for $j \\neq y _ { n }$ . Also note that $\\tilde { e } ^ { - \\tilde { q } _ { n } } \\geq e ^ { - q _ { n } }$ . So $s _ { n j } \\geq q _ { n } \\geq$ $\\tilde { q } _ { n }$ . Then we have ", + "bbox": [ + 173, + 575, + 825, + 630 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/894d2b9a3b10d6d26717c4b6e7efa95bda05ec6943f4ea768d45a6c7cd5321bc.jpg", + "text": "$$\n\\nu ( t ) \\geq \\sum _ { n = 1 } ^ { N } \\frac { \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } } { 1 + \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } } \\cdot \\tilde { q } _ { n } = \\sum _ { n = 1 } ^ { N } \\frac { e ^ { - \\tilde { q } _ { n } } } { 1 + e ^ { - \\tilde { q } _ { n } } } \\cdot \\tilde { q } _ { n } = \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\tilde { q } _ { n } ) } f ^ { \\prime } ( \\tilde { q } _ { n } ) \\tilde { q } _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 636, + 764, + 680 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Note that $\\begin{array} { r } { \\mathcal { L } = \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\tilde { q } _ { n } ) } } \\end{array}$ . Then using Lemma B.5 for logistic loss can conclude that $\\nu ( t ) \\geq$ $\\begin{array} { r } { \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } } \\end{array}$ ", + "bbox": [ + 174, + 688, + 823, + 729 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "The rest of the proof for this lemma is exactly the same as that for Lemma 5.1. ", + "bbox": [ + 174, + 734, + 689, + 751 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Gradient Descent. For gradient descent, we only need to replace (M1) with (S1) and make the assumption (S5) on the learning rate. ", + "bbox": [ + 174, + 768, + 826, + 799 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "(S1). (Smoothness). For any fixed $_ { \\textbf { \\em x } }$ and $j \\in [ C ] , \\Phi _ { j } ( \\cdot ; x )$ is $\\mathcal { C } ^ { 2 }$ -smooth; \n(S5). (Learing rate condition). $\\begin{array} { r } { \\sum _ { t \\geq 0 } \\eta ( t ) = + \\infty } \\end{array}$ and $\\eta ( t ) \\leq H ( \\mathcal { L } ( \\pmb { \\theta } ( t ) ) )$ . ", + "bbox": [ + 187, + 809, + 684, + 848 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Here $H ( x ) : = \\mu ( x ) / ( C _ { \\eta } \\kappa ( x ) )$ is defined to be the same as in Appendix $\\mathrm { F }$ (when $\\ell ( \\cdot )$ is set to logistic loss) except that we use another $C _ { \\eta }$ which will be specified later. ", + "bbox": [ + 176, + 858, + 823, + 890 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "We only need to show that Lemma F.2 and Lemma F.3 continue to hold. Using the same argument as we do for gradient flow, we can show that $\\gamma ( t )$ and $\\lambda ( x )$ do satisfy the propositions in Lemma F.2. ", + "bbox": [ + 173, + 895, + 823, + 925 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "For Lemma F.3, we first note the original definition of $C _ { \\eta }$ involves $B _ { 0 } , B _ { 1 } , B _ { 2 }$ , which are undefined in the multi-class setting. So now we redefine them as ", + "bbox": [ + 169, + 102, + 825, + 132 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/b799d068e64b7aa07d6cb0056e98a70266e4fc15361e1caaef279a10ffd5cd44.jpg", + "text": "$$\n\\begin{array} { r l } & { B _ { 0 } : = \\operatorname* { s u p } \\left. s _ { n j } : \\pmb \\theta \\in { \\cal S } ^ { d - 1 } , n \\in [ N ] , j \\in [ C ] \\right. , } \\\\ & { B _ { 1 } : = \\operatorname* { s u p } \\left. \\left\\| \\nabla s _ { n j } \\right\\| _ { 2 } : \\pmb \\theta \\in { \\cal S } ^ { d - 1 } , n \\in [ N ] , j \\in [ C ] \\right. , } \\\\ & { B _ { 2 } : = \\operatorname* { s u p } \\left. \\left\\| \\nabla ^ { 2 } s _ { n j } \\right\\| _ { 2 } : \\pmb \\theta \\in { \\cal S } ^ { d - 1 } , n \\in [ N ] , j \\in [ C ] \\right. . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 141, + 683, + 207 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "By the property of LSE, we can use $B _ { 0 } , B _ { 1 } , B _ { 2 }$ to bound $\\tilde { q } _ { n } , \\nabla \\tilde { q } _ { n } , \\nabla ^ { 2 } \\tilde { q } _ { n }$ . ", + "bbox": [ + 173, + 215, + 655, + 233 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/f28750a72d31d71ed4c06b244441f500142fa238d4f842c59ebcd7ab4ade0c1e.jpg", + "text": "$$\n\\begin{array} { c } { \\displaystyle \\tilde { q } _ { n } \\leq q _ { n } \\leq B _ { 0 } \\rho ^ { L } , } \\\\ { \\| \\nabla \\tilde { q } _ { n } \\| _ { 2 } = \\left\\| \\sum _ { j = 1 } ^ { C } \\frac { e ^ { - s _ { n j } } } { \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } \\nabla s _ { n j } \\right\\| _ { 2 } \\leq \\sum _ { j = 1 } ^ { C } \\frac { e ^ { - s _ { n j } } } { \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } \\| \\nabla s _ { n j } \\| _ { 2 } \\leq B _ { 1 } \\rho ^ { L - 1 } , } \\\\ { \\| \\nabla ^ { 2 } \\tilde { q } _ { n } \\| _ { 2 } = \\left\\| \\sum _ { j = 1 } ^ { C } \\left( \\frac { e ^ { - s _ { n j } } } { \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } ( \\nabla ^ { 2 } s _ { n j } - \\nabla s _ { n j } \\nabla s _ { n j } ^ { \\top } ) + \\frac { e ^ { - 2 s _ { n j } } } { \\left( \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } \\right) ^ { 2 } } \\nabla s _ { n j } \\nabla s _ { n j } ^ { \\top } \\right) \\right\\| _ { 2 } } \\\\ { \\leq B _ { 2 } \\rho ^ { L - 2 } + 2 B _ { 1 } ^ { 2 } \\rho ^ { 2 L - 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 194, + 239, + 805, + 392 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Using these bounds, we can prove with the constant $C _ { \\eta }$ defined as follows: ", + "bbox": [ + 174, + 400, + 665, + 416 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/f2df2ed0c09873a866782ce039c4fd60775ef759223ef50e9af242ee12758847.jpg", + "text": "$$\n\\begin{array} { r l } & { C _ { \\eta } : = \\frac { 1 } { 2 } \\left( \\left( \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } + \\frac { p 2 ^ { p + 1 } } { \\log { \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } } \\right) B _ { 1 } ^ { 2 } + ( 2 \\log { 2 } ) \\left( \\frac { B _ { 2 } } { \\rho ( t _ { 0 } ) ^ { L } } + 2 B _ { 1 } ^ { 2 } \\right) \\right) \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } M , } \\\\ & { \\textsf { r e } M : = \\operatorname* { m i n } \\Big \\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\Big \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 173, + 425, + 792, + 508 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "H EXTENSION: MULTI-HOMOGENEOUS MODELS ", + "text_level": 1, + "bbox": [ + 173, + 526, + 596, + 544 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "In this section, we extend our results to multi-homogeneous models. For this, the main difference from the proof for homogeneous models is that now we have to separate the norm of each homogeneous parts of the parameter, rather than consider them as a whole. So only a small part to proof needs to be changed. We focus on gradient flow, but it is worth to note that following the same argument, it is not hard to extend the results to gradient descent. ", + "bbox": [ + 173, + 560, + 825, + 632 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Let smo $\\Phi ( { \\pmb w } _ { 1 } , \\ldots , { \\pmb w } _ { m } ; { \\pmb x } )$ be arg $\\left( k _ { 1 } , \\ldots , k _ { m } \\right)$ -homogeneous. Let (5) can be rewritten a $\\rho _ { i } = \\| \\pmb { w } _ { i } \\| _ { 2 }$ and $\\begin{array} { r } { \\hat { \\pmb { w } } _ { i } = \\frac { { \\pmb w } _ { i } } { \\| { \\pmb w } _ { i } \\| _ { 2 } } } \\end{array}$ . The ", + "bbox": [ + 173, + 637, + 821, + 670 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Definition H.1. For a multi-homogeneous model with loss function $\\ell ( \\cdot )$ satisfying (B3), the smoothed normalized margin $\\widetilde \\gamma ( \\pmb \\theta )$ of $\\pmb \\theta$ is defined as ", + "bbox": [ + 171, + 674, + 825, + 704 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/3ae3606dfb09b2840c65f58b06091b37967bf8963d073a84af9b3575f8217724.jpg", + "text": "$$\n{ \\tilde { \\gamma } } ( \\pmb { \\theta } ) : = \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 714, + 607, + 755 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "We only prove the generalized version of Lemma 5.1 here. The other proofs are almost the same. ", + "bbox": [ + 171, + 770, + 807, + 786 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Lemma H.2. For all $t > t _ { 0 }$ , ", + "bbox": [ + 173, + 791, + 362, + 806 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/dcfeeec80720f64ad1fbe77ac3c3da3e230fc876b8d8a3344fc5780ef1a15457.jpg", + "text": "$$\n{ \\frac { d } { d t } } \\log \\rho _ { i } > 0 f o r a l l i \\in [ m ] \\quad a n d \\quad { \\frac { d } { d t } } \\log { \\tilde { \\gamma } } \\geq \\sum _ { i = 1 } ^ { m } k _ { i } \\left( { \\frac { d } { d t } } \\log \\rho _ { i } \\right) ^ { - 1 } \\left\\| { \\frac { d { \\hat { w } } _ { i } } { d t } } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 210, + 815, + 758, + 857 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Proof. Note that ddt log ρi = 12ρ2 dρidt $\\begin{array} { r } { \\frac { d } { d t } \\log \\rho _ { i } = \\frac { 1 } { 2 \\rho _ { i } ^ { 2 } } \\frac { d \\rho _ { i } ^ { 2 } } { d t } = \\frac { k _ { i } \\nu ( t ) } { \\rho _ { i } ^ { 2 } } } \\end{array}$ by Theorem B.4. It simply follows from Lemma B.5 that $\\begin{array} { r } { \\frac { d } { d t } \\log \\rho > 0 } \\end{array}$ for a.e. $t > t _ { 0 }$ . And it is easy to see that $\\begin{array} { r } { \\log { \\tilde { \\gamma } } = \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\rho ^ { L } \\right) } \\end{array}$ exists for all ", + "bbox": [ + 174, + 885, + 823, + 926 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "$t \\geq t _ { 0 }$ . By the chain rule and Lemma B.5, we have ", + "bbox": [ + 173, + 103, + 508, + 118 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/15b8d64da4d0d441df55c40893b3efafb4a77051c8c500dd022e79b71d155104.jpg", + "text": "$$\n\\begin{array} { r } { \\frac { d } { d t } \\log \\tilde { \\gamma } = \\frac { d } { d t } \\left( \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\right) - \\displaystyle \\sum _ { i = 1 } ^ { m } k _ { i } \\log \\rho _ { i } \\right) = \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\cdot \\frac { 1 } { \\mathcal { L } } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - \\displaystyle \\sum _ { i = 1 } ^ { m } \\frac { k _ { i } ^ { 2 } \\nu ( t ) } { \\rho _ { i } ^ { 2 } } } \\\\ { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - \\displaystyle \\sum _ { i = 1 } ^ { m } \\frac { k _ { i } ^ { 2 } \\nu ( t ) } { \\rho _ { i } ^ { 2 } } } \\\\ { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } - \\displaystyle \\sum _ { i = 1 } ^ { m } \\frac { k _ { i } ^ { 2 } \\nu ( t ) ^ { 2 } } { \\rho _ { i } ^ { 2 } } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 121, + 807, + 250 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "On the one hand, $\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } = \\sum _ { i = 1 } ^ { m } \\left\\| \\frac { d \\pmb { w } _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } } \\end{array}$ for a.e. $t > 0$ by Lemma I.3; on the other hand, $k _ { i } \\nu ( t ) =$ $\\begin{array} { r } { \\left. w _ { i } , \\frac { d w _ { i } } { d t } \\right. } \\end{array}$ dt i=1 \r dt \r2 by Theorem B.4. Combining these together yields ", + "bbox": [ + 173, + 251, + 826, + 289 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/47f7987bb5760861c83e4842fce0b8ecdec07f1d936900cac1ee117fb7476335.jpg", + "text": "$$\n\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq \\frac { 1 } { \\nu ( t ) } \\sum _ { i = 1 } ^ { m } \\left( \\left\\| \\frac { d w _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } - \\left. \\hat { w } _ { i } , \\frac { d w _ { i } } { d t } \\right. ^ { 2 } \\right) = \\frac { 1 } { \\nu ( t ) } \\left\\| ( I - \\hat { w } _ { i } \\hat { w } _ { i } ^ { \\top } ) \\frac { d w _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 230, + 291, + 767, + 334 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "By the chain rule, $\\begin{array} { r } { \\frac { d \\hat { \\pmb { w } } _ { i } } { d t } = \\frac { 1 } { \\rho _ { i } } ( \\pmb { I } - \\hat { \\pmb { w } _ { i } } \\hat { \\pmb { w } _ { i } } ^ { \\top } ) \\frac { d \\pmb { w } _ { i } } { d t } } \\end{array}$ i>) dwidt for a.e. t > 0. So we have ", + "bbox": [ + 174, + 344, + 653, + 364 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/41c70e65772082b7eff9d9af185f6da7de896636485fd823cf76e8e7a892004d.jpg", + "text": "$$\n\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq \\sum _ { i = 1 } ^ { m } \\frac { \\rho _ { i } ^ { 2 } } { \\nu ( t ) } \\left\\| \\frac { d \\hat { w } _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } = \\sum _ { i = 1 } ^ { m } k _ { i } \\left( \\frac { d } { d t } \\log \\rho _ { i } \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { w } _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 367, + 714, + 409 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "For cross-entropy loss, we can combine the proofs in Appendix $\\mathbf { G }$ to show that Lemma H.2 holds if we use the following definition of the smoothed normalized margin: ", + "bbox": [ + 173, + 440, + 823, + 469 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Definition H.3. For a multi-homogeneous model with cross-entropy, the smoothed normalized margin $\\widetilde \\gamma ( \\pmb \\theta )$ of $\\pmb { \\theta }$ is defined as ", + "bbox": [ + 173, + 472, + 820, + 500 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/c5a3f3fe44de61c42ae4a8bc947e189166dd41676dd27f985a02670e35472171.jpg", + "text": "$$\n\\widetilde { \\gamma } = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } = \\frac { - \\log ( e ^ { \\mathcal { L } } - 1 ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 498, + 611, + 536 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "where $\\ell ^ { - 1 } ( \\cdot )$ is the inverse function of the logistic loss $\\ell ( \\cdot )$ . ", + "bbox": [ + 173, + 536, + 565, + 553 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "The only place we need to change in the proof for Lemma H.2 is that instead of using Lemma B.5, we need to prove $\\begin{array} { r } { \\nu ( t ) \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } } \\end{array}$ g(log 1L )g0(log 1 ) L in a similar way as in Lemma G.2. The other parts of the proof are exactly the same as before. ", + "bbox": [ + 173, + 561, + 826, + 613 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "I CHAIN RULES FOR NON-DIFFERENTIABLE FUNCTIONS ", + "text_level": 1, + "bbox": [ + 176, + 632, + 658, + 648 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "In this section, we provide some background on the chain rule for non-differentiable functions. The ordinary chain rule for differentiable functions is a very useful formula for computing derivatives in calculus. However, for non-differentiable functions, it is difficult to find a natural definition of subdifferential so that the chain rule equation holds exactly. To solve this issue, Clarke proposed Clarke’s subdifferential (Clarke, 1975; 1990; Clarke et al., 2008) for locally Lipschitz functions, for which the chain rule holds as an inclusion rather than an equation: ", + "bbox": [ + 173, + 662, + 826, + 747 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Theorem I.1 (Theorem 2.3.9 and 2.3.10 of (Clarke, 1990)). Let $z _ { 1 } , \\dots , z _ { n } : \\mathbb { R } ^ { d } \\to \\mathbb { R }$ and $f :$ $\\mathbb { R } ^ { n } \\to \\mathbb { R }$ be locally Lipschitz functions. Let $( f \\circ z ) ( { \\pmb x } ) = f ( z _ { 1 } ( { \\pmb x } ) , \\ldots , z _ { n } ( { \\pmb x } ) )$ be the composition of $f$ and $_ z$ . Then, ", + "bbox": [ + 174, + 750, + 826, + 791 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/525ea9af94868c51dd6fd748596ad09fb133baa862186595e71700a703df6b97.jpg", + "text": "$$\n\\partial ^ { \\circ } ( f \\circ z ) ( \\pmb { x } ) \\subseteq \\mathrm { c o n v } \\left\\{ \\sum _ { i = 1 } ^ { n } \\alpha _ { i } \\pmb { h } _ { i } : \\pmb { \\alpha } \\in \\partial ^ { \\circ } f ( z _ { 1 } ( \\pmb { x } ) , \\ldots , z _ { n } ( \\pmb { x } ) ) , \\pmb { h } _ { i } \\in \\partial ^ { \\circ } z _ { i } ( \\pmb { x } ) \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 230, + 794, + 767, + 837 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "For analyzing gradient flow, the chain rule is crucial. For a differentiable loss function $\\mathcal { L } ( \\pmb { \\theta } )$ , we the chain rule that the function value keeps decreasing along the gradient flow : $\\begin{array} { r l r } { \\mathrm { ~ } } & { { } } & { \\frac { d \\pmb \\theta ( t ) } { d t } = } \\end{array}$ $- \\nabla \\mathcal { L } ( \\pmb \\theta ( t ) )$ ", + "bbox": [ + 173, + 845, + 825, + 892 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/58b648788b9f4aa77f760b8e37d496e4fd0939dfd021f3ff35649e1fcd6be781.jpg", + "text": "$$\n\\frac { d \\mathcal { L } ( \\pmb { \\theta } ( t ) ) } { d t } = \\left. \\nabla \\mathcal { L } ( \\pmb { \\theta } ( t ) ) , \\frac { d \\pmb { \\theta } ( t ) } { d t } \\right. = - \\left\\| \\frac { d \\pmb { \\theta } ( t ) } { d t } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 330, + 890, + 668, + 929 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "But for locally Lipschitz functions which could be non-differentiable, (25) may not hold in general since Theorem I.1 only holds for an inclusion. ", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Following (Davis et al., 2020; Drusvyatskiy et al., 2015), we consider the functions that admit a chain rule for any arc. ", + "bbox": [ + 171, + 138, + 823, + 167 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Definition I.2 (Chain Rule). A locally Lipschitz function $f : { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }$ admits a chain rule if for any arc $z : [ 0 , + \\infty ) \\to \\mathbb { R } ^ { d }$ , ", + "bbox": [ + 173, + 171, + 823, + 202 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/15165a0145e9b766471106cd7edd1e4318a24b5556a8fb6e0abc1338004f8c1b.jpg", + "text": "$$\n\\forall h \\in \\partial ^ { \\circ } f ( z ( t ) ) : ( f \\circ z ) ^ { \\prime } ( t ) = \\langle h , z ^ { \\prime } ( t ) \\rangle\n$$", + "text_format": "latex", + "bbox": [ + 356, + 203, + 640, + 222 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "holds for a.e. $t > 0$ . ", + "bbox": [ + 173, + 227, + 307, + 242 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "It is shown in (Davis et al., 2020; Drusvyatskiy et al., 2015) that a generalized version of (25) holds for such functions: ", + "bbox": [ + 171, + 255, + 825, + 284 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Lemma I.3 (Lemma 5.2 (Davis et al., 2020)). Let $\\mathcal { L } : \\mathbb { R } ^ { d } \\mathbb { R }$ be a locally Lipschitz function that admits a chain rule. Let $\\pmb \\theta : [ 0 , + \\infty ) \\mathbb R ^ { d }$ be the gradient flow on $\\mathcal { L }$ : ", + "bbox": [ + 171, + 287, + 823, + 319 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/5d8820c1aac7a673b6fdad51023cc575b991266002b1818644a2261fd1c7cbfa.jpg", + "text": "$$\n\\frac { d \\pmb { \\theta } ( t ) } { d t } \\in - \\partial ^ { \\circ } \\mathcal { L } ( \\pmb { \\theta } ( t ) ) \\qquad f o r a . e . \\ t \\geq 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 328, + 632, + 359 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 367, + 209, + 382 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/0bc592a7d66b49bc959f304957e4f98563f200683921f779299506d906db8a28.jpg", + "text": "$$\n{ \\frac { d { \\mathcal { L } } ( \\pmb { \\theta } ( t ) ) } { d t } } = - \\left\\| { \\frac { d \\pmb { \\theta } ( t ) } { d t } } \\right\\| _ { 2 } ^ { 2 } = - \\operatorname* { m i n } \\{ \\| \\pmb { h } \\| _ { 2 } ^ { 2 } : \\pmb { h } \\in \\partial ^ { \\circ } { \\mathcal { L } } ( \\pmb { \\theta } ( t ) ) \\}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 381, + 699, + 420 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "holds for a.e. $t > 0$ . ", + "bbox": [ + 174, + 425, + 305, + 440 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "We can see that ${ \\mathcal { C } } ^ { 1 }$ -smooth functions admit chain rules. As shown in (Davis et al., 2020), if a locally Lipschitz function is subdifferentiablly regular or Whitney $\\mathcal { C } ^ { 1 }$ -stratifiable, then it admits a chain rule. The latter one includes a large family of functions, e.g., semi-algebraic functions, semianalytic functions, and definable functions in an o-minimal structure (Coste, 2002; van den Dries & Miller, 1996). ", + "bbox": [ + 173, + 452, + 826, + 523 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "It is worth noting that the class of functions that admits chain rules is closed under composition. \nThis is indeed a simple corollary of Theorem I.1. ", + "bbox": [ + 173, + 530, + 821, + 559 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Theorem I.4. Let $z _ { 1 } , \\dots , z _ { n } : \\mathbb { R } ^ { d } \\to \\mathbb { R }$ and $f : \\mathbb { R } ^ { n } \\mathbb { R }$ be locally Lipschitz functions and assume all of them admit chain rules. Let $( f \\circ z ) ( \\pmb { x } ) = f ( z _ { 1 } ( \\pmb { x } ) , \\dots , z _ { n } ( \\pmb { x } ) )$ be the composition of $f$ and $_ z$ . Then $f \\circ z$ also admits a chain rule. ", + "bbox": [ + 173, + 563, + 825, + 607 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Proof. We can see that $f \\circ z$ is locally Lipschitz. Let $\\pmb { x } : [ 0 , + \\infty ) \\mathbb { R } ^ { d } , t \\mapsto \\pmb { x } ( t )$ be an arc. First, we show that $z \\circ \\dot { \\pmb x } : [ 0 , + \\infty ) \\mathbb { R } ^ { d } , t \\mapsto z ( { \\pmb x } ( t ) )$ is also an arc. For any closed sub-interval $I$ , $z ( \\pmb { x } ( I ) )$ must be contained in a compact set $U$ . Then it can be shown that the locally Lipschitz continuous function $_ z$ is (globally) Lipschitz continuous on $U$ . By the fact that the composition of a Lipschitz continuous and an absolutely continuous function is absolutely continuous, $z \\circ x$ is absolutely continuous on $I$ , and thus it is an arc. ", + "bbox": [ + 173, + 628, + 825, + 715 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Since $f$ and $_ z$ admit chain rules on arcs $_ { z \\mathrm { ~ O ~ } \\pmb { x } }$ and $_ { \\textbf { \\em x } }$ respectively, the following holds for a.e. $t > 0$ ", + "bbox": [ + 173, + 722, + 823, + 737 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/7ab99dc6f7bef7a94477202573d304224d964786573688b98bd1c2a53ba4ed72.jpg", + "text": "$$\n\\begin{array} { r l } { \\forall \\pmb { \\alpha } \\in \\partial ^ { \\circ } f ( \\pmb { z } ( \\pmb { x } ( t ) ) ) : } & { ( f \\circ ( \\pmb { z } \\circ \\pmb { x } ) ) ^ { \\prime } ( t ) = \\langle \\pmb { \\alpha } , ( \\pmb { z } \\circ \\pmb { x } ) ^ { \\prime } ( t ) \\rangle , } \\\\ { \\forall \\pmb { h } _ { i } \\in \\partial ^ { \\circ } z _ { i } ( \\pmb { x } ( t ) ) : } & { ( z _ { i } \\circ \\pmb { x } ) ^ { \\prime } ( t ) = \\langle \\pmb { h } _ { i } , \\pmb { x } ^ { \\prime } ( t ) \\rangle . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 744, + 700, + 785 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Combining these we obtain that for a.e. $t > 0$ , ", + "bbox": [ + 173, + 791, + 478, + 806 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/1ff7032281da7fde9bc23e878b2c698dd5d49689f0be7f0d2170438550c190ad.jpg", + "text": "$$\n( f \\circ z \\circ x ) ^ { \\prime } ( t ) = \\sum _ { i = 1 } ^ { n } \\alpha _ { i } \\left. h _ { i } , x ^ { \\prime } ( t ) \\right. ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 815, + 620, + 857 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "for all $\\alpha ~ \\in ~ \\partial ^ { \\circ } f ( z ( { \\pmb x } ( t ) ) )$ and for all $\\begin{array} { r } { \\pmb { h } _ { i } \\in \\partial ^ { \\circ } z _ { i } ( \\pmb { x } ( t ) ) } \\end{array}$ . The RHS can be rewritten as $\\textstyle \\langle \\sum _ { i = 1 } ^ { n } \\alpha _ { i } \\pmb { h } _ { i } , \\pmb { x } ^ { \\prime } ( t ) \\rangle$ . By Theorem I.1, every $k \\in \\partial ^ { \\circ } ( f \\circ z ) ( { \\pmb x } ( t ) )$ can be written as a convex combination of a finite set of points in the form of $\\scriptstyle \\sum _ { i = 1 } ^ { n } \\alpha _ { i } h _ { i }$ . So $( f \\circ z \\circ x ) ^ { \\prime } ( t ) = \\langle k , x ^ { \\prime } ( t ) \\rangle$ holds for a.e. $t > 0$ . □ ", + "bbox": [ + 173, + 864, + 825, + 924 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "J MEXICAN HAT ", + "text_level": 1, + "bbox": [ + 174, + 102, + 328, + 118 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "In this section, we give an example to illustrate that gradient flow does not necessarily converge in direction, even for $C ^ { \\infty }$ -smooth homogeneous models. ", + "bbox": [ + 173, + 132, + 825, + 161 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "It is known that gradient flow (or gradient descent) may not converge to any point even when optimizing an $C ^ { \\infty }$ function (Curry, 1944; Zoutendijk, 1976; Palis & De Melo, 2012; Absil et al., 2005). One famous counterexample is the “Mexican Hat” function described in (Absil et al., 2005): ", + "bbox": [ + 173, + 167, + 826, + 210 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/3b091b743632542554d9606ad91468856cfbd26acbf370ee6eb9ce8e13409653.jpg", + "text": "$$\nf ( u , v ) : = f ( r \\cos \\varphi , r \\sin \\varphi ) : = \\left\\{ { \\begin{array} { l l } { e ^ { - { \\frac { 1 } { 1 - r ^ { 2 } } } } \\left( 1 - C ( r ) \\sin \\left( \\varphi - { \\frac { 1 } { 1 - r ^ { 2 } } } \\right) \\right) \\quad } & { r < 1 } \\\\ { 0 \\quad } & { r \\geq 1 } \\end{array} } \\right.\n$$", + "text_format": "latex", + "bbox": [ + 223, + 210, + 774, + 253 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { C ( r ) = \\frac { 4 r ^ { 4 } } { 4 r ^ { 4 } + ( 1 - r ^ { 2 } ) ^ { 4 } } \\in [ 0 , 1 ] } \\end{array}$ . It can be shown that $f$ is ${ \\mathcal { C } } ^ { \\infty }$ -smooth on $\\mathbb { R } ^ { 2 }$ but not analytic. \nSee Figure 2 for a plot for $f ( u , v )$ . ", + "bbox": [ + 173, + 256, + 825, + 291 + ], + "page_idx": 44 + }, + { + "type": "image", + "img_path": "images/fdd8b960a9c32bef3e12b8b8208114e0c2be68507e93e7f18e6e76881a178ab2.jpg", + "image_caption": [ + "Figure 2: A plot for the Mexican Hat function $f ( u , v )$ . " + ], + "image_footnote": [], + "bbox": [ + 348, + 314, + 633, + 541 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "However, the Maxican Hat function is not homogeneous, and Absil et al. (2005) did not consider the directional convergence, either. To make it homogeneous, we introduce an extra variable $z$ , and normalize the parameter before evaluate $f$ . In particular, we fix $L > 0$ and define ", + "bbox": [ + 174, + 582, + 825, + 625 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/4ba558e383b22d3ba772f532746acd869b33691841b2fb0b2b8e76532b201413.jpg", + "text": "$$\nh ( \\theta ) = h ( x , y , z ) = \\rho ^ { L } ( 1 - f ( u , v ) ) \\qquad \\mathrm { w h e r e } \\quad u = x / \\rho , \\quad v = y / \\rho , \\quad \\rho = \\sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 174, + 627, + 815, + 645 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 646, + 415, + 661 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Theorem J.1. Consider gradient flow on $\\begin{array} { r } { { \\mathcal L } ( \\pmb { \\theta } ) = \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( \\pmb { \\theta } ) } } \\end{array}$ , where $q _ { n } ( \\pmb \\theta ) \\ = \\ h ( \\pmb \\theta )$ for all $n \\in [ N ]$ . Suppose the polar representation of $( u , v )$ is $( r \\cos \\varphi , r \\sin \\varphi )$ . If $0 ~ < ~ r ~ < ~ 1$ and $\\textstyle \\varphi = { \\frac { 1 } { 1 - r ^ { 2 } } }$ holds at time $t = 0$ , then $\\frac { \\pmb { \\theta } ( t ) } { \\| \\pmb { \\theta } ( t ) \\| _ { 2 } }$ does not converge to any point, and the limit points of $\\big \\{ \\frac { \\pmb \\theta ( t ) } { \\lVert \\pmb \\theta ( t ) \\rVert _ { 2 } } : t > 0 \\big \\}$ form a circle $\\left\\{ \\left( x , y , z \\right) \\in S ^ { 2 } : x ^ { 2 } + y ^ { 2 } = 1 , z = 0 \\right\\}$ . ", + "bbox": [ + 173, + 662, + 826, + 734 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Proof. Define $\\textstyle \\psi = \\varphi - { \\frac { 1 } { 1 - r ^ { 2 } } }$ . Our proof consists of two parts, following from the idea in (Absil et al., 2005). First, we show that $\\begin{array} { r } { { \\frac { d \\psi } { d t } } = 0 } \\end{array}$ as long as $\\psi = 0$ . Then we can infer that $\\psi = 0$ for all $t \\geq 0$ . Next, we show that $r 1$ as $t \\to + \\infty$ . Using $\\psi = 0$ , we know that the polar angle $\\varphi + \\infty$ as $t \\to + \\infty$ . Therefore, $( u , v )$ circles around $\\{ ( u , v ) : u ^ { 2 } + v ^ { 2 } = 1 \\}$ , and thus it does not converge. ", + "bbox": [ + 173, + 748, + 825, + 813 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Proof for $\\begin{array} { r } { \\frac { d \\psi } { d t } = 0 } \\end{array}$ . For convenience, we use $w$ to denote $z / \\rho$ . By simple calculation, we have the following formulas for partial derivatives: ", + "bbox": [ + 173, + 819, + 823, + 848 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/7256c675819920b017f2ef7258f04927dbaa92c7af1376f5458ea52079a674e2.jpg", + "text": "$$\n\\left\\{ \\begin{array} { l l } { \\frac { \\partial h } { \\partial x } = \\rho ^ { L - 1 } \\left( L u ( 1 - f ) - ( 1 - u ^ { 2 } ) \\frac { \\partial f } { \\partial u } + u v \\frac { \\partial f } { \\partial v } \\right) } \\\\ { \\frac { \\partial h } { \\partial y } = \\rho ^ { L - 1 } \\left( L v ( 1 - f ) + u v \\frac { \\partial f } { \\partial u } - ( 1 - v ^ { 2 } ) \\frac { \\partial f } { \\partial v } \\right) } \\\\ { \\frac { \\partial h } { \\partial z } = \\rho ^ { L - 1 } \\left( L w ( 1 - f ) + u w \\frac { \\partial f } { \\partial u } + v w \\frac { \\partial f } { \\partial v } \\right) } \\end{array} \\right. \\qquad \\left\\{ \\begin{array} { l l } { \\frac { \\partial f } { \\partial u } = \\frac { u } { r } \\cdot \\frac { \\partial f } { \\partial r } - \\frac { v } { r ^ { 2 } } \\cdot \\frac { \\partial f } { \\partial \\varphi } } \\\\ { \\frac { \\partial f } { \\partial v } = \\frac { v } { r } \\cdot \\frac { \\partial f } { \\partial r } + \\frac { u } { r ^ { 2 } } \\cdot \\frac { \\partial f } { \\partial \\varphi } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 210, + 849, + 772, + 922 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "For gradient flow, we have ", + "bbox": [ + 174, + 103, + 349, + 117 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/1316172d676f8122217c655e87f03702f8b0fb555e4ff8bbdf9e28a84b2146f6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { d \\theta } { d t } = N e ^ { - h } \\nabla h } \\\\ & { \\frac { d u } { d t } = \\frac { 1 } { \\rho } \\left( ( 1 - u ^ { 2 } ) \\frac { d x } { d t } - u v \\frac { d y } { d t } - u w \\frac { d z } { d t } \\right) = N e ^ { - h } \\rho ^ { L - 2 } \\left( - ( 1 - u ^ { 2 } ) \\frac { \\partial f } { \\partial u } + u v \\frac { \\partial f } { \\partial v } \\right) } \\\\ & { \\frac { d v } { d t } = \\frac { 1 } { \\rho } \\left( ( 1 - v ^ { 2 } ) \\frac { d y } { d t } - u v \\frac { d x } { d t } - v w \\frac { d z } { d t } \\right) = N e ^ { - h } \\rho ^ { L - 2 } \\left( - ( 1 - v ^ { 2 } ) \\frac { \\partial f } { \\partial v } + u v \\frac { \\partial f } { \\partial u } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 121, + 784, + 220 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "By writing down the movement of $( u , v )$ in the polar coordinate system, we have ", + "bbox": [ + 173, + 220, + 705, + 236 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/f51b92421b0d6fcacec36086749500283c8d5d8b2eb9784a0cd755900843bdbd.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { d r } { d t } = \\frac { u } { r } \\cdot \\frac { d u } { d t } + \\frac { v } { r } \\cdot \\frac { d v } { d t } } \\\\ & { \\quad = - \\frac { N e ^ { - h } \\rho ^ { L - 2 } } { r } \\left( u ( 1 - r ^ { 2 } ) \\frac { \\partial f } { \\partial u } + v ( 1 - r ^ { 2 } ) \\frac { \\partial f } { \\partial v } \\right) } \\\\ & { \\quad = - N e ^ { - h } \\rho ^ { L - 2 } ( 1 - r ^ { 2 } ) \\frac { \\partial f } { \\partial r } } \\\\ & { \\frac { d \\varphi } { d t } = - \\frac { v } { r ^ { 2 } } \\cdot \\frac { d u } { d t } + \\frac { u } { r ^ { 2 } } \\cdot \\frac { d v } { d t } } \\\\ & { \\quad = \\frac { N e ^ { - h } \\rho ^ { L - 2 } } { r ^ { 2 } } \\left( v \\frac { \\partial f } { \\partial u } - u \\frac { \\partial f } { \\partial v } \\right) } \\\\ & { \\quad = - N e ^ { - h } \\rho ^ { L - 2 } \\frac { 1 } { r ^ { 2 } } \\cdot \\frac { \\partial f } { \\partial \\varphi } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 315, + 238, + 679, + 434 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "For $\\psi = 0$ , the partial derivatives of $f$ with respect to $r$ and $\\varphi$ can be evaluated as follows: ", + "bbox": [ + 166, + 434, + 763, + 449 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/bd526aaa270ec74261c2e9754b774c77b6fa1033556ac90dd5d54c026d698207.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { \\partial f } { \\partial r } = \\frac { d } { d r } \\left( e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } \\right) - e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ^ { \\prime } ( r ) \\sin \\psi - e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ( r ) \\cos \\psi \\cdot \\frac { \\partial \\psi } { \\partial r } } \\\\ { = \\frac { d } { d r } \\left( e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } \\right) + e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ( r ) \\frac { d } { d r } \\left( \\frac { 1 } { 1 - r ^ { 2 } } \\right) } \\\\ { = - \\frac { 2 r } { ( 1 - r ^ { 2 } ) ^ { 2 } } ( 1 - C ( r ) ) e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } . } \\\\ { \\displaystyle \\frac { \\partial f } { \\partial \\varphi } = - e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ( r ) \\cos \\psi \\cdot \\frac { \\partial \\psi } { \\partial \\varphi } } \\\\ { = - e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ( r ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 450, + 730, + 604 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "So if $\\psi = 0$ , then dψdt = 0 by the direct calculation below: ", + "bbox": [ + 173, + 607, + 552, + 623 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/459daec14a824dcb47fcdc0cb6a5e579578c374c42bc1237767ddfaf64fbbe9b.jpg", + "text": "$$\n{ \\begin{array} { r l } & { { \\frac { d \\psi } { d t } } = { \\frac { d \\varphi } { d t } } - { \\frac { d } { d r } } \\left( { \\frac { 1 } { 1 - r ^ { 2 } } } \\right) \\cdot { \\frac { d r } { d t } } } \\\\ & { \\qquad = N e ^ { - h } \\rho ^ { L - 2 } \\left( - { \\frac { 1 } { r ^ { 2 } } } { \\frac { \\partial f } { \\partial \\varphi } } + { \\frac { 2 r } { ( 1 - r ^ { 2 } ) ^ { 2 } } } { \\frac { \\partial f } { \\partial r } } \\right) } \\\\ & { \\qquad = N e ^ { - h } \\rho ^ { L - 2 } e ^ { - { \\frac { 1 } { 1 - r ^ { 2 } } } } \\left( \\left( { \\frac { 1 } { r ^ { 2 } } } + { \\frac { 4 r ^ { 2 } } { ( 1 - r ^ { 2 } ) ^ { 4 } } } \\right) C ( r ) - { \\frac { 4 r ^ { 2 } } { ( 1 - r ^ { 2 } ) ^ { 4 } } } \\right) = 0 . } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 254, + 626, + 743, + 733 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Proof for $r 1$ . Let $( \\bar { u } , \\bar { v } )$ be a convergent point of $\\{ ( u , v ) : t \\geq 0 \\}$ . Define $\\bar { r } = \\sqrt { u ^ { 2 } + v ^ { 2 } }$ . It is easy to see that $r \\leq 1$ from the normalization of $\\pmb { \\theta }$ in the definition. According to Theorem 4.4, we know that $( \\bar { u } , \\bar { v } )$ is a stationary point of $\\bar { \\gamma } ( u ( t ) , v ( t ) ) \\ : = \\ : 1 \\ : - \\ : f ( u ( t ) , v ( t ) )$ . If $\\bar { r } \\ = \\ 0$ , then $f ( \\bar { u } , \\bar { v } ) > f ( u ( 0 ) , v ( 0 ) )$ , which contradicts to the monotonicity of $\\tilde { \\gamma } ( t ) = \\bar { \\gamma } ( t ) = 1 - f ( u ( t ) , v ( t ) )$ . If $\\bar { r } < 1 $ , then $\\bar { \\psi } = 0$ (defined as $\\psi$ for $( \\bar { u } , \\bar { v } ) .$ ). So − 2r(1−r2)2 (1 − C(r))e− 1−r2 6= 0, which again leads to a contradiction. Therefore, $\\bar { r } = 1$ , and thus $r 1$ . □ ", + "bbox": [ + 173, + 739, + 826, + 833 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "K EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 851, + 331, + 867 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "To validate our theoretical results, we conduct several experiments. We mainly focus on MNIST dataset. We trained two models with Tensorflow. The first one (called the CNN with bias) is a standard 4-layer CNN with exactly the same architecture as that used in MNIST Adversarial Examples ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Challenge5. The layers of this model can be described as conv-32 with filter size $5 \\times 5$ , max-pool, conv-64 with filter size $3 \\times 3$ , max-pool, fc-1024, fc-10 in order. Notice that this model has bias terms in each layer, and thus does not satisfy homogeneity. To make its outputs homogeneous to its parameters, we also trained this model after removing all the bias terms except those in the first layer (the modified model is called the CNN without bias). Note that keeping the bias terms in the first layer prevents the model to be homogeneous in the input data while retains the homogeneity in parameters. We initialize all layer weights by He normal initializer (He et al., 2015) and all bias terms by zero. In training the models, we use SGD with batch size 100 without momentum. We normalize all the images to $[ 0 , 1 ] ^ { 3 2 \\times 3 2 }$ by dividing 255 for each pixel. ", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "K.1 EVALUATION FOR NORMALIZED MARGIN ", + "text_level": 1, + "bbox": [ + 174, + 247, + 506, + 261 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "In the first part of our experiments, we evaluate the normalized margin every few epochs to see how it changes over time. From now on, we view the bias term in the first layer as a part of the weight in the first layer for convenience. Observe that the CNN without bias is multi-homogeneous in layer weights (see (4) in Section 4.4). So for the CNN without bias, we define the normalized margin $\\bar { \\gamma }$ as the margin divided by the product of the $L ^ { 2 }$ -norm of all layer weights. Here we compute the $L ^ { 2 }$ -norm of a layer weight parameter after flattening it into a one-dimensional vector. For the CNN with bias, we still compute the smoothed normalized margin in this way. When computing the $L ^ { 2 }$ - norm of every layer weight, we simply ignore the bias terms if they are not in the first layer. For completeness, we include the plots for the normalized margin using the original definition (2) in Figure 3 and 4. ", + "bbox": [ + 173, + 273, + 825, + 412 + ], + "page_idx": 46 + }, + { + "type": "image", + "img_path": "images/537667231ed5f10711f6cfa0e044280666fd050879ce6aa6d81b43708667e02e.jpg", + "image_caption": [ + "Figure 3: Training CNNs with and without bias on MNIST, using SGD with learning rate 0.01. The training accuracy (left) increases to $1 0 0 \\%$ after about 100 epochs, and the normalized margin with the original definition (right) keeps increasing after the model is fitted. " + ], + "image_footnote": [], + "bbox": [ + 271, + 431, + 715, + 587 + ], + "page_idx": 46 + }, + { + "type": "image", + "img_path": "images/6a977df3714686b1cdc63609ca27ce2478cbf361e5fd64ab3db15c2a4a1067fc.jpg", + "image_caption": [ + "Figure 4: Training CNNs with and without bias on MNIST, using SGD with the loss-based learning rate scheduler. The training accuracy (left) increases to $1 0 0 \\%$ after about 20 epochs, and the normalized margin with the original definition (middle) increases rapidly after the model is fitted. The right figure shows the change of the relative learning rate $\\alpha ( t )$ (see (27) for its definition) during training. " + ], + "image_footnote": [], + "bbox": [ + 205, + 672, + 802, + 810 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "SGD with Constant Learning Rate. We first train the CNNs using SGD with constant learning rate 0.01. After about 100 epochs, both CNNs have fitted the training set. After that, we can see that the normalized margins of both CNNs increase. However, the growth rate of the normalized margin is rather slow. The results are shown in Figure 1 in Section 1. We also tried other learning rates other than 0.01, and similar phenomena can be observed. ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "SGD with Loss-based Learning Rate. Indeed, we can speed up the training by using a proper scheduling of learning rates for SGD. We propose a heuristic learning rate scheduling method, called the loss-based learning rate scheduling. The basic idea is to find the maximum possible learning rate at each epoch based on the current training loss (in a similar way as the line search method). See Appendix L.1 for the details. As shown in Figure 1, SGD with loss-based learning rate scheduling decreases the training loss exponentially faster than SGD with constant learning rate. Also, a rapid growth of normalized margin is observed for both CNNs. Note that with this scheduling the training loss can be as small as $\\overline { { 1 0 ^ { - 8 0 0 } } }$ , which may lead to numerical issues. To address such issues, we applied some re-parameterization tricks and numerical tricks in our implementation. See Appendix L.2 for the details. ", + "bbox": [ + 173, + 180, + 825, + 319 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Experiments on CIFAR-10. To verify whether the normalized margin is increasing in practice, we also conduct experiments on CIFAR-10. We use a modified version of VGGNet-16. The layers of this model can be described as conv- $6 4 \\times 2$ , max-pool, conv- $1 2 8 \\times 2$ , max-pool, conv-256 $\\times 3$ , max-pool, conv- $. 5 1 2 \\times 3$ , $\\mathtt { m a x - p o o l }$ , $\\mathsf { c o n v } { - } 5 1 2 \\times 3$ , $\\mathtt { m a x - p o o l }$ , $\\pm _ { \\mathrm { C - 1 0 } }$ in order, where each conv has filter size $3 \\times 3$ . We train two networks: one is exactly the same as the VGGNet we described, and the other one is the VGGNet without any bias terms except those in the first layer (similar as in the experiments on MNIST). The experiment results are shown in Figure 5 and 6. We can see that the normalize margin is increasing over time. ", + "bbox": [ + 173, + 325, + 825, + 438 + ], + "page_idx": 47 + }, + { + "type": "image", + "img_path": "images/58ff1562f78c324281e18d2aa99b831e549164308703f02119c458659d4d2e3c.jpg", + "image_caption": [ + "Figure 5: Training VGGNet with and without bias on CIFAR-10, using SGD with learning rate 0.1. " + ], + "image_footnote": [], + "bbox": [ + 266, + 454, + 728, + 614 + ], + "page_idx": 47 + }, + { + "type": "image", + "img_path": "images/495a9fd0459b12209dbeccc75dc0894e2a64a5daed8bfc4e279791f66c40feb6.jpg", + "image_caption": [ + "Figure 6: Training VGGNet with and without bias on CIFAR-10, using SGD with the loss-based learning rate scheduler. " + ], + "image_footnote": [], + "bbox": [ + 253, + 672, + 732, + 833 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Test Accuracy. Previous works on margin-based generalization bounds (Neyshabur et al., 2018; Bartlett et al., 2017; Golowich et al., 2018; Li et al., 2018a; Wei et al., 2019; Banburski et al., ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "2019) usually suggest that a larger margin implies a better generalization bound. To see whether the generalization error also gets smaller in practice, we plot train and test accuracy for both MNIST and CIFAR-10. As shown in Figure 7, the test accuracy changes only slightly after training with lossbased learning rate scheduling for 10000 epochs, although the normalized margin does increase a lot. We leave it as a future work to study this interesting gap between margin-based generalization bound and generalization error. Concurrent to this work, Wei & Ma (2020) proposed a generalization bound based on a new notion of margin called all-layer margin, and showed via experiments that enlarging all-layer margin can indeed improve generalization. It would be an interesting research direction to study how different definitions of margin may lead to different generalization abilities. ", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 48 + }, + { + "type": "image", + "img_path": "images/30c454b39b58b3eb361944408c107dca51e6a4f80df7e6e0667b0ce844855b9b.jpg", + "image_caption": [ + "Figure 7: (Left). Training and test accuracy during training CNNs without bias on MNIST, using SGD with the loss-based learning rate scheduler. Every number is averaged over 10 runs. (Right). Training and test accuracy during training VGGNet without bias on CIFAR-10, using SGD with the loss-based learning rate scheduler. Every number is averaged over 3 runs. " + ], + "image_footnote": [], + "bbox": [ + 271, + 247, + 725, + 404 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "K.2 EVALUATION FOR ROBUSTNESS ", + "text_level": 1, + "bbox": [ + 176, + 505, + 437, + 520 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Recently, robustness of deep learning has received considerable attention (Szegedy et al., 2013; Biggio et al., 2013; Athalye et al., 2018), since most state-of-the-arts deep neural networks are found to be very vulnerable against small but adversarial perturbations of the input points. In our experiments, we found that enlarging the normalized margin can improve the robustness. In particular, by simply training the neural network for a longer time with our loss-based learning rate, we observe noticeable improvements of $L ^ { 2 }$ -robustness on both the training set and test set. ", + "bbox": [ + 174, + 531, + 825, + 616 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "We first elaborate the relationship between the normalized margin and the robustness from a theoretical perspective. For a data point $\\boldsymbol { z } = ( \\boldsymbol { x } , y )$ , we can define the robustness (with respect to some norm $\\| \\cdot \\| )$ of a neural network $\\Phi$ for $_ z$ to be ", + "bbox": [ + 174, + 622, + 825, + 665 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/d0f4d23a57fce2b2b32f43b75f804e39c37bf29b1e16d9831407e64ddd9245cb.jpg", + "text": "$$\nR _ { \\theta } ( z ) : = \\operatorname* { i n f } _ { x ^ { \\prime } \\in X } \\left\\{ \\| x - x ^ { \\prime } \\| : ( x ^ { \\prime } , y ) { \\mathrm { ~ i s ~ m i s c l a s s i f i e d } } \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 672, + 674, + 696 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "where $X$ is the data domain (which is $[ 0 , 1 ] ^ { 3 2 \\times 3 2 }$ for MNIST). It is well-known that the normalized margin is a lower bound of $L ^ { 2 }$ -robustness for fully-connected networks (See, e.g., Theorem 4 in (Sokolic et al., 2017)). Indeed, a general relationship between those two quantities can be easily shown. Note that a data point $_ { z }$ is correctly classified iff the margin for $_ z$ , denoted as $q _ { \\pmb { \\theta } } ( z )$ , is larger than 0. For homogeneous models, the margin $q _ { \\pmb { \\theta } } ( z )$ and the normalized margin $q _ { \\hat { \\theta } } ( z )$ for $_ { \\textbf { \\em x } }$ have the same sign. If $q _ { \\hat { \\theta } } ( \\cdot ) : \\mathbb { R } ^ { d _ { \\mathrm { x } } } \\mathbb { R }$ is $\\beta$ -Lipschitz (with respect to some norm $\\| \\cdot \\| )$ , then it is easy to see that $R _ { \\pmb \\theta } ( z ) \\ge q _ { \\hat { \\pmb \\theta } } ( z ) / \\beta$ . This suggests that improving the normalize margin on the training set can improve the robustness on the training set. Therefore, our theoretical analysis suggests that training longer can improve the robustness of the model on the training set. ", + "bbox": [ + 173, + 704, + 825, + 834 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "This observation does match with our experiment results. In the experiments, we measure the $L ^ { 2 }$ - robustness of the CNN without bias for the first time its loss decreases below $1 0 ^ { - 1 0 }$ , $1 0 ^ { - 1 5 }$ , $1 0 ^ { - 2 0 }$ , $1 0 ^ { - 1 2 0 }$ (labelled as $\\mathtt { m o d e l - 1 }$ to $\\mathtt { m o d e l - 4 }$ respectively). We also measure the $L ^ { 2 }$ -robustness for the final model after training for 10000 epochs (labelled as mode $\\boldsymbol { \\perp } - \\boldsymbol { 5 }$ ), whose training loss is about $1 0 ^ { - 8 8 2 }$ . The normalized margin of each model is monotone increasing with respect to the number of epochs, as shown in Table 1. ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 48 + }, + { + "type": "image", + "img_path": "images/a031802da08e0e1c7aacc79fcc3841f0f08566e90e8c918e7dfc0c538827da67.jpg", + "image_caption": [ + "Figure 8: $L ^ { 2 }$ -robustness of the models of CNNs without bias trained for different number of epochs (see Table 1 for the statistics of each model). Figures on the first row show the robust accuracy on the training set, and figures on the second row show that on the test set. On every row, the left figure and the right figure plot the same curves but they are in different scales. From model-1 to model-4, noticeable robust accuracy improvements can be observed. The improvement of model-5 upon model-4 is marginal or nonexistent for some \u000f, but the improvement upon model-1 is always significant. " + ], + "image_footnote": [], + "bbox": [ + 223, + 104, + 740, + 406 + ], + "page_idx": 49 + }, + { + "type": "table", + "img_path": "images/1bae5f426ef5e1234f4dceca54a906ef81ac10291a2f70b9187b03b85446f257.jpg", + "table_caption": [ + "Table 1: Statistics of the CNN without bias after training for different number of epochs. " + ], + "table_footnote": [], + "table_body": "
model namenumber of epochstrain lossnormalized margintrain acctest acc
model-13810-10.045.65× 10-5100%99.3%
model-27510-15.129.50 ×10-5100%99.3%
model-310710-20.071.30 × 10-4100%99.3%
model-493510- -120.014.61 × 10-4100%99.2%
model-51000010 -881.511.18 × 10-3100%99.1%
", + "bbox": [ + 192, + 564, + 802, + 665 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "We use the standard method for evaluating $L ^ { 2 }$ -robustness in (Carlini & Wagner, 2017) and the source code from the authors with default hyperparameters6. We plot the robust accuracy (the percentage of data with robustness $> \\epsilon \\bar { }$ ) for the training set in the figures on the first row of Figure 8. It can be seen from the figures that for small $\\epsilon$ (e.g., $\\epsilon < 0 . 3 )$ ), the relative order of robust accuracy is just the order of model-1 to model-5. For relatively large $\\epsilon$ (e.g., $\\epsilon > 0 . 3$ ), the improvement of model-5 upon model-2 to model-4 becomes marginal or nonexistent in certain intervals of $\\epsilon$ , but model-1 to model-4 still have an increasing order of robust accuracy and the improvement of model-5 upon model-1 is always significant. This shows that training longer can help to improve the $L ^ { 2 }$ -robust accuracy on the training set. ", + "bbox": [ + 173, + 691, + 825, + 818 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "We also evaluate the robustness on the test set, in which a misclassified test sample is considered to have robustness 0, and plot the robust accuracy in the figures on the second row of Figure 8. It can be seen from the figures that for small $\\epsilon$ (e.g., $\\epsilon < 0 . 2 $ ), the curves of the robust accuracy of model-1 to model-5 are almost indistinguishable. However, for relatively large $\\epsilon$ (e.g., $\\epsilon > 0 . 2$ ), again, model-1 to model-4 have an increasing order of robust accuracy and the improvement of model-5 upon model-1 is always significant. This shows that training longer can also help to improve the $\\bar { L } ^ { 2 }$ -robust accuracy on the test set. ", + "bbox": [ + 173, + 825, + 825, + 895 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "We tried various different settings of hyperparameters for the evaluation method (including different learning rates, different binary search steps, etc.) and we observed that the shapes and relative positions of the curves in Figure 8 are stable across different hyperparameter settings. ", + "bbox": [ + 174, + 138, + 823, + 181 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "It is worth to note that the normalized margin and robustness do not grow in the same speed in our experiments, although the theory suggests $R _ { \\pmb \\theta } ( \\pmb z ) \\geq q _ { \\hat { \\pmb \\theta } } ( \\pmb z ) / \\beta$ . This may be because the Lipschitz constant $\\beta$ (if defined locally) is also changing during training. Combining training longer with existing techniques for constraining Lipschitz number (Anil et al., 2019; Cisse et al., 2017) could potentially alleviate this issue, and we leave it as a future work. ", + "bbox": [ + 174, + 188, + 825, + 257 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "L ADDITIONAL EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 277, + 534, + 294 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "In this section, we provide additional details of our experiments. ", + "bbox": [ + 173, + 308, + 594, + 323 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "L.1 LOSS-BASED LEARNING RATE SCHEDULING ", + "text_level": 1, + "bbox": [ + 173, + 338, + 526, + 354 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "The intuition of the loss-based learning rate scheduling is as follows. If the training loss is $\\alpha$ - smooth, then optimization theory suggests that we should set the learning rate to roughly $1 / \\alpha$ . For a homogeneous model with cross-entropy loss, if the training accuracy is $1 0 0 \\%$ at $\\pmb \\theta$ , then a simple calculation can show that the smoothness (the $L ^ { 2 }$ -norm of the Hessian matrix) at $\\pmb \\theta$ is $\\mathcal { \\dot { O } } ( \\bar { \\mathcal { L } } { \\cdot } \\mathrm { p o l y } ( \\bar { \\rho } ) )$ , where $\\bar { \\mathcal { L } }$ is the average training loss and $\\operatorname { p o l y } ( \\rho )$ is some polynomial. Motivated by this fact, we parameterize the learning rate $\\eta ( t )$ at epoch $t$ as ", + "bbox": [ + 173, + 364, + 825, + 449 + ], + "page_idx": 50 + }, + { + "type": "equation", + "img_path": "images/e1acbfda557de355cf9063f958080240c4483fab266230d0857617ac51c19f2a.jpg", + "text": "$$\n\\eta ( t ) : = \\frac { \\alpha ( t ) } { \\bar { \\mathcal { L } } ( t - 1 ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 436, + 453, + 560, + 487 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "where $\\bar { \\mathcal { L } } ( t - 1 )$ is the average training loss at epoch $t - 1$ , and $\\alpha ( t )$ is a relative learning rate to be tuned (Similar parameterization has been considiered in (Nacson et al., 2019b) for linear model). The loss-based learning rate scheduling is indeed a variant of line search. In particular, we initialize $\\alpha ( 0 )$ by some value, and do the following at each epoch $t$ : ", + "bbox": [ + 173, + 491, + 825, + 549 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Step 1. Initially $\\alpha ( t ) \\alpha ( t - 1 )$ ; Let $\\bar { \\mathcal { L } } ( t - 1 )$ be the training loss at the end of the last epoch; \nStep 2. Run SGD through the whole training set with learning rate $\\eta ( t ) : = \\alpha ( t ) / \\mathcal { L } ( t - 1 )$ ; \nStep 3. Evaluate the training loss $\\bar { \\mathcal { L } } ( t )$ on the whole training set; \nStep 4. If $\\bar { \\mathcal { L } } ( t ) < \\bar { \\mathcal { L } } ( t - 1 )$ , $\\alpha ( t ) \\gets \\alpha ( t ) \\cdot r _ { u }$ and end this epoch; otherwise, $\\alpha ( t ) \\gets \\alpha ( t ) / r _ { d }$ and go to Step 2. ", + "bbox": [ + 173, + 558, + 823, + 643 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "In all our experiments, we set $\\alpha ( 0 ) : = 0 . 1 , r _ { u } : = 2 ^ { 1 / 5 } \\approx 1 . 1 4 9 , r _ { d } : = 2 ^ { 1 / 1 0 } \\approx 1 . 0 7 2 .$ This specific choice of those hyperparameters is not important; other choices can only affact the computational efficiency, but not the overall tendency of normalized margin. ", + "bbox": [ + 174, + 654, + 825, + 698 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "L.2 ADDRESSING NUMERICAL ISSUES ", + "text_level": 1, + "bbox": [ + 176, + 713, + 454, + 728 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Since we are dealing with extremely small loss (as small as $1 0 ^ { - 8 0 0 }$ ), the current Tensorflow implementation would run into numerical issues. To address the issues, we work as follows. Let $\\bar { \\mathcal { L } } _ { B } ( \\pmb { \\theta } )$ be the (average) training loss within a batch $B \\subseteq [ N ]$ . We use the notations $C , s _ { n j } , \\tilde { q } _ { n } , q _ { n }$ from Appendix G. We only need to show how to perform forward and backward passes for $\\bar { \\mathcal { L } } _ { B } ( \\pmb { \\theta } )$ . ", + "bbox": [ + 174, + 739, + 825, + 797 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Forward Pass. Suppose we have a good estimate $\\widetilde { \\mathcal F }$ for $\\log \\bar { \\mathcal { L } } _ { B } ( \\pmb { \\theta } )$ in the sense that ", + "bbox": [ + 174, + 805, + 730, + 821 + ], + "page_idx": 50 + }, + { + "type": "equation", + "img_path": "images/5c2a3726840477c1ca9c014ae159497fd235ae8d17968e9ed3a468c1f67e544f.jpg", + "text": "$$\n\\mathcal { R } _ { B } ( \\pmb { \\theta } ) : = \\bar { \\mathcal { L } } _ { B } ( \\pmb { \\theta } ) e ^ { - \\widetilde { \\mathcal { F } } } = \\frac { 1 } { B } \\sum _ { n \\in B } \\log \\left( 1 + \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } ( \\pmb { \\theta } ) } \\right) e ^ { - \\widetilde { \\mathcal { F } } }\n$$", + "text_format": "latex", + "bbox": [ + 285, + 825, + 712, + 876 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "is in the range of float64. ${ \\mathcal { R } } _ { B } ( \\theta )$ can be thought of a relative training loss with respect to $\\widetilde { \\mathcal F }$ . Instead of evaluating the training loss $\\bar { \\mathcal { L } } _ { B } ( \\pmb { \\theta } )$ directly, we turn to evaluate this relative training loss in a numerically stable way: ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Step 1. Perform forward pass to compute the values of $s _ { n j }$ with float32, and convert them into float64; ", + "bbox": [ + 173, + 103, + 825, + 133 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Step 2. Let $Q : = 3 0$ . If $q _ { n } ( \\pmb { \\theta } ) > Q$ for all $n \\in B$ , then we compute ", + "bbox": [ + 173, + 136, + 620, + 151 + ], + "page_idx": 51 + }, + { + "type": "equation", + "img_path": "images/8c874cd618b49964d4eb2317212fc9c6e2cfc506277a81535a328ab2667e57a0.jpg", + "text": "$$\n\\mathcal { R } _ { B } ( \\pmb { \\theta } ) = \\frac { 1 } { B } \\sum _ { n \\in B } e ^ { - ( \\tilde { q } _ { n } ( \\pmb { \\theta } ) + \\widetilde { \\mathcal { F } } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 419, + 157, + 629, + 196 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\widetilde { q } _ { n } ( \\pmb { \\theta } ) : = - \\mathrm { L S E } ( \\{ - s _ { n j } : j \\neq y _ { n } \\} ) = - \\log \\left( \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } \\right) } \\end{array}$ is evaluated in a numerically stable way; otherwise, we compute ", + "bbox": [ + 225, + 203, + 825, + 241 + ], + "page_idx": 51 + }, + { + "type": "equation", + "img_path": "images/ceb3a3b01e95a4b027ea59982e6535985f5b4dff449371bc0c488b3ecaab308c.jpg", + "text": "$$\n\\mathcal { R } _ { B } ( \\pmb { \\theta } ) = \\frac { 1 } { B } \\sum _ { n \\in B } \\log 1 \\mathrm { p } \\left( \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } ( \\pmb { \\theta } ) } \\right) e ^ { - \\widetilde { \\mathcal { F } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 366, + 246, + 684, + 297 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "where $\\log 1 \\mathrm { p } ( x )$ is a numerical stable implementation of $\\log ( 1 + x )$ . ", + "bbox": [ + 227, + 303, + 674, + 319 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "This algorithm can be explained as follows. Step 1 is numerically stable because we observe from the experiments that the layer weights and layer outputs grow slowly. Now we consider Step 2. If $q _ { n } ( \\pmb \\theta ) \\leq Q$ for some $n \\in [ B ]$ , then $\\bar { \\mathcal { L } } _ { B } ( \\pmb { \\theta } ) \\dot { = } \\Omega ( \\dot { e } ^ { - Q } )$ is in the range of float64, so we can compute ${ \\mathcal { R } } _ { B } ( \\theta )$ by (28) directly except that we need to use a numerical stable implementation of $\\log ( 1 + x )$ . For $q _ { n } ( \\pmb { \\theta } ) > Q$ , arithmetic underflow can occur. By Taylor expansion of $\\log ( 1 + x )$ , we know that when $x$ is small enough $\\log ( 1 + x ) \\approx x$ in the sense that the relative error $\\textstyle { \\frac { | \\log ( 1 + x ) - x | } { \\log ( 1 + x ) } } =$ $O ( x )$ . Thus, we can do the following approximation ", + "bbox": [ + 173, + 328, + 826, + 435 + ], + "page_idx": 51 + }, + { + "type": "equation", + "img_path": "images/0b0f8f7a74e0ebb8111a5fcaba2c23c8c7f4fa8332e8c5e6ba2ebecfbad2b04c.jpg", + "text": "$$\n\\log \\left( 1 + \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } ( \\theta ) } \\right) e ^ { - \\widetilde { \\mathcal { F } } } \\approx \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } ( \\theta ) } \\cdot e ^ { - \\widetilde { \\mathcal { F } } }\n$$", + "text_format": "latex", + "bbox": [ + 320, + 440, + 679, + 491 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "for $q _ { n } ( \\pmb { \\theta } ) > Q$ , and only introduce a relative error of $O ( C e ^ { - Q } )$ (recall that $C$ is the number of classes). Using a numerical stable implementation of LSE, we can compute $\\tilde { q } _ { n }$ easily. Then the RHS of (29) can be rewritten as $e ^ { - ( \\tilde { q } _ { n } ( \\pmb { \\theta } ) + \\widetilde { \\mathcal { F } } ) }$ . Note that computing $e ^ { - ( \\tilde { q } _ { n } ( \\pmb { \\theta } ) + \\widetilde { \\mathcal { F } } ) }$ does not have underflow or overflow problems if $\\widetilde { \\mathcal F }$ is a good approximation for $\\log \\bar { \\mathcal { L } } _ { B } ( \\pmb { \\theta } )$ . ", + "bbox": [ + 173, + 497, + 826, + 561 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Backward Pass. To perform backward pass, we build a computation graph in Tensorflow for the above forward pass for the relative training loss and use the automatic differentiation. We parameterize the learning rate as $\\boldsymbol \\eta = \\boldsymbol \\hat { \\eta } \\cdot \\boldsymbol e ^ { \\widetilde { \\mathcal F } }$ . Then it is easy to see that taking a step of gradient descent for $\\mathcal { L } _ { B } ( \\pmb { \\theta } )$ with learning rate $\\eta$ is equivalent to taking a step for ${ \\mathcal { R } } _ { B } ( \\theta )$ with $\\hat { \\eta }$ . Thus, as long as $\\hat { \\eta }$ can fit into float64, we can perform gradient descent on ${ \\mathcal { R } } _ { B } ( \\theta )$ to ensure numerical stability. ", + "bbox": [ + 173, + 566, + 825, + 641 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "The Choice of $\\widetilde { \\mathcal F }$ . The only question remains is how to choose $\\widetilde { \\mathcal F }$ . In our experiments, we set $\\widetilde { \\mathcal { F } } ( t ) : = \\log \\bar { \\mathcal { L } } ( t - 1 )$ to be the training loss at the end of the last epoch, since the training loss cannot change a lot within one single epoch. For this, we need to maintain $\\log \\bar { \\mathcal { L } } ( t )$ during training. This can be done as follows: after evaluating the relative training loss $\\mathcal { R } ( t )$ on the whole training set, we can obtain $\\log \\bar { \\mathcal { L } } ( t )$ by adding $\\widetilde { \\mathcal { F } } ( t )$ and $\\log \\mathcal { R } ( t )$ together. ", + "bbox": [ + 173, + 646, + 825, + 727 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "It is worth noting that with this choice of $\\widetilde { \\mathcal F }$ , $\\hat { \\eta } ( t ) = \\alpha ( t )$ in the loss-based learning rate scheduling. As shown in the right figure of Figure 4, $\\alpha ( t )$ is always between $1 0 ^ { - 9 }$ and $1 0 ^ { 0 }$ , which ensures the numerical stability of backward pass. ", + "bbox": [ + 174, + 732, + 825, + 779 + ], + "page_idx": 51 + } +] \ No newline at end of file diff --git a/parse/train/SJeLIgBKPS/SJeLIgBKPS_middle.json b/parse/train/SJeLIgBKPS/SJeLIgBKPS_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..15bda6e167a135ea273b202574a214335e855ddc --- /dev/null +++ b/parse/train/SJeLIgBKPS/SJeLIgBKPS_middle.json @@ -0,0 +1,185296 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 506, + 98 + ], + "score": 1.0, + "content": "GRADIENT DESCENT MAXIMIZES THE MARGIN OF", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 98, + 385, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 385, + 118 + ], + "score": 1.0, + "content": "HOMOGENEOUS NEURAL NETWORKS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 135, + 407, + 190 + ], + "lines": [ + { + "bbox": [ + 112, + 136, + 211, + 147 + ], + "spans": [ + { + "bbox": [ + 112, + 136, + 211, + 147 + ], + "score": 1.0, + "content": "Kaifeng Lyu & Jian Li", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 145, + 317, + 158 + ], + "spans": [ + { + "bbox": [ + 112, + 145, + 317, + 158 + ], + "score": 1.0, + "content": "Institute for Interdisciplinary Information Sciences", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 111, + 156, + 195, + 169 + ], + "spans": [ + { + "bbox": [ + 111, + 156, + 195, + 169 + ], + "score": 1.0, + "content": "Tsinghua University", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 111, + 168, + 174, + 180 + ], + "spans": [ + { + "bbox": [ + 111, + 168, + 174, + 180 + ], + "score": 1.0, + "content": "Beijing, China", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 111, + 178, + 408, + 191 + ], + "spans": [ + { + "bbox": [ + 111, + 178, + 408, + 191 + ], + "score": 1.0, + "content": "vfleaking@gmail.com,lijian83@mail.tsinghua.edu.cn", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 278, + 219, + 333, + 231 + ], + "lines": [ + { + "bbox": [ + 276, + 218, + 336, + 234 + ], + "spans": [ + { + "bbox": [ + 276, + 218, + 336, + 234 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 143, + 243, + 468, + 429 + ], + "lines": [ + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "score": 1.0, + "content": "In this paper, we study the implicit regularization of the gradient descent algorithm", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "score": 1.0, + "content": "in homogeneous neural networks, including fully-connected and convolutional", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 265, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 469, + 277 + ], + "score": 1.0, + "content": "neural networks with ReLU or LeakyReLU activations. In particular, we study", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 275, + 470, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 470, + 289 + ], + "score": 1.0, + "content": "the gradient descent or gradient flow (i.e., gradient descent with infinitesimal step", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "score": 1.0, + "content": "size) optimizing the logistic loss or cross-entropy loss of any homogeneous model", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "score": 1.0, + "content": "(possibly non-smooth), and show that if the training loss decreases below a certain", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "score": 1.0, + "content": "threshold, then we can define a smoothed version of the normalized margin which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 320, + 469, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 469, + 332 + ], + "score": 1.0, + "content": "increases over time. We also formulate a natural constrained optimization problem", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 331, + 469, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 469, + 343 + ], + "score": 1.0, + "content": "related to margin maximization, and prove that both the normalized margin and its", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 340, + 469, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 469, + 355 + ], + "score": 1.0, + "content": "smoothed version converge to the objective value at a KKT point of the optimiza-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 352, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 352, + 469, + 366 + ], + "score": 1.0, + "content": "tion problem. 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We conduct several experiments to justify our theoretical", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 397, + 469, + 409 + ], + "spans": [ + { + "bbox": [ + 142, + 397, + 469, + 409 + ], + "score": 1.0, + "content": "finding on MNIST and CIFAR-10 datasets. 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To achieve a better understanding,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "previous works have studied the implicit bias of gradient descent in different settings. One simple", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "but insightful setting is linear logistic regression on linearly separable data. In this setting, the model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 104, + 517, + 253, + 531 + ], + "score": 1.0, + "content": "is parameterized by a weight vector", + "type": "text" + }, + { + "bbox": [ + 253, + 519, + 262, + 528 + ], + "score": 0.72, + "content": "\\textbf { \\em w }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 517, + 438, + 531 + ], + "score": 1.0, + "content": ", and the class prediction for any data point", + "type": "text" + }, + { + "bbox": [ + 439, + 519, + 447, + 528 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "is determined", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 167, + 543 + ], + "score": 1.0, + "content": "by the sign of", + "type": "text" + }, + { + "bbox": [ + 168, + 528, + 190, + 539 + ], + "score": 0.9, + "content": "{ \\pmb w } ^ { \\top } { \\pmb x }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 528, + 318, + 543 + ], + "score": 1.0, + "content": ". Therefore, only the direction", + "type": "text" + }, + { + "bbox": [ + 319, + 528, + 357, + 541 + ], + "score": 0.93, + "content": "{ w / \\| \\boldsymbol { w } \\| _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 528, + 505, + 543 + ], + "score": 1.0, + "content": "is important for making prediction.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "Soudry et al. (2018a;b); Ji & Telgarsky (2018; 2019c); Nacson et al. 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A nat-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "ural and important question is to what extent gradient descent has similiar implicit bias for modern", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "deep neural networks. For theoretical analysis, a natural candidate is to consider homogeneous neu-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 255, + 668 + ], + "score": 1.0, + "content": "ral networks. Here a neural network", + "type": "text" + }, + { + "bbox": [ + 256, + 657, + 264, + 666 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "is said to be (positively) homogeneous if there is a number", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 107, + 667, + 133, + 678 + ], + "score": 0.89, + "content": "L > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 667, + 319, + 680 + ], + "score": 1.0, + "content": "(called the order) such that the network output", + "type": "text" + }, + { + "bbox": [ + 320, + 667, + 352, + 679 + ], + "score": 0.94, + "content": "\\Phi ( \\pmb \\theta ; \\pmb x )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 667, + 383, + 680 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 383, + 668, + 390, + 677 + ], + "score": 0.79, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "stands for the parameter and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 678, + 290, + 692 + ], + "spans": [ + { + "bbox": [ + 107, + 680, + 114, + 688 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 678, + 290, + 692 + ], + "score": 1.0, + "content": "stands for the input, satisfies the following:", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 693, + 403, + 707 + ], + "lines": [ + { + "bbox": [ + 208, + 693, + 403, + 707 + ], + "spans": [ + { + "bbox": [ + 208, + 693, + 403, + 707 + ], + "score": 0.83, + "content": "\\forall c > 0 : \\Phi ( c \\pmb \\theta ; \\pmb x ) = c ^ { L } \\Phi ( \\pmb \\theta ; \\pmb x ) \\mathrm { f o r ~ a l l ~ } \\pmb \\theta \\mathrm { ~ a n d ~ } \\pmb x .", + "type": "interline_equation", + "image_path": "e3ce4a0c9a7c979bfaa42d4d19d9cc3e042356de9fed3b5d78284d8b31c44bbd.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 208, + 693, + 403, + 707 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 507, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "It is important to note that many neural networks are homogeneous (Neyshabur et al., 2015a; Du", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "et al., 2018). 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In particular, we study", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 275, + 470, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 470, + 289 + ], + "score": 1.0, + "content": "the gradient descent or gradient flow (i.e., gradient descent with infinitesimal step", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "score": 1.0, + "content": "size) optimizing the logistic loss or cross-entropy loss of any homogeneous model", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "score": 1.0, + "content": "(possibly non-smooth), and show that if the training loss decreases below a certain", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "score": 1.0, + "content": "threshold, then we can define a smoothed version of the normalized margin which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 320, + 469, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 469, + 332 + ], + "score": 1.0, + "content": "increases over time. We also formulate a natural constrained optimization problem", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 331, + 469, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 469, + 343 + ], + "score": 1.0, + "content": "related to margin maximization, and prove that both the normalized margin and its", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 340, + 469, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 469, + 355 + ], + "score": 1.0, + "content": "smoothed version converge to the objective value at a KKT point of the optimiza-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 352, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 352, + 469, + 366 + ], + "score": 1.0, + "content": "tion problem. Our results generalize the previous results for logistic regression", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 364, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 364, + 469, + 376 + ], + "score": 1.0, + "content": "with one-layer or multi-layer linear networks, and provide more quantitative con-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 376, + 469, + 387 + ], + "spans": [ + { + "bbox": [ + 142, + 376, + 469, + 387 + ], + "score": 1.0, + "content": "vergence results with weaker assumptions than previous results for homogeneous", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 386, + 470, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 470, + 398 + ], + "score": 1.0, + "content": "smooth neural networks. We conduct several experiments to justify our theoretical", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 397, + 469, + 409 + ], + "spans": [ + { + "bbox": [ + 142, + 397, + 469, + 409 + ], + "score": 1.0, + "content": "finding on MNIST and CIFAR-10 datasets. Finally, as margin is closely related", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 407, + 469, + 420 + ], + "spans": [ + { + "bbox": [ + 141, + 407, + 469, + 420 + ], + "score": 1.0, + "content": "to robustness, we discuss potential benefits of training longer for improving the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 419, + 242, + 430 + ], + "spans": [ + { + "bbox": [ + 142, + 419, + 242, + 430 + ], + "score": 1.0, + "content": "robustness of the model.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 16, + "bbox_fs": [ + 141, + 243, + 470, + 430 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 449, + 206, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 208, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 208, + 465 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "A major open question in deep learning is why gradient descent or its variants, are biased towards", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "solutions with good generalization performance on the test set. To achieve a better understanding,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "previous works have studied the implicit bias of gradient descent in different settings. One simple", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "but insightful setting is linear logistic regression on linearly separable data. 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Therefore, only the direction", + "type": "text" + }, + { + "bbox": [ + 319, + 528, + 357, + 541 + ], + "score": 0.93, + "content": "{ w / \\| \\boldsymbol { w } \\| _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 528, + 505, + 543 + ], + "score": 1.0, + "content": "is important for making prediction.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "Soudry et al. (2018a;b); Ji & Telgarsky (2018; 2019c); Nacson et al. (2019c) investigated this prob-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 258, + 564 + ], + "score": 1.0, + "content": "lem and proved that the direction of", + "type": "text" + }, + { + "bbox": [ + 258, + 552, + 268, + 560 + ], + "score": 0.69, + "content": "\\pmb { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 549, + 460, + 564 + ], + "score": 1.0, + "content": "converges to the direction that maximizes the", + "type": "text" + }, + { + "bbox": [ + 460, + 550, + 472, + 560 + ], + "score": 0.87, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "-margin", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 182, + 574 + ], + "score": 1.0, + "content": "while the norm of", + "type": "text" + }, + { + "bbox": [ + 183, + 563, + 192, + 571 + ], + "score": 0.75, + "content": "\\pmb { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 560, + 241, + 574 + ], + "score": 1.0, + "content": "diverges to", + "type": "text" + }, + { + "bbox": [ + 241, + 563, + 260, + 572 + ], + "score": 0.88, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 560, + 310, + 574 + ], + "score": 1.0, + "content": ", if we train", + "type": "text" + }, + { + "bbox": [ + 310, + 563, + 320, + 571 + ], + "score": 0.69, + "content": "\\pmb { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "with (stochastic) gradient descent on logistic", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "loss. 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A nat-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "ural and important question is to what extent gradient descent has similiar implicit bias for modern", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "deep neural networks. For theoretical analysis, a natural candidate is to consider homogeneous neu-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 255, + 668 + ], + "score": 1.0, + "content": "ral networks. Here a neural network", + "type": "text" + }, + { + "bbox": [ + 256, + 657, + 264, + 666 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "is said to be (positively) homogeneous if there is a number", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 107, + 667, + 133, + 678 + ], + "score": 0.89, + "content": "L > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 667, + 319, + 680 + ], + "score": 1.0, + "content": "(called the order) such that the network output", + "type": "text" + }, + { + "bbox": [ + 320, + 667, + 352, + 679 + ], + "score": 0.94, + "content": "\\Phi ( \\pmb \\theta ; \\pmb x )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 667, + 383, + 680 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 383, + 668, + 390, + 677 + ], + "score": 0.79, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "stands for the parameter and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 678, + 290, + 692 + ], + "spans": [ + { + "bbox": [ + 107, + 680, + 114, + 688 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 678, + 290, + 692 + ], + "score": 1.0, + "content": "stands for the input, satisfies the following:", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 624, + 506, + 692 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 693, + 403, + 707 + ], + "lines": [ + { + "bbox": [ + 208, + 693, + 403, + 707 + ], + "spans": [ + { + "bbox": [ + 208, + 693, + 403, + 707 + ], + "score": 0.83, + "content": "\\forall c > 0 : \\Phi ( c \\pmb \\theta ; \\pmb x ) = c ^ { L } \\Phi ( \\pmb \\theta ; \\pmb x ) \\mathrm { f o r ~ a l l ~ } \\pmb \\theta \\mathrm { ~ a n d ~ } \\pmb x .", + "type": "interline_equation", + "image_path": "e3ce4a0c9a7c979bfaa42d4d19d9cc3e042356de9fed3b5d78284d8b31c44bbd.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 208, + 693, + 403, + 707 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 507, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "It is important to note that many neural networks are homogeneous (Neyshabur et al., 2015a; Du", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "et al., 2018). For example, deep fully-connected neural networks or deep CNNs with ReLU or", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 709, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 504, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 495, + 96 + ], + "score": 1.0, + "content": "LeakyReLU activations can be made homogeneous if we remove all the bias terms, and the order", + "type": "text" + }, + { + "bbox": [ + 496, + 83, + 504, + 92 + ], + "score": 0.59, + "content": "L", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 267, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 267, + 106 + ], + "score": 1.0, + "content": "is exactly equal to the number of layers.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "In (Wei et al., 2019), it is shown that the regularization path does converge to the max-margin di-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "rection for homogeneous neural networks with cross-entropy or logistic loss. This result suggests", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "that gradient descent or gradient flow may also converges to the max-margin direction by assuming", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "homogeneity, and this is indeed true for some sub-classes of homogeneous neural networks. For gra-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "dient flow, this convergent direction is proven for linear fully-connected networks (Ji & Telgarsky,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "2019a). For gradient descent on linear fully-connected and convolutional networks, (Gunasekar", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "et al., 2018b) formulate a constrained optimization problem related to margin maximization and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "prove that gradient descent converges to the direction of a KKT point or even the max-margin di-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "rection, under various assumptions including the convergence of loss and gradient directions. In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "an independent work, (Nacson et al., 2019a) generalize the result in (Gunasekar et al., 2018b) to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 452, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 452, + 232 + ], + "score": 1.0, + "content": "smooth homogeneous models (we will discuss this work in more details in Section 2).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 245, + 198, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 199, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 199, + 258 + ], + "score": 1.0, + "content": "1.1 MAIN RESULTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 267, + 504, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 267, + 504, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 504, + 279 + ], + "score": 1.0, + "content": "In this paper, we identify a minimal set of assumptions for proving our theoretical results for ho-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "mogeneous neural networks on classification tasks. Besides homogeneity, we make two additional", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 289, + 162, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 162, + 301 + ], + "score": 1.0, + "content": "assumptions:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 128, + 309, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 130, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 130, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "1. Exponential-type Loss Function. We require the loss function to have certain exponential", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "tail (see Appendix A for the details). This assumption is not restrictive as it includes the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 332, + 492, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 492, + 345 + ], + "score": 1.0, + "content": "most popular classfication losses: exponential loss, logistic loss and cross-entropy loss.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 128, + 347, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 128, + 347, + 505, + 361 + ], + "score": 1.0, + "content": "2. Separability. The neural network can separate the training data during training (i.e., the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 358, + 361, + 372 + ], + "spans": [ + { + "bbox": [ + 141, + 358, + 254, + 372 + ], + "score": 1.0, + "content": "neural network can achieve", + "type": "text" + }, + { + "bbox": [ + 254, + 358, + 279, + 369 + ], + "score": 0.88, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 358, + 361, + 372 + ], + "score": 1.0, + "content": "training accuracy)1.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "While the first assumption is natural, the second requires some explanation. In fact, we assume", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 178, + 403 + ], + "score": 1.0, + "content": "that at some time", + "type": "text" + }, + { + "bbox": [ + 179, + 392, + 189, + 402 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 391, + 505, + 403 + ], + "score": 1.0, + "content": ", the training loss is smaller than a threshold, and the threshold here is chosen", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 355, + 415 + ], + "score": 1.0, + "content": "to be so small that the training accuracy is guaranteed to be", + "type": "text" + }, + { + "bbox": [ + 356, + 402, + 380, + 413 + ], + "score": 0.88, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "(e.g., for the logistic loss and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 412, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 294, + 427 + ], + "score": 1.0, + "content": "cross-entropy loss, the threshold can be set to", + "type": "text" + }, + { + "bbox": [ + 295, + 414, + 312, + 423 + ], + "score": 0.48, + "content": "\\mathrm { l n } 2", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 412, + 506, + 427 + ], + "score": 1.0, + "content": "). Empirically, state-of-the-art CNNs for image", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "classification can even fit randomly labeled data easily (Zhang et al., 2017). Recent theoretical work", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "on over-parameterized neural networks (Allen-Zhu et al., 2019; Zou et al., 2018) show that gradient", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 446, + 504, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 504, + 458 + ], + "score": 1.0, + "content": "descent can fit the training data if the width is large enough. Furthermore, in order to study the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "margin, ensuring the training data can be separated is inevitable; otherwise, there is no positive", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 468, + 300, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 300, + 481 + ], + "score": 1.0, + "content": "margin between the data and decision boundary.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 108, + 484, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "Our Contribution. Similar to linear models, for homogeneous models, only the direction of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 148, + 508 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 149, + 496, + 155, + 506 + ], + "score": 0.76, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 496, + 424, + 508 + ], + "score": 1.0, + "content": "is important for making predictions, and one can see that the margin", + "type": "text" + }, + { + "bbox": [ + 425, + 496, + 445, + 507 + ], + "score": 0.92, + "content": "\\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "scales linearly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 127, + 520 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 506, + 150, + 519 + ], + "score": 0.92, + "content": "\\lVert \\pmb { \\theta } \\rVert _ { 2 } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 505, + 269, + 520 + ], + "score": 1.0, + "content": ", when fixing the direction of", + "type": "text" + }, + { + "bbox": [ + 269, + 507, + 276, + 516 + ], + "score": 0.66, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 505, + 396, + 520 + ], + "score": 1.0, + "content": ". To compare margins among", + "type": "text" + }, + { + "bbox": [ + 396, + 507, + 403, + 516 + ], + "score": 0.76, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "in different directions, it", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 516, + 374, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 288, + 531 + ], + "score": 1.0, + "content": "makes sense to study the normalized margin,", + "type": "text" + }, + { + "bbox": [ + 288, + 517, + 371, + 530 + ], + "score": 0.93, + "content": "\\bar { \\gamma } ( \\pmb \\theta ) : = \\gamma ( \\pmb \\theta ) / \\| \\pmb \\theta \\| _ { 2 } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 516, + 374, + 531 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 395, + 547 + ], + "score": 1.0, + "content": "In this paper, we focus on the training dynamics of the network after", + "type": "text" + }, + { + "bbox": [ + 396, + 536, + 405, + 546 + ], + "score": 0.86, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 534, + 454, + 547 + ], + "score": 1.0, + "content": "(recall that", + "type": "text" + }, + { + "bbox": [ + 455, + 536, + 464, + 546 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "is a time", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "score": 1.0, + "content": "that the training loss is less than the threshold). Our theoretical results can answer the following", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 282, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 282, + 570 + ], + "score": 1.0, + "content": "questions regarding the normalized margin.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "First, how does the normalized margin change during training? The answer may seem complicated", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 336, + 596 + ], + "score": 1.0, + "content": "since one can easily come up with examples in which", + "type": "text" + }, + { + "bbox": [ + 336, + 586, + 344, + 596 + ], + "score": 0.83, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "increases or decreases in a short time", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "interval. However, we can show that the overall trend of the normalized margin is to increase in the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 455, + 618 + ], + "score": 1.0, + "content": "following sense: there exists a smoothed version of the normalized margin, denoted as", + "type": "text" + }, + { + "bbox": [ + 455, + 607, + 462, + 618 + ], + "score": 0.85, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 606, + 505, + 618 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 617, + 374, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 120, + 630 + ], + "score": 1.0, + "content": "(1)", + "type": "text" + }, + { + "bbox": [ + 121, + 617, + 172, + 629 + ], + "score": 0.9, + "content": "| \\tilde { \\gamma } - \\bar { \\tilde { \\gamma } } | \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 617, + 183, + 630 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 183, + 618, + 214, + 627 + ], + "score": 0.89, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 617, + 235, + 630 + ], + "score": 1.0, + "content": "; and", + "type": "text" + }, + { + "bbox": [ + 235, + 618, + 256, + 629 + ], + "score": 0.39, + "content": "( 2 ) \\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 617, + 343, + 630 + ], + "score": 1.0, + "content": "is non-decreasing for", + "type": "text" + }, + { + "bbox": [ + 344, + 618, + 370, + 628 + ], + "score": 0.91, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 617, + 374, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "Second, how large is the normalized margin at convergence? To answer this question, we formulate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "a natural constrained optimization problem which aims to directly maximize the margin. We show", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 204, + 669 + ], + "score": 1.0, + "content": "that every limit point of", + "type": "text" + }, + { + "bbox": [ + 205, + 656, + 303, + 668 + ], + "score": 0.92, + "content": "\\{ \\pmb \\theta ( t ) / \\| \\pmb \\theta ( t ) \\| _ { 2 } : t > 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "is along the direction of a KKT point of the max-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "margin problem. This indicates that gradient descent/gradient flow performs margin maximization", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "implicitly in deep homogeneous networks. This result can be seen as a significant generalization of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 329, + 702 + ], + "score": 1.0, + "content": "previous works (Soudry et al., 2018a;b; Ji & Telgarsky,", + "type": "text" + }, + { + "bbox": [ + 329, + 689, + 354, + 700 + ], + "score": 0.69, + "content": "2 0 1 9 \\mathrm { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "; Gunasekar et al., 2018b) from linear", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 260, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 260, + 713 + ], + "score": 1.0, + "content": "classifiers to homogeneous classifiers.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 309, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 311, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 311, + 733 + ], + "score": 1.0, + "content": "1Note that this does NOT mean the training loss is 0.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 504, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 495, + 96 + ], + "score": 1.0, + "content": "LeakyReLU activations can be made homogeneous if we remove all the bias terms, and the order", + "type": "text" + }, + { + "bbox": [ + 496, + 83, + 504, + 92 + ], + "score": 0.59, + "content": "L", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 267, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 267, + 106 + ], + "score": 1.0, + "content": "is exactly equal to the number of layers.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 504, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "In (Wei et al., 2019), it is shown that the regularization path does converge to the max-margin di-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "rection for homogeneous neural networks with cross-entropy or logistic loss. This result suggests", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "that gradient descent or gradient flow may also converges to the max-margin direction by assuming", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "homogeneity, and this is indeed true for some sub-classes of homogeneous neural networks. For gra-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "dient flow, this convergent direction is proven for linear fully-connected networks (Ji & Telgarsky,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "2019a). For gradient descent on linear fully-connected and convolutional networks, (Gunasekar", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "et al., 2018b) formulate a constrained optimization problem related to margin maximization and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "prove that gradient descent converges to the direction of a KKT point or even the max-margin di-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "rection, under various assumptions including the convergence of loss and gradient directions. In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "an independent work, (Nacson et al., 2019a) generalize the result in (Gunasekar et al., 2018b) to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 452, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 452, + 232 + ], + "score": 1.0, + "content": "smooth homogeneous models (we will discuss this work in more details in Section 2).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 111, + 505, + 232 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 245, + 198, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 199, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 199, + 258 + ], + "score": 1.0, + "content": "1.1 MAIN RESULTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 267, + 504, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 267, + 504, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 504, + 279 + ], + "score": 1.0, + "content": "In this paper, we identify a minimal set of assumptions for proving our theoretical results for ho-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "mogeneous neural networks on classification tasks. Besides homogeneity, we make two additional", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 289, + 162, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 162, + 301 + ], + "score": 1.0, + "content": "assumptions:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 267, + 505, + 301 + ] + }, + { + "type": "list", + "bbox": [ + 128, + 309, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 130, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 130, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "1. Exponential-type Loss Function. We require the loss function to have certain exponential", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "tail (see Appendix A for the details). This assumption is not restrictive as it includes the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 332, + 492, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 492, + 345 + ], + "score": 1.0, + "content": "most popular classfication losses: exponential loss, logistic loss and cross-entropy loss.", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 347, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 128, + 347, + 505, + 361 + ], + "score": 1.0, + "content": "2. Separability. The neural network can separate the training data during training (i.e., the", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 358, + 361, + 372 + ], + "spans": [ + { + "bbox": [ + 141, + 358, + 254, + 372 + ], + "score": 1.0, + "content": "neural network can achieve", + "type": "text" + }, + { + "bbox": [ + 254, + 358, + 279, + 369 + ], + "score": 0.88, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 358, + 361, + 372 + ], + "score": 1.0, + "content": "training accuracy)1.", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + } + ], + "index": 19, + "bbox_fs": [ + 128, + 310, + 505, + 372 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "While the first assumption is natural, the second requires some explanation. In fact, we assume", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 178, + 403 + ], + "score": 1.0, + "content": "that at some time", + "type": "text" + }, + { + "bbox": [ + 179, + 392, + 189, + 402 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 391, + 505, + 403 + ], + "score": 1.0, + "content": ", the training loss is smaller than a threshold, and the threshold here is chosen", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 355, + 415 + ], + "score": 1.0, + "content": "to be so small that the training accuracy is guaranteed to be", + "type": "text" + }, + { + "bbox": [ + 356, + 402, + 380, + 413 + ], + "score": 0.88, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "(e.g., for the logistic loss and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 412, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 294, + 427 + ], + "score": 1.0, + "content": "cross-entropy loss, the threshold can be set to", + "type": "text" + }, + { + "bbox": [ + 295, + 414, + 312, + 423 + ], + "score": 0.48, + "content": "\\mathrm { l n } 2", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 412, + 506, + 427 + ], + "score": 1.0, + "content": "). Empirically, state-of-the-art CNNs for image", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "classification can even fit randomly labeled data easily (Zhang et al., 2017). Recent theoretical work", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "on over-parameterized neural networks (Allen-Zhu et al., 2019; Zou et al., 2018) show that gradient", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 446, + 504, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 504, + 458 + ], + "score": 1.0, + "content": "descent can fit the training data if the width is large enough. Furthermore, in order to study the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "margin, ensuring the training data can be separated is inevitable; otherwise, there is no positive", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 468, + 300, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 300, + 481 + ], + "score": 1.0, + "content": "margin between the data and decision boundary.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 379, + 506, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 484, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "Our Contribution. Similar to linear models, for homogeneous models, only the direction of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 148, + 508 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 149, + 496, + 155, + 506 + ], + "score": 0.76, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 496, + 424, + 508 + ], + "score": 1.0, + "content": "is important for making predictions, and one can see that the margin", + "type": "text" + }, + { + "bbox": [ + 425, + 496, + 445, + 507 + ], + "score": 0.92, + "content": "\\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "scales linearly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 127, + 520 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 506, + 150, + 519 + ], + "score": 0.92, + "content": "\\lVert \\pmb { \\theta } \\rVert _ { 2 } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 505, + 269, + 520 + ], + "score": 1.0, + "content": ", when fixing the direction of", + "type": "text" + }, + { + "bbox": [ + 269, + 507, + 276, + 516 + ], + "score": 0.66, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 505, + 396, + 520 + ], + "score": 1.0, + "content": ". To compare margins among", + "type": "text" + }, + { + "bbox": [ + 396, + 507, + 403, + 516 + ], + "score": 0.76, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "in different directions, it", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 516, + 374, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 288, + 531 + ], + "score": 1.0, + "content": "makes sense to study the normalized margin,", + "type": "text" + }, + { + "bbox": [ + 288, + 517, + 371, + 530 + ], + "score": 0.93, + "content": "\\bar { \\gamma } ( \\pmb \\theta ) : = \\gamma ( \\pmb \\theta ) / \\| \\pmb \\theta \\| _ { 2 } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 516, + 374, + 531 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 484, + 506, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 395, + 547 + ], + "score": 1.0, + "content": "In this paper, we focus on the training dynamics of the network after", + "type": "text" + }, + { + "bbox": [ + 396, + 536, + 405, + 546 + ], + "score": 0.86, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 534, + 454, + 547 + ], + "score": 1.0, + "content": "(recall that", + "type": "text" + }, + { + "bbox": [ + 455, + 536, + 464, + 546 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "is a time", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "score": 1.0, + "content": "that the training loss is less than the threshold). Our theoretical results can answer the following", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 282, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 282, + 570 + ], + "score": 1.0, + "content": "questions regarding the normalized margin.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 534, + 505, + 570 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "First, how does the normalized margin change during training? The answer may seem complicated", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 336, + 596 + ], + "score": 1.0, + "content": "since one can easily come up with examples in which", + "type": "text" + }, + { + "bbox": [ + 336, + 586, + 344, + 596 + ], + "score": 0.83, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "increases or decreases in a short time", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "interval. However, we can show that the overall trend of the normalized margin is to increase in the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 455, + 618 + ], + "score": 1.0, + "content": "following sense: there exists a smoothed version of the normalized margin, denoted as", + "type": "text" + }, + { + "bbox": [ + 455, + 607, + 462, + 618 + ], + "score": 0.85, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 606, + 505, + 618 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 617, + 374, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 120, + 630 + ], + "score": 1.0, + "content": "(1)", + "type": "text" + }, + { + "bbox": [ + 121, + 617, + 172, + 629 + ], + "score": 0.9, + "content": "| \\tilde { \\gamma } - \\bar { \\tilde { \\gamma } } | \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 617, + 183, + 630 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 183, + 618, + 214, + 627 + ], + "score": 0.89, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 617, + 235, + 630 + ], + "score": 1.0, + "content": "; and", + "type": "text" + }, + { + "bbox": [ + 235, + 618, + 256, + 629 + ], + "score": 0.39, + "content": "( 2 ) \\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 617, + 343, + 630 + ], + "score": 1.0, + "content": "is non-decreasing for", + "type": "text" + }, + { + "bbox": [ + 344, + 618, + 370, + 628 + ], + "score": 0.91, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 617, + 374, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 573, + 505, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "Second, how large is the normalized margin at convergence? To answer this question, we formulate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "a natural constrained optimization problem which aims to directly maximize the margin. We show", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 204, + 669 + ], + "score": 1.0, + "content": "that every limit point of", + "type": "text" + }, + { + "bbox": [ + 205, + 656, + 303, + 668 + ], + "score": 0.92, + "content": "\\{ \\pmb \\theta ( t ) / \\| \\pmb \\theta ( t ) \\| _ { 2 } : t > 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "is along the direction of a KKT point of the max-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "margin problem. This indicates that gradient descent/gradient flow performs margin maximization", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "implicitly in deep homogeneous networks. This result can be seen as a significant generalization of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 329, + 702 + ], + "score": 1.0, + "content": "previous works (Soudry et al., 2018a;b; Ji & Telgarsky,", + "type": "text" + }, + { + "bbox": [ + 329, + 689, + 354, + 700 + ], + "score": 0.69, + "content": "2 0 1 9 \\mathrm { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "; Gunasekar et al., 2018b) from linear", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 260, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 260, + 713 + ], + "score": 1.0, + "content": "classifiers to homogeneous classifiers.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 633, + 506, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 86, + 498, + 188 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 86, + 498, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 86, + 498, + 188 + ], + "spans": [ + { + "bbox": [ + 113, + 86, + 498, + 188 + ], + "score": 0.971, + "type": "image", + "image_path": "2e818eb552978f56e83f486e74331e74a063c35463a07ca1e458303572110c47.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 86, + 498, + 120.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 120.0, + 498, + 154.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 154.0, + 498, + 188.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 200, + 505, + 278 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 200, + 504, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 504, + 213 + ], + "score": 1.0, + "content": "Figure 1: (a) Training CNNs with and without bias on MNIST, using SGD with learning rate 0.01.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "The training loss (left) decreases over time, and the normalized margin (right) keeps increasing after", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 221, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 301, + 236 + ], + "score": 1.0, + "content": "the model is fitted, but the growth rate is slow", + "type": "text" + }, + { + "bbox": [ + 301, + 222, + 363, + 233 + ], + "score": 0.9, + "content": "( \\approx 1 . 8 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 221, + 506, + 236 + ], + "score": 1.0, + "content": "after 10000 epochs). (b) Training", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "CNNs with and without bias on MNIST, using SGD with the loss-based learning rate scheduler. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 243, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 318, + 258 + ], + "score": 1.0, + "content": "training loss (left) decreases exponentially over time", + "type": "text" + }, + { + "bbox": [ + 319, + 244, + 360, + 255 + ], + "score": 0.91, + "content": "( < 1 0 ^ { - 8 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 243, + 506, + 258 + ], + "score": 1.0, + "content": "after 9000 epochs), and the normal-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 254, + 504, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 347, + 268 + ], + "score": 1.0, + "content": "ized margin (right) increases rapidly after the model is fitted", + "type": "text" + }, + { + "bbox": [ + 348, + 255, + 404, + 266 + ], + "score": 0.9, + "content": "( \\approx 1 . 2 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 254, + 484, + 268 + ], + "score": 1.0, + "content": "after 10000 epochs,", + "type": "text" + }, + { + "bbox": [ + 485, + 256, + 504, + 267 + ], + "score": 0.85, + "content": "1 0 \\times", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 266, + 465, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 453, + 279 + ], + "score": 1.0, + "content": "larger than that of SGD with learning rate 0.01). Experimental details are in Appendix", + "type": "text" + }, + { + "bbox": [ + 453, + 267, + 461, + 276 + ], + "score": 0.26, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 266, + 465, + 279 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "As by-products of the above results, we derive tight asymptotic convergence/growth rates of the loss", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "and weights. It is shown in (Soudry et al., 2018a;b; Ji & Telgarsky, 2018; 2019c) that the loss de-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 188, + 333 + ], + "score": 1.0, + "content": "creases at the rate of", + "type": "text" + }, + { + "bbox": [ + 189, + 320, + 218, + 333 + ], + "score": 0.92, + "content": "O ( 1 / t )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 320, + 325, + 333 + ], + "score": 1.0, + "content": ", the weight norm grows as", + "type": "text" + }, + { + "bbox": [ + 326, + 321, + 360, + 333 + ], + "score": 0.92, + "content": "O ( \\log t )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "for linear logistic regression. In this", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 332, + 503, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 416, + 347 + ], + "score": 1.0, + "content": "work, we generalize the result by showing that the loss decreases at the rate of", + "type": "text" + }, + { + "bbox": [ + 416, + 332, + 503, + 345 + ], + "score": 0.92, + "content": "O ( 1 / ( t ( \\log t ) ^ { 2 - 2 / L } ) )", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 344, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 234, + 359 + ], + "score": 1.0, + "content": "and the weight norm grows as", + "type": "text" + }, + { + "bbox": [ + 234, + 344, + 290, + 357 + ], + "score": 0.93, + "content": "O ( ( \\log t ) ^ { 1 / L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 344, + 505, + 359 + ], + "score": 1.0, + "content": "for homogeneous neural networks with exponential", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 356, + 267, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 267, + 369 + ], + "score": 1.0, + "content": "loss, logistic loss, or cross-entropy loss.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "Experiments.2 The main practical implication of our theoretical result is that training longer can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "enlarge the normalized margin. To justify this claim empiricaly, we train CNNs on MNIST and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "CIFAR-10 with SGD (see Section K.1). Results on MNIST are presented in Figure 1. For constant", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "step size, we can see that the normalized margin keeps increasing, but the growth rate is rather slow", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "(because the gradient gets smaller and smaller). Inspired by our convergence results for gradient", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "descent, we use a learning rate scheduling method which enlarges the learning rate according to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "current training loss, then the training loss decreases exponentially faster and the normalized margin", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 450, + 255, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 255, + 462 + ], + "score": 1.0, + "content": "increases significantly faster as well.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "For feedforward neural networks with ReLU activation, the normalized margin on a training sample", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 206, + 490 + ], + "score": 1.0, + "content": "is closely related to the", + "type": "text" + }, + { + "bbox": [ + 207, + 477, + 219, + 488 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 477, + 286, + 490 + ], + "score": 1.0, + "content": "-robustness (the", + "type": "text" + }, + { + "bbox": [ + 286, + 477, + 298, + 488 + ], + "score": 0.89, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "-distance from the training sample to the decision", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "boundary). Indeed, the former divided by a Lipschitz constant is a lower bound for the latter. For", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 331, + 512 + ], + "score": 1.0, + "content": "example, the normalized margin is a lower bound for the", + "type": "text" + }, + { + "bbox": [ + 331, + 499, + 343, + 510 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "-robustness on fully-connected networks", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 509, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 524 + ], + "score": 1.0, + "content": "with ReLU activation (see, e.g., Theorem 4 in (Sokolic et al., 2017)). This fact suggests that training", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "longer may have potential benefits on improving the robustness of the model. In our experiments, we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 533, + 502, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 253, + 545 + ], + "score": 1.0, + "content": "observe noticeable improvements of", + "type": "text" + }, + { + "bbox": [ + 253, + 533, + 266, + 542 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 533, + 502, + 545 + ], + "score": 1.0, + "content": "-robustness on both training and test sets (see Section K.2).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 560, + 211, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 213, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 213, + 575 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 585, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "Implicit Bias in Training Linear Classifiers. For linear logistic regression on linearly separable", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "data, Soudry et al. (2018a;b) showed that full-batch gradient descent converges in the direction of the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 126, + 620 + ], + "score": 1.0, + "content": "max", + "type": "text" + }, + { + "bbox": [ + 126, + 607, + 138, + 618 + ], + "score": 0.86, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "-margin solution of the corresponding hard-margin Support Vector Machine (SVM). Subse-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "quent works extended this result in several ways: Nacson et al. (2019c) extended the results to the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "case of stochastic gradient descent; Gunasekar et al. (2018a) considered other optimization methods;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "Nacson et al. (2019b) considered other loss functions including those with poly-exponential tails;", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 651, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 506, + 665 + ], + "score": 1.0, + "content": "Ji & Telgarsky (2018; 2019c) characterized the convergence of weight direction without assuming", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 662, + 486, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 486, + 675 + ], + "score": 1.0, + "content": "separability; Ji & Telgarsky (2019b) proved a tighter convergence rate for the weight direction.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 108, + 679, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 692 + ], + "score": 1.0, + "content": "Those results on linear logistic regression have been generalized to deep linear networks. Ji & Tel-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 690, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 704 + ], + "score": 1.0, + "content": "garsky (2019a) showed that the product of weights in a deep linear network with strictly decreasing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 281, + 713 + ], + "score": 1.0, + "content": "loss converges in the direction of the max", + "type": "text" + }, + { + "bbox": [ + 282, + 701, + 294, + 711 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "-margin solution. Gunasekar et al. (2018b) showed", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 118, + 722, + 393, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 394, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 394, + 734 + ], + "score": 1.0, + "content": "2Code available: https://github.com/vfleaking/max-margin", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 86, + 498, + 188 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 86, + 498, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 86, + 498, + 188 + ], + "spans": [ + { + "bbox": [ + 113, + 86, + 498, + 188 + ], + "score": 0.971, + "type": "image", + "image_path": "2e818eb552978f56e83f486e74331e74a063c35463a07ca1e458303572110c47.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 86, + 498, + 120.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 120.0, + 498, + 154.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 154.0, + 498, + 188.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 200, + 505, + 278 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 200, + 504, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 504, + 213 + ], + "score": 1.0, + "content": "Figure 1: (a) Training CNNs with and without bias on MNIST, using SGD with learning rate 0.01.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "The training loss (left) decreases over time, and the normalized margin (right) keeps increasing after", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 221, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 301, + 236 + ], + "score": 1.0, + "content": "the model is fitted, but the growth rate is slow", + "type": "text" + }, + { + "bbox": [ + 301, + 222, + 363, + 233 + ], + "score": 0.9, + "content": "( \\approx 1 . 8 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 221, + 506, + 236 + ], + "score": 1.0, + "content": "after 10000 epochs). (b) Training", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "CNNs with and without bias on MNIST, using SGD with the loss-based learning rate scheduler. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 243, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 318, + 258 + ], + "score": 1.0, + "content": "training loss (left) decreases exponentially over time", + "type": "text" + }, + { + "bbox": [ + 319, + 244, + 360, + 255 + ], + "score": 0.91, + "content": "( < 1 0 ^ { - 8 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 243, + 506, + 258 + ], + "score": 1.0, + "content": "after 9000 epochs), and the normal-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 254, + 504, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 347, + 268 + ], + "score": 1.0, + "content": "ized margin (right) increases rapidly after the model is fitted", + "type": "text" + }, + { + "bbox": [ + 348, + 255, + 404, + 266 + ], + "score": 0.9, + "content": "( \\approx 1 . 2 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 254, + 484, + 268 + ], + "score": 1.0, + "content": "after 10000 epochs,", + "type": "text" + }, + { + "bbox": [ + 485, + 256, + 504, + 267 + ], + "score": 0.85, + "content": "1 0 \\times", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 266, + 465, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 453, + 279 + ], + "score": 1.0, + "content": "larger than that of SGD with learning rate 0.01). Experimental details are in Appendix", + "type": "text" + }, + { + "bbox": [ + 453, + 267, + 461, + 276 + ], + "score": 0.26, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 266, + 465, + 279 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "As by-products of the above results, we derive tight asymptotic convergence/growth rates of the loss", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "and weights. It is shown in (Soudry et al., 2018a;b; Ji & Telgarsky, 2018; 2019c) that the loss de-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 188, + 333 + ], + "score": 1.0, + "content": "creases at the rate of", + "type": "text" + }, + { + "bbox": [ + 189, + 320, + 218, + 333 + ], + "score": 0.92, + "content": "O ( 1 / t )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 320, + 325, + 333 + ], + "score": 1.0, + "content": ", the weight norm grows as", + "type": "text" + }, + { + "bbox": [ + 326, + 321, + 360, + 333 + ], + "score": 0.92, + "content": "O ( \\log t )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "for linear logistic regression. In this", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 332, + 503, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 416, + 347 + ], + "score": 1.0, + "content": "work, we generalize the result by showing that the loss decreases at the rate of", + "type": "text" + }, + { + "bbox": [ + 416, + 332, + 503, + 345 + ], + "score": 0.92, + "content": "O ( 1 / ( t ( \\log t ) ^ { 2 - 2 / L } ) )", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 344, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 234, + 359 + ], + "score": 1.0, + "content": "and the weight norm grows as", + "type": "text" + }, + { + "bbox": [ + 234, + 344, + 290, + 357 + ], + "score": 0.93, + "content": "O ( ( \\log t ) ^ { 1 / L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 344, + 505, + 359 + ], + "score": 1.0, + "content": "for homogeneous neural networks with exponential", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 356, + 267, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 267, + 369 + ], + "score": 1.0, + "content": "loss, logistic loss, or cross-entropy loss.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 299, + 505, + 369 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "Experiments.2 The main practical implication of our theoretical result is that training longer can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "enlarge the normalized margin. To justify this claim empiricaly, we train CNNs on MNIST and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "CIFAR-10 with SGD (see Section K.1). Results on MNIST are presented in Figure 1. For constant", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "step size, we can see that the normalized margin keeps increasing, but the growth rate is rather slow", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "(because the gradient gets smaller and smaller). Inspired by our convergence results for gradient", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "descent, we use a learning rate scheduling method which enlarges the learning rate according to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "current training loss, then the training loss decreases exponentially faster and the normalized margin", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 450, + 255, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 255, + 462 + ], + "score": 1.0, + "content": "increases significantly faster as well.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 372, + 505, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "For feedforward neural networks with ReLU activation, the normalized margin on a training sample", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 206, + 490 + ], + "score": 1.0, + "content": "is closely related to the", + "type": "text" + }, + { + "bbox": [ + 207, + 477, + 219, + 488 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 477, + 286, + 490 + ], + "score": 1.0, + "content": "-robustness (the", + "type": "text" + }, + { + "bbox": [ + 286, + 477, + 298, + 488 + ], + "score": 0.89, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "-distance from the training sample to the decision", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "boundary). Indeed, the former divided by a Lipschitz constant is a lower bound for the latter. For", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 331, + 512 + ], + "score": 1.0, + "content": "example, the normalized margin is a lower bound for the", + "type": "text" + }, + { + "bbox": [ + 331, + 499, + 343, + 510 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "-robustness on fully-connected networks", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 509, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 524 + ], + "score": 1.0, + "content": "with ReLU activation (see, e.g., Theorem 4 in (Sokolic et al., 2017)). This fact suggests that training", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "longer may have potential benefits on improving the robustness of the model. In our experiments, we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 533, + 502, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 253, + 545 + ], + "score": 1.0, + "content": "observe noticeable improvements of", + "type": "text" + }, + { + "bbox": [ + 253, + 533, + 266, + 542 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 533, + 502, + 545 + ], + "score": 1.0, + "content": "-robustness on both training and test sets (see Section K.2).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 466, + 505, + 545 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 560, + 211, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 213, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 213, + 575 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 585, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "Implicit Bias in Training Linear Classifiers. For linear logistic regression on linearly separable", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "data, Soudry et al. (2018a;b) showed that full-batch gradient descent converges in the direction of the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 126, + 620 + ], + "score": 1.0, + "content": "max", + "type": "text" + }, + { + "bbox": [ + 126, + 607, + 138, + 618 + ], + "score": 0.86, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "-margin solution of the corresponding hard-margin Support Vector Machine (SVM). Subse-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "quent works extended this result in several ways: Nacson et al. (2019c) extended the results to the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "case of stochastic gradient descent; Gunasekar et al. (2018a) considered other optimization methods;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "Nacson et al. (2019b) considered other loss functions including those with poly-exponential tails;", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 651, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 506, + 665 + ], + "score": 1.0, + "content": "Ji & Telgarsky (2018; 2019c) characterized the convergence of weight direction without assuming", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 662, + 486, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 486, + 675 + ], + "score": 1.0, + "content": "separability; Ji & Telgarsky (2019b) proved a tighter convergence rate for the weight direction.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 586, + 506, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 679, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 692 + ], + "score": 1.0, + "content": "Those results on linear logistic regression have been generalized to deep linear networks. Ji & Tel-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 690, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 704 + ], + "score": 1.0, + "content": "garsky (2019a) showed that the product of weights in a deep linear network with strictly decreasing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 281, + 713 + ], + "score": 1.0, + "content": "loss converges in the direction of the max", + "type": "text" + }, + { + "bbox": [ + 282, + 701, + 294, + 711 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "-margin solution. Gunasekar et al. (2018b) showed", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "more general results for gradient descent on linear fully-connected and convolutional networks with", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 95, + 499, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 499, + 106 + ], + "score": 1.0, + "content": "exponential loss, under various assumptions on the convergence of the loss and gradient direction.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 678, + 505, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "more general results for gradient descent on linear fully-connected and convolutional networks with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 95, + 499, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 499, + 106 + ], + "score": 1.0, + "content": "exponential loss, under various assumptions on the convergence of the loss and gradient direction.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 109, + 110, + 503, + 144 + ], + "lines": [ + { + "bbox": [ + 107, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 107, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "Margin maximization phenomenon is also studied for boosting methods (Schapire et al., 1998;", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 121, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 107, + 121, + 505, + 133 + ], + "score": 1.0, + "content": "Schapire & Freund, 2012; Shalev-Shwartz & Singer, 2010; Telgarsky, 2013) and Normalized Per-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 133, + 239, + 144 + ], + "spans": [ + { + "bbox": [ + 107, + 133, + 239, + 144 + ], + "score": 1.0, + "content": "ceptron (Ramdas & Pena, 2016).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 149, + 505, + 193 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "Implicit Bias in Training Nonlinear Classifiers. Soudry et al. (2018a) analyzed the case where", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "there is only one trainable layer of a ReLU network. Xu et al. (2018) characterized the implicit bias", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "for the model consisting of one single ReLU unit. Our work is closely related to a recent independent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 372, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 372, + 195 + ], + "score": 1.0, + "content": "work by (Nacson et al., 2019a) which we discuss in details below.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 199, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 212 + ], + "score": 1.0, + "content": "Comparison with (Nacson et al., 2019a). Very recently, (Nacson et al., 2019a) analyzed gradient", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "descent for smooth homogeneous models and proved the convergence of parameter direction to a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "score": 1.0, + "content": "KKT point of the aforementioned max-margin problem. Compared with their work, our work adopt", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "much weaker assumptions: (1) They assume the training loss converges to 0, but in our work we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 457, + 256 + ], + "score": 1.0, + "content": "only require that the training loss is lower than a small threshold value at some time", + "type": "text" + }, + { + "bbox": [ + 458, + 244, + 468, + 254 + ], + "score": 0.86, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "(and we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 300, + 267 + ], + "score": 1.0, + "content": "prove the exact convergence rate of the loss after", + "type": "text" + }, + { + "bbox": [ + 301, + 255, + 310, + 265 + ], + "score": 0.81, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "); (2) They assume the convergence of parameter", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 406, + 278 + ], + "score": 1.0, + "content": "direction3, while we prove that KKT conditions hold for all limit points of", + "type": "text" + }, + { + "bbox": [ + 406, + 264, + 501, + 277 + ], + "score": 0.92, + "content": "\\{ \\pmb \\theta ( t ) \\bar { / } \\lVert \\pmb \\theta ( t ) \\rVert _ { 2 } : t > 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 264, + 505, + 278 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "without requiring any convergence assumption; (3) They assume the convergence of the direction", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "score": 1.0, + "content": "of losses (the direction of the vector whose entries are loss values on every data point) and Linear", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "Independence Constraint Qualification (LICQ) for the max-margin problem, while we do not need", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "such assumptions. Besides the above differences in assumptions, we also prove the monotonicity of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "score": 1.0, + "content": "the normalized margin and provide tight convergence rate for training loss. We believe both results", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 239, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 239, + 343 + ], + "score": 1.0, + "content": "are interesting in their own right.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "Another technical difference is that their work analyzes discrete gradient descent on smooth homo-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "geneous models (which fails to capture ReLU networks). In our work, we analyze both gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "descent on smooth homogeneous models and also gradient flow on homogeneous models which", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 194, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 194, + 391 + ], + "score": 1.0, + "content": "could be non-smooth.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "Other Works on Implicit Bias. Banburski et al. (2019) also studied the dynamics of gradient flow", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "and among other things, provided mathematical insights to the implicit bias towards max margin", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 419, + 504, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 504, + 432 + ], + "score": 1.0, + "content": "solution for homogeneous networks. We note that their analysis of gradient flow decomposes the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "dynamics to the tangent component and radial component, which is similar to our proof of Theo-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "rem 4.1 in spirit. Wilson et al. (2017); Ali et al. (2019); Gunasekar et al. 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Compared with their work, our work adopt", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "much weaker assumptions: (1) They assume the training loss converges to 0, but in our work we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 457, + 256 + ], + "score": 1.0, + "content": "only require that the training loss is lower than a small threshold value at some time", + "type": "text" + }, + { + "bbox": [ + 458, + 244, + 468, + 254 + ], + "score": 0.86, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "(and we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 300, + 267 + ], + "score": 1.0, + "content": "prove the exact convergence rate of the loss after", + "type": "text" + }, + { + "bbox": [ + 301, + 255, + 310, + 265 + ], + "score": 0.81, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "); (2) They assume the convergence of parameter", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 406, + 278 + ], + "score": 1.0, + "content": "direction3, while we prove that KKT conditions hold for all limit points of", + "type": "text" + }, + { + "bbox": [ + 406, + 264, + 501, + 277 + ], + "score": 0.92, + "content": "\\{ \\pmb \\theta ( t ) \\bar { / } \\lVert \\pmb \\theta ( t ) \\rVert _ { 2 } : t > 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 264, + 505, + 278 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "without requiring any convergence assumption; (3) They assume the convergence of the direction", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "score": 1.0, + "content": "of losses (the direction of the vector whose entries are loss values on every data point) and Linear", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "Independence Constraint Qualification (LICQ) for the max-margin problem, while we do not need", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "such assumptions. Besides the above differences in assumptions, we also prove the monotonicity of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "score": 1.0, + "content": "the normalized margin and provide tight convergence rate for training loss. We believe both results", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 239, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 239, + 343 + ], + "score": 1.0, + "content": "are interesting in their own right.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 198, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "Another technical difference is that their work analyzes discrete gradient descent on smooth homo-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "geneous models (which fails to capture ReLU networks). In our work, we analyze both gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "descent on smooth homogeneous models and also gradient flow on homogeneous models which", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 194, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 194, + 391 + ], + "score": 1.0, + "content": "could be non-smooth.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 347, + 506, + 391 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "Other Works on Implicit Bias. Banburski et al. (2019) also studied the dynamics of gradient flow", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "and among other things, provided mathematical insights to the implicit bias towards max margin", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 419, + 504, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 504, + 432 + ], + "score": 1.0, + "content": "solution for homogeneous networks. We note that their analysis of gradient flow decomposes the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "dynamics to the tangent component and radial component, which is similar to our proof of Theo-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "rem 4.1 in spirit. Wilson et al. (2017); Ali et al. (2019); Gunasekar et al. 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(Exponential Loss).", + "type": "text" + }, + { + "bbox": [ + 224, + 523, + 270, + 536 + ], + "score": 0.95, + "content": "\\ell ( q ) = e ^ { - q }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 524, + 275, + 536 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 115, + 538, + 389, + 552 + ], + "spans": [ + { + "bbox": [ + 115, + 538, + 280, + 552 + ], + "score": 1.0, + "content": "(A4). (Separability). 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As shown in (Davis et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "2020), the output of almost every neural network admits a chain rule (as long as the neural network", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 486, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 486, + 596 + ], + "score": 1.0, + "content": "is composed by definable pieces in an o-minimal structure, e.g., ReLU, sigmoid, LeakyReLU).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 560, + 505, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "(A2) assumes the homogeneity, the main property we assume in this work. (A3), (A4) correspond to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "the two conditions introduced in Section 1. 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See Appendix E.1 for the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 719, + 174, + 732 + ], + "spans": [ + { + "bbox": [ + 141, + 719, + 174, + 732 + ], + "score": 1.0, + "content": "details.", + "type": "text" + } + ], + "index": 44, + "is_list_end_line": true + } + ], + "index": 42.5, + "bbox_fs": [ + 116, + 680, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "(S5) is natural since deep neural networks are usually trained with constant learning rates. (S1)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 211, + 106 + ], + "score": 1.0, + "content": "ensures the smoothness of", + "type": "text" + }, + { + "bbox": [ + 211, + 94, + 219, + 104 + ], + "score": 0.78, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 93, + 505, + 106 + ], + "score": 1.0, + "content": ", which is often assumed in the optimization literature in order to analyze", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "gradient descent. 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So we consider the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 257, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 257, + 229 + ], + "score": 1.0, + "content": "normalized margin defined as below:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 228, + 378, + 256 + ], + "lines": [ + { + "bbox": [ + 232, + 228, + 378, + 256 + ], + "spans": [ + { + "bbox": [ + 232, + 228, + 378, + 256 + ], + "score": 0.95, + "content": "\\bar { \\gamma } ( \\pmb \\theta ) : = q _ { \\mathrm { m i n } } \\left( \\frac { \\pmb \\theta } { \\lVert \\pmb \\theta \\rVert _ { 2 } } \\right) = \\frac { q _ { \\mathrm { m i n } } ( \\pmb \\theta ) } { \\lVert \\pmb \\theta \\rVert _ { 2 } ^ { L } } .", + "type": "interline_equation", + "image_path": "d2e15873428b21f2f19058c3a10617e16b49f92c60197383dac6ac706aebf063.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 232, + 228, + 378, + 256 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 506, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 140, + 270 + ], + "score": 1.0, + "content": "We say", + "type": "text" + }, + { + "bbox": [ + 140, + 257, + 147, + 268 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 255, + 173, + 270 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 173, + 259, + 178, + 266 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 255, + 399, + 270 + ], + "score": 1.0, + "content": "-additive approximation for the normalized margin if", + "type": "text" + }, + { + "bbox": [ + 400, + 257, + 471, + 268 + ], + "score": 0.92, + "content": "\\bar { \\gamma } \\mathrm { ~ - ~ } \\epsilon \\le f \\le \\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 255, + 495, + 270 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 495, + 259, + 501, + 266 + ], + "score": 0.65, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 255, + 505, + 270 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 287, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 233, + 281 + ], + "score": 1.0, + "content": "multiplicative approximation if", + "type": "text" + }, + { + "bbox": [ + 234, + 268, + 282, + 279 + ], + "score": 0.92, + "content": "c \\bar { \\gamma } \\leq f \\leq \\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 266, + 287, + 281 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 432, + 296 + ], + "score": 1.0, + "content": "Gradient Flow. Our first result is on the overall trend of the normalized margin", + "type": "text" + }, + { + "bbox": [ + 432, + 284, + 464, + 296 + ], + "score": 0.93, + "content": "\\bar { \\gamma } ( \\pmb \\theta ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 284, + 505, + 296 + ], + "score": 1.0, + "content": ". For both", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "score": 1.0, + "content": "gradient flow and gradient descent, we identify a smoothed version of the normalized margin, and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "show that it is non-decreasing during training. More specifically, we have the following theorem for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 317, + 164, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 164, + 329 + ], + "score": 1.0, + "content": "gradient flow.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 382, + 343 + ], + "score": 1.0, + "content": "Theorem 4.1 (Corollary of Theorem A.7). Under assumptions", + "type": "text" + }, + { + "bbox": [ + 383, + 331, + 433, + 342 + ], + "score": 0.58, + "content": "( A I ) ~ - ~ ( A 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 330, + 505, + 343 + ], + "score": 1.0, + "content": ", there exists an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 340, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 107, + 341, + 151, + 354 + ], + "score": 0.93, + "content": "O ( \\| \\pmb { \\theta } \\| _ { 2 } ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 340, + 283, + 356 + ], + "score": 1.0, + "content": "-additive approximation function", + "type": "text" + }, + { + "bbox": [ + 284, + 342, + 304, + 354 + ], + "score": 0.91, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 340, + 506, + 356 + ], + "score": 1.0, + "content": "for the normalized margin such that the following", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 354, + 257, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 257, + 365 + ], + "score": 1.0, + "content": "statements are true for gradient flow:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 131, + 371, + 270, + 385 + ], + "lines": [ + { + "bbox": [ + 129, + 370, + 270, + 387 + ], + "spans": [ + { + "bbox": [ + 129, + 370, + 175, + 387 + ], + "score": 1.0, + "content": "1. For a.e.", + "type": "text" + }, + { + "bbox": [ + 175, + 372, + 201, + 384 + ], + "score": 0.63, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 370, + 205, + 387 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 370, + 266, + 385 + ], + "score": 0.64, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\tilde { \\gamma } ( \\pmb { \\theta } ( t ) ) \\geq 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 370, + 270, + 387 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 130, + 391, + 366, + 409 + ], + "lines": [ + { + "bbox": [ + 175, + 391, + 361, + 409 + ], + "spans": [ + { + "bbox": [ + 175, + 395, + 201, + 405 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 393, + 291, + 407 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\tilde { \\gamma } ( \\pmb { \\theta } ( t ) ) > 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 391, + 361, + 409 + ], + "score": 0.66, + "content": "{ \\frac { d } { d t } } { \\frac { \\pmb { \\theta } ( t ) } { \\| \\pmb { \\theta } ( t ) \\| _ { 2 } } } = 0", + "type": "inline_equation" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 127, + 414, + 472, + 428 + ], + "lines": [ + { + "bbox": [ + 128, + 414, + 473, + 429 + ], + "spans": [ + { + "bbox": [ + 128, + 414, + 142, + 429 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 142, + 415, + 196, + 427 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 414, + 215, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 415, + 273, + 427 + ], + "score": 0.91, + "content": "\\| \\pmb \\theta ( t ) \\| _ { 2 } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 414, + 285, + 429 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 286, + 416, + 324, + 426 + ], + "score": 0.9, + "content": "t \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 414, + 369, + 429 + ], + "score": 1.0, + "content": "; therefore,", + "type": "text" + }, + { + "bbox": [ + 369, + 415, + 469, + 428 + ], + "score": 0.9, + "content": "| \\bar { \\gamma } ( \\pmb \\theta ( t ) ) - \\tilde { \\gamma } ( \\pmb \\theta ( t ) ) | 0", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 414, + 473, + 429 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 433, + 364, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 365, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 226, + 447 + ], + "score": 1.0, + "content": "More concretely, the function", + "type": "text" + }, + { + "bbox": [ + 227, + 434, + 247, + 446 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 433, + 365, + 447 + ], + "score": 1.0, + "content": "in Theorem 4.1 is defined as", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 447, + 398, + 481 + ], + "lines": [ + { + "bbox": [ + 212, + 447, + 398, + 481 + ], + "spans": [ + { + "bbox": [ + 212, + 447, + 398, + 481 + ], + "score": 0.94, + "content": "\\widetilde { \\gamma } ( \\pmb \\theta ) : = \\frac { \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } } { \\| \\pmb \\theta \\| _ { 2 } ^ { L } } = \\frac { - \\log \\left( \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( \\pmb \\theta ) } \\right) } { \\| \\pmb \\theta \\| _ { 2 } ^ { L } } .", + "type": "interline_equation", + "image_path": "951e8ac53f1ca499cb06974b9c9a00d287116613be412b3e7d5ef1bf5a2a049f.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 447, + 398, + 464.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 212, + 464.0, + 398, + 481.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 267, + 495 + ], + "score": 1.0, + "content": "Note that the only difference between", + "type": "text" + }, + { + "bbox": [ + 267, + 482, + 288, + 494 + ], + "score": 0.89, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 481, + 309, + 495 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 482, + 330, + 494 + ], + "score": 0.9, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 481, + 410, + 495 + ], + "score": 1.0, + "content": "is that the margin", + "type": "text" + }, + { + "bbox": [ + 410, + 482, + 443, + 494 + ], + "score": 0.91, + "content": "q _ { \\mathrm { m i n } } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 481, + 457, + 495 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 457, + 482, + 478, + 493 + ], + "score": 0.9, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 481, + 505, + 495 + ], + "score": 1.0, + "content": "is re-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 491, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 491, + 150, + 508 + ], + "score": 1.0, + "content": "placed by", + "type": "text" + }, + { + "bbox": [ + 150, + 492, + 320, + 507 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\log \\frac { 1 } { \\mathcal { L } ( \\pmb { \\theta } ) } = - \\mathrm { L S E } ( - q _ { 1 } ( \\pmb { \\theta } ) , \\dots , - q _ { N } ( \\pmb { \\theta } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 491, + 353, + 508 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 354, + 493, + 505, + 505 + ], + "score": 0.89, + "content": "\\operatorname { L S E } ( a _ { 1 } , \\dots , a _ { N } ) = \\log ( \\exp ( a _ { 1 } ) +", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 173, + 519 + ], + "score": 0.88, + "content": "\\cdot \\cdot \\cdot + \\exp ( a _ { N } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 505, + 505, + 520 + ], + "score": 1.0, + "content": "is the LogSumExp function. This is indeed a very natural idea, and previous", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 464, + 532 + ], + "score": 1.0, + "content": "works on linear models (e.g., (Telgarsky, 2013; Nacson et al., 2019b)) also approximate", + "type": "text" + }, + { + "bbox": [ + 464, + 519, + 483, + 529 + ], + "score": 0.87, + "content": "q _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 516, + 505, + 532 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 528, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 361, + 544 + ], + "score": 1.0, + "content": "LogSumExp in the analysis of margin. It is easy to see why", + "type": "text" + }, + { + "bbox": [ + 361, + 529, + 382, + 542 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 528, + 407, + 544 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 408, + 529, + 452, + 542 + ], + "score": 0.93, + "content": "O ( \\| \\pmb { \\theta } \\| _ { 2 } ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 528, + 506, + 544 + ], + "score": 1.0, + "content": "-additive ap-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 102, + 540, + 508, + 561 + ], + "spans": [ + { + "bbox": [ + 102, + 540, + 174, + 561 + ], + "score": 1.0, + "content": "proximation for", + "type": "text" + }, + { + "bbox": [ + 174, + 542, + 194, + 555 + ], + "score": 0.72, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 540, + 196, + 561 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 196, + 541, + 333, + 555 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\mathrm { ~ : ~ } e ^ { a _ { \\mathrm { m a x } } } \\leq \\sum _ { n = 1 } ^ { N } e ^ { a _ { n } } \\leq N e ^ { a _ { \\mathrm { m a x } } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 540, + 376, + 561 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + }, + { + "bbox": [ + 376, + 543, + 489, + 555 + ], + "score": 0.9, + "content": "a _ { \\mathrm { { m a x } } } = \\operatorname* { m a x } \\{ a _ { 1 } , \\dots , a _ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 540, + 508, + 561 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 295, + 566 + ], + "score": 0.84, + "content": "a _ { \\mathrm { m a x } } \\leq \\mathrm { L S E } ( a _ { 1 } , \\ldots , a _ { N } ) \\leq a _ { \\mathrm { m a x } } \\bar { + } \\log N", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 553, + 458, + 567 + ], + "score": 1.0, + "content": "; combining this with the definition of", + "type": "text" + }, + { + "bbox": [ + 459, + 555, + 479, + 566 + ], + "score": 0.91, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "gives", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 565, + 262, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 258, + 579 + ], + "score": 0.89, + "content": "\\bar { \\gamma } ( \\pmb \\theta ) - \\lVert \\pmb \\theta \\rVert _ { 2 } ^ { - L } \\log N \\leq \\tilde { \\gamma } ( \\pmb \\theta ) \\leq \\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 565, + 262, + 580 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 503, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "Gradient Descent. For gradient descent, Theorem 4.1 holds similarly with a slightly different", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 594, + 414, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 142, + 607 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 595, + 163, + 606 + ], + "score": 0.91, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 595, + 237, + 607 + ], + "score": 1.0, + "content": "that approximates", + "type": "text" + }, + { + "bbox": [ + 238, + 594, + 258, + 606 + ], + "score": 0.91, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 595, + 414, + 607 + ], + "score": 1.0, + "content": "multiplicatively rather than additively.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 363, + 621 + ], + "score": 1.0, + "content": "Theorem 4.2 (Corollary of Theorem E.2). Under assumptions", + "type": "text" + }, + { + "bbox": [ + 363, + 608, + 381, + 618 + ], + "score": 0.33, + "content": "( S I )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 606, + 505, + 621 + ], + "score": 1.0, + "content": ", (A2) - (A4), (S5), there exists", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 617, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 119, + 632 + ], + "score": 1.0, + "content": "an", + "type": "text" + }, + { + "bbox": [ + 120, + 618, + 199, + 631 + ], + "score": 0.9, + "content": "( 1 - O ( 1 / ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 617, + 355, + 632 + ], + "score": 1.0, + "content": "-multiplicative approximation function", + "type": "text" + }, + { + "bbox": [ + 356, + 619, + 376, + 631 + ], + "score": 0.91, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 617, + 506, + 632 + ], + "score": 1.0, + "content": "for the normalized margin such", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 629, + 344, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 344, + 642 + ], + "score": 1.0, + "content": "that the following statements are true for gradient descent:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 667, + 417, + 685 + ], + "lines": [], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 227, + 667, + 417, + 685 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 374, + 722 + ], + "score": 1.0, + "content": "Due to the discreteness of gradient descent, the explicit formula for", + "type": "text" + }, + { + "bbox": [ + 375, + 710, + 395, + 722 + ], + "score": 0.92, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is somewhat technical, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 309, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 309, + 732 + ], + "score": 1.0, + "content": "we refer the readers to Appendix E for full details.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "(S5) is natural since deep neural networks are usually trained with constant learning rates. (S1)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 211, + 106 + ], + "score": 1.0, + "content": "ensures the smoothness of", + "type": "text" + }, + { + "bbox": [ + 211, + 94, + 219, + 104 + ], + "score": 0.78, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 93, + 505, + 106 + ], + "score": 1.0, + "content": ", which is often assumed in the optimization literature in order to analyze", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "gradient descent. While (S1) does not hold for neural networks with ReLU, it does hold for neural", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 419, + 127 + ], + "score": 1.0, + "content": "networks with smooth homogeneous activation such as the quadratic activation", + "type": "text" + }, + { + "bbox": [ + 420, + 115, + 466, + 127 + ], + "score": 0.92, + "content": "\\phi ( x ) : = x ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "(Li et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 292, + 140 + ], + "score": 1.0, + "content": "2018b; Du & Lee, 2018) or powers of ReLU", + "type": "text" + }, + { + "bbox": [ + 292, + 126, + 375, + 138 + ], + "score": 0.92, + "content": "\\phi ( x ) : = \\mathrm { R e L U } ( x ) ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 125, + 392, + 140 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 392, + 127, + 420, + 137 + ], + "score": 0.89, + "content": "\\alpha > 2", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "(Zhong et al., 2017;", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 284, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 284, + 150 + ], + "score": 1.0, + "content": "Klusowski & Barron, 2018; Li et al., 2019).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 505, + 150 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 160, + 398, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 161, + 399, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 399, + 173 + ], + "score": 1.0, + "content": "4.2 MAIN THEOREM: MONOTONICITY OF NORMALIZED MARGINS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 181, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 241, + 194 + ], + "score": 1.0, + "content": "The margin for a single data point", + "type": "text" + }, + { + "bbox": [ + 242, + 182, + 276, + 194 + ], + "score": 0.91, + "content": "( { \\pmb x } _ { n } , y _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 181, + 339, + 194 + ], + "score": 1.0, + "content": "is defined to be", + "type": "text" + }, + { + "bbox": [ + 339, + 182, + 427, + 194 + ], + "score": 0.92, + "content": "q _ { n } ( \\pmb \\theta ) : = y _ { n } \\Phi ( \\pmb \\theta ; \\pmb x _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 181, + 505, + 194 + ], + "score": 1.0, + "content": ", and the margin for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 504, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 240, + 207 + ], + "score": 1.0, + "content": "the entire dataset is defined to be", + "type": "text" + }, + { + "bbox": [ + 241, + 194, + 354, + 206 + ], + "score": 0.91, + "content": "q _ { \\operatorname* { m i n } } ( \\pmb { \\theta } ) : = \\operatorname* { m i n } _ { n \\in [ N ] } q _ { n } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 192, + 470, + 207 + ], + "score": 1.0, + "content": ". By homogenity, the margin", + "type": "text" + }, + { + "bbox": [ + 471, + 193, + 504, + 205 + ], + "score": 0.94, + "content": "q _ { \\mathrm { m i n } } ( \\pmb { \\theta } )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 187, + 219 + ], + "score": 1.0, + "content": "scales linearly with", + "type": "text" + }, + { + "bbox": [ + 187, + 205, + 210, + 218 + ], + "score": 0.92, + "content": "\\lVert \\pmb { \\theta } \\rVert _ { 2 } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 205, + 327, + 219 + ], + "score": 1.0, + "content": "for any fixed direction since", + "type": "text" + }, + { + "bbox": [ + 327, + 205, + 421, + 218 + ], + "score": 0.92, + "content": "q _ { \\mathrm { m i n } } ( c \\pmb \\theta ) = c ^ { L } q _ { \\mathrm { m i n } } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 205, + 506, + 219 + ], + "score": 1.0, + "content": ". So we consider the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 257, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 257, + 229 + ], + "score": 1.0, + "content": "normalized margin defined as below:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 181, + 506, + 229 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 228, + 378, + 256 + ], + "lines": [ + { + "bbox": [ + 232, + 228, + 378, + 256 + ], + "spans": [ + { + "bbox": [ + 232, + 228, + 378, + 256 + ], + "score": 0.95, + "content": "\\bar { \\gamma } ( \\pmb \\theta ) : = q _ { \\mathrm { m i n } } \\left( \\frac { \\pmb \\theta } { \\lVert \\pmb \\theta \\rVert _ { 2 } } \\right) = \\frac { q _ { \\mathrm { m i n } } ( \\pmb \\theta ) } { \\lVert \\pmb \\theta \\rVert _ { 2 } ^ { L } } .", + "type": "interline_equation", + "image_path": "d2e15873428b21f2f19058c3a10617e16b49f92c60197383dac6ac706aebf063.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 232, + 228, + 378, + 256 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 506, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 140, + 270 + ], + "score": 1.0, + "content": "We say", + "type": "text" + }, + { + "bbox": [ + 140, + 257, + 147, + 268 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 255, + 173, + 270 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 173, + 259, + 178, + 266 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 255, + 399, + 270 + ], + "score": 1.0, + "content": "-additive approximation for the normalized margin if", + "type": "text" + }, + { + "bbox": [ + 400, + 257, + 471, + 268 + ], + "score": 0.92, + "content": "\\bar { \\gamma } \\mathrm { ~ - ~ } \\epsilon \\le f \\le \\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 255, + 495, + 270 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 495, + 259, + 501, + 266 + ], + "score": 0.65, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 255, + 505, + 270 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 287, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 233, + 281 + ], + "score": 1.0, + "content": "multiplicative approximation if", + "type": "text" + }, + { + "bbox": [ + 234, + 268, + 282, + 279 + ], + "score": 0.92, + "content": "c \\bar { \\gamma } \\leq f \\leq \\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 266, + 287, + 281 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 255, + 505, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 432, + 296 + ], + "score": 1.0, + "content": "Gradient Flow. Our first result is on the overall trend of the normalized margin", + "type": "text" + }, + { + "bbox": [ + 432, + 284, + 464, + 296 + ], + "score": 0.93, + "content": "\\bar { \\gamma } ( \\pmb \\theta ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 284, + 505, + 296 + ], + "score": 1.0, + "content": ". For both", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "score": 1.0, + "content": "gradient flow and gradient descent, we identify a smoothed version of the normalized margin, and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "show that it is non-decreasing during training. More specifically, we have the following theorem for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 317, + 164, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 164, + 329 + ], + "score": 1.0, + "content": "gradient flow.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 284, + 506, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 382, + 343 + ], + "score": 1.0, + "content": "Theorem 4.1 (Corollary of Theorem A.7). Under assumptions", + "type": "text" + }, + { + "bbox": [ + 383, + 331, + 433, + 342 + ], + "score": 0.58, + "content": "( A I ) ~ - ~ ( A 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 330, + 505, + 343 + ], + "score": 1.0, + "content": ", there exists an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 340, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 107, + 341, + 151, + 354 + ], + "score": 0.93, + "content": "O ( \\| \\pmb { \\theta } \\| _ { 2 } ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 340, + 283, + 356 + ], + "score": 1.0, + "content": "-additive approximation function", + "type": "text" + }, + { + "bbox": [ + 284, + 342, + 304, + 354 + ], + "score": 0.91, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 340, + 506, + 356 + ], + "score": 1.0, + "content": "for the normalized margin such that the following", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 354, + 257, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 257, + 365 + ], + "score": 1.0, + "content": "statements are true for gradient flow:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 330, + 506, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 371, + 270, + 385 + ], + "lines": [ + { + "bbox": [ + 129, + 370, + 270, + 387 + ], + "spans": [ + { + "bbox": [ + 129, + 370, + 175, + 387 + ], + "score": 1.0, + "content": "1. For a.e.", + "type": "text" + }, + { + "bbox": [ + 175, + 372, + 201, + 384 + ], + "score": 0.63, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 370, + 205, + 387 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 370, + 266, + 385 + ], + "score": 0.64, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\tilde { \\gamma } ( \\pmb { \\theta } ( t ) ) \\geq 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 370, + 270, + 387 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 129, + 370, + 270, + 387 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 391, + 366, + 409 + ], + "lines": [ + { + "bbox": [ + 175, + 391, + 361, + 409 + ], + "spans": [ + { + "bbox": [ + 175, + 395, + 201, + 405 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 393, + 291, + 407 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\tilde { \\gamma } ( \\pmb { \\theta } ( t ) ) > 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 391, + 361, + 409 + ], + "score": 0.66, + "content": "{ \\frac { d } { d t } } { \\frac { \\pmb { \\theta } ( t ) } { \\| \\pmb { \\theta } ( t ) \\| _ { 2 } } } = 0", + "type": "inline_equation" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 175, + 391, + 361, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 127, + 414, + 472, + 428 + ], + "lines": [ + { + "bbox": [ + 128, + 414, + 473, + 429 + ], + "spans": [ + { + "bbox": [ + 128, + 414, + 142, + 429 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 142, + 415, + 196, + 427 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 414, + 215, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 415, + 273, + 427 + ], + "score": 0.91, + "content": "\\| \\pmb \\theta ( t ) \\| _ { 2 } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 414, + 285, + 429 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 286, + 416, + 324, + 426 + ], + "score": 0.9, + "content": "t \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 414, + 369, + 429 + ], + "score": 1.0, + "content": "; therefore,", + "type": "text" + }, + { + "bbox": [ + 369, + 415, + 469, + 428 + ], + "score": 0.9, + "content": "| \\bar { \\gamma } ( \\pmb \\theta ( t ) ) - \\tilde { \\gamma } ( \\pmb \\theta ( t ) ) | 0", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 414, + 473, + 429 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 128, + 414, + 473, + 429 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 433, + 364, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 365, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 226, + 447 + ], + "score": 1.0, + "content": "More concretely, the function", + "type": "text" + }, + { + "bbox": [ + 227, + 434, + 247, + 446 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 433, + 365, + 447 + ], + "score": 1.0, + "content": "in Theorem 4.1 is defined as", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 433, + 365, + 447 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 447, + 398, + 481 + ], + "lines": [ + { + "bbox": [ + 212, + 447, + 398, + 481 + ], + "spans": [ + { + "bbox": [ + 212, + 447, + 398, + 481 + ], + "score": 0.94, + "content": "\\widetilde { \\gamma } ( \\pmb \\theta ) : = \\frac { \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } } { \\| \\pmb \\theta \\| _ { 2 } ^ { L } } = \\frac { - \\log \\left( \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( \\pmb \\theta ) } \\right) } { \\| \\pmb \\theta \\| _ { 2 } ^ { L } } .", + "type": "interline_equation", + "image_path": "951e8ac53f1ca499cb06974b9c9a00d287116613be412b3e7d5ef1bf5a2a049f.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 447, + 398, + 464.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 212, + 464.0, + 398, + 481.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 267, + 495 + ], + "score": 1.0, + "content": "Note that the only difference between", + "type": "text" + }, + { + "bbox": [ + 267, + 482, + 288, + 494 + ], + "score": 0.89, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 481, + 309, + 495 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 482, + 330, + 494 + ], + "score": 0.9, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 481, + 410, + 495 + ], + "score": 1.0, + "content": "is that the margin", + "type": "text" + }, + { + "bbox": [ + 410, + 482, + 443, + 494 + ], + "score": 0.91, + "content": "q _ { \\mathrm { m i n } } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 481, + 457, + 495 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 457, + 482, + 478, + 493 + ], + "score": 0.9, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 481, + 505, + 495 + ], + "score": 1.0, + "content": "is re-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 491, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 491, + 150, + 508 + ], + "score": 1.0, + "content": "placed by", + "type": "text" + }, + { + "bbox": [ + 150, + 492, + 320, + 507 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\log \\frac { 1 } { \\mathcal { L } ( \\pmb { \\theta } ) } = - \\mathrm { L S E } ( - q _ { 1 } ( \\pmb { \\theta } ) , \\dots , - q _ { N } ( \\pmb { \\theta } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 491, + 353, + 508 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 354, + 493, + 505, + 505 + ], + "score": 0.89, + "content": "\\operatorname { L S E } ( a _ { 1 } , \\dots , a _ { N } ) = \\log ( \\exp ( a _ { 1 } ) +", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 173, + 519 + ], + "score": 0.88, + "content": "\\cdot \\cdot \\cdot + \\exp ( a _ { N } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 505, + 505, + 520 + ], + "score": 1.0, + "content": "is the LogSumExp function. This is indeed a very natural idea, and previous", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 464, + 532 + ], + "score": 1.0, + "content": "works on linear models (e.g., (Telgarsky, 2013; Nacson et al., 2019b)) also approximate", + "type": "text" + }, + { + "bbox": [ + 464, + 519, + 483, + 529 + ], + "score": 0.87, + "content": "q _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 516, + 505, + 532 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 528, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 361, + 544 + ], + "score": 1.0, + "content": "LogSumExp in the analysis of margin. It is easy to see why", + "type": "text" + }, + { + "bbox": [ + 361, + 529, + 382, + 542 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 528, + 407, + 544 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 408, + 529, + 452, + 542 + ], + "score": 0.93, + "content": "O ( \\| \\pmb { \\theta } \\| _ { 2 } ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 528, + 506, + 544 + ], + "score": 1.0, + "content": "-additive ap-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 102, + 540, + 508, + 561 + ], + "spans": [ + { + "bbox": [ + 102, + 540, + 174, + 561 + ], + "score": 1.0, + "content": "proximation for", + "type": "text" + }, + { + "bbox": [ + 174, + 542, + 194, + 555 + ], + "score": 0.72, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 540, + 196, + 561 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 196, + 541, + 333, + 555 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\mathrm { ~ : ~ } e ^ { a _ { \\mathrm { m a x } } } \\leq \\sum _ { n = 1 } ^ { N } e ^ { a _ { n } } \\leq N e ^ { a _ { \\mathrm { m a x } } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 540, + 376, + 561 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + }, + { + "bbox": [ + 376, + 543, + 489, + 555 + ], + "score": 0.9, + "content": "a _ { \\mathrm { { m a x } } } = \\operatorname* { m a x } \\{ a _ { 1 } , \\dots , a _ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 540, + 508, + 561 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 295, + 566 + ], + "score": 0.84, + "content": "a _ { \\mathrm { m a x } } \\leq \\mathrm { L S E } ( a _ { 1 } , \\ldots , a _ { N } ) \\leq a _ { \\mathrm { m a x } } \\bar { + } \\log N", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 553, + 458, + 567 + ], + "score": 1.0, + "content": "; combining this with the definition of", + "type": "text" + }, + { + "bbox": [ + 459, + 555, + 479, + 566 + ], + "score": 0.91, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "gives", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 565, + 262, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 258, + 579 + ], + "score": 0.89, + "content": "\\bar { \\gamma } ( \\pmb \\theta ) - \\lVert \\pmb \\theta \\rVert _ { 2 } ^ { - L } \\log N \\leq \\tilde { \\gamma } ( \\pmb \\theta ) \\leq \\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 565, + 262, + 580 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 102, + 481, + 508, + 580 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 503, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "Gradient Descent. For gradient descent, Theorem 4.1 holds similarly with a slightly different", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 594, + 414, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 142, + 607 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 595, + 163, + 606 + ], + "score": 0.91, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 595, + 237, + 607 + ], + "score": 1.0, + "content": "that approximates", + "type": "text" + }, + { + "bbox": [ + 238, + 594, + 258, + 606 + ], + "score": 0.91, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 595, + 414, + 607 + ], + "score": 1.0, + "content": "multiplicatively rather than additively.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 582, + 505, + 607 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 363, + 621 + ], + "score": 1.0, + "content": "Theorem 4.2 (Corollary of Theorem E.2). Under assumptions", + "type": "text" + }, + { + "bbox": [ + 363, + 608, + 381, + 618 + ], + "score": 0.33, + "content": "( S I )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 606, + 505, + 621 + ], + "score": 1.0, + "content": ", (A2) - (A4), (S5), there exists", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 617, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 119, + 632 + ], + "score": 1.0, + "content": "an", + "type": "text" + }, + { + "bbox": [ + 120, + 618, + 199, + 631 + ], + "score": 0.9, + "content": "( 1 - O ( 1 / ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 617, + 355, + 632 + ], + "score": 1.0, + "content": "-multiplicative approximation function", + "type": "text" + }, + { + "bbox": [ + 356, + 619, + 376, + 631 + ], + "score": 0.91, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 617, + 506, + 632 + ], + "score": 1.0, + "content": "for the normalized margin such", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 629, + 344, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 344, + 642 + ], + "score": 1.0, + "content": "that the following statements are true for gradient descent:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 606, + 506, + 642 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 667, + 417, + 685 + ], + "lines": [], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 227, + 667, + 417, + 685 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 374, + 722 + ], + "score": 1.0, + "content": "Due to the discreteness of gradient descent, the explicit formula for", + "type": "text" + }, + { + "bbox": [ + 375, + 710, + 395, + 722 + ], + "score": 0.92, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is somewhat technical, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 309, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 309, + 732 + ], + "score": 1.0, + "content": "we refer the readers to Appendix E for full details.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 81, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 341, + 95 + ], + "score": 1.0, + "content": "Convergence Rates. It is shown in Theorem 4.1, 4.2 that", + "type": "text" + }, + { + "bbox": [ + 342, + 82, + 395, + 95 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 82, + 412, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 413, + 82, + 470, + 95 + ], + "score": 0.95, + "content": "\\| \\pmb \\theta ( t ) \\| _ { 2 } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". In fact,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "with a more refined analysis, we can prove tight loss convergence and weight growth rates using the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 257, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 257, + 117 + ], + "score": 1.0, + "content": "monotonicity of normalized margins.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 506, + 153 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 505, + 132 + ], + "score": 1.0, + "content": "Theorem 4.3 (Corollary of Theorem A.10 and E.5). For gradient flow under assumptions (A1) -", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 130, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 505, + 142 + ], + "score": 1.0, + "content": "(A4) or gradient descent under assumptions (S1), (A2) - (A4), (S5), we have the following tight", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 142, + 278, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 278, + 154 + ], + "score": 1.0, + "content": "bounds for training loss and weight norm:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 158, + 457, + 186 + ], + "lines": [ + { + "bbox": [ + 152, + 158, + 457, + 186 + ], + "spans": [ + { + "bbox": [ + 152, + 158, + 457, + 186 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) = \\Theta \\left( \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } \\right) \\qquad a n d \\qquad \\| \\pmb \\theta ( t ) \\| _ { 2 } = \\Theta \\left( ( \\log T ) ^ { 1 / L } \\right) ,", + "type": "interline_equation", + "image_path": "bf296f66e9bbf0a5e8d9ae747033439ecb368de26010005c8aa49638bcdecef3.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 152, + 158, + 457, + 167.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 152, + 167.33333333333334, + 457, + 176.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 152, + 176.66666666666669, + 457, + 186.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 191, + 401, + 207 + ], + "lines": [ + { + "bbox": [ + 104, + 187, + 405, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 133, + 211 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 194, + 158, + 204 + ], + "score": 0.88, + "content": "T = t", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 187, + 246, + 211 + ], + "score": 1.0, + "content": "for gradient flow and", + "type": "text" + }, + { + "bbox": [ + 246, + 191, + 316, + 207 + ], + "score": 0.92, + "content": "\\begin{array} { r } { T = \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 187, + 405, + 211 + ], + "score": 1.0, + "content": "for gradient descent.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 106, + 217, + 349, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 349, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 349, + 231 + ], + "score": 1.0, + "content": "4.3 MAIN THEOREM: CONVERGENCE TO KKT POINTS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 180, + 252 + ], + "score": 1.0, + "content": "For gradient flow,", + "type": "text" + }, + { + "bbox": [ + 181, + 240, + 188, + 250 + ], + "score": 0.85, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 238, + 274, + 252 + ], + "score": 1.0, + "content": "is upper-bounded by", + "type": "text" + }, + { + "bbox": [ + 275, + 238, + 414, + 251 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } \\le \\bar { \\gamma } \\le \\operatorname* { s u p } \\{ q _ { n } ( \\pmb \\theta ) : \\lVert \\pmb \\theta \\rVert _ { 2 } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 238, + 505, + 252 + ], + "score": 1.0, + "content": ". Combining this with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 431, + 263 + ], + "score": 1.0, + "content": "Theorem 4.1 and the monotone convergence theorem, it is not hard to see that", + "type": "text" + }, + { + "bbox": [ + 431, + 250, + 504, + 262 + ], + "score": 0.91, + "content": "\\scriptstyle \\operatorname* { l i m } _ { t \\to + \\infty } { \\bar { \\gamma } } ( \\pmb { \\theta } ( t ) )", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 123, + 273 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 261, + 196, + 273 + ], + "score": 0.92, + "content": "\\scriptstyle \\operatorname* { l i m } _ { t \\to + \\infty } { \\tilde { \\gamma } } ( \\pmb { \\theta } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "exist and equal to the same value. Using a similar argument, we can draw the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 259, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 259, + 285 + ], + "score": 1.0, + "content": "same conclusion for gradient descent.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 505, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 303 + ], + "score": 1.0, + "content": "To understand the implicit regularization effect, a natural question arises: what optimality property", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "does the limit of normalized margin have? To this end, we identify a natural constrained optimization", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 340, + 322 + ], + "score": 1.0, + "content": "problem related to margin maximization, and prove that", + "type": "text" + }, + { + "bbox": [ + 340, + 310, + 358, + 322 + ], + "score": 0.91, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 311, + 505, + 322 + ], + "score": 1.0, + "content": "directionally converges to its KKT", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 333 + ], + "score": 1.0, + "content": "points, as shown below. We note that we can extend this result to the finite time case, and show", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "that gradient flow or gradient descent passes through an approximate KKT point after a certain", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 344, + 504, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 504, + 355 + ], + "score": 1.0, + "content": "amount of time. See Theorem A.9 in Appendix A and Theorem E.4 in Appendix E for the details.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "We will briefly review the definition of KKT points and approximate KKT points for a constraint", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 266, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 266, + 377 + ], + "score": 1.0, + "content": "optimization problem in Appendix C.1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "Theorem 4.4 (Corollary of Theorem A.8 and E.3). For gradient flow under assumptions (A1) - (A4)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 504, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 412, + 414 + ], + "score": 1.0, + "content": "or gradient descent under assumptions (S1), (A2) - (A4), (S5), any limit point", + "type": "text" + }, + { + "bbox": [ + 413, + 394, + 420, + 405 + ], + "score": 0.61, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 387, + 431, + 414 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 432, + 391, + 504, + 411 + ], + "score": 0.88, + "content": "\\left\\{ \\frac { \\pmb \\theta ( t ) } { \\| \\pmb \\theta ( t ) \\| _ { 2 } } : t \\geq 0 \\right\\}", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 409, + 476, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 458, + 423 + ], + "score": 1.0, + "content": "is along the direction of a KKT point of the following constrained optimization problem", + "type": "text" + }, + { + "bbox": [ + 458, + 411, + 471, + 421 + ], + "score": 0.75, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 409, + 476, + 423 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 425, + 410, + 450 + ], + "lines": [ + { + "bbox": [ + 199, + 425, + 410, + 450 + ], + "spans": [ + { + "bbox": [ + 199, + 425, + 410, + 450 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } \\quad \\frac { 1 } { 2 } \\| \\pmb { \\theta } \\| _ { 2 } ^ { 2 } \\qquad \\quad s . t . \\quad q _ { n } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ]", + "type": "interline_equation", + "image_path": "b658d9b248bc89608d0277c369d86ef3ba9b1253168693340ce7ae371032716e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 199, + 425, + 410, + 450 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 504, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 212, + 469 + ], + "score": 1.0, + "content": "That is, for any limit point", + "type": "text" + }, + { + "bbox": [ + 212, + 455, + 219, + 466 + ], + "score": 0.78, + "content": "\\bar { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 454, + 332, + 469 + ], + "score": 1.0, + "content": ", there exists a scaling factor", + "type": "text" + }, + { + "bbox": [ + 333, + 456, + 358, + 466 + ], + "score": 0.89, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 454, + 397, + 469 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 397, + 455, + 411, + 466 + ], + "score": 0.86, + "content": "\\alpha \\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 454, + 505, + 469 + ], + "score": 1.0, + "content": "satisfies Karush-Kuhn-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 466, + 236, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 218, + 479 + ], + "score": 1.0, + "content": "Tucker (KKT) conditions of", + "type": "text" + }, + { + "bbox": [ + 218, + 467, + 232, + 478 + ], + "score": 0.79, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 466, + 236, + 479 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "Minimizing (P) over its feasible region is equivalent to maximizing the normalized margin over all", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "possible directions. The proof is as follows. Note that we only need to consider all feasible points", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 114, + 521 + ], + "score": 0.64, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 509, + 136, + 524 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 137, + 510, + 190, + 523 + ], + "score": 0.93, + "content": "q _ { \\mathrm { m i n } } ( \\pmb \\theta ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 509, + 243, + 524 + ], + "score": 1.0, + "content": ". For a fixed", + "type": "text" + }, + { + "bbox": [ + 243, + 511, + 250, + 520 + ], + "score": 0.45, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 509, + 254, + 524 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 255, + 511, + 268, + 521 + ], + "score": 0.47, + "content": "\\alpha \\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 509, + 383, + 524 + ], + "score": 1.0, + "content": "is a feasible point of (P) iff", + "type": "text" + }, + { + "bbox": [ + 384, + 509, + 458, + 522 + ], + "score": 0.93, + "content": "\\alpha \\ge q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { - 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 509, + 506, + 524 + ], + "score": 1.0, + "content": ". Thus, the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 520, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 398, + 538 + ], + "score": 1.0, + "content": "minimum objective value over all feasible points of (P) in the direction of", + "type": "text" + }, + { + "bbox": [ + 398, + 523, + 405, + 533 + ], + "score": 0.77, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 520, + 415, + 538 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 415, + 522, + 506, + 536 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\frac { 1 } { 2 } \\| \\pmb { \\theta } / q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { 1 / L } \\| _ { 2 } ^ { 2 } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 534, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 153, + 550 + ], + "score": 0.93, + "content": "\\frac { 1 } { 2 } \\bar { \\gamma } ( \\pmb { \\theta } ) ^ { - 2 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 534, + 506, + 551 + ], + "score": 1.0, + "content": ". Taking minimum over all possible directions, we can conclude that if the maximum", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 548, + 393, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 548, + 193, + 565 + ], + "score": 1.0, + "content": "normalized margin is", + "type": "text" + }, + { + "bbox": [ + 193, + 552, + 204, + 563 + ], + "score": 0.87, + "content": "\\bar { \\gamma } _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 548, + 357, + 565 + ], + "score": 1.0, + "content": ", then the minimum objective of (P) is", + "type": "text" + }, + { + "bbox": [ + 357, + 548, + 390, + 564 + ], + "score": 0.92, + "content": "\\frac { 1 } { 2 } \\bar { \\gamma } _ { * } ^ { - 2 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 548, + 393, + 565 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "It can be proved that (P) satisfies the Mangasarian-Fromovitz Constraint Qualification (MFCQ) (See", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "score": 1.0, + "content": "Lemma C.7). Thus, KKT conditions are first-order necessary conditions for global optimality. For", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "linear models, KKT conditions are also sufficient for ensuring global optimality; however, for deep", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 205, + 613 + ], + "score": 1.0, + "content": "homogeneous networks,", + "type": "text" + }, + { + "bbox": [ + 205, + 601, + 230, + 613 + ], + "score": 0.92, + "content": "q _ { n } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "can be highly non-convex. Indeed, as gradient descent is a first-order", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 367, + 624 + ], + "score": 1.0, + "content": "optimization method, if we do not make further assumptions on", + "type": "text" + }, + { + "bbox": [ + 367, + 612, + 392, + 624 + ], + "score": 0.92, + "content": "q _ { n } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 612, + 505, + 624 + ], + "score": 1.0, + "content": ", then it is easy to construct", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "examples that gradient descent does not lead to a normalized margin that is globally optimal. Thus,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "proving the convergence to KKT points is perhaps the best we can hope for in our setting, and it is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 645, + 497, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 497, + 658 + ], + "score": 1.0, + "content": "an interesting future work to prove stronger convergence results with further natural assumptions.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 661, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "score": 1.0, + "content": "Moreover, we can prove the following corollary, which characterizes the optimality of the normal-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "ized margin using SVM with Neural Tangent Kernel (NTK, introduced in (Jacot et al., 2018)) defined", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 684, + 325, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 325, + 696 + ], + "score": 1.0, + "content": "at limit points. The proof is deferred to Appendix C.6.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "Corollary 4.5 (Corollary of Theorem 4.4). Assume (S1). Then for gradient flow under assump-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 130, + 723 + ], + "score": 1.0, + "content": "tions", + "type": "text" + }, + { + "bbox": [ + 130, + 710, + 178, + 720 + ], + "score": 0.41, + "content": "( A 2 ) \\textrm { - } ( A 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 709, + 485, + 723 + ], + "score": 1.0, + "content": "or gradient descent under assumptions (A2) - (A4), (S5), any limit point", + "type": "text" + }, + { + "bbox": [ + 485, + 709, + 493, + 720 + ], + "score": 0.71, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 107, + 719, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 208, + 733 + ], + "score": 0.92, + "content": "\\{ \\pmb \\theta ( t ) / \\| \\pmb \\theta ( t ) \\| _ { 2 } : t \\geq 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 719, + 507, + 734 + ], + "score": 1.0, + "content": "is along the max-margin direction for the hard-margin SVM with kernel", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 81, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 341, + 95 + ], + "score": 1.0, + "content": "Convergence Rates. It is shown in Theorem 4.1, 4.2 that", + "type": "text" + }, + { + "bbox": [ + 342, + 82, + 395, + 95 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 82, + 412, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 413, + 82, + 470, + 95 + ], + "score": 0.95, + "content": "\\| \\pmb \\theta ( t ) \\| _ { 2 } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". In fact,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "with a more refined analysis, we can prove tight loss convergence and weight growth rates using the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 257, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 257, + 117 + ], + "score": 1.0, + "content": "monotonicity of normalized margins.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 82, + 505, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 506, + 153 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 505, + 132 + ], + "score": 1.0, + "content": "Theorem 4.3 (Corollary of Theorem A.10 and E.5). For gradient flow under assumptions (A1) -", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 130, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 505, + 142 + ], + "score": 1.0, + "content": "(A4) or gradient descent under assumptions (S1), (A2) - (A4), (S5), we have the following tight", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 142, + 278, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 278, + 154 + ], + "score": 1.0, + "content": "bounds for training loss and weight norm:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 118, + 505, + 154 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 158, + 457, + 186 + ], + "lines": [ + { + "bbox": [ + 152, + 158, + 457, + 186 + ], + "spans": [ + { + "bbox": [ + 152, + 158, + 457, + 186 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) = \\Theta \\left( \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } \\right) \\qquad a n d \\qquad \\| \\pmb \\theta ( t ) \\| _ { 2 } = \\Theta \\left( ( \\log T ) ^ { 1 / L } \\right) ,", + "type": "interline_equation", + "image_path": "bf296f66e9bbf0a5e8d9ae747033439ecb368de26010005c8aa49638bcdecef3.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 152, + 158, + 457, + 167.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 152, + 167.33333333333334, + 457, + 176.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 152, + 176.66666666666669, + 457, + 186.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 191, + 401, + 207 + ], + "lines": [ + { + "bbox": [ + 104, + 187, + 405, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 133, + 211 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 194, + 158, + 204 + ], + "score": 0.88, + "content": "T = t", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 187, + 246, + 211 + ], + "score": 1.0, + "content": "for gradient flow and", + "type": "text" + }, + { + "bbox": [ + 246, + 191, + 316, + 207 + ], + "score": 0.92, + "content": "\\begin{array} { r } { T = \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 187, + 405, + 211 + ], + "score": 1.0, + "content": "for gradient descent.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 187, + 405, + 211 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 217, + 349, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 349, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 349, + 231 + ], + "score": 1.0, + "content": "4.3 MAIN THEOREM: CONVERGENCE TO KKT POINTS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 180, + 252 + ], + "score": 1.0, + "content": "For gradient flow,", + "type": "text" + }, + { + "bbox": [ + 181, + 240, + 188, + 250 + ], + "score": 0.85, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 238, + 274, + 252 + ], + "score": 1.0, + "content": "is upper-bounded by", + "type": "text" + }, + { + "bbox": [ + 275, + 238, + 414, + 251 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } \\le \\bar { \\gamma } \\le \\operatorname* { s u p } \\{ q _ { n } ( \\pmb \\theta ) : \\lVert \\pmb \\theta \\rVert _ { 2 } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 238, + 505, + 252 + ], + "score": 1.0, + "content": ". Combining this with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 431, + 263 + ], + "score": 1.0, + "content": "Theorem 4.1 and the monotone convergence theorem, it is not hard to see that", + "type": "text" + }, + { + "bbox": [ + 431, + 250, + 504, + 262 + ], + "score": 0.91, + "content": "\\scriptstyle \\operatorname* { l i m } _ { t \\to + \\infty } { \\bar { \\gamma } } ( \\pmb { \\theta } ( t ) )", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 123, + 273 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 261, + 196, + 273 + ], + "score": 0.92, + "content": "\\scriptstyle \\operatorname* { l i m } _ { t \\to + \\infty } { \\tilde { \\gamma } } ( \\pmb { \\theta } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "exist and equal to the same value. Using a similar argument, we can draw the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 259, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 259, + 285 + ], + "score": 1.0, + "content": "same conclusion for gradient descent.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 238, + 505, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 505, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 303 + ], + "score": 1.0, + "content": "To understand the implicit regularization effect, a natural question arises: what optimality property", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "does the limit of normalized margin have? To this end, we identify a natural constrained optimization", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 340, + 322 + ], + "score": 1.0, + "content": "problem related to margin maximization, and prove that", + "type": "text" + }, + { + "bbox": [ + 340, + 310, + 358, + 322 + ], + "score": 0.91, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 311, + 505, + 322 + ], + "score": 1.0, + "content": "directionally converges to its KKT", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 333 + ], + "score": 1.0, + "content": "points, as shown below. We note that we can extend this result to the finite time case, and show", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "that gradient flow or gradient descent passes through an approximate KKT point after a certain", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 344, + 504, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 504, + 355 + ], + "score": 1.0, + "content": "amount of time. See Theorem A.9 in Appendix A and Theorem E.4 in Appendix E for the details.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "We will briefly review the definition of KKT points and approximate KKT points for a constraint", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 266, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 266, + 377 + ], + "score": 1.0, + "content": "optimization problem in Appendix C.1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 287, + 506, + 377 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "Theorem 4.4 (Corollary of Theorem A.8 and E.3). For gradient flow under assumptions (A1) - (A4)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 504, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 412, + 414 + ], + "score": 1.0, + "content": "or gradient descent under assumptions (S1), (A2) - (A4), (S5), any limit point", + "type": "text" + }, + { + "bbox": [ + 413, + 394, + 420, + 405 + ], + "score": 0.61, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 387, + 431, + 414 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 432, + 391, + 504, + 411 + ], + "score": 0.88, + "content": "\\left\\{ \\frac { \\pmb \\theta ( t ) } { \\| \\pmb \\theta ( t ) \\| _ { 2 } } : t \\geq 0 \\right\\}", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 409, + 476, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 458, + 423 + ], + "score": 1.0, + "content": "is along the direction of a KKT point of the following constrained optimization problem", + "type": "text" + }, + { + "bbox": [ + 458, + 411, + 471, + 421 + ], + "score": 0.75, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 409, + 476, + 423 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 379, + 506, + 423 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 425, + 410, + 450 + ], + "lines": [ + { + "bbox": [ + 199, + 425, + 410, + 450 + ], + "spans": [ + { + "bbox": [ + 199, + 425, + 410, + 450 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } \\quad \\frac { 1 } { 2 } \\| \\pmb { \\theta } \\| _ { 2 } ^ { 2 } \\qquad \\quad s . t . \\quad q _ { n } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ]", + "type": "interline_equation", + "image_path": "b658d9b248bc89608d0277c369d86ef3ba9b1253168693340ce7ae371032716e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 199, + 425, + 410, + 450 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 504, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 212, + 469 + ], + "score": 1.0, + "content": "That is, for any limit point", + "type": "text" + }, + { + "bbox": [ + 212, + 455, + 219, + 466 + ], + "score": 0.78, + "content": "\\bar { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 454, + 332, + 469 + ], + "score": 1.0, + "content": ", there exists a scaling factor", + "type": "text" + }, + { + "bbox": [ + 333, + 456, + 358, + 466 + ], + "score": 0.89, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 454, + 397, + 469 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 397, + 455, + 411, + 466 + ], + "score": 0.86, + "content": "\\alpha \\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 454, + 505, + 469 + ], + "score": 1.0, + "content": "satisfies Karush-Kuhn-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 466, + 236, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 218, + 479 + ], + "score": 1.0, + "content": "Tucker (KKT) conditions of", + "type": "text" + }, + { + "bbox": [ + 218, + 467, + 232, + 478 + ], + "score": 0.79, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 466, + 236, + 479 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 454, + 505, + 479 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "Minimizing (P) over its feasible region is equivalent to maximizing the normalized margin over all", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "possible directions. The proof is as follows. Note that we only need to consider all feasible points", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 114, + 521 + ], + "score": 0.64, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 509, + 136, + 524 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 137, + 510, + 190, + 523 + ], + "score": 0.93, + "content": "q _ { \\mathrm { m i n } } ( \\pmb \\theta ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 509, + 243, + 524 + ], + "score": 1.0, + "content": ". For a fixed", + "type": "text" + }, + { + "bbox": [ + 243, + 511, + 250, + 520 + ], + "score": 0.45, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 509, + 254, + 524 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 255, + 511, + 268, + 521 + ], + "score": 0.47, + "content": "\\alpha \\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 509, + 383, + 524 + ], + "score": 1.0, + "content": "is a feasible point of (P) iff", + "type": "text" + }, + { + "bbox": [ + 384, + 509, + 458, + 522 + ], + "score": 0.93, + "content": "\\alpha \\ge q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { - 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 509, + 506, + 524 + ], + "score": 1.0, + "content": ". Thus, the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 520, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 398, + 538 + ], + "score": 1.0, + "content": "minimum objective value over all feasible points of (P) in the direction of", + "type": "text" + }, + { + "bbox": [ + 398, + 523, + 405, + 533 + ], + "score": 0.77, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 520, + 415, + 538 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 415, + 522, + 506, + 536 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\frac { 1 } { 2 } \\| \\pmb { \\theta } / q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { 1 / L } \\| _ { 2 } ^ { 2 } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 534, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 153, + 550 + ], + "score": 0.93, + "content": "\\frac { 1 } { 2 } \\bar { \\gamma } ( \\pmb { \\theta } ) ^ { - 2 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 534, + 506, + 551 + ], + "score": 1.0, + "content": ". Taking minimum over all possible directions, we can conclude that if the maximum", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 548, + 393, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 548, + 193, + 565 + ], + "score": 1.0, + "content": "normalized margin is", + "type": "text" + }, + { + "bbox": [ + 193, + 552, + 204, + 563 + ], + "score": 0.87, + "content": "\\bar { \\gamma } _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 548, + 357, + 565 + ], + "score": 1.0, + "content": ", then the minimum objective of (P) is", + "type": "text" + }, + { + "bbox": [ + 357, + 548, + 390, + 564 + ], + "score": 0.92, + "content": "\\frac { 1 } { 2 } \\bar { \\gamma } _ { * } ^ { - 2 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 548, + 393, + 565 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 488, + 506, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "It can be proved that (P) satisfies the Mangasarian-Fromovitz Constraint Qualification (MFCQ) (See", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "score": 1.0, + "content": "Lemma C.7). Thus, KKT conditions are first-order necessary conditions for global optimality. For", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "linear models, KKT conditions are also sufficient for ensuring global optimality; however, for deep", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 205, + 613 + ], + "score": 1.0, + "content": "homogeneous networks,", + "type": "text" + }, + { + "bbox": [ + 205, + 601, + 230, + 613 + ], + "score": 0.92, + "content": "q _ { n } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "can be highly non-convex. Indeed, as gradient descent is a first-order", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 367, + 624 + ], + "score": 1.0, + "content": "optimization method, if we do not make further assumptions on", + "type": "text" + }, + { + "bbox": [ + 367, + 612, + 392, + 624 + ], + "score": 0.92, + "content": "q _ { n } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 612, + 505, + 624 + ], + "score": 1.0, + "content": ", then it is easy to construct", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "examples that gradient descent does not lead to a normalized margin that is globally optimal. Thus,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "proving the convergence to KKT points is perhaps the best we can hope for in our setting, and it is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 645, + 497, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 497, + 658 + ], + "score": 1.0, + "content": "an interesting future work to prove stronger convergence results with further natural assumptions.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 568, + 506, + 658 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 661, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "score": 1.0, + "content": "Moreover, we can prove the following corollary, which characterizes the optimality of the normal-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "ized margin using SVM with Neural Tangent Kernel (NTK, introduced in (Jacot et al., 2018)) defined", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 684, + 325, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 325, + 696 + ], + "score": 1.0, + "content": "at limit points. The proof is deferred to Appendix C.6.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 661, + 505, + 696 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "Corollary 4.5 (Corollary of Theorem 4.4). Assume (S1). Then for gradient flow under assump-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 130, + 723 + ], + "score": 1.0, + "content": "tions", + "type": "text" + }, + { + "bbox": [ + 130, + 710, + 178, + 720 + ], + "score": 0.41, + "content": "( A 2 ) \\textrm { - } ( A 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 709, + 485, + 723 + ], + "score": 1.0, + "content": "or gradient descent under assumptions (A2) - (A4), (S5), any limit point", + "type": "text" + }, + { + "bbox": [ + 485, + 709, + 493, + 720 + ], + "score": 0.71, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 107, + 719, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 208, + 733 + ], + "score": 0.92, + "content": "\\{ \\pmb \\theta ( t ) / \\| \\pmb \\theta ( t ) \\| _ { 2 } : t \\geq 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 719, + 507, + 734 + ], + "score": 1.0, + "content": "is along the max-margin direction for the hard-margin SVM with kernel", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 698, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 107, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 107, + 82, + 250, + 96 + ], + "score": 0.91, + "content": "K _ { \\bar { \\theta } } ( x , x ^ { \\prime } ) = \\left. \\nabla \\Phi _ { x } ( \\bar { \\theta } ) , \\nabla \\Phi _ { x ^ { \\prime } } ( \\bar { \\theta } ) \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 81, + 281, + 96 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 281, + 82, + 358, + 95 + ], + "score": 0.92, + "content": "\\Phi _ { x } ( \\pmb { \\theta } ) : = \\Phi ( \\pmb { \\theta } ; \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 81, + 434, + 96 + ], + "score": 1.0, + "content": ". That is, for some", + "type": "text" + }, + { + "bbox": [ + 434, + 83, + 461, + 93 + ], + "score": 0.82, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 81, + 466, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 466, + 82, + 479, + 93 + ], + "score": 0.78, + "content": "\\alpha \\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 383, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 383, + 107 + ], + "score": 1.0, + "content": "optimal solution for the following constrained optimization problem:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 108, + 435, + 131 + ], + "lines": [ + { + "bbox": [ + 175, + 108, + 435, + 131 + ], + "spans": [ + { + "bbox": [ + 175, + 108, + 435, + 131 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } \\quad \\frac { 1 } { 2 } \\| \\pmb { \\theta } \\| _ { 2 } ^ { 2 } \\qquad \\quad s . t . \\quad y _ { n } \\left. \\pmb { \\theta } , \\nabla \\Phi _ { \\pmb { x } _ { n } } ( \\bar { \\pmb { \\theta } } ) \\right. \\geq 1 \\qquad \\forall n \\in [ N ]", + "type": "interline_equation", + "image_path": "4c459abaf9318ba5d7a7ad87e0c53f28e8a58b443c42245c1d4e9e6b3243bd74.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 175, + 108, + 435, + 131 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 135, + 503, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 134, + 504, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 162, + 149 + ], + "score": 1.0, + "content": "If we assume", + "type": "text" + }, + { + "bbox": [ + 162, + 136, + 181, + 147 + ], + "score": 0.53, + "content": "( A I )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 134, + 225, + 149 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 225, + 136, + 243, + 147 + ], + "score": 0.47, + "content": "( S I )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 134, + 430, + 149 + ], + "score": 1.0, + "content": "for gradient flow, then there exists a mapping", + "type": "text" + }, + { + "bbox": [ + 431, + 134, + 504, + 147 + ], + "score": 0.93, + "content": "h ( x ) \\in \\partial ^ { \\circ } \\Phi _ { x } ( \\bar { \\theta } )", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 146, + 383, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 266, + 159 + ], + "score": 1.0, + "content": "such that the same conclusion holds for", + "type": "text" + }, + { + "bbox": [ + 267, + 146, + 379, + 158 + ], + "score": 0.93, + "content": "K _ { \\bar { \\theta } } ( x , x ^ { \\prime } ) = \\langle h ( x ) , h ( x ^ { \\prime } ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 146, + 383, + 159 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 171, + 231, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 232, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 232, + 184 + ], + "score": 1.0, + "content": "4.4 OTHER MAIN RESULTS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 191, + 378, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 379, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 379, + 205 + ], + "score": 1.0, + "content": "The above results can be extended to other settings as shown below.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 208, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "Other Binary Classification Loss. The results on exponential loss can be generalized to a much", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "broader class of binary classification loss. The class includes the logistic loss which is one of the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 220, + 243 + ], + "score": 1.0, + "content": "most popular loss functions,", + "type": "text" + }, + { + "bbox": [ + 221, + 230, + 303, + 242 + ], + "score": 0.9, + "content": "\\ell ( q ) = \\log ( 1 + e ^ { - q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 229, + 505, + 243 + ], + "score": 1.0, + "content": ". The function class also includes other losses with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 193, + 258 + ], + "score": 1.0, + "content": "exponential tail, e.g.,", + "type": "text" + }, + { + "bbox": [ + 194, + 242, + 338, + 255 + ], + "score": 0.9, + "content": "\\ell ( q ) = e ^ { - q ^ { 3 } } , \\ell ( q ) = \\log ( 1 + e ^ { - q ^ { 3 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 241, + 506, + 258 + ], + "score": 1.0, + "content": ". For all those loss functions, we can use", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 253, + 416, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 185, + 267 + ], + "score": 1.0, + "content": "its inverse function", + "type": "text" + }, + { + "bbox": [ + 185, + 254, + 201, + 264 + ], + "score": 0.92, + "content": "\\ell ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 253, + 416, + 267 + ], + "score": 1.0, + "content": "to define the smoothed normalized margin as follows", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 268, + 347, + 297 + ], + "lines": [ + { + "bbox": [ + 263, + 268, + 347, + 297 + ], + "spans": [ + { + "bbox": [ + 263, + 268, + 347, + 297 + ], + "score": 0.94, + "content": "\\tilde { \\gamma } ( \\pmb { \\theta } ) = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ( \\pmb { \\theta } ) ) } { \\| \\pmb { \\theta } \\| _ { 2 } ^ { L } } .", + "type": "interline_equation", + "image_path": "21749494740c2b50e962f8915088cb2872acdb2696ccd7a5a5173548dddf35db.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 263, + 268, + 347, + 282.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 263, + 282.5, + 347, + 297.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 299, + 504, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "Then all our results for gradient flow continue to hold (Appendix A). Using a similar modification,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 310, + 328, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 328, + 323 + ], + "score": 1.0, + "content": "we can also extend it to gradient descent (Appendix F).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 375, + 339 + ], + "score": 1.0, + "content": "Cross-entropy Loss. In multi-class classification, we can define", + "type": "text" + }, + { + "bbox": [ + 375, + 330, + 387, + 339 + ], + "score": 0.86, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "to be the difference between", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 337, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 352 + ], + "score": 1.0, + "content": "the classification score for the true label and the maximum score for the other labels, then the margin", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 103, + 348, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 103, + 348, + 370, + 368 + ], + "score": 1.0, + "content": "qmin := minn∈[N] qn and the normalized margin γ¯(θ) := qmin(θ)kθkL", + "type": "text" + }, + { + "bbox": [ + 367, + 350, + 505, + 365 + ], + "score": 1.0, + "content": "can be similarly defined as before.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "In Appendix G, we define the smoothed normalized margin for cross-entropy loss to be the same as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "score": 1.0, + "content": "that for logistic loss (See Remark A.4). Then we show that Theorem 4.1 and Theorem 4.4 still hold", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "(but with a slightly different definition of (P)) for gradient flow, and we also extend the results to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 176, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 176, + 410 + ], + "score": 1.0, + "content": "gradient descent.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "Multi-homogeneous Models. Some neural networks indeed possess a stronger property than", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "homogeneity, which we call multi-homogeneity. For example, the output of a CNN (without bias", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "terms) is 1-homogeneous with respect to the weights of each layer. 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The results on exponential loss can be generalized to a much", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "broader class of binary classification loss. The class includes the logistic loss which is one of the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 220, + 243 + ], + "score": 1.0, + "content": "most popular loss functions,", + "type": "text" + }, + { + "bbox": [ + 221, + 230, + 303, + 242 + ], + "score": 0.9, + "content": "\\ell ( q ) = \\log ( 1 + e ^ { - q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 229, + 505, + 243 + ], + "score": 1.0, + "content": ". The function class also includes other losses with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 193, + 258 + ], + "score": 1.0, + "content": "exponential tail, e.g.,", + "type": "text" + }, + { + "bbox": [ + 194, + 242, + 338, + 255 + ], + "score": 0.9, + "content": "\\ell ( q ) = e ^ { - q ^ { 3 } } , \\ell ( q ) = \\log ( 1 + e ^ { - q ^ { 3 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 241, + 506, + 258 + ], + "score": 1.0, + "content": ". For all those loss functions, we can use", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 253, + 416, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 185, + 267 + ], + "score": 1.0, + "content": "its inverse function", + "type": "text" + }, + { + "bbox": [ + 185, + 254, + 201, + 264 + ], + "score": 0.92, + "content": "\\ell ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 253, + 416, + 267 + ], + "score": 1.0, + "content": "to define the smoothed normalized margin as follows", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 209, + 506, + 267 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 268, + 347, + 297 + ], + "lines": [ + { + "bbox": [ + 263, + 268, + 347, + 297 + ], + "spans": [ + { + "bbox": [ + 263, + 268, + 347, + 297 + ], + "score": 0.94, + "content": "\\tilde { \\gamma } ( \\pmb { \\theta } ) = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ( \\pmb { \\theta } ) ) } { \\| \\pmb { \\theta } \\| _ { 2 } ^ { L } } .", + "type": "interline_equation", + "image_path": "21749494740c2b50e962f8915088cb2872acdb2696ccd7a5a5173548dddf35db.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 263, + 268, + 347, + 282.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 263, + 282.5, + 347, + 297.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 299, + 504, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "Then all our results for gradient flow continue to hold (Appendix A). Using a similar modification,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 310, + 328, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 328, + 323 + ], + "score": 1.0, + "content": "we can also extend it to gradient descent (Appendix F).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 299, + 506, + 323 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 375, + 339 + ], + "score": 1.0, + "content": "Cross-entropy Loss. In multi-class classification, we can define", + "type": "text" + }, + { + "bbox": [ + 375, + 330, + 387, + 339 + ], + "score": 0.86, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "to be the difference between", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 337, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 352 + ], + "score": 1.0, + "content": "the classification score for the true label and the maximum score for the other labels, then the margin", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 103, + 348, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 103, + 348, + 370, + 368 + ], + "score": 1.0, + "content": "qmin := minn∈[N] qn and the normalized margin γ¯(θ) := qmin(θ)kθkL", + "type": "text" + }, + { + "bbox": [ + 367, + 350, + 505, + 365 + ], + "score": 1.0, + "content": "can be similarly defined as before.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "In Appendix G, we define the smoothed normalized margin for cross-entropy loss to be the same as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "score": 1.0, + "content": "that for logistic loss (See Remark A.4). Then we show that Theorem 4.1 and Theorem 4.4 still hold", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "(but with a slightly different definition of (P)) for gradient flow, and we also extend the results to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 176, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 176, + 410 + ], + "score": 1.0, + "content": "gradient descent.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 103, + 328, + 506, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "Multi-homogeneous Models. Some neural networks indeed possess a stronger property than", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "homogeneity, which we call multi-homogeneity. For example, the output of a CNN (without bias", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "terms) is 1-homogeneous with respect to the weights of each layer. In general, we say that a neu-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 156, + 462 + ], + "score": 1.0, + "content": "ral network", + "type": "text" + }, + { + "bbox": [ + 156, + 448, + 189, + 460 + ], + "score": 0.93, + "content": "\\Phi ( \\pmb \\theta ; \\pmb x )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 447, + 212, + 462 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 212, + 448, + 293, + 460 + ], + "score": 0.91, + "content": "\\pmb \\theta = ( \\pmb w _ { 1 } , \\dots , \\pmb w _ { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 447, + 306, + 462 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 306, + 448, + 358, + 460 + ], + "score": 0.9, + "content": "\\left( k _ { 1 } , \\ldots , k _ { m } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 447, + 461, + 462 + ], + "score": 1.0, + "content": "-homogeneous if for any", + "type": "text" + }, + { + "bbox": [ + 469, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "and any", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 453, + 510, + 480 + ], + "spans": [ + { + "bbox": [ + 107, + 461, + 169, + 473 + ], + "score": 0.9, + "content": "c _ { 1 } , \\ldots , c _ { m } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 453, + 207, + 480 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 208, + 460, + 439, + 473 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\Phi \\big ( c _ { 1 } \\pmb { w } _ { 1 } , \\ldots , c _ { m } \\pmb { w } _ { m } ; \\pmb { x } \\big ) = \\prod _ { i = 1 } ^ { m } c _ { i } ^ { k _ { i } } \\cdot \\Phi \\big ( \\pmb { w } _ { 1 } , \\ldots , \\pmb { w } _ { m } ; \\pmb { x } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 453, + 510, + 480 + ], + "score": 1.0, + "content": ". 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1 } ( \\pmb { \\mathscr { L } } ) } { \\prod _ { i = 1 } ^ { m } \\| \\pmb { w } _ { i } \\| _ { 2 } ^ { k _ { i } } } ,", + "type": "interline_equation", + "image_path": "6f61b68c966b00aa003d66d58ce28f342b2d5d2a6005bf234a614e207c517d7f.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 578, + 376, + 593.5 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 234, + 593.5, + 376, + 609.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 104, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 192, + 624 + ], + "score": 1.0, + "content": "It can be shown that", + "type": "text" + }, + { + "bbox": [ + 192, + 612, + 200, + 623 + ], + "score": 0.86, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "is also non-decreasing during training when the loss is small enough (Ap-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 137, + 635 + ], + "score": 1.0, + "content": "pendix", + "type": "text" + }, + { + "bbox": [ + 137, + 623, + 146, + 632 + ], + "score": 0.27, + "content": "_ \\mathrm { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 622, + 372, + 635 + ], + "score": 1.0, + "content": "). In the case of cross-entropy loss, we can still define", + "type": "text" + }, + { + "bbox": [ + 372, + 623, + 380, + 634 + ], + "score": 0.84, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 622, + 436, + 635 + ], + "score": 1.0, + "content": "by (5) while", + "type": "text" + }, + { + "bbox": [ + 436, + 622, + 452, + 634 + ], + "score": 0.9, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "is set to the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 633, + 218, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 218, + 645 + ], + "score": 1.0, + "content": "logistic loss in the formula.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 610, + 506, + 645 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 659, + 478, + 687 + ], + "lines": [ + { + "bbox": [ + 104, + 659, + 480, + 674 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 480, + 674 + ], + "score": 1.0, + "content": "5 PROOF SKETCH: GRADIENT FLOW ON HOMOGENEOUS MODEL WITH", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 125, + 675, + 232, + 687 + ], + "spans": [ + { + "bbox": [ + 125, + 675, + 232, + 687 + ], + "score": 1.0, + "content": "EXPONENTIAL LOSS", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "In this section, we present a proof sketch in the case of gradient flow on homogeneous model with", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "exponential loss to illustrate our proof ideas. Due to space limit, the proof for the main theorems on", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 485, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 485, + 734 + ], + "score": 1.0, + "content": "gradient flow and gradient descent in Section 4 are deferred to Appendix A and E respectively.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 698, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 338, + 94 + ], + "score": 1.0, + "content": "For convenience, we introduce a few more notations for a", + "type": "text" + }, + { + "bbox": [ + 338, + 83, + 346, + 92 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 82, + 468, + 94 + ], + "score": 1.0, + "content": "-homogeneous neural network", + "type": "text" + }, + { + "bbox": [ + 468, + 82, + 501, + 94 + ], + "score": 0.92, + "content": "\\Phi ( \\pmb \\theta ; \\pmb x )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 82, + 505, + 94 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 103, + 89, + 504, + 108 + ], + "spans": [ + { + "bbox": [ + 103, + 89, + 123, + 108 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 93, + 258, + 106 + ], + "score": 0.91, + "content": "S ^ { d - 1 } = \\{ \\pmb \\theta \\in \\mathbb { R } ^ { d } : \\| \\pmb \\theta \\| _ { 2 } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 89, + 313, + 108 + ], + "score": 1.0, + "content": "be the set of", + "type": "text" + }, + { + "bbox": [ + 314, + 93, + 326, + 104 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 89, + 458, + 108 + ], + "score": 1.0, + "content": "-normalized parameters. Define", + "type": "text" + }, + { + "bbox": [ + 458, + 94, + 504, + 106 + ], + "score": 0.91, + "content": "\\rho : = \\| \\pmb \\theta \\| _ { 2 }", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 104, + 506, + 121 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 125, + 120 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 105, + 214, + 121 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\Vert \\pmb \\theta \\Vert _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 105, + 360, + 120 + ], + "score": 1.0, + "content": "to be the length and direction of", + "type": "text" + }, + { + "bbox": [ + 360, + 107, + 367, + 117 + ], + "score": 0.68, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 105, + 506, + 120 + ], + "score": 1.0, + "content": ". For both gradient descent and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 166, + 132 + ], + "score": 1.0, + "content": "gradient flow,", + "type": "text" + }, + { + "bbox": [ + 167, + 122, + 174, + 130 + ], + "score": 0.73, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 119, + 264, + 132 + ], + "score": 1.0, + "content": "is a function of time", + "type": "text" + }, + { + "bbox": [ + 265, + 121, + 270, + 130 + ], + "score": 0.59, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 119, + 479, + 132 + ], + "score": 1.0, + "content": ". For convenience, we also view the functions of", + "type": "text" + }, + { + "bbox": [ + 479, + 120, + 486, + 130 + ], + "score": 0.72, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 119, + 505, + 132 + ], + "score": 1.0, + "content": ", in-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 130, + 503, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 140, + 144 + ], + "score": 1.0, + "content": "cluding", + "type": "text" + }, + { + "bbox": [ + 141, + 131, + 227, + 143 + ], + "score": 0.9, + "content": "\\mathcal { L } ( \\pmb { \\theta } ) , q _ { n } ( \\pmb { \\theta } ) , q _ { \\mathrm { m i n } } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 130, + 299, + 144 + ], + "score": 1.0, + "content": ", as functions of", + "type": "text" + }, + { + "bbox": [ + 299, + 132, + 304, + 141 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 130, + 387, + 144 + ], + "score": 1.0, + "content": ". So we can write", + "type": "text" + }, + { + "bbox": [ + 388, + 131, + 503, + 144 + ], + "score": 0.92, + "content": "\\mathscr { L } ( t ) : = \\mathscr { L } ( \\pmb \\theta ( t ) ) , q _ { n } ( t ) : =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 141, + 240, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 235, + 154 + ], + "score": 0.82, + "content": "q _ { n } ( \\pmb \\theta ( t ) ) , q _ { \\mathrm { m i n } } ( t ) : = q _ { \\mathrm { m i n } } ( \\pmb \\theta ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 141, + 240, + 155 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 158, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 504, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 504, + 171 + ], + "score": 1.0, + "content": "Lemma 5.1 below is the key lemma in our proof. It decomposes the growth of the smoothed normal-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "ized margin into the ratio of two quantities related to the radial and tangential velocity components", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 181, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 118, + 192 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 181, + 126, + 191 + ], + "score": 0.74, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 181, + 505, + 192 + ], + "score": 1.0, + "content": "respectively. We will give a proof sketch for this later in this section. We believe that this", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 193, + 241, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 241, + 204 + ], + "score": 1.0, + "content": "lemma is of independent interest.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 332, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 332, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 302, + 219 + ], + "score": 1.0, + "content": "Lemma 5.1 (Corollary of Lemma B.1). For a.e.", + "type": "text" + }, + { + "bbox": [ + 302, + 206, + 328, + 216 + ], + "score": 0.88, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 203, + 332, + 219 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 218, + 424, + 253 + ], + "lines": [ + { + "bbox": [ + 186, + 218, + 424, + 253 + ], + "spans": [ + { + "bbox": [ + 186, + 218, + 424, + 253 + ], + "score": 0.9, + "content": "\\frac { d } { d t } \\log \\rho > 0 \\quad a n d \\quad \\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "965649f468c15c8e964a4520f5c3cbf7c57a6c5b2f4a7a2a6b09815e4dc23ac0.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 218, + 424, + 235.5 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 186, + 235.5, + 424, + 253.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 506, + 304 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "Using Lemma 5.1, the first two claims in Theorem 4.1 can be directly proved. For the third claim,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 271, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 475, + 282 + ], + "score": 1.0, + "content": "we make use of the monotonicity of the margin to lower bound the gradient, and then show", + "type": "text" + }, + { + "bbox": [ + 475, + 271, + 504, + 281 + ], + "score": 0.89, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 124, + 294 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 282, + 165, + 293 + ], + "score": 0.9, + "content": "\\rho + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 281, + 217, + 294 + ], + "score": 1.0, + "content": ". Recall that", + "type": "text" + }, + { + "bbox": [ + 218, + 282, + 225, + 293 + ], + "score": 0.86, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 281, + 248, + 294 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 249, + 281, + 282, + 293 + ], + "score": 0.93, + "content": "O ( \\rho ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 281, + 395, + 294 + ], + "score": 1.0, + "content": "-additive approximation for", + "type": "text" + }, + { + "bbox": [ + 396, + 282, + 402, + 293 + ], + "score": 0.83, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 281, + 506, + 294 + ], + "score": 1.0, + "content": ". So this proves the third", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 292, + 309, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 309, + 304 + ], + "score": 1.0, + "content": "claim. We defer the detailed proof to Appendix B.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 506, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 507, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 343, + 323 + ], + "score": 1.0, + "content": "To show Theorem 4.4, we first change the time measure to", + "type": "text" + }, + { + "bbox": [ + 343, + 310, + 364, + 321 + ], + "score": 0.88, + "content": "\\log \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 308, + 434, + 323 + ], + "score": 1.0, + "content": ", i.e., now we see", + "type": "text" + }, + { + "bbox": [ + 434, + 311, + 439, + 319 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 308, + 507, + 323 + ], + "score": 1.0, + "content": "as a function of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 320, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 128, + 335 + ], + "score": 0.86, + "content": "\\log \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 320, + 370, + 336 + ], + "score": 1.0, + "content": ". So the second inequality in Lemma 5.1 can be rewritten as", + "type": "text" + }, + { + "bbox": [ + 370, + 320, + 454, + 337 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d \\log \\tilde { \\gamma } } { d \\log \\rho } \\geq L \\| \\frac { d \\hat { \\pmb \\theta } } { d \\log \\rho } \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 321, + 505, + 336 + ], + "score": 1.0, + "content": ". Integrating", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 334, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 104, + 334, + 227, + 349 + ], + "score": 1.0, + "content": "on both sides and noting that", + "type": "text" + }, + { + "bbox": [ + 227, + 336, + 235, + 347 + ], + "score": 0.85, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 334, + 482, + 349 + ], + "score": 1.0, + "content": "is upper-bounded, we know that there must be many instant", + "type": "text" + }, + { + "bbox": [ + 483, + 336, + 504, + 347 + ], + "score": 0.85, + "content": "\\log \\rho", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 508, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 145, + 364 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 145, + 347, + 184, + 364 + ], + "score": 0.94, + "content": "\\big \\| \\frac { d \\hat { \\pmb { \\theta } } } { d \\log \\rho } \\big \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 347, + 508, + 364 + ], + "score": 1.0, + "content": "is small. By analyzing the landscape of training loss, we show that these points", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 504, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 445, + 378 + ], + "score": 1.0, + "content": "are “approximate” KKT points. Then we show that every convergent sub-sequence of", + "type": "text" + }, + { + "bbox": [ + 446, + 362, + 504, + 377 + ], + "score": 0.93, + "content": "\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\}", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "can be modified to be a sequence of “approximate” KKT points which converges to the same limit.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "Then we conclude the proof by applying a theorem from (Dutta et al., 2013) to show that the limit of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "this convergent sequence of “approximate” KKT points is a KKT point. We defer the detailed proof", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 408, + 170, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 170, + 422 + ], + "score": 1.0, + "content": "to Appendix C.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 430, + 438 + ], + "score": 1.0, + "content": "Now we give a proof sketch for Lemma 5.1, in which we derive the formula of", + "type": "text" + }, + { + "bbox": [ + 430, + 426, + 438, + 437 + ], + "score": 0.86, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "step by step. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "the proof, we obtain several clean close form formulas for several relevant quantities, by using the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 392, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 392, + 460 + ], + "score": 1.0, + "content": "chain rule and Euler’s theorem for homogeneous functions extensively.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 484 + ], + "score": 1.0, + "content": "Proof Sketch of Lemma 5.1. For ease of presentation, we ignore the regularity issues of taking", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 480, + 504, + 502 + ], + "spans": [ + { + "bbox": [ + 102, + 480, + 348, + 502 + ], + "score": 1.0, + "content": "derivatives in this proof sketch. We start from the equation", + "type": "text" + }, + { + "bbox": [ + 348, + 482, + 504, + 498 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { d \\mathcal { L } } { d t } = - \\left. \\partial ^ { \\circ } \\mathcal { L } ( \\pmb { \\theta } ( t ) ) , \\frac { d \\pmb { \\theta } } { d t } \\right. = - \\left\\| \\frac { d \\pmb { \\theta } } { d t } \\right\\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 494, + 507, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 411, + 515 + ], + "score": 1.0, + "content": "which follows from the chain rule (see also Lemma I.3). Then we note that", + "type": "text" + }, + { + "bbox": [ + 411, + 498, + 423, + 511 + ], + "score": 0.89, + "content": "\\textstyle { \\frac { d \\theta } { d t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 494, + 507, + 515 + ], + "score": 1.0, + "content": "can be decomposed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 505, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 104, + 505, + 254, + 532 + ], + "score": 1.0, + "content": "into two parts: the radial component", + "type": "text" + }, + { + "bbox": [ + 254, + 510, + 306, + 525 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { v } : = \\hat { \\pmb { \\theta } } \\hat { \\pmb { \\theta } } ^ { \\top } \\frac { d \\pmb { \\theta } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 505, + 418, + 532 + ], + "score": 1.0, + "content": "and the tangent component", + "type": "text" + }, + { + "bbox": [ + 419, + 510, + 497, + 525 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbf { \\hat { \\boldsymbol { u } } } : = ( \\mathbf { \\boldsymbol { I } } - \\hat { \\theta } \\hat { \\theta } ^ { \\top } ) \\frac { d \\theta } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 505, + 506, + 532 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 504, + 562 + ], + "lines": [ + { + "bbox": [ + 104, + 530, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 346, + 547 + ], + "score": 1.0, + "content": "The radial component is easier to analyze. By the chain rule,", + "type": "text" + }, + { + "bbox": [ + 346, + 530, + 501, + 549 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| \\pmb { v } \\| _ { 2 } = \\hat { \\pmb { \\theta } } ^ { \\top } \\frac { d \\pmb { \\theta } } { d t } = \\frac { 1 } { \\rho } \\left. \\pmb { \\theta } , \\frac { d \\pmb { \\theta } } { d t } \\right. = \\frac { 1 } { \\rho } \\cdot \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 538, + 505, + 549 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 547, + 258, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 124, + 561 + ], + "score": 1.0, + "content": "Fo", + "type": "text" + }, + { + "bbox": [ + 118, + 547, + 258, + 565 + ], + "score": 1.0, + "content": "r 12 dρ2dt , we have an exact formula:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 563, + 429, + 598 + ], + "lines": [ + { + "bbox": [ + 182, + 563, + 429, + 598 + ], + "spans": [ + { + "bbox": [ + 182, + 563, + 429, + 598 + ], + "score": 0.94, + "content": "{ \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } = \\left. \\theta , { \\frac { d \\theta } { d t } } \\right. = \\left. \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\partial ^ { \\circ } q _ { n } , \\theta \\right. = L \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { n } .", + "type": "interline_equation", + "image_path": "35a1eb0d3944fb2de6f893f0d95c9248a8a4b75bbc7d53bb93d1bb4fc0121251.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 182, + 563, + 429, + 574.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 182, + 574.6666666666666, + 429, + 586.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 182, + 586.3333333333333, + 429, + 597.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 211, + 613 + ], + "score": 1.0, + "content": "The last equality is due to", + "type": "text" + }, + { + "bbox": [ + 212, + 599, + 280, + 611 + ], + "score": 0.92, + "content": "\\langle \\partial ^ { \\circ } q _ { n } , \\theta \\rangle = L q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 598, + 358, + 613 + ], + "score": 1.0, + "content": "by homogeneity of", + "type": "text" + }, + { + "bbox": [ + 359, + 600, + 370, + 610 + ], + "score": 0.87, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 598, + 506, + 613 + ], + "score": 1.0, + "content": ". This is sometimes called Euler’s", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 401, + 623 + ], + "score": 1.0, + "content": "theorem for homogeneous functions (see Theorem B.2). For differentiable", + "type": "text" + }, + { + "bbox": [ + 401, + 612, + 412, + 621 + ], + "score": 0.86, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 609, + 505, + 623 + ], + "score": 1.0, + "content": ", it can be easily proved", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 620, + 446, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 222, + 633 + ], + "score": 1.0, + "content": "by taking the derivative over", + "type": "text" + }, + { + "bbox": [ + 223, + 623, + 228, + 631 + ], + "score": 0.76, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 620, + 295, + 633 + ], + "score": 1.0, + "content": "on both sides of", + "type": "text" + }, + { + "bbox": [ + 295, + 620, + 371, + 633 + ], + "score": 0.94, + "content": "q _ { n } ( c \\pmb { \\theta } ) = c ^ { L } q _ { n } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 620, + 417, + 633 + ], + "score": 1.0, + "content": "and letting", + "type": "text" + }, + { + "bbox": [ + 417, + 621, + 441, + 631 + ], + "score": 0.89, + "content": "c = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 620, + 446, + 633 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 108, + 639, + 263, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 262, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 228, + 656 + ], + "score": 1.0, + "content": "With (6), we can lower bound", + "type": "text" + }, + { + "bbox": [ + 228, + 637, + 250, + 654 + ], + "score": 0.92, + "content": "\\textstyle { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 639, + 262, + 653 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 654, + 423, + 689 + ], + "lines": [ + { + "bbox": [ + 188, + 654, + 423, + 689 + ], + "spans": [ + { + "bbox": [ + 188, + 654, + 423, + 689 + ], + "score": 0.93, + "content": "\\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { n } \\ge L \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { \\operatorname* { m i n } } \\ge L \\cdot \\mathcal { L } \\log \\frac { 1 } { \\mathcal { L } } ,", + "type": "interline_equation", + "image_path": "aa90c976b89dda2f0cccb3a4dbadb19b4be6f7b1e3486c23865443d0cea35669.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 188, + 654, + 423, + 671.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 188, + 671.5, + 423, + 689.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 691, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 690, + 504, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 279, + 709 + ], + "score": 1.0, + "content": "where the last inequality uses the fact that", + "type": "text" + }, + { + "bbox": [ + 279, + 693, + 329, + 704 + ], + "score": 0.9, + "content": "e ^ { - q _ { \\mathrm { m i n } } } \\leq { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 691, + 418, + 709 + ], + "score": 1.0, + "content": ". (7) also implies that", + "type": "text" + }, + { + "bbox": [ + 419, + 690, + 460, + 706 + ], + "score": 0.93, + "content": "\\textstyle { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 691, + 476, + 709 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 476, + 694, + 504, + 705 + ], + "score": 0.88, + "content": "t > t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 103, + 704, + 507, + 727 + ], + "spans": [ + { + "bbox": [ + 103, + 704, + 129, + 727 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 129, + 707, + 172, + 720 + ], + "score": 0.93, + "content": "\\mathcal { L } ( t _ { 0 } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 704, + 190, + 727 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 190, + 708, + 198, + 718 + ], + "score": 0.82, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 704, + 288, + 727 + ], + "score": 1.0, + "content": "is non-increasing. As", + "type": "text" + }, + { + "bbox": [ + 288, + 705, + 361, + 722 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho = \\frac { 1 } { 2 \\rho ^ { 2 } } \\frac { d \\rho ^ { 2 } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 704, + 507, + 727 + ], + "score": 1.0, + "content": ", this also proves the first inequality", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 169, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 169, + 732 + ], + "score": 1.0, + "content": "of Lemma 5.1.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 338, + 94 + ], + "score": 1.0, + "content": "For convenience, we introduce a few more notations for a", + "type": "text" + }, + { + "bbox": [ + 338, + 83, + 346, + 92 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 82, + 468, + 94 + ], + "score": 1.0, + "content": "-homogeneous neural network", + "type": "text" + }, + { + "bbox": [ + 468, + 82, + 501, + 94 + ], + "score": 0.92, + "content": "\\Phi ( \\pmb \\theta ; \\pmb x )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 82, + 505, + 94 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 103, + 89, + 504, + 108 + ], + "spans": [ + { + "bbox": [ + 103, + 89, + 123, + 108 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 93, + 258, + 106 + ], + "score": 0.91, + "content": "S ^ { d - 1 } = \\{ \\pmb \\theta \\in \\mathbb { R } ^ { d } : \\| \\pmb \\theta \\| _ { 2 } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 89, + 313, + 108 + ], + "score": 1.0, + "content": "be the set of", + "type": "text" + }, + { + "bbox": [ + 314, + 93, + 326, + 104 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 89, + 458, + 108 + ], + "score": 1.0, + "content": "-normalized parameters. Define", + "type": "text" + }, + { + "bbox": [ + 458, + 94, + 504, + 106 + ], + "score": 0.91, + "content": "\\rho : = \\| \\pmb \\theta \\| _ { 2 }", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 104, + 506, + 121 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 125, + 120 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 105, + 214, + 121 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\Vert \\pmb \\theta \\Vert _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 105, + 360, + 120 + ], + "score": 1.0, + "content": "to be the length and direction of", + "type": "text" + }, + { + "bbox": [ + 360, + 107, + 367, + 117 + ], + "score": 0.68, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 105, + 506, + 120 + ], + "score": 1.0, + "content": ". For both gradient descent and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 166, + 132 + ], + "score": 1.0, + "content": "gradient flow,", + "type": "text" + }, + { + "bbox": [ + 167, + 122, + 174, + 130 + ], + "score": 0.73, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 119, + 264, + 132 + ], + "score": 1.0, + "content": "is a function of time", + "type": "text" + }, + { + "bbox": [ + 265, + 121, + 270, + 130 + ], + "score": 0.59, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 119, + 479, + 132 + ], + "score": 1.0, + "content": ". For convenience, we also view the functions of", + "type": "text" + }, + { + "bbox": [ + 479, + 120, + 486, + 130 + ], + "score": 0.72, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 119, + 505, + 132 + ], + "score": 1.0, + "content": ", in-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 130, + 503, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 140, + 144 + ], + "score": 1.0, + "content": "cluding", + "type": "text" + }, + { + "bbox": [ + 141, + 131, + 227, + 143 + ], + "score": 0.9, + "content": "\\mathcal { L } ( \\pmb { \\theta } ) , q _ { n } ( \\pmb { \\theta } ) , q _ { \\mathrm { m i n } } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 130, + 299, + 144 + ], + "score": 1.0, + "content": ", as functions of", + "type": "text" + }, + { + "bbox": [ + 299, + 132, + 304, + 141 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 130, + 387, + 144 + ], + "score": 1.0, + "content": ". So we can write", + "type": "text" + }, + { + "bbox": [ + 388, + 131, + 503, + 144 + ], + "score": 0.92, + "content": "\\mathscr { L } ( t ) : = \\mathscr { L } ( \\pmb \\theta ( t ) ) , q _ { n } ( t ) : =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 141, + 240, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 235, + 154 + ], + "score": 0.82, + "content": "q _ { n } ( \\pmb \\theta ( t ) ) , q _ { \\mathrm { m i n } } ( t ) : = q _ { \\mathrm { m i n } } ( \\pmb \\theta ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 141, + 240, + 155 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 103, + 82, + 506, + 155 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 158, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 504, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 504, + 171 + ], + "score": 1.0, + "content": "Lemma 5.1 below is the key lemma in our proof. It decomposes the growth of the smoothed normal-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "ized margin into the ratio of two quantities related to the radial and tangential velocity components", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 181, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 118, + 192 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 181, + 126, + 191 + ], + "score": 0.74, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 181, + 505, + 192 + ], + "score": 1.0, + "content": "respectively. We will give a proof sketch for this later in this section. We believe that this", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 193, + 241, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 241, + 204 + ], + "score": 1.0, + "content": "lemma is of independent interest.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 159, + 505, + 204 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 332, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 332, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 302, + 219 + ], + "score": 1.0, + "content": "Lemma 5.1 (Corollary of Lemma B.1). For a.e.", + "type": "text" + }, + { + "bbox": [ + 302, + 206, + 328, + 216 + ], + "score": 0.88, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 203, + 332, + 219 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 203, + 332, + 219 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 218, + 424, + 253 + ], + "lines": [ + { + "bbox": [ + 186, + 218, + 424, + 253 + ], + "spans": [ + { + "bbox": [ + 186, + 218, + 424, + 253 + ], + "score": 0.9, + "content": "\\frac { d } { d t } \\log \\rho > 0 \\quad a n d \\quad \\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "965649f468c15c8e964a4520f5c3cbf7c57a6c5b2f4a7a2a6b09815e4dc23ac0.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 218, + 424, + 235.5 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 186, + 235.5, + 424, + 253.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 506, + 304 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "Using Lemma 5.1, the first two claims in Theorem 4.1 can be directly proved. For the third claim,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 271, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 475, + 282 + ], + "score": 1.0, + "content": "we make use of the monotonicity of the margin to lower bound the gradient, and then show", + "type": "text" + }, + { + "bbox": [ + 475, + 271, + 504, + 281 + ], + "score": 0.89, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 124, + 294 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 282, + 165, + 293 + ], + "score": 0.9, + "content": "\\rho + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 281, + 217, + 294 + ], + "score": 1.0, + "content": ". Recall that", + "type": "text" + }, + { + "bbox": [ + 218, + 282, + 225, + 293 + ], + "score": 0.86, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 281, + 248, + 294 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 249, + 281, + 282, + 293 + ], + "score": 0.93, + "content": "O ( \\rho ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 281, + 395, + 294 + ], + "score": 1.0, + "content": "-additive approximation for", + "type": "text" + }, + { + "bbox": [ + 396, + 282, + 402, + 293 + ], + "score": 0.83, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 281, + 506, + 294 + ], + "score": 1.0, + "content": ". So this proves the third", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 292, + 309, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 309, + 304 + ], + "score": 1.0, + "content": "claim. We defer the detailed proof to Appendix B.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 259, + 506, + 304 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 506, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 507, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 343, + 323 + ], + "score": 1.0, + "content": "To show Theorem 4.4, we first change the time measure to", + "type": "text" + }, + { + "bbox": [ + 343, + 310, + 364, + 321 + ], + "score": 0.88, + "content": "\\log \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 308, + 434, + 323 + ], + "score": 1.0, + "content": ", i.e., now we see", + "type": "text" + }, + { + "bbox": [ + 434, + 311, + 439, + 319 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 308, + 507, + 323 + ], + "score": 1.0, + "content": "as a function of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 320, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 128, + 335 + ], + "score": 0.86, + "content": "\\log \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 320, + 370, + 336 + ], + "score": 1.0, + "content": ". So the second inequality in Lemma 5.1 can be rewritten as", + "type": "text" + }, + { + "bbox": [ + 370, + 320, + 454, + 337 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d \\log \\tilde { \\gamma } } { d \\log \\rho } \\geq L \\| \\frac { d \\hat { \\pmb \\theta } } { d \\log \\rho } \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 321, + 505, + 336 + ], + "score": 1.0, + "content": ". Integrating", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 334, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 104, + 334, + 227, + 349 + ], + "score": 1.0, + "content": "on both sides and noting that", + "type": "text" + }, + { + "bbox": [ + 227, + 336, + 235, + 347 + ], + "score": 0.85, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 334, + 482, + 349 + ], + "score": 1.0, + "content": "is upper-bounded, we know that there must be many instant", + "type": "text" + }, + { + "bbox": [ + 483, + 336, + 504, + 347 + ], + "score": 0.85, + "content": "\\log \\rho", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 508, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 145, + 364 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 145, + 347, + 184, + 364 + ], + "score": 0.94, + "content": "\\big \\| \\frac { d \\hat { \\pmb { \\theta } } } { d \\log \\rho } \\big \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 347, + 508, + 364 + ], + "score": 1.0, + "content": "is small. By analyzing the landscape of training loss, we show that these points", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 504, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 445, + 378 + ], + "score": 1.0, + "content": "are “approximate” KKT points. Then we show that every convergent sub-sequence of", + "type": "text" + }, + { + "bbox": [ + 446, + 362, + 504, + 377 + ], + "score": 0.93, + "content": "\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\}", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "can be modified to be a sequence of “approximate” KKT points which converges to the same limit.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "Then we conclude the proof by applying a theorem from (Dutta et al., 2013) to show that the limit of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "this convergent sequence of “approximate” KKT points is a KKT point. We defer the detailed proof", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 408, + 170, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 170, + 422 + ], + "score": 1.0, + "content": "to Appendix C.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 308, + 508, + 422 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 430, + 438 + ], + "score": 1.0, + "content": "Now we give a proof sketch for Lemma 5.1, in which we derive the formula of", + "type": "text" + }, + { + "bbox": [ + 430, + 426, + 438, + 437 + ], + "score": 0.86, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "step by step. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "the proof, we obtain several clean close form formulas for several relevant quantities, by using the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 392, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 392, + 460 + ], + "score": 1.0, + "content": "chain rule and Euler’s theorem for homogeneous functions extensively.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 425, + 505, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 484 + ], + "score": 1.0, + "content": "Proof Sketch of Lemma 5.1. For ease of presentation, we ignore the regularity issues of taking", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 480, + 504, + 502 + ], + "spans": [ + { + "bbox": [ + 102, + 480, + 348, + 502 + ], + "score": 1.0, + "content": "derivatives in this proof sketch. We start from the equation", + "type": "text" + }, + { + "bbox": [ + 348, + 482, + 504, + 498 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { d \\mathcal { L } } { d t } = - \\left. \\partial ^ { \\circ } \\mathcal { L } ( \\pmb { \\theta } ( t ) ) , \\frac { d \\pmb { \\theta } } { d t } \\right. = - \\left\\| \\frac { d \\pmb { \\theta } } { d t } \\right\\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 494, + 507, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 411, + 515 + ], + "score": 1.0, + "content": "which follows from the chain rule (see also Lemma I.3). Then we note that", + "type": "text" + }, + { + "bbox": [ + 411, + 498, + 423, + 511 + ], + "score": 0.89, + "content": "\\textstyle { \\frac { d \\theta } { d t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 494, + 507, + 515 + ], + "score": 1.0, + "content": "can be decomposed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 505, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 104, + 505, + 254, + 532 + ], + "score": 1.0, + "content": "into two parts: the radial component", + "type": "text" + }, + { + "bbox": [ + 254, + 510, + 306, + 525 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { v } : = \\hat { \\pmb { \\theta } } \\hat { \\pmb { \\theta } } ^ { \\top } \\frac { d \\pmb { \\theta } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 505, + 418, + 532 + ], + "score": 1.0, + "content": "and the tangent component", + "type": "text" + }, + { + "bbox": [ + 419, + 510, + 497, + 525 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbf { \\hat { \\boldsymbol { u } } } : = ( \\mathbf { \\boldsymbol { I } } - \\hat { \\theta } \\hat { \\theta } ^ { \\top } ) \\frac { d \\theta } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 505, + 506, + 532 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 102, + 469, + 507, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 504, + 562 + ], + "lines": [ + { + "bbox": [ + 104, + 530, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 346, + 547 + ], + "score": 1.0, + "content": "The radial component is easier to analyze. By the chain rule,", + "type": "text" + }, + { + "bbox": [ + 346, + 530, + 501, + 549 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| \\pmb { v } \\| _ { 2 } = \\hat { \\pmb { \\theta } } ^ { \\top } \\frac { d \\pmb { \\theta } } { d t } = \\frac { 1 } { \\rho } \\left. \\pmb { \\theta } , \\frac { d \\pmb { \\theta } } { d t } \\right. = \\frac { 1 } { \\rho } \\cdot \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 538, + 505, + 549 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 547, + 258, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 124, + 561 + ], + "score": 1.0, + "content": "Fo", + "type": "text" + }, + { + "bbox": [ + 118, + 547, + 258, + 565 + ], + "score": 1.0, + "content": "r 12 dρ2dt , we have an exact formula:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 530, + 505, + 565 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 563, + 429, + 598 + ], + "lines": [ + { + "bbox": [ + 182, + 563, + 429, + 598 + ], + "spans": [ + { + "bbox": [ + 182, + 563, + 429, + 598 + ], + "score": 0.94, + "content": "{ \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } = \\left. \\theta , { \\frac { d \\theta } { d t } } \\right. = \\left. \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\partial ^ { \\circ } q _ { n } , \\theta \\right. = L \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { n } .", + "type": "interline_equation", + "image_path": "35a1eb0d3944fb2de6f893f0d95c9248a8a4b75bbc7d53bb93d1bb4fc0121251.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 182, + 563, + 429, + 574.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 182, + 574.6666666666666, + 429, + 586.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 182, + 586.3333333333333, + 429, + 597.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 211, + 613 + ], + "score": 1.0, + "content": "The last equality is due to", + "type": "text" + }, + { + "bbox": [ + 212, + 599, + 280, + 611 + ], + "score": 0.92, + "content": "\\langle \\partial ^ { \\circ } q _ { n } , \\theta \\rangle = L q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 598, + 358, + 613 + ], + "score": 1.0, + "content": "by homogeneity of", + "type": "text" + }, + { + "bbox": [ + 359, + 600, + 370, + 610 + ], + "score": 0.87, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 598, + 506, + 613 + ], + "score": 1.0, + "content": ". This is sometimes called Euler’s", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 401, + 623 + ], + "score": 1.0, + "content": "theorem for homogeneous functions (see Theorem B.2). For differentiable", + "type": "text" + }, + { + "bbox": [ + 401, + 612, + 412, + 621 + ], + "score": 0.86, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 609, + 505, + 623 + ], + "score": 1.0, + "content": ", it can be easily proved", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 620, + 446, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 222, + 633 + ], + "score": 1.0, + "content": "by taking the derivative over", + "type": "text" + }, + { + "bbox": [ + 223, + 623, + 228, + 631 + ], + "score": 0.76, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 620, + 295, + 633 + ], + "score": 1.0, + "content": "on both sides of", + "type": "text" + }, + { + "bbox": [ + 295, + 620, + 371, + 633 + ], + "score": 0.94, + "content": "q _ { n } ( c \\pmb { \\theta } ) = c ^ { L } q _ { n } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 620, + 417, + 633 + ], + "score": 1.0, + "content": "and letting", + "type": "text" + }, + { + "bbox": [ + 417, + 621, + 441, + 631 + ], + "score": 0.89, + "content": "c = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 620, + 446, + 633 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 598, + 506, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 639, + 263, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 262, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 228, + 656 + ], + "score": 1.0, + "content": "With (6), we can lower bound", + "type": "text" + }, + { + "bbox": [ + 228, + 637, + 250, + 654 + ], + "score": 0.92, + "content": "\\textstyle { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 639, + 262, + 653 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 635, + 262, + 656 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 654, + 423, + 689 + ], + "lines": [ + { + "bbox": [ + 188, + 654, + 423, + 689 + ], + "spans": [ + { + "bbox": [ + 188, + 654, + 423, + 689 + ], + "score": 0.93, + "content": "\\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { n } \\ge L \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } q _ { \\operatorname* { m i n } } \\ge L \\cdot \\mathcal { L } \\log \\frac { 1 } { \\mathcal { L } } ,", + "type": "interline_equation", + "image_path": "aa90c976b89dda2f0cccb3a4dbadb19b4be6f7b1e3486c23865443d0cea35669.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 188, + 654, + 423, + 671.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 188, + 671.5, + 423, + 689.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 691, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 690, + 504, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 279, + 709 + ], + "score": 1.0, + "content": "where the last inequality uses the fact that", + "type": "text" + }, + { + "bbox": [ + 279, + 693, + 329, + 704 + ], + "score": 0.9, + "content": "e ^ { - q _ { \\mathrm { m i n } } } \\leq { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 691, + 418, + 709 + ], + "score": 1.0, + "content": ". (7) also implies that", + "type": "text" + }, + { + "bbox": [ + 419, + 690, + 460, + 706 + ], + "score": 0.93, + "content": "\\textstyle { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 691, + 476, + 709 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 476, + 694, + 504, + 705 + ], + "score": 0.88, + "content": "t > t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 103, + 704, + 507, + 727 + ], + "spans": [ + { + "bbox": [ + 103, + 704, + 129, + 727 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 129, + 707, + 172, + 720 + ], + "score": 0.93, + "content": "\\mathcal { L } ( t _ { 0 } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 704, + 190, + 727 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 190, + 708, + 198, + 718 + ], + "score": 0.82, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 704, + 288, + 727 + ], + "score": 1.0, + "content": "is non-increasing. As", + "type": "text" + }, + { + "bbox": [ + 288, + 705, + 361, + 722 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho = \\frac { 1 } { 2 \\rho ^ { 2 } } \\frac { d \\rho ^ { 2 } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 704, + 507, + 727 + ], + "score": 1.0, + "content": ", this also proves the first inequality", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 169, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 169, + 732 + ], + "score": 1.0, + "content": "of Lemma 5.1.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 103, + 690, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 505, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 508, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 167, + 101 + ], + "score": 1.0, + "content": "Now, we have", + "type": "text" + }, + { + "bbox": [ + 167, + 80, + 328, + 102 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\left\\| \\pmb { v } \\right\\| _ { 2 } ^ { 2 } = \\frac { 1 } { \\rho ^ { 2 } } \\left( \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } \\right) ^ { 2 } = \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } \\cdot \\frac { d } { d t } \\log \\rho } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 80, + 508, + 104 + ], + "score": 1.0, + "content": "on the one hand; on the other hand, by the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 98, + 447, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 184, + 123 + ], + "score": 1.0, + "content": "chain rule we have", + "type": "text" + }, + { + "bbox": [ + 184, + 102, + 391, + 119 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { d \\hat { \\pmb { \\theta } } } { d t } = \\frac { 1 } { \\rho ^ { 2 } } ( \\rho \\frac { d \\pmb { \\theta } } { d t } - \\frac { d \\rho } { d t } \\pmb { \\theta } ) = \\frac { 1 } { \\rho ^ { 2 } } ( \\rho \\frac { d \\pmb { \\theta } } { d t } - ( \\hat { \\pmb { \\theta } } ^ { \\top } \\frac { d \\pmb { \\theta } } { d t } ) \\pmb { \\theta } ) = \\frac { \\pmb { u } } { \\rho } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 98, + 447, + 123 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 124, + 439, + 161 + ], + "lines": [ + { + "bbox": [ + 171, + 124, + 439, + 161 + ], + "spans": [ + { + "bbox": [ + 171, + 124, + 439, + 161 + ], + "score": 0.92, + "content": "- { \\frac { d { \\mathcal { L } } } { d t } } = \\left\\| { \\frac { d \\theta } { d t } } \\right\\| _ { 2 } ^ { 2 } = \\left\\| v \\right\\| _ { 2 } ^ { 2 } + \\left\\| u \\right\\| _ { 2 } ^ { 2 } = { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } \\cdot { \\frac { d } { d t } } \\log \\rho + \\rho ^ { 2 } \\left\\| { \\frac { d \\hat { \\theta } } { d t } } \\right\\| _ { 2 } ^ { 2 }", + "type": "interline_equation", + "image_path": "ca355ccf7e1a696811895fc82cec9e80f2776d9d6e70d92ff473a3d270ec5f78.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 171, + 124, + 439, + 136.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 171, + 136.33333333333334, + 439, + 148.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 171, + 148.66666666666669, + 439, + 161.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 168, + 347, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 348, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 167, + 186 + ], + "score": 1.0, + "content": "Dividing 12 dρ2dt", + "type": "text" + }, + { + "bbox": [ + 167, + 166, + 348, + 183 + ], + "score": 1.0, + "content": "on the leftmost and rightmost sides, we have", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 190, + 422, + 226 + ], + "lines": [ + { + "bbox": [ + 186, + 190, + 422, + 226 + ], + "spans": [ + { + "bbox": [ + 186, + 190, + 422, + 226 + ], + "score": 0.92, + "content": "- { \\frac { d { \\mathcal { L } } } { d t } } \\cdot \\left( { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } \\right) ^ { - 1 } = { \\frac { d } { d t } } \\log \\rho + \\left( { \\frac { d } { d t } } \\log \\rho \\right) ^ { - 1 } \\left\\| { \\frac { d { \\hat { \\pmb { \\theta } } } } { d t } } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "d4f906716e4572d65d93729f4d450c357f426bc223e5dc8c9ba6ad398322e017.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 190, + 422, + 208.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 186, + 208.0, + 422, + 226.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 233, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 104, + 231, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 231, + 121, + 252 + ], + "score": 1.0, + "content": "By", + "type": "text" + }, + { + "bbox": [ + 121, + 234, + 162, + 249 + ], + "score": 0.92, + "content": "\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } \\ge 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 231, + 307, + 252 + ], + "score": 1.0, + "content": "and (7), the LHS is no greater than", + "type": "text" + }, + { + "bbox": [ + 308, + 232, + 463, + 249 + ], + "score": 0.92, + "content": "\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } \\cdot \\left( L \\cdot \\mathcal { L } \\log \\frac { 1 } { \\mathcal { L } } \\right) ^ { - 1 } = \\frac { 1 } { L } \\log \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 231, + 506, + 252 + ], + "score": 1.0, + "content": ". Thus we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 316, + 240, + 330, + 257 + ], + "spans": [ + { + "bbox": [ + 316, + 240, + 330, + 257 + ], + "score": 1.0, + "content": "dt 2", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 248, + 499, + 270 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 128, + 269 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 128, + 248, + 327, + 270 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\log \\frac { 1 } { \\mathcal { L } } - L \\frac { d } { d t } \\log \\rho \\geq L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left. \\frac { d \\hat { \\theta } } { d t } \\right. _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 250, + 435, + 269 + ], + "score": 1.0, + "content": ", where the LHS is exactly", + "type": "text" + }, + { + "bbox": [ + 435, + 252, + 467, + 267 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\tilde { \\gamma } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 250, + 473, + 269 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 257, + 499, + 262 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 107, + 284, + 329, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 330, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 330, + 300 + ], + "score": 1.0, + "content": "6 DISCUSSION AND FUTURE DIRECTIONS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 310, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "In this paper, we analyze the dynamics of gradient flow/descent of homogeneous neural networks", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "under a minimal set of assumptions. The main technical contribution of our work is to prove rigor-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "ously that for gradient flow/descent, the normalized margin is increasing and converges to a KKT", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 344, + 473, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 473, + 356 + ], + "score": 1.0, + "content": "point of a natural max-margin problem. Our results leads to some natural further questions:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 133, + 365, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 133, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 133, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "• Can we generalize our results for gradient descent on smooth neural networks to non-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "smooth ones? In the smooth case, we can lower bound the decrement of training loss by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 388, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 388, + 505, + 399 + ], + "score": 1.0, + "content": "the gradient norm squared, multiplied by a factor related to learning rate. However, in the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 398, + 447, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 447, + 410 + ], + "score": 1.0, + "content": "non-smooth case, no such inequality is known in the optimization literature.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 132, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 132, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "• Can we make more structural assumptions on the neural network to prove stronger results?", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 141, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "In this work, we use a minimal set of assumptions to show that the convergent direction of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 141, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "parameters is a KKT point. A potential research direction is to identify more key properties", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 446, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 446, + 505, + 460 + ], + "score": 1.0, + "content": "of modern neural networks and show that the normalized margin at convergence is locally", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 458, + 336, + 471 + ], + "spans": [ + { + "bbox": [ + 141, + 458, + 336, + 471 + ], + "score": 1.0, + "content": "or globally optimal (in terms of optimizing (P)).", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 138, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 138, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "Can we extend our results to neural networks with bias terms? In our experiments, the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "normalized margin of the CNN with bias also increases during training despite that its", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 141, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "output is non-homogeneous. It is very interesting (and technically challenging) to provide", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 508, + 260, + 520 + ], + "spans": [ + { + "bbox": [ + 141, + 508, + 260, + 520 + ], + "score": 1.0, + "content": "a rigorous proof for this fact.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 537, + 218, + 549 + ], + "lines": [ + { + "bbox": [ + 107, + 537, + 219, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 537, + 219, + 551 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 562, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "score": 1.0, + "content": "The research is supported in part by the National Natural Science Foundation of China Grant", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "61822203, 61772297, 61632016, 61761146003and the Zhongguancun Haihua Institute for Fron-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "tier Information Technology and Turing AI Institute of Nanjing. We thank Liwei Wang for helpful", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "suggestions on the connection between margin and robustness. We thank Sanjeev Arora, Tianle Cai,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 606, + 467, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 467, + 619 + ], + "score": 1.0, + "content": "Simon Du, Jason D. Lee, Zhiyuan Li, Tengyu Ma, Ruosong Wang for helpful discussions.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 175, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 176, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 176, + 649 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Pierre-Antoine Absil, Robert Mahony, and Benjamin Andrews. Convergence of the iterates of de-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 666, + 503, + 678 + ], + "spans": [ + { + "bbox": [ + 115, + 666, + 503, + 678 + ], + "score": 1.0, + "content": "scent methods for analytic cost functions. SIAM Journal on Optimization, 16(2):531–547, 2005.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "Alnur Ali, J. Zico Kolter, and Ryan J. Tibshirani. A continuous-time view of early stopping for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 115, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 115, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "least squares regression. In Kamalika Chaudhuri and Masashi Sugiyama (eds.), Proceedings", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 115, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 115, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "of Machine Learning Research, volume 89 of Proceedings of Machine Learning Research, pp.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 116, + 721, + 268, + 732 + ], + "spans": [ + { + "bbox": [ + 116, + 721, + 268, + 732 + ], + "score": 1.0, + "content": "1370–1378. PMLR, 16–18 Apr 2019.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 505, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 508, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 167, + 101 + ], + "score": 1.0, + "content": "Now, we have", + "type": "text" + }, + { + "bbox": [ + 167, + 80, + 328, + 102 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\left\\| \\pmb { v } \\right\\| _ { 2 } ^ { 2 } = \\frac { 1 } { \\rho ^ { 2 } } \\left( \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } \\right) ^ { 2 } = \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } \\cdot \\frac { d } { d t } \\log \\rho } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 80, + 508, + 104 + ], + "score": 1.0, + "content": "on the one hand; on the other hand, by the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 98, + 447, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 184, + 123 + ], + "score": 1.0, + "content": "chain rule we have", + "type": "text" + }, + { + "bbox": [ + 184, + 102, + 391, + 119 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { d \\hat { \\pmb { \\theta } } } { d t } = \\frac { 1 } { \\rho ^ { 2 } } ( \\rho \\frac { d \\pmb { \\theta } } { d t } - \\frac { d \\rho } { d t } \\pmb { \\theta } ) = \\frac { 1 } { \\rho ^ { 2 } } ( \\rho \\frac { d \\pmb { \\theta } } { d t } - ( \\hat { \\pmb { \\theta } } ^ { \\top } \\frac { d \\pmb { \\theta } } { d t } ) \\pmb { \\theta } ) = \\frac { \\pmb { u } } { \\rho } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 98, + 447, + 123 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 80, + 508, + 123 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 124, + 439, + 161 + ], + "lines": [ + { + "bbox": [ + 171, + 124, + 439, + 161 + ], + "spans": [ + { + "bbox": [ + 171, + 124, + 439, + 161 + ], + "score": 0.92, + "content": "- { \\frac { d { \\mathcal { L } } } { d t } } = \\left\\| { \\frac { d \\theta } { d t } } \\right\\| _ { 2 } ^ { 2 } = \\left\\| v \\right\\| _ { 2 } ^ { 2 } + \\left\\| u \\right\\| _ { 2 } ^ { 2 } = { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } \\cdot { \\frac { d } { d t } } \\log \\rho + \\rho ^ { 2 } \\left\\| { \\frac { d \\hat { \\theta } } { d t } } \\right\\| _ { 2 } ^ { 2 }", + "type": "interline_equation", + "image_path": "ca355ccf7e1a696811895fc82cec9e80f2776d9d6e70d92ff473a3d270ec5f78.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 171, + 124, + 439, + 136.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 171, + 136.33333333333334, + 439, + 148.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 171, + 148.66666666666669, + 439, + 161.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 168, + 347, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 348, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 167, + 186 + ], + "score": 1.0, + "content": "Dividing 12 dρ2dt", + "type": "text" + }, + { + "bbox": [ + 167, + 166, + 348, + 183 + ], + "score": 1.0, + "content": "on the leftmost and rightmost sides, we have", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 166, + 348, + 186 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 190, + 422, + 226 + ], + "lines": [ + { + "bbox": [ + 186, + 190, + 422, + 226 + ], + "spans": [ + { + "bbox": [ + 186, + 190, + 422, + 226 + ], + "score": 0.92, + "content": "- { \\frac { d { \\mathcal { L } } } { d t } } \\cdot \\left( { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } \\right) ^ { - 1 } = { \\frac { d } { d t } } \\log \\rho + \\left( { \\frac { d } { d t } } \\log \\rho \\right) ^ { - 1 } \\left\\| { \\frac { d { \\hat { \\pmb { \\theta } } } } { d t } } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "d4f906716e4572d65d93729f4d450c357f426bc223e5dc8c9ba6ad398322e017.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 190, + 422, + 208.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 186, + 208.0, + 422, + 226.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 233, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 104, + 231, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 231, + 121, + 252 + ], + "score": 1.0, + "content": "By", + "type": "text" + }, + { + "bbox": [ + 121, + 234, + 162, + 249 + ], + "score": 0.92, + "content": "\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } \\ge 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 231, + 307, + 252 + ], + "score": 1.0, + "content": "and (7), the LHS is no greater than", + "type": "text" + }, + { + "bbox": [ + 308, + 232, + 463, + 249 + ], + "score": 0.92, + "content": "\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } \\cdot \\left( L \\cdot \\mathcal { L } \\log \\frac { 1 } { \\mathcal { L } } \\right) ^ { - 1 } = \\frac { 1 } { L } \\log \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 231, + 506, + 252 + ], + "score": 1.0, + "content": ". Thus we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 316, + 240, + 330, + 257 + ], + "spans": [ + { + "bbox": [ + 316, + 240, + 330, + 257 + ], + "score": 1.0, + "content": "dt 2", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 248, + 499, + 270 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 128, + 269 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 128, + 248, + 327, + 270 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\log \\frac { 1 } { \\mathcal { L } } - L \\frac { d } { d t } \\log \\rho \\geq L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left. \\frac { d \\hat { \\theta } } { d t } \\right. _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 250, + 435, + 269 + ], + "score": 1.0, + "content": ", where the LHS is exactly", + "type": "text" + }, + { + "bbox": [ + 435, + 252, + 467, + 267 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\tilde { \\gamma } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 250, + 473, + 269 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 257, + 499, + 262 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 231, + 506, + 270 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 284, + 329, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 330, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 330, + 300 + ], + "score": 1.0, + "content": "6 DISCUSSION AND FUTURE DIRECTIONS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 310, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "In this paper, we analyze the dynamics of gradient flow/descent of homogeneous neural networks", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "under a minimal set of assumptions. The main technical contribution of our work is to prove rigor-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "ously that for gradient flow/descent, the normalized margin is increasing and converges to a KKT", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 344, + 473, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 473, + 356 + ], + "score": 1.0, + "content": "point of a natural max-margin problem. Our results leads to some natural further questions:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 311, + 506, + 356 + ] + }, + { + "type": "list", + "bbox": [ + 133, + 365, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 133, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 133, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "• Can we generalize our results for gradient descent on smooth neural networks to non-", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "smooth ones? 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However, in the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 398, + 447, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 447, + 410 + ], + "score": 1.0, + "content": "non-smooth case, no such inequality is known in the optimization literature.", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 132, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "• Can we make more structural assumptions on the neural network to prove stronger results?", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 141, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "In this work, we use a minimal set of assumptions to show that the convergent direction of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 141, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "parameters is a KKT point. A potential research direction is to identify more key properties", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 446, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 446, + 505, + 460 + ], + "score": 1.0, + "content": "of modern neural networks and show that the normalized margin at convergence is locally", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 458, + 336, + 471 + ], + "spans": [ + { + "bbox": [ + 141, + 458, + 336, + 471 + ], + "score": 1.0, + "content": "or globally optimal (in terms of optimizing (P)).", + "type": "text" + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 138, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 138, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "Can we extend our results to neural networks with bias terms? In our experiments, the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "normalized margin of the CNN with bias also increases during training despite that its", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 141, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "output is non-homogeneous. It is very interesting (and technically challenging) to provide", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 508, + 260, + 520 + ], + "spans": [ + { + "bbox": [ + 141, + 508, + 260, + 520 + ], + "score": 1.0, + "content": "a rigorous proof for this fact.", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + } + ], + "index": 22, + "bbox_fs": [ + 132, + 365, + 506, + 520 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 537, + 218, + 549 + ], + "lines": [ + { + "bbox": [ + 107, + 537, + 219, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 537, + 219, + 551 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 562, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "score": 1.0, + "content": "The research is supported in part by the National Natural Science Foundation of China Grant", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "61822203, 61772297, 61632016, 61761146003and the Zhongguancun Haihua Institute for Fron-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "tier Information Technology and Turing AI Institute of Nanjing. We thank Liwei Wang for helpful", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "suggestions on the connection between margin and robustness. We thank Sanjeev Arora, Tianle Cai,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 606, + 467, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 467, + 619 + ], + "score": 1.0, + "content": "Simon Du, Jason D. Lee, Zhiyuan Li, Tengyu Ma, Ruosong Wang for helpful discussions.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 562, + 506, + 619 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 175, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 176, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 176, + 649 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Pierre-Antoine Absil, Robert Mahony, and Benjamin Andrews. Convergence of the iterates of de-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 666, + 503, + 678 + ], + "spans": [ + { + "bbox": [ + 115, + 666, + 503, + 678 + ], + "score": 1.0, + "content": "scent methods for analytic cost functions. SIAM Journal on Optimization, 16(2):531–547, 2005.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 655, + 505, + 678 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "Alnur Ali, J. Zico Kolter, and Ryan J. Tibshirani. A continuous-time view of early stopping for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 115, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 115, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "least squares regression. In Kamalika Chaudhuri and Masashi Sugiyama (eds.), Proceedings", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 115, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 115, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "of Machine Learning Research, volume 89 of Proceedings of Machine Learning Research, pp.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 116, + 721, + 268, + 732 + ], + "spans": [ + { + "bbox": [ + 116, + 721, + 268, + 732 + ], + "score": 1.0, + "content": "1370–1378. PMLR, 16–18 Apr 2019.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 686, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. 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In Doina Precup and Yee Whye Teh (eds.), Proceedings of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 115, + 307, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 115, + 307, + 505, + 322 + ], + "score": 1.0, + "content": "the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 116, + 320, + 504, + 332 + ], + "spans": [ + { + "bbox": [ + 116, + 320, + 504, + 332 + ], + "score": 1.0, + "content": "Learning Research, pp. 4140–4149, International Convention Centre, Sydney, Australia, 06–11", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 115, + 329, + 193, + 344 + ], + "spans": [ + { + "bbox": [ + 115, + 329, + 193, + 344 + ], + "score": 1.0, + "content": "Aug 2017. PMLR.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 286, + 506, + 344 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 504, + 372 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 504, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 504, + 362 + ], + "score": 1.0, + "content": "Difan Zou, Yuan Cao, Dongruo Zhou, and Quanquan Gu. Stochastic gradient descent optimizes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 116, + 361, + 440, + 372 + ], + "spans": [ + { + "bbox": [ + 116, + 361, + 440, + 372 + ], + "score": 1.0, + "content": "over-parameterized deep relu networks. arXiv preprint arXiv:1811.08888, 2018.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 349, + 504, + 372 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 358, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 358, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 358, + 393 + ], + "score": 1.0, + "content": "Guus Zoutendijk. Mathematical programming methods. 1976.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 379, + 358, + 393 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 282, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 284, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 284, + 96 + ], + "score": 1.0, + "content": "A RESULTS FOR GENERAL LOSS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 153 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "In this section, we state our results for a broad class of binary classification loss. A major con-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "sequence of this generalization is that the logistic loss, one of the most popular loss functions,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 107, + 127, + 190, + 140 + ], + "score": 0.91, + "content": "\\ell ( \\bar { q } ) = \\log ( 1 + e ^ { - \\bar { q } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "is included. The function class also includes other losses with exponential tail,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 139, + 271, + 154 + ], + "spans": [ + { + "bbox": [ + 104, + 140, + 126, + 153 + ], + "score": 1.0, + "content": "e.g.,", + "type": "text" + }, + { + "bbox": [ + 126, + 139, + 267, + 154 + ], + "score": 0.78, + "content": "\\ell ( q ) = e ^ { - q ^ { 3 } } , \\ell ( q ) = \\log ( 1 + e ^ { - q ^ { 3 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 140, + 271, + 153 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 107, + 165, + 197, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 164, + 198, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 198, + 178 + ], + "score": 1.0, + "content": "A.1 ASSUMPTIONS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 109, + 186, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 200 + ], + "score": 1.0, + "content": "We first focus on gradient flow. We assume (A1), (A2) as we do for exponential loss. For (A3), (A4),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 197, + 491, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 491, + 210 + ], + "score": 1.0, + "content": "we replace them with two weaker assumptions (B3), (B4). All the assumptions are listed below:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 115, + 216, + 507, + 357 + ], + "lines": [ + { + "bbox": [ + 115, + 216, + 474, + 231 + ], + "spans": [ + { + "bbox": [ + 115, + 216, + 251, + 231 + ], + "score": 1.0, + "content": "(A1). (Regularity). For any fixed", + "type": "text" + }, + { + "bbox": [ + 252, + 219, + 259, + 227 + ], + "score": 0.52, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 216, + 263, + 231 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 263, + 217, + 296, + 230 + ], + "score": 0.9, + "content": "\\Phi ( \\cdot ; x )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 216, + 474, + 231 + ], + "score": 1.0, + "content": "is locally Lipschitz and admits a chain rule;", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 115, + 231, + 465, + 244 + ], + "spans": [ + { + "bbox": [ + 115, + 232, + 259, + 244 + ], + "score": 1.0, + "content": "(A2). (Homogeneity). There exists", + "type": "text" + }, + { + "bbox": [ + 260, + 232, + 286, + 243 + ], + "score": 0.9, + "content": "L > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 232, + 325, + 244 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 325, + 231, + 461, + 244 + ], + "score": 0.93, + "content": "\\forall \\alpha > 0 : \\Phi ( \\alpha \\pmb { \\theta } ; \\pmb { x } ) = \\alpha ^ { L } \\Phi ( \\pmb { \\theta } ; \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 232, + 465, + 244 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 115, + 246, + 412, + 261 + ], + "spans": [ + { + "bbox": [ + 115, + 247, + 214, + 261 + ], + "score": 1.0, + "content": "(B3). The loss function", + "type": "text" + }, + { + "bbox": [ + 214, + 248, + 232, + 260 + ], + "score": 0.89, + "content": "\\ell ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 247, + 313, + 261 + ], + "score": 1.0, + "content": "can be expressed as", + "type": "text" + }, + { + "bbox": [ + 314, + 246, + 371, + 260 + ], + "score": 0.93, + "content": "\\ell ( q ) = e ^ { - f ( q ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 247, + 412, + 261 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 262, + 279, + 274 + ], + "spans": [ + { + "bbox": [ + 142, + 262, + 175, + 274 + ], + "score": 1.0, + "content": "(B3.1).", + "type": "text" + }, + { + "bbox": [ + 175, + 262, + 221, + 274 + ], + "score": 0.91, + "content": "f : \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 262, + 231, + 274 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 231, + 262, + 243, + 272 + ], + "score": 0.87, + "content": "{ \\mathcal { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 262, + 279, + 274 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 275, + 273, + 288 + ], + "spans": [ + { + "bbox": [ + 142, + 275, + 175, + 288 + ], + "score": 1.0, + "content": "(B3.2).", + "type": "text" + }, + { + "bbox": [ + 176, + 275, + 216, + 287 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 275, + 244, + 288 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 244, + 275, + 269, + 286 + ], + "score": 0.89, + "content": "q \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 275, + 273, + 288 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 287, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 142, + 287, + 228, + 301 + ], + "score": 1.0, + "content": "(B3.3). There exists", + "type": "text" + }, + { + "bbox": [ + 228, + 288, + 261, + 300 + ], + "score": 0.91, + "content": "b _ { f } \\ \\geq \\ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 287, + 303, + 301 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 303, + 288, + 331, + 300 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q ) q", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 287, + 422, + 301 + ], + "score": 1.0, + "content": "is non-decreasing for", + "type": "text" + }, + { + "bbox": [ + 422, + 288, + 483, + 300 + ], + "score": 0.92, + "content": "q \\in ( b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 287, + 505, + 301 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 176, + 298, + 291, + 312 + ], + "spans": [ + { + "bbox": [ + 176, + 299, + 236, + 311 + ], + "score": 0.9, + "content": "f ^ { \\prime } ( q ) q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 298, + 248, + 312 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 248, + 300, + 288, + 310 + ], + "score": 0.88, + "content": "q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 298, + 291, + 312 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 142, + 311, + 192, + 325 + ], + "score": 1.0, + "content": "(B3.4). Let", + "type": "text" + }, + { + "bbox": [ + 192, + 311, + 322, + 324 + ], + "score": 0.91, + "content": "g \\ : \\ [ f ( b _ { f } ) , + \\infty ) \\ \\ [ b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 311, + 433, + 325 + ], + "score": 1.0, + "content": "be the inverse function of", + "type": "text" + }, + { + "bbox": [ + 434, + 312, + 441, + 323 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "on the domain", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 176, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 176, + 323, + 214, + 334 + ], + "score": 0.85, + "content": "[ b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 321, + 275, + 336 + ], + "score": 1.0, + "content": ". There exists", + "type": "text" + }, + { + "bbox": [ + 275, + 322, + 427, + 334 + ], + "score": 0.87, + "content": "b _ { g } ~ \\overset { \\cdot } { = } ~ \\operatorname* { m a x } \\{ 2 f ( b _ { f } ) , f ( 2 b _ { f } ) \\} , K \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 321, + 469, + 336 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 469, + 322, + 505, + 334 + ], + "score": 0.9, + "content": "g ^ { \\prime } ( x ) \\leq", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 176, + 332, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 176, + 334, + 212, + 345 + ], + "score": 0.82, + "content": "K g ^ { \\prime } ( \\theta x )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 332, + 233, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 334, + 311, + 345 + ], + "score": 0.89, + "content": "f ^ { \\prime } ( y ) ~ \\leq ~ \\breve { K } f ^ { \\prime } ( \\theta y )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 332, + 343, + 347 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 343, + 334, + 485, + 346 + ], + "score": 0.87, + "content": "\\dot { x } \\in ( \\bar { b _ { g } } , + \\infty ) , y \\in ( g ( b _ { g } ) , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 332, + 505, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 176, + 342, + 230, + 358 + ], + "spans": [ + { + "bbox": [ + 176, + 345, + 225, + 356 + ], + "score": 0.88, + "content": "\\theta \\in [ 1 / 2 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 342, + 230, + 358 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 116, + 359, + 434, + 372 + ], + "lines": [ + { + "bbox": [ + 115, + 358, + 435, + 375 + ], + "spans": [ + { + "bbox": [ + 115, + 358, + 280, + 375 + ], + "score": 1.0, + "content": "(B4). (Separability). There exists a time", + "type": "text" + }, + { + "bbox": [ + 280, + 362, + 290, + 371 + ], + "score": 0.87, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 358, + 329, + 375 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 330, + 359, + 432, + 372 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t _ { 0 } ) < e ^ { - f ( b _ { f } ) } = \\ell ( b _ { f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 358, + 435, + 375 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 503, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 390, + 394 + ], + "score": 1.0, + "content": "(A1) and (A2) remain unchanged. (B3) is satisfied by exponential loss", + "type": "text" + }, + { + "bbox": [ + 391, + 380, + 437, + 392 + ], + "score": 0.92, + "content": "\\ell ( q ) = e ^ { - q }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 379, + 462, + 394 + ], + "score": 1.0, + "content": "(with", + "type": "text" + }, + { + "bbox": [ + 462, + 380, + 503, + 393 + ], + "score": 0.9, + "content": "f ( q ) = q )", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 174, + 404 + ], + "score": 1.0, + "content": "and logistic loss", + "type": "text" + }, + { + "bbox": [ + 175, + 391, + 261, + 403 + ], + "score": 0.93, + "content": "\\ell ( q ) = \\log ( \\bar { 1 } + e ^ { - q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 391, + 287, + 404 + ], + "score": 1.0, + "content": "(with", + "type": "text" + }, + { + "bbox": [ + 288, + 391, + 402, + 403 + ], + "score": 0.92, + "content": "f ( q ) \\stackrel { \\cdot } { = } - \\log \\log ( 1 + e ^ { - q } ) \\stackrel { \\cdot } { ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 391, + 505, + 404 + ], + "score": 1.0, + "content": ". (B4) are essentially the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 401, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 506, + 416 + ], + "score": 1.0, + "content": "same as (A4) but (B4) uses a threshold value that depends on the loss function. Assuming (B3), it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 412, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 349, + 428 + ], + "score": 1.0, + "content": "is easy to see that (B4) ensures the separability of data since", + "type": "text" + }, + { + "bbox": [ + 349, + 413, + 415, + 426 + ], + "score": 0.93, + "content": "\\ell ( q _ { n } ) < e ^ { - f ( b _ { f } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 412, + 448, + 428 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 449, + 415, + 501, + 426 + ], + "score": 0.92, + "content": "q _ { n } > b _ { f } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 412, + 506, + 428 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 218, + 439 + ], + "score": 1.0, + "content": "For logistic loss, we can set", + "type": "text" + }, + { + "bbox": [ + 218, + 425, + 248, + 438 + ], + "score": 0.92, + "content": "b _ { f } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 424, + 506, + 439 + ], + "score": 1.0, + "content": "(see Remark A.2). So the corresponding threshold value in (B4)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 435, + 172, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 115, + 451 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 116, + 436, + 167, + 449 + ], + "score": 0.92, + "content": "\\ell ( 0 ) = \\log 2", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 435, + 172, + 451 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "Now we discuss each of the assumptions in (B3). (B3.1) is a natural assumption on smoothness.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 170, + 478 + ], + "score": 1.0, + "content": "(B3.2) requires", + "type": "text" + }, + { + "bbox": [ + 171, + 464, + 186, + 476 + ], + "score": 0.9, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 463, + 413, + 478 + ], + "score": 1.0, + "content": "to be monotone decreasing, which is also natural since", + "type": "text" + }, + { + "bbox": [ + 414, + 464, + 429, + 476 + ], + "score": 0.88, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 463, + 505, + 478 + ], + "score": 1.0, + "content": "is used for binary", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 474, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 440, + 489 + ], + "score": 1.0, + "content": "classification. The rest of two assumptions in (B3) characterize the properties of", + "type": "text" + }, + { + "bbox": [ + 440, + 475, + 461, + 487 + ], + "score": 0.92, + "content": "\\ell ^ { \\prime } ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 474, + 487, + 489 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 488, + 477, + 494, + 487 + ], + "score": 0.77, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 474, + 506, + 489 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "large enough. (B3.3) is an assumption that appears naturally from the proof. For exponential loss,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 155, + 509 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q ) q = q", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 497, + 322, + 511 + ], + "score": 1.0, + "content": "is always non-decreasing, so we can set", + "type": "text" + }, + { + "bbox": [ + 322, + 497, + 354, + 510 + ], + "score": 0.92, + "content": "b _ { f } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 497, + 487, + 511 + ], + "score": 1.0, + "content": ". In (B3.4), the inverse function", + "type": "text" + }, + { + "bbox": [ + 487, + 499, + 494, + 509 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 308, + 522 + ], + "score": 1.0, + "content": "defined. It is guaranteed by (B3.1) and (B3.2) that", + "type": "text" + }, + { + "bbox": [ + 308, + 511, + 315, + 520 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 508, + 388, + 522 + ], + "score": 1.0, + "content": "always exists and", + "type": "text" + }, + { + "bbox": [ + 388, + 511, + 394, + 520 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 508, + 423, + 522 + ], + "score": 1.0, + "content": "is also", + "type": "text" + }, + { + "bbox": [ + 423, + 509, + 434, + 519 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "-smooth. Though", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 325, + 533 + ], + "score": 1.0, + "content": "(B3.4) looks very complicated, it essentially says that", + "type": "text" + }, + { + "bbox": [ + 325, + 520, + 504, + 532 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( \\Theta ( q ) ) = \\Theta ( \\bar { f } ^ { \\prime } ( q ) ) , g ^ { \\prime } ( \\Theta ( q ) ) = \\Theta ( g ^ { \\prime } ( \\bar { q } ) )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 118, + 544 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 532, + 154, + 542 + ], + "score": 0.89, + "content": "q \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 531, + 506, + 544 + ], + "score": 1.0, + "content": ". (B3.4) is indeed a technical assumption that enables us to asymptotically compare", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "the loss or the length of gradient at different data points. It is possible to base our results on weaker", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 552, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 505, + 567 + ], + "score": 1.0, + "content": "assumptions than (B3.4), but we use (B3.4) for simplicity since it has already been satisfied by many", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 564, + 317, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 317, + 577 + ], + "score": 1.0, + "content": "loss functions such as the aforementioned examples.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 465, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 466, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 242, + 595 + ], + "score": 1.0, + "content": "We summarize the corresponding", + "type": "text" + }, + { + "bbox": [ + 243, + 581, + 259, + 593 + ], + "score": 0.9, + "content": "f , g", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 579, + 276, + 595 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 277, + 581, + 287, + 593 + ], + "score": 0.88, + "content": "b _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 579, + 466, + 595 + ], + "score": 1.0, + "content": "for exponential loss and logistic loss below:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 359, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 357, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 235, + 609 + ], + "score": 1.0, + "content": "Remark A.1. Exponential loss", + "type": "text" + }, + { + "bbox": [ + 235, + 595, + 281, + 608 + ], + "score": 0.92, + "content": "\\ell ( q ) = e ^ { - q }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 594, + 357, + 609 + ], + "score": 1.0, + "content": "satisfies (B3) with", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 609, + 467, + 624 + ], + "lines": [ + { + "bbox": [ + 144, + 609, + 467, + 624 + ], + "spans": [ + { + "bbox": [ + 144, + 609, + 467, + 624 + ], + "score": 0.84, + "content": "f ( q ) = q \\qquad f ^ { \\prime } ( q ) = 1 \\qquad g ( q ) = q \\qquad g ^ { \\prime } ( q ) = 1 \\qquad b _ { f } = 0 \\qquad \\ell ( b _ { f } ) = 1 .", + "type": "interline_equation", + "image_path": "a25fb0e480632ea559ad84b365b194aa5460fcbd8ffffe15715d67bf92d7e7d7.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 144, + 609, + 467, + 624 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 378, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 378, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 219, + 641 + ], + "score": 1.0, + "content": "Remark A.2. Logistic loss", + "type": "text" + }, + { + "bbox": [ + 219, + 627, + 303, + 640 + ], + "score": 0.93, + "content": "\\ell ( q ) = \\log ( 1 + e ^ { - q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 627, + 338, + 641 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 338, + 628, + 357, + 639 + ], + "score": 0.46, + "content": "( B 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 627, + 378, + 641 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 644, + 483, + 713 + ], + "lines": [ + { + "bbox": [ + 127, + 644, + 483, + 713 + ], + "spans": [ + { + "bbox": [ + 127, + 644, + 483, + 713 + ], + "score": 0.92, + "content": "\\begin{array} { l l } { { f ( q ) = - \\log \\log ( 1 + e ^ { - q } ) = \\Theta ( q ) ~ } } & { { ~ f ^ { \\prime } ( q ) = \\displaystyle \\frac { e ^ { - q } } { ( 1 + e ^ { - q } ) \\log ( 1 + e ^ { - q } ) } = \\Theta ( 1 ) } } \\\\ { { { } } } & { { { } } } \\\\ { { g ( q ) = - \\log ( e ^ { e ^ { - q } } - 1 ) = \\Theta ( q ) ~ } } & { { ~ g ^ { \\prime } ( q ) = \\displaystyle \\frac { e ^ { - q } } { e ^ { e ^ { - q } } - 1 } = \\Theta ( 1 ) } } \\\\ { { { } } } & { { { } } } \\\\ { { b _ { f } = 0 ~ } } & { { ~ \\ell ( b _ { f } ) = \\log 2 . } } \\end{array}", + "type": "interline_equation", + "image_path": "9aeacebecb22545acf7006232f10f6e95a6f872953d3e0452bcf59b04b4d5f1c.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 127, + 644, + 483, + 667.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 127, + 667.0, + 483, + 690.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 127, + 690.0, + 483, + 713.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 414, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 414, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 414, + 734 + ], + "score": 1.0, + "content": "The proof for Remark A.1 is trivial. For Remark A.2, we give a proof below.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 282, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 284, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 284, + 96 + ], + "score": 1.0, + "content": "A RESULTS FOR GENERAL LOSS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 153 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "In this section, we state our results for a broad class of binary classification loss. A major con-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "sequence of this generalization is that the logistic loss, one of the most popular loss functions,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 107, + 127, + 190, + 140 + ], + "score": 0.91, + "content": "\\ell ( \\bar { q } ) = \\log ( 1 + e ^ { - \\bar { q } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "is included. The function class also includes other losses with exponential tail,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 139, + 271, + 154 + ], + "spans": [ + { + "bbox": [ + 104, + 140, + 126, + 153 + ], + "score": 1.0, + "content": "e.g.,", + "type": "text" + }, + { + "bbox": [ + 126, + 139, + 267, + 154 + ], + "score": 0.78, + "content": "\\ell ( q ) = e ^ { - q ^ { 3 } } , \\ell ( q ) = \\log ( 1 + e ^ { - q ^ { 3 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 140, + 271, + 153 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 105, + 505, + 154 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 165, + 197, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 164, + 198, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 198, + 178 + ], + "score": 1.0, + "content": "A.1 ASSUMPTIONS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 109, + 186, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 200 + ], + "score": 1.0, + "content": "We first focus on gradient flow. We assume (A1), (A2) as we do for exponential loss. For (A3), (A4),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 197, + 491, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 491, + 210 + ], + "score": 1.0, + "content": "we replace them with two weaker assumptions (B3), (B4). All the assumptions are listed below:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 106, + 184, + 505, + 210 + ] + }, + { + "type": "list", + "bbox": [ + 115, + 216, + 507, + 357 + ], + "lines": [ + { + "bbox": [ + 115, + 216, + 474, + 231 + ], + "spans": [ + { + "bbox": [ + 115, + 216, + 251, + 231 + ], + "score": 1.0, + "content": "(A1). (Regularity). For any fixed", + "type": "text" + }, + { + "bbox": [ + 252, + 219, + 259, + 227 + ], + "score": 0.52, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 216, + 263, + 231 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 263, + 217, + 296, + 230 + ], + "score": 0.9, + "content": "\\Phi ( \\cdot ; x )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 216, + 474, + 231 + ], + "score": 1.0, + "content": "is locally Lipschitz and admits a chain rule;", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 115, + 231, + 465, + 244 + ], + "spans": [ + { + "bbox": [ + 115, + 232, + 259, + 244 + ], + "score": 1.0, + "content": "(A2). (Homogeneity). There exists", + "type": "text" + }, + { + "bbox": [ + 260, + 232, + 286, + 243 + ], + "score": 0.9, + "content": "L > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 232, + 325, + 244 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 325, + 231, + 461, + 244 + ], + "score": 0.93, + "content": "\\forall \\alpha > 0 : \\Phi ( \\alpha \\pmb { \\theta } ; \\pmb { x } ) = \\alpha ^ { L } \\Phi ( \\pmb { \\theta } ; \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 232, + 465, + 244 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 115, + 246, + 412, + 261 + ], + "spans": [ + { + "bbox": [ + 115, + 247, + 214, + 261 + ], + "score": 1.0, + "content": "(B3). The loss function", + "type": "text" + }, + { + "bbox": [ + 214, + 248, + 232, + 260 + ], + "score": 0.89, + "content": "\\ell ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 247, + 313, + 261 + ], + "score": 1.0, + "content": "can be expressed as", + "type": "text" + }, + { + "bbox": [ + 314, + 246, + 371, + 260 + ], + "score": 0.93, + "content": "\\ell ( q ) = e ^ { - f ( q ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 247, + 412, + 261 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 142, + 262, + 279, + 274 + ], + "spans": [ + { + "bbox": [ + 142, + 262, + 175, + 274 + ], + "score": 1.0, + "content": "(B3.1).", + "type": "text" + }, + { + "bbox": [ + 175, + 262, + 221, + 274 + ], + "score": 0.91, + "content": "f : \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 262, + 231, + 274 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 231, + 262, + 243, + 272 + ], + "score": 0.87, + "content": "{ \\mathcal { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 262, + 279, + 274 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 11, + "is_list_end_line": true + }, + { + "bbox": [ + 142, + 275, + 273, + 288 + ], + "spans": [ + { + "bbox": [ + 142, + 275, + 175, + 288 + ], + "score": 1.0, + "content": "(B3.2).", + "type": "text" + }, + { + "bbox": [ + 176, + 275, + 216, + 287 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 275, + 244, + 288 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 244, + 275, + 269, + 286 + ], + "score": 0.89, + "content": "q \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 275, + 273, + 288 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12, + "is_list_end_line": true + }, + { + "bbox": [ + 142, + 287, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 142, + 287, + 228, + 301 + ], + "score": 1.0, + "content": "(B3.3). There exists", + "type": "text" + }, + { + "bbox": [ + 228, + 288, + 261, + 300 + ], + "score": 0.91, + "content": "b _ { f } \\ \\geq \\ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 287, + 303, + 301 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 303, + 288, + 331, + 300 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q ) q", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 287, + 422, + 301 + ], + "score": 1.0, + "content": "is non-decreasing for", + "type": "text" + }, + { + "bbox": [ + 422, + 288, + 483, + 300 + ], + "score": 0.92, + "content": "q \\in ( b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 287, + 505, + 301 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 176, + 298, + 291, + 312 + ], + "spans": [ + { + "bbox": [ + 176, + 299, + 236, + 311 + ], + "score": 0.9, + "content": "f ^ { \\prime } ( q ) q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 298, + 248, + 312 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 248, + 300, + 288, + 310 + ], + "score": 0.88, + "content": "q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 298, + 291, + 312 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 142, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 142, + 311, + 192, + 325 + ], + "score": 1.0, + "content": "(B3.4). Let", + "type": "text" + }, + { + "bbox": [ + 192, + 311, + 322, + 324 + ], + "score": 0.91, + "content": "g \\ : \\ [ f ( b _ { f } ) , + \\infty ) \\ \\ [ b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 311, + 433, + 325 + ], + "score": 1.0, + "content": "be the inverse function of", + "type": "text" + }, + { + "bbox": [ + 434, + 312, + 441, + 323 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "on the domain", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 176, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 176, + 323, + 214, + 334 + ], + "score": 0.85, + "content": "[ b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 321, + 275, + 336 + ], + "score": 1.0, + "content": ". There exists", + "type": "text" + }, + { + "bbox": [ + 275, + 322, + 427, + 334 + ], + "score": 0.87, + "content": "b _ { g } ~ \\overset { \\cdot } { = } ~ \\operatorname* { m a x } \\{ 2 f ( b _ { f } ) , f ( 2 b _ { f } ) \\} , K \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 321, + 469, + 336 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 469, + 322, + 505, + 334 + ], + "score": 0.9, + "content": "g ^ { \\prime } ( x ) \\leq", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 176, + 332, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 176, + 334, + 212, + 345 + ], + "score": 0.82, + "content": "K g ^ { \\prime } ( \\theta x )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 332, + 233, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 334, + 311, + 345 + ], + "score": 0.89, + "content": "f ^ { \\prime } ( y ) ~ \\leq ~ \\breve { K } f ^ { \\prime } ( \\theta y )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 332, + 343, + 347 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 343, + 334, + 485, + 346 + ], + "score": 0.87, + "content": "\\dot { x } \\in ( \\bar { b _ { g } } , + \\infty ) , y \\in ( g ( b _ { g } ) , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 332, + 505, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 176, + 342, + 230, + 358 + ], + "spans": [ + { + "bbox": [ + 176, + 345, + 225, + 356 + ], + "score": 0.88, + "content": "\\theta \\in [ 1 / 2 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 342, + 230, + 358 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + } + ], + "index": 13, + "bbox_fs": [ + 115, + 216, + 505, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 116, + 359, + 434, + 372 + ], + "lines": [ + { + "bbox": [ + 115, + 358, + 435, + 375 + ], + "spans": [ + { + "bbox": [ + 115, + 358, + 280, + 375 + ], + "score": 1.0, + "content": "(B4). (Separability). There exists a time", + "type": "text" + }, + { + "bbox": [ + 280, + 362, + 290, + 371 + ], + "score": 0.87, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 358, + 329, + 375 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 330, + 359, + 432, + 372 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t _ { 0 } ) < e ^ { - f ( b _ { f } ) } = \\ell ( b _ { f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 358, + 435, + 375 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 115, + 358, + 435, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 503, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 390, + 394 + ], + "score": 1.0, + "content": "(A1) and (A2) remain unchanged. (B3) is satisfied by exponential loss", + "type": "text" + }, + { + "bbox": [ + 391, + 380, + 437, + 392 + ], + "score": 0.92, + "content": "\\ell ( q ) = e ^ { - q }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 379, + 462, + 394 + ], + "score": 1.0, + "content": "(with", + "type": "text" + }, + { + "bbox": [ + 462, + 380, + 503, + 393 + ], + "score": 0.9, + "content": "f ( q ) = q )", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 174, + 404 + ], + "score": 1.0, + "content": "and logistic loss", + "type": "text" + }, + { + "bbox": [ + 175, + 391, + 261, + 403 + ], + "score": 0.93, + "content": "\\ell ( q ) = \\log ( \\bar { 1 } + e ^ { - q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 391, + 287, + 404 + ], + "score": 1.0, + "content": "(with", + "type": "text" + }, + { + "bbox": [ + 288, + 391, + 402, + 403 + ], + "score": 0.92, + "content": "f ( q ) \\stackrel { \\cdot } { = } - \\log \\log ( 1 + e ^ { - q } ) \\stackrel { \\cdot } { ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 391, + 505, + 404 + ], + "score": 1.0, + "content": ". (B4) are essentially the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 401, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 506, + 416 + ], + "score": 1.0, + "content": "same as (A4) but (B4) uses a threshold value that depends on the loss function. Assuming (B3), it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 412, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 349, + 428 + ], + "score": 1.0, + "content": "is easy to see that (B4) ensures the separability of data since", + "type": "text" + }, + { + "bbox": [ + 349, + 413, + 415, + 426 + ], + "score": 0.93, + "content": "\\ell ( q _ { n } ) < e ^ { - f ( b _ { f } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 412, + 448, + 428 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 449, + 415, + 501, + 426 + ], + "score": 0.92, + "content": "q _ { n } > b _ { f } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 412, + 506, + 428 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 218, + 439 + ], + "score": 1.0, + "content": "For logistic loss, we can set", + "type": "text" + }, + { + "bbox": [ + 218, + 425, + 248, + 438 + ], + "score": 0.92, + "content": "b _ { f } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 424, + 506, + 439 + ], + "score": 1.0, + "content": "(see Remark A.2). So the corresponding threshold value in (B4)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 435, + 172, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 115, + 451 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 116, + 436, + 167, + 449 + ], + "score": 0.92, + "content": "\\ell ( 0 ) = \\log 2", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 435, + 172, + 451 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 379, + 506, + 451 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "Now we discuss each of the assumptions in (B3). (B3.1) is a natural assumption on smoothness.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 463, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 170, + 478 + ], + "score": 1.0, + "content": "(B3.2) requires", + "type": "text" + }, + { + "bbox": [ + 171, + 464, + 186, + 476 + ], + "score": 0.9, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 463, + 413, + 478 + ], + "score": 1.0, + "content": "to be monotone decreasing, which is also natural since", + "type": "text" + }, + { + "bbox": [ + 414, + 464, + 429, + 476 + ], + "score": 0.88, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 463, + 505, + 478 + ], + "score": 1.0, + "content": "is used for binary", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 474, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 440, + 489 + ], + "score": 1.0, + "content": "classification. The rest of two assumptions in (B3) characterize the properties of", + "type": "text" + }, + { + "bbox": [ + 440, + 475, + 461, + 487 + ], + "score": 0.92, + "content": "\\ell ^ { \\prime } ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 474, + 487, + 489 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 488, + 477, + 494, + 487 + ], + "score": 0.77, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 474, + 506, + 489 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "large enough. (B3.3) is an assumption that appears naturally from the proof. For exponential loss,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 155, + 509 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q ) q = q", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 497, + 322, + 511 + ], + "score": 1.0, + "content": "is always non-decreasing, so we can set", + "type": "text" + }, + { + "bbox": [ + 322, + 497, + 354, + 510 + ], + "score": 0.92, + "content": "b _ { f } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 497, + 487, + 511 + ], + "score": 1.0, + "content": ". In (B3.4), the inverse function", + "type": "text" + }, + { + "bbox": [ + 487, + 499, + 494, + 509 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 308, + 522 + ], + "score": 1.0, + "content": "defined. It is guaranteed by (B3.1) and (B3.2) that", + "type": "text" + }, + { + "bbox": [ + 308, + 511, + 315, + 520 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 508, + 388, + 522 + ], + "score": 1.0, + "content": "always exists and", + "type": "text" + }, + { + "bbox": [ + 388, + 511, + 394, + 520 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 508, + 423, + 522 + ], + "score": 1.0, + "content": "is also", + "type": "text" + }, + { + "bbox": [ + 423, + 509, + 434, + 519 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "-smooth. Though", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 325, + 533 + ], + "score": 1.0, + "content": "(B3.4) looks very complicated, it essentially says that", + "type": "text" + }, + { + "bbox": [ + 325, + 520, + 504, + 532 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( \\Theta ( q ) ) = \\Theta ( \\bar { f } ^ { \\prime } ( q ) ) , g ^ { \\prime } ( \\Theta ( q ) ) = \\Theta ( g ^ { \\prime } ( \\bar { q } ) )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 118, + 544 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 532, + 154, + 542 + ], + "score": 0.89, + "content": "q \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 531, + 506, + 544 + ], + "score": 1.0, + "content": ". (B3.4) is indeed a technical assumption that enables us to asymptotically compare", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "the loss or the length of gradient at different data points. It is possible to base our results on weaker", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 552, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 505, + 567 + ], + "score": 1.0, + "content": "assumptions than (B3.4), but we use (B3.4) for simplicity since it has already been satisfied by many", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 564, + 317, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 317, + 577 + ], + "score": 1.0, + "content": "loss functions such as the aforementioned examples.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 452, + 506, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 465, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 466, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 242, + 595 + ], + "score": 1.0, + "content": "We summarize the corresponding", + "type": "text" + }, + { + "bbox": [ + 243, + 581, + 259, + 593 + ], + "score": 0.9, + "content": "f , g", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 579, + 276, + 595 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 277, + 581, + 287, + 593 + ], + "score": 0.88, + "content": "b _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 579, + 466, + 595 + ], + "score": 1.0, + "content": "for exponential loss and logistic loss below:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 579, + 466, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 359, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 357, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 235, + 609 + ], + "score": 1.0, + "content": "Remark A.1. Exponential loss", + "type": "text" + }, + { + "bbox": [ + 235, + 595, + 281, + 608 + ], + "score": 0.92, + "content": "\\ell ( q ) = e ^ { - q }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 594, + 357, + 609 + ], + "score": 1.0, + "content": "satisfies (B3) with", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 594, + 357, + 609 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 609, + 467, + 624 + ], + "lines": [ + { + "bbox": [ + 144, + 609, + 467, + 624 + ], + "spans": [ + { + "bbox": [ + 144, + 609, + 467, + 624 + ], + "score": 0.84, + "content": "f ( q ) = q \\qquad f ^ { \\prime } ( q ) = 1 \\qquad g ( q ) = q \\qquad g ^ { \\prime } ( q ) = 1 \\qquad b _ { f } = 0 \\qquad \\ell ( b _ { f } ) = 1 .", + "type": "interline_equation", + "image_path": "a25fb0e480632ea559ad84b365b194aa5460fcbd8ffffe15715d67bf92d7e7d7.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 144, + 609, + 467, + 624 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 378, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 378, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 219, + 641 + ], + "score": 1.0, + "content": "Remark A.2. Logistic loss", + "type": "text" + }, + { + "bbox": [ + 219, + 627, + 303, + 640 + ], + "score": 0.93, + "content": "\\ell ( q ) = \\log ( 1 + e ^ { - q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 627, + 338, + 641 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 338, + 628, + 357, + 639 + ], + "score": 0.46, + "content": "( B 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 627, + 378, + 641 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 106, + 627, + 378, + 641 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 644, + 483, + 713 + ], + "lines": [ + { + "bbox": [ + 127, + 644, + 483, + 713 + ], + "spans": [ + { + "bbox": [ + 127, + 644, + 483, + 713 + ], + "score": 0.92, + "content": "\\begin{array} { l l } { { f ( q ) = - \\log \\log ( 1 + e ^ { - q } ) = \\Theta ( q ) ~ } } & { { ~ f ^ { \\prime } ( q ) = \\displaystyle \\frac { e ^ { - q } } { ( 1 + e ^ { - q } ) \\log ( 1 + e ^ { - q } ) } = \\Theta ( 1 ) } } \\\\ { { { } } } & { { { } } } \\\\ { { g ( q ) = - \\log ( e ^ { e ^ { - q } } - 1 ) = \\Theta ( q ) ~ } } & { { ~ g ^ { \\prime } ( q ) = \\displaystyle \\frac { e ^ { - q } } { e ^ { e ^ { - q } } - 1 } = \\Theta ( 1 ) } } \\\\ { { { } } } & { { { } } } \\\\ { { b _ { f } = 0 ~ } } & { { ~ \\ell ( b _ { f } ) = \\log 2 . } } \\end{array}", + "type": "interline_equation", + "image_path": "9aeacebecb22545acf7006232f10f6e95a6f872953d3e0452bcf59b04b4d5f1c.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 127, + 644, + 483, + 667.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 127, + 667.0, + 483, + 690.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 127, + 690.0, + 483, + 713.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 414, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 414, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 414, + 734 + ], + "score": 1.0, + "content": "The proof for Remark A.1 is trivial. For Remark A.2, we give a proof below.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 106, + 719, + 414, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 124 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 363, + 95 + ], + "score": 1.0, + "content": "Proof for Remark A.2. By simple calculations, the formulas for", + "type": "text" + }, + { + "bbox": [ + 364, + 82, + 456, + 95 + ], + "score": 0.92, + "content": "f ( q ) , f ^ { \\prime } ( q ) , g ( q ) , g ^ { \\prime } ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "are correct.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 179, + 114 + ], + "score": 1.0, + "content": "(B3.1) is trivial.", + "type": "text" + }, + { + "bbox": [ + 180, + 94, + 317, + 111 + ], + "score": 0.93, + "content": "f ^ { \\prime } ( q ) ~ = ~ \\frac { e ^ { - q } } { ( 1 + e ^ { - q } ) \\log ( 1 + e ^ { - q } ) } ~ > ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 93, + 506, + 114 + ], + "score": 1.0, + "content": ", so (B3.2) is satisfied. For (B3.3), note that", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 411, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 109, + 215, + 125 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f ^ { \\prime } ( q ) q = \\frac { q } { ( 1 + e ^ { q } ) \\log ( 1 + e ^ { - q } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 106, + 411, + 130 + ], + "score": 1.0, + "content": ". The denominator is a decreasing function since", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 127, + 451, + 153 + ], + "lines": [ + { + "bbox": [ + 160, + 127, + 451, + 153 + ], + "spans": [ + { + "bbox": [ + 160, + 127, + 451, + 153 + ], + "score": 0.91, + "content": "{ \\frac { d } { d q } } \\left( ( 1 + e ^ { q } ) \\log ( 1 + e ^ { - q } ) \\right) = e ^ { q } \\log ( 1 + e ^ { - q } ) - 1 < e ^ { q } \\cdot e ^ { - q } - 1 = 0 .", + "type": "interline_equation", + "image_path": "ba9ead046cd453e239d2c935dfde05a2254e3890d255d93b27a1cbeb40f1e0e0.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 160, + 127, + 451, + 153 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 506, + 194 + ], + "lines": [ + { + "bbox": [ + 106, + 155, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 132, + 169 + ], + "score": 1.0, + "content": "Thus,", + "type": "text" + }, + { + "bbox": [ + 133, + 156, + 160, + 169 + ], + "score": 0.93, + "content": "f ^ { \\prime } ( q ) q", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 155, + 305, + 169 + ], + "score": 1.0, + "content": "is a strictly increasing function on", + "type": "text" + }, + { + "bbox": [ + 306, + 157, + 313, + 166 + ], + "score": 0.82, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 155, + 335, + 169 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 335, + 157, + 346, + 169 + ], + "score": 0.88, + "content": "b _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 155, + 506, + 169 + ], + "score": 1.0, + "content": "is required to be non-negative, we set", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 168, + 504, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 136, + 182 + ], + "score": 0.9, + "content": "b _ { f } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 169, + 210, + 185 + ], + "score": 1.0, + "content": ". For proving that", + "type": "text" + }, + { + "bbox": [ + 210, + 170, + 271, + 182 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q ) q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 169, + 426, + 185 + ], + "score": 1.0, + "content": "and (B4), we only need to notice that", + "type": "text" + }, + { + "bbox": [ + 426, + 168, + 504, + 183 + ], + "score": 0.89, + "content": "\\begin{array} { r } { f ^ { \\prime } ( q ) \\sim \\frac { e ^ { - q } } { 1 \\cdot e ^ { - q } } = 1 } \\end{array}", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 123, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 181, + 222, + 193 + ], + "score": 0.91, + "content": "g ^ { \\prime } ( x ) = \\bar { 1 } / f ^ { \\prime } ( \\bar { g ( x ) } ) \\sim 1", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 181, + 227, + 194 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 181, + 506, + 192 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 108, + 205, + 284, + 217 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 285, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 285, + 218 + ], + "score": 1.0, + "content": "A.2 SMOOTHED NORMALIZED MARGIN", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 504, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 504, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 184, + 240 + ], + "score": 1.0, + "content": "For a loss function", + "type": "text" + }, + { + "bbox": [ + 184, + 226, + 200, + 238 + ], + "score": 0.9, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 225, + 477, + 240 + ], + "score": 1.0, + "content": "satisfying (B3), it is easy to see from (B3.2) that its inverse function", + "type": "text" + }, + { + "bbox": [ + 478, + 226, + 504, + 239 + ], + "score": 0.91, + "content": "\\ell ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 236, + 497, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 497, + 250 + ], + "score": 1.0, + "content": "must exist. For this kind of loss functions, we define the smoothed normalized margin as follows:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 105, + 250, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 249, + 263 + ], + "score": 1.0, + "content": "Definition A.3. For a loss function", + "type": "text" + }, + { + "bbox": [ + 249, + 251, + 265, + 263 + ], + "score": 0.85, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 250, + 464, + 263 + ], + "score": 1.0, + "content": "satisfying (B3), the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 464, + 251, + 484, + 263 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 250, + 497, + 263 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 497, + 252, + 504, + 261 + ], + "score": 0.76, + "content": "\\pmb \\theta", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 260, + 159, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 159, + 275 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 274, + 437, + 309 + ], + "lines": [ + { + "bbox": [ + 173, + 274, + 437, + 309 + ], + "spans": [ + { + "bbox": [ + 173, + 274, + 437, + 309 + ], + "score": 0.94, + "content": "\\widetilde { \\gamma } ( \\pmb { \\theta } ) : = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\rho ^ { L } } = \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\rho ^ { L } } = \\frac { g \\left( - \\log \\left( \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ( \\pmb { \\theta } ) ) } \\right) \\right) } { \\rho ^ { L } } ,", + "type": "interline_equation", + "image_path": "8b913b60edcebfcd8526eedfbe5f446b272c22f815e963b107ad97eb2188564a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 173, + 274, + 437, + 285.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 173, + 285.6666666666667, + 437, + 297.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 173, + 297.33333333333337, + 437, + 309.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 312, + 342, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 339, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 133, + 327 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 311, + 160, + 325 + ], + "score": 0.93, + "content": "\\ell ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 309, + 262, + 327 + ], + "score": 1.0, + "content": "is the inverse function of", + "type": "text" + }, + { + "bbox": [ + 262, + 312, + 278, + 324 + ], + "score": 0.88, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 309, + 296, + 327 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 296, + 312, + 339, + 325 + ], + "score": 0.93, + "content": "\\rho : = \\| \\pmb \\theta \\| _ { 2 }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 503, + 354 + ], + "lines": [ + { + "bbox": [ + 104, + 323, + 507, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 323, + 231, + 344 + ], + "score": 1.0, + "content": "Remark A.4. For logistic loss", + "type": "text" + }, + { + "bbox": [ + 231, + 326, + 312, + 339 + ], + "score": 0.84, + "content": "\\ell ( q ) = \\log ( 1 + e ^ { - q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 323, + 316, + 344 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 316, + 325, + 420, + 341 + ], + "score": 0.68, + "content": "\\begin{array} { r } { \\widetilde \\gamma ( \\pmb \\theta ) = \\rho ^ { - L } \\log \\frac { 1 } { \\exp ( \\mathcal L ) - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 323, + 507, + 344 + ], + "score": 1.0, + "content": "; for exponential loss", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 339, + 336, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 152, + 353 + ], + "score": 0.76, + "content": "\\ell ( q ) = e ^ { - q }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 339, + 156, + 356 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 157, + 340, + 232, + 354 + ], + "score": 0.62, + "content": "\\begin{array} { r } { \\tilde { \\gamma } ( \\pmb { \\theta } ) = \\rho ^ { - L } \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 339, + 336, + 356 + ], + "score": 1.0, + "content": ", which is the same as (3).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 502, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 276, + 374 + ], + "score": 1.0, + "content": "Now we give some insights on how well", + "type": "text" + }, + { + "bbox": [ + 276, + 361, + 297, + 373 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 360, + 356, + 374 + ], + "score": 1.0, + "content": "approximates", + "type": "text" + }, + { + "bbox": [ + 356, + 361, + 376, + 373 + ], + "score": 0.92, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 360, + 505, + 374 + ], + "score": 1.0, + "content": "using a similar argument as in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 371, + 504, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 433, + 384 + ], + "score": 1.0, + "content": "Section 4.2. Using the LogSumExp function, the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 434, + 372, + 454, + 384 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 371, + 504, + 384 + ], + "score": 1.0, + "content": "can also be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 149, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 149, + 395 + ], + "score": 1.0, + "content": "written as", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 390, + 392, + 417 + ], + "lines": [ + { + "bbox": [ + 218, + 390, + 392, + 417 + ], + "spans": [ + { + "bbox": [ + 218, + 390, + 392, + 417 + ], + "score": 0.94, + "content": "\\tilde { \\gamma } ( \\pmb \\theta ) = \\frac { g ( - \\mathrm { L S E } ( - f ( q _ { 1 } ) , \\dots , - f ( q _ { N } ) ) ) } { \\rho ^ { L } } .", + "type": "interline_equation", + "image_path": "6da21a0eaa2407d08cd6f7318527e50b01a85acc2a2d07edbc7204f22ecf0aee.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 218, + 390, + 392, + 417 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 418, + 471, + 430 + ], + "lines": [ + { + "bbox": [ + 104, + 415, + 474, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 144, + 433 + ], + "score": 1.0, + "content": "LSE is a", + "type": "text" + }, + { + "bbox": [ + 144, + 418, + 176, + 430 + ], + "score": 0.91, + "content": "( \\log N )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 415, + 440, + 433 + ], + "score": 1.0, + "content": "-additive approximation for max. So we can roughly approximate", + "type": "text" + }, + { + "bbox": [ + 441, + 418, + 461, + 430 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 415, + 474, + 433 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 432, + 441, + 460 + ], + "lines": [ + { + "bbox": [ + 170, + 432, + 441, + 460 + ], + "spans": [ + { + "bbox": [ + 170, + 432, + 441, + 460 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( \\pmb \\theta ) \\approx \\frac { g ( - \\operatorname* { m a x } \\{ - f ( q _ { 1 } ) , \\dots , - f ( q _ { N } ) \\} ) } { \\rho ^ { L } } = \\frac { g ( f ( q _ { \\operatorname* { m i n } } ) ) } { \\rho ^ { L } } = \\bar { \\gamma } ( \\pmb \\theta ) .", + "type": "interline_equation", + "image_path": "cb28978f2607d2b1cc9719bfbe8f62a00a8937b9042e17c82b0c0a2147c13a22.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 170, + 432, + 441, + 460 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 504, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "score": 1.0, + "content": "Note that (B3.3) is crucial to make the above approximation reasonable. Similar to exponential loss,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 472, + 429, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 300, + 485 + ], + "score": 1.0, + "content": "we can show the following lemma asserting that", + "type": "text" + }, + { + "bbox": [ + 301, + 473, + 308, + 484 + ], + "score": 0.86, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 472, + 419, + 485 + ], + "score": 1.0, + "content": "is a good approximation of", + "type": "text" + }, + { + "bbox": [ + 419, + 474, + 426, + 484 + ], + "score": 0.85, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 472, + 429, + 485 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 108, + 486, + 440, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 438, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 205, + 500 + ], + "score": 1.0, + "content": "Lemma A.5. Assuming", + "type": "text" + }, + { + "bbox": [ + 205, + 486, + 227, + 498 + ], + "score": 0.89, + "content": "( B 3 ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 485, + 438, + 500 + ], + "score": 1.0, + "content": ", we have the following properties about the margin:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 108, + 505, + 275, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 503, + 275, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 123, + 522 + ], + "score": 1.0, + "content": "(a)", + "type": "text" + }, + { + "bbox": [ + 123, + 504, + 275, + 520 + ], + "score": 0.87, + "content": "\\begin{array} { r } { f ( q _ { \\mathrm { m i n } } ) - \\log N \\leq \\log \\frac { 1 } { \\mathcal { L } } \\leq f ( q _ { \\mathrm { m i n } } ) . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 525, + 482, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 483, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 123, + 541 + ], + "score": 1.0, + "content": "(b)", + "type": "text" + }, + { + "bbox": [ + 123, + 525, + 191, + 540 + ], + "score": 0.83, + "content": "\\begin{array} { r } { I f \\log { \\frac { 1 } { \\mathcal { L } } } > f ( b _ { f } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 525, + 262, + 541 + ], + "score": 1.0, + "content": ", then there exists", + "type": "text" + }, + { + "bbox": [ + 262, + 526, + 442, + 539 + ], + "score": 0.92, + "content": "\\xi \\in \\left( f ( q _ { \\operatorname* { m i n } } ) - \\log N , f ( q _ { \\operatorname* { m i n } } ) \\right) \\cap \\left( b _ { f } , + \\infty \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 525, + 483, + 541 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 542, + 367, + 569 + ], + "lines": [ + { + "bbox": [ + 258, + 542, + 367, + 569 + ], + "spans": [ + { + "bbox": [ + 258, + 542, + 367, + 569 + ], + "score": 0.77, + "content": "\\bar { \\gamma } - \\frac { g ^ { \\prime } ( \\xi ) \\log N } { \\rho ^ { L } } \\leq \\tilde { \\gamma } \\leq \\bar { \\gamma } .", + "type": "interline_equation", + "image_path": "47307422b79dc5f6ccec5c3ed6587478b29e621b522846f60e153ef29ed620f4.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 258, + 542, + 367, + 569 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 577, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 240, + 593 + ], + "score": 1.0, + "content": "(c) For a seuqnce of parameters", + "type": "text" + }, + { + "bbox": [ + 240, + 578, + 324, + 591 + ], + "score": 0.87, + "content": "\\{ \\pmb \\theta _ { m } \\in \\mathbb { R } ^ { d } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 577, + 335, + 593 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 336, + 579, + 385, + 591 + ], + "score": 0.82, + "content": "\\mathcal { L } ( \\pmb { \\theta } _ { m } ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 577, + 409, + 593 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 409, + 578, + 501, + 591 + ], + "score": 0.9, + "content": "\\lvert \\tilde { \\gamma } ( \\pmb { \\theta } _ { m } ) - \\bar { \\gamma } ( \\pmb { \\theta } _ { m } ) \\rvert 0", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 577, + 505, + 593 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 270, + 617 + ], + "score": 1.0, + "content": "Proof. (a) can be easily deduced from", + "type": "text" + }, + { + "bbox": [ + 270, + 603, + 399, + 614 + ], + "score": 0.89, + "content": "e ^ { - f ( q _ { \\mathrm { m i n } } ) } \\leq \\mathcal { L } \\leq N e ^ { - f ( q _ { \\mathrm { m i n } } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 601, + 456, + 617 + ], + "score": 1.0, + "content": ". Combining", + "type": "text" + }, + { + "bbox": [ + 457, + 605, + 470, + 615 + ], + "score": 0.47, + "content": "( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 601, + 506, + 617 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 613, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 174, + 628 + ], + "score": 1.0, + "content": "monotonicity of", + "type": "text" + }, + { + "bbox": [ + 174, + 615, + 191, + 627 + ], + "score": 0.9, + "content": "g ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 613, + 264, + 628 + ], + "score": 1.0, + "content": ", we further have", + "type": "text" + }, + { + "bbox": [ + 264, + 615, + 368, + 627 + ], + "score": 0.89, + "content": "g ( s ) \\leq g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\leq q _ { \\operatorname* { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 613, + 385, + 628 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 386, + 615, + 505, + 626 + ], + "score": 0.85, + "content": "s : = \\operatorname* { m a x } \\{ f ( b _ { f } ) , f ( q _ { \\operatorname* { m i n } } ) \\ : -", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 624, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 136, + 637 + ], + "score": 0.9, + "content": "\\log N \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 624, + 308, + 639 + ], + "score": 1.0, + "content": ". By the mean value Theorem, there exists", + "type": "text" + }, + { + "bbox": [ + 309, + 626, + 375, + 638 + ], + "score": 0.88, + "content": "\\xi \\in \\widetilde { ( s , f ( q _ { \\operatorname* { m i n } } ) ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 624, + 416, + 639 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 416, + 626, + 505, + 637 + ], + "score": 0.87, + "content": "g ( s ) = { \\dot { g } } ( f ( q _ { \\mathrm { m i n } } ) ) -", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 636, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 282, + 649 + ], + "score": 0.88, + "content": "g ^ { \\prime } ( \\xi ) ( f ( q _ { \\mathrm { m i n } } ) - s ) \\geq q _ { \\mathrm { m i n } } - g ^ { \\prime } ( \\xi ) \\log N", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 636, + 329, + 650 + ], + "score": 1.0, + "content": ". Dividing", + "type": "text" + }, + { + "bbox": [ + 329, + 637, + 342, + 649 + ], + "score": 0.88, + "content": "\\rho ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 636, + 412, + 650 + ], + "score": 1.0, + "content": "on each side of", + "type": "text" + }, + { + "bbox": [ + 412, + 637, + 504, + 649 + ], + "score": 0.91, + "content": "q _ { \\operatorname* { m i n } } - g ^ { \\prime } ( \\xi ) \\log N \\le", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 648, + 223, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 174, + 661 + ], + "score": 0.8, + "content": "g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) \\leq q _ { \\operatorname* { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 648, + 204, + 660 + ], + "score": 1.0, + "content": "proves", + "type": "text" + }, + { + "bbox": [ + 205, + 649, + 217, + 659 + ], + "score": 0.34, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 648, + 223, + 660 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 102, + 663, + 507, + 686 + ], + "spans": [ + { + "bbox": [ + 102, + 663, + 167, + 686 + ], + "score": 1.0, + "content": "Now we prove", + "type": "text" + }, + { + "bbox": [ + 168, + 668, + 180, + 678 + ], + "score": 0.61, + "content": "( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 663, + 354, + 686 + ], + "score": 1.0, + "content": ". 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It follows", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 678, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 104, + 678, + 128, + 695 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 128, + 681, + 141, + 691 + ], + "score": 0.75, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 678, + 197, + 695 + ], + "score": 1.0, + "content": "that for every", + "type": "text" + }, + { + "bbox": [ + 197, + 680, + 211, + 691 + ], + "score": 0.89, + "content": "\\pmb { \\theta } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 678, + 259, + 695 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 259, + 680, + 483, + 693 + ], + "score": 0.89, + "content": "\\xi _ { m } \\in ( f ( q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) ) - \\log N , f ( q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) ) ) \\cap ( b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 678, + 506, + 695 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 690, + 126, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 126, + 703 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 700, + 416, + 715 + ], + "lines": [ + { + "bbox": [ + 195, + 700, + 416, + 715 + ], + "spans": [ + { + "bbox": [ + 195, + 700, + 416, + 715 + ], + "score": 0.86, + "content": "\\bar { \\gamma } ( \\pmb \\theta _ { m } ) - ( g ^ { \\prime } ( \\xi _ { m } ) \\log N ) / \\rho ( \\pmb \\theta _ { m } ) ^ { L } \\leq \\widetilde \\gamma ( \\pmb \\theta _ { m } ) \\leq \\bar { \\gamma } ( \\pmb \\theta _ { m } ) .", + "type": "interline_equation", + "image_path": "5bd289c0893766d5b9254ca13097eee09973bf064d805c5311650e27a2430b9b.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 195, + 700, + 416, + 715 + ], + "spans": [], + "index": 39 + } + ] + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 118, + 721, + 384, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 385, + 732 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 385, + 732 + ], + "score": 1.0, + "content": "4Indeed, (B3.4) is not needed for showing Lemma A.5 and Theorem A.7.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 124 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 363, + 95 + ], + "score": 1.0, + "content": "Proof for Remark A.2. By simple calculations, the formulas for", + "type": "text" + }, + { + "bbox": [ + 364, + 82, + 456, + 95 + ], + "score": 0.92, + "content": "f ( q ) , f ^ { \\prime } ( q ) , g ( q ) , g ^ { \\prime } ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "are correct.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 179, + 114 + ], + "score": 1.0, + "content": "(B3.1) is trivial.", + "type": "text" + }, + { + "bbox": [ + 180, + 94, + 317, + 111 + ], + "score": 0.93, + "content": "f ^ { \\prime } ( q ) ~ = ~ \\frac { e ^ { - q } } { ( 1 + e ^ { - q } ) \\log ( 1 + e ^ { - q } ) } ~ > ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 93, + 506, + 114 + ], + "score": 1.0, + "content": ", so (B3.2) is satisfied. For (B3.3), note that", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 411, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 109, + 215, + 125 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f ^ { \\prime } ( q ) q = \\frac { q } { ( 1 + e ^ { q } ) \\log ( 1 + e ^ { - q } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 106, + 411, + 130 + ], + "score": 1.0, + "content": ". The denominator is a decreasing function since", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 130 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 127, + 451, + 153 + ], + "lines": [ + { + "bbox": [ + 160, + 127, + 451, + 153 + ], + "spans": [ + { + "bbox": [ + 160, + 127, + 451, + 153 + ], + "score": 0.91, + "content": "{ \\frac { d } { d q } } \\left( ( 1 + e ^ { q } ) \\log ( 1 + e ^ { - q } ) \\right) = e ^ { q } \\log ( 1 + e ^ { - q } ) - 1 < e ^ { q } \\cdot e ^ { - q } - 1 = 0 .", + "type": "interline_equation", + "image_path": "ba9ead046cd453e239d2c935dfde05a2254e3890d255d93b27a1cbeb40f1e0e0.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 160, + 127, + 451, + 153 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 506, + 194 + ], + "lines": [ + { + "bbox": [ + 106, + 155, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 132, + 169 + ], + "score": 1.0, + "content": "Thus,", + "type": "text" + }, + { + "bbox": [ + 133, + 156, + 160, + 169 + ], + "score": 0.93, + "content": "f ^ { \\prime } ( q ) q", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 155, + 305, + 169 + ], + "score": 1.0, + "content": "is a strictly increasing function on", + "type": "text" + }, + { + "bbox": [ + 306, + 157, + 313, + 166 + ], + "score": 0.82, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 155, + 335, + 169 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 335, + 157, + 346, + 169 + ], + "score": 0.88, + "content": "b _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 155, + 506, + 169 + ], + "score": 1.0, + "content": "is required to be non-negative, we set", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 168, + 504, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 136, + 182 + ], + "score": 0.9, + "content": "b _ { f } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 169, + 210, + 185 + ], + "score": 1.0, + "content": ". For proving that", + "type": "text" + }, + { + "bbox": [ + 210, + 170, + 271, + 182 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q ) q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 169, + 426, + 185 + ], + "score": 1.0, + "content": "and (B4), we only need to notice that", + "type": "text" + }, + { + "bbox": [ + 426, + 168, + 504, + 183 + ], + "score": 0.89, + "content": "\\begin{array} { r } { f ^ { \\prime } ( q ) \\sim \\frac { e ^ { - q } } { 1 \\cdot e ^ { - q } } = 1 } \\end{array}", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 123, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 181, + 222, + 193 + ], + "score": 0.91, + "content": "g ^ { \\prime } ( x ) = \\bar { 1 } / f ^ { \\prime } ( \\bar { g ( x ) } ) \\sim 1", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 181, + 227, + 194 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 181, + 506, + 192 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 155, + 506, + 194 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 205, + 284, + 217 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 285, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 285, + 218 + ], + "score": 1.0, + "content": "A.2 SMOOTHED NORMALIZED MARGIN", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 504, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 504, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 184, + 240 + ], + "score": 1.0, + "content": "For a loss function", + "type": "text" + }, + { + "bbox": [ + 184, + 226, + 200, + 238 + ], + "score": 0.9, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 225, + 477, + 240 + ], + "score": 1.0, + "content": "satisfying (B3), it is easy to see from (B3.2) that its inverse function", + "type": "text" + }, + { + "bbox": [ + 478, + 226, + 504, + 239 + ], + "score": 0.91, + "content": "\\ell ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 236, + 497, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 497, + 250 + ], + "score": 1.0, + "content": "must exist. For this kind of loss functions, we define the smoothed normalized margin as follows:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 225, + 504, + 250 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 250, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 249, + 263 + ], + "score": 1.0, + "content": "Definition A.3. For a loss function", + "type": "text" + }, + { + "bbox": [ + 249, + 251, + 265, + 263 + ], + "score": 0.85, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 250, + 464, + 263 + ], + "score": 1.0, + "content": "satisfying (B3), the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 464, + 251, + 484, + 263 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 250, + 497, + 263 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 497, + 252, + 504, + 261 + ], + "score": 0.76, + "content": "\\pmb \\theta", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 260, + 159, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 159, + 275 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 250, + 504, + 275 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 274, + 437, + 309 + ], + "lines": [ + { + "bbox": [ + 173, + 274, + 437, + 309 + ], + "spans": [ + { + "bbox": [ + 173, + 274, + 437, + 309 + ], + "score": 0.94, + "content": "\\widetilde { \\gamma } ( \\pmb { \\theta } ) : = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\rho ^ { L } } = \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\rho ^ { L } } = \\frac { g \\left( - \\log \\left( \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ( \\pmb { \\theta } ) ) } \\right) \\right) } { \\rho ^ { L } } ,", + "type": "interline_equation", + "image_path": "8b913b60edcebfcd8526eedfbe5f446b272c22f815e963b107ad97eb2188564a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 173, + 274, + 437, + 285.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 173, + 285.6666666666667, + 437, + 297.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 173, + 297.33333333333337, + 437, + 309.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 312, + 342, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 339, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 133, + 327 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 311, + 160, + 325 + ], + "score": 0.93, + "content": "\\ell ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 309, + 262, + 327 + ], + "score": 1.0, + "content": "is the inverse function of", + "type": "text" + }, + { + "bbox": [ + 262, + 312, + 278, + 324 + ], + "score": 0.88, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 309, + 296, + 327 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 296, + 312, + 339, + 325 + ], + "score": 0.93, + "content": "\\rho : = \\| \\pmb \\theta \\| _ { 2 }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 309, + 339, + 327 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 503, + 354 + ], + "lines": [ + { + "bbox": [ + 104, + 323, + 507, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 323, + 231, + 344 + ], + "score": 1.0, + "content": "Remark A.4. For logistic loss", + "type": "text" + }, + { + "bbox": [ + 231, + 326, + 312, + 339 + ], + "score": 0.84, + "content": "\\ell ( q ) = \\log ( 1 + e ^ { - q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 323, + 316, + 344 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 316, + 325, + 420, + 341 + ], + "score": 0.68, + "content": "\\begin{array} { r } { \\widetilde \\gamma ( \\pmb \\theta ) = \\rho ^ { - L } \\log \\frac { 1 } { \\exp ( \\mathcal L ) - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 323, + 507, + 344 + ], + "score": 1.0, + "content": "; for exponential loss", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 339, + 336, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 152, + 353 + ], + "score": 0.76, + "content": "\\ell ( q ) = e ^ { - q }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 339, + 156, + 356 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 157, + 340, + 232, + 354 + ], + "score": 0.62, + "content": "\\begin{array} { r } { \\tilde { \\gamma } ( \\pmb { \\theta } ) = \\rho ^ { - L } \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 339, + 336, + 356 + ], + "score": 1.0, + "content": ", which is the same as (3).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 323, + 507, + 356 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 502, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 276, + 374 + ], + "score": 1.0, + "content": "Now we give some insights on how well", + "type": "text" + }, + { + "bbox": [ + 276, + 361, + 297, + 373 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 360, + 356, + 374 + ], + "score": 1.0, + "content": "approximates", + "type": "text" + }, + { + "bbox": [ + 356, + 361, + 376, + 373 + ], + "score": 0.92, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 360, + 505, + 374 + ], + "score": 1.0, + "content": "using a similar argument as in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 371, + 504, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 433, + 384 + ], + "score": 1.0, + "content": "Section 4.2. Using the LogSumExp function, the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 434, + 372, + 454, + 384 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 371, + 504, + 384 + ], + "score": 1.0, + "content": "can also be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 149, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 149, + 395 + ], + "score": 1.0, + "content": "written as", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 360, + 505, + 395 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 390, + 392, + 417 + ], + "lines": [ + { + "bbox": [ + 218, + 390, + 392, + 417 + ], + "spans": [ + { + "bbox": [ + 218, + 390, + 392, + 417 + ], + "score": 0.94, + "content": "\\tilde { \\gamma } ( \\pmb \\theta ) = \\frac { g ( - \\mathrm { L S E } ( - f ( q _ { 1 } ) , \\dots , - f ( q _ { N } ) ) ) } { \\rho ^ { L } } .", + "type": "interline_equation", + "image_path": "6da21a0eaa2407d08cd6f7318527e50b01a85acc2a2d07edbc7204f22ecf0aee.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 218, + 390, + 392, + 417 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 418, + 471, + 430 + ], + "lines": [ + { + "bbox": [ + 104, + 415, + 474, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 144, + 433 + ], + "score": 1.0, + "content": "LSE is a", + "type": "text" + }, + { + "bbox": [ + 144, + 418, + 176, + 430 + ], + "score": 0.91, + "content": "( \\log N )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 415, + 440, + 433 + ], + "score": 1.0, + "content": "-additive approximation for max. So we can roughly approximate", + "type": "text" + }, + { + "bbox": [ + 441, + 418, + 461, + 430 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 415, + 474, + 433 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 415, + 474, + 433 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 432, + 441, + 460 + ], + "lines": [ + { + "bbox": [ + 170, + 432, + 441, + 460 + ], + "spans": [ + { + "bbox": [ + 170, + 432, + 441, + 460 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( \\pmb \\theta ) \\approx \\frac { g ( - \\operatorname* { m a x } \\{ - f ( q _ { 1 } ) , \\dots , - f ( q _ { N } ) \\} ) } { \\rho ^ { L } } = \\frac { g ( f ( q _ { \\operatorname* { m i n } } ) ) } { \\rho ^ { L } } = \\bar { \\gamma } ( \\pmb \\theta ) .", + "type": "interline_equation", + "image_path": "cb28978f2607d2b1cc9719bfbe8f62a00a8937b9042e17c82b0c0a2147c13a22.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 170, + 432, + 441, + 460 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 504, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "score": 1.0, + "content": "Note that (B3.3) is crucial to make the above approximation reasonable. Similar to exponential loss,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 472, + 429, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 300, + 485 + ], + "score": 1.0, + "content": "we can show the following lemma asserting that", + "type": "text" + }, + { + "bbox": [ + 301, + 473, + 308, + 484 + ], + "score": 0.86, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 472, + 419, + 485 + ], + "score": 1.0, + "content": "is a good approximation of", + "type": "text" + }, + { + "bbox": [ + 419, + 474, + 426, + 484 + ], + "score": 0.85, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 472, + 429, + 485 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 460, + 505, + 485 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 486, + 440, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 438, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 205, + 500 + ], + "score": 1.0, + "content": "Lemma A.5. Assuming", + "type": "text" + }, + { + "bbox": [ + 205, + 486, + 227, + 498 + ], + "score": 0.89, + "content": "( B 3 ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 485, + 438, + 500 + ], + "score": 1.0, + "content": ", we have the following properties about the margin:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 485, + 438, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 505, + 275, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 503, + 275, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 123, + 522 + ], + "score": 1.0, + "content": "(a)", + "type": "text" + }, + { + "bbox": [ + 123, + 504, + 275, + 520 + ], + "score": 0.87, + "content": "\\begin{array} { r } { f ( q _ { \\mathrm { m i n } } ) - \\log N \\leq \\log \\frac { 1 } { \\mathcal { L } } \\leq f ( q _ { \\mathrm { m i n } } ) . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 503, + 275, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 525, + 482, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 483, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 123, + 541 + ], + "score": 1.0, + "content": "(b)", + "type": "text" + }, + { + "bbox": [ + 123, + 525, + 191, + 540 + ], + "score": 0.83, + "content": "\\begin{array} { r } { I f \\log { \\frac { 1 } { \\mathcal { L } } } > f ( b _ { f } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 525, + 262, + 541 + ], + "score": 1.0, + "content": ", then there exists", + "type": "text" + }, + { + "bbox": [ + 262, + 526, + 442, + 539 + ], + "score": 0.92, + "content": "\\xi \\in \\left( f ( q _ { \\operatorname* { m i n } } ) - \\log N , f ( q _ { \\operatorname* { m i n } } ) \\right) \\cap \\left( b _ { f } , + \\infty \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 525, + 483, + 541 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 525, + 483, + 541 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 542, + 367, + 569 + ], + "lines": [ + { + "bbox": [ + 258, + 542, + 367, + 569 + ], + "spans": [ + { + "bbox": [ + 258, + 542, + 367, + 569 + ], + "score": 0.77, + "content": "\\bar { \\gamma } - \\frac { g ^ { \\prime } ( \\xi ) \\log N } { \\rho ^ { L } } \\leq \\tilde { \\gamma } \\leq \\bar { \\gamma } .", + "type": "interline_equation", + "image_path": "47307422b79dc5f6ccec5c3ed6587478b29e621b522846f60e153ef29ed620f4.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 258, + 542, + 367, + 569 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 577, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 240, + 593 + ], + "score": 1.0, + "content": "(c) For a seuqnce of parameters", + "type": "text" + }, + { + "bbox": [ + 240, + 578, + 324, + 591 + ], + "score": 0.87, + "content": "\\{ \\pmb \\theta _ { m } \\in \\mathbb { R } ^ { d } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 577, + 335, + 593 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 336, + 579, + 385, + 591 + ], + "score": 0.82, + "content": "\\mathcal { L } ( \\pmb { \\theta } _ { m } ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 577, + 409, + 593 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 409, + 578, + 501, + 591 + ], + "score": 0.9, + "content": "\\lvert \\tilde { \\gamma } ( \\pmb { \\theta } _ { m } ) - \\bar { \\gamma } ( \\pmb { \\theta } _ { m } ) \\rvert 0", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 577, + 505, + 593 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 577, + 505, + 593 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 270, + 617 + ], + "score": 1.0, + "content": "Proof. (a) can be easily deduced from", + "type": "text" + }, + { + "bbox": [ + 270, + 603, + 399, + 614 + ], + "score": 0.89, + "content": "e ^ { - f ( q _ { \\mathrm { m i n } } ) } \\leq \\mathcal { L } \\leq N e ^ { - f ( q _ { \\mathrm { m i n } } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 601, + 456, + 617 + ], + "score": 1.0, + "content": ". Combining", + "type": "text" + }, + { + "bbox": [ + 457, + 605, + 470, + 615 + ], + "score": 0.47, + "content": "( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 601, + 506, + 617 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 613, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 174, + 628 + ], + "score": 1.0, + "content": "monotonicity of", + "type": "text" + }, + { + "bbox": [ + 174, + 615, + 191, + 627 + ], + "score": 0.9, + "content": "g ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 613, + 264, + 628 + ], + "score": 1.0, + "content": ", we further have", + "type": "text" + }, + { + "bbox": [ + 264, + 615, + 368, + 627 + ], + "score": 0.89, + "content": "g ( s ) \\leq g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\leq q _ { \\operatorname* { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 613, + 385, + 628 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 386, + 615, + 505, + 626 + ], + "score": 0.85, + "content": "s : = \\operatorname* { m a x } \\{ f ( b _ { f } ) , f ( q _ { \\operatorname* { m i n } } ) \\ : -", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 624, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 136, + 637 + ], + "score": 0.9, + "content": "\\log N \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 624, + 308, + 639 + ], + "score": 1.0, + "content": ". By the mean value Theorem, there exists", + "type": "text" + }, + { + "bbox": [ + 309, + 626, + 375, + 638 + ], + "score": 0.88, + "content": "\\xi \\in \\widetilde { ( s , f ( q _ { \\operatorname* { m i n } } ) ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 624, + 416, + 639 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 416, + 626, + 505, + 637 + ], + "score": 0.87, + "content": "g ( s ) = { \\dot { g } } ( f ( q _ { \\mathrm { m i n } } ) ) -", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 636, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 282, + 649 + ], + "score": 0.88, + "content": "g ^ { \\prime } ( \\xi ) ( f ( q _ { \\mathrm { m i n } } ) - s ) \\geq q _ { \\mathrm { m i n } } - g ^ { \\prime } ( \\xi ) \\log N", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 636, + 329, + 650 + ], + "score": 1.0, + "content": ". Dividing", + "type": "text" + }, + { + "bbox": [ + 329, + 637, + 342, + 649 + ], + "score": 0.88, + "content": "\\rho ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 636, + 412, + 650 + ], + "score": 1.0, + "content": "on each side of", + "type": "text" + }, + { + "bbox": [ + 412, + 637, + 504, + 649 + ], + "score": 0.91, + "content": "q _ { \\operatorname* { m i n } } - g ^ { \\prime } ( \\xi ) \\log N \\le", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 648, + 223, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 174, + 661 + ], + "score": 0.8, + "content": "g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) \\leq q _ { \\operatorname* { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 648, + 204, + 660 + ], + "score": 1.0, + "content": "proves", + "type": "text" + }, + { + "bbox": [ + 205, + 649, + 217, + 659 + ], + "score": 0.34, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 648, + 223, + 660 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 601, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 102, + 663, + 507, + 686 + ], + "spans": [ + { + "bbox": [ + 102, + 663, + 167, + 686 + ], + "score": 1.0, + "content": "Now we prove", + "type": "text" + }, + { + "bbox": [ + 168, + 668, + 180, + 678 + ], + "score": 0.61, + "content": "( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 663, + 354, + 686 + ], + "score": 1.0, + "content": ". Without loss of generality, we assume log", + "type": "text" + }, + { + "bbox": [ + 354, + 666, + 416, + 681 + ], + "score": 0.91, + "content": "\\frac { 1 } { \\mathcal { L } ( \\pmb { \\theta } _ { m } ) } > f ( b _ { f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 663, + 445, + 686 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 445, + 667, + 459, + 678 + ], + "score": 0.88, + "content": "\\pmb { \\theta } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 663, + 507, + 686 + ], + "score": 1.0, + "content": ". It follows", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 678, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 104, + 678, + 128, + 695 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 128, + 681, + 141, + 691 + ], + "score": 0.75, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 678, + 197, + 695 + ], + "score": 1.0, + "content": "that for every", + "type": "text" + }, + { + "bbox": [ + 197, + 680, + 211, + 691 + ], + "score": 0.89, + "content": "\\pmb { \\theta } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 678, + 259, + 695 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 259, + 680, + 483, + 693 + ], + "score": 0.89, + "content": "\\xi _ { m } \\in ( f ( q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) ) - \\log N , f ( q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) ) ) \\cap ( b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 678, + 506, + 695 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 690, + 126, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 126, + 703 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 102, + 663, + 507, + 703 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 700, + 416, + 715 + ], + "lines": [ + { + "bbox": [ + 195, + 700, + 416, + 715 + ], + "spans": [ + { + "bbox": [ + 195, + 700, + 416, + 715 + ], + "score": 0.86, + "content": "\\bar { \\gamma } ( \\pmb \\theta _ { m } ) - ( g ^ { \\prime } ( \\xi _ { m } ) \\log N ) / \\rho ( \\pmb \\theta _ { m } ) ^ { L } \\leq \\widetilde \\gamma ( \\pmb \\theta _ { m } ) \\leq \\bar { \\gamma } ( \\pmb \\theta _ { m } ) .", + "type": "interline_equation", + "image_path": "5bd289c0893766d5b9254ca13097eee09973bf064d805c5311650e27a2430b9b.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 195, + 700, + 416, + 715 + ], + "spans": [], + "index": 39 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 506, + 119 + ], + "lines": [ + { + "bbox": [ + 102, + 78, + 504, + 102 + ], + "spans": [ + { + "bbox": [ + 102, + 78, + 151, + 102 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 151, + 81, + 432, + 97 + ], + "score": 0.86, + "content": "\\begin{array} { r c l } { \\xi _ { m } } & { \\ge } & { f \\big ( q _ { \\mathrm { m i n } } ( \\pmb \\theta _ { m } ) \\big ) - \\log N \\ge \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta _ { m } ) } - \\log N + \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 78, + 461, + 102 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 461, + 80, + 504, + 97 + ], + "score": 0.89, + "content": "\\begin{array} { r l } { \\frac { g ( \\xi _ { m } ) } { g ^ { \\prime } ( \\xi _ { m } ) } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 95, + 504, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 207, + 108 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( g ( \\xi _ { m } ) ) g ( \\xi _ { m } ) \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 96, + 398, + 109 + ], + "score": 1.0, + "content": "by (B3.3). Also note that there exists a constant", + "type": "text" + }, + { + "bbox": [ + 399, + 97, + 412, + 107 + ], + "score": 0.88, + "content": "B _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 96, + 451, + 109 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 451, + 97, + 504, + 108 + ], + "score": 0.9, + "content": "\\bar { \\gamma } ( \\pmb { \\theta } _ { m } ) \\leq B _ { 0 }", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 105, + 381, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 133, + 120 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 133, + 110, + 143, + 117 + ], + "score": 0.8, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 105, + 167, + 120 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 167, + 109, + 174, + 119 + ], + "score": 0.85, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 105, + 304, + 120 + ], + "score": 1.0, + "content": "is continuous on the unit sphere", + "type": "text" + }, + { + "bbox": [ + 304, + 106, + 327, + 118 + ], + "score": 0.91, + "content": "S ^ { d - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 105, + 381, + 120 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 121, + 483, + 149 + ], + "lines": [ + { + "bbox": [ + 126, + 121, + 483, + 149 + ], + "spans": [ + { + "bbox": [ + 126, + 121, + 483, + 149 + ], + "score": 0.92, + "content": "\\frac { g ^ { \\prime } ( \\xi _ { m } ) } { \\rho ( \\theta _ { m } ) ^ { L } } = \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\frac { g ( \\xi _ { m } ) } { \\rho ( \\theta _ { m } ) ^ { L } } \\le \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\frac { q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) } { \\rho ( \\theta _ { m } ) ^ { L } } = \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\cdot \\bar { \\gamma } ( \\theta _ { m } ) \\le \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\cdot B \\to 0 ,", + "type": "interline_equation", + "image_path": "12d6697634102808eb7462e498cc1d2cbbc1209f368febcaea930ba6d9f02bdd.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 126, + 121, + 483, + 149 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 151, + 505, + 175 + ], + "lines": [ + { + "bbox": [ + 104, + 147, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 147, + 266, + 168 + ], + "score": 1.0, + "content": "where the first inequality follows since", + "type": "text" + }, + { + "bbox": [ + 266, + 151, + 346, + 164 + ], + "score": 0.92, + "content": "\\xi _ { m } \\leq f ( q _ { \\operatorname* { m i n } } ( \\pmb \\theta _ { m } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 147, + 463, + 168 + ], + "score": 1.0, + "content": ". Together with (8), we have", + "type": "text" + }, + { + "bbox": [ + 463, + 152, + 505, + 164 + ], + "score": 0.86, + "content": "\\tilde { \\gamma } ( \\pmb { \\theta } _ { m } ) -", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 161, + 165, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 160, + 175 + ], + "score": 0.91, + "content": "\\bar { \\gamma } ( \\pmb \\theta _ { m } ) | ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 161, + 165, + 176 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 506, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 179, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 423, + 193 + ], + "score": 1.0, + "content": "Remark A.6. For exponential loss, we have already shown in Section 4.2 that", + "type": "text" + }, + { + "bbox": [ + 423, + 181, + 443, + 192 + ], + "score": 0.9, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 179, + 466, + 193 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 466, + 180, + 500, + 192 + ], + "score": 0.92, + "content": "O ( \\rho ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 179, + 506, + 193 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 217, + 205 + ], + "score": 1.0, + "content": "additive approximation for", + "type": "text" + }, + { + "bbox": [ + 218, + 192, + 237, + 203 + ], + "score": 0.88, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 191, + 402, + 205 + ], + "score": 1.0, + "content": ". For logistic loss, it follows easily from", + "type": "text" + }, + { + "bbox": [ + 403, + 192, + 460, + 204 + ], + "score": 0.91, + "content": "g ^ { \\prime } ( q ) = \\Theta ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 191, + 480, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 480, + 193, + 493, + 203 + ], + "score": 0.42, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 202, + 482, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 172, + 216 + ], + "score": 1.0, + "content": "Lemma A.5 that", + "type": "text" + }, + { + "bbox": [ + 173, + 203, + 193, + 214 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 202, + 216, + 216 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 216, + 203, + 250, + 214 + ], + "score": 0.91, + "content": "O ( \\rho ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 202, + 361, + 216 + ], + "score": 1.0, + "content": "-additive approximation for", + "type": "text" + }, + { + "bbox": [ + 362, + 203, + 382, + 214 + ], + "score": 0.91, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 202, + 392, + 216 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 392, + 203, + 400, + 213 + ], + "score": 0.59, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 202, + 482, + 216 + ], + "score": 1.0, + "content": "is sufficiently small.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 227, + 183, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 183, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 183, + 240 + ], + "score": 1.0, + "content": "A.3 THEOREMS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 247, + 504, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "score": 1.0, + "content": "Now we state our main theorems. For the monotonicity of the normalized margin, we have the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 258, + 339, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 339, + 271 + ], + "score": 1.0, + "content": "following theorem. The proof is provided in Appendix B.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 503, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 300, + 286 + ], + "score": 1.0, + "content": "Theorem A.7. Under assumptions (A1), (A2),", + "type": "text" + }, + { + "bbox": [ + 301, + 272, + 323, + 284 + ], + "score": 0.81, + "content": "( B 3 ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 271, + 329, + 286 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 329, + 272, + 347, + 284 + ], + "score": 0.52, + "content": "( B 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 271, + 505, + 286 + ], + "score": 1.0, + "content": ", the following statements are true for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 283, + 166, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 166, + 297 + ], + "score": 1.0, + "content": "gradient flow:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 129, + 302, + 474, + 361 + ], + "lines": [ + { + "bbox": [ + 129, + 302, + 270, + 317 + ], + "spans": [ + { + "bbox": [ + 129, + 302, + 175, + 317 + ], + "score": 1.0, + "content": "1. For a.e.", + "type": "text" + }, + { + "bbox": [ + 175, + 303, + 201, + 315 + ], + "score": 0.64, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 302, + 205, + 317 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 302, + 265, + 317 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\tilde { \\gamma } ( \\pmb { \\theta } ( t ) ) \\geq 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 302, + 270, + 317 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 126, + 321, + 366, + 343 + ], + "spans": [ + { + "bbox": [ + 126, + 321, + 346, + 343 + ], + "score": 1.0, + "content": "2. For a.e. t > t0, either ddt γ˜(θ(t)) > 0 or ddt θ(t)kθ(t)k2", + "type": "text" + }, + { + "bbox": [ + 361, + 324, + 366, + 338 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 129, + 347, + 473, + 360 + ], + "spans": [ + { + "bbox": [ + 129, + 347, + 142, + 360 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 142, + 347, + 196, + 360 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 347, + 215, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 347, + 273, + 360 + ], + "score": 0.9, + "content": "\\| \\pmb \\theta ( t ) \\| _ { 2 } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 347, + 285, + 360 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 286, + 348, + 324, + 358 + ], + "score": 0.86, + "content": "t \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 347, + 369, + 360 + ], + "score": 1.0, + "content": "; therefore,", + "type": "text" + }, + { + "bbox": [ + 369, + 347, + 470, + 360 + ], + "score": 0.9, + "content": "| \\bar { \\gamma } ( \\pmb \\theta ( t ) ) - \\tilde { \\gamma } ( \\pmb \\theta ( t ) ) | 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 347, + 473, + 360 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 108, + 367, + 503, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "For the normalized margin at convergence, we have two theorems, one for infinite-time limiting", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 379, + 501, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 493, + 391 + ], + "score": 1.0, + "content": "case, and the other being a finite-time quantitative result. Their proofs can be found in Appendix", + "type": "text" + }, + { + "bbox": [ + 493, + 379, + 501, + 389 + ], + "score": 0.3, + "content": "\\textrm { C }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 389, + 487, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 487, + 402 + ], + "score": 1.0, + "content": "As in the exponential loss case, we define the constrained optimization problem (P) as follows:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 403, + 370, + 442 + ], + "lines": [ + { + "bbox": [ + 240, + 403, + 370, + 442 + ], + "spans": [ + { + "bbox": [ + 240, + 403, + 370, + 442 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\mathrm { s . t . } } & { q _ { n } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ] } \\end{array}", + "type": "interline_equation", + "image_path": "c75aa4259ad04a173fdd66e7e2e38ab68f8bcd8c2839aca4e7d52cbd97febe84.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 403, + 370, + 422.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 240, + 422.5, + 370, + 442.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 444, + 400, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 400, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 290, + 459 + ], + "score": 1.0, + "content": "First, we show the directional convergence of", + "type": "text" + }, + { + "bbox": [ + 290, + 444, + 308, + 457 + ], + "score": 0.92, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 442, + 400, + 459 + ], + "score": 1.0, + "content": "to a KKT point of (P).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 502, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 394, + 472 + ], + "score": 1.0, + "content": "Theorem A.8. Consider gradient flow under assumptions (A1), (A2),", + "type": "text" + }, + { + "bbox": [ + 394, + 459, + 411, + 470 + ], + "score": 0.39, + "content": "( B 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 457, + 417, + 472 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 417, + 459, + 435, + 470 + ], + "score": 0.43, + "content": "( B 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 457, + 505, + 472 + ], + "score": 1.0, + "content": ". For every limit", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 469, + 363, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 129, + 488 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 472, + 137, + 483 + ], + "score": 0.6, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 470, + 148, + 488 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 149, + 469, + 209, + 489 + ], + "score": 0.87, + "content": "\\left\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 470, + 214, + 488 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 214, + 471, + 272, + 488 + ], + "score": 0.79, + "content": "\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 470, + 344, + 488 + ], + "score": 1.0, + "content": "is a KKT point of", + "type": "text" + }, + { + "bbox": [ + 345, + 474, + 358, + 484 + ], + "score": 0.8, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 470, + 363, + 488 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 496, + 498, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 499, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 499, + 510 + ], + "score": 1.0, + "content": "Second, we show that after finite time, gradient flow can pass through an approximate KKT point.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 388, + 524 + ], + "score": 1.0, + "content": "Theorem A.9. Consider gradient flow under assumptions (A1), (A2),", + "type": "text" + }, + { + "bbox": [ + 388, + 511, + 406, + 521 + ], + "score": 0.25, + "content": "( B 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 509, + 411, + 524 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 411, + 511, + 429, + 522 + ], + "score": 0.49, + "content": "( B 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 509, + 468, + 524 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 468, + 510, + 501, + 522 + ], + "score": 0.89, + "content": "\\epsilon , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 509, + 505, + 524 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 520, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 154, + 535 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 155, + 521, + 223, + 533 + ], + "score": 0.92, + "content": "r : = \\Theta ( \\log \\delta ^ { \\bar { - } 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 520, + 244, + 535 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 521, + 301, + 533 + ], + "score": 0.91, + "content": "\\bar { \\Delta } : = \\Theta ( \\epsilon ^ { - 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 520, + 343, + 535 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 343, + 521, + 401, + 533 + ], + "score": 0.92, + "content": "\\theta / \\dot { q } _ { \\mathrm { m i n } } ( \\theta ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 520, + 425, + 535 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 426, + 522, + 447, + 533 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 520, + 506, + 535 + ], + "score": 1.0, + "content": "-KKT point at", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 533, + 309, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 149, + 545 + ], + "score": 1.0, + "content": "some time", + "type": "text" + }, + { + "bbox": [ + 149, + 533, + 158, + 543 + ], + "score": 0.84, + "content": "t _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 533, + 200, + 545 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 200, + 533, + 306, + 545 + ], + "score": 0.9, + "content": "\\log \\| \\pmb { \\theta } ( t _ { * } ) \\| _ { 2 } \\in ( r , r + \\Delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 533, + 309, + 545 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 504, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 566 + ], + "score": 1.0, + "content": "For the definitions for KKT points and approximate KKT points, we refer the readers to Ap-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 563, + 221, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 221, + 576 + ], + "score": 1.0, + "content": "pendix C.1 for more details.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 504, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 593 + ], + "score": 1.0, + "content": "With a refined analysis, we can also provide tight rates for loss convergence and weight growth. The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 591, + 228, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 228, + 604 + ], + "score": 1.0, + "content": "proof is given in Appendix D.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 504, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 298, + 618 + ], + "score": 1.0, + "content": "Theorem A.10. Under assumptions (A1), (A2),", + "type": "text" + }, + { + "bbox": [ + 299, + 606, + 315, + 616 + ], + "score": 0.36, + "content": "( B 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 604, + 320, + 618 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 320, + 606, + 338, + 616 + ], + "score": 0.51, + "content": "( B 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 604, + 505, + 618 + ], + "score": 1.0, + "content": ", we have the following tight rates for loss", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 617, + 238, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 238, + 628 + ], + "score": 1.0, + "content": "convergence and weight growth:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 630, + 438, + 659 + ], + "lines": [ + { + "bbox": [ + 173, + 630, + 438, + 659 + ], + "spans": [ + { + "bbox": [ + 173, + 630, + 438, + 659 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) = \\Theta \\left( \\frac { g ( \\log t ) ^ { 2 / L } } { t ( \\log t ) ^ { 2 } } \\right) \\quad a n d \\quad \\| \\pmb \\theta ( t ) \\| _ { 2 } = \\Theta ( g ( \\log t ) ^ { 1 / L } ) .", + "type": "interline_equation", + "image_path": "ef93c471a7a9aa2b5d7129d9b54e552aa5cdb46846903a7a95270900ab83f5cd.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 173, + 630, + 438, + 659 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 667, + 504, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 394, + 681 + ], + "score": 1.0, + "content": "Applying Theorem A.10 to exponential loss and logistic loss, in which", + "type": "text" + }, + { + "bbox": [ + 395, + 668, + 450, + 680 + ], + "score": 0.93, + "content": "g ( x ) = \\Theta ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 666, + 505, + 681 + ], + "score": 1.0, + "content": ", we have the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 678, + 189, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 189, + 693 + ], + "score": 1.0, + "content": "following corollary:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 693, + 361, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 362, + 708 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 186, + 708 + ], + "score": 1.0, + "content": "Corollary A.11. If", + "type": "text" + }, + { + "bbox": [ + 187, + 693, + 202, + 705 + ], + "score": 0.61, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 690, + 362, + 708 + ], + "score": 1.0, + "content": "is the exponential or logistic loss, then,", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 708, + 439, + 736 + ], + "lines": [ + { + "bbox": [ + 171, + 708, + 439, + 736 + ], + "spans": [ + { + "bbox": [ + 171, + 708, + 439, + 736 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) = \\Theta \\left( \\frac { 1 } { t ( \\log t ) ^ { 2 - 2 / L } } \\right) \\quad a n d \\quad \\| \\pmb \\theta ( t ) \\| _ { 2 } = \\Theta ( ( \\log t ) ^ { 1 / L } ) .", + "type": "interline_equation", + "image_path": "225daa3015c64476ab96ce5ea5eb64367ea28e66eb24ea655f9955ca65643b41.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 171, + 708, + 439, + 736 + ], + "spans": [], + "index": 39 + } + ] + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 506, + 119 + ], + "lines": [ + { + "bbox": [ + 102, + 78, + 504, + 102 + ], + "spans": [ + { + "bbox": [ + 102, + 78, + 151, + 102 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 151, + 81, + 432, + 97 + ], + "score": 0.86, + "content": "\\begin{array} { r c l } { \\xi _ { m } } & { \\ge } & { f \\big ( q _ { \\mathrm { m i n } } ( \\pmb \\theta _ { m } ) \\big ) - \\log N \\ge \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta _ { m } ) } - \\log N + \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 78, + 461, + 102 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 461, + 80, + 504, + 97 + ], + "score": 0.89, + "content": "\\begin{array} { r l } { \\frac { g ( \\xi _ { m } ) } { g ^ { \\prime } ( \\xi _ { m } ) } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 95, + 504, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 207, + 108 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( g ( \\xi _ { m } ) ) g ( \\xi _ { m } ) \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 96, + 398, + 109 + ], + "score": 1.0, + "content": "by (B3.3). Also note that there exists a constant", + "type": "text" + }, + { + "bbox": [ + 399, + 97, + 412, + 107 + ], + "score": 0.88, + "content": "B _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 96, + 451, + 109 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 451, + 97, + 504, + 108 + ], + "score": 0.9, + "content": "\\bar { \\gamma } ( \\pmb { \\theta } _ { m } ) \\leq B _ { 0 }", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 105, + 381, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 133, + 120 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 133, + 110, + 143, + 117 + ], + "score": 0.8, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 105, + 167, + 120 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 167, + 109, + 174, + 119 + ], + "score": 0.85, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 105, + 304, + 120 + ], + "score": 1.0, + "content": "is continuous on the unit sphere", + "type": "text" + }, + { + "bbox": [ + 304, + 106, + 327, + 118 + ], + "score": 0.91, + "content": "S ^ { d - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 105, + 381, + 120 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 102, + 78, + 504, + 120 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 121, + 483, + 149 + ], + "lines": [ + { + "bbox": [ + 126, + 121, + 483, + 149 + ], + "spans": [ + { + "bbox": [ + 126, + 121, + 483, + 149 + ], + "score": 0.92, + "content": "\\frac { g ^ { \\prime } ( \\xi _ { m } ) } { \\rho ( \\theta _ { m } ) ^ { L } } = \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\frac { g ( \\xi _ { m } ) } { \\rho ( \\theta _ { m } ) ^ { L } } \\le \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\frac { q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) } { \\rho ( \\theta _ { m } ) ^ { L } } = \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\cdot \\bar { \\gamma } ( \\theta _ { m } ) \\le \\frac { g ^ { \\prime } ( \\xi _ { m } ) } { g ( \\xi _ { m } ) } \\cdot B \\to 0 ,", + "type": "interline_equation", + "image_path": "12d6697634102808eb7462e498cc1d2cbbc1209f368febcaea930ba6d9f02bdd.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 126, + 121, + 483, + 149 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 151, + 505, + 175 + ], + "lines": [ + { + "bbox": [ + 104, + 147, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 147, + 266, + 168 + ], + "score": 1.0, + "content": "where the first inequality follows since", + "type": "text" + }, + { + "bbox": [ + 266, + 151, + 346, + 164 + ], + "score": 0.92, + "content": "\\xi _ { m } \\leq f ( q _ { \\operatorname* { m i n } } ( \\pmb \\theta _ { m } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 147, + 463, + 168 + ], + "score": 1.0, + "content": ". Together with (8), we have", + "type": "text" + }, + { + "bbox": [ + 463, + 152, + 505, + 164 + ], + "score": 0.86, + "content": "\\tilde { \\gamma } ( \\pmb { \\theta } _ { m } ) -", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 161, + 165, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 160, + 175 + ], + "score": 0.91, + "content": "\\bar { \\gamma } ( \\pmb \\theta _ { m } ) | ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 161, + 165, + 176 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 147, + 505, + 176 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 506, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 179, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 423, + 193 + ], + "score": 1.0, + "content": "Remark A.6. For exponential loss, we have already shown in Section 4.2 that", + "type": "text" + }, + { + "bbox": [ + 423, + 181, + 443, + 192 + ], + "score": 0.9, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 179, + 466, + 193 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 466, + 180, + 500, + 192 + ], + "score": 0.92, + "content": "O ( \\rho ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 179, + 506, + 193 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 217, + 205 + ], + "score": 1.0, + "content": "additive approximation for", + "type": "text" + }, + { + "bbox": [ + 218, + 192, + 237, + 203 + ], + "score": 0.88, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 191, + 402, + 205 + ], + "score": 1.0, + "content": ". For logistic loss, it follows easily from", + "type": "text" + }, + { + "bbox": [ + 403, + 192, + 460, + 204 + ], + "score": 0.91, + "content": "g ^ { \\prime } ( q ) = \\Theta ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 191, + 480, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 480, + 193, + 493, + 203 + ], + "score": 0.42, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 202, + 482, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 172, + 216 + ], + "score": 1.0, + "content": "Lemma A.5 that", + "type": "text" + }, + { + "bbox": [ + 173, + 203, + 193, + 214 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 202, + 216, + 216 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 216, + 203, + 250, + 214 + ], + "score": 0.91, + "content": "O ( \\rho ^ { - L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 202, + 361, + 216 + ], + "score": 1.0, + "content": "-additive approximation for", + "type": "text" + }, + { + "bbox": [ + 362, + 203, + 382, + 214 + ], + "score": 0.91, + "content": "\\bar { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 202, + 392, + 216 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 392, + 203, + 400, + 213 + ], + "score": 0.59, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 202, + 482, + 216 + ], + "score": 1.0, + "content": "is sufficiently small.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 179, + 506, + 216 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 227, + 183, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 183, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 183, + 240 + ], + "score": 1.0, + "content": "A.3 THEOREMS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 247, + 504, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "score": 1.0, + "content": "Now we state our main theorems. For the monotonicity of the normalized margin, we have the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 258, + 339, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 339, + 271 + ], + "score": 1.0, + "content": "following theorem. The proof is provided in Appendix B.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 246, + 505, + 271 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 503, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 300, + 286 + ], + "score": 1.0, + "content": "Theorem A.7. Under assumptions (A1), (A2),", + "type": "text" + }, + { + "bbox": [ + 301, + 272, + 323, + 284 + ], + "score": 0.81, + "content": "( B 3 ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 271, + 329, + 286 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 329, + 272, + 347, + 284 + ], + "score": 0.52, + "content": "( B 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 271, + 505, + 286 + ], + "score": 1.0, + "content": ", the following statements are true for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 283, + 166, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 166, + 297 + ], + "score": 1.0, + "content": "gradient flow:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 271, + 505, + 297 + ] + }, + { + "type": "index", + "bbox": [ + 129, + 302, + 474, + 361 + ], + "lines": [ + { + "bbox": [ + 129, + 302, + 270, + 317 + ], + "spans": [ + { + "bbox": [ + 129, + 302, + 175, + 317 + ], + "score": 1.0, + "content": "1. For a.e.", + "type": "text" + }, + { + "bbox": [ + 175, + 303, + 201, + 315 + ], + "score": 0.64, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 302, + 205, + 317 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 302, + 265, + 317 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\tilde { \\gamma } ( \\pmb { \\theta } ( t ) ) \\geq 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 302, + 270, + 317 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 126, + 321, + 366, + 343 + ], + "spans": [ + { + "bbox": [ + 126, + 321, + 346, + 343 + ], + "score": 1.0, + "content": "2. For a.e. t > t0, either ddt γ˜(θ(t)) > 0 or ddt θ(t)kθ(t)k2", + "type": "text" + }, + { + "bbox": [ + 361, + 324, + 366, + 338 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 129, + 347, + 473, + 360 + ], + "spans": [ + { + "bbox": [ + 129, + 347, + 142, + 360 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 142, + 347, + 196, + 360 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 347, + 215, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 347, + 273, + 360 + ], + "score": 0.9, + "content": "\\| \\pmb \\theta ( t ) \\| _ { 2 } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 347, + 285, + 360 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 286, + 348, + 324, + 358 + ], + "score": 0.86, + "content": "t \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 347, + 369, + 360 + ], + "score": 1.0, + "content": "; therefore,", + "type": "text" + }, + { + "bbox": [ + 369, + 347, + 470, + 360 + ], + "score": 0.9, + "content": "| \\bar { \\gamma } ( \\pmb \\theta ( t ) ) - \\tilde { \\gamma } ( \\pmb \\theta ( t ) ) | 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 347, + 473, + 360 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + } + ], + "index": 15, + "bbox_fs": [ + 126, + 302, + 473, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 367, + 503, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "For the normalized margin at convergence, we have two theorems, one for infinite-time limiting", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 379, + 501, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 493, + 391 + ], + "score": 1.0, + "content": "case, and the other being a finite-time quantitative result. Their proofs can be found in Appendix", + "type": "text" + }, + { + "bbox": [ + 493, + 379, + 501, + 389 + ], + "score": 0.3, + "content": "\\textrm { C }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 389, + 487, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 487, + 402 + ], + "score": 1.0, + "content": "As in the exponential loss case, we define the constrained optimization problem (P) as follows:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 367, + 505, + 402 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 403, + 370, + 442 + ], + "lines": [ + { + "bbox": [ + 240, + 403, + 370, + 442 + ], + "spans": [ + { + "bbox": [ + 240, + 403, + 370, + 442 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\mathrm { s . t . } } & { q _ { n } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ] } \\end{array}", + "type": "interline_equation", + "image_path": "c75aa4259ad04a173fdd66e7e2e38ab68f8bcd8c2839aca4e7d52cbd97febe84.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 403, + 370, + 422.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 240, + 422.5, + 370, + 442.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 444, + 400, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 400, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 290, + 459 + ], + "score": 1.0, + "content": "First, we show the directional convergence of", + "type": "text" + }, + { + "bbox": [ + 290, + 444, + 308, + 457 + ], + "score": 0.92, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 442, + 400, + 459 + ], + "score": 1.0, + "content": "to a KKT point of (P).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 442, + 400, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 502, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 394, + 472 + ], + "score": 1.0, + "content": "Theorem A.8. Consider gradient flow under assumptions (A1), (A2),", + "type": "text" + }, + { + "bbox": [ + 394, + 459, + 411, + 470 + ], + "score": 0.39, + "content": "( B 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 457, + 417, + 472 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 417, + 459, + 435, + 470 + ], + "score": 0.43, + "content": "( B 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 457, + 505, + 472 + ], + "score": 1.0, + "content": ". For every limit", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 469, + 363, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 129, + 488 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 472, + 137, + 483 + ], + "score": 0.6, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 470, + 148, + 488 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 149, + 469, + 209, + 489 + ], + "score": 0.87, + "content": "\\left\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 470, + 214, + 488 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 214, + 471, + 272, + 488 + ], + "score": 0.79, + "content": "\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 470, + 344, + 488 + ], + "score": 1.0, + "content": "is a KKT point of", + "type": "text" + }, + { + "bbox": [ + 345, + 474, + 358, + 484 + ], + "score": 0.8, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 470, + 363, + 488 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 457, + 505, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 496, + 498, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 499, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 499, + 510 + ], + "score": 1.0, + "content": "Second, we show that after finite time, gradient flow can pass through an approximate KKT point.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 495, + 499, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 388, + 524 + ], + "score": 1.0, + "content": "Theorem A.9. Consider gradient flow under assumptions (A1), (A2),", + "type": "text" + }, + { + "bbox": [ + 388, + 511, + 406, + 521 + ], + "score": 0.25, + "content": "( B 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 509, + 411, + 524 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 411, + 511, + 429, + 522 + ], + "score": 0.49, + "content": "( B 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 509, + 468, + 524 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 468, + 510, + 501, + 522 + ], + "score": 0.89, + "content": "\\epsilon , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 509, + 505, + 524 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 520, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 154, + 535 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 155, + 521, + 223, + 533 + ], + "score": 0.92, + "content": "r : = \\Theta ( \\log \\delta ^ { \\bar { - } 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 520, + 244, + 535 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 521, + 301, + 533 + ], + "score": 0.91, + "content": "\\bar { \\Delta } : = \\Theta ( \\epsilon ^ { - 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 520, + 343, + 535 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 343, + 521, + 401, + 533 + ], + "score": 0.92, + "content": "\\theta / \\dot { q } _ { \\mathrm { m i n } } ( \\theta ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 520, + 425, + 535 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 426, + 522, + 447, + 533 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 520, + 506, + 535 + ], + "score": 1.0, + "content": "-KKT point at", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 533, + 309, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 149, + 545 + ], + "score": 1.0, + "content": "some time", + "type": "text" + }, + { + "bbox": [ + 149, + 533, + 158, + 543 + ], + "score": 0.84, + "content": "t _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 533, + 200, + 545 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 200, + 533, + 306, + 545 + ], + "score": 0.9, + "content": "\\log \\| \\pmb { \\theta } ( t _ { * } ) \\| _ { 2 } \\in ( r , r + \\Delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 533, + 309, + 545 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 509, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 504, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 566 + ], + "score": 1.0, + "content": "For the definitions for KKT points and approximate KKT points, we refer the readers to Ap-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 563, + 221, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 221, + 576 + ], + "score": 1.0, + "content": "pendix C.1 for more details.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 550, + 505, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 504, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 593 + ], + "score": 1.0, + "content": "With a refined analysis, we can also provide tight rates for loss convergence and weight growth. The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 591, + 228, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 228, + 604 + ], + "score": 1.0, + "content": "proof is given in Appendix D.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 578, + 505, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 504, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 298, + 618 + ], + "score": 1.0, + "content": "Theorem A.10. Under assumptions (A1), (A2),", + "type": "text" + }, + { + "bbox": [ + 299, + 606, + 315, + 616 + ], + "score": 0.36, + "content": "( B 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 604, + 320, + 618 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 320, + 606, + 338, + 616 + ], + "score": 0.51, + "content": "( B 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 604, + 505, + 618 + ], + "score": 1.0, + "content": ", we have the following tight rates for loss", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 617, + 238, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 238, + 628 + ], + "score": 1.0, + "content": "convergence and weight growth:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 604, + 505, + 628 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 630, + 438, + 659 + ], + "lines": [ + { + "bbox": [ + 173, + 630, + 438, + 659 + ], + "spans": [ + { + "bbox": [ + 173, + 630, + 438, + 659 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) = \\Theta \\left( \\frac { g ( \\log t ) ^ { 2 / L } } { t ( \\log t ) ^ { 2 } } \\right) \\quad a n d \\quad \\| \\pmb \\theta ( t ) \\| _ { 2 } = \\Theta ( g ( \\log t ) ^ { 1 / L } ) .", + "type": "interline_equation", + "image_path": "ef93c471a7a9aa2b5d7129d9b54e552aa5cdb46846903a7a95270900ab83f5cd.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 173, + 630, + 438, + 659 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 667, + 504, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 394, + 681 + ], + "score": 1.0, + "content": "Applying Theorem A.10 to exponential loss and logistic loss, in which", + "type": "text" + }, + { + "bbox": [ + 395, + 668, + 450, + 680 + ], + "score": 0.93, + "content": "g ( x ) = \\Theta ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 666, + 505, + 681 + ], + "score": 1.0, + "content": ", we have the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 678, + 189, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 189, + 693 + ], + "score": 1.0, + "content": "following corollary:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 666, + 505, + 693 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 693, + 361, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 362, + 708 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 186, + 708 + ], + "score": 1.0, + "content": "Corollary A.11. If", + "type": "text" + }, + { + "bbox": [ + 187, + 693, + 202, + 705 + ], + "score": 0.61, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 690, + 362, + 708 + ], + "score": 1.0, + "content": "is the exponential or logistic loss, then,", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 690, + 362, + 708 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 708, + 439, + 736 + ], + "lines": [ + { + "bbox": [ + 171, + 708, + 439, + 736 + ], + "spans": [ + { + "bbox": [ + 171, + 708, + 439, + 736 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb \\theta ( t ) ) = \\Theta \\left( \\frac { 1 } { t ( \\log t ) ^ { 2 - 2 / L } } \\right) \\quad a n d \\quad \\| \\pmb \\theta ( t ) \\| _ { 2 } = \\Theta ( ( \\log t ) ^ { 1 / L } ) .", + "type": "interline_equation", + "image_path": "225daa3015c64476ab96ce5ea5eb64367ea28e66eb24ea655f9955ca65643b41.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 171, + 708, + 439, + 736 + ], + "spans": [], + "index": 39 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 367, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 367, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 367, + 96 + ], + "score": 1.0, + "content": "B MARGIN MONOTONICITY FOR GENERAL LOSS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 119 + ], + "score": 1.0, + "content": "In this section, we consider gradient flow and prove Theorem A.7. We assume (A1), (A2), (B3),", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 116, + 246, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 246, + 129 + ], + "score": 1.0, + "content": "(B4) as mentioned in Appendix A.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 108, + 133, + 504, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 295, + 150 + ], + "score": 1.0, + "content": "We follow the notations in Section 5 to define", + "type": "text" + }, + { + "bbox": [ + 295, + 135, + 340, + 147 + ], + "score": 0.93, + "content": "\\rho : = \\| \\pmb \\theta \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 131, + 359, + 150 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 133, + 438, + 150 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\| \\pmb \\theta \\| _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 132, + 506, + 149 + ], + "score": 1.0, + "content": ", and sometimes", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 286, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 207, + 161 + ], + "score": 1.0, + "content": "we view the functions of", + "type": "text" + }, + { + "bbox": [ + 208, + 149, + 214, + 159 + ], + "score": 0.78, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 149, + 277, + 161 + ], + "score": 1.0, + "content": "as functions of", + "type": "text" + }, + { + "bbox": [ + 277, + 150, + 282, + 159 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 149, + 286, + 161 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 172, + 281, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 280, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 280, + 185 + ], + "score": 1.0, + "content": "B.1 PROOF FOR PROPOSITION 1 AND 2", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 102, + 193, + 482, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 485, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 485, + 209 + ], + "score": 1.0, + "content": "To prove the first two propositions, we generalize our key lemma (Lemma 5.1) to general loss.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 108, + 208, + 429, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 206, + 426, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 179, + 223 + ], + "score": 1.0, + "content": "Lemma B.1. For", + "type": "text" + }, + { + "bbox": [ + 180, + 209, + 187, + 220 + ], + "score": 0.78, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 206, + 396, + 223 + ], + "score": 1.0, + "content": "defined in Definition A.3, the following holds for all", + "type": "text" + }, + { + "bbox": [ + 396, + 209, + 422, + 219 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 206, + 426, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 224, + 424, + 259 + ], + "lines": [ + { + "bbox": [ + 187, + 224, + 424, + 259 + ], + "spans": [ + { + "bbox": [ + 187, + 224, + 424, + 259 + ], + "score": 0.92, + "content": "\\frac { d } { d t } \\log \\rho > 0 \\quad a n d \\quad \\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "e8a1cd862b6a3eb2c0ce049b244bfce75b01b358d707f0e8d5e497fa60007b6f.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 224, + 424, + 241.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 187, + 241.5, + 424, + 259.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "Before proving Lemma B.1, we review two important properties of homogeneous functions. Note", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 279, + 504, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 504, + 291 + ], + "score": 1.0, + "content": "that these two properties are usually shown for smooth functions. By considering Clarke’s subdif-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 289, + 438, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 438, + 303 + ], + "score": 1.0, + "content": "ferential, we can generalize it to locally Lipschitz functions that admit chain rules:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 303, + 504, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 186, + 317 + ], + "score": 1.0, + "content": "Theorem B.2. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 303, + 243, + 315 + ], + "score": 0.9, + "content": "F : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 302, + 485, + 317 + ], + "score": 1.0, + "content": "be a locally Lipschitz function that admits a chain rule. If", + "type": "text" + }, + { + "bbox": [ + 485, + 305, + 494, + 315 + ], + "score": 0.75, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 302, + 506, + 317 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 315, + 195, + 327 + ], + "spans": [ + { + "bbox": [ + 107, + 316, + 113, + 325 + ], + "score": 0.73, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 315, + 195, + 327 + ], + "score": 1.0, + "content": "-homogeneous, then", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 367, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 232, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 152, + 349 + ], + "score": 1.0, + "content": "(a) For all", + "type": "text" + }, + { + "bbox": [ + 153, + 334, + 184, + 346 + ], + "score": 0.89, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 334, + 203, + 349 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 203, + 336, + 229, + 346 + ], + "score": 0.82, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 334, + 232, + 349 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 259, + 344, + 367, + 358 + ], + "spans": [ + { + "bbox": [ + 259, + 344, + 367, + 358 + ], + "score": 0.33, + "content": "\\partial ^ { \\circ } F ( \\alpha x ) = \\alpha ^ { k - 1 } \\partial ^ { \\circ } F ( x )", + "type": "inline_equation", + "image_path": "233643ed866e7bafb8dfb99a8e0f8e450ece79eb7d483477c9a05e113df1a8da.jpg" + } + ], + "index": 16 + }, + { + "bbox": [ + 122, + 357, + 309, + 375 + ], + "spans": [ + { + "bbox": [ + 122, + 357, + 155, + 375 + ], + "score": 1.0, + "content": "That is,", + "type": "text" + }, + { + "bbox": [ + 156, + 359, + 306, + 373 + ], + "score": 0.89, + "content": "\\partial ^ { \\circ } F ( \\alpha x ) = \\{ \\alpha ^ { k - 1 } h : h \\in \\partial ^ { \\circ } F ( x ) \\} .", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 357, + 309, + 375 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 105, + 378, + 382, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 383, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 347, + 393 + ], + "score": 1.0, + "content": "(b) (Euler’s Theorem for Homogeneous Functions). For all", + "type": "text" + }, + { + "bbox": [ + 347, + 378, + 378, + 389 + ], + "score": 0.91, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 377, + 383, + 393 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 121, + 394, + 365, + 424 + ], + "lines": [ + { + "bbox": [ + 262, + 394, + 365, + 408 + ], + "spans": [ + { + "bbox": [ + 262, + 394, + 365, + 408 + ], + "score": 0.87, + "content": "\\langle { \\pmb x } , \\partial ^ { \\circ } F ( { \\pmb x } ) \\rangle = k \\cdot F ( { \\pmb x } )", + "type": "inline_equation", + "image_path": "a6a4e16f4c7e2057dadf2607a70a1249bee3fd7e5398234a18d48ccfe005202b.jpg" + } + ], + "index": 19 + }, + { + "bbox": [ + 122, + 410, + 315, + 425 + ], + "spans": [ + { + "bbox": [ + 122, + 410, + 156, + 425 + ], + "score": 1.0, + "content": "That is,", + "type": "text" + }, + { + "bbox": [ + 156, + 411, + 231, + 424 + ], + "score": 0.92, + "content": "\\langle { \\pmb x } , { \\pmb h } \\rangle = { \\boldsymbol k } \\cdot { \\boldsymbol F } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 410, + 258, + 425 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 259, + 411, + 311, + 424 + ], + "score": 0.91, + "content": "\\pmb { h } \\in \\partial ^ { \\circ } F ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 410, + 315, + 425 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 504, + 458 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 151, + 448 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 436, + 162, + 445 + ], + "score": 0.83, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 435, + 240, + 448 + ], + "score": 1.0, + "content": "be the set of points", + "type": "text" + }, + { + "bbox": [ + 241, + 438, + 248, + 445 + ], + "score": 0.79, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 435, + 287, + 448 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 288, + 436, + 297, + 445 + ], + "score": 0.86, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 435, + 372, + 448 + ], + "score": 1.0, + "content": "is differentiable at", + "type": "text" + }, + { + "bbox": [ + 372, + 438, + 380, + 445 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 435, + 505, + 448 + ], + "score": 1.0, + "content": ". 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We assume (A1), (A2), (B3),", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 116, + 246, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 246, + 129 + ], + "score": 1.0, + "content": "(B4) as mentioned in Appendix A.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 104, + 505, + 129 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 133, + 504, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 295, + 150 + ], + "score": 1.0, + "content": "We follow the notations in Section 5 to define", + "type": "text" + }, + { + "bbox": [ + 295, + 135, + 340, + 147 + ], + "score": 0.93, + "content": "\\rho : = \\| \\pmb \\theta \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 131, + 359, + 150 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 133, + 438, + 150 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\| \\pmb \\theta \\| _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 132, + 506, + 149 + ], + "score": 1.0, + "content": ", and sometimes", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 286, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 207, + 161 + ], + "score": 1.0, + "content": "we view the functions of", + "type": "text" + }, + { + "bbox": [ + 208, + 149, + 214, + 159 + ], + "score": 0.78, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 149, + 277, + 161 + ], + "score": 1.0, + "content": "as functions of", + "type": "text" + }, + { + "bbox": [ + 277, + 150, + 282, + 159 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 149, + 286, + 161 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 131, + 506, + 161 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 172, + 281, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 280, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 280, + 185 + ], + "score": 1.0, + "content": "B.1 PROOF FOR PROPOSITION 1 AND 2", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 102, + 193, + 482, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 485, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 485, + 209 + ], + "score": 1.0, + "content": "To prove the first two propositions, we generalize our key lemma (Lemma 5.1) to general loss.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 191, + 485, + 209 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 208, + 429, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 206, + 426, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 179, + 223 + ], + "score": 1.0, + "content": "Lemma B.1. For", + "type": "text" + }, + { + "bbox": [ + 180, + 209, + 187, + 220 + ], + "score": 0.78, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 206, + 396, + 223 + ], + "score": 1.0, + "content": "defined in Definition A.3, the following holds for all", + "type": "text" + }, + { + "bbox": [ + 396, + 209, + 422, + 219 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 206, + 426, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 206, + 426, + 223 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 224, + 424, + 259 + ], + "lines": [ + { + "bbox": [ + 187, + 224, + 424, + 259 + ], + "spans": [ + { + "bbox": [ + 187, + 224, + 424, + 259 + ], + "score": 0.92, + "content": "\\frac { d } { d t } \\log \\rho > 0 \\quad a n d \\quad \\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "e8a1cd862b6a3eb2c0ce049b244bfce75b01b358d707f0e8d5e497fa60007b6f.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 224, + 424, + 241.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 187, + 241.5, + 424, + 259.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "Before proving Lemma B.1, we review two important properties of homogeneous functions. Note", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 279, + 504, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 504, + 291 + ], + "score": 1.0, + "content": "that these two properties are usually shown for smooth functions. By considering Clarke’s subdif-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 289, + 438, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 438, + 303 + ], + "score": 1.0, + "content": "ferential, we can generalize it to locally Lipschitz functions that admit chain rules:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 267, + 505, + 303 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 303, + 504, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 186, + 317 + ], + "score": 1.0, + "content": "Theorem B.2. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 303, + 243, + 315 + ], + "score": 0.9, + "content": "F : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 302, + 485, + 317 + ], + "score": 1.0, + "content": "be a locally Lipschitz function that admits a chain rule. If", + "type": "text" + }, + { + "bbox": [ + 485, + 305, + 494, + 315 + ], + "score": 0.75, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 302, + 506, + 317 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 315, + 195, + 327 + ], + "spans": [ + { + "bbox": [ + 107, + 316, + 113, + 325 + ], + "score": 0.73, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 315, + 195, + 327 + ], + "score": 1.0, + "content": "-homogeneous, then", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 302, + 506, + 327 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 367, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 232, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 152, + 349 + ], + "score": 1.0, + "content": "(a) For all", + "type": "text" + }, + { + "bbox": [ + 153, + 334, + 184, + 346 + ], + "score": 0.89, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 334, + 203, + 349 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 203, + 336, + 229, + 346 + ], + "score": 0.82, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 334, + 232, + 349 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 259, + 344, + 367, + 358 + ], + "spans": [ + { + "bbox": [ + 259, + 344, + 367, + 358 + ], + "score": 0.33, + "content": "\\partial ^ { \\circ } F ( \\alpha x ) = \\alpha ^ { k - 1 } \\partial ^ { \\circ } F ( x )", + "type": "inline_equation", + "image_path": "233643ed866e7bafb8dfb99a8e0f8e450ece79eb7d483477c9a05e113df1a8da.jpg" + } + ], + "index": 16 + }, + { + "bbox": [ + 122, + 357, + 309, + 375 + ], + "spans": [ + { + "bbox": [ + 122, + 357, + 155, + 375 + ], + "score": 1.0, + "content": "That is,", + "type": "text" + }, + { + "bbox": [ + 156, + 359, + 306, + 373 + ], + "score": 0.89, + "content": "\\partial ^ { \\circ } F ( \\alpha x ) = \\{ \\alpha ^ { k - 1 } h : h \\in \\partial ^ { \\circ } F ( x ) \\} .", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 357, + 309, + 375 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 334, + 367, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 378, + 382, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 383, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 347, + 393 + ], + "score": 1.0, + "content": "(b) (Euler’s Theorem for Homogeneous Functions). 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Let", + "type": "text" + }, + { + "bbox": [ + 152, + 436, + 162, + 445 + ], + "score": 0.83, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 435, + 240, + 448 + ], + "score": 1.0, + "content": "be the set of points", + "type": "text" + }, + { + "bbox": [ + 241, + 438, + 248, + 445 + ], + "score": 0.79, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 435, + 287, + 448 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 288, + 436, + 297, + 445 + ], + "score": 0.86, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 435, + 372, + 448 + ], + "score": 1.0, + "content": "is differentiable at", + "type": "text" + }, + { + "bbox": [ + 372, + 438, + 380, + 445 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 435, + 505, + 448 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 179, + 487, + 189, + 496 + ], + "score": 0.83, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 484, + 279, + 498 + ], + "score": 1.0, + "content": "be a neighborhood of", + "type": "text" + }, + { + "bbox": [ + 280, + 488, + 291, + 497 + ], + "score": 0.87, + "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 484, + 453, + 498 + ], + "score": 1.0, + "content": ". 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This proves (10).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 544, + 505, + 580 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 146, + 596 + ], + "score": 1.0, + "content": "To prove", + "type": "text" + }, + { + "bbox": [ + 147, + 584, + 159, + 595 + ], + "score": 0.79, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 581, + 194, + 596 + ], + "score": 1.0, + "content": ", we fix", + "type": "text" + }, + { + "bbox": [ + 195, + 582, + 231, + 594 + ], + "score": 0.9, + "content": "\\textbf { \\em x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 581, + 256, + 596 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 257, + 582, + 387, + 595 + ], + "score": 0.91, + "content": "z \\ : \\ [ 0 , + \\infty ) \\ \\to \\ \\mathbb { R } ^ { d } , \\alpha \\ \\mapsto \\ \\alpha x", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 581, + 506, + 596 + ], + "score": 1.0, + "content": "be an arc. 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Pick an arbitrary", + "type": "text" + }, + { + "bbox": [ + 264, + 640, + 291, + 649 + ], + "score": 0.89, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 637, + 405, + 652 + ], + "score": 1.0, + "content": "making (11) hold. Then by", + "type": "text" + }, + { + "bbox": [ + 405, + 640, + 418, + 650 + ], + "score": 0.69, + "content": "( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 637, + 506, + 652 + ], + "score": 1.0, + "content": ", (11) is equivalent to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 648, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 284, + 663 + ], + "score": 0.91, + "content": "\\forall { \\pmb { h } } \\in \\partial ^ { \\circ } F ( { \\pmb x } ) : \\langle { \\pmb x } , \\alpha ^ { k - 1 } { \\pmb h } \\rangle = k \\alpha ^ { k - 1 } F ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 648, + 343, + 663 + ], + "score": 1.0, + "content": ", which proves", + "type": "text" + }, + { + "bbox": [ + 343, + 650, + 356, + 661 + ], + "score": 0.79, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 648, + 361, + 663 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 651, + 505, + 660 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 637, + 506, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 673, + 470, + 685 + ], + "lines": [ + { + "bbox": [ + 104, + 671, + 472, + 688 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 472, + 688 + ], + "score": 1.0, + "content": "Applying Theorem B.2 to homogeneous neural networks, we have the following corollary:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 671, + 472, + 688 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 447, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 447, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 358, + 701 + ], + "score": 1.0, + "content": "Corollary B.3. Under the assumptions (A1) and (A2), for any", + "type": "text" + }, + { + "bbox": [ + 358, + 687, + 389, + 698 + ], + "score": 0.91, + "content": "\\pmb \\theta \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 687, + 407, + 701 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 408, + 687, + 443, + 698 + ], + "score": 0.9, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d _ { \\mathbf { x } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 687, + 447, + 701 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 687, + 447, + 701 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 703, + 361, + 717 + ], + "lines": [ + { + "bbox": [ + 248, + 703, + 361, + 717 + ], + "spans": [ + { + "bbox": [ + 248, + 703, + 361, + 717 + ], + "score": 0.9, + "content": "\\langle \\pmb { \\theta } , \\partial ^ { \\circ } \\Phi _ { \\pmb { x } } ( \\pmb { \\theta } ) \\rangle = L \\cdot \\Phi _ { \\pmb { x } } ( \\pmb { \\theta } ) ,", + "type": "interline_equation", + "image_path": "1b68bb55bdd66175a9f968c7093368496ce7d564841a625caa7b53c16b5eb5d8.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 248, + 703, + 361, + 717 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 370, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 719, + 371, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 133, + 734 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 720, + 206, + 732 + ], + "score": 0.93, + "content": "\\Phi _ { x } ( \\pmb { \\theta } ) = \\Phi ( \\pmb { \\theta } ; \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 719, + 359, + 734 + ], + "score": 1.0, + "content": "is the network output for a fixed input", + "type": "text" + }, + { + "bbox": [ + 360, + 723, + 367, + 730 + ], + "score": 0.77, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 719, + 371, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 719, + 371, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 472, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 473, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 473, + 97 + ], + "score": 1.0, + "content": "Corollary B.3 can be used to derive an exact formula for the weight growth during training.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 97, + 228, + 109 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 228, + 110 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 202, + 110 + ], + "score": 1.0, + "content": "Theorem B.4. For a.e.", + "type": "text" + }, + { + "bbox": [ + 202, + 97, + 225, + 108 + ], + "score": 0.9, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 96, + 228, + 110 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 112, + 373, + 147 + ], + "lines": [ + { + "bbox": [ + 237, + 112, + 373, + 147 + ], + "spans": [ + { + "bbox": [ + 237, + 112, + 373, + 147 + ], + "score": 0.94, + "content": "\\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } .", + "type": "interline_equation", + "image_path": "ac3c6044b8852109467d58ccdc24c96868de854235173601b5edb1f564158ec7.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 112, + 373, + 129.5 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 237, + 129.5, + 373, + 147.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 506, + 194 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "Proof. The proof idea is to use Corollary B.3 and chain rules (See Appendix I for chain rules in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 168, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 295, + 187 + ], + "score": 1.0, + "content": "Clarke’s sense). Applying the chain rule on", + "type": "text" + }, + { + "bbox": [ + 295, + 169, + 369, + 183 + ], + "score": 0.92, + "content": "t \\mapsto \\rho ^ { 2 } = \\| \\pmb \\theta \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 169, + 399, + 187 + ], + "score": 1.0, + "content": "yields", + "type": "text" + }, + { + "bbox": [ + 400, + 168, + 474, + 184 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } \\ = \\ - \\langle \\theta , h \\rangle } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 181, + 404, + 194 + ], + "spans": [ + { + "bbox": [ + 107, + 182, + 144, + 192 + ], + "score": 0.91, + "content": "h \\in \\partial ^ { \\circ } { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 181, + 178, + 194 + ], + "score": 1.0, + "content": "and a.e.", + "type": "text" + }, + { + "bbox": [ + 178, + 183, + 201, + 192 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 181, + 335, + 194 + ], + "score": 1.0, + "content": ". Then applying the chain rule on", + "type": "text" + }, + { + "bbox": [ + 335, + 183, + 365, + 192 + ], + "score": 0.88, + "content": "\\pmb \\theta \\mapsto \\mathcal L", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 181, + 404, + 194 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 198, + 459, + 233 + ], + "lines": [ + { + "bbox": [ + 150, + 198, + 459, + 233 + ], + "spans": [ + { + "bbox": [ + 150, + 198, + 459, + 233 + ], + "score": 0.93, + "content": "- \\partial ^ { \\circ } { \\mathcal { L } } \\subseteq \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) \\partial ^ { \\circ } q _ { n } = \\left\\{ \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) { h _ { n } } : { h _ { n } } \\in \\partial ^ { \\circ } q _ { n } \\right\\} .", + "type": "interline_equation", + "image_path": "9109f516043faf884f5cb79eabea68297191ae253402f05e16f9eaa28c025c6d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 150, + 198, + 459, + 209.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 150, + 209.66666666666666, + 459, + 221.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 150, + 221.33333333333331, + 459, + 232.99999999999997 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 423, + 254 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 425, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 181, + 256 + ], + "score": 1.0, + "content": "By Corollary B.3,", + "type": "text" + }, + { + "bbox": [ + 181, + 240, + 241, + 252 + ], + "score": 0.93, + "content": "\\langle \\pmb \\theta , \\pmb h _ { n } \\rangle = L q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 237, + 282, + 256 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + }, + { + "bbox": [ + 282, + 237, + 420, + 254 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 236, + 425, + 256 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 264, + 505, + 294 + ], + "lines": [ + { + "bbox": [ + 102, + 263, + 508, + 281 + ], + "spans": [ + { + "bbox": [ + 102, + 263, + 219, + 281 + ], + "score": 1.0, + "content": "For convenience, we define", + "type": "text" + }, + { + "bbox": [ + 220, + 265, + 350, + 280 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nu ( t ) : = \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 263, + 379, + 281 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 379, + 267, + 404, + 279 + ], + "score": 0.89, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 263, + 508, + 281 + ], + "score": 1.0, + "content": ". Then Theorem B.4 can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 276, + 288, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 171, + 297 + ], + "score": 1.0, + "content": "be rephrased as", + "type": "text" + }, + { + "bbox": [ + 171, + 279, + 229, + 295 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\nu ( t ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 280, + 261, + 294 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 261, + 282, + 284, + 293 + ], + "score": 0.89, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 280, + 288, + 294 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 221, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 218, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 192, + 310 + ], + "score": 1.0, + "content": "Lemma B.5. For all", + "type": "text" + }, + { + "bbox": [ + 192, + 297, + 218, + 308 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 305, + 347, + 336 + ], + "lines": [ + { + "bbox": [ + 263, + 305, + 347, + 336 + ], + "spans": [ + { + "bbox": [ + 263, + 305, + 347, + 336 + ], + "score": 0.93, + "content": "\\nu ( t ) \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } .", + "type": "interline_equation", + "image_path": "3e0297bdd5cee400141a2f7def5edba66393efe1503e7488e9133cb6347f0a36.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 263, + 305, + 347, + 336 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 438, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 439, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 203, + 363 + ], + "score": 1.0, + "content": "Proof. By Lemma A.5,", + "type": "text" + }, + { + "bbox": [ + 203, + 347, + 262, + 361 + ], + "score": 0.93, + "content": "q _ { n } \\geq g ( \\log { \\frac { 1 } { \\mathcal { L } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 345, + 290, + 363 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 291, + 348, + 324, + 360 + ], + "score": 0.93, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 345, + 439, + 363 + ], + "score": 1.0, + "content": ". Then by Assumption (B3),", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 365, + 408, + 396 + ], + "lines": [ + { + "bbox": [ + 202, + 365, + 408, + 396 + ], + "spans": [ + { + "bbox": [ + 202, + 365, + 408, + 396 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q _ { n } ) q _ { n } \\geq f ^ { \\prime } ( g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ) \\cdot g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) = { \\frac { g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } { g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } .", + "type": "interline_equation", + "image_path": "ecaf2c6b9571079c80d2249dbf4350e91c9ad62abf14b593e8e7b88e63685840.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 202, + 365, + 408, + 396 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 330, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 330, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 261, + 414 + ], + "score": 1.0, + "content": "Combining this with the definitions of", + "type": "text" + }, + { + "bbox": [ + 261, + 399, + 279, + 412 + ], + "score": 0.92, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 398, + 297, + 414 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 298, + 400, + 306, + 410 + ], + "score": 0.83, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 398, + 330, + 414 + ], + "score": 1.0, + "content": "gives", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 416, + 446, + 450 + ], + "lines": [ + { + "bbox": [ + 164, + 416, + 446, + 450 + ], + "spans": [ + { + "bbox": [ + 164, + 416, + 446, + 450 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nu ( t ) = \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } \\geq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } \\displaystyle \\frac { g ( \\log \\frac { 1 } { \\mathscr { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathscr { L } } ) } = \\displaystyle \\frac { g ( \\log \\frac { 1 } { \\mathscr { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathscr { L } } ) } \\mathscr { L } . } \\end{array}", + "type": "interline_equation", + "image_path": "21914080f13a4faa6dd5c62496274d03818eccbde3f4225e09449f6f978b16f0.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 164, + 416, + 446, + 450 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 506, + 556 + ], + "lines": [ + { + "bbox": [ + 101, + 473, + 509, + 497 + ], + "spans": [ + { + "bbox": [ + 101, + 473, + 240, + 497 + ], + "score": 1.0, + "content": "Proof for Lemma B.1. Note that", + "type": "text" + }, + { + "bbox": [ + 241, + 475, + 361, + 493 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho = \\frac { 1 } { 2 \\rho ^ { 2 } } \\frac { d \\rho ^ { 2 } } { d t } = L \\frac { \\nu ( t ) } { \\rho ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 473, + 509, + 497 + ], + "score": 1.0, + "content": "by Theorem B.4. Then it simply", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 228, + 506 + ], + "score": 1.0, + "content": "follows from Lemma B.5 that", + "type": "text" + }, + { + "bbox": [ + 229, + 492, + 279, + 505 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho > 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 491, + 311, + 506 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 311, + 494, + 337, + 504 + ], + "score": 0.89, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 491, + 506, + 506 + ], + "score": 1.0, + "content": ". For the second inequality, we first prove", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 504, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 124, + 519 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 505, + 321, + 518 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\log \\tilde { \\gamma } = \\log \\left( \\ell ^ { - 1 } ( \\mathcal { L } ) / \\rho ^ { L } \\right) = \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\rho ^ { L } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 504, + 377, + 519 + ], + "score": 1.0, + "content": "exists for all", + "type": "text" + }, + { + "bbox": [ + 377, + 506, + 406, + 517 + ], + "score": 0.88, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 504, + 412, + 519 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 412, + 505, + 432, + 518 + ], + "score": 0.86, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 504, + 505, + 519 + ], + "score": 1.0, + "content": "is non-increasing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 516, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 127, + 533 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 520, + 132, + 529 + ], + "score": 0.69, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 516, + 150, + 533 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 151, + 518, + 214, + 531 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t ) < e ^ { - f ( b _ { f } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 516, + 243, + 533 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 243, + 519, + 270, + 530 + ], + "score": 0.89, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 516, + 360, + 533 + ], + "score": 1.0, + "content": ". This implies that (1)", + "type": "text" + }, + { + "bbox": [ + 360, + 518, + 384, + 532 + ], + "score": 0.86, + "content": "\\log { \\frac { 1 } { \\mathcal { L } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 516, + 495, + 533 + ], + "score": 1.0, + "content": "is always in the domain of", + "type": "text" + }, + { + "bbox": [ + 495, + 521, + 501, + 530 + ], + "score": 0.71, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 516, + 505, + 533 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 529, + 504, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 120, + 546 + ], + "score": 1.0, + "content": "(2)", + "type": "text" + }, + { + "bbox": [ + 121, + 532, + 147, + 544 + ], + "score": 0.9, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 529, + 193, + 546 + ], + "score": 1.0, + "content": "(otherwise", + "type": "text" + }, + { + "bbox": [ + 194, + 531, + 295, + 543 + ], + "score": 0.91, + "content": "\\mathcal { L } \\geq N e ^ { - f ( 0 ) } > e ^ { - f ( b _ { f } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 529, + 428, + 546 + ], + "score": 1.0, + "content": ", contradicting (B4)). Therefore,", + "type": "text" + }, + { + "bbox": [ + 428, + 531, + 504, + 545 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\widetilde { \\gamma } : = g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\rho ^ { L } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 542, + 425, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 248, + 557 + ], + "score": 1.0, + "content": "exists and is always positive for all", + "type": "text" + }, + { + "bbox": [ + 248, + 544, + 273, + 555 + ], + "score": 0.88, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 542, + 398, + 557 + ], + "score": 1.0, + "content": ", which proves the existence of", + "type": "text" + }, + { + "bbox": [ + 398, + 544, + 420, + 555 + ], + "score": 0.9, + "content": "\\log { \\tilde { \\gamma } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 542, + 425, + 557 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 281, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 282, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 282, + 573 + ], + "score": 1.0, + "content": "By the chain rule and Lemma B.5, we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 576, + 480, + 664 + ], + "lines": [ + { + "bbox": [ + 131, + 576, + 480, + 664 + ], + "spans": [ + { + "bbox": [ + 131, + 576, + 480, + 664 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } & { \\frac { d } { d t } \\log \\tilde { \\gamma } = \\frac { d } { d t } \\left( \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\right) - L \\log \\rho \\right) = \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\cdot \\frac { 1 } { \\mathcal { L } } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - L ^ { 2 } \\cdot \\frac { \\nu ( t ) } { \\rho ^ { 2 } } } & \\\\ & { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - L ^ { 2 } \\cdot \\frac { \\nu ( t ) } { \\rho ^ { 2 } } } & \\\\ & { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } - \\frac { L ^ { 2 } \\nu ( t ) ^ { 2 } } { \\rho ^ { 2 } } \\right) . } & \\end{array}", + "type": "interline_equation", + "image_path": "63b24edc3f36a85ec0a4c3e33860a27487032315f67b9dc1305ab9c184118bf3.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 131, + 576, + 480, + 605.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 131, + 605.3333333333334, + 480, + 634.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 131, + 634.6666666666667, + 480, + 664.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 102, + 663, + 504, + 695 + ], + "spans": [ + { + "bbox": [ + 102, + 663, + 178, + 695 + ], + "score": 1.0, + "content": "On the one hand, by Theorem B.4.", + "type": "text" + }, + { + "bbox": [ + 178, + 667, + 239, + 684 + ], + "score": 0.94, + "content": "\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } = \\left. \\frac { d \\pmb { \\theta } } { d t } \\right. _ { 2 } ^ { 2 } } \\\\ { \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 663, + 271, + 695 + ], + "score": 1.0, + "content": "for a.e. se toget", + "type": "text" + }, + { + "bbox": [ + 272, + 671, + 295, + 681 + ], + "score": 0.88, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 663, + 434, + 695 + ], + "score": 1.0, + "content": "by Lemma I.3; on the other hand, lds", + "type": "text" + }, + { + "bbox": [ + 434, + 669, + 504, + 684 + ], + "score": 0.93, + "content": "\\begin{array} { r } { L \\nu ( t ) = \\left. \\theta , \\frac { d \\theta } { d t } \\right. } \\end{array}", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 697, + 447, + 731 + ], + "lines": [ + { + "bbox": [ + 164, + 697, + 447, + 731 + ], + "spans": [ + { + "bbox": [ + 164, + 697, + 447, + 731 + ], + "score": 0.93, + "content": "{ \\frac { d } { d t } } \\log { \\tilde { \\gamma } } \\geq { \\frac { 1 } { \\nu ( t ) } } \\left( \\left\\| { \\frac { d \\theta } { d t } } \\right\\| _ { 2 } ^ { 2 } - \\left. { \\hat { \\theta } } , { \\frac { d \\theta } { d t } } \\right. ^ { 2 } \\right) = { \\frac { 1 } { \\nu ( t ) } } \\left\\| ( I - { \\hat { \\theta } } { \\hat { \\theta } } ^ { \\top } ) { \\frac { d \\theta } { d t } } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "f88bde66b96035a463935ca77b3e84afdbaba801adc29f134a065671b81f43ec.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 164, + 697, + 447, + 731 + ], + "spans": [], + "index": 30 + } + ] + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 312, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "20", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 240, + 505, + 251 + ], + "lines": [ + { + "bbox": [ + 495, + 242, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 495, + 242, + 505, + 251 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 454, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 496, + 456, + 504, + 465 + ], + "spans": [ + { + "bbox": [ + 496, + 456, + 504, + 465 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 472, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 473, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 473, + 97 + ], + "score": 1.0, + "content": "Corollary B.3 can be used to derive an exact formula for the weight growth during training.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 473, + 97 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 97, + 228, + 109 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 228, + 110 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 202, + 110 + ], + "score": 1.0, + "content": "Theorem B.4. For a.e.", + "type": "text" + }, + { + "bbox": [ + 202, + 97, + 225, + 108 + ], + "score": 0.9, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 96, + 228, + 110 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 96, + 228, + 110 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 112, + 373, + 147 + ], + "lines": [ + { + "bbox": [ + 237, + 112, + 373, + 147 + ], + "spans": [ + { + "bbox": [ + 237, + 112, + 373, + 147 + ], + "score": 0.94, + "content": "\\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } .", + "type": "interline_equation", + "image_path": "ac3c6044b8852109467d58ccdc24c96868de854235173601b5edb1f564158ec7.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 112, + 373, + 129.5 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 237, + 129.5, + 373, + 147.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 506, + 194 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "Proof. The proof idea is to use Corollary B.3 and chain rules (See Appendix I for chain rules in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 168, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 295, + 187 + ], + "score": 1.0, + "content": "Clarke’s sense). Applying the chain rule on", + "type": "text" + }, + { + "bbox": [ + 295, + 169, + 369, + 183 + ], + "score": 0.92, + "content": "t \\mapsto \\rho ^ { 2 } = \\| \\pmb \\theta \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 169, + 399, + 187 + ], + "score": 1.0, + "content": "yields", + "type": "text" + }, + { + "bbox": [ + 400, + 168, + 474, + 184 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } \\ = \\ - \\langle \\theta , h \\rangle } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 181, + 404, + 194 + ], + "spans": [ + { + "bbox": [ + 107, + 182, + 144, + 192 + ], + "score": 0.91, + "content": "h \\in \\partial ^ { \\circ } { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 181, + 178, + 194 + ], + "score": 1.0, + "content": "and a.e.", + "type": "text" + }, + { + "bbox": [ + 178, + 183, + 201, + 192 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 181, + 335, + 194 + ], + "score": 1.0, + "content": ". Then applying the chain rule on", + "type": "text" + }, + { + "bbox": [ + 335, + 183, + 365, + 192 + ], + "score": 0.88, + "content": "\\pmb \\theta \\mapsto \\mathcal L", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 181, + 404, + 194 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 156, + 505, + 194 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 198, + 459, + 233 + ], + "lines": [ + { + "bbox": [ + 150, + 198, + 459, + 233 + ], + "spans": [ + { + "bbox": [ + 150, + 198, + 459, + 233 + ], + "score": 0.93, + "content": "- \\partial ^ { \\circ } { \\mathcal { L } } \\subseteq \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) \\partial ^ { \\circ } q _ { n } = \\left\\{ \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) { h _ { n } } : { h _ { n } } \\in \\partial ^ { \\circ } q _ { n } \\right\\} .", + "type": "interline_equation", + "image_path": "9109f516043faf884f5cb79eabea68297191ae253402f05e16f9eaa28c025c6d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 150, + 198, + 459, + 209.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 150, + 209.66666666666666, + 459, + 221.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 150, + 221.33333333333331, + 459, + 232.99999999999997 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 423, + 254 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 425, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 181, + 256 + ], + "score": 1.0, + "content": "By Corollary B.3,", + "type": "text" + }, + { + "bbox": [ + 181, + 240, + 241, + 252 + ], + "score": 0.93, + "content": "\\langle \\pmb \\theta , \\pmb h _ { n } \\rangle = L q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 237, + 282, + 256 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + }, + { + "bbox": [ + 282, + 237, + 420, + 254 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 236, + 425, + 256 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 236, + 425, + 256 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 264, + 505, + 294 + ], + "lines": [ + { + "bbox": [ + 102, + 263, + 508, + 281 + ], + "spans": [ + { + "bbox": [ + 102, + 263, + 219, + 281 + ], + "score": 1.0, + "content": "For convenience, we define", + "type": "text" + }, + { + "bbox": [ + 220, + 265, + 350, + 280 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nu ( t ) : = \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 263, + 379, + 281 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 379, + 267, + 404, + 279 + ], + "score": 0.89, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 263, + 508, + 281 + ], + "score": 1.0, + "content": ". Then Theorem B.4 can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 276, + 288, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 171, + 297 + ], + "score": 1.0, + "content": "be rephrased as", + "type": "text" + }, + { + "bbox": [ + 171, + 279, + 229, + 295 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\frac { d \\rho ^ { 2 } } { d t } = L \\nu ( t ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 280, + 261, + 294 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 261, + 282, + 284, + 293 + ], + "score": 0.89, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 280, + 288, + 294 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 102, + 263, + 508, + 297 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 221, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 218, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 192, + 310 + ], + "score": 1.0, + "content": "Lemma B.5. For all", + "type": "text" + }, + { + "bbox": [ + 192, + 297, + 218, + 308 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 295, + 218, + 310 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 305, + 347, + 336 + ], + "lines": [ + { + "bbox": [ + 263, + 305, + 347, + 336 + ], + "spans": [ + { + "bbox": [ + 263, + 305, + 347, + 336 + ], + "score": 0.93, + "content": "\\nu ( t ) \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } .", + "type": "interline_equation", + "image_path": "3e0297bdd5cee400141a2f7def5edba66393efe1503e7488e9133cb6347f0a36.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 263, + 305, + 347, + 336 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 438, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 439, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 203, + 363 + ], + "score": 1.0, + "content": "Proof. By Lemma A.5,", + "type": "text" + }, + { + "bbox": [ + 203, + 347, + 262, + 361 + ], + "score": 0.93, + "content": "q _ { n } \\geq g ( \\log { \\frac { 1 } { \\mathcal { L } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 345, + 290, + 363 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 291, + 348, + 324, + 360 + ], + "score": 0.93, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 345, + 439, + 363 + ], + "score": 1.0, + "content": ". Then by Assumption (B3),", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 345, + 439, + 363 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 365, + 408, + 396 + ], + "lines": [ + { + "bbox": [ + 202, + 365, + 408, + 396 + ], + "spans": [ + { + "bbox": [ + 202, + 365, + 408, + 396 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q _ { n } ) q _ { n } \\geq f ^ { \\prime } ( g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ) \\cdot g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) = { \\frac { g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } { g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } .", + "type": "interline_equation", + "image_path": "ecaf2c6b9571079c80d2249dbf4350e91c9ad62abf14b593e8e7b88e63685840.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 202, + 365, + 408, + 396 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 330, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 330, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 261, + 414 + ], + "score": 1.0, + "content": "Combining this with the definitions of", + "type": "text" + }, + { + "bbox": [ + 261, + 399, + 279, + 412 + ], + "score": 0.92, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 398, + 297, + 414 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 298, + 400, + 306, + 410 + ], + "score": 0.83, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 398, + 330, + 414 + ], + "score": 1.0, + "content": "gives", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 398, + 330, + 414 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 416, + 446, + 450 + ], + "lines": [ + { + "bbox": [ + 164, + 416, + 446, + 450 + ], + "spans": [ + { + "bbox": [ + 164, + 416, + 446, + 450 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nu ( t ) = \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } \\geq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } \\displaystyle \\frac { g ( \\log \\frac { 1 } { \\mathscr { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathscr { L } } ) } = \\displaystyle \\frac { g ( \\log \\frac { 1 } { \\mathscr { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathscr { L } } ) } \\mathscr { L } . } \\end{array}", + "type": "interline_equation", + "image_path": "21914080f13a4faa6dd5c62496274d03818eccbde3f4225e09449f6f978b16f0.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 164, + 416, + 446, + 450 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 506, + 556 + ], + "lines": [ + { + "bbox": [ + 101, + 473, + 509, + 497 + ], + "spans": [ + { + "bbox": [ + 101, + 473, + 240, + 497 + ], + "score": 1.0, + "content": "Proof for Lemma B.1. Note that", + "type": "text" + }, + { + "bbox": [ + 241, + 475, + 361, + 493 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho = \\frac { 1 } { 2 \\rho ^ { 2 } } \\frac { d \\rho ^ { 2 } } { d t } = L \\frac { \\nu ( t ) } { \\rho ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 473, + 509, + 497 + ], + "score": 1.0, + "content": "by Theorem B.4. Then it simply", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 228, + 506 + ], + "score": 1.0, + "content": "follows from Lemma B.5 that", + "type": "text" + }, + { + "bbox": [ + 229, + 492, + 279, + 505 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho > 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 491, + 311, + 506 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 311, + 494, + 337, + 504 + ], + "score": 0.89, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 491, + 506, + 506 + ], + "score": 1.0, + "content": ". For the second inequality, we first prove", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 504, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 124, + 519 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 505, + 321, + 518 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\log \\tilde { \\gamma } = \\log \\left( \\ell ^ { - 1 } ( \\mathcal { L } ) / \\rho ^ { L } \\right) = \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\rho ^ { L } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 504, + 377, + 519 + ], + "score": 1.0, + "content": "exists for all", + "type": "text" + }, + { + "bbox": [ + 377, + 506, + 406, + 517 + ], + "score": 0.88, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 504, + 412, + 519 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 412, + 505, + 432, + 518 + ], + "score": 0.86, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 504, + 505, + 519 + ], + "score": 1.0, + "content": "is non-increasing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 516, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 127, + 533 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 520, + 132, + 529 + ], + "score": 0.69, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 516, + 150, + 533 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 151, + 518, + 214, + 531 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t ) < e ^ { - f ( b _ { f } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 516, + 243, + 533 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 243, + 519, + 270, + 530 + ], + "score": 0.89, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 516, + 360, + 533 + ], + "score": 1.0, + "content": ". This implies that (1)", + "type": "text" + }, + { + "bbox": [ + 360, + 518, + 384, + 532 + ], + "score": 0.86, + "content": "\\log { \\frac { 1 } { \\mathcal { L } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 516, + 495, + 533 + ], + "score": 1.0, + "content": "is always in the domain of", + "type": "text" + }, + { + "bbox": [ + 495, + 521, + 501, + 530 + ], + "score": 0.71, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 516, + 505, + 533 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 529, + 504, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 120, + 546 + ], + "score": 1.0, + "content": "(2)", + "type": "text" + }, + { + "bbox": [ + 121, + 532, + 147, + 544 + ], + "score": 0.9, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 529, + 193, + 546 + ], + "score": 1.0, + "content": "(otherwise", + "type": "text" + }, + { + "bbox": [ + 194, + 531, + 295, + 543 + ], + "score": 0.91, + "content": "\\mathcal { L } \\geq N e ^ { - f ( 0 ) } > e ^ { - f ( b _ { f } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 529, + 428, + 546 + ], + "score": 1.0, + "content": ", contradicting (B4)). Therefore,", + "type": "text" + }, + { + "bbox": [ + 428, + 531, + 504, + 545 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\widetilde { \\gamma } : = g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\rho ^ { L } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 542, + 425, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 248, + 557 + ], + "score": 1.0, + "content": "exists and is always positive for all", + "type": "text" + }, + { + "bbox": [ + 248, + 544, + 273, + 555 + ], + "score": 0.88, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 542, + 398, + 557 + ], + "score": 1.0, + "content": ", which proves the existence of", + "type": "text" + }, + { + "bbox": [ + 398, + 544, + 420, + 555 + ], + "score": 0.9, + "content": "\\log { \\tilde { \\gamma } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 542, + 425, + 557 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 101, + 473, + 509, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 281, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 282, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 282, + 573 + ], + "score": 1.0, + "content": "By the chain rule and Lemma B.5, we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 559, + 282, + 573 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 576, + 480, + 664 + ], + "lines": [ + { + "bbox": [ + 131, + 576, + 480, + 664 + ], + "spans": [ + { + "bbox": [ + 131, + 576, + 480, + 664 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } & { \\frac { d } { d t } \\log \\tilde { \\gamma } = \\frac { d } { d t } \\left( \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\right) - L \\log \\rho \\right) = \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\cdot \\frac { 1 } { \\mathcal { L } } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - L ^ { 2 } \\cdot \\frac { \\nu ( t ) } { \\rho ^ { 2 } } } & \\\\ & { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - L ^ { 2 } \\cdot \\frac { \\nu ( t ) } { \\rho ^ { 2 } } } & \\\\ & { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } - \\frac { L ^ { 2 } \\nu ( t ) ^ { 2 } } { \\rho ^ { 2 } } \\right) . } & \\end{array}", + "type": "interline_equation", + "image_path": "63b24edc3f36a85ec0a4c3e33860a27487032315f67b9dc1305ab9c184118bf3.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 131, + 576, + 480, + 605.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 131, + 605.3333333333334, + 480, + 634.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 131, + 634.6666666666667, + 480, + 664.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 102, + 663, + 504, + 695 + ], + "spans": [ + { + "bbox": [ + 102, + 663, + 178, + 695 + ], + "score": 1.0, + "content": "On the one hand, by Theorem B.4.", + "type": "text" + }, + { + "bbox": [ + 178, + 667, + 239, + 684 + ], + "score": 0.94, + "content": "\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } = \\left. \\frac { d \\pmb { \\theta } } { d t } \\right. _ { 2 } ^ { 2 } } \\\\ { \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 663, + 271, + 695 + ], + "score": 1.0, + "content": "for a.e. se toget", + "type": "text" + }, + { + "bbox": [ + 272, + 671, + 295, + 681 + ], + "score": 0.88, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 663, + 434, + 695 + ], + "score": 1.0, + "content": "by Lemma I.3; on the other hand, lds", + "type": "text" + }, + { + "bbox": [ + 434, + 669, + 504, + 684 + ], + "score": 0.93, + "content": "\\begin{array} { r } { L \\nu ( t ) = \\left. \\theta , \\frac { d \\theta } { d t } \\right. } \\end{array}", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 102, + 663, + 504, + 695 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 697, + 447, + 731 + ], + "lines": [ + { + "bbox": [ + 164, + 697, + 447, + 731 + ], + "spans": [ + { + "bbox": [ + 164, + 697, + 447, + 731 + ], + "score": 0.93, + "content": "{ \\frac { d } { d t } } \\log { \\tilde { \\gamma } } \\geq { \\frac { 1 } { \\nu ( t ) } } \\left( \\left\\| { \\frac { d \\theta } { d t } } \\right\\| _ { 2 } ^ { 2 } - \\left. { \\hat { \\theta } } , { \\frac { d \\theta } { d t } } \\right. ^ { 2 } \\right) = { \\frac { 1 } { \\nu ( t ) } } \\left\\| ( I - { \\hat { \\theta } } { \\hat { \\theta } } ^ { \\top } ) { \\frac { d \\theta } { d t } } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "f88bde66b96035a463935ca77b3e84afdbaba801adc29f134a065671b81f43ec.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 164, + 697, + 447, + 731 + ], + "spans": [], + "index": 30 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 79, + 375, + 97 + ], + "lines": [ + { + "bbox": [ + 102, + 77, + 379, + 101 + ], + "spans": [ + { + "bbox": [ + 102, + 77, + 180, + 101 + ], + "score": 1.0, + "content": "By the chain rule,", + "type": "text" + }, + { + "bbox": [ + 181, + 80, + 267, + 97 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d \\hat { \\pmb { \\theta } } } { d t } = \\frac { 1 } { \\rho } ( \\pmb { I } - \\hat { \\pmb { \\theta } } \\hat { \\pmb { \\theta } } ^ { \\top } ) \\frac { d \\pmb { \\theta } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 77, + 299, + 101 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 299, + 83, + 322, + 94 + ], + "score": 0.88, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 77, + 379, + 101 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 99, + 412, + 136 + ], + "lines": [ + { + "bbox": [ + 198, + 99, + 412, + 136 + ], + "spans": [ + { + "bbox": [ + 198, + 99, + 412, + 136 + ], + "score": 0.93, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq \\frac { \\rho ^ { 2 } } { \\nu ( t ) } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } = L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "ebf30250bc461ae48bbd6db054e790d690872f7ce0600982803024f156831d8d.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 99, + 412, + 117.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 198, + 117.5, + 412, + 136.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 161, + 251, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 161, + 250, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 250, + 173 + ], + "score": 1.0, + "content": "B.2 PROOF FOR PROPOSITION 3", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 506, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 417, + 195 + ], + "score": 1.0, + "content": "To prove the third proposition, we prove the following lemma to show that", + "type": "text" + }, + { + "bbox": [ + 417, + 183, + 449, + 192 + ], + "score": 0.9, + "content": "\\mathcal { L } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "by giving an", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 174, + 205 + ], + "score": 1.0, + "content": "upper bound for", + "type": "text" + }, + { + "bbox": [ + 174, + 194, + 182, + 203 + ], + "score": 0.8, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 193, + 212, + 205 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 213, + 194, + 221, + 203 + ], + "score": 0.8, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 193, + 335, + 205 + ], + "score": 1.0, + "content": "can never be 0 for bounded", + "type": "text" + }, + { + "bbox": [ + 335, + 195, + 342, + 205 + ], + "score": 0.61, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 193, + 345, + 205 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 345, + 193, + 376, + 204 + ], + "score": 0.79, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 193, + 442, + 205 + ], + "score": 1.0, + "content": "directly implies", + "type": "text" + }, + { + "bbox": [ + 443, + 194, + 483, + 204 + ], + "score": 0.9, + "content": "\\rho + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 193, + 505, + 205 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 204, + 491, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 143, + 217 + ], + "score": 1.0, + "content": "showing", + "type": "text" + }, + { + "bbox": [ + 143, + 204, + 193, + 216 + ], + "score": 0.93, + "content": "| \\bar { \\gamma } - \\tilde { \\gamma } | \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 204, + 287, + 217 + ], + "score": 1.0, + "content": ", we only need to apply", + "type": "text" + }, + { + "bbox": [ + 288, + 205, + 300, + 216 + ], + "score": 0.58, + "content": "( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 204, + 459, + 217 + ], + "score": 1.0, + "content": "in Lemma A.5, which shows this when", + "type": "text" + }, + { + "bbox": [ + 459, + 205, + 487, + 214 + ], + "score": 0.89, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 204, + 491, + 217 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 221, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 218, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 192, + 231 + ], + "score": 1.0, + "content": "Lemma B.6. For all", + "type": "text" + }, + { + "bbox": [ + 193, + 218, + 218, + 229 + ], + "score": 0.91, + "content": "t > t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 230, + 464, + 260 + ], + "lines": [ + { + "bbox": [ + 147, + 230, + 464, + 260 + ], + "spans": [ + { + "bbox": [ + 147, + 230, + 464, + 260 + ], + "score": 0.91, + "content": "G ( 1 / \\mathcal { L } ( t ) ) \\geq L ^ { 2 } \\widetilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } ( t - t _ { 0 } ) \\qquad f o r G ( x ) : = \\int _ { 1 / \\mathcal { L } ( t _ { 0 } ) } ^ { x } \\frac { g ^ { \\prime } ( \\log u ) ^ { 2 } } { g ( \\log u ) ^ { 2 - 2 / L } } d u .", + "type": "interline_equation", + "image_path": "28b25dce057457ac09c7824cc0ec94de27dcfe36225c937c7724662c01a68540.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 147, + 230, + 464, + 240.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 147, + 240.0, + 464, + 250.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 147, + 250.0, + 464, + 260.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 306, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 261, + 306, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 150, + 276 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 150, + 262, + 190, + 275 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 261, + 208, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 209, + 262, + 259, + 275 + ], + "score": 0.88, + "content": "\\rho ( t ) \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 261, + 272, + 276 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 272, + 263, + 302, + 273 + ], + "score": 0.82, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 261, + 306, + 276 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 333, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 333, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 333, + 299 + ], + "score": 1.0, + "content": "Proof for Lemma B.6. By Lemma I.3 and Theorem B.4,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 300, + 395, + 330 + ], + "lines": [ + { + "bbox": [ + 214, + 300, + 395, + 330 + ], + "spans": [ + { + "bbox": [ + 214, + 300, + 395, + 330 + ], + "score": 0.94, + "content": "- { \\frac { d { \\mathcal { L } } } { d t } } = \\left\\| { \\frac { d \\pmb { \\theta } } { d t } } \\right\\| _ { 2 } ^ { 2 } \\geq \\left. { \\hat { \\pmb { \\theta } } } , { \\frac { d \\pmb { \\theta } } { d t } } \\right. ^ { 2 } = L ^ { 2 } \\cdot { \\frac { \\nu ( t ) ^ { 2 } } { \\rho ^ { 2 } } }", + "type": "interline_equation", + "image_path": "49aded740ec06573e6614b08c4628bebfd2d5809f147ca242767773616a537af.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 214, + 300, + 395, + 315.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 214, + 315.0, + 395, + 330.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 504, + 358 + ], + "lines": [ + { + "bbox": [ + 103, + 330, + 501, + 352 + ], + "spans": [ + { + "bbox": [ + 103, + 330, + 247, + 352 + ], + "score": 1.0, + "content": "Using Lemma B.5 to lower bound", + "type": "text" + }, + { + "bbox": [ + 248, + 338, + 254, + 346 + ], + "score": 0.8, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 330, + 313, + 352 + ], + "score": 1.0, + "content": "and replacing", + "type": "text" + }, + { + "bbox": [ + 313, + 338, + 320, + 348 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 330, + 342, + 352 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 343, + 332, + 412, + 349 + ], + "score": 0.93, + "content": "\\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\tilde { \\gamma } \\right) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 330, + 494, + 352 + ], + "score": 1.0, + "content": "by the definition of", + "type": "text" + }, + { + "bbox": [ + 495, + 337, + 501, + 348 + ], + "score": 0.84, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 346, + 142, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 142, + 359 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 358, + 473, + 391 + ], + "lines": [ + { + "bbox": [ + 136, + 358, + 473, + 391 + ], + "spans": [ + { + "bbox": [ + 136, + 358, + 473, + 391 + ], + "score": 0.93, + "content": "- \\frac { d \\mathcal { L } } { d t } \\geq L ^ { 2 } \\cdot \\bigg ( \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } \\bigg ) ^ { 2 } \\cdot \\bigg ( \\frac { \\widetilde { \\gamma } ( t ) } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\bigg ) ^ { 2 / L } \\geq L ^ { 2 } \\widetilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\cdot \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 - 2 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\mathcal { L } ,", + "type": "interline_equation", + "image_path": "86ca346aa77772599b25aa77d54282aa3bf8cea82f6e930d3a4bc68b7efff397.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 136, + 358, + 473, + 369.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 136, + 369.0, + 473, + 380.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 136, + 380.0, + 473, + 391.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 471, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 468, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 307, + 406 + ], + "score": 1.0, + "content": "where the last inequality uses the monotonicity of", + "type": "text" + }, + { + "bbox": [ + 308, + 393, + 314, + 404 + ], + "score": 0.85, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 390, + 442, + 406 + ], + "score": 1.0, + "content": ". So the following holds for a.e.", + "type": "text" + }, + { + "bbox": [ + 442, + 393, + 468, + 403 + ], + "score": 0.89, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 406, + 383, + 436 + ], + "lines": [ + { + "bbox": [ + 228, + 406, + 383, + 436 + ], + "spans": [ + { + "bbox": [ + 228, + 406, + 383, + 436 + ], + "score": 0.94, + "content": "\\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 - 2 / L } } \\cdot \\frac { d } { d t } \\frac { 1 } { \\mathcal { L } } \\ge L ^ { 2 } \\tilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } .", + "type": "interline_equation", + "image_path": "aef9ca675b70538a2a335a8f196ae2fcc98b67ee53e099398a97e520273bbca5.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 228, + 406, + 383, + 421.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 228, + 421.0, + 383, + 436.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 346, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 346, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 229, + 450 + ], + "score": 1.0, + "content": "Integrating on both sides from", + "type": "text" + }, + { + "bbox": [ + 230, + 439, + 239, + 449 + ], + "score": 0.87, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 437, + 250, + 450 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 250, + 439, + 255, + 448 + ], + "score": 0.79, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 437, + 346, + 450 + ], + "score": 1.0, + "content": ", we can conclude that", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 451, + 370, + 467 + ], + "lines": [ + { + "bbox": [ + 241, + 451, + 370, + 467 + ], + "spans": [ + { + "bbox": [ + 241, + 451, + 370, + 467 + ], + "score": 0.93, + "content": "G ( 1 / \\mathcal { L } ) \\geq L ^ { 2 } \\tilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } ( t - t _ { 0 } ) .", + "type": "interline_equation", + "image_path": "d72e2b18dad626c7ea80f7b77b588e5ee43c630c2afaca93c9909af530bf576a.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 241, + 451, + 370, + 467 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 469, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 146, + 482 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 469, + 164, + 481 + ], + "score": 0.9, + "content": "1 / \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 468, + 251, + 482 + ], + "score": 1.0, + "content": "is non-decreasing. If", + "type": "text" + }, + { + "bbox": [ + 252, + 469, + 270, + 481 + ], + "score": 0.9, + "content": "1 / \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 468, + 340, + 482 + ], + "score": 1.0, + "content": "does not grow to", + "type": "text" + }, + { + "bbox": [ + 341, + 470, + 360, + 480 + ], + "score": 0.88, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 468, + 434, + 482 + ], + "score": 1.0, + "content": ", then neither does", + "type": "text" + }, + { + "bbox": [ + 435, + 469, + 468, + 481 + ], + "score": 0.95, + "content": "G ( 1 / \\mathcal { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 468, + 505, + 482 + ], + "score": 1.0, + "content": ". But the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 480, + 359, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 165, + 493 + ], + "score": 1.0, + "content": "RHS grows to", + "type": "text" + }, + { + "bbox": [ + 165, + 482, + 184, + 491 + ], + "score": 0.88, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 480, + 326, + 493 + ], + "score": 1.0, + "content": ", which leads to a contradiction. 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So", + "type": "text" + }, + { + "bbox": [ + 303, + 498, + 343, + 509 + ], + "score": 0.82, + "content": "\\rho \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 495, + 347, + 511 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 524, + 381, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 382, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 382, + 538 + ], + "score": 1.0, + "content": "C CONVERGENCE TO THE MAX-MARGIN SOLUTION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 548, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 104, + 546, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 323, + 562 + ], + "score": 1.0, + "content": "In this section, we analyze the convergent direction of", + "type": "text" + }, + { + "bbox": [ + 324, + 549, + 330, + 559 + ], + "score": 0.78, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 546, + 506, + 562 + ], + "score": 1.0, + "content": "and prove Theorem A.8 and A.9, assuming", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 560, + 308, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 308, + 571 + ], + "score": 1.0, + "content": "(A1), (A2), (B3), (B4) as mentioned in Section A.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 108, + 576, + 504, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 295, + 592 + ], + "score": 1.0, + "content": "We follow the notations in Section 5 to define", + "type": "text" + }, + { + "bbox": [ + 295, + 577, + 340, + 590 + ], + "score": 0.93, + "content": "\\rho : = \\| \\pmb \\theta \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 574, + 359, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 576, + 438, + 593 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\| \\pmb \\theta \\| _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 574, + 506, + 592 + ], + "score": 1.0, + "content": ", and sometimes", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 591, + 286, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 207, + 603 + ], + "score": 1.0, + "content": "we view the functions of", + "type": "text" + }, + { + "bbox": [ + 208, + 591, + 214, + 601 + ], + "score": 0.76, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 591, + 277, + 603 + ], + "score": 1.0, + "content": "as functions of", + "type": "text" + }, + { + "bbox": [ + 277, + 592, + 282, + 601 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 591, + 286, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 106, + 615, + 304, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 304, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 304, + 628 + ], + "score": 1.0, + "content": "C.1 PRELIMINARIES FOR KKT CONDITIONS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 504, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 504, + 649 + ], + "score": 1.0, + "content": "We first review the definition of Karush-Kuhn-Tucker (KKT) conditions for non-smooth optimiza-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 647, + 307, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 307, + 659 + ], + "score": 1.0, + "content": "tion problems following from (Dutta et al., 2013).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 354, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 354, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 318, + 677 + ], + "score": 1.0, + "content": "Consider the following optimization problem (P) for", + "type": "text" + }, + { + "bbox": [ + 318, + 663, + 349, + 674 + ], + "score": 0.9, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 662, + 354, + 677 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 678, + 370, + 706 + ], + "lines": [ + { + "bbox": [ + 240, + 678, + 370, + 706 + ], + "spans": [ + { + "bbox": [ + 240, + 678, + 370, + 706 + ], + "score": 0.9, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { { } f ( { \\pmb x } ) } \\\\ { \\mathrm { s . t . } } & { { } g _ { n } ( { \\pmb x } ) \\le 0 \\qquad \\forall n \\in [ N ] } \\end{array}", + "type": "interline_equation", + "image_path": "35be983d7db429f620f565512bad71d74b7d66612203f761ae8e1879acade459.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 240, + 678, + 370, + 706 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 502, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 134, + 722 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 709, + 235, + 721 + ], + "score": 0.92, + "content": "f , g _ { 1 } , \\ldots , g _ { n } : \\mathbb { R } ^ { d } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 708, + 418, + 722 + ], + "score": 1.0, + "content": "are locally Lipschitz functions. 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So we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 102, + 77, + 379, + 101 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 99, + 412, + 136 + ], + "lines": [ + { + "bbox": [ + 198, + 99, + 412, + 136 + ], + "spans": [ + { + "bbox": [ + 198, + 99, + 412, + 136 + ], + "score": 0.93, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq \\frac { \\rho ^ { 2 } } { \\nu ( t ) } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } = L \\left( \\frac { d } { d t } \\log \\rho \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { \\pmb { \\theta } } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "ebf30250bc461ae48bbd6db054e790d690872f7ce0600982803024f156831d8d.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 99, + 412, + 117.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 198, + 117.5, + 412, + 136.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 161, + 251, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 161, + 250, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 250, + 173 + ], + "score": 1.0, + "content": "B.2 PROOF FOR PROPOSITION 3", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 506, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 417, + 195 + ], + "score": 1.0, + "content": "To prove the third proposition, we prove the following lemma to show that", + "type": "text" + }, + { + "bbox": [ + 417, + 183, + 449, + 192 + ], + "score": 0.9, + "content": "\\mathcal { L } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "by giving an", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 174, + 205 + ], + "score": 1.0, + "content": "upper bound for", + "type": "text" + }, + { + "bbox": [ + 174, + 194, + 182, + 203 + ], + "score": 0.8, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 193, + 212, + 205 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 213, + 194, + 221, + 203 + ], + "score": 0.8, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 193, + 335, + 205 + ], + "score": 1.0, + "content": "can never be 0 for bounded", + "type": "text" + }, + { + "bbox": [ + 335, + 195, + 342, + 205 + ], + "score": 0.61, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 193, + 345, + 205 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 345, + 193, + 376, + 204 + ], + "score": 0.79, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 193, + 442, + 205 + ], + "score": 1.0, + "content": "directly implies", + "type": "text" + }, + { + "bbox": [ + 443, + 194, + 483, + 204 + ], + "score": 0.9, + "content": "\\rho + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 193, + 505, + 205 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 204, + 491, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 143, + 217 + ], + "score": 1.0, + "content": "showing", + "type": "text" + }, + { + "bbox": [ + 143, + 204, + 193, + 216 + ], + "score": 0.93, + "content": "| \\bar { \\gamma } - \\tilde { \\gamma } | \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 204, + 287, + 217 + ], + "score": 1.0, + "content": ", we only need to apply", + "type": "text" + }, + { + "bbox": [ + 288, + 205, + 300, + 216 + ], + "score": 0.58, + "content": "( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 204, + 459, + 217 + ], + "score": 1.0, + "content": "in Lemma A.5, which shows this when", + "type": "text" + }, + { + "bbox": [ + 459, + 205, + 487, + 214 + ], + "score": 0.89, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 204, + 491, + 217 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 181, + 506, + 217 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 221, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 218, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 192, + 231 + ], + "score": 1.0, + "content": "Lemma B.6. For all", + "type": "text" + }, + { + "bbox": [ + 193, + 218, + 218, + 229 + ], + "score": 0.91, + "content": "t > t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 216, + 218, + 231 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 230, + 464, + 260 + ], + "lines": [ + { + "bbox": [ + 147, + 230, + 464, + 260 + ], + "spans": [ + { + "bbox": [ + 147, + 230, + 464, + 260 + ], + "score": 0.91, + "content": "G ( 1 / \\mathcal { L } ( t ) ) \\geq L ^ { 2 } \\widetilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } ( t - t _ { 0 } ) \\qquad f o r G ( x ) : = \\int _ { 1 / \\mathcal { L } ( t _ { 0 } ) } ^ { x } \\frac { g ^ { \\prime } ( \\log u ) ^ { 2 } } { g ( \\log u ) ^ { 2 - 2 / L } } d u .", + "type": "interline_equation", + "image_path": "28b25dce057457ac09c7824cc0ec94de27dcfe36225c937c7724662c01a68540.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 147, + 230, + 464, + 240.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 147, + 240.0, + 464, + 250.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 147, + 250.0, + 464, + 260.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 306, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 261, + 306, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 150, + 276 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 150, + 262, + 190, + 275 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 261, + 208, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 209, + 262, + 259, + 275 + ], + "score": 0.88, + "content": "\\rho ( t ) \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 261, + 272, + 276 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 272, + 263, + 302, + 273 + ], + "score": 0.82, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 261, + 306, + 276 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 261, + 306, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 333, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 333, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 333, + 299 + ], + "score": 1.0, + "content": "Proof for Lemma B.6. By Lemma I.3 and Theorem B.4,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 285, + 333, + 299 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 300, + 395, + 330 + ], + "lines": [ + { + "bbox": [ + 214, + 300, + 395, + 330 + ], + "spans": [ + { + "bbox": [ + 214, + 300, + 395, + 330 + ], + "score": 0.94, + "content": "- { \\frac { d { \\mathcal { L } } } { d t } } = \\left\\| { \\frac { d \\pmb { \\theta } } { d t } } \\right\\| _ { 2 } ^ { 2 } \\geq \\left. { \\hat { \\pmb { \\theta } } } , { \\frac { d \\pmb { \\theta } } { d t } } \\right. ^ { 2 } = L ^ { 2 } \\cdot { \\frac { \\nu ( t ) ^ { 2 } } { \\rho ^ { 2 } } }", + "type": "interline_equation", + "image_path": "49aded740ec06573e6614b08c4628bebfd2d5809f147ca242767773616a537af.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 214, + 300, + 395, + 315.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 214, + 315.0, + 395, + 330.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 504, + 358 + ], + "lines": [ + { + "bbox": [ + 103, + 330, + 501, + 352 + ], + "spans": [ + { + "bbox": [ + 103, + 330, + 247, + 352 + ], + "score": 1.0, + "content": "Using Lemma B.5 to lower bound", + "type": "text" + }, + { + "bbox": [ + 248, + 338, + 254, + 346 + ], + "score": 0.8, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 330, + 313, + 352 + ], + "score": 1.0, + "content": "and replacing", + "type": "text" + }, + { + "bbox": [ + 313, + 338, + 320, + 348 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 330, + 342, + 352 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 343, + 332, + 412, + 349 + ], + "score": 0.93, + "content": "\\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\tilde { \\gamma } \\right) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 330, + 494, + 352 + ], + "score": 1.0, + "content": "by the definition of", + "type": "text" + }, + { + "bbox": [ + 495, + 337, + 501, + 348 + ], + "score": 0.84, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 346, + 142, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 142, + 359 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 103, + 330, + 501, + 359 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 358, + 473, + 391 + ], + "lines": [ + { + "bbox": [ + 136, + 358, + 473, + 391 + ], + "spans": [ + { + "bbox": [ + 136, + 358, + 473, + 391 + ], + "score": 0.93, + "content": "- \\frac { d \\mathcal { L } } { d t } \\geq L ^ { 2 } \\cdot \\bigg ( \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } \\bigg ) ^ { 2 } \\cdot \\bigg ( \\frac { \\widetilde { \\gamma } ( t ) } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\bigg ) ^ { 2 / L } \\geq L ^ { 2 } \\widetilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\cdot \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 - 2 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\mathcal { L } ,", + "type": "interline_equation", + "image_path": "86ca346aa77772599b25aa77d54282aa3bf8cea82f6e930d3a4bc68b7efff397.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 136, + 358, + 473, + 369.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 136, + 369.0, + 473, + 380.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 136, + 380.0, + 473, + 391.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 471, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 468, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 307, + 406 + ], + "score": 1.0, + "content": "where the last inequality uses the monotonicity of", + "type": "text" + }, + { + "bbox": [ + 308, + 393, + 314, + 404 + ], + "score": 0.85, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 390, + 442, + 406 + ], + "score": 1.0, + "content": ". So the following holds for a.e.", + "type": "text" + }, + { + "bbox": [ + 442, + 393, + 468, + 403 + ], + "score": 0.89, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 390, + 468, + 406 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 406, + 383, + 436 + ], + "lines": [ + { + "bbox": [ + 228, + 406, + 383, + 436 + ], + "spans": [ + { + "bbox": [ + 228, + 406, + 383, + 436 + ], + "score": 0.94, + "content": "\\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 - 2 / L } } \\cdot \\frac { d } { d t } \\frac { 1 } { \\mathcal { L } } \\ge L ^ { 2 } \\tilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } .", + "type": "interline_equation", + "image_path": "aef9ca675b70538a2a335a8f196ae2fcc98b67ee53e099398a97e520273bbca5.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 228, + 406, + 383, + 421.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 228, + 421.0, + 383, + 436.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 346, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 346, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 229, + 450 + ], + "score": 1.0, + "content": "Integrating on both sides from", + "type": "text" + }, + { + "bbox": [ + 230, + 439, + 239, + 449 + ], + "score": 0.87, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 437, + 250, + 450 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 250, + 439, + 255, + 448 + ], + "score": 0.79, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 437, + 346, + 450 + ], + "score": 1.0, + "content": ", we can conclude that", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 437, + 346, + 450 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 451, + 370, + 467 + ], + "lines": [ + { + "bbox": [ + 241, + 451, + 370, + 467 + ], + "spans": [ + { + "bbox": [ + 241, + 451, + 370, + 467 + ], + "score": 0.93, + "content": "G ( 1 / \\mathcal { L } ) \\geq L ^ { 2 } \\tilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } ( t - t _ { 0 } ) .", + "type": "interline_equation", + "image_path": "d72e2b18dad626c7ea80f7b77b588e5ee43c630c2afaca93c9909af530bf576a.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 241, + 451, + 370, + 467 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 469, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 146, + 482 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 469, + 164, + 481 + ], + "score": 0.9, + "content": "1 / \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 468, + 251, + 482 + ], + "score": 1.0, + "content": "is non-decreasing. If", + "type": "text" + }, + { + "bbox": [ + 252, + 469, + 270, + 481 + ], + "score": 0.9, + "content": "1 / \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 468, + 340, + 482 + ], + "score": 1.0, + "content": "does not grow to", + "type": "text" + }, + { + "bbox": [ + 341, + 470, + 360, + 480 + ], + "score": 0.88, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 468, + 434, + 482 + ], + "score": 1.0, + "content": ", then neither does", + "type": "text" + }, + { + "bbox": [ + 435, + 469, + 468, + 481 + ], + "score": 0.95, + "content": "G ( 1 / \\mathcal { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 468, + 505, + 482 + ], + "score": 1.0, + "content": ". But the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 480, + 359, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 165, + 493 + ], + "score": 1.0, + "content": "RHS grows to", + "type": "text" + }, + { + "bbox": [ + 165, + 482, + 184, + 491 + ], + "score": 0.88, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 480, + 326, + 493 + ], + "score": 1.0, + "content": ", which leads to a contradiction. So", + "type": "text" + }, + { + "bbox": [ + 326, + 481, + 354, + 491 + ], + "score": 0.9, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 480, + 359, + 493 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 468, + 505, + 493 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 346, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 347, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 143, + 511 + ], + "score": 1.0, + "content": "To make", + "type": "text" + }, + { + "bbox": [ + 144, + 497, + 172, + 507 + ], + "score": 0.86, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 495, + 175, + 511 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 176, + 499, + 195, + 509 + ], + "score": 0.87, + "content": "q _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 495, + 266, + 511 + ], + "score": 1.0, + "content": "must converge to", + "type": "text" + }, + { + "bbox": [ + 267, + 499, + 285, + 508 + ], + "score": 0.83, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 495, + 302, + 511 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 303, + 498, + 343, + 509 + ], + "score": 0.82, + "content": "\\rho \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 495, + 347, + 511 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 495, + 347, + 511 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 524, + 381, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 382, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 382, + 538 + ], + "score": 1.0, + "content": "C CONVERGENCE TO THE MAX-MARGIN SOLUTION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 548, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 104, + 546, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 323, + 562 + ], + "score": 1.0, + "content": "In this section, we analyze the convergent direction of", + "type": "text" + }, + { + "bbox": [ + 324, + 549, + 330, + 559 + ], + "score": 0.78, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 546, + 506, + 562 + ], + "score": 1.0, + "content": "and prove Theorem A.8 and A.9, assuming", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 560, + 308, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 308, + 571 + ], + "score": 1.0, + "content": "(A1), (A2), (B3), (B4) as mentioned in Section A.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 546, + 506, + 571 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 576, + 504, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 295, + 592 + ], + "score": 1.0, + "content": "We follow the notations in Section 5 to define", + "type": "text" + }, + { + "bbox": [ + 295, + 577, + 340, + 590 + ], + "score": 0.93, + "content": "\\rho : = \\| \\pmb \\theta \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 574, + 359, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 576, + 438, + 593 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\| \\pmb \\theta \\| _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 574, + 506, + 592 + ], + "score": 1.0, + "content": ", and sometimes", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 591, + 286, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 207, + 603 + ], + "score": 1.0, + "content": "we view the functions of", + "type": "text" + }, + { + "bbox": [ + 208, + 591, + 214, + 601 + ], + "score": 0.76, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 591, + 277, + 603 + ], + "score": 1.0, + "content": "as functions of", + "type": "text" + }, + { + "bbox": [ + 277, + 592, + 282, + 601 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 591, + 286, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 574, + 506, + 603 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 615, + 304, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 304, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 304, + 628 + ], + "score": 1.0, + "content": "C.1 PRELIMINARIES FOR KKT CONDITIONS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 504, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 504, + 649 + ], + "score": 1.0, + "content": "We first review the definition of Karush-Kuhn-Tucker (KKT) conditions for non-smooth optimiza-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 647, + 307, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 307, + 659 + ], + "score": 1.0, + "content": "tion problems following from (Dutta et al., 2013).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 635, + 504, + 659 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 354, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 354, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 318, + 677 + ], + "score": 1.0, + "content": "Consider the following optimization problem (P) for", + "type": "text" + }, + { + "bbox": [ + 318, + 663, + 349, + 674 + ], + "score": 0.9, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 662, + 354, + 677 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 662, + 354, + 677 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 678, + 370, + 706 + ], + "lines": [ + { + "bbox": [ + 240, + 678, + 370, + 706 + ], + "spans": [ + { + "bbox": [ + 240, + 678, + 370, + 706 + ], + "score": 0.9, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { { } f ( { \\pmb x } ) } \\\\ { \\mathrm { s . t . } } & { { } g _ { n } ( { \\pmb x } ) \\le 0 \\qquad \\forall n \\in [ N ] } \\end{array}", + "type": "interline_equation", + "image_path": "35be983d7db429f620f565512bad71d74b7d66612203f761ae8e1879acade459.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 240, + 678, + 370, + 706 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 502, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 134, + 722 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 709, + 235, + 721 + ], + "score": 0.92, + "content": "f , g _ { 1 } , \\ldots , g _ { n } : \\mathbb { R } ^ { d } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 708, + 418, + 722 + ], + "score": 1.0, + "content": "are locally Lipschitz functions. We say that", + "type": "text" + }, + { + "bbox": [ + 418, + 709, + 452, + 720 + ], + "score": 0.92, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 708, + 504, + 722 + ], + "score": 1.0, + "content": "is a feasible", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 719, + 317, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 163, + 734 + ], + "score": 1.0, + "content": "point of (P) if", + "type": "text" + }, + { + "bbox": [ + 164, + 722, + 171, + 730 + ], + "score": 0.77, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 719, + 206, + 734 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 207, + 721, + 251, + 732 + ], + "score": 0.92, + "content": "g _ { n } ( { \\pmb x } ) \\le 0", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 719, + 278, + 734 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 279, + 721, + 312, + 732 + ], + "score": 0.92, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 719, + 317, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 708, + 504, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 290, + 95 + ], + "score": 1.0, + "content": "Definition C.1 (KKT Point). A feasible point", + "type": "text" + }, + { + "bbox": [ + 290, + 85, + 298, + 92 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 82, + 393, + 95 + ], + "score": 1.0, + "content": "of (P) is a KKT point if", + "type": "text" + }, + { + "bbox": [ + 393, + 84, + 401, + 92 + ], + "score": 0.74, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "satisfies KKT conditions:", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 259, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 154, + 106 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 154, + 94, + 219, + 105 + ], + "score": 0.92, + "content": "\\lambda _ { 1 } , \\ldots , \\lambda _ { N } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 93, + 259, + 106 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 134, + 245, + 147 + ], + "lines": [ + { + "bbox": [ + 132, + 134, + 245, + 147 + ], + "spans": [ + { + "bbox": [ + 132, + 134, + 245, + 147 + ], + "score": 0.59, + "content": "2 . \\ \\forall n \\in [ N ] : \\lambda _ { n } g _ { n } ( { \\pmb x } ) = 0 .", + "type": "interline_equation", + "image_path": "74aedca2a3f5a1b8a3694e9f58b86a2e1cbae12024ea5703aa8b45f248601048.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 132, + 134, + 245, + 147 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 155, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 506, + 169 + ], + "score": 1.0, + "content": "It is important to note that a global minimum of (P) may not be a KKT point, but under some", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "score": 1.0, + "content": "regularity assumptions, the KKT conditions become a necessary condition for global optimality. The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "regularity condition we shall use in this paper is the non-smooth version of Mangasarian-Fromovitz", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "Constraint Qualification (MFCQ) (see, e.g., the constraint qualification (C.Q.5) in (Giorgi et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 198, + 139, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 139, + 213 + ], + "score": 1.0, + "content": "2004)):", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 504, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 296, + 228 + ], + "score": 1.0, + "content": "Definition C.2 (MFCQ). For a feasible point", + "type": "text" + }, + { + "bbox": [ + 296, + 216, + 304, + 224 + ], + "score": 0.79, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 212, + 463, + 228 + ], + "score": 1.0, + "content": "of (P), (P) is said to satisfy MFCQ at", + "type": "text" + }, + { + "bbox": [ + 463, + 216, + 471, + 224 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 212, + 505, + 228 + ], + "score": 1.0, + "content": "if there", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 224, + 333, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 131, + 238 + ], + "score": 1.0, + "content": "exists", + "type": "text" + }, + { + "bbox": [ + 132, + 225, + 163, + 235 + ], + "score": 0.92, + "content": "\\pmb { v } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 224, + 228, + 238 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 229, + 225, + 263, + 237 + ], + "score": 0.93, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 224, + 284, + 238 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 284, + 225, + 328, + 237 + ], + "score": 0.93, + "content": "g _ { n } ( { \\pmb x } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 224, + 333, + 238 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 241, + 363, + 254 + ], + "lines": [ + { + "bbox": [ + 247, + 241, + 363, + 254 + ], + "spans": [ + { + "bbox": [ + 247, + 241, + 363, + 254 + ], + "score": 0.89, + "content": "\\forall h \\in \\partial ^ { \\circ } g _ { n } ( { \\pmb x } ) : \\langle h , { \\pmb v } \\rangle > 0 .", + "type": "interline_equation", + "image_path": "deaf377e4b6acae33659aab9cfeaa4d726fedbd90548f9a387f6e3833667c928.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 247, + 241, + 363, + 254 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "Following from (Dutta et al., 2013), we define an approximate version of KKT point, as shown", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 344, + 288 + ], + "score": 1.0, + "content": "below. Note that this definition is essentially the modified", + "type": "text" + }, + { + "bbox": [ + 344, + 277, + 349, + 286 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "-KKT point defined in their paper, but", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "these two definitions differ in the following two ways: (1) First, in their paper, the subdifferential is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 293, + 310 + ], + "score": 1.0, + "content": "allowed to be evaluated in a neighborhood of", + "type": "text" + }, + { + "bbox": [ + 293, + 300, + 300, + 307 + ], + "score": 0.71, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 297, + 505, + 310 + ], + "score": 1.0, + "content": ", so our definition is slightly stronger; (2) Second,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 307, + 401, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 173, + 322 + ], + "score": 1.0, + "content": "their paper fixes", + "type": "text" + }, + { + "bbox": [ + 173, + 308, + 200, + 319 + ], + "score": 0.91, + "content": "\\delta = \\epsilon ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 307, + 401, + 322 + ], + "score": 1.0, + "content": ", but in our definition we make them independent.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 105, + 322, + 502, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 299, + 336 + ], + "score": 1.0, + "content": "Definition C.3 (Approximate KKT Point). For", + "type": "text" + }, + { + "bbox": [ + 299, + 323, + 333, + 334 + ], + "score": 0.89, + "content": "\\epsilon , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 322, + 401, + 336 + ], + "score": 1.0, + "content": ", a feasible point", + "type": "text" + }, + { + "bbox": [ + 401, + 325, + 409, + 333 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 322, + 458, + 336 + ], + "score": 1.0, + "content": "of (P) is an", + "type": "text" + }, + { + "bbox": [ + 458, + 323, + 480, + 335 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "-KKT", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 333, + 435, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 186, + 347 + ], + "score": 1.0, + "content": "point if there exists", + "type": "text" + }, + { + "bbox": [ + 186, + 334, + 333, + 346 + ], + "score": 0.91, + "content": "\\lambda _ { n } \\geq 0 , \\pmb { k } \\in \\partial ^ { \\circ } f ( \\pmb { x } ) , \\pmb { h } _ { n } \\in \\partial ^ { \\circ } g _ { n } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 333, + 361, + 347 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 361, + 334, + 395, + 346 + ], + "score": 0.93, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 333, + 435, + 347 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 353, + 270, + 376 + ], + "lines": [ + { + "bbox": [ + 131, + 353, + 270, + 376 + ], + "spans": [ + { + "bbox": [ + 131, + 353, + 270, + 376 + ], + "score": 0.55, + "content": "\\begin{array} { r } { 1 . \\ \\left\\| k + \\sum _ { n \\in [ N ] } \\lambda _ { n } \\pmb { h } _ { n } ( \\pmb { x } ) \\right\\| _ { 2 } \\leq \\epsilon ; } \\end{array}", + "type": "interline_equation", + "image_path": "51e96b6f10fffba71794ea436b968fb2994e9a0dba5a53fc1931e5073c9efe1e.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 131, + 353, + 270, + 376 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 380, + 254, + 394 + ], + "lines": [ + { + "bbox": [ + 129, + 380, + 254, + 394 + ], + "spans": [ + { + "bbox": [ + 129, + 380, + 254, + 394 + ], + "score": 0.59, + "content": "2 . \\forall n \\in [ N ] : \\lambda _ { n } g _ { n } ( { \\pmb x } ) \\geq - \\delta .", + "type": "interline_equation", + "image_path": "7ea9732df11d18814c1b4d7f5de20ba948f8a9480b5d152a38bf9fbec31459b5.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 129, + 380, + 254, + 394 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 243, + 415 + ], + "score": 1.0, + "content": "As shown in (Dutta et al., 2013),", + "type": "text" + }, + { + "bbox": [ + 243, + 402, + 265, + 414 + ], + "score": 0.9, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "-KKT point is an approximate version of KKT point in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 194, + 426 + ], + "score": 1.0, + "content": "sense that a series of", + "type": "text" + }, + { + "bbox": [ + 194, + 414, + 215, + 425 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "-KKT points can converge to a KKT point. We restate their theorem in", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 424, + 155, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 155, + 438 + ], + "score": 1.0, + "content": "our setting:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 507, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 394, + 451 + ], + "score": 1.0, + "content": "Theorem C.4 (Corollary of Theorem 3.6 in (Dutta et al., 2013)). Let", + "type": "text" + }, + { + "bbox": [ + 394, + 438, + 482, + 451 + ], + "score": 0.91, + "content": "\\{ \\pmb { x } _ { k } \\in \\mathbb { R } ^ { d } : k \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 437, + 507, + 451 + ], + "score": 1.0, + "content": "be a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 448, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 226, + 463 + ], + "score": 1.0, + "content": "sequence of feasible points of", + "type": "text" + }, + { + "bbox": [ + 227, + 450, + 240, + 461 + ], + "score": 0.68, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 448, + 244, + 463 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 245, + 450, + 315, + 461 + ], + "score": 0.91, + "content": "\\{ \\epsilon _ { k } > 0 : k \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 448, + 334, + 463 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 335, + 450, + 406, + 461 + ], + "score": 0.91, + "content": "\\{ \\delta _ { k } > 0 : k \\in \\tilde { \\mathbb { N } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 448, + 482, + 463 + ], + "score": 1.0, + "content": "be two sequences.", + "type": "text" + }, + { + "bbox": [ + 482, + 452, + 495, + 461 + ], + "score": 0.86, + "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 448, + 506, + 463 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 119, + 473 + ], + "score": 1.0, + "content": "an", + "type": "text" + }, + { + "bbox": [ + 120, + 461, + 150, + 473 + ], + "score": 0.91, + "content": "( \\epsilon _ { k } , \\delta _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 460, + 235, + 473 + ], + "score": 1.0, + "content": "-KKT point for every", + "type": "text" + }, + { + "bbox": [ + 236, + 462, + 242, + 471 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 460, + 264, + 473 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 264, + 462, + 329, + 472 + ], + "score": 0.89, + "content": "\\epsilon _ { k } 0 , \\delta _ { k } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 460, + 343, + 473 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 343, + 462, + 378, + 472 + ], + "score": 0.88, + "content": "\\mathbfit { x } _ { k } \\to \\mathbfit { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 460, + 390, + 473 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 391, + 462, + 432, + 471 + ], + "score": 0.9, + "content": "k \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "and MFCQ holds", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 471, + 245, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 117, + 485 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 474, + 124, + 482 + ], + "score": 0.51, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 471, + 148, + 485 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 148, + 474, + 156, + 482 + ], + "score": 0.46, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 471, + 228, + 485 + ], + "score": 1.0, + "content": "is a KKT point of", + "type": "text" + }, + { + "bbox": [ + 228, + 472, + 242, + 483 + ], + "score": 0.77, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 471, + 245, + 485 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 107, + 496, + 249, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 494, + 250, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 250, + 510 + ], + "score": 1.0, + "content": "C.2 KKT CONDITIONS FOR (P)", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 516, + 499, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 501, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 501, + 531 + ], + "score": 1.0, + "content": "Recall that for a homogeneous neural network, the optimization problem (P) is defined as follows:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 532, + 370, + 572 + ], + "lines": [ + { + "bbox": [ + 240, + 532, + 370, + 572 + ], + "spans": [ + { + "bbox": [ + 240, + 532, + 370, + 572 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\mathrm { s . t . } } & { q _ { n } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ] } \\end{array}", + "type": "interline_equation", + "image_path": "6fb4f7835d0f9d076ad7003b4eb71d439d4eaceac823d4a752c73979f1032d90.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 532, + 370, + 552.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 240, + 552.0, + 370, + 572.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 470, + 594 + ], + "score": 1.0, + "content": "Using the terminologies and notations in Appendix C.1, the objective and constraints are", + "type": "text" + }, + { + "bbox": [ + 471, + 581, + 505, + 593 + ], + "score": 0.9, + "content": "f ( { \\pmb x } ) =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 590, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 107, + 591, + 135, + 605 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { { { \\frac { 1 } { 2 } } \\| \\pmb { x } \\| _ { 2 } ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 590, + 153, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 154, + 591, + 235, + 604 + ], + "score": 0.92, + "content": "g _ { n } ( \\pmb { x } ) = 1 - q _ { n } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 590, + 506, + 606 + ], + "score": 1.0, + "content": ". The KKT points and approximate KKT points for (P) are defined", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 601, + 153, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 153, + 615 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 104, + 616, + 504, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 337, + 630 + ], + "score": 1.0, + "content": "Definition C.5 (KKT Point of (P)). A feasible point", + "type": "text" + }, + { + "bbox": [ + 337, + 618, + 345, + 627 + ], + "score": 0.75, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "of (P) is a KKT point if there exist", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 627, + 212, + 641 + ], + "spans": [ + { + "bbox": [ + 107, + 628, + 172, + 640 + ], + "score": 0.91, + "content": "\\lambda _ { 1 } , \\dots , \\lambda _ { N } \\ge 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 627, + 212, + 641 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 129, + 648, + 426, + 681 + ], + "lines": [ + { + "bbox": [ + 126, + 642, + 421, + 669 + ], + "spans": [ + { + "bbox": [ + 126, + 642, + 142, + 669 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 648, + 231, + 663 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\pmb { \\theta } - \\sum _ { n = 1 } ^ { N } \\lambda _ { n } \\pmb { h } _ { n } = \\mathbf { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 642, + 271, + 669 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 271, + 651, + 319, + 662 + ], + "score": 0.92, + "content": "h _ { 1 } , \\ldots , h _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 642, + 362, + 669 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 362, + 650, + 421, + 662 + ], + "score": 0.92, + "content": "h _ { n } \\in \\partial ^ { \\circ } q _ { n } ( \\theta )", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 129, + 668, + 270, + 683 + ], + "spans": [ + { + "bbox": [ + 129, + 668, + 139, + 683 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 140, + 668, + 270, + 682 + ], + "score": 0.73, + "content": "\\forall n \\in [ N ] : \\lambda _ { n } ( q _ { n } ( \\pmb \\theta ) - 1 ) = 0 .", + "type": "inline_equation" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 105, + 683, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 377, + 697 + ], + "score": 1.0, + "content": "Definition C.6 (Approximate KKT Point of (P)). A feasible point", + "type": "text" + }, + { + "bbox": [ + 378, + 684, + 384, + 694 + ], + "score": 0.8, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 682, + 398, + 697 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 398, + 684, + 411, + 695 + ], + "score": 0.25, + "content": "( \\mathrm { P } )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 682, + 434, + 697 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 435, + 684, + 456, + 695 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 682, + 505, + 697 + ], + "score": 1.0, + "content": "-KKT point", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 694, + 293, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 188, + 707 + ], + "score": 1.0, + "content": "of (P) if there exists", + "type": "text" + }, + { + "bbox": [ + 188, + 695, + 253, + 706 + ], + "score": 0.92, + "content": "\\lambda _ { 1 } , \\ldots , \\lambda _ { N } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 694, + 293, + 707 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 290, + 95 + ], + "score": 1.0, + "content": "Definition C.1 (KKT Point). A feasible point", + "type": "text" + }, + { + "bbox": [ + 290, + 85, + 298, + 92 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 82, + 393, + 95 + ], + "score": 1.0, + "content": "of (P) is a KKT point if", + "type": "text" + }, + { + "bbox": [ + 393, + 84, + 401, + 92 + ], + "score": 0.74, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "satisfies KKT conditions:", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 259, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 154, + 106 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 154, + 94, + 219, + 105 + ], + "score": 0.92, + "content": "\\lambda _ { 1 } , \\ldots , \\lambda _ { N } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 93, + 259, + 106 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 505, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 134, + 245, + 147 + ], + "lines": [ + { + "bbox": [ + 132, + 134, + 245, + 147 + ], + "spans": [ + { + "bbox": [ + 132, + 134, + 245, + 147 + ], + "score": 0.59, + "content": "2 . \\ \\forall n \\in [ N ] : \\lambda _ { n } g _ { n } ( { \\pmb x } ) = 0 .", + "type": "interline_equation", + "image_path": "74aedca2a3f5a1b8a3694e9f58b86a2e1cbae12024ea5703aa8b45f248601048.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 132, + 134, + 245, + 147 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 155, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 506, + 169 + ], + "score": 1.0, + "content": "It is important to note that a global minimum of (P) may not be a KKT point, but under some", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "score": 1.0, + "content": "regularity assumptions, the KKT conditions become a necessary condition for global optimality. The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "regularity condition we shall use in this paper is the non-smooth version of Mangasarian-Fromovitz", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "Constraint Qualification (MFCQ) (see, e.g., the constraint qualification (C.Q.5) in (Giorgi et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 198, + 139, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 139, + 213 + ], + "score": 1.0, + "content": "2004)):", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 155, + 506, + 213 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 504, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 296, + 228 + ], + "score": 1.0, + "content": "Definition C.2 (MFCQ). For a feasible point", + "type": "text" + }, + { + "bbox": [ + 296, + 216, + 304, + 224 + ], + "score": 0.79, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 212, + 463, + 228 + ], + "score": 1.0, + "content": "of (P), (P) is said to satisfy MFCQ at", + "type": "text" + }, + { + "bbox": [ + 463, + 216, + 471, + 224 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 212, + 505, + 228 + ], + "score": 1.0, + "content": "if there", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 224, + 333, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 131, + 238 + ], + "score": 1.0, + "content": "exists", + "type": "text" + }, + { + "bbox": [ + 132, + 225, + 163, + 235 + ], + "score": 0.92, + "content": "\\pmb { v } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 224, + 228, + 238 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 229, + 225, + 263, + 237 + ], + "score": 0.93, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 224, + 284, + 238 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 284, + 225, + 328, + 237 + ], + "score": 0.93, + "content": "g _ { n } ( { \\pmb x } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 224, + 333, + 238 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 212, + 505, + 238 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 241, + 363, + 254 + ], + "lines": [ + { + "bbox": [ + 247, + 241, + 363, + 254 + ], + "spans": [ + { + "bbox": [ + 247, + 241, + 363, + 254 + ], + "score": 0.89, + "content": "\\forall h \\in \\partial ^ { \\circ } g _ { n } ( { \\pmb x } ) : \\langle h , { \\pmb v } \\rangle > 0 .", + "type": "interline_equation", + "image_path": "deaf377e4b6acae33659aab9cfeaa4d726fedbd90548f9a387f6e3833667c928.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 247, + 241, + 363, + 254 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "Following from (Dutta et al., 2013), we define an approximate version of KKT point, as shown", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 344, + 288 + ], + "score": 1.0, + "content": "below. Note that this definition is essentially the modified", + "type": "text" + }, + { + "bbox": [ + 344, + 277, + 349, + 286 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "-KKT point defined in their paper, but", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "these two definitions differ in the following two ways: (1) First, in their paper, the subdifferential is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 293, + 310 + ], + "score": 1.0, + "content": "allowed to be evaluated in a neighborhood of", + "type": "text" + }, + { + "bbox": [ + 293, + 300, + 300, + 307 + ], + "score": 0.71, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 297, + 505, + 310 + ], + "score": 1.0, + "content": ", so our definition is slightly stronger; (2) Second,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 307, + 401, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 173, + 322 + ], + "score": 1.0, + "content": "their paper fixes", + "type": "text" + }, + { + "bbox": [ + 173, + 308, + 200, + 319 + ], + "score": 0.91, + "content": "\\delta = \\epsilon ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 307, + 401, + 322 + ], + "score": 1.0, + "content": ", but in our definition we make them independent.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 264, + 505, + 322 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 322, + 502, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 299, + 336 + ], + "score": 1.0, + "content": "Definition C.3 (Approximate KKT Point). For", + "type": "text" + }, + { + "bbox": [ + 299, + 323, + 333, + 334 + ], + "score": 0.89, + "content": "\\epsilon , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 322, + 401, + 336 + ], + "score": 1.0, + "content": ", a feasible point", + "type": "text" + }, + { + "bbox": [ + 401, + 325, + 409, + 333 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 322, + 458, + 336 + ], + "score": 1.0, + "content": "of (P) is an", + "type": "text" + }, + { + "bbox": [ + 458, + 323, + 480, + 335 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "-KKT", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 333, + 435, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 186, + 347 + ], + "score": 1.0, + "content": "point if there exists", + "type": "text" + }, + { + "bbox": [ + 186, + 334, + 333, + 346 + ], + "score": 0.91, + "content": "\\lambda _ { n } \\geq 0 , \\pmb { k } \\in \\partial ^ { \\circ } f ( \\pmb { x } ) , \\pmb { h } _ { n } \\in \\partial ^ { \\circ } g _ { n } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 333, + 361, + 347 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 361, + 334, + 395, + 346 + ], + "score": 0.93, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 333, + 435, + 347 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 322, + 505, + 347 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 353, + 270, + 376 + ], + "lines": [ + { + "bbox": [ + 131, + 353, + 270, + 376 + ], + "spans": [ + { + "bbox": [ + 131, + 353, + 270, + 376 + ], + "score": 0.55, + "content": "\\begin{array} { r } { 1 . \\ \\left\\| k + \\sum _ { n \\in [ N ] } \\lambda _ { n } \\pmb { h } _ { n } ( \\pmb { x } ) \\right\\| _ { 2 } \\leq \\epsilon ; } \\end{array}", + "type": "interline_equation", + "image_path": "51e96b6f10fffba71794ea436b968fb2994e9a0dba5a53fc1931e5073c9efe1e.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 131, + 353, + 270, + 376 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 380, + 254, + 394 + ], + "lines": [ + { + "bbox": [ + 129, + 380, + 254, + 394 + ], + "spans": [ + { + "bbox": [ + 129, + 380, + 254, + 394 + ], + "score": 0.59, + "content": "2 . \\forall n \\in [ N ] : \\lambda _ { n } g _ { n } ( { \\pmb x } ) \\geq - \\delta .", + "type": "interline_equation", + "image_path": "7ea9732df11d18814c1b4d7f5de20ba948f8a9480b5d152a38bf9fbec31459b5.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 129, + 380, + 254, + 394 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 243, + 415 + ], + "score": 1.0, + "content": "As shown in (Dutta et al., 2013),", + "type": "text" + }, + { + "bbox": [ + 243, + 402, + 265, + 414 + ], + "score": 0.9, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "-KKT point is an approximate version of KKT point in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 194, + 426 + ], + "score": 1.0, + "content": "sense that a series of", + "type": "text" + }, + { + "bbox": [ + 194, + 414, + 215, + 425 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "-KKT points can converge to a KKT point. We restate their theorem in", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 424, + 155, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 155, + 438 + ], + "score": 1.0, + "content": "our setting:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 401, + 505, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 507, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 394, + 451 + ], + "score": 1.0, + "content": "Theorem C.4 (Corollary of Theorem 3.6 in (Dutta et al., 2013)). Let", + "type": "text" + }, + { + "bbox": [ + 394, + 438, + 482, + 451 + ], + "score": 0.91, + "content": "\\{ \\pmb { x } _ { k } \\in \\mathbb { R } ^ { d } : k \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 437, + 507, + 451 + ], + "score": 1.0, + "content": "be a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 448, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 226, + 463 + ], + "score": 1.0, + "content": "sequence of feasible points of", + "type": "text" + }, + { + "bbox": [ + 227, + 450, + 240, + 461 + ], + "score": 0.68, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 448, + 244, + 463 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 245, + 450, + 315, + 461 + ], + "score": 0.91, + "content": "\\{ \\epsilon _ { k } > 0 : k \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 448, + 334, + 463 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 335, + 450, + 406, + 461 + ], + "score": 0.91, + "content": "\\{ \\delta _ { k } > 0 : k \\in \\tilde { \\mathbb { N } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 448, + 482, + 463 + ], + "score": 1.0, + "content": "be two sequences.", + "type": "text" + }, + { + "bbox": [ + 482, + 452, + 495, + 461 + ], + "score": 0.86, + "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 448, + 506, + 463 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 119, + 473 + ], + "score": 1.0, + "content": "an", + "type": "text" + }, + { + "bbox": [ + 120, + 461, + 150, + 473 + ], + "score": 0.91, + "content": "( \\epsilon _ { k } , \\delta _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 460, + 235, + 473 + ], + "score": 1.0, + "content": "-KKT point for every", + "type": "text" + }, + { + "bbox": [ + 236, + 462, + 242, + 471 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 460, + 264, + 473 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 264, + 462, + 329, + 472 + ], + "score": 0.89, + "content": "\\epsilon _ { k } 0 , \\delta _ { k } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 460, + 343, + 473 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 343, + 462, + 378, + 472 + ], + "score": 0.88, + "content": "\\mathbfit { x } _ { k } \\to \\mathbfit { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 460, + 390, + 473 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 391, + 462, + 432, + 471 + ], + "score": 0.9, + "content": "k \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "and MFCQ holds", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 471, + 245, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 117, + 485 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 474, + 124, + 482 + ], + "score": 0.51, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 471, + 148, + 485 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 148, + 474, + 156, + 482 + ], + "score": 0.46, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 471, + 228, + 485 + ], + "score": 1.0, + "content": "is a KKT point of", + "type": "text" + }, + { + "bbox": [ + 228, + 472, + 242, + 483 + ], + "score": 0.77, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 471, + 245, + 485 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 437, + 507, + 485 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 496, + 249, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 494, + 250, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 250, + 510 + ], + "score": 1.0, + "content": "C.2 KKT CONDITIONS FOR (P)", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 516, + 499, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 501, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 501, + 531 + ], + "score": 1.0, + "content": "Recall that for a homogeneous neural network, the optimization problem (P) is defined as follows:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 515, + 501, + 531 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 532, + 370, + 572 + ], + "lines": [ + { + "bbox": [ + 240, + 532, + 370, + 572 + ], + "spans": [ + { + "bbox": [ + 240, + 532, + 370, + 572 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\mathrm { s . t . } } & { q _ { n } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ] } \\end{array}", + "type": "interline_equation", + "image_path": "6fb4f7835d0f9d076ad7003b4eb71d439d4eaceac823d4a752c73979f1032d90.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 532, + 370, + 552.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 240, + 552.0, + 370, + 572.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 470, + 594 + ], + "score": 1.0, + "content": "Using the terminologies and notations in Appendix C.1, the objective and constraints are", + "type": "text" + }, + { + "bbox": [ + 471, + 581, + 505, + 593 + ], + "score": 0.9, + "content": "f ( { \\pmb x } ) =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 590, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 107, + 591, + 135, + 605 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { { { \\frac { 1 } { 2 } } \\| \\pmb { x } \\| _ { 2 } ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 590, + 153, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 154, + 591, + 235, + 604 + ], + "score": 0.92, + "content": "g _ { n } ( \\pmb { x } ) = 1 - q _ { n } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 590, + 506, + 606 + ], + "score": 1.0, + "content": ". The KKT points and approximate KKT points for (P) are defined", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 601, + 153, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 153, + 615 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 579, + 506, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 616, + 504, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 337, + 630 + ], + "score": 1.0, + "content": "Definition C.5 (KKT Point of (P)). A feasible point", + "type": "text" + }, + { + "bbox": [ + 337, + 618, + 345, + 627 + ], + "score": 0.75, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "of (P) is a KKT point if there exist", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 627, + 212, + 641 + ], + "spans": [ + { + "bbox": [ + 107, + 628, + 172, + 640 + ], + "score": 0.91, + "content": "\\lambda _ { 1 } , \\dots , \\lambda _ { N } \\ge 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 627, + 212, + 641 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 615, + 506, + 641 + ] + }, + { + "type": "index", + "bbox": [ + 129, + 648, + 426, + 681 + ], + "lines": [ + { + "bbox": [ + 126, + 642, + 421, + 669 + ], + "spans": [ + { + "bbox": [ + 126, + 642, + 142, + 669 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 648, + 231, + 663 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\pmb { \\theta } - \\sum _ { n = 1 } ^ { N } \\lambda _ { n } \\pmb { h } _ { n } = \\mathbf { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 642, + 271, + 669 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 271, + 651, + 319, + 662 + ], + "score": 0.92, + "content": "h _ { 1 } , \\ldots , h _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 642, + 362, + 669 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 362, + 650, + 421, + 662 + ], + "score": 0.92, + "content": "h _ { n } \\in \\partial ^ { \\circ } q _ { n } ( \\theta )", + "type": "inline_equation" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 129, + 668, + 270, + 683 + ], + "spans": [ + { + "bbox": [ + 129, + 668, + 139, + 683 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 140, + 668, + 270, + 682 + ], + "score": 0.73, + "content": "\\forall n \\in [ N ] : \\lambda _ { n } ( q _ { n } ( \\pmb \\theta ) - 1 ) = 0 .", + "type": "inline_equation" + } + ], + "index": 37, + "is_list_start_line": true + } + ], + "index": 36.5, + "bbox_fs": [ + 126, + 642, + 421, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 683, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 377, + 697 + ], + "score": 1.0, + "content": "Definition C.6 (Approximate KKT Point of (P)). 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Let", + "type": "text" + }, + { + "bbox": [ + 179, + 415, + 208, + 427 + ], + "score": 0.91, + "content": "C _ { 1 } , C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 413, + 321, + 429 + ], + "score": 1.0, + "content": "be two constants defined as", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 432, + 421, + 462 + ], + "lines": [ + { + "bbox": [ + 189, + 432, + 421, + 462 + ], + "spans": [ + { + "bbox": [ + 189, + 432, + 421, + 462 + ], + "score": 0.93, + "content": "C _ { 1 } : = \\frac { \\sqrt { 2 } } { \\tilde { \\gamma } ( t _ { 0 } ) ^ { 1 / L } } , \\qquad C _ { 2 } : = \\frac { 2 e N K ^ { 2 } } { L \\tilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } } \\left( \\frac { B _ { 1 } } { \\tilde { \\gamma } ( t _ { 0 } ) } \\right) ^ { \\log _ { 2 } K } ,", + "type": "interline_equation", + "image_path": "1d690195c3487763fde4df2134558983f5a2e2d36a8d53fccef60d9395966b76.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 189, + 432, + 421, + 462 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 503, + 495 + ], + "lines": [ + { + "bbox": [ + 104, + 465, + 504, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 465, + 132, + 483 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 469, + 156, + 481 + ], + "score": 0.9, + "content": "K , b _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 465, + 290, + 483 + ], + "score": 1.0, + "content": "are constants specified in (B3.4).", + "type": "text" + }, + { + "bbox": [ + 291, + 468, + 344, + 482 + ], + "score": 0.85, + "content": "I f \\log { \\frac { 1 } { \\mathcal { L } } } \\geq b _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 465, + 374, + 483 + ], + "score": 1.0, + "content": "at time", + "type": "text" + }, + { + "bbox": [ + 375, + 469, + 401, + 480 + ], + "score": 0.89, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 465, + 424, + 483 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 424, + 467, + 504, + 481 + ], + "score": 0.94, + "content": "\\tilde { \\pmb { \\theta } } : = \\pmb { \\theta } / q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 480, + 407, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 128, + 495 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 129, + 482, + 150, + 494 + ], + "score": 0.89, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 480, + 207, + 495 + ], + "score": 1.0, + "content": "-KKT point of", + "type": "text" + }, + { + "bbox": [ + 207, + 482, + 221, + 493 + ], + "score": 0.79, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 480, + 251, + 495 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 252, + 481, + 316, + 493 + ], + "score": 0.92, + "content": "\\epsilon : = C _ { 1 } \\sqrt { 1 - \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 480, + 335, + 495 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 335, + 481, + 401, + 495 + ], + "score": 0.9, + "content": "\\delta : = C _ { 2 } / ( \\log \\frac { 1 } { \\mathcal { L } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 480, + 407, + 495 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 104, + 503, + 508, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 503, + 153, + 524 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 154, + 506, + 216, + 521 + ], + "score": 0.94, + "content": "\\begin{array} { r } \\begin{array} { r } { \\begin{array} { r } { \\boldsymbol { h } ( t ) : = \\frac { d \\theta } { d t } ( t ) } \\end{array} } \\end{array} \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 503, + 252, + 524 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 252, + 508, + 280, + 518 + ], + "score": 0.88, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 503, + 414, + 524 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 326, + 519, + 453, + 535 + ], + "score": 0.92, + "content": "\\tilde { \\pmb { h } } _ { n } : = \\pmb { h } _ { n } / q _ { \\mathrm { m i n } } ^ { 1 - 1 / L } \\in \\partial ^ { \\circ } q _ { n } ( \\tilde { \\pmb { \\theta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 518, + 506, + 536 + ], + "score": 1.0, + "content": "q1−1/Lmin ∈ ∂◦qn(θ˜) (Recall that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 534, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 287, + 553 + ], + "score": 1.0, + "content": "θ˜ := θ/qmin(θ)1/L). 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Define", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 229, + 508, + 277 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 280, + 422, + 370 + ], + "lines": [ + { + "bbox": [ + 188, + 280, + 422, + 370 + ], + "spans": [ + { + "bbox": [ + 188, + 280, + 422, + 370 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { B _ { 0 } : = \\operatorname* { s u p } \\{ \\cfrac { q _ { n } } { \\rho ^ { L } } : \\theta \\in \\mathbb { R } ^ { d } \\setminus \\{ \\mathbf { 0 } \\} \\} } \\\\ & { \\quad = \\operatorname* { s u p } \\{ q _ { n } : \\theta \\in \\mathcal { S } ^ { d - 1 } \\} < \\infty . } \\\\ & { B _ { 1 } : = \\operatorname* { s u p } \\{ \\cfrac { \\| h \\| _ { 2 } } { \\rho ^ { L - 1 } } : \\theta \\in \\mathbb { R } ^ { d } \\setminus \\{ \\mathbf { 0 } \\} , h \\in \\partial ^ { \\circ } q _ { n } , n \\in [ N ] \\} } \\\\ & { \\quad = \\operatorname* { s u p } \\{ \\| h \\| _ { 2 } : \\theta \\in \\mathcal { S } ^ { d - 1 } , \\boldsymbol { h } \\in \\partial ^ { \\circ } q _ { n } , n \\in [ N ] \\} < \\infty . } \\end{array}", + "type": "interline_equation", + "image_path": "ea5c88a2428113a373c5ff93faae948c1cec85843257137e416126bfacc8b677.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 188, + 280, + 422, + 298.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 188, + 298.0, + 422, + 316.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 188, + 316.0, + 422, + 334.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 188, + 334.0, + 422, + 352.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 188, + 352.0, + 422, + 370.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 507, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 390 + ], + "score": 1.0, + "content": "For showing Theorem A.8 and Theorem A.9, we first prove Lemma C.8. In light of this lemma, if", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 390, + 504, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 191, + 401 + ], + "score": 1.0, + "content": "we aim to show that", + "type": "text" + }, + { + "bbox": [ + 191, + 390, + 199, + 399 + ], + "score": 0.8, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 390, + 504, + 401 + ], + "score": 1.0, + "content": "is along the direction of an approximate KKT point, we only need to show", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 401, + 362, + 412 + ], + "spans": [ + { + "bbox": [ + 107, + 401, + 134, + 412 + ], + "score": 0.89, + "content": "\\beta 1", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 401, + 194, + 412 + ], + "score": 1.0, + "content": "(which makes", + "type": "text" + }, + { + "bbox": [ + 194, + 401, + 219, + 411 + ], + "score": 0.88, + "content": "\\epsilon \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 401, + 241, + 412 + ], + "score": 1.0, + "content": ") and", + "type": "text" + }, + { + "bbox": [ + 241, + 401, + 270, + 411 + ], + "score": 0.89, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 401, + 329, + 412 + ], + "score": 1.0, + "content": "(which makes", + "type": "text" + }, + { + "bbox": [ + 329, + 401, + 355, + 411 + ], + "score": 0.89, + "content": "\\delta 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 401, + 362, + 412 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 379, + 506, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 321, + 427 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 321, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 179, + 429 + ], + "score": 1.0, + "content": "Lemma C.8. Let", + "type": "text" + }, + { + "bbox": [ + 179, + 415, + 208, + 427 + ], + "score": 0.91, + "content": "C _ { 1 } , C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 413, + 321, + 429 + ], + "score": 1.0, + "content": "be two constants defined as", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 413, + 321, + 429 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 432, + 421, + 462 + ], + "lines": [ + { + "bbox": [ + 189, + 432, + 421, + 462 + ], + "spans": [ + { + "bbox": [ + 189, + 432, + 421, + 462 + ], + "score": 0.93, + "content": "C _ { 1 } : = \\frac { \\sqrt { 2 } } { \\tilde { \\gamma } ( t _ { 0 } ) ^ { 1 / L } } , \\qquad C _ { 2 } : = \\frac { 2 e N K ^ { 2 } } { L \\tilde { \\gamma } ( t _ { 0 } ) ^ { 2 / L } } \\left( \\frac { B _ { 1 } } { \\tilde { \\gamma } ( t _ { 0 } ) } \\right) ^ { \\log _ { 2 } K } ,", + "type": "interline_equation", + "image_path": "1d690195c3487763fde4df2134558983f5a2e2d36a8d53fccef60d9395966b76.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 189, + 432, + 421, + 462 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 503, + 495 + ], + "lines": [ + { + "bbox": [ + 104, + 465, + 504, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 465, + 132, + 483 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 469, + 156, + 481 + ], + "score": 0.9, + "content": "K , b _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 465, + 290, + 483 + ], + "score": 1.0, + "content": "are constants specified in (B3.4).", + "type": "text" + }, + { + "bbox": [ + 291, + 468, + 344, + 482 + ], + "score": 0.85, + "content": "I f \\log { \\frac { 1 } { \\mathcal { L } } } \\geq b _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 465, + 374, + 483 + ], + "score": 1.0, + "content": "at time", + "type": "text" + }, + { + "bbox": [ + 375, + 469, + 401, + 480 + ], + "score": 0.89, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 465, + 424, + 483 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 424, + 467, + 504, + 481 + ], + "score": 0.94, + "content": "\\tilde { \\pmb { \\theta } } : = \\pmb { \\theta } / q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 480, + 407, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 128, + 495 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 129, + 482, + 150, + 494 + ], + "score": 0.89, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 480, + 207, + 495 + ], + "score": 1.0, + "content": "-KKT point of", + "type": "text" + }, + { + "bbox": [ + 207, + 482, + 221, + 493 + ], + "score": 0.79, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 480, + 251, + 495 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 252, + 481, + 316, + 493 + ], + "score": 0.92, + "content": "\\epsilon : = C _ { 1 } \\sqrt { 1 - \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 480, + 335, + 495 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 335, + 481, + 401, + 495 + ], + "score": 0.9, + "content": "\\delta : = C _ { 2 } / ( \\log \\frac { 1 } { \\mathcal { L } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 480, + 407, + 495 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 465, + 504, + 495 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 104, + 503, + 508, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 503, + 153, + 524 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 154, + 506, + 216, + 521 + ], + "score": 0.94, + "content": "\\begin{array} { r } \\begin{array} { r } { \\begin{array} { r } { \\boldsymbol { h } ( t ) : = \\frac { d \\theta } { d t } ( t ) } \\end{array} } \\end{array} \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 503, + 252, + 524 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 252, + 508, + 280, + 518 + ], + "score": 0.88, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 503, + 414, + 524 + ], + "score": 1.0, + "content": ". By the chain rule, there exist", + "type": "text" + }, + { + "bbox": [ + 414, + 508, + 463, + 519 + ], + "score": 0.92, + "content": "\\pmb { h } _ { 1 } , \\dots , \\pmb { h } _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 503, + 508, + 524 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 518, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 155, + 534 + ], + "score": 0.91, + "content": "h _ { n } \\in \\partial ^ { \\circ } q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 519, + 175, + 539 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 521, + 302, + 537 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { h } = \\sum _ { \\pmb { n } \\in [ N ] } e ^ { - \\pmb { f } ( \\ v { q } _ { n } ) } f ^ { \\prime } ( \\ v { q } _ { n } ) \\pmb { h } _ { n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 519, + 326, + 539 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 326, + 519, + 453, + 535 + ], + "score": 0.92, + "content": "\\tilde { \\pmb { h } } _ { n } : = \\pmb { h } _ { n } / q _ { \\mathrm { m i n } } ^ { 1 - 1 / L } \\in \\partial ^ { \\circ } q _ { n } ( \\tilde { \\pmb { \\theta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 518, + 506, + 536 + ], + "score": 1.0, + "content": "q1−1/Lmin ∈ ∂◦qn(θ˜) (Recall that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 534, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 287, + 553 + ], + "score": 1.0, + "content": "θ˜ := θ/qmin(θ)1/L). Construct λn := q1−2/min", + "type": "text" + }, + { + "bbox": [ + 236, + 536, + 390, + 551 + ], + "score": 0.92, + "content": "\\lambda _ { n } : = q _ { \\mathrm { m i n } } ^ { 1 - 2 / L } \\rho \\cdot e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) / \\left\\| h \\right\\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 534, + 506, + 552 + ], + "score": 1.0, + "content": ". Now we only need to show", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 503, + 508, + 553 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 556, + 424, + 629 + ], + "lines": [ + { + "bbox": [ + 187, + 556, + 424, + 629 + ], + "spans": [ + { + "bbox": [ + 187, + 556, + 424, + 629 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\left\\| \\tilde { \\theta } - \\displaystyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } \\tilde { h } _ { n } \\right\\| _ { 2 } ^ { 2 } \\leq \\frac { 2 } { \\tilde { \\gamma } ^ { 2 / L } } ( 1 - \\beta ) } \\\\ & { \\displaystyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) \\leq \\left( \\frac { 2 e N K ^ { 2 } } { L \\tilde { \\gamma } ^ { 2 / L } } \\left( \\frac { B _ { 1 } } { \\tilde { \\gamma } } \\right) ^ { \\log _ { 2 } K } \\right) \\frac { 1 } { f ( \\tilde { \\gamma } \\rho ^ { L } ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "419bad4bb3a0475390aabacd1d37e592e7f903894afed489d1a7f03ffe6932ec.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 556, + 424, + 574.25 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 187, + 574.25, + 424, + 592.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 187, + 592.5, + 424, + 610.75 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 187, + 610.75, + 424, + 629.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 634, + 479, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 476, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 129, + 649 + ], + "score": 1.0, + "content": "Then", + "type": "text" + }, + { + "bbox": [ + 130, + 633, + 137, + 645 + ], + "score": 0.81, + "content": "\\tilde { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 632, + 228, + 649 + ], + "score": 1.0, + "content": "can be shown to be an", + "type": "text" + }, + { + "bbox": [ + 229, + 635, + 250, + 647 + ], + "score": 0.92, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 632, + 381, + 649 + ], + "score": 1.0, + "content": "-KKT point by the monotonicity", + "type": "text" + }, + { + "bbox": [ + 381, + 635, + 434, + 647 + ], + "score": 0.95, + "content": "\\tilde { \\gamma } ( t ) \\geq \\tilde { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 632, + 450, + 649 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 450, + 636, + 476, + 646 + ], + "score": 0.91, + "content": "t > t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 632, + 476, + 649 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 658, + 415, + 675 + ], + "lines": [ + { + "bbox": [ + 101, + 653, + 419, + 680 + ], + "spans": [ + { + "bbox": [ + 101, + 653, + 266, + 680 + ], + "score": 1.0, + "content": "Proof of (12). From our construction,", + "type": "text" + }, + { + "bbox": [ + 267, + 658, + 396, + 675 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { n = 1 } ^ { N } \\lambda _ { n } \\tilde { \\pmb { h } } _ { n } = q _ { \\mathrm { m i n } } ^ { - 1 / L } \\rho \\pmb { h } / \\left\\| \\pmb { h } \\right\\| _ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 653, + 419, + 680 + ], + "score": 1.0, + "content": ". So", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 101, + 653, + 419, + 680 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 678, + 473, + 716 + ], + "lines": [ + { + "bbox": [ + 136, + 678, + 473, + 716 + ], + "spans": [ + { + "bbox": [ + 136, + 678, + 473, + 716 + ], + "score": 0.93, + "content": "\\left\\| \\tilde { \\theta } - \\sum _ { n = 1 } ^ { N } \\lambda _ { n } \\tilde { h } _ { n } \\right\\| _ { 2 } ^ { 2 } = q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } \\left\\| \\hat { \\theta } - \\frac { h } { \\| h \\| _ { 2 } } \\right\\| _ { 2 } ^ { 2 } = q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } ( 2 - 2 \\beta ) \\leq \\frac { 2 } { \\tilde { \\gamma } ^ { 2 / L } } ( 1 - \\beta ) ,", + "type": "interline_equation", + "image_path": "19035e35b79d57b0e90e8ef3adc8c5c9923b6bcf01a857920e8c1dec3f761136.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 136, + 678, + 473, + 690.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 136, + 690.6666666666666, + 473, + 703.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 136, + 703.3333333333333, + 473, + 715.9999999999999 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 720, + 272, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 273, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 273, + 734 + ], + "score": 1.0, + "content": "where the last equality is by Lemma A.5.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 720, + 273, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 300, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 302, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 302, + 96 + ], + "score": 1.0, + "content": "Proof for (13). According to our construction,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 98, + 431, + 133 + ], + "lines": [ + { + "bbox": [ + 178, + 98, + 431, + 133 + ], + "spans": [ + { + "bbox": [ + 178, + 98, + 431, + 133 + ], + "score": 0.94, + "content": "\\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) = \\frac { q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho } { \\| h \\| _ { 2 } } \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) .", + "type": "interline_equation", + "image_path": "a57c568281e51a23a2ff29bba9b5b1f407d0bd6541d8850dfe0f80d7b78f18a2.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 178, + 98, + 431, + 109.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 178, + 109.66666666666667, + 431, + 121.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 178, + 121.33333333333334, + 431, + 133.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 136, + 419, + 155 + ], + "lines": [ + { + "bbox": [ + 104, + 136, + 422, + 157 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 146, + 155 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 136, + 246, + 157 + ], + "score": 0.95, + "content": "\\| h \\| _ { 2 } \\geq \\left. h , \\hat { \\theta } \\right. = L \\nu / \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 136, + 422, + 155 + ], + "score": 1.0, + "content": ". By Lemma B.5 and Lemma D.1, we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 160, + 424, + 191 + ], + "lines": [ + { + "bbox": [ + 186, + 160, + 424, + 191 + ], + "spans": [ + { + "bbox": [ + 186, + 160, + 424, + 191 + ], + "score": 0.93, + "content": "\\nu \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } \\geq \\frac { 1 } { 2 K } \\log \\frac { 1 } { \\mathcal { L } } \\cdot \\mathcal { L } \\geq \\frac { 1 } { 2 K } f ( \\widetilde { \\gamma } \\rho ^ { L } ) e ^ { - f ( q _ { \\operatorname* { m i n } } ) } ,", + "type": "interline_equation", + "image_path": "affb1be95e3ec6eca81bc912efabcc91b8769aec9401d359158fed6344f9955d.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 160, + 424, + 175.5 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 186, + 175.5, + 424, + 191.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 194, + 468, + 209 + ], + "lines": [ + { + "bbox": [ + 104, + 192, + 471, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 226, + 212 + ], + "score": 1.0, + "content": "where the last inequality uses", + "type": "text" + }, + { + "bbox": [ + 227, + 195, + 294, + 209 + ], + "score": 0.93, + "content": "\\begin{array} { r } { f ( \\tilde { \\gamma } \\rho ^ { L } ) = \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 192, + 312, + 212 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 195, + 371, + 207 + ], + "score": 0.92, + "content": "\\mathcal { L } \\geq e ^ { - f ( q _ { \\mathrm { m i n } } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 192, + 471, + 212 + ], + "score": 1.0, + "content": ". Combining these gives", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 213, + 453, + 247 + ], + "lines": [ + { + "bbox": [ + 156, + 213, + 453, + 247 + ], + "spans": [ + { + "bbox": [ + 156, + 213, + 453, + 247 + ], + "score": 0.94, + "content": "\\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) \\leq \\frac { 2 K q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } } { L f ( \\tilde { \\gamma } \\rho ^ { L } ) } \\sum _ { n = 1 } ^ { N } e ^ { f ( q _ { \\operatorname* { m i n } } ) - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) .", + "type": "interline_equation", + "image_path": "15cb0d557a767c7431bba1921d27bb346a1ecdf6ff2f0721ac08ec3a1923f999.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 156, + 213, + 453, + 224.33333333333334 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 156, + 224.33333333333334, + 453, + 235.66666666666669 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 156, + 235.66666666666669, + 453, + 247.00000000000003 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 251, + 504, + 288 + ], + "lines": [ + { + "bbox": [ + 104, + 250, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 250, + 117, + 266 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 117, + 253, + 164, + 263 + ], + "score": 0.89, + "content": "q _ { n } > q _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 250, + 357, + 266 + ], + "score": 1.0, + "content": ", then by the mean value theorem there exists", + "type": "text" + }, + { + "bbox": [ + 357, + 252, + 425, + 263 + ], + "score": 0.92, + "content": "\\xi _ { n } ~ \\in ~ ( q _ { \\operatorname* { m i n } } , q _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 250, + 468, + 266 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 468, + 252, + 505, + 263 + ], + "score": 0.9, + "content": "f ( q _ { n } ) \\mathrm { ~ - ~ }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 261, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 223, + 276 + ], + "score": 0.89, + "content": "f ( q _ { \\mathrm { m i n } } ) = f ^ { \\prime } ( \\xi _ { n } ) ( q _ { n } - q _ { \\mathrm { m i n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 261, + 376, + 277 + ], + "score": 1.0, + "content": ". By Assumption (B3.4), we know that", + "type": "text" + }, + { + "bbox": [ + 376, + 263, + 501, + 276 + ], + "score": 0.9, + "content": "f ^ { \\prime } ( q _ { n } ) \\leq K ^ { \\lceil \\log _ { 2 } ( q _ { n } / \\xi _ { n } ) \\rceil } f ^ { \\prime } ( \\xi _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 261, + 506, + 277 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 273, + 425, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 148, + 288 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 148, + 274, + 362, + 288 + ], + "score": 0.45, + "content": "\\begin{array} { r } { \\lceil \\log _ { 2 } ( q _ { n } / \\xi _ { n } ) \\rceil \\leq \\log _ { 2 } ( 2 B _ { 1 } \\rho ^ { L } / q _ { \\mathrm { m i n } } ) \\leq \\log _ { 2 } ( 2 B _ { 1 } / \\tilde { \\gamma } ) . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 273, + 425, + 288 + ], + "score": 1.0, + "content": ". Then we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 292, + 496, + 356 + ], + "lines": [ + { + "bbox": [ + 112, + 292, + 496, + 356 + ], + "spans": [ + { + "bbox": [ + 112, + 292, + 496, + 356 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\displaystyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) \\le \\frac { 2 K q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } } { L f ( \\tilde { \\gamma } \\rho ^ { L } ) } K ^ { \\log _ { 2 } ( 2 B _ { 1 } / \\tilde { \\gamma } ) } \\displaystyle \\sum _ { n \\colon { q _ { n } \\ne q _ { \\operatorname* { m i n } } } } e ^ { - f ^ { \\prime } ( \\xi _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) } f ^ { \\prime } ( \\xi _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) } \\\\ & { \\displaystyle \\le \\frac { 2 K \\tilde { \\gamma } ^ { - 2 / L } \\rho ^ { 2 } } { L f ( \\tilde { \\gamma } \\rho ^ { L } ) } K ^ { \\log _ { 2 } ( 2 B _ { 1 } / \\tilde { \\gamma } ) } \\cdot N e } \\end{array}", + "type": "interline_equation", + "image_path": "440dc95ad072a4d575fa597b8aa69c00b5ac0cde52c186d30ecd36b3b4fa4b86.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 112, + 292, + 496, + 313.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 112, + 313.3333333333333, + 496, + 334.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 112, + 334.66666666666663, + 496, + 355.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 360, + 503, + 386 + ], + "lines": [ + { + "bbox": [ + 151, + 352, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 151, + 352, + 164, + 385 + ], + "score": 1.0, + "content": "sec", + "type": "text" + }, + { + "bbox": [ + 200, + 352, + 243, + 385 + ], + "score": 1.0, + "content": "uality uses", + "type": "text" + }, + { + "bbox": [ + 243, + 360, + 320, + 375 + ], + "score": 0.93, + "content": "q _ { \\mathrm { m i n } } ^ { - 2 / L } \\rho ^ { 2 } \\le \\tilde { \\gamma } ^ { - 2 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 352, + 328, + 385 + ], + "score": 1.0, + "content": "b", + "type": "text" + }, + { + "bbox": [ + 353, + 352, + 505, + 385 + ], + "score": 1.0, + "content": "ma A.5 and the fact that the function", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 374, + 353, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 151, + 384 + ], + "score": 0.91, + "content": "x \\mapsto e ^ { - x } x", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 374, + 200, + 386 + ], + "score": 0.91, + "content": "( 0 , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 374, + 311, + 386 + ], + "score": 1.0, + "content": "attains the maximum value", + "type": "text" + }, + { + "bbox": [ + 312, + 377, + 317, + 384 + ], + "score": 0.77, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 375, + 353, + 384 + ], + "score": 0.88, + "content": "x = 1", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 297, + 411 + ], + "score": 1.0, + "content": "By Theorem A.7, we have already known that", + "type": "text" + }, + { + "bbox": [ + 297, + 398, + 327, + 408 + ], + "score": 0.9, + "content": "\\mathcal { L } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 396, + 429, + 411 + ], + "score": 1.0, + "content": ". So it remains to bound", + "type": "text" + }, + { + "bbox": [ + 429, + 398, + 448, + 410 + ], + "score": 0.92, + "content": "\\beta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 396, + 506, + 411 + ], + "score": 1.0, + "content": ". For this, we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 408, + 354, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 330, + 421 + ], + "score": 1.0, + "content": "first prove the following lemma to bound the integral of", + "type": "text" + }, + { + "bbox": [ + 331, + 409, + 349, + 421 + ], + "score": 0.91, + "content": "\\beta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 408, + 354, + 421 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 423, + 248, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 249, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 193, + 438 + ], + "score": 1.0, + "content": "Lemma C.9. For all", + "type": "text" + }, + { + "bbox": [ + 193, + 424, + 245, + 435 + ], + "score": 0.9, + "content": "t _ { 2 } > t _ { 1 } \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 421, + 249, + 438 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 439, + 413, + 467 + ], + "lines": [ + { + "bbox": [ + 196, + 439, + 413, + 467 + ], + "spans": [ + { + "bbox": [ + 196, + 439, + 413, + 467 + ], + "score": 0.91, + "content": "\\int _ { t _ { 1 } } ^ { t _ { 2 } } \\left( \\beta ( \\tau ) ^ { - 2 } - 1 \\right) \\cdot \\frac { d } { d \\tau } \\log \\rho ( \\tau ) \\cdot d \\tau \\leq \\frac { 1 } { L } \\log \\frac { \\tilde { \\gamma } ( t _ { 2 } ) } { \\tilde { \\gamma } ( t _ { 1 } ) } .", + "type": "interline_equation", + "image_path": "dac78b74f96b4e463429a84c11ed07ee6980b2a764b99211ea13d8234d00f21a.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 196, + 439, + 413, + 467 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 504, + 496 + ], + "lines": [ + { + "bbox": [ + 99, + 475, + 501, + 502 + ], + "spans": [ + { + "bbox": [ + 99, + 475, + 202, + 502 + ], + "score": 1.0, + "content": "Proof. By Lemma B.4,", + "type": "text" + }, + { + "bbox": [ + 202, + 479, + 302, + 496 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho = \\frac 1 { 2 \\rho ^ { 2 } } \\frac { d \\rho ^ { 2 } } { d t } = \\frac { L \\nu } { \\rho ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 475, + 333, + 502 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 333, + 482, + 356, + 492 + ], + "score": 0.87, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 475, + 456, + 502 + ], + "score": 1.0, + "content": ". By Lemma B.1, for a.e.", + "type": "text" + }, + { + "bbox": [ + 456, + 481, + 501, + 493 + ], + "score": 0.93, + "content": "t \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 499, + 380, + 536 + ], + "lines": [ + { + "bbox": [ + 231, + 499, + 380, + 536 + ], + "spans": [ + { + "bbox": [ + 231, + 499, + 380, + 536 + ], + "score": 0.95, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\cdot \\left. \\frac { \\rho ^ { 2 } } { L \\nu } \\frac { d \\hat { \\pmb \\theta } } { d t } \\right. _ { 2 } ^ { 2 } \\cdot \\frac { d } { d t } \\log \\rho .", + "type": "interline_equation", + "image_path": "6508e6afe1e2ffd15241a4c742d42e547fb849be66ea1cf6e12ad431825d7fdd.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 231, + 499, + 380, + 517.5 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 231, + 517.5, + 380, + 536.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 321, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 321, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 180, + 556 + ], + "score": 1.0, + "content": "By the chain rule,", + "type": "text" + }, + { + "bbox": [ + 181, + 540, + 267, + 558 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d \\hat { \\pmb { \\theta } } } { d t } = \\frac { 1 } { \\rho } ( \\pmb { I } - \\hat { \\pmb { \\theta } } \\hat { \\pmb { \\theta } } ^ { \\top } ) \\frac { d \\pmb { \\theta } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 540, + 321, + 556 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 561, + 457, + 608 + ], + "lines": [ + { + "bbox": [ + 154, + 561, + 457, + 608 + ], + "spans": [ + { + "bbox": [ + 154, + 561, + 457, + 608 + ], + "score": 0.94, + "content": "\\left\\| \\frac { \\rho ^ { 2 } } { L \\nu } \\frac { d \\hat { \\theta } } { d t } \\right\\| _ { 2 } ^ { 2 } = \\left\\| \\frac { \\rho } { L \\nu ( t ) } ( I - \\hat { \\theta } \\hat { \\theta } ^ { \\top } ) \\frac { d \\theta } { d t } \\right\\| _ { 2 } ^ { 2 } = \\frac { \\left\\| \\frac { d \\theta } { d t } \\right\\| _ { 2 } ^ { 2 } - \\left. \\frac { d \\theta } { d t } , \\hat { \\theta } \\right. ^ { 2 } } { \\left. \\frac { d \\theta } { d t } , \\hat { \\theta } \\right. ^ { 2 } } = \\beta ^ { - 2 } - 1 .", + "type": "interline_equation", + "image_path": "210dfa36e4abcc2585e666992efad29ac29c5f3292a0db70948a091544ed1133.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 154, + 561, + 457, + 576.6666666666666 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 154, + 576.6666666666666, + 457, + 592.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 154, + 592.3333333333333, + 457, + 607.9999999999999 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 457, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 458, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 319, + 625 + ], + "score": 1.0, + "content": "where the last equality follows from the definition of", + "type": "text" + }, + { + "bbox": [ + 319, + 612, + 326, + 623 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 610, + 458, + 625 + ], + "score": 1.0, + "content": ". Combining 14 and 15, we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 627, + 377, + 651 + ], + "lines": [ + { + "bbox": [ + 234, + 627, + 377, + 651 + ], + "spans": [ + { + "bbox": [ + 234, + 627, + 377, + 651 + ], + "score": 0.93, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\left( \\beta ^ { - 2 } - 1 \\right) \\cdot \\frac { d } { d t } \\log \\rho .", + "type": "interline_equation", + "image_path": "bfd5cbfa6b4b2fa170dd08b40dc378686678d761b74ae96f5f573362ffca551e.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 234, + 627, + 377, + 651 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 335, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 335, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 229, + 668 + ], + "score": 1.0, + "content": "Integrating on both sides from", + "type": "text" + }, + { + "bbox": [ + 230, + 656, + 239, + 666 + ], + "score": 0.88, + "content": "t _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 654, + 250, + 668 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 250, + 655, + 259, + 666 + ], + "score": 0.87, + "content": "t _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 654, + 335, + 668 + ], + "score": 1.0, + "content": "proves the lemma.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 677, + 508, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 385, + 690 + ], + "score": 1.0, + "content": "A direct corollary of Lemma C.9 is the upper bound for the minimum", + "type": "text" + }, + { + "bbox": [ + 385, + 678, + 414, + 690 + ], + "score": 0.91, + "content": "\\beta ^ { 2 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "within a time interval:", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 691, + 420, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 207, + 705 + ], + "score": 1.0, + "content": "Corollary C.10. For all", + "type": "text" + }, + { + "bbox": [ + 207, + 693, + 258, + 704 + ], + "score": 0.91, + "content": "t _ { 2 } > t _ { 1 } \\ge t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 691, + 329, + 705 + ], + "score": 1.0, + "content": ", then there exists", + "type": "text" + }, + { + "bbox": [ + 329, + 693, + 379, + 705 + ], + "score": 0.93, + "content": "t _ { * } \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 691, + 420, + 705 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 709, + 391, + 736 + ], + "lines": [ + { + "bbox": [ + 219, + 709, + 391, + 736 + ], + "spans": [ + { + "bbox": [ + 219, + 709, + 391, + 736 + ], + "score": 0.94, + "content": "\\beta ( t _ { * } ) ^ { - 2 } - 1 \\le \\frac { 1 } { L } \\cdot \\frac { \\log \\tilde { \\gamma } ( t _ { 2 } ) - \\log \\tilde { \\gamma } ( t _ { 1 } ) } { \\log \\rho ( t _ { 2 } ) - \\log \\rho ( t _ { 1 } ) } .", + "type": "interline_equation", + "image_path": "8316fe2ff9b5754e266daa69050f0d52f561584e2ad688a3dfc2fd823952a4e6.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 219, + 709, + 391, + 736 + ], + "spans": [], + "index": 35 + } + ] + } + ], + "page_idx": 23, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 16, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 654, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 495, + 656, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 495, + 656, + 505, + 666 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 300, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 302, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 302, + 96 + ], + "score": 1.0, + "content": "Proof for (13). According to our construction,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 302, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 98, + 431, + 133 + ], + "lines": [ + { + "bbox": [ + 178, + 98, + 431, + 133 + ], + "spans": [ + { + "bbox": [ + 178, + 98, + 431, + 133 + ], + "score": 0.94, + "content": "\\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) = \\frac { q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho } { \\| h \\| _ { 2 } } \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) .", + "type": "interline_equation", + "image_path": "a57c568281e51a23a2ff29bba9b5b1f407d0bd6541d8850dfe0f80d7b78f18a2.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 178, + 98, + 431, + 109.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 178, + 109.66666666666667, + 431, + 121.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 178, + 121.33333333333334, + 431, + 133.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 136, + 419, + 155 + ], + "lines": [ + { + "bbox": [ + 104, + 136, + 422, + 157 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 146, + 155 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 136, + 246, + 157 + ], + "score": 0.95, + "content": "\\| h \\| _ { 2 } \\geq \\left. h , \\hat { \\theta } \\right. = L \\nu / \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 136, + 422, + 155 + ], + "score": 1.0, + "content": ". By Lemma B.5 and Lemma D.1, we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 136, + 422, + 157 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 160, + 424, + 191 + ], + "lines": [ + { + "bbox": [ + 186, + 160, + 424, + 191 + ], + "spans": [ + { + "bbox": [ + 186, + 160, + 424, + 191 + ], + "score": 0.93, + "content": "\\nu \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } \\geq \\frac { 1 } { 2 K } \\log \\frac { 1 } { \\mathcal { L } } \\cdot \\mathcal { L } \\geq \\frac { 1 } { 2 K } f ( \\widetilde { \\gamma } \\rho ^ { L } ) e ^ { - f ( q _ { \\operatorname* { m i n } } ) } ,", + "type": "interline_equation", + "image_path": "affb1be95e3ec6eca81bc912efabcc91b8769aec9401d359158fed6344f9955d.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 160, + 424, + 175.5 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 186, + 175.5, + 424, + 191.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 194, + 468, + 209 + ], + "lines": [ + { + "bbox": [ + 104, + 192, + 471, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 226, + 212 + ], + "score": 1.0, + "content": "where the last inequality uses", + "type": "text" + }, + { + "bbox": [ + 227, + 195, + 294, + 209 + ], + "score": 0.93, + "content": "\\begin{array} { r } { f ( \\tilde { \\gamma } \\rho ^ { L } ) = \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 192, + 312, + 212 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 195, + 371, + 207 + ], + "score": 0.92, + "content": "\\mathcal { L } \\geq e ^ { - f ( q _ { \\mathrm { m i n } } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 192, + 471, + 212 + ], + "score": 1.0, + "content": ". Combining these gives", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 192, + 471, + 212 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 213, + 453, + 247 + ], + "lines": [ + { + "bbox": [ + 156, + 213, + 453, + 247 + ], + "spans": [ + { + "bbox": [ + 156, + 213, + 453, + 247 + ], + "score": 0.94, + "content": "\\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) \\leq \\frac { 2 K q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } } { L f ( \\tilde { \\gamma } \\rho ^ { L } ) } \\sum _ { n = 1 } ^ { N } e ^ { f ( q _ { \\operatorname* { m i n } } ) - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) .", + "type": "interline_equation", + "image_path": "15cb0d557a767c7431bba1921d27bb346a1ecdf6ff2f0721ac08ec3a1923f999.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 156, + 213, + 453, + 224.33333333333334 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 156, + 224.33333333333334, + 453, + 235.66666666666669 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 156, + 235.66666666666669, + 453, + 247.00000000000003 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 251, + 504, + 288 + ], + "lines": [ + { + "bbox": [ + 104, + 250, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 250, + 117, + 266 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 117, + 253, + 164, + 263 + ], + "score": 0.89, + "content": "q _ { n } > q _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 250, + 357, + 266 + ], + "score": 1.0, + "content": ", then by the mean value theorem there exists", + "type": "text" + }, + { + "bbox": [ + 357, + 252, + 425, + 263 + ], + "score": 0.92, + "content": "\\xi _ { n } ~ \\in ~ ( q _ { \\operatorname* { m i n } } , q _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 250, + 468, + 266 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 468, + 252, + 505, + 263 + ], + "score": 0.9, + "content": "f ( q _ { n } ) \\mathrm { ~ - ~ }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 261, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 223, + 276 + ], + "score": 0.89, + "content": "f ( q _ { \\mathrm { m i n } } ) = f ^ { \\prime } ( \\xi _ { n } ) ( q _ { n } - q _ { \\mathrm { m i n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 261, + 376, + 277 + ], + "score": 1.0, + "content": ". By Assumption (B3.4), we know that", + "type": "text" + }, + { + "bbox": [ + 376, + 263, + 501, + 276 + ], + "score": 0.9, + "content": "f ^ { \\prime } ( q _ { n } ) \\leq K ^ { \\lceil \\log _ { 2 } ( q _ { n } / \\xi _ { n } ) \\rceil } f ^ { \\prime } ( \\xi _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 261, + 506, + 277 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 273, + 425, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 148, + 288 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 148, + 274, + 362, + 288 + ], + "score": 0.45, + "content": "\\begin{array} { r } { \\lceil \\log _ { 2 } ( q _ { n } / \\xi _ { n } ) \\rceil \\leq \\log _ { 2 } ( 2 B _ { 1 } \\rho ^ { L } / q _ { \\mathrm { m i n } } ) \\leq \\log _ { 2 } ( 2 B _ { 1 } / \\tilde { \\gamma } ) . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 273, + 425, + 288 + ], + "score": 1.0, + "content": ". Then we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 250, + 506, + 288 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 292, + 496, + 356 + ], + "lines": [ + { + "bbox": [ + 112, + 292, + 496, + 356 + ], + "spans": [ + { + "bbox": [ + 112, + 292, + 496, + 356 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\displaystyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } ( q _ { n } ( \\tilde { \\theta } ) - 1 ) \\le \\frac { 2 K q _ { \\operatorname* { m i n } } ^ { - 2 / L } \\rho ^ { 2 } } { L f ( \\tilde { \\gamma } \\rho ^ { L } ) } K ^ { \\log _ { 2 } ( 2 B _ { 1 } / \\tilde { \\gamma } ) } \\displaystyle \\sum _ { n \\colon { q _ { n } \\ne q _ { \\operatorname* { m i n } } } } e ^ { - f ^ { \\prime } ( \\xi _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) } f ^ { \\prime } ( \\xi _ { n } ) ( q _ { n } - q _ { \\operatorname* { m i n } } ) } \\\\ & { \\displaystyle \\le \\frac { 2 K \\tilde { \\gamma } ^ { - 2 / L } \\rho ^ { 2 } } { L f ( \\tilde { \\gamma } \\rho ^ { L } ) } K ^ { \\log _ { 2 } ( 2 B _ { 1 } / \\tilde { \\gamma } ) } \\cdot N e } \\end{array}", + "type": "interline_equation", + "image_path": "440dc95ad072a4d575fa597b8aa69c00b5ac0cde52c186d30ecd36b3b4fa4b86.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 112, + 292, + 496, + 313.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 112, + 313.3333333333333, + 496, + 334.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 112, + 334.66666666666663, + 496, + 355.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 360, + 503, + 386 + ], + "lines": [ + { + "bbox": [ + 151, + 352, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 151, + 352, + 164, + 385 + ], + "score": 1.0, + "content": "sec", + "type": "text" + }, + { + "bbox": [ + 200, + 352, + 243, + 385 + ], + "score": 1.0, + "content": "uality uses", + "type": "text" + }, + { + "bbox": [ + 243, + 360, + 320, + 375 + ], + "score": 0.93, + "content": "q _ { \\mathrm { m i n } } ^ { - 2 / L } \\rho ^ { 2 } \\le \\tilde { \\gamma } ^ { - 2 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 352, + 328, + 385 + ], + "score": 1.0, + "content": "b", + "type": "text" + }, + { + "bbox": [ + 353, + 352, + 505, + 385 + ], + "score": 1.0, + "content": "ma A.5 and the fact that the function", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 374, + 353, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 151, + 384 + ], + "score": 0.91, + "content": "x \\mapsto e ^ { - x } x", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 374, + 200, + 386 + ], + "score": 0.91, + "content": "( 0 , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 374, + 311, + 386 + ], + "score": 1.0, + "content": "attains the maximum value", + "type": "text" + }, + { + "bbox": [ + 312, + 377, + 317, + 384 + ], + "score": 0.77, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 375, + 353, + 384 + ], + "score": 0.88, + "content": "x = 1", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 106, + 352, + 505, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 297, + 411 + ], + "score": 1.0, + "content": "By Theorem A.7, we have already known that", + "type": "text" + }, + { + "bbox": [ + 297, + 398, + 327, + 408 + ], + "score": 0.9, + "content": "\\mathcal { L } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 396, + 429, + 411 + ], + "score": 1.0, + "content": ". So it remains to bound", + "type": "text" + }, + { + "bbox": [ + 429, + 398, + 448, + 410 + ], + "score": 0.92, + "content": "\\beta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 396, + 506, + 411 + ], + "score": 1.0, + "content": ". For this, we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 408, + 354, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 330, + 421 + ], + "score": 1.0, + "content": "first prove the following lemma to bound the integral of", + "type": "text" + }, + { + "bbox": [ + 331, + 409, + 349, + 421 + ], + "score": 0.91, + "content": "\\beta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 408, + 354, + 421 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 396, + 506, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 423, + 248, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 249, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 193, + 438 + ], + "score": 1.0, + "content": "Lemma C.9. For all", + "type": "text" + }, + { + "bbox": [ + 193, + 424, + 245, + 435 + ], + "score": 0.9, + "content": "t _ { 2 } > t _ { 1 } \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 421, + 249, + 438 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 421, + 249, + 438 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 439, + 413, + 467 + ], + "lines": [ + { + "bbox": [ + 196, + 439, + 413, + 467 + ], + "spans": [ + { + "bbox": [ + 196, + 439, + 413, + 467 + ], + "score": 0.91, + "content": "\\int _ { t _ { 1 } } ^ { t _ { 2 } } \\left( \\beta ( \\tau ) ^ { - 2 } - 1 \\right) \\cdot \\frac { d } { d \\tau } \\log \\rho ( \\tau ) \\cdot d \\tau \\leq \\frac { 1 } { L } \\log \\frac { \\tilde { \\gamma } ( t _ { 2 } ) } { \\tilde { \\gamma } ( t _ { 1 } ) } .", + "type": "interline_equation", + "image_path": "dac78b74f96b4e463429a84c11ed07ee6980b2a764b99211ea13d8234d00f21a.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 196, + 439, + 413, + 467 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 504, + 496 + ], + "lines": [ + { + "bbox": [ + 99, + 475, + 501, + 502 + ], + "spans": [ + { + "bbox": [ + 99, + 475, + 202, + 502 + ], + "score": 1.0, + "content": "Proof. By Lemma B.4,", + "type": "text" + }, + { + "bbox": [ + 202, + 479, + 302, + 496 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho = \\frac 1 { 2 \\rho ^ { 2 } } \\frac { d \\rho ^ { 2 } } { d t } = \\frac { L \\nu } { \\rho ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 475, + 333, + 502 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 333, + 482, + 356, + 492 + ], + "score": 0.87, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 475, + 456, + 502 + ], + "score": 1.0, + "content": ". By Lemma B.1, for a.e.", + "type": "text" + }, + { + "bbox": [ + 456, + 481, + 501, + 493 + ], + "score": 0.93, + "content": "t \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 99, + 475, + 501, + 502 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 499, + 380, + 536 + ], + "lines": [ + { + "bbox": [ + 231, + 499, + 380, + 536 + ], + "spans": [ + { + "bbox": [ + 231, + 499, + 380, + 536 + ], + "score": 0.95, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\cdot \\left. \\frac { \\rho ^ { 2 } } { L \\nu } \\frac { d \\hat { \\pmb \\theta } } { d t } \\right. _ { 2 } ^ { 2 } \\cdot \\frac { d } { d t } \\log \\rho .", + "type": "interline_equation", + "image_path": "6508e6afe1e2ffd15241a4c742d42e547fb849be66ea1cf6e12ad431825d7fdd.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 231, + 499, + 380, + 517.5 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 231, + 517.5, + 380, + 536.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 321, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 321, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 180, + 556 + ], + "score": 1.0, + "content": "By the chain rule,", + "type": "text" + }, + { + "bbox": [ + 181, + 540, + 267, + 558 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d \\hat { \\pmb { \\theta } } } { d t } = \\frac { 1 } { \\rho } ( \\pmb { I } - \\hat { \\pmb { \\theta } } \\hat { \\pmb { \\theta } } ^ { \\top } ) \\frac { d \\pmb { \\theta } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 540, + 321, + 556 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 540, + 321, + 558 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 561, + 457, + 608 + ], + "lines": [ + { + "bbox": [ + 154, + 561, + 457, + 608 + ], + "spans": [ + { + "bbox": [ + 154, + 561, + 457, + 608 + ], + "score": 0.94, + "content": "\\left\\| \\frac { \\rho ^ { 2 } } { L \\nu } \\frac { d \\hat { \\theta } } { d t } \\right\\| _ { 2 } ^ { 2 } = \\left\\| \\frac { \\rho } { L \\nu ( t ) } ( I - \\hat { \\theta } \\hat { \\theta } ^ { \\top } ) \\frac { d \\theta } { d t } \\right\\| _ { 2 } ^ { 2 } = \\frac { \\left\\| \\frac { d \\theta } { d t } \\right\\| _ { 2 } ^ { 2 } - \\left. \\frac { d \\theta } { d t } , \\hat { \\theta } \\right. ^ { 2 } } { \\left. \\frac { d \\theta } { d t } , \\hat { \\theta } \\right. ^ { 2 } } = \\beta ^ { - 2 } - 1 .", + "type": "interline_equation", + "image_path": "210dfa36e4abcc2585e666992efad29ac29c5f3292a0db70948a091544ed1133.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 154, + 561, + 457, + 576.6666666666666 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 154, + 576.6666666666666, + 457, + 592.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 154, + 592.3333333333333, + 457, + 607.9999999999999 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 457, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 458, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 319, + 625 + ], + "score": 1.0, + "content": "where the last equality follows from the definition of", + "type": "text" + }, + { + "bbox": [ + 319, + 612, + 326, + 623 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 610, + 458, + 625 + ], + "score": 1.0, + "content": ". Combining 14 and 15, we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 610, + 458, + 625 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 627, + 377, + 651 + ], + "lines": [ + { + "bbox": [ + 234, + 627, + 377, + 651 + ], + "spans": [ + { + "bbox": [ + 234, + 627, + 377, + 651 + ], + "score": 0.93, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq L \\left( \\beta ^ { - 2 } - 1 \\right) \\cdot \\frac { d } { d t } \\log \\rho .", + "type": "interline_equation", + "image_path": "bfd5cbfa6b4b2fa170dd08b40dc378686678d761b74ae96f5f573362ffca551e.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 234, + 627, + 377, + 651 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 335, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 335, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 229, + 668 + ], + "score": 1.0, + "content": "Integrating on both sides from", + "type": "text" + }, + { + "bbox": [ + 230, + 656, + 239, + 666 + ], + "score": 0.88, + "content": "t _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 654, + 250, + 668 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 250, + 655, + 259, + 666 + ], + "score": 0.87, + "content": "t _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 654, + 335, + 668 + ], + "score": 1.0, + "content": "proves the lemma.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 654, + 335, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 677, + 508, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 385, + 690 + ], + "score": 1.0, + "content": "A direct corollary of Lemma C.9 is the upper bound for the minimum", + "type": "text" + }, + { + "bbox": [ + 385, + 678, + 414, + 690 + ], + "score": 0.91, + "content": "\\beta ^ { 2 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "within a time interval:", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 691, + 420, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 207, + 705 + ], + "score": 1.0, + "content": "Corollary C.10. For all", + "type": "text" + }, + { + "bbox": [ + 207, + 693, + 258, + 704 + ], + "score": 0.91, + "content": "t _ { 2 } > t _ { 1 } \\ge t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 691, + 329, + 705 + ], + "score": 1.0, + "content": ", then there exists", + "type": "text" + }, + { + "bbox": [ + 329, + 693, + 379, + 705 + ], + "score": 0.93, + "content": "t _ { * } \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 691, + 420, + 705 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 677, + 505, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 709, + 391, + 736 + ], + "lines": [ + { + "bbox": [ + 219, + 709, + 391, + 736 + ], + "spans": [ + { + "bbox": [ + 219, + 709, + 391, + 736 + ], + "score": 0.94, + "content": "\\beta ( t _ { * } ) ^ { - 2 } - 1 \\le \\frac { 1 } { L } \\cdot \\frac { \\log \\tilde { \\gamma } ( t _ { 2 } ) - \\log \\tilde { \\gamma } ( t _ { 1 } ) } { \\log \\rho ( t _ { 2 } ) - \\log \\rho ( t _ { 1 } ) } .", + "type": "interline_equation", + "image_path": "8316fe2ff9b5754e266daa69050f0d52f561584e2ad688a3dfc2fd823952a4e6.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 219, + 709, + 391, + 736 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 214, + 96 + ], + "score": 1.0, + "content": "Proof. Denote the RHS as", + "type": "text" + }, + { + "bbox": [ + 214, + 83, + 223, + 92 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 79, + 339, + 96 + ], + "score": 1.0, + "content": ". Assume to the contrary that", + "type": "text" + }, + { + "bbox": [ + 340, + 82, + 408, + 95 + ], + "score": 0.92, + "content": "\\beta ( \\tau ) ^ { - 2 } - 1 > C", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 79, + 439, + 96 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 439, + 82, + 486, + 95 + ], + "score": 0.93, + "content": "\\tau \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 79, + 506, + 96 + ], + "score": 1.0, + "content": ". By", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 417, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 158, + 106 + ], + "score": 1.0, + "content": "Lemma B.1,", + "type": "text" + }, + { + "bbox": [ + 159, + 93, + 211, + 106 + ], + "score": 0.92, + "content": "\\log \\rho ( \\tau ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 93, + 242, + 106 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 243, + 93, + 289, + 106 + ], + "score": 0.93, + "content": "\\tau \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 93, + 417, + 106 + ], + "score": 1.0, + "content": ". Then by Lemma C.9, we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 110, + 480, + 140 + ], + "lines": [ + { + "bbox": [ + 131, + 110, + 480, + 140 + ], + "spans": [ + { + "bbox": [ + 131, + 110, + 480, + 140 + ], + "score": 0.92, + "content": "\\frac { 1 } { L } \\log \\frac { \\tilde { \\gamma } ( t _ { 2 } ) } { \\tilde { \\gamma } ( t _ { 1 } ) } > \\int _ { t _ { 1 } } ^ { t _ { 2 } } { C \\cdot \\frac { d } { d \\tau } \\log \\rho ( \\tau ) \\cdot d \\tau } = C \\cdot ( \\log \\rho ( t _ { 2 } ) - \\log \\rho ( t _ { 1 } ) ) = \\frac { 1 } { L } \\log \\frac { \\tilde { \\gamma } ( t _ { 2 } ) } { \\tilde { \\gamma } ( t _ { 1 } ) } ,", + "type": "interline_equation", + "image_path": "70b4604217f9bf52fea8326ade97d9ae6b35ab4a1782b21ccbdc1610c54ae7f2.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 131, + 110, + 480, + 140 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 230, + 157 + ], + "lines": [ + { + "bbox": [ + 106, + 145, + 231, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 231, + 157 + ], + "score": 1.0, + "content": "which leads to a contradiction.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 105, + 170, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "In the rest of this section, we present both asymptotic and non-asymptotic analyses for the directional", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 181, + 317, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 295, + 193 + ], + "score": 1.0, + "content": "convergence by using Corollary C.10 to bound", + "type": "text" + }, + { + "bbox": [ + 295, + 181, + 313, + 193 + ], + "score": 0.91, + "content": "\\beta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 181, + 317, + 193 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 206, + 237, + 218 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 237, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 237, + 219 + ], + "score": 1.0, + "content": "C.4 ASYMPTOTIC ANALYSIS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 439, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 440, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 429, + 240 + ], + "score": 1.0, + "content": "We first prove an auxiliary lemma which gives an upper bound for the change of", + "type": "text" + }, + { + "bbox": [ + 429, + 226, + 436, + 237 + ], + "score": 0.83, + "content": "\\hat { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 225, + 440, + 240 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 231, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 232, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 201, + 257 + ], + "score": 1.0, + "content": "Lemma C.11. For a.e.", + "type": "text" + }, + { + "bbox": [ + 202, + 243, + 228, + 254 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 240, + 232, + 257 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 252, + 356, + 286 + ], + "lines": [ + { + "bbox": [ + 252, + 252, + 356, + 286 + ], + "spans": [ + { + "bbox": [ + 252, + 252, + 356, + 286 + ], + "score": 0.94, + "content": "\\left\\| \\frac { d \\hat { \\pmb \\theta } } { d t } \\right\\| _ { 2 } \\le \\frac { B _ { 1 } } { L \\tilde { \\gamma } } \\cdot \\frac { d } { d t } \\log \\rho .", + "type": "interline_equation", + "image_path": "800af00bec61077132a4a2cdac85a1e820897f5661056be725fa7507524b6f78.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 252, + 252, + 356, + 269.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 252, + 269.0, + 356, + 286.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 298, + 506, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 508, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 189, + 322 + ], + "score": 1.0, + "content": "Proof. Observe that", + "type": "text" + }, + { + "bbox": [ + 190, + 299, + 359, + 319 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\left\\| \\frac { d \\hat { \\pmb \\theta } } { d t } \\right\\| _ { 2 } = \\frac { 1 } { \\rho } \\left\\| ( \\pmb I - \\hat { \\pmb \\theta } \\hat { \\pmb \\theta } ^ { \\top } ) \\frac { d \\pmb \\theta } { d t } \\right\\| _ { 2 } \\leq \\frac { 1 } { \\rho } \\left\\| \\frac { d \\pmb \\theta } { d t } \\right\\| _ { 2 } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 295, + 459, + 322 + ], + "score": 1.0, + "content": ". It is sufficient to bound", + "type": "text" + }, + { + "bbox": [ + 459, + 301, + 486, + 317 + ], + "score": 0.93, + "content": "\\left\\| { \\frac { d \\theta } { d t } } \\right\\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 295, + 508, + 322 + ], + "score": 1.0, + "content": ". By", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 101, + 316, + 509, + 356 + ], + "spans": [ + { + "bbox": [ + 101, + 324, + 124, + 356 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 105, + 316, + 212, + 334 + ], + "score": 1.0, + "content": "the chain rule, there exists", + "type": "text" + }, + { + "bbox": [ + 124, + 330, + 265, + 345 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d \\theta } { d t } = \\sum _ { n \\in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) { h _ { n } ( t ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 320, + 330, + 331 + ], + "score": 0.9, + "content": "h _ { 1 } , \\ldots , h _ { N } : [ 0 , + \\infty ) \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 324, + 337, + 356 + ], + "score": 1.0, + "content": ". By definition of", + "type": "text" + }, + { + "bbox": [ + 330, + 316, + 420, + 334 + ], + "score": 1.0, + "content": "satisfying that for a.e.", + "type": "text" + }, + { + "bbox": [ + 337, + 332, + 350, + 343 + ], + "score": 0.8, + "content": "B _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 324, + 355, + 356 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 355, + 330, + 430, + 344 + ], + "score": 0.91, + "content": "\\| h _ { n } \\| _ { 2 } \\leq B _ { 1 } \\rho ^ { L - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 319, + 443, + 330 + ], + "score": 0.81, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 324, + 462, + 356 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 443, + 316, + 446, + 334 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 447, + 318, + 504, + 331 + ], + "score": 0.82, + "content": "h _ { n } ( t ) \\in \\partial ^ { \\circ } q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 332, + 487, + 342 + ], + "score": 0.88, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 324, + 509, + 356 + ], + "score": 1.0, + "content": ". So", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 359, + 405, + 425 + ], + "lines": [ + { + "bbox": [ + 204, + 359, + 405, + 425 + ], + "spans": [ + { + "bbox": [ + 204, + 359, + 405, + 425 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\left\\| \\frac { d \\pmb { \\theta } } { d t } \\right\\| _ { 2 } \\leq \\sum _ { n \\in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) \\left\\| \\pmb { h } _ { n } \\right\\| _ { 2 } } \\\\ { \\displaystyle \\qquad \\leq \\sum _ { n \\in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } \\cdot \\frac { 1 } { q _ { n } } \\cdot B _ { 1 } \\rho ^ { L - 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "952e58f4d575390690d4caad3e44e31d48b480a262f8961ef774e48e9cfa3e69.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 359, + 405, + 375.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 204, + 375.5, + 405, + 392.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 204, + 392.0, + 405, + 408.5 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 204, + 408.5, + 405, + 425.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 429, + 502, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 340, + 443 + ], + "score": 1.0, + "content": "Note that every summand is positive. By Lemma A.5,", + "type": "text" + }, + { + "bbox": [ + 340, + 431, + 352, + 441 + ], + "score": 0.84, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 427, + 442, + 443 + ], + "score": 1.0, + "content": "is lower-bounded by", + "type": "text" + }, + { + "bbox": [ + 443, + 430, + 504, + 441 + ], + "score": 0.87, + "content": "q _ { n } \\geq q _ { \\mathrm { m i n } } \\geq", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 439, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 142, + 453 + ], + "score": 0.91, + "content": "g ( \\log { \\frac { 1 } { \\mathcal { L } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 439, + 220, + 455 + ], + "score": 1.0, + "content": ", so we can replace", + "type": "text" + }, + { + "bbox": [ + 220, + 442, + 231, + 452 + ], + "score": 0.85, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 439, + 253, + 455 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 253, + 440, + 289, + 453 + ], + "score": 0.92, + "content": "g ( \\log { \\frac { 1 } { \\mathcal { L } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 439, + 506, + 455 + ], + "score": 1.0, + "content": "in the above inequality. Combining with the fact that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 451, + 284, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 209, + 468 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { n \\in [ N ] } { \\stackrel { \\sim } { e } } ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } \\quad } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 451, + 236, + 468 + ], + "score": 1.0, + "content": "is just", + "type": "text" + }, + { + "bbox": [ + 237, + 456, + 243, + 464 + ], + "score": 0.76, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 451, + 284, + 468 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 473, + 386, + 502 + ], + "lines": [ + { + "bbox": [ + 225, + 473, + 386, + 502 + ], + "spans": [ + { + "bbox": [ + 225, + 473, + 386, + 502 + ], + "score": 0.91, + "content": "\\left\\| \\frac { d \\pmb { \\theta } } { d t } \\right\\| _ { 2 } \\leq \\frac { \\nu } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\cdot B _ { 1 } \\rho ^ { L - 1 } = \\frac { B _ { 1 } \\nu } { \\tilde { \\gamma } \\rho } .", + "type": "interline_equation", + "image_path": "4bd63c906f7dc11a45aa59373ba0d8180344b4ec650568da529f5954ad5d2bfa.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 225, + 473, + 386, + 502 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 343, + 529 + ], + "lines": [ + { + "bbox": [ + 102, + 504, + 342, + 531 + ], + "spans": [ + { + "bbox": [ + 102, + 504, + 155, + 531 + ], + "score": 1.0, + "content": "So we have", + "type": "text" + }, + { + "bbox": [ + 155, + 508, + 342, + 528 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\left\\| \\frac { d \\hat { \\pmb \\theta } } { d t } \\right\\| _ { 2 } \\le \\frac { 1 } { \\rho } \\left\\| \\frac { d \\pmb \\theta } { d t } \\right\\| _ { 2 } \\le \\frac { B _ { 1 } } { L \\tilde { \\gamma } } \\cdot \\frac { L \\nu } { \\rho ^ { 2 } } = \\frac { B _ { 1 } } { L \\tilde { \\gamma } } \\cdot \\frac { d } { d t } \\log \\rho . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 104, + 539, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 325, + 558 + ], + "score": 1.0, + "content": "To prove Theorem A.8, we consider each limit point", + "type": "text" + }, + { + "bbox": [ + 325, + 541, + 382, + 555 + ], + "score": 0.93, + "content": "\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 539, + 505, + 558 + ], + "score": 1.0, + "content": ", and construct a series of ap-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 552, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 296, + 570 + ], + "score": 1.0, + "content": "proximate KKT points converging to it. Then", + "type": "text" + }, + { + "bbox": [ + 297, + 554, + 354, + 567 + ], + "score": 0.92, + "content": "\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 552, + 506, + 570 + ], + "score": 1.0, + "content": "can be shown to be a KKT point by", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 566, + 403, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 403, + 579 + ], + "score": 1.0, + "content": "Theorem C.4. The following lemma ensures that such construction exists.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 104, + 581, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 254, + 600 + ], + "score": 1.0, + "content": "Lemma C.12. For every limit point", + "type": "text" + }, + { + "bbox": [ + 254, + 584, + 261, + 595 + ], + "score": 0.62, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 581, + 273, + 600 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 273, + 581, + 334, + 601 + ], + "score": 0.93, + "content": "\\left\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 581, + 444, + 600 + ], + "score": 1.0, + "content": ", there exists a sequence of", + "type": "text" + }, + { + "bbox": [ + 444, + 585, + 504, + 597 + ], + "score": 0.91, + "content": "\\{ t _ { m } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 600, + 309, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 145, + 615 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 145, + 603, + 157, + 613 + ], + "score": 0.83, + "content": "t _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 600, + 236, + 614 + ], + "score": 0.8, + "content": "\\cdot + \\infty , \\hat { \\pmb { \\theta } } ( t _ { m } ) \\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 600, + 258, + 615 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 258, + 602, + 304, + 614 + ], + "score": 0.91, + "content": "\\beta ( t _ { m } ) 1", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 600, + 309, + 615 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 152, + 640 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 153, + 627, + 232, + 639 + ], + "score": 0.92, + "content": "\\{ \\epsilon _ { m } > 0 : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 626, + 354, + 640 + ], + "score": 1.0, + "content": "be an arbitrary sequence with", + "type": "text" + }, + { + "bbox": [ + 354, + 628, + 388, + 638 + ], + "score": 0.9, + "content": "\\epsilon _ { m } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 626, + 469, + 640 + ], + "score": 1.0, + "content": ". Now we construct", + "type": "text" + }, + { + "bbox": [ + 469, + 627, + 491, + 639 + ], + "score": 0.92, + "content": "\\left\\{ { { t } _ { m } } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 187, + 651 + ], + "score": 1.0, + "content": "induction. Suppose", + "type": "text" + }, + { + "bbox": [ + 187, + 639, + 280, + 649 + ], + "score": 0.88, + "content": "t _ { 1 } < t _ { 2 } < \\dots < t _ { m - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 637, + 435, + 651 + ], + "score": 1.0, + "content": "have already been constructed. Since", + "type": "text" + }, + { + "bbox": [ + 435, + 637, + 442, + 648 + ], + "score": 0.81, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "is a limit point", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 649, + 444, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 123, + 661 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 649, + 165, + 661 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( t ) \\uparrow \\tilde { \\gamma } _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 649, + 212, + 661 + ], + "score": 1.0, + "content": "(recall that", + "type": "text" + }, + { + "bbox": [ + 213, + 650, + 304, + 661 + ], + "score": 0.87, + "content": "\\tilde { \\gamma } _ { \\infty } : = \\mathrm { l i m } _ { t + \\infty } \\tilde { \\gamma } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 649, + 356, + 661 + ], + "score": 1.0, + "content": ", there exists", + "type": "text" + }, + { + "bbox": [ + 356, + 650, + 403, + 661 + ], + "score": 0.92, + "content": "s _ { m } > t _ { m - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 649, + 444, + 661 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 666, + 406, + 693 + ], + "lines": [ + { + "bbox": [ + 202, + 666, + 406, + 693 + ], + "spans": [ + { + "bbox": [ + 202, + 666, + 406, + 693 + ], + "score": 0.91, + "content": "\\left\\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\right\\| _ { 2 } \\leq \\epsilon _ { m } \\quad \\mathrm { a n d } \\quad \\frac { 1 } { L } \\log \\frac { \\widetilde { \\gamma } _ { \\infty } } { \\widetilde { \\gamma } ( s _ { m } ) } \\leq \\epsilon _ { m } ^ { 3 } .", + "type": "interline_equation", + "image_path": "bcb8951d33d8c5dbf3c118211ee2e04e3fe19fe8f13713f580a9a223007cb6f8.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 202, + 666, + 406, + 693 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 123, + 712 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 124, + 698, + 168, + 711 + ], + "score": 0.91, + "content": "s _ { m } ^ { \\prime } > s _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 697, + 254, + 712 + ], + "score": 1.0, + "content": "be a time such that", + "type": "text" + }, + { + "bbox": [ + 254, + 698, + 379, + 711 + ], + "score": 0.9, + "content": "\\log { \\rho ( s _ { m } ^ { \\prime } ) } = \\log { \\rho ( s _ { m } ) } + \\epsilon _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 697, + 506, + 712 + ], + "score": 1.0, + "content": ". According to Theorem A.7,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 706, + 507, + 725 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 160, + 721 + ], + "score": 0.88, + "content": "\\log \\rho \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 706, + 176, + 725 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 176, + 710, + 190, + 722 + ], + "score": 0.9, + "content": "s _ { m } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 706, + 292, + 725 + ], + "score": 1.0, + "content": "must exist. We construct", + "type": "text" + }, + { + "bbox": [ + 292, + 711, + 353, + 722 + ], + "score": 0.9, + "content": "t _ { m } \\in ( s _ { m } , s _ { m } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 706, + 422, + 725 + ], + "score": 1.0, + "content": "to be a time that", + "type": "text" + }, + { + "bbox": [ + 422, + 709, + 501, + 722 + ], + "score": 0.92, + "content": "\\bar { \\beta } ( t _ { m } ) ^ { - 2 } - 1 \\leq \\epsilon _ { m } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 706, + 507, + 725 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 720, + 321, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 321, + 732 + ], + "score": 1.0, + "content": "where the existence can be shown by Corollary C.10.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + } + ], + "page_idx": 24, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 16, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 511, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 495, + 513, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 495, + 513, + 505, + 523 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 145, + 505, + 156 + ], + "lines": [ + { + "bbox": [ + 495, + 146, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 495, + 146, + 505, + 156 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 214, + 96 + ], + "score": 1.0, + "content": "Proof. Denote the RHS as", + "type": "text" + }, + { + "bbox": [ + 214, + 83, + 223, + 92 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 79, + 339, + 96 + ], + "score": 1.0, + "content": ". Assume to the contrary that", + "type": "text" + }, + { + "bbox": [ + 340, + 82, + 408, + 95 + ], + "score": 0.92, + "content": "\\beta ( \\tau ) ^ { - 2 } - 1 > C", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 79, + 439, + 96 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 439, + 82, + 486, + 95 + ], + "score": 0.93, + "content": "\\tau \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 79, + 506, + 96 + ], + "score": 1.0, + "content": ". By", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 417, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 158, + 106 + ], + "score": 1.0, + "content": "Lemma B.1,", + "type": "text" + }, + { + "bbox": [ + 159, + 93, + 211, + 106 + ], + "score": 0.92, + "content": "\\log \\rho ( \\tau ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 93, + 242, + 106 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 243, + 93, + 289, + 106 + ], + "score": 0.93, + "content": "\\tau \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 93, + 417, + 106 + ], + "score": 1.0, + "content": ". Then by Lemma C.9, we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 79, + 506, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 110, + 480, + 140 + ], + "lines": [ + { + "bbox": [ + 131, + 110, + 480, + 140 + ], + "spans": [ + { + "bbox": [ + 131, + 110, + 480, + 140 + ], + "score": 0.92, + "content": "\\frac { 1 } { L } \\log \\frac { \\tilde { \\gamma } ( t _ { 2 } ) } { \\tilde { \\gamma } ( t _ { 1 } ) } > \\int _ { t _ { 1 } } ^ { t _ { 2 } } { C \\cdot \\frac { d } { d \\tau } \\log \\rho ( \\tau ) \\cdot d \\tau } = C \\cdot ( \\log \\rho ( t _ { 2 } ) - \\log \\rho ( t _ { 1 } ) ) = \\frac { 1 } { L } \\log \\frac { \\tilde { \\gamma } ( t _ { 2 } ) } { \\tilde { \\gamma } ( t _ { 1 } ) } ,", + "type": "interline_equation", + "image_path": "70b4604217f9bf52fea8326ade97d9ae6b35ab4a1782b21ccbdc1610c54ae7f2.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 131, + 110, + 480, + 140 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 230, + 157 + ], + "lines": [ + { + "bbox": [ + 106, + 145, + 231, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 231, + 157 + ], + "score": 1.0, + "content": "which leads to a contradiction.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 106, + 145, + 231, + 157 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 170, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "In the rest of this section, we present both asymptotic and non-asymptotic analyses for the directional", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 181, + 317, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 295, + 193 + ], + "score": 1.0, + "content": "convergence by using Corollary C.10 to bound", + "type": "text" + }, + { + "bbox": [ + 295, + 181, + 313, + 193 + ], + "score": 0.91, + "content": "\\beta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 181, + 317, + 193 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 170, + 505, + 193 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 206, + 237, + 218 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 237, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 237, + 219 + ], + "score": 1.0, + "content": "C.4 ASYMPTOTIC ANALYSIS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 439, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 440, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 429, + 240 + ], + "score": 1.0, + "content": "We first prove an auxiliary lemma which gives an upper bound for the change of", + "type": "text" + }, + { + "bbox": [ + 429, + 226, + 436, + 237 + ], + "score": 0.83, + "content": "\\hat { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 225, + 440, + 240 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 225, + 440, + 240 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 231, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 232, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 201, + 257 + ], + "score": 1.0, + "content": "Lemma C.11. For a.e.", + "type": "text" + }, + { + "bbox": [ + 202, + 243, + 228, + 254 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 240, + 232, + 257 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 240, + 232, + 257 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 252, + 356, + 286 + ], + "lines": [ + { + "bbox": [ + 252, + 252, + 356, + 286 + ], + "spans": [ + { + "bbox": [ + 252, + 252, + 356, + 286 + ], + "score": 0.94, + "content": "\\left\\| \\frac { d \\hat { \\pmb \\theta } } { d t } \\right\\| _ { 2 } \\le \\frac { B _ { 1 } } { L \\tilde { \\gamma } } \\cdot \\frac { d } { d t } \\log \\rho .", + "type": "interline_equation", + "image_path": "800af00bec61077132a4a2cdac85a1e820897f5661056be725fa7507524b6f78.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 252, + 252, + 356, + 269.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 252, + 269.0, + 356, + 286.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 298, + 506, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 508, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 189, + 322 + ], + "score": 1.0, + "content": "Proof. Observe that", + "type": "text" + }, + { + "bbox": [ + 190, + 299, + 359, + 319 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\left\\| \\frac { d \\hat { \\pmb \\theta } } { d t } \\right\\| _ { 2 } = \\frac { 1 } { \\rho } \\left\\| ( \\pmb I - \\hat { \\pmb \\theta } \\hat { \\pmb \\theta } ^ { \\top } ) \\frac { d \\pmb \\theta } { d t } \\right\\| _ { 2 } \\leq \\frac { 1 } { \\rho } \\left\\| \\frac { d \\pmb \\theta } { d t } \\right\\| _ { 2 } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 295, + 459, + 322 + ], + "score": 1.0, + "content": ". It is sufficient to bound", + "type": "text" + }, + { + "bbox": [ + 459, + 301, + 486, + 317 + ], + "score": 0.93, + "content": "\\left\\| { \\frac { d \\theta } { d t } } \\right\\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 295, + 508, + 322 + ], + "score": 1.0, + "content": ". By", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 101, + 316, + 509, + 356 + ], + "spans": [ + { + "bbox": [ + 101, + 324, + 124, + 356 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 105, + 316, + 212, + 334 + ], + "score": 1.0, + "content": "the chain rule, there exists", + "type": "text" + }, + { + "bbox": [ + 124, + 330, + 265, + 345 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d \\theta } { d t } = \\sum _ { n \\in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) { h _ { n } ( t ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 320, + 330, + 331 + ], + "score": 0.9, + "content": "h _ { 1 } , \\ldots , h _ { N } : [ 0 , + \\infty ) \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 324, + 337, + 356 + ], + "score": 1.0, + "content": ". By definition of", + "type": "text" + }, + { + "bbox": [ + 330, + 316, + 420, + 334 + ], + "score": 1.0, + "content": "satisfying that for a.e.", + "type": "text" + }, + { + "bbox": [ + 337, + 332, + 350, + 343 + ], + "score": 0.8, + "content": "B _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 324, + 355, + 356 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 355, + 330, + 430, + 344 + ], + "score": 0.91, + "content": "\\| h _ { n } \\| _ { 2 } \\leq B _ { 1 } \\rho ^ { L - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 319, + 443, + 330 + ], + "score": 0.81, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 324, + 462, + 356 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 443, + 316, + 446, + 334 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 447, + 318, + 504, + 331 + ], + "score": 0.82, + "content": "h _ { n } ( t ) \\in \\partial ^ { \\circ } q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 332, + 487, + 342 + ], + "score": 0.88, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 324, + 509, + 356 + ], + "score": 1.0, + "content": ". So", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 101, + 295, + 509, + 356 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 359, + 405, + 425 + ], + "lines": [ + { + "bbox": [ + 204, + 359, + 405, + 425 + ], + "spans": [ + { + "bbox": [ + 204, + 359, + 405, + 425 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\left\\| \\frac { d \\pmb { \\theta } } { d t } \\right\\| _ { 2 } \\leq \\sum _ { n \\in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) \\left\\| \\pmb { h } _ { n } \\right\\| _ { 2 } } \\\\ { \\displaystyle \\qquad \\leq \\sum _ { n \\in [ N ] } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } \\cdot \\frac { 1 } { q _ { n } } \\cdot B _ { 1 } \\rho ^ { L - 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "952e58f4d575390690d4caad3e44e31d48b480a262f8961ef774e48e9cfa3e69.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 359, + 405, + 375.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 204, + 375.5, + 405, + 392.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 204, + 392.0, + 405, + 408.5 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 204, + 408.5, + 405, + 425.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 429, + 502, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 340, + 443 + ], + "score": 1.0, + "content": "Note that every summand is positive. By Lemma A.5,", + "type": "text" + }, + { + "bbox": [ + 340, + 431, + 352, + 441 + ], + "score": 0.84, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 427, + 442, + 443 + ], + "score": 1.0, + "content": "is lower-bounded by", + "type": "text" + }, + { + "bbox": [ + 443, + 430, + 504, + 441 + ], + "score": 0.87, + "content": "q _ { n } \\geq q _ { \\mathrm { m i n } } \\geq", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 439, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 142, + 453 + ], + "score": 0.91, + "content": "g ( \\log { \\frac { 1 } { \\mathcal { L } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 439, + 220, + 455 + ], + "score": 1.0, + "content": ", so we can replace", + "type": "text" + }, + { + "bbox": [ + 220, + 442, + 231, + 452 + ], + "score": 0.85, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 439, + 253, + 455 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 253, + 440, + 289, + 453 + ], + "score": 0.92, + "content": "g ( \\log { \\frac { 1 } { \\mathcal { L } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 439, + 506, + 455 + ], + "score": 1.0, + "content": "in the above inequality. Combining with the fact that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 451, + 284, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 209, + 468 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { n \\in [ N ] } { \\stackrel { \\sim } { e } } ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) q _ { n } \\quad } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 451, + 236, + 468 + ], + "score": 1.0, + "content": "is just", + "type": "text" + }, + { + "bbox": [ + 237, + 456, + 243, + 464 + ], + "score": 0.76, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 451, + 284, + 468 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 427, + 506, + 468 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 473, + 386, + 502 + ], + "lines": [ + { + "bbox": [ + 225, + 473, + 386, + 502 + ], + "spans": [ + { + "bbox": [ + 225, + 473, + 386, + 502 + ], + "score": 0.91, + "content": "\\left\\| \\frac { d \\pmb { \\theta } } { d t } \\right\\| _ { 2 } \\leq \\frac { \\nu } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\cdot B _ { 1 } \\rho ^ { L - 1 } = \\frac { B _ { 1 } \\nu } { \\tilde { \\gamma } \\rho } .", + "type": "interline_equation", + "image_path": "4bd63c906f7dc11a45aa59373ba0d8180344b4ec650568da529f5954ad5d2bfa.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 225, + 473, + 386, + 502 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 343, + 529 + ], + "lines": [ + { + "bbox": [ + 102, + 504, + 342, + 531 + ], + "spans": [ + { + "bbox": [ + 102, + 504, + 155, + 531 + ], + "score": 1.0, + "content": "So we have", + "type": "text" + }, + { + "bbox": [ + 155, + 508, + 342, + 528 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\left\\| \\frac { d \\hat { \\pmb \\theta } } { d t } \\right\\| _ { 2 } \\le \\frac { 1 } { \\rho } \\left\\| \\frac { d \\pmb \\theta } { d t } \\right\\| _ { 2 } \\le \\frac { B _ { 1 } } { L \\tilde { \\gamma } } \\cdot \\frac { L \\nu } { \\rho ^ { 2 } } = \\frac { B _ { 1 } } { L \\tilde { \\gamma } } \\cdot \\frac { d } { d t } \\log \\rho . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 102, + 504, + 342, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 104, + 539, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 325, + 558 + ], + "score": 1.0, + "content": "To prove Theorem A.8, we consider each limit point", + "type": "text" + }, + { + "bbox": [ + 325, + 541, + 382, + 555 + ], + "score": 0.93, + "content": "\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 539, + 505, + 558 + ], + "score": 1.0, + "content": ", and construct a series of ap-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 552, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 296, + 570 + ], + "score": 1.0, + "content": "proximate KKT points converging to it. Then", + "type": "text" + }, + { + "bbox": [ + 297, + 554, + 354, + 567 + ], + "score": 0.92, + "content": "\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 552, + 506, + 570 + ], + "score": 1.0, + "content": "can be shown to be a KKT point by", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 566, + 403, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 403, + 579 + ], + "score": 1.0, + "content": "Theorem C.4. The following lemma ensures that such construction exists.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 539, + 506, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 104, + 581, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 254, + 600 + ], + "score": 1.0, + "content": "Lemma C.12. For every limit point", + "type": "text" + }, + { + "bbox": [ + 254, + 584, + 261, + 595 + ], + "score": 0.62, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 581, + 273, + 600 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 273, + 581, + 334, + 601 + ], + "score": 0.93, + "content": "\\left\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 581, + 444, + 600 + ], + "score": 1.0, + "content": ", there exists a sequence of", + "type": "text" + }, + { + "bbox": [ + 444, + 585, + 504, + 597 + ], + "score": 0.91, + "content": "\\{ t _ { m } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 600, + 309, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 145, + 615 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 145, + 603, + 157, + 613 + ], + "score": 0.83, + "content": "t _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 600, + 236, + 614 + ], + "score": 0.8, + "content": "\\cdot + \\infty , \\hat { \\pmb { \\theta } } ( t _ { m } ) \\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 600, + 258, + 615 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 258, + 602, + 304, + 614 + ], + "score": 0.91, + "content": "\\beta ( t _ { m } ) 1", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 600, + 309, + 615 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 581, + 504, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 152, + 640 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 153, + 627, + 232, + 639 + ], + "score": 0.92, + "content": "\\{ \\epsilon _ { m } > 0 : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 626, + 354, + 640 + ], + "score": 1.0, + "content": "be an arbitrary sequence with", + "type": "text" + }, + { + "bbox": [ + 354, + 628, + 388, + 638 + ], + "score": 0.9, + "content": "\\epsilon _ { m } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 626, + 469, + 640 + ], + "score": 1.0, + "content": ". Now we construct", + "type": "text" + }, + { + "bbox": [ + 469, + 627, + 491, + 639 + ], + "score": 0.92, + "content": "\\left\\{ { { t } _ { m } } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 187, + 651 + ], + "score": 1.0, + "content": "induction. Suppose", + "type": "text" + }, + { + "bbox": [ + 187, + 639, + 280, + 649 + ], + "score": 0.88, + "content": "t _ { 1 } < t _ { 2 } < \\dots < t _ { m - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 637, + 435, + 651 + ], + "score": 1.0, + "content": "have already been constructed. Since", + "type": "text" + }, + { + "bbox": [ + 435, + 637, + 442, + 648 + ], + "score": 0.81, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "is a limit point", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 649, + 444, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 123, + 661 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 649, + 165, + 661 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( t ) \\uparrow \\tilde { \\gamma } _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 649, + 212, + 661 + ], + "score": 1.0, + "content": "(recall that", + "type": "text" + }, + { + "bbox": [ + 213, + 650, + 304, + 661 + ], + "score": 0.87, + "content": "\\tilde { \\gamma } _ { \\infty } : = \\mathrm { l i m } _ { t + \\infty } \\tilde { \\gamma } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 649, + 356, + 661 + ], + "score": 1.0, + "content": ", there exists", + "type": "text" + }, + { + "bbox": [ + 356, + 650, + 403, + 661 + ], + "score": 0.92, + "content": "s _ { m } > t _ { m - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 649, + 444, + 661 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 626, + 506, + 661 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 666, + 406, + 693 + ], + "lines": [ + { + "bbox": [ + 202, + 666, + 406, + 693 + ], + "spans": [ + { + "bbox": [ + 202, + 666, + 406, + 693 + ], + "score": 0.91, + "content": "\\left\\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\right\\| _ { 2 } \\leq \\epsilon _ { m } \\quad \\mathrm { a n d } \\quad \\frac { 1 } { L } \\log \\frac { \\widetilde { \\gamma } _ { \\infty } } { \\widetilde { \\gamma } ( s _ { m } ) } \\leq \\epsilon _ { m } ^ { 3 } .", + "type": "interline_equation", + "image_path": "bcb8951d33d8c5dbf3c118211ee2e04e3fe19fe8f13713f580a9a223007cb6f8.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 202, + 666, + 406, + 693 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 123, + 712 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 124, + 698, + 168, + 711 + ], + "score": 0.91, + "content": "s _ { m } ^ { \\prime } > s _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 697, + 254, + 712 + ], + "score": 1.0, + "content": "be a time such that", + "type": "text" + }, + { + "bbox": [ + 254, + 698, + 379, + 711 + ], + "score": 0.9, + "content": "\\log { \\rho ( s _ { m } ^ { \\prime } ) } = \\log { \\rho ( s _ { m } ) } + \\epsilon _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 697, + 506, + 712 + ], + "score": 1.0, + "content": ". According to Theorem A.7,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 706, + 507, + 725 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 160, + 721 + ], + "score": 0.88, + "content": "\\log \\rho \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 706, + 176, + 725 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 176, + 710, + 190, + 722 + ], + "score": 0.9, + "content": "s _ { m } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 706, + 292, + 725 + ], + "score": 1.0, + "content": "must exist. We construct", + "type": "text" + }, + { + "bbox": [ + 292, + 711, + 353, + 722 + ], + "score": 0.9, + "content": "t _ { m } \\in ( s _ { m } , s _ { m } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 706, + 422, + 725 + ], + "score": 1.0, + "content": "to be a time that", + "type": "text" + }, + { + "bbox": [ + 422, + 709, + 501, + 722 + ], + "score": 0.92, + "content": "\\bar { \\beta } ( t _ { m } ) ^ { - 2 } - 1 \\leq \\epsilon _ { m } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 706, + 507, + 725 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 720, + 321, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 321, + 732 + ], + "score": 1.0, + "content": "where the existence can be shown by Corollary C.10.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 697, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 81, + 505, + 107 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 408, + 96 + ], + "score": 1.0, + "content": "Now we show that this construction meets our requirement. It follows from", + "type": "text" + }, + { + "bbox": [ + 408, + 82, + 487, + 95 + ], + "score": 0.92, + "content": "\\beta ( t _ { m } ) ^ { - 2 } - 1 \\leq \\epsilon _ { m } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 366, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 216, + 108 + ], + "score": 0.91, + "content": "\\beta ( t _ { m } ) \\geq 1 / \\sqrt { 1 + \\epsilon _ { m } ^ { 2 } } \\to 1", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 93, + 366, + 109 + ], + "score": 1.0, + "content": ". By Lemma C.11, we also know that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 111, + 473, + 138 + ], + "lines": [ + { + "bbox": [ + 137, + 111, + 473, + 138 + ], + "spans": [ + { + "bbox": [ + 137, + 111, + 473, + 138 + ], + "score": 0.92, + "content": "\\biggl \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\bar { \\pmb \\theta } \\biggl \\| _ { 2 } \\leq \\biggl \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\hat { \\pmb \\theta } ( s _ { m } ) \\biggr \\| _ { 2 } + \\biggl \\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\biggl \\| _ { 2 } \\leq \\frac { B _ { 1 } } { L \\tilde { \\gamma } ( t _ { 0 } ) } \\cdot \\epsilon _ { m } + \\epsilon _ { m } 0 .", + "type": "interline_equation", + "image_path": "e1a5de3dc6b09edbe64e933fc2b31000581c0b9e36d779af1d2de2ae88d2a1c8.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 137, + 111, + 473, + 138 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 141, + 210, + 153 + ], + "lines": [ + { + "bbox": [ + 106, + 140, + 211, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 211, + 154 + ], + "score": 1.0, + "content": "This completes the proof.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 216, + 180 + ], + "score": 1.0, + "content": "Proof of Theorem A.8. Let", + "type": "text" + }, + { + "bbox": [ + 217, + 165, + 297, + 178 + ], + "score": 0.92, + "content": "\\tilde { \\pmb { \\theta } } : = \\bar { \\pmb { \\theta } } / q _ { \\mathrm { m i n } } ( \\bar { \\pmb { \\theta } } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 164, + 356, + 180 + ], + "score": 1.0, + "content": "for short. Let", + "type": "text" + }, + { + "bbox": [ + 356, + 166, + 417, + 178 + ], + "score": 0.92, + "content": "\\{ t _ { m } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 164, + 505, + 180 + ], + "score": 1.0, + "content": "be the sequence con-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 257, + 191 + ], + "score": 1.0, + "content": "structed as in Lemma C.12. For each", + "type": "text" + }, + { + "bbox": [ + 257, + 179, + 269, + 189 + ], + "score": 0.88, + "content": "t _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 177, + 300, + 191 + ], + "score": 1.0, + "content": ", define", + "type": "text" + }, + { + "bbox": [ + 300, + 177, + 324, + 190 + ], + "score": 0.92, + "content": "\\epsilon ( t _ { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 177, + 343, + 191 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 343, + 178, + 367, + 190 + ], + "score": 0.92, + "content": "\\delta ( t _ { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 177, + 505, + 191 + ], + "score": 1.0, + "content": "as in Lemma C.8. Then we know", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 187, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 124, + 204 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 189, + 206, + 202 + ], + "score": 0.92, + "content": "\\pmb { \\theta } ( t _ { m } ) / q _ { \\mathrm { m i n } } ( t _ { m } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 187, + 231, + 204 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 231, + 190, + 290, + 202 + ], + "score": 0.91, + "content": "( \\epsilon ( t _ { m } ) , \\delta ( t _ { m } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 187, + 358, + 204 + ], + "score": 1.0, + "content": "-KKT point and", + "type": "text" + }, + { + "bbox": [ + 358, + 189, + 505, + 202 + ], + "score": 0.91, + "content": "\\pmb { \\theta } ( t _ { m } ) / q _ { \\mathrm { m i n } } ( t _ { m } ) ^ { 1 / L } \\tilde { \\pmb { \\theta } } , \\epsilon ( t _ { m } ) ", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 201, + 503, + 214 + ], + "spans": [ + { + "bbox": [ + 107, + 202, + 161, + 213 + ], + "score": 0.92, + "content": "0 , \\delta ( t _ { m } ) \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 201, + 503, + 214 + ], + "score": 1.0, + "content": ". By Lemma C.7, (P) satisfies MFCQ. Applying Theorem C.4 proves the theorem.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 225, + 259, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 260, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 260, + 238 + ], + "score": 1.0, + "content": "C.5 NON-ASYMPTOTIC ANALYSIS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 503, + 273 + ], + "lines": [ + { + "bbox": [ + 103, + 240, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 103, + 241, + 298, + 263 + ], + "score": 1.0, + "content": "Proof of Theorem A.9. Let C0 := 1L log γ˜∞γ˜(t0) .", + "type": "text" + }, + { + "bbox": [ + 295, + 240, + 457, + 266 + ], + "score": 1.0, + "content": "Without loss of generality, we assume", + "type": "text" + }, + { + "bbox": [ + 458, + 244, + 504, + 260 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\epsilon < \\frac { \\sqrt { 6 } } { 2 } C _ { 1 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 257, + 186, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 123, + 273 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 259, + 182, + 272 + ], + "score": 0.92, + "content": "\\delta < C _ { 2 } / f ( b _ { f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 257, + 186, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 102, + 276, + 507, + 299 + ], + "spans": [ + { + "bbox": [ + 102, + 276, + 122, + 299 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 282, + 132, + 292 + ], + "score": 0.86, + "content": "t _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 276, + 219, + 299 + ], + "score": 1.0, + "content": "be the time such that", + "type": "text" + }, + { + "bbox": [ + 219, + 277, + 389, + 295 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\log \\rho ( t _ { 1 } ) = \\frac { 1 } { L } \\log \\frac { g ( C _ { 2 } \\delta ^ { - 1 } ) } { \\tilde { \\gamma } ( t _ { 0 } ) } = \\Theta ( \\log \\delta ^ { - 1 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 276, + 507, + 295 + ], + "score": 1.0, + "content": "g g(C2δ−1) = Θ(log δ−1) and t2 be the time such that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 297, + 308 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\log \\rho ( t _ { 2 } ) - \\log \\rho ( t _ { 1 } ) = \\frac { 1 } { 2 } C _ { 0 } C _ { 1 } ^ { 2 } \\epsilon ^ { - 2 } = \\Theta ( \\epsilon ^ { - 2 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 293, + 431, + 308 + ], + "score": 1.0, + "content": ". By Corollary C.10, there exists", + "type": "text" + }, + { + "bbox": [ + 431, + 295, + 482, + 308 + ], + "score": 0.93, + "content": "t _ { * } \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 307, + 225, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 307, + 124, + 322 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 308, + 220, + 321 + ], + "score": 0.91, + "content": "\\beta ( t _ { * } ) ^ { - 2 } - 1 \\le 2 \\epsilon ^ { 2 } C _ { 1 } ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 307, + 225, + 322 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 506, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 194, + 340 + ], + "score": 1.0, + "content": "Now we argue that", + "type": "text" + }, + { + "bbox": [ + 194, + 326, + 217, + 340 + ], + "score": 0.92, + "content": "\\tilde { \\pmb { \\theta } } ( t _ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 326, + 248, + 340 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 248, + 327, + 269, + 340 + ], + "score": 0.88, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "-KKT point. By Lemma C.8, we only need to show", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 339, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 186, + 354 + ], + "score": 0.91, + "content": "C _ { 1 } \\sqrt { 1 - \\beta ( t _ { * } ) } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 339, + 204, + 354 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 204, + 341, + 302, + 354 + ], + "score": 0.91, + "content": "C _ { 2 } / f ( \\tilde { \\gamma } ( t _ { * } ) \\rho ( t _ { * } ) ^ { L } ) \\leq \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 339, + 458, + 354 + ], + "score": 1.0, + "content": ". For the first inequality, by assumption", + "type": "text" + }, + { + "bbox": [ + 459, + 339, + 501, + 354 + ], + "score": 0.93, + "content": "\\epsilon < \\frac { \\sqrt { 6 } } { 2 } C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 339, + 505, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 168, + 369 + ], + "score": 1.0, + "content": "we know that", + "type": "text" + }, + { + "bbox": [ + 168, + 354, + 248, + 366 + ], + "score": 0.91, + "content": "\\beta ( t _ { * } ) ^ { - 2 } - 1 < 3", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 353, + 316, + 369 + ], + "score": 1.0, + "content": ", which implies", + "type": "text" + }, + { + "bbox": [ + 317, + 353, + 372, + 368 + ], + "score": 0.93, + "content": "| \\beta ( t _ { * } ) | ~ < ~ \\frac { 1 } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 353, + 446, + 369 + ], + "score": 1.0, + "content": ". Then we have", + "type": "text" + }, + { + "bbox": [ + 446, + 355, + 505, + 367 + ], + "score": 0.9, + "content": "1 - \\beta ( t _ { * } ) \\ \\leq", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 364, + 508, + 386 + ], + "spans": [ + { + "bbox": [ + 103, + 364, + 180, + 383 + ], + "score": 1.0, + "content": "2\u000f2C −2 · β(t∗)2", + "type": "text" + }, + { + "bbox": [ + 148, + 367, + 224, + 384 + ], + "score": 0.82, + "content": "\\frac { \\beta ( t _ { * } ) ^ { 2 } } { 1 + \\beta ( t _ { * } ) } ~ \\leq ~ \\epsilon ^ { 2 } C _ { 1 } ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 366, + 278, + 386 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 279, + 368, + 363, + 383 + ], + "score": 0.94, + "content": "C _ { 1 } \\sqrt { 1 - \\beta ( t _ { * } ) } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 366, + 508, + 386 + ], + "score": 1.0, + "content": "holds. For the second inequality,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 384, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 287, + 402 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\tilde { \\gamma } ( t _ { * } ) \\rho ( t _ { * } ) ^ { L } \\geq \\tilde { \\gamma } ( t _ { * } ) \\cdot \\frac { g ( C _ { 2 } \\delta ^ { - 1 } ) } { \\tilde { \\gamma } ( t _ { 0 } ) } \\geq g ( C _ { 2 } \\delta ^ { - 1 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 384, + 336, + 402 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 336, + 386, + 434, + 400 + ], + "score": 0.93, + "content": "C _ { 2 } / f ( \\tilde { \\gamma } ( t _ { * } ) \\rho ( t _ { * } ) ^ { L } ) \\leq \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 384, + 463, + 402 + ], + "score": 1.0, + "content": "holds.", + "type": "text" + }, + { + "bbox": [ + 494, + 388, + 505, + 399 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 255, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 255, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 255, + 426 + ], + "score": 1.0, + "content": "C.6 PROOF FOR COROLLARY 4.5", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 424, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 425, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 200, + 449 + ], + "score": 1.0, + "content": "By the homogeneity of", + "type": "text" + }, + { + "bbox": [ + 201, + 437, + 211, + 446 + ], + "score": 0.85, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 433, + 425, + 449 + ], + "score": 1.0, + "content": ", we can characterize KKT points using kernel SVM.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 105, + 448, + 503, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 178, + 462 + ], + "score": 1.0, + "content": "Lemma C.13. If", + "type": "text" + }, + { + "bbox": [ + 178, + 449, + 189, + 460 + ], + "score": 0.82, + "content": "\\pmb { \\theta } _ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 448, + 255, + 462 + ], + "score": 1.0, + "content": "is KKT point of", + "type": "text" + }, + { + "bbox": [ + 256, + 450, + 270, + 460 + ], + "score": 0.78, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 448, + 341, + 462 + ], + "score": 1.0, + "content": ", then there exists", + "type": "text" + }, + { + "bbox": [ + 342, + 449, + 414, + 461 + ], + "score": 0.86, + "content": "h _ { n } \\in \\partial ^ { \\circ } \\Phi _ { x _ { n } } ( \\theta _ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 448, + 429, + 462 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 429, + 449, + 464, + 461 + ], + "score": 0.9, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 459, + 442, + 474 + ], + "spans": [ + { + "bbox": [ + 107, + 459, + 126, + 473 + ], + "score": 0.91, + "content": "\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 459, + 422, + 474 + ], + "score": 1.0, + "content": "is an optimal solution for the following constrained optimization problem", + "type": "text" + }, + { + "bbox": [ + 423, + 461, + 437, + 471 + ], + "score": 0.81, + "content": "( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 459, + 442, + 474 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 476, + 379, + 515 + ], + "lines": [ + { + "bbox": [ + 230, + 476, + 379, + 515 + ], + "spans": [ + { + "bbox": [ + 230, + 476, + 379, + 515 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { s . t . } & { y _ { n } \\left. \\pmb { \\theta } , \\pmb { h } _ { n } \\right. \\geq 1 \\qquad \\forall n \\in [ N ] } \\end{array}", + "type": "interline_equation", + "image_path": "0aada129d097b964a4f06db44f35cd70871add69082af91fec96a1d54ad6e955.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 230, + 476, + 379, + 495.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 230, + 495.5, + 379, + 515.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 384, + 541 + ], + "score": 1.0, + "content": "Proof. It is easy to see that (Q) is a convex optimization problem. For", + "type": "text" + }, + { + "bbox": [ + 385, + 526, + 423, + 540 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { \\theta } = \\frac { 2 } { L } \\pmb { \\theta } _ { * } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 525, + 505, + 541 + ], + "score": 1.0, + "content": ", from Theorem B.2,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 172, + 551 + ], + "score": 1.0, + "content": "we can see that", + "type": "text" + }, + { + "bbox": [ + 172, + 538, + 303, + 550 + ], + "score": 0.9, + "content": "y _ { n _ { \\cdot } } \\langle \\theta , h _ { n } \\rangle = 2 q _ { n } ( \\theta _ { \\ast } ) \\geq 2 > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 538, + 505, + 551 + ], + "score": 1.0, + "content": ", which implies Slater’s condition. Thus, we only", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 549, + 332, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 179, + 562 + ], + "score": 1.0, + "content": "need to show that", + "type": "text" + }, + { + "bbox": [ + 179, + 549, + 198, + 563 + ], + "score": 0.89, + "content": "\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 549, + 332, + 562 + ], + "score": 1.0, + "content": "satisfies KKT conditions for (Q).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 104, + 565, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 223, + 580 + ], + "score": 1.0, + "content": "By the KKT conditions for", + "type": "text" + }, + { + "bbox": [ + 223, + 567, + 235, + 577 + ], + "score": 0.38, + "content": "( \\mathrm { P } )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 564, + 314, + 580 + ], + "score": 1.0, + "content": ", we can construct", + "type": "text" + }, + { + "bbox": [ + 314, + 566, + 382, + 578 + ], + "score": 0.93, + "content": "h _ { n } \\in \\partial ^ { \\circ } q _ { n } ( \\theta _ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 564, + 399, + 580 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 400, + 566, + 438, + 578 + ], + "score": 0.92, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 564, + 481, + 580 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 482, + 566, + 505, + 577 + ], + "score": 0.82, + "content": "\\theta _ { * } \\mathrm { ~ - ~ }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 571, + 482, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 188, + 592 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } y _ { n } h _ { n } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 571, + 227, + 598 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 227, + 579, + 293, + 591 + ], + "score": 0.93, + "content": "\\lambda _ { 1 } , \\dots , \\lambda _ { N } \\ge 0", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 571, + 310, + 598 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 579, + 394, + 591 + ], + "score": 0.92, + "content": "\\lambda _ { n } ( q _ { n } ( \\pmb { \\theta } _ { * } ) - 1 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 571, + 423, + 598 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + }, + { + "bbox": [ + 424, + 579, + 442, + 592 + ], + "score": 0.92, + "content": "\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 571, + 482, + 598 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 624, + 351, + 640 + ], + "lines": [ + { + "bbox": [ + 138, + 624, + 351, + 640 + ], + "spans": [ + { + "bbox": [ + 138, + 624, + 351, + 640 + ], + "score": 0.62, + "content": "\\begin{array} { r } { \\frac { 1 } { L } \\lambda _ { n } \\left( y _ { n } \\left. \\frac { 1 } { L } \\pmb { \\theta } _ { * } , \\pmb { h } _ { n } \\right. - 1 \\right) = \\frac { 1 } { L } \\lambda _ { n } \\left( q _ { n } ( \\pmb { \\theta } ) - 1 \\right) \\geq 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "4d818b9aa35a65ee65b2d1dee1ff8fe779e64efd59d0d4a14cd445774a1ba0c2.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 138, + 624, + 351, + 640 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 273, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 274, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 120, + 665 + ], + "score": 1.0, + "content": "So", + "type": "text" + }, + { + "bbox": [ + 120, + 650, + 138, + 664 + ], + "score": 0.9, + "content": "\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 648, + 274, + 665 + ], + "score": 1.0, + "content": "satisfies KKT conditions for (Q).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 108, + 674, + 281, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 674, + 283, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 283, + 688 + ], + "score": 1.0, + "content": "Now we prove Corollary 4.5 in Section 4.3.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 275, + 713 + ], + "score": 1.0, + "content": "Proof. By Theorem A.8, every limit point", + "type": "text" + }, + { + "bbox": [ + 276, + 698, + 282, + 708 + ], + "score": 0.83, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 696, + 505, + 713 + ], + "score": 1.0, + "content": "is along the direction of a KKT point of (P). Combining", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 335, + 722 + ], + "score": 1.0, + "content": "this with Lemma C.13, we know that every limit point", + "type": "text" + }, + { + "bbox": [ + 335, + 709, + 342, + 720 + ], + "score": 0.85, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is also along the max-margin direction", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 137, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 137, + 733 + ], + "score": 1.0, + "content": "of (Q).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + } + ], + "page_idx": 25, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "26", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 651, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 495, + 653, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 495, + 653, + 505, + 663 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 141, + 505, + 152 + ], + "lines": [ + { + "bbox": [ + 495, + 143, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 495, + 143, + 505, + 151 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 81, + 505, + 107 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 408, + 96 + ], + "score": 1.0, + "content": "Now we show that this construction meets our requirement. It follows from", + "type": "text" + }, + { + "bbox": [ + 408, + 82, + 487, + 95 + ], + "score": 0.92, + "content": "\\beta ( t _ { m } ) ^ { - 2 } - 1 \\leq \\epsilon _ { m } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 366, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 216, + 108 + ], + "score": 0.91, + "content": "\\beta ( t _ { m } ) \\geq 1 / \\sqrt { 1 + \\epsilon _ { m } ^ { 2 } } \\to 1", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 93, + 366, + 109 + ], + "score": 1.0, + "content": ". By Lemma C.11, we also know that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 109 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 111, + 473, + 138 + ], + "lines": [ + { + "bbox": [ + 137, + 111, + 473, + 138 + ], + "spans": [ + { + "bbox": [ + 137, + 111, + 473, + 138 + ], + "score": 0.92, + "content": "\\biggl \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\bar { \\pmb \\theta } \\biggl \\| _ { 2 } \\leq \\biggl \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\hat { \\pmb \\theta } ( s _ { m } ) \\biggr \\| _ { 2 } + \\biggl \\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\biggl \\| _ { 2 } \\leq \\frac { B _ { 1 } } { L \\tilde { \\gamma } ( t _ { 0 } ) } \\cdot \\epsilon _ { m } + \\epsilon _ { m } 0 .", + "type": "interline_equation", + "image_path": "e1a5de3dc6b09edbe64e933fc2b31000581c0b9e36d779af1d2de2ae88d2a1c8.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 137, + 111, + 473, + 138 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 141, + 210, + 153 + ], + "lines": [ + { + "bbox": [ + 106, + 140, + 211, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 211, + 154 + ], + "score": 1.0, + "content": "This completes the proof.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 106, + 140, + 211, + 154 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 216, + 180 + ], + "score": 1.0, + "content": "Proof of Theorem A.8. Let", + "type": "text" + }, + { + "bbox": [ + 217, + 165, + 297, + 178 + ], + "score": 0.92, + "content": "\\tilde { \\pmb { \\theta } } : = \\bar { \\pmb { \\theta } } / q _ { \\mathrm { m i n } } ( \\bar { \\pmb { \\theta } } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 164, + 356, + 180 + ], + "score": 1.0, + "content": "for short. Let", + "type": "text" + }, + { + "bbox": [ + 356, + 166, + 417, + 178 + ], + "score": 0.92, + "content": "\\{ t _ { m } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 164, + 505, + 180 + ], + "score": 1.0, + "content": "be the sequence con-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 257, + 191 + ], + "score": 1.0, + "content": "structed as in Lemma C.12. For each", + "type": "text" + }, + { + "bbox": [ + 257, + 179, + 269, + 189 + ], + "score": 0.88, + "content": "t _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 177, + 300, + 191 + ], + "score": 1.0, + "content": ", define", + "type": "text" + }, + { + "bbox": [ + 300, + 177, + 324, + 190 + ], + "score": 0.92, + "content": "\\epsilon ( t _ { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 177, + 343, + 191 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 343, + 178, + 367, + 190 + ], + "score": 0.92, + "content": "\\delta ( t _ { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 177, + 505, + 191 + ], + "score": 1.0, + "content": "as in Lemma C.8. Then we know", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 187, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 124, + 204 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 189, + 206, + 202 + ], + "score": 0.92, + "content": "\\pmb { \\theta } ( t _ { m } ) / q _ { \\mathrm { m i n } } ( t _ { m } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 187, + 231, + 204 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 231, + 190, + 290, + 202 + ], + "score": 0.91, + "content": "( \\epsilon ( t _ { m } ) , \\delta ( t _ { m } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 187, + 358, + 204 + ], + "score": 1.0, + "content": "-KKT point and", + "type": "text" + }, + { + "bbox": [ + 358, + 189, + 505, + 202 + ], + "score": 0.91, + "content": "\\pmb { \\theta } ( t _ { m } ) / q _ { \\mathrm { m i n } } ( t _ { m } ) ^ { 1 / L } \\tilde { \\pmb { \\theta } } , \\epsilon ( t _ { m } ) ", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 201, + 503, + 214 + ], + "spans": [ + { + "bbox": [ + 107, + 202, + 161, + 213 + ], + "score": 0.92, + "content": "0 , \\delta ( t _ { m } ) \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 201, + 503, + 214 + ], + "score": 1.0, + "content": ". By Lemma C.7, (P) satisfies MFCQ. Applying Theorem C.4 proves the theorem.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 164, + 505, + 214 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 225, + 259, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 260, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 260, + 238 + ], + "score": 1.0, + "content": "C.5 NON-ASYMPTOTIC ANALYSIS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 503, + 273 + ], + "lines": [ + { + "bbox": [ + 103, + 240, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 103, + 241, + 298, + 263 + ], + "score": 1.0, + "content": "Proof of Theorem A.9. Let C0 := 1L log γ˜∞γ˜(t0) .", + "type": "text" + }, + { + "bbox": [ + 295, + 240, + 457, + 266 + ], + "score": 1.0, + "content": "Without loss of generality, we assume", + "type": "text" + }, + { + "bbox": [ + 458, + 244, + 504, + 260 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\epsilon < \\frac { \\sqrt { 6 } } { 2 } C _ { 1 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 257, + 186, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 123, + 273 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 259, + 182, + 272 + ], + "score": 0.92, + "content": "\\delta < C _ { 2 } / f ( b _ { f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 257, + 186, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 103, + 240, + 504, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 102, + 276, + 507, + 299 + ], + "spans": [ + { + "bbox": [ + 102, + 276, + 122, + 299 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 282, + 132, + 292 + ], + "score": 0.86, + "content": "t _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 276, + 219, + 299 + ], + "score": 1.0, + "content": "be the time such that", + "type": "text" + }, + { + "bbox": [ + 219, + 277, + 389, + 295 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\log \\rho ( t _ { 1 } ) = \\frac { 1 } { L } \\log \\frac { g ( C _ { 2 } \\delta ^ { - 1 } ) } { \\tilde { \\gamma } ( t _ { 0 } ) } = \\Theta ( \\log \\delta ^ { - 1 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 276, + 507, + 295 + ], + "score": 1.0, + "content": "g g(C2δ−1) = Θ(log δ−1) and t2 be the time such that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 297, + 308 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\log \\rho ( t _ { 2 } ) - \\log \\rho ( t _ { 1 } ) = \\frac { 1 } { 2 } C _ { 0 } C _ { 1 } ^ { 2 } \\epsilon ^ { - 2 } = \\Theta ( \\epsilon ^ { - 2 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 293, + 431, + 308 + ], + "score": 1.0, + "content": ". By Corollary C.10, there exists", + "type": "text" + }, + { + "bbox": [ + 431, + 295, + 482, + 308 + ], + "score": 0.93, + "content": "t _ { * } \\in ( t _ { 1 } , t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 307, + 225, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 307, + 124, + 322 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 308, + 220, + 321 + ], + "score": 0.91, + "content": "\\beta ( t _ { * } ) ^ { - 2 } - 1 \\le 2 \\epsilon ^ { 2 } C _ { 1 } ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 307, + 225, + 322 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 102, + 276, + 507, + 322 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 506, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 194, + 340 + ], + "score": 1.0, + "content": "Now we argue that", + "type": "text" + }, + { + "bbox": [ + 194, + 326, + 217, + 340 + ], + "score": 0.92, + "content": "\\tilde { \\pmb { \\theta } } ( t _ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 326, + 248, + 340 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 248, + 327, + 269, + 340 + ], + "score": 0.88, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "-KKT point. By Lemma C.8, we only need to show", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 339, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 186, + 354 + ], + "score": 0.91, + "content": "C _ { 1 } \\sqrt { 1 - \\beta ( t _ { * } ) } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 339, + 204, + 354 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 204, + 341, + 302, + 354 + ], + "score": 0.91, + "content": "C _ { 2 } / f ( \\tilde { \\gamma } ( t _ { * } ) \\rho ( t _ { * } ) ^ { L } ) \\leq \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 339, + 458, + 354 + ], + "score": 1.0, + "content": ". For the first inequality, by assumption", + "type": "text" + }, + { + "bbox": [ + 459, + 339, + 501, + 354 + ], + "score": 0.93, + "content": "\\epsilon < \\frac { \\sqrt { 6 } } { 2 } C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 339, + 505, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 168, + 369 + ], + "score": 1.0, + "content": "we know that", + "type": "text" + }, + { + "bbox": [ + 168, + 354, + 248, + 366 + ], + "score": 0.91, + "content": "\\beta ( t _ { * } ) ^ { - 2 } - 1 < 3", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 353, + 316, + 369 + ], + "score": 1.0, + "content": ", which implies", + "type": "text" + }, + { + "bbox": [ + 317, + 353, + 372, + 368 + ], + "score": 0.93, + "content": "| \\beta ( t _ { * } ) | ~ < ~ \\frac { 1 } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 353, + 446, + 369 + ], + "score": 1.0, + "content": ". Then we have", + "type": "text" + }, + { + "bbox": [ + 446, + 355, + 505, + 367 + ], + "score": 0.9, + "content": "1 - \\beta ( t _ { * } ) \\ \\leq", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 364, + 508, + 386 + ], + "spans": [ + { + "bbox": [ + 103, + 364, + 180, + 383 + ], + "score": 1.0, + "content": "2\u000f2C −2 · β(t∗)2", + "type": "text" + }, + { + "bbox": [ + 148, + 367, + 224, + 384 + ], + "score": 0.82, + "content": "\\frac { \\beta ( t _ { * } ) ^ { 2 } } { 1 + \\beta ( t _ { * } ) } ~ \\leq ~ \\epsilon ^ { 2 } C _ { 1 } ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 366, + 278, + 386 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 279, + 368, + 363, + 383 + ], + "score": 0.94, + "content": "C _ { 1 } \\sqrt { 1 - \\beta ( t _ { * } ) } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 366, + 508, + 386 + ], + "score": 1.0, + "content": "holds. For the second inequality,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 384, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 287, + 402 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\tilde { \\gamma } ( t _ { * } ) \\rho ( t _ { * } ) ^ { L } \\geq \\tilde { \\gamma } ( t _ { * } ) \\cdot \\frac { g ( C _ { 2 } \\delta ^ { - 1 } ) } { \\tilde { \\gamma } ( t _ { 0 } ) } \\geq g ( C _ { 2 } \\delta ^ { - 1 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 384, + 336, + 402 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 336, + 386, + 434, + 400 + ], + "score": 0.93, + "content": "C _ { 2 } / f ( \\tilde { \\gamma } ( t _ { * } ) \\rho ( t _ { * } ) ^ { L } ) \\leq \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 384, + 463, + 402 + ], + "score": 1.0, + "content": "holds.", + "type": "text" + }, + { + "bbox": [ + 494, + 388, + 505, + 399 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 103, + 326, + 508, + 402 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 255, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 255, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 255, + 426 + ], + "score": 1.0, + "content": "C.6 PROOF FOR COROLLARY 4.5", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 424, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 425, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 200, + 449 + ], + "score": 1.0, + "content": "By the homogeneity of", + "type": "text" + }, + { + "bbox": [ + 201, + 437, + 211, + 446 + ], + "score": 0.85, + "content": "q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 433, + 425, + 449 + ], + "score": 1.0, + "content": ", we can characterize KKT points using kernel SVM.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 433, + 425, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 448, + 503, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 178, + 462 + ], + "score": 1.0, + "content": "Lemma C.13. If", + "type": "text" + }, + { + "bbox": [ + 178, + 449, + 189, + 460 + ], + "score": 0.82, + "content": "\\pmb { \\theta } _ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 448, + 255, + 462 + ], + "score": 1.0, + "content": "is KKT point of", + "type": "text" + }, + { + "bbox": [ + 256, + 450, + 270, + 460 + ], + "score": 0.78, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 448, + 341, + 462 + ], + "score": 1.0, + "content": ", then there exists", + "type": "text" + }, + { + "bbox": [ + 342, + 449, + 414, + 461 + ], + "score": 0.86, + "content": "h _ { n } \\in \\partial ^ { \\circ } \\Phi _ { x _ { n } } ( \\theta _ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 448, + 429, + 462 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 429, + 449, + 464, + 461 + ], + "score": 0.9, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 459, + 442, + 474 + ], + "spans": [ + { + "bbox": [ + 107, + 459, + 126, + 473 + ], + "score": 0.91, + "content": "\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 459, + 422, + 474 + ], + "score": 1.0, + "content": "is an optimal solution for the following constrained optimization problem", + "type": "text" + }, + { + "bbox": [ + 423, + 461, + 437, + 471 + ], + "score": 0.81, + "content": "( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 459, + 442, + 474 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 448, + 506, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 476, + 379, + 515 + ], + "lines": [ + { + "bbox": [ + 230, + 476, + 379, + 515 + ], + "spans": [ + { + "bbox": [ + 230, + 476, + 379, + 515 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { s . t . } & { y _ { n } \\left. \\pmb { \\theta } , \\pmb { h } _ { n } \\right. \\geq 1 \\qquad \\forall n \\in [ N ] } \\end{array}", + "type": "interline_equation", + "image_path": "0aada129d097b964a4f06db44f35cd70871add69082af91fec96a1d54ad6e955.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 230, + 476, + 379, + 495.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 230, + 495.5, + 379, + 515.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 384, + 541 + ], + "score": 1.0, + "content": "Proof. It is easy to see that (Q) is a convex optimization problem. For", + "type": "text" + }, + { + "bbox": [ + 385, + 526, + 423, + 540 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { \\theta } = \\frac { 2 } { L } \\pmb { \\theta } _ { * } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 525, + 505, + 541 + ], + "score": 1.0, + "content": ", from Theorem B.2,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 172, + 551 + ], + "score": 1.0, + "content": "we can see that", + "type": "text" + }, + { + "bbox": [ + 172, + 538, + 303, + 550 + ], + "score": 0.9, + "content": "y _ { n _ { \\cdot } } \\langle \\theta , h _ { n } \\rangle = 2 q _ { n } ( \\theta _ { \\ast } ) \\geq 2 > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 538, + 505, + 551 + ], + "score": 1.0, + "content": ", which implies Slater’s condition. Thus, we only", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 549, + 332, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 179, + 562 + ], + "score": 1.0, + "content": "need to show that", + "type": "text" + }, + { + "bbox": [ + 179, + 549, + 198, + 563 + ], + "score": 0.89, + "content": "\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 549, + 332, + 562 + ], + "score": 1.0, + "content": "satisfies KKT conditions for (Q).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 525, + 505, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 565, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 223, + 580 + ], + "score": 1.0, + "content": "By the KKT conditions for", + "type": "text" + }, + { + "bbox": [ + 223, + 567, + 235, + 577 + ], + "score": 0.38, + "content": "( \\mathrm { P } )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 564, + 314, + 580 + ], + "score": 1.0, + "content": ", we can construct", + "type": "text" + }, + { + "bbox": [ + 314, + 566, + 382, + 578 + ], + "score": 0.93, + "content": "h _ { n } \\in \\partial ^ { \\circ } q _ { n } ( \\theta _ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 564, + 399, + 580 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 400, + 566, + 438, + 578 + ], + "score": 0.92, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 564, + 481, + 580 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 482, + 566, + 505, + 577 + ], + "score": 0.82, + "content": "\\theta _ { * } \\mathrm { ~ - ~ }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 571, + 482, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 188, + 592 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { n = 1 } ^ { N } \\lambda _ { n } y _ { n } h _ { n } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 571, + 227, + 598 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 227, + 579, + 293, + 591 + ], + "score": 0.93, + "content": "\\lambda _ { 1 } , \\dots , \\lambda _ { N } \\ge 0", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 571, + 310, + 598 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 579, + 394, + 591 + ], + "score": 0.92, + "content": "\\lambda _ { n } ( q _ { n } ( \\pmb { \\theta } _ { * } ) - 1 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 571, + 423, + 598 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + }, + { + "bbox": [ + 424, + 579, + 442, + 592 + ], + "score": 0.92, + "content": "\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 571, + 482, + 598 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 564, + 505, + 598 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 624, + 351, + 640 + ], + "lines": [ + { + "bbox": [ + 138, + 624, + 351, + 640 + ], + "spans": [ + { + "bbox": [ + 138, + 624, + 351, + 640 + ], + "score": 0.62, + "content": "\\begin{array} { r } { \\frac { 1 } { L } \\lambda _ { n } \\left( y _ { n } \\left. \\frac { 1 } { L } \\pmb { \\theta } _ { * } , \\pmb { h } _ { n } \\right. - 1 \\right) = \\frac { 1 } { L } \\lambda _ { n } \\left( q _ { n } ( \\pmb { \\theta } ) - 1 \\right) \\geq 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "4d818b9aa35a65ee65b2d1dee1ff8fe779e64efd59d0d4a14cd445774a1ba0c2.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 138, + 624, + 351, + 640 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 273, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 274, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 120, + 665 + ], + "score": 1.0, + "content": "So", + "type": "text" + }, + { + "bbox": [ + 120, + 650, + 138, + 664 + ], + "score": 0.9, + "content": "\\scriptstyle { \\frac { 1 } { L } } \\theta _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 648, + 274, + 665 + ], + "score": 1.0, + "content": "satisfies KKT conditions for (Q).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 648, + 274, + 665 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 674, + 281, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 674, + 283, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 283, + 688 + ], + "score": 1.0, + "content": "Now we prove Corollary 4.5 in Section 4.3.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 674, + 283, + 688 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 275, + 713 + ], + "score": 1.0, + "content": "Proof. By Theorem A.8, every limit point", + "type": "text" + }, + { + "bbox": [ + 276, + 698, + 282, + 708 + ], + "score": 0.83, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 696, + 505, + 713 + ], + "score": 1.0, + "content": "is along the direction of a KKT point of (P). Combining", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 335, + 722 + ], + "score": 1.0, + "content": "this with Lemma C.13, we know that every limit point", + "type": "text" + }, + { + "bbox": [ + 335, + 709, + 342, + 720 + ], + "score": 0.85, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is also along the max-margin direction", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 137, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 137, + 733 + ], + "score": 1.0, + "content": "of (Q).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 696, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 187, + 96 + ], + "score": 1.0, + "content": "For smooth models,", + "type": "text" + }, + { + "bbox": [ + 187, + 83, + 201, + 93 + ], + "score": 0.87, + "content": "h _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 80, + 212, + 96 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 212, + 83, + 227, + 93 + ], + "score": 0.33, + "content": "( \\mathrm { Q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 80, + 317, + 96 + ], + "score": 1.0, + "content": "is exactly the gradient", + "type": "text" + }, + { + "bbox": [ + 317, + 82, + 357, + 95 + ], + "score": 0.93, + "content": "\\nabla \\Phi _ { x _ { n } } ( { \\bar { \\theta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 80, + 505, + 96 + ], + "score": 1.0, + "content": ". So, (Q) is the optimization problem", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 191, + 106 + ], + "score": 1.0, + "content": "for SVM with kernel", + "type": "text" + }, + { + "bbox": [ + 191, + 93, + 333, + 107 + ], + "score": 0.91, + "content": "K _ { \\bar { \\theta } } ( x , x ^ { \\prime } ) = \\left. \\nabla \\Phi _ { x } ( \\bar { \\theta } ) , \\nabla \\Phi _ { x ^ { \\prime } } ( \\bar { \\theta } ) \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 94, + 506, + 106 + ], + "score": 1.0, + "content": ". For non-smooth models, we can construct", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 106, + 193, + 120 + ], + "score": 1.0, + "content": "an arbitrary function", + "type": "text" + }, + { + "bbox": [ + 194, + 107, + 267, + 119 + ], + "score": 0.89, + "content": "h ( x ) \\in \\partial ^ { \\circ } \\Phi _ { x } ( \\bar { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 106, + 321, + 120 + ], + "score": 1.0, + "content": "that ensures", + "type": "text" + }, + { + "bbox": [ + 321, + 107, + 376, + 119 + ], + "score": 0.92, + "content": "\\pmb { h } ( \\pmb { x } _ { n } ) = \\pmb { h } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 106, + 506, + 120 + ], + "score": 1.0, + "content": ". Then, (Q) is the optimization", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 118, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 228, + 132 + ], + "score": 1.0, + "content": "problem for SVM with kernel", + "type": "text" + }, + { + "bbox": [ + 228, + 118, + 340, + 130 + ], + "score": 0.93, + "content": "K _ { \\bar { \\theta } } ( x , x ^ { \\prime } ) = \\langle h ( x ) , h ( x ^ { \\prime } ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 118, + 345, + 132 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 119, + 505, + 129 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 105, + 146, + 467, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 146, + 469, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 469, + 160 + ], + "score": 1.0, + "content": "D TIGHT BOUNDS FOR LOSS CONVERGENCE AND WEIGHT GROWTH", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 505, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "score": 1.0, + "content": "In this section, we give proof for Theorem A.10, which gives tight bounds for loss convergence and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 183, + 336, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 336, + 195 + ], + "score": 1.0, + "content": "weight growth under Assumption (A1), (A2), (B3), (B4).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 208, + 249, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 250, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 250, + 223 + ], + "score": 1.0, + "content": "D.1 CONSEQUENCES OF (B3.4)", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 391, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 392, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 392, + 244 + ], + "score": 1.0, + "content": "Before proving Theorem A.10, we show some consequences of (B3.4).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 272, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 272, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 180, + 258 + ], + "score": 1.0, + "content": "Lemma D.1. For", + "type": "text" + }, + { + "bbox": [ + 180, + 244, + 198, + 257 + ], + "score": 0.9, + "content": "f ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 244, + 216, + 258 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 217, + 244, + 233, + 257 + ], + "score": 0.89, + "content": "g ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 244, + 272, + 258 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 129, + 266, + 323, + 284 + ], + "lines": [ + { + "bbox": [ + 129, + 265, + 321, + 285 + ], + "spans": [ + { + "bbox": [ + 129, + 266, + 172, + 284 + ], + "score": 1.0, + "content": "1. For all", + "type": "text" + }, + { + "bbox": [ + 172, + 267, + 228, + 282 + ], + "score": 0.69, + "content": "x \\in [ b _ { g } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 266, + 233, + 284 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 233, + 266, + 316, + 284 + ], + "score": 0.69, + "content": "\\begin{array} { r } { \\frac { g ( x ) } { g ^ { \\prime } ( x ) } \\in [ \\frac { 1 } { 2 K } x , 2 K x ] , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 265, + 321, + 285 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 129, + 291, + 330, + 308 + ], + "lines": [ + { + "bbox": [ + 126, + 289, + 331, + 310 + ], + "spans": [ + { + "bbox": [ + 126, + 289, + 172, + 310 + ], + "score": 1.0, + "content": "2. 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Combin-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 543, + 343, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 343, + 557 + ], + "score": 1.0, + "content": "ing this with Lemma D.1, we have the following corollary:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 361, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 361, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 172, + 573 + ], + "score": 1.0, + "content": "Corollary D.2.", + "type": "text" + }, + { + "bbox": [ + 172, + 558, + 256, + 571 + ], + "score": 0.92, + "content": "f ( \\Theta ( x ) ) = \\Theta ( f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 558, + 275, + 573 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 276, + 558, + 358, + 572 + ], + "score": 0.8, + "content": "g ( \\Theta ( x ) ) = \\Theta ( g ( x ) ) .", + "type": "inline_equation" + }, 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So upper bounding", + "type": "text" + }, + { + "bbox": [ + 241, + 707, + 260, + 718 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 704, + 369, + 723 + ], + "score": 1.0, + "content": "reduces to lower bounding", + "type": "text" + }, + { + "bbox": [ + 369, + 706, + 388, + 717 + ], + "score": 0.89, + "content": "G ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 704, + 507, + 723 + ], + "score": 1.0, + "content": ". 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So, (Q) is the optimization problem", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 191, + 106 + ], + "score": 1.0, + "content": "for SVM with kernel", + "type": "text" + }, + { + "bbox": [ + 191, + 93, + 333, + 107 + ], + "score": 0.91, + "content": "K _ { \\bar { \\theta } } ( x , x ^ { \\prime } ) = \\left. \\nabla \\Phi _ { x } ( \\bar { \\theta } ) , \\nabla \\Phi _ { x ^ { \\prime } } ( \\bar { \\theta } ) \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 94, + 506, + 106 + ], + "score": 1.0, + "content": ". For non-smooth models, we can construct", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 106, + 193, + 120 + ], + "score": 1.0, + "content": "an arbitrary function", + "type": "text" + }, + { + "bbox": [ + 194, + 107, + 267, + 119 + ], + "score": 0.89, + "content": "h ( x ) \\in \\partial ^ { \\circ } \\Phi _ { x } ( \\bar { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 106, + 321, + 120 + ], + "score": 1.0, + "content": "that ensures", + "type": "text" + }, + { + "bbox": [ + 321, + 107, + 376, + 119 + ], + "score": 0.92, + "content": "\\pmb { h } ( \\pmb { x } _ { n } ) = \\pmb { h } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 106, + 506, + 120 + ], + "score": 1.0, + "content": ". 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For", + "type": "text" + }, + { + "bbox": [ + 180, + 244, + 198, + 257 + ], + "score": 0.9, + "content": "f ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 244, + 216, + 258 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 217, + 244, + 233, + 257 + ], + "score": 0.89, + "content": "g ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 244, + 272, + 258 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 244, + 272, + 258 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 266, + 323, + 284 + ], + "lines": [ + { + "bbox": [ + 129, + 265, + 321, + 285 + ], + "spans": [ + { + "bbox": [ + 129, + 266, + 172, + 284 + ], + "score": 1.0, + "content": "1. 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Combin-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 543, + 343, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 343, + 557 + ], + "score": 1.0, + "content": "ing this with Lemma D.1, we have the following corollary:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 531, + 506, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 361, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 361, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 172, + 573 + ], + "score": 1.0, + "content": "Corollary D.2.", + "type": "text" + }, + { + "bbox": [ + 172, + 558, + 256, + 571 + ], + "score": 0.92, + "content": "f ( \\Theta ( x ) ) = \\Theta ( f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 558, + 275, + 573 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 276, + 558, + 358, + 572 + ], + "score": 0.8, + "content": "g ( \\Theta ( x ) ) = \\Theta ( 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"score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 617, + 302, + 631 + ], + "score": 0.92, + "content": "g ( x ) = x ^ { \\Theta ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 616, + 306, + 632 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 616, + 306, + 632 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 644, + 253, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 254, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 254, + 657 + ], + "score": 1.0, + "content": "D.2 PROOF FOR THEOREM A.10", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 102, + 660, + 509, + 685 + ], + "spans": [ + { + "bbox": [ + 102, + 660, + 295, + 685 + ], + "score": 1.0, + "content": "We follow the notations in Section 5 to define", + "type": "text" + }, + { + "bbox": [ + 295, + 665, + 340, + 677 + ], + "score": 0.93, + "content": "\\rho : = \\| \\pmb \\theta \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 660, + 359, + 685 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 664, + 438, + 679 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\hat { \\pmb \\theta } : = \\frac { \\pmb \\theta } { \\| \\pmb \\theta \\| _ { 2 } } \\in \\mathcal { S } ^ { d - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 660, + 509, + 685 + ], + "score": 1.0, + "content": ", and sometimes", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 677, + 503, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 207, + 691 + ], + "score": 1.0, + "content": "we view the functions of", + "type": "text" + }, + { + "bbox": [ + 215, + 677, + 277, + 691 + ], + "score": 1.0, + "content": "as functions of", + "type": "text" + }, + { + "bbox": [ + 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And we use the notations", + "type": "text" + }, + { + "bbox": [ + 389, + 679, + 419, + 690 + ], + "score": 0.89, + "content": "B _ { 0 } , B _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 677, + 503, + 691 + ], + "score": 1.0, + "content": "from Appendix C.3.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 102, + 660, + 509, + 691 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 694, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 694, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 390, + 708 + ], + "score": 1.0, + "content": "The key idea to prove Theorem A.10 is to utilize Lemma B.6, in which", + "type": "text" + }, + { + "bbox": [ + 391, + 695, + 410, + 707 + ], + "score": 0.9, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 694, + 505, + 708 + ], + "score": 1.0, + "content": "is bounded from above", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 103, + 704, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 103, + 704, + 119, + 723 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 706, + 159, + 721 + ], + "score": 0.93, + "content": "\\frac { 1 } { G ^ { - 1 } ( \\Omega ( t ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 704, + 241, + 723 + ], + "score": 1.0, + "content": ". So upper bounding", + "type": "text" + }, + { + "bbox": [ + 241, + 707, + 260, + 718 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 704, + 369, + 723 + ], + "score": 1.0, + "content": "reduces to lower bounding", + "type": "text" + }, + { + "bbox": [ + 369, + 706, + 388, + 717 + ], + "score": 0.89, + "content": "G ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 704, + 507, + 723 + ], + "score": 1.0, + "content": ". In the following lemma, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 719, + 318, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 246, + 734 + ], + "score": 1.0, + "content": "obtain tight asymptotic bounds for", + "type": "text" + }, + { + "bbox": [ + 247, + 720, + 266, + 732 + ], + "score": 0.91, + "content": "G ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 719, + 284, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 284, + 720, + 314, + 733 + ], + "score": 0.91, + "content": "G ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 719, + 318, + 734 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 103, + 694, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 102, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 215, + 95 + ], + "score": 1.0, + "content": "Lemma D.4. For function", + "type": "text" + }, + { + "bbox": [ + 216, + 82, + 235, + 95 + ], + "score": 0.9, + "content": "G ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 82, + 421, + 95 + ], + "score": 1.0, + "content": "defined in Lemma B.6 and its inverse function", + "type": "text" + }, + { + "bbox": [ + 421, + 82, + 451, + 95 + ], + "score": 0.92, + "content": "G ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", we have the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 181, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 181, + 106 + ], + "score": 1.0, + "content": "following bounds:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 110, + 446, + 140 + ], + "lines": [ + { + "bbox": [ + 165, + 110, + 446, + 140 + ], + "spans": [ + { + "bbox": [ + 165, + 110, + 446, + 140 + ], + "score": 0.93, + "content": "G ( x ) = \\Theta \\left( { \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } } x \\right) \\quad a n d \\quad G ^ { - 1 } ( y ) = \\Theta \\left( { \\frac { ( \\log y ) ^ { 2 } } { g ( \\log y ) ^ { 2 / L } } } y \\right) .", + "type": "interline_equation", + "image_path": "873b38301550e74e24092b0508febd59e7b0e41111c53a61297064017b84fe92.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 165, + 110, + 446, + 140 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 438, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 439, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 254, + 168 + ], + "score": 1.0, + "content": "Proof. We first prove the bounds for", + "type": "text" + }, + { + "bbox": [ + 255, + 154, + 277, + 167 + ], + "score": 0.92, + "content": "G ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 153, + 402, + 168 + ], + "score": 1.0, + "content": ", and then prove the bounds for", + "type": "text" + }, + { + "bbox": [ + 403, + 154, + 435, + 167 + ], + "score": 0.93, + "content": "G ^ { - 1 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 153, + 439, + 168 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 416, + 198 + ], + "lines": [ + { + "bbox": [ + 102, + 177, + 419, + 202 + ], + "spans": [ + { + "bbox": [ + 102, + 177, + 165, + 202 + ], + "score": 1.0, + "content": "Bounding for", + "type": "text" + }, + { + "bbox": [ + 166, + 183, + 187, + 195 + ], + "score": 0.89, + "content": "G ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 177, + 215, + 202 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 215, + 179, + 339, + 198 + ], + "score": 0.62, + "content": "\\begin{array} { r } { C _ { G } = \\int _ { 1 / \\mathcal { L } ( t _ { 0 } ) } ^ { \\exp ( b _ { g } ) } \\frac { g ^ { \\prime } ( \\log u ) ^ { 2 } } { g ( \\log u ) ^ { 2 - 2 / L } } d u } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 177, + 419, + 202 + ], + "score": 1.0, + "content": ". For x ≥ exp(bg),", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 203, + 426, + 266 + ], + "lines": [ + { + "bbox": [ + 184, + 203, + 426, + 266 + ], + "spans": [ + { + "bbox": [ + 184, + 203, + 426, + 266 + ], + "score": 0.91, + "content": "\\begin{array} { l } { { \\displaystyle G ( x ) = C _ { G } + \\int _ { \\exp ( b _ { g } ) } ^ { x } \\frac { g ^ { \\prime } ( \\log u ) ^ { 2 } } { g ( \\log u ) ^ { 2 - 2 / L } } d u } } \\\\ { { \\displaystyle \\ = C _ { G } + \\int _ { \\exp ( b _ { g } ) } ^ { x } \\left( \\frac { g ^ { \\prime } ( \\log u ) \\log u } { g ( \\log u ) } \\right) ^ { 2 } \\frac { g ( \\log u ) ^ { 2 / L } } { ( \\log u ) ^ { 2 } } d u , } } \\end{array}", + "type": "interline_equation", + "image_path": "b2882dceff5a604fa691927c857b52c892df903f7f7c1f9d0bcd280c7430f693.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 203, + 426, + 218.75 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 184, + 218.75, + 426, + 234.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 184, + 234.5, + 426, + 250.25 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 184, + 250.25, + 426, + 266.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 172, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 173, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 173, + 284 + ], + "score": 1.0, + "content": "By Lemma D.1,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 288, + 458, + 318 + ], + "lines": [ + { + "bbox": [ + 151, + 288, + 458, + 318 + ], + "spans": [ + { + "bbox": [ + 151, + 288, + 458, + 318 + ], + "score": 0.93, + "content": "G ( x ) \\leq C _ { G } + 4 K ^ { 2 } g ( \\log x ) ^ { 2 / L } \\int _ { \\exp ( b _ { g } ) } ^ { x } \\frac { 1 } { ( \\log u ) ^ { 2 } } d u = O \\left( \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } x \\right) .", + "type": "interline_equation", + "image_path": "a4c8c22821d169edde56026779bc3a844badf68a7be988c5f59c33afe1aa9e6d.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 151, + 288, + 458, + 318 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 292, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 323, + 293, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 197, + 338 + ], + "score": 1.0, + "content": "On the other hand, for", + "type": "text" + }, + { + "bbox": [ + 198, + 324, + 254, + 337 + ], + "score": 0.93, + "content": "x \\ge \\exp ( 2 b _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 323, + 293, + 338 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 342, + 496, + 372 + ], + "lines": [ + { + "bbox": [ + 114, + 342, + 496, + 372 + ], + "spans": [ + { + "bbox": [ + 114, + 342, + 496, + 372 + ], + "score": 0.94, + "content": "G ( x ) \\geq \\frac { 1 } { 4 K ^ { 2 } } \\int _ { \\sqrt x } ^ { x } \\frac { g ( \\log u ) ^ { 2 / L } } { ( \\log u ) ^ { 2 } } d u \\geq \\frac { g ( ( \\log x ) / 2 ) ^ { 2 / L } } { 4 K ^ { 2 } } \\int _ { \\sqrt x } ^ { x } \\frac { 1 } { ( \\log u ) ^ { 2 } } d u = \\Omega \\left( \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } x \\right) .", + "type": "interline_equation", + "image_path": "c364f8bc4059ba52e67ee43cb4cf3b48d3b83aa6c8a7800e421a994a9223c02a.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 114, + 342, + 496, + 372 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 167, + 398 + ], + "score": 1.0, + "content": "Bounding for", + "type": "text" + }, + { + "bbox": [ + 167, + 385, + 200, + 397 + ], + "score": 0.92, + "content": "G ^ { - 1 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 384, + 228, + 398 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 228, + 385, + 283, + 397 + ], + "score": 0.91, + "content": "x = G ^ { - 1 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 384, + 300, + 398 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 300, + 385, + 329, + 397 + ], + "score": 0.84, + "content": "y \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 384, + 335, + 398 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 335, + 385, + 358, + 397 + ], + "score": 0.86, + "content": "G ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "always has a finite value whenever", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 114, + 406 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 396, + 171, + 409 + ], + "score": 1.0, + "content": "is finite. So", + "type": "text" + }, + { + "bbox": [ + 171, + 398, + 216, + 407 + ], + "score": 0.88, + "content": "x \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 396, + 244, + 409 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 244, + 398, + 289, + 408 + ], + "score": 0.9, + "content": "y + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 396, + 505, + 409 + ], + "score": 1.0, + "content": ". According to the first part of the proof, we know", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 406, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 125, + 426 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 126, + 407, + 218, + 427 + ], + "score": 0.92, + "content": "\\begin{array} { r } { y \\ = \\ \\Theta \\left( { \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } } x \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 406, + 506, + 426 + ], + "score": 1.0, + "content": ". Taking logarithm on both sides and using Corollary D.3, we have", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 424, + 462, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 176, + 438 + ], + "score": 0.91, + "content": "\\log y = \\Theta ( \\log x )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 424, + 255, + 440 + ], + "score": 1.0, + "content": ". By Corollary D.2,", + "type": "text" + }, + { + "bbox": [ + 255, + 425, + 412, + 438 + ], + "score": 0.93, + "content": "g ( \\log y ) = g ( \\Theta ( \\log x ) ) = \\Theta ( g ( \\log x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 424, + 462, + 440 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 443, + 398, + 472 + ], + "lines": [ + { + "bbox": [ + 212, + 443, + 398, + 472 + ], + "spans": [ + { + "bbox": [ + 212, + 443, + 398, + 472 + ], + "score": 0.92, + "content": "{ \\frac { ( \\log y ) ^ { 2 } } { g ( \\log y ) ^ { 2 / L } } } y = \\Theta \\left( { \\frac { ( \\log x ) ^ { 2 } } { g ( \\log x ) ^ { 2 / L } } } y \\right) = \\Theta ( x ) .", + "type": "interline_equation", + "image_path": "d2a2e49e397f77c07e6594ee7b1c4475ae2912cf9503dba747bc1f13e391440c.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 212, + 443, + 398, + 472 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 478, + 266, + 498 + ], + "lines": [ + { + "bbox": [ + 100, + 470, + 263, + 505 + ], + "spans": [ + { + "bbox": [ + 100, + 470, + 176, + 505 + ], + "score": 1.0, + "content": "This implies that", + "type": "text" + }, + { + "bbox": [ + 176, + 478, + 263, + 498 + ], + "score": 0.93, + "content": "\\begin{array} { r } { x = \\Theta \\left( \\frac { ( \\log y ) ^ { 2 } } { g ( \\log y ) ^ { 2 / L } } y \\right) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 512, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 378, + 527 + ], + "score": 1.0, + "content": "For other bounds, we derive them as follows. We first show that", + "type": "text" + }, + { + "bbox": [ + 378, + 513, + 458, + 527 + ], + "score": 0.92, + "content": "\\begin{array} { r } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) = \\Theta ( \\rho ^ { L } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 513, + 506, + 527 + ], + "score": 1.0, + "content": ". With this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 396, + 537 + ], + "score": 1.0, + "content": "equivalence, we derive an upper bound for the gradient at each time", + "type": "text" + }, + { + "bbox": [ + 396, + 527, + 402, + 535 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 525, + 454, + 537 + ], + "score": 1.0, + "content": "in terms of", + "type": "text" + }, + { + "bbox": [ + 454, + 526, + 462, + 535 + ], + "score": 0.82, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 525, + 505, + 537 + ], + "score": 1.0, + "content": ", and take", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 205, + 549 + ], + "score": 1.0, + "content": "an integration to bound", + "type": "text" + }, + { + "bbox": [ + 206, + 536, + 225, + 547 + ], + "score": 0.91, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 536, + 482, + 549 + ], + "score": 1.0, + "content": "from below. Now we have both lower and upper bounds for", + "type": "text" + }, + { + "bbox": [ + 482, + 536, + 501, + 548 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 536, + 505, + 549 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 546, + 477, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 227, + 560 + ], + "score": 1.0, + "content": "Plugging these two bounds to", + "type": "text" + }, + { + "bbox": [ + 227, + 546, + 303, + 560 + ], + "score": 0.94, + "content": "\\begin{array} { r } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) = \\Theta ( \\rho ^ { L } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 546, + 455, + 560 + ], + "score": 1.0, + "content": "gives the lower and upper bounds for", + "type": "text" + }, + { + "bbox": [ + 455, + 547, + 472, + 559 + ], + "score": 0.92, + "content": "\\rho ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 546, + 477, + 560 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 354, + 587 + ], + "score": 1.0, + "content": "Proof for Theorem A.10. We first prove the upper bound for", + "type": "text" + }, + { + "bbox": [ + 355, + 574, + 363, + 583 + ], + "score": 0.78, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 572, + 506, + 587 + ], + "score": 1.0, + "content": ". Then we derive lower and upper", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 152, + 597 + ], + "score": 1.0, + "content": "bounds for", + "type": "text" + }, + { + "bbox": [ + 153, + 587, + 159, + 596 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 585, + 207, + 597 + ], + "score": 1.0, + "content": "in terms of", + "type": "text" + }, + { + "bbox": [ + 208, + 585, + 216, + 595 + ], + "score": 0.79, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 585, + 413, + 597 + ], + "score": 1.0, + "content": ", and use these bounds to give a lower bound for", + "type": "text" + }, + { + "bbox": [ + 413, + 585, + 421, + 595 + ], + "score": 0.76, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 585, + 505, + 597 + ], + "score": 1.0, + "content": ". Finally, we plug in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 595, + 424, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 187, + 608 + ], + "score": 1.0, + "content": "the tight bounds for", + "type": "text" + }, + { + "bbox": [ + 187, + 596, + 195, + 605 + ], + "score": 0.83, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 595, + 361, + 608 + ], + "score": 1.0, + "content": "to obtain the lower and upper bounds for", + "type": "text" + }, + { + "bbox": [ + 361, + 597, + 368, + 607 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 595, + 415, + 608 + ], + "score": 1.0, + "content": "in terms of", + "type": "text" + }, + { + "bbox": [ + 415, + 597, + 419, + 605 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 595, + 424, + 608 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 182, + 634 + ], + "score": 1.0, + "content": "Upper Bounding", + "type": "text" + }, + { + "bbox": [ + 182, + 621, + 190, + 630 + ], + "score": 0.67, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 618, + 308, + 634 + ], + "score": 1.0, + "content": ". By Lemma B.6, we have", + "type": "text" + }, + { + "bbox": [ + 308, + 619, + 379, + 633 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { \\mathcal { L } } \\geq G ^ { - 1 } ( \\Omega ( t ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 618, + 506, + 634 + ], + "score": 1.0, + "content": ". Using Lemma D.4, we have", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 136, + 631, + 311, + 654 + ], + "spans": [ + { + "bbox": [ + 136, + 631, + 311, + 654 + ], + "score": 1.0, + "content": "\u0010 (log t)2g(log t)2/L t\u0011, which completes the proof.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 505, + 706 + ], + "lines": [ + { + "bbox": [ + 104, + 663, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 663, + 151, + 681 + ], + "score": 1.0, + "content": "Bounding", + "type": "text" + }, + { + "bbox": [ + 151, + 668, + 159, + 677 + ], + "score": 0.78, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 663, + 215, + 681 + ], + "score": 1.0, + "content": "in Terms of", + "type": "text" + }, + { + "bbox": [ + 216, + 666, + 224, + 676 + ], + "score": 0.73, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 663, + 234, + 681 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 235, + 665, + 293, + 678 + ], + "score": 0.91, + "content": "\\tilde { \\gamma } ( t ) \\ \\geq \\ \\tilde { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 663, + 311, + 681 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 311, + 664, + 465, + 680 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\rho ^ { L } \\ \\leq \\ \\frac { 1 } { \\tilde { \\gamma } ( t _ { 0 } ) } g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\ = \\ O ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 663, + 506, + 681 + ], + "score": 1.0, + "content": ". On the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 678, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 155, + 695 + ], + "score": 1.0, + "content": "other hand,", + "type": "text" + }, + { + "bbox": [ + 155, + 679, + 263, + 693 + ], + "score": 0.92, + "content": "g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) \\leq q _ { \\operatorname* { m i n } } \\leq B _ { 0 } \\rho ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 678, + 283, + 695 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 284, + 680, + 360, + 693 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\rho ^ { L } = \\Omega ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 678, + 506, + 695 + ], + "score": 1.0, + "content": ". Therefore, we have the following", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 690, + 264, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 191, + 707 + ], + "score": 1.0, + "content": "relationship between", + "type": "text" + }, + { + "bbox": [ + 192, + 693, + 204, + 705 + ], + "score": 0.89, + "content": "\\rho ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 690, + 222, + 707 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 222, + 693, + 258, + 707 + ], + "score": 0.91, + "content": "g ( \\log { \\frac { 1 } { \\mathcal { L } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 690, + 264, + 707 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 712, + 347, + 736 + ], + "lines": [ + { + "bbox": [ + 264, + 712, + 347, + 736 + ], + "spans": [ + { + "bbox": [ + 264, + 712, + 347, + 736 + ], + "score": 0.93, + "content": "\\rho ^ { L } = \\Theta ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) .", + "type": "interline_equation", + "image_path": "fef86a533fa4ff25f7e96e530b5cb9179fd506dfc2e2a0c22279830c787169ef.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 264, + 712, + 347, + 736 + ], + "spans": [], + "index": 31 + } + ] + } + ], + "page_idx": 27, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "28", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 482, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 495, + 484, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 495, + 484, + 505, + 493 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 102, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 215, + 95 + ], + "score": 1.0, + "content": "Lemma D.4. For function", + "type": "text" + }, + { + "bbox": [ + 216, + 82, + 235, + 95 + ], + "score": 0.9, + "content": "G ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 82, + 421, + 95 + ], + "score": 1.0, + "content": "defined in Lemma B.6 and its inverse function", + "type": "text" + }, + { + "bbox": [ + 421, + 82, + 451, + 95 + ], + "score": 0.92, + "content": "G ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", we have the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 181, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 181, + 106 + ], + "score": 1.0, + "content": "following bounds:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 82, + 505, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 110, + 446, + 140 + ], + "lines": [ + { + "bbox": [ + 165, + 110, + 446, + 140 + ], + "spans": [ + { + "bbox": [ + 165, + 110, + 446, + 140 + ], + "score": 0.93, + "content": "G ( x ) = \\Theta \\left( { \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } } x \\right) \\quad a n d \\quad G ^ { - 1 } ( y ) = \\Theta \\left( { \\frac { ( \\log y ) ^ { 2 } } { g ( \\log y ) ^ { 2 / L } } } y \\right) .", + "type": "interline_equation", + "image_path": "873b38301550e74e24092b0508febd59e7b0e41111c53a61297064017b84fe92.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 165, + 110, + 446, + 140 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 438, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 439, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 254, + 168 + ], + "score": 1.0, + "content": "Proof. We first prove the bounds for", + "type": "text" + }, + { + "bbox": [ + 255, + 154, + 277, + 167 + ], + "score": 0.92, + "content": "G ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 153, + 402, + 168 + ], + "score": 1.0, + "content": ", and then prove the bounds for", + "type": "text" + }, + { + "bbox": [ + 403, + 154, + 435, + 167 + ], + "score": 0.93, + "content": "G ^ { - 1 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 153, + 439, + 168 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 153, + 439, + 168 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 416, + 198 + ], + "lines": [ + { + "bbox": [ + 102, + 177, + 419, + 202 + ], + "spans": [ + { + "bbox": [ + 102, + 177, + 165, + 202 + ], + "score": 1.0, + "content": "Bounding for", + "type": "text" + }, + { + "bbox": [ + 166, + 183, + 187, + 195 + ], + "score": 0.89, + "content": "G ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 177, + 215, + 202 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 215, + 179, + 339, + 198 + ], + "score": 0.62, + "content": "\\begin{array} { r } { C _ { G } = \\int _ { 1 / \\mathcal { L } ( t _ { 0 } ) } ^ { \\exp ( b _ { g } ) } \\frac { g ^ { \\prime } ( \\log u ) ^ { 2 } } { g ( \\log u ) ^ { 2 - 2 / L } } d u } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 177, + 419, + 202 + ], + "score": 1.0, + "content": ". For x ≥ exp(bg),", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 102, + 177, + 419, + 202 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 203, + 426, + 266 + ], + "lines": [ + { + "bbox": [ + 184, + 203, + 426, + 266 + ], + "spans": [ + { + "bbox": [ + 184, + 203, + 426, + 266 + ], + "score": 0.91, + "content": "\\begin{array} { l } { { \\displaystyle G ( x ) = C _ { G } + \\int _ { \\exp ( b _ { g } ) } ^ { x } \\frac { g ^ { \\prime } ( \\log u ) ^ { 2 } } { g ( \\log u ) ^ { 2 - 2 / L } } d u } } \\\\ { { \\displaystyle \\ = C _ { G } + \\int _ { \\exp ( b _ { g } ) } ^ { x } \\left( \\frac { g ^ { \\prime } ( \\log u ) \\log u } { g ( \\log u ) } \\right) ^ { 2 } \\frac { g ( \\log u ) ^ { 2 / L } } { ( \\log u ) ^ { 2 } } d u , } } \\end{array}", + "type": "interline_equation", + "image_path": "b2882dceff5a604fa691927c857b52c892df903f7f7c1f9d0bcd280c7430f693.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 203, + 426, + 218.75 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 184, + 218.75, + 426, + 234.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 184, + 234.5, + 426, + 250.25 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 184, + 250.25, + 426, + 266.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 172, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 173, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 173, + 284 + ], + "score": 1.0, + "content": "By Lemma D.1,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 269, + 173, + 284 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 288, + 458, + 318 + ], + "lines": [ + { + "bbox": [ + 151, + 288, + 458, + 318 + ], + "spans": [ + { + "bbox": [ + 151, + 288, + 458, + 318 + ], + "score": 0.93, + "content": "G ( x ) \\leq C _ { G } + 4 K ^ { 2 } g ( \\log x ) ^ { 2 / L } \\int _ { \\exp ( b _ { g } ) } ^ { x } \\frac { 1 } { ( \\log u ) ^ { 2 } } d u = O \\left( \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } x \\right) .", + "type": "interline_equation", + "image_path": "a4c8c22821d169edde56026779bc3a844badf68a7be988c5f59c33afe1aa9e6d.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 151, + 288, + 458, + 318 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 292, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 323, + 293, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 197, + 338 + ], + "score": 1.0, + "content": "On the other hand, for", + "type": "text" + }, + { + "bbox": [ + 198, + 324, + 254, + 337 + ], + "score": 0.93, + "content": "x \\ge \\exp ( 2 b _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 323, + 293, + 338 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 323, + 293, + 338 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 342, + 496, + 372 + ], + "lines": [ + { + "bbox": [ + 114, + 342, + 496, + 372 + ], + "spans": [ + { + "bbox": [ + 114, + 342, + 496, + 372 + ], + "score": 0.94, + "content": "G ( x ) \\geq \\frac { 1 } { 4 K ^ { 2 } } \\int _ { \\sqrt x } ^ { x } \\frac { g ( \\log u ) ^ { 2 / L } } { ( \\log u ) ^ { 2 } } d u \\geq \\frac { g ( ( \\log x ) / 2 ) ^ { 2 / L } } { 4 K ^ { 2 } } \\int _ { \\sqrt x } ^ { x } \\frac { 1 } { ( \\log u ) ^ { 2 } } d u = \\Omega \\left( \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } x \\right) .", + "type": "interline_equation", + "image_path": "c364f8bc4059ba52e67ee43cb4cf3b48d3b83aa6c8a7800e421a994a9223c02a.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 114, + 342, + 496, + 372 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 167, + 398 + ], + "score": 1.0, + "content": "Bounding for", + "type": "text" + }, + { + "bbox": [ + 167, + 385, + 200, + 397 + ], + "score": 0.92, + "content": "G ^ { - 1 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 384, + 228, + 398 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 228, + 385, + 283, + 397 + ], + "score": 0.91, + "content": "x = G ^ { - 1 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 384, + 300, + 398 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 300, + 385, + 329, + 397 + ], + "score": 0.84, + "content": "y \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 384, + 335, + 398 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 335, + 385, + 358, + 397 + ], + "score": 0.86, + "content": "G ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "always has a finite value whenever", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 114, + 406 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 396, + 171, + 409 + ], + "score": 1.0, + "content": "is finite. So", + "type": "text" + }, + { + "bbox": [ + 171, + 398, + 216, + 407 + ], + "score": 0.88, + "content": "x \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 396, + 244, + 409 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 244, + 398, + 289, + 408 + ], + "score": 0.9, + "content": "y + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 396, + 505, + 409 + ], + "score": 1.0, + "content": ". According to the first part of the proof, we know", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 406, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 125, + 426 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 126, + 407, + 218, + 427 + ], + "score": 0.92, + "content": "\\begin{array} { r } { y \\ = \\ \\Theta \\left( { \\frac { g ( \\log x ) ^ { 2 / L } } { ( \\log x ) ^ { 2 } } } x \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 406, + 506, + 426 + ], + "score": 1.0, + "content": ". Taking logarithm on both sides and using Corollary D.3, we have", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 424, + 462, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 176, + 438 + ], + "score": 0.91, + "content": "\\log y = \\Theta ( \\log x )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 424, + 255, + 440 + ], + "score": 1.0, + "content": ". By Corollary D.2,", + "type": "text" + }, + { + "bbox": [ + 255, + 425, + 412, + 438 + ], + "score": 0.93, + "content": "g ( \\log y ) = g ( \\Theta ( \\log x ) ) = \\Theta ( g ( \\log x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 424, + 462, + 440 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 384, + 506, + 440 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 443, + 398, + 472 + ], + "lines": [ + { + "bbox": [ + 212, + 443, + 398, + 472 + ], + "spans": [ + { + "bbox": [ + 212, + 443, + 398, + 472 + ], + "score": 0.92, + "content": "{ \\frac { ( \\log y ) ^ { 2 } } { g ( \\log y ) ^ { 2 / L } } } y = \\Theta \\left( { \\frac { ( \\log x ) ^ { 2 } } { g ( \\log x ) ^ { 2 / L } } } y \\right) = \\Theta ( x ) .", + "type": "interline_equation", + "image_path": "d2a2e49e397f77c07e6594ee7b1c4475ae2912cf9503dba747bc1f13e391440c.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 212, + 443, + 398, + 472 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 478, + 266, + 498 + ], + "lines": [ + { + "bbox": [ + 100, + 470, + 263, + 505 + ], + "spans": [ + { + "bbox": [ + 100, + 470, + 176, + 505 + ], + "score": 1.0, + "content": "This implies that", + "type": "text" + }, + { + "bbox": [ + 176, + 478, + 263, + 498 + ], + "score": 0.93, + "content": "\\begin{array} { r } { x = \\Theta \\left( \\frac { ( \\log y ) ^ { 2 } } { g ( \\log y ) ^ { 2 / L } } y \\right) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 100, + 470, + 263, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 512, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 378, + 527 + ], + "score": 1.0, + "content": "For other bounds, we derive them as follows. We first show that", + "type": "text" + }, + { + "bbox": [ + 378, + 513, + 458, + 527 + ], + "score": 0.92, + "content": "\\begin{array} { r } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) = \\Theta ( \\rho ^ { L } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 513, + 506, + 527 + ], + "score": 1.0, + "content": ". With this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 396, + 537 + ], + "score": 1.0, + "content": "equivalence, we derive an upper bound for the gradient at each time", + "type": "text" + }, + { + "bbox": [ + 396, + 527, + 402, + 535 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 525, + 454, + 537 + ], + "score": 1.0, + "content": "in terms of", + "type": "text" + }, + { + "bbox": [ + 454, + 526, + 462, + 535 + ], + "score": 0.82, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 525, + 505, + 537 + ], + "score": 1.0, + "content": ", and take", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 205, + 549 + ], + "score": 1.0, + "content": "an integration to bound", + "type": "text" + }, + { + "bbox": [ + 206, + 536, + 225, + 547 + ], + "score": 0.91, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 536, + 482, + 549 + ], + "score": 1.0, + "content": "from below. Now we have both lower and upper bounds for", + "type": "text" + }, + { + "bbox": [ + 482, + 536, + 501, + 548 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 536, + 505, + 549 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 546, + 477, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 227, + 560 + ], + "score": 1.0, + "content": "Plugging these two bounds to", + "type": "text" + }, + { + "bbox": [ + 227, + 546, + 303, + 560 + ], + "score": 0.94, + "content": "\\begin{array} { r } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) = \\Theta ( \\rho ^ { L } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 546, + 455, + 560 + ], + "score": 1.0, + "content": "gives the lower and upper bounds for", + "type": "text" + }, + { + "bbox": [ + 455, + 547, + 472, + 559 + ], + "score": 0.92, + "content": "\\rho ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 546, + 477, + 560 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 513, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 354, + 587 + ], + "score": 1.0, + "content": "Proof for Theorem A.10. We first prove the upper bound for", + "type": "text" + }, + { + "bbox": [ + 355, + 574, + 363, + 583 + ], + "score": 0.78, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 572, + 506, + 587 + ], + "score": 1.0, + "content": ". Then we derive lower and upper", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 152, + 597 + ], + "score": 1.0, + "content": "bounds for", + "type": "text" + }, + { + "bbox": [ + 153, + 587, + 159, + 596 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 585, + 207, + 597 + ], + "score": 1.0, + "content": "in terms of", + "type": "text" + }, + { + "bbox": [ + 208, + 585, + 216, + 595 + ], + "score": 0.79, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 585, + 413, + 597 + ], + "score": 1.0, + "content": ", and use these bounds to give a lower bound for", + "type": "text" + }, + { + "bbox": [ + 413, + 585, + 421, + 595 + ], + "score": 0.76, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 585, + 505, + 597 + ], + "score": 1.0, + "content": ". Finally, we plug in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 595, + 424, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 187, + 608 + ], + "score": 1.0, + "content": "the tight bounds for", + "type": "text" + }, + { + "bbox": [ + 187, + 596, + 195, + 605 + ], + "score": 0.83, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 595, + 361, + 608 + ], + "score": 1.0, + "content": "to obtain the lower and upper bounds for", + "type": "text" + }, + { + "bbox": [ + 361, + 597, + 368, + 607 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 595, + 415, + 608 + ], + "score": 1.0, + "content": "in terms of", + "type": "text" + }, + { + "bbox": [ + 415, + 597, + 419, + 605 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 595, + 424, + 608 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 572, + 506, + 608 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 182, + 634 + ], + "score": 1.0, + "content": "Upper Bounding", + "type": "text" + }, + { + "bbox": [ + 182, + 621, + 190, + 630 + ], + "score": 0.67, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 618, + 308, + 634 + ], + "score": 1.0, + "content": ". By Lemma B.6, we have", + "type": "text" + }, + { + "bbox": [ + 308, + 619, + 379, + 633 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { \\mathcal { L } } \\geq G ^ { - 1 } ( \\Omega ( t ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 618, + 506, + 634 + ], + "score": 1.0, + "content": ". Using Lemma D.4, we have", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 136, + 631, + 311, + 654 + ], + "spans": [ + { + "bbox": [ + 136, + 631, + 311, + 654 + ], + "score": 1.0, + "content": "\u0010 (log t)2g(log t)2/L t\u0011, which completes the proof.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 618, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 505, + 706 + ], + "lines": [ + { + "bbox": [ + 104, + 663, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 663, + 151, + 681 + ], + "score": 1.0, + "content": "Bounding", + "type": "text" + }, + { + "bbox": [ + 151, + 668, + 159, + 677 + ], + "score": 0.78, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 663, + 215, + 681 + ], + "score": 1.0, + "content": "in Terms of", + "type": "text" + }, + { + "bbox": [ + 216, + 666, + 224, + 676 + ], + "score": 0.73, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 663, + 234, + 681 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 235, + 665, + 293, + 678 + ], + "score": 0.91, + "content": "\\tilde { \\gamma } ( t ) \\ \\geq \\ \\tilde { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 663, + 311, + 681 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 311, + 664, + 465, + 680 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\rho ^ { L } \\ \\leq \\ \\frac { 1 } { \\tilde { \\gamma } ( t _ { 0 } ) } g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\ = \\ O ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 663, + 506, + 681 + ], + "score": 1.0, + "content": ". On the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 678, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 155, + 695 + ], + "score": 1.0, + "content": "other hand,", + "type": "text" + }, + { + "bbox": [ + 155, + 679, + 263, + 693 + ], + "score": 0.92, + "content": "g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) \\leq q _ { \\operatorname* { m i n } } \\leq B _ { 0 } \\rho ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 678, + 283, + 695 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 284, + 680, + 360, + 693 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\rho ^ { L } = \\Omega ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 678, + 506, + 695 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 216, + 83, + 264, + 94 + ], + "score": 0.91, + "content": "\\pmb { h } _ { 1 } , \\dots , \\pmb { h } _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 82, + 378, + 95 + ], + "score": 1.0, + "content": "be a set of vectors such that", + "type": "text" + }, + { + "bbox": [ + 378, + 81, + 419, + 96 + ], + "score": 0.94, + "content": "\\begin{array} { r } { h _ { n } \\in { \\frac { \\partial q _ { n } } { \\partial \\theta } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 82, + 438, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 102, + 364, + 137 + ], + "lines": [ + { + "bbox": [ + 246, + 102, + 364, + 137 + ], + "spans": [ + { + "bbox": [ + 246, + 102, + 364, + 137 + ], + "score": 0.94, + "content": "\\frac { d \\pmb { \\theta } } { d t } = \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) \\pmb { h } _ { n } .", + "type": "interline_equation", + "image_path": "2046a4f56e514d4773a2bb499f195f078901c66a9903fe6b96090cff56a524e5.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 102, + 364, + 119.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 246, + 119.5, + 364, + 137.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 504, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 225, + 158 + ], + "score": 1.0, + "content": "By (17) and Corollary D.2,", + "type": "text" + }, + { + "bbox": [ + 226, + 144, + 504, + 158 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q _ { n } ) \\ = \\ f ^ { \\prime } ( { \\cal O } ( \\rho ^ { L } ) ) \\ = \\ f ^ { \\prime } ( { \\cal O } ( g ( \\log \\frac { 1 } { \\cal L } ) ) ) \\ = \\ { \\cal O } ( f ^ { \\prime } ( g ( \\log \\frac { 1 } { \\cal L } ) ) ) \\ =", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 155, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 107, + 157, + 170, + 171 + ], + "score": 0.9, + "content": "\\begin{array} { r } { O ( 1 / g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 155, + 280, + 173 + ], + "score": 1.0, + "content": ". 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Combining", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 168, + 354, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 354, + 182 + ], + "score": 1.0, + "content": "these two bounds together, it follows from Corollary D.2 that", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 187, + 430, + 221 + ], + "lines": [ + { + "bbox": [ + 180, + 187, + 430, + 221 + ], + "spans": [ + { + "bbox": [ + 180, + 187, + 430, + 221 + ], + "score": 0.94, + "content": "f ^ { \\prime } ( q _ { n } ) \\left\\| h _ { n } \\right\\| _ { 2 } = O \\left( \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 1 - 1 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\right) = O \\left( \\frac { \\log \\frac { 1 } { \\mathcal { L } } } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 1 / L } } \\right) .", + "type": "interline_equation", + "image_path": "e681658330e304224bcd2459bc8a4ed0e20d67e8f32100d55977931866412fa7.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 180, + 187, + 430, + 198.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 180, + 198.33333333333334, + 430, + 209.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 180, + 209.66666666666669, + 430, + 221.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 131, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 134, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 134, + 241 + ], + "score": 1.0, + "content": "Thus,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 245, + 483, + 320 + ], + "lines": [ + { + "bbox": [ + 126, + 245, + 483, + 320 + ], + "spans": [ + { + "bbox": [ + 126, + 245, + 483, + 320 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle - \\frac { d \\mathcal { L } } { d t } = \\left\\| \\frac { d \\theta } { d t } \\right\\| _ { 2 } ^ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) h _ { n } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\displaystyle \\qquad \\leq \\left( \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } \\cdot \\displaystyle \\operatorname* { m a x } _ { n \\in [ N ] } \\left\\{ f ^ { \\prime } ( q _ { n } ) \\left\\| h _ { n } \\right\\| _ { 2 } \\right\\} \\right) ^ { 2 } \\leq \\mathcal { L } ^ { 2 } \\cdot O \\left( \\frac { ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 / L } } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "bbd5fc34dcd9fca31f99dd860b1c191455438422e5ee18f8f4a82a98bc029080.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 126, + 245, + 483, + 270.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 126, + 270.0, + 483, + 295.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 126, + 295.0, + 483, + 320.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 504, + 351 + ], + "lines": [ + { + "bbox": [ + 104, + 324, + 507, + 345 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 172, + 345 + ], + "score": 1.0, + "content": "By definition of", + "type": "text" + }, + { + "bbox": [ + 172, + 327, + 191, + 339 + ], + "score": 0.91, + "content": "G ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 324, + 350, + 345 + ], + "score": 1.0, + "content": ", this implies that there exists a constant", + "type": "text" + }, + { + "bbox": [ + 350, + 330, + 356, + 338 + ], + "score": 0.74, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 324, + 395, + 345 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 396, + 326, + 447, + 341 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d } { d t } G ( \\frac { 1 } { \\mathcal { L } } ) \\leq c } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 324, + 478, + 345 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 479, + 328, + 487, + 338 + ], + "score": 0.81, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 324, + 507, + 345 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 338, + 388, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 388, + 351 + ], + "score": 1.0, + "content": "is small enough. 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By (17) and the tight bound for", + "type": "text" + }, + { + "bbox": [ + 379, + 364, + 398, + 376 + ], + "score": 0.7, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 363, + 505, + 377 + ], + "score": 0.77, + "content": ", ~ \\rho ^ { L } ~ = ~ \\Theta ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) ~ =", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 376, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 107, + 377, + 168, + 390 + ], + "score": 0.91, + "content": "\\Theta ( g ( \\Theta ( \\log t ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 376, + 347, + 391 + ], + "score": 1.0, + "content": "). Using Corollary D.2, we can conclude that", + "type": "text" + }, + { + "bbox": [ + 347, + 376, + 417, + 389 + ], + "score": 0.93, + "content": "\\rho ^ { L } = \\Theta ( g ( \\log t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 376, + 423, + 391 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 379, + 505, + 388 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 406, + 451, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 452, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 452, + 421 + ], + "score": 1.0, + "content": "E GRADIENT DESCENT: SMOOTH HOMOGENEOUS MODELS WITH", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 125, + 420, + 234, + 434 + ], + "spans": [ + { + "bbox": [ + 125, + 420, + 234, + 434 + ], + "score": 1.0, + "content": "EXPONENTIAL LOSS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 503, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "In this section, we discretize our proof to prove similar results for gradient descent on smooth ho-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 458, + 420, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 420, + 470 + ], + "score": 1.0, + "content": "mogeneous models. As usual, the update rule of gradient descent is defined as", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 476, + 367, + 491 + ], + "lines": [ + { + "bbox": [ + 244, + 476, + 367, + 491 + ], + "spans": [ + { + "bbox": [ + 244, + 476, + 367, + 491 + ], + "score": 0.91, + "content": "\\pmb \\theta ( t + 1 ) = \\pmb \\theta ( t ) - \\eta ( t ) \\nabla \\mathcal L ( t )", + "type": "interline_equation", + "image_path": "6fb50094d1e04ee2654af4670b3920e27bf088cd4cd48b37572f80637967a50e.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 244, + 476, + 367, + 491 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 438, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 439, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 128, + 512 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 497, + 146, + 510 + ], + "score": 0.92, + "content": "\\eta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 495, + 243, + 512 + ], + "score": 1.0, + "content": "is the learning rate, and", + "type": "text" + }, + { + "bbox": [ + 244, + 497, + 327, + 510 + ], + "score": 0.93, + "content": "\\nabla \\mathcal { L } ( t ) : = \\nabla \\mathcal { L } ( \\pmb { \\theta } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 495, + 398, + 512 + ], + "score": 1.0, + "content": "is the gradient of", + "type": "text" + }, + { + "bbox": [ + 398, + 498, + 406, + 507 + ], + "score": 0.82, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 495, + 417, + 512 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 417, + 497, + 435, + 510 + ], + "score": 0.92, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 495, + 439, + 512 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "We first focus on the exponential loss. At the end of this section (Appendix F), we discuss how to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 525, + 413, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 413, + 538 + ], + "score": 1.0, + "content": "extend the proof to general loss functions with a similar assumption as (B3).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 620 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "The main difficulty for discretizing our previous analysis comes from the fact that the original ver-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 271, + 567 + ], + "score": 1.0, + "content": "sion of the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 271, + 552, + 349, + 565 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\tilde { \\gamma } ( \\pmb { \\theta } ) : = \\rho ^ { - \\tilde { L } } \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 552, + 461, + 567 + ], + "score": 1.0, + "content": "becomes less smooth when", + "type": "text" + }, + { + "bbox": [ + 461, + 555, + 501, + 565 + ], + "score": 0.9, + "content": "\\rho \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 552, + 506, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 271, + 577 + ], + "score": 1.0, + "content": "Thus, if we take a Taylor expansion for", + "type": "text" + }, + { + "bbox": [ + 272, + 565, + 321, + 576 + ], + "score": 0.9, + "content": "\\tilde { \\gamma } ( \\pmb { \\theta } ( t + 1 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 564, + 385, + 577 + ], + "score": 1.0, + "content": "from the point", + "type": "text" + }, + { + "bbox": [ + 386, + 564, + 404, + 576 + ], + "score": 0.9, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 564, + 505, + 577 + ], + "score": 1.0, + "content": ", although one can show", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "that the first-order term is positive as in the gradient flow analysis, the second-order term is unlikely", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "score": 1.0, + "content": "to be bounded during gradient descent with a constant step size. 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Let", + "type": "text" + }, + { + "bbox": [ + 216, + 83, + 264, + 94 + ], + "score": 0.91, + "content": "\\pmb { h } _ { 1 } , \\dots , \\pmb { h } _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 82, + 378, + 95 + ], + "score": 1.0, + "content": "be a set of vectors such that", + "type": "text" + }, + { + "bbox": [ + 378, + 81, + 419, + 96 + ], + "score": 0.94, + "content": "\\begin{array} { r } { h _ { n } \\in { \\frac { \\partial q _ { n } } { \\partial \\theta } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 82, + 438, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 438, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 102, + 364, + 137 + ], + "lines": [ + { + "bbox": [ + 246, + 102, + 364, + 137 + ], + "spans": [ + { + "bbox": [ + 246, + 102, + 364, + 137 + ], + "score": 0.94, + "content": "\\frac { d \\pmb { \\theta } } { d t } = \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) \\pmb { h } _ { n } .", + "type": "interline_equation", + "image_path": "2046a4f56e514d4773a2bb499f195f078901c66a9903fe6b96090cff56a524e5.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 102, + 364, + 119.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 246, + 119.5, + 364, + 137.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 504, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 225, + 158 + ], + "score": 1.0, + "content": "By (17) and Corollary D.2,", + "type": "text" + }, + { + "bbox": [ + 226, + 144, + 504, + 158 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q _ { n } ) \\ = \\ f ^ { \\prime } ( { \\cal O } ( \\rho ^ { L } ) ) \\ = \\ f ^ { \\prime } ( { \\cal O } ( g ( \\log \\frac { 1 } { \\cal L } ) ) ) \\ = \\ { \\cal O } ( f ^ { \\prime } ( g ( \\log \\frac { 1 } { \\cal L } ) ) ) \\ =", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 155, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 107, + 157, + 170, + 171 + ], + "score": 0.9, + "content": "\\begin{array} { r } { O ( 1 / g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 155, + 280, + 173 + ], + "score": 1.0, + "content": ". Again by (17), we have", + "type": "text" + }, + { + "bbox": [ + 280, + 158, + 450, + 171 + ], + "score": 0.89, + "content": "\\| h _ { n } \\| _ { 2 } \\le B _ { 1 } \\rho ^ { L - 1 } = O ( g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ^ { 1 - 1 / L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 155, + 506, + 173 + ], + "score": 1.0, + "content": ". Combining", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 168, + 354, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 354, + 182 + ], + "score": 1.0, + "content": "these two bounds together, it follows from Corollary D.2 that", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 143, + 506, + 182 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 187, + 430, + 221 + ], + "lines": [ + { + "bbox": [ + 180, + 187, + 430, + 221 + ], + "spans": [ + { + "bbox": [ + 180, + 187, + 430, + 221 + ], + "score": 0.94, + "content": "f ^ { \\prime } ( q _ { n } ) \\left\\| h _ { n } \\right\\| _ { 2 } = O \\left( \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 1 - 1 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\right) = O \\left( \\frac { \\log \\frac { 1 } { \\mathcal { L } } } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 1 / L } } \\right) .", + "type": "interline_equation", + "image_path": "e681658330e304224bcd2459bc8a4ed0e20d67e8f32100d55977931866412fa7.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 180, + 187, + 430, + 198.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 180, + 198.33333333333334, + 430, + 209.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 180, + 209.66666666666669, + 430, + 221.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 131, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 134, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 134, + 241 + ], + "score": 1.0, + "content": "Thus,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 226, + 134, + 241 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 245, + 483, + 320 + ], + "lines": [ + { + "bbox": [ + 126, + 245, + 483, + 320 + ], + "spans": [ + { + "bbox": [ + 126, + 245, + 483, + 320 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle - \\frac { d \\mathcal { L } } { d t } = \\left\\| \\frac { d \\theta } { d t } \\right\\| _ { 2 } ^ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } f ^ { \\prime } ( q _ { n } ) h _ { n } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\displaystyle \\qquad \\leq \\left( \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ) } \\cdot \\displaystyle \\operatorname* { m a x } _ { n \\in [ N ] } \\left\\{ f ^ { \\prime } ( q _ { n } ) \\left\\| h _ { n } \\right\\| _ { 2 } \\right\\} \\right) ^ { 2 } \\leq \\mathcal { L } ^ { 2 } \\cdot O \\left( \\frac { ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 / L } } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "bbd5fc34dcd9fca31f99dd860b1c191455438422e5ee18f8f4a82a98bc029080.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 126, + 245, + 483, + 270.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 126, + 270.0, + 483, + 295.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 126, + 295.0, + 483, + 320.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 504, + 351 + ], + "lines": [ + { + "bbox": [ + 104, + 324, + 507, + 345 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 172, + 345 + ], + "score": 1.0, + "content": "By definition of", + "type": "text" + }, + { + "bbox": [ + 172, + 327, + 191, + 339 + ], + "score": 0.91, + "content": "G ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 324, + 350, + 345 + ], + "score": 1.0, + "content": ", this implies that there exists a constant", + "type": "text" + }, + { + "bbox": [ + 350, + 330, + 356, + 338 + ], + "score": 0.74, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 324, + 395, + 345 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 396, + 326, + 447, + 341 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d } { d t } G ( \\frac { 1 } { \\mathcal { L } } ) \\leq c } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 324, + 478, + 345 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 479, + 328, + 487, + 338 + ], + "score": 0.81, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 324, + 507, + 345 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 338, + 388, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 388, + 351 + ], + "score": 1.0, + "content": "is small enough. We can complete our proof by applying Lemma D.4.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 324, + 507, + 351 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 363, + 504, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 152, + 379 + ], + "score": 1.0, + "content": "Bounding", + "type": "text" + }, + { + "bbox": [ + 153, + 367, + 159, + 376 + ], + "score": 0.76, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 361, + 219, + 379 + ], + "score": 1.0, + "content": "in Terms of", + "type": "text" + }, + { + "bbox": [ + 220, + 366, + 225, + 374 + ], + "score": 0.65, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 361, + 378, + 379 + ], + "score": 1.0, + "content": ". By (17) and the tight bound for", + "type": "text" + }, + { + "bbox": [ + 379, + 364, + 398, + 376 + ], + "score": 0.7, + "content": "\\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 363, + 505, + 377 + ], + "score": 0.77, + "content": ", ~ \\rho ^ { L } ~ = ~ \\Theta ( g ( \\log \\frac { 1 } { \\mathcal { L } } ) ) ~ =", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 376, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 107, + 377, + 168, + 390 + ], + "score": 0.91, + "content": "\\Theta ( g ( \\Theta ( \\log t ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 376, + 347, + 391 + ], + "score": 1.0, + "content": "). Using Corollary D.2, we can conclude that", + "type": "text" + }, + { + "bbox": [ + 347, + 376, + 417, + 389 + ], + "score": 0.93, + "content": "\\rho ^ { L } = \\Theta ( g ( \\log t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 376, + 423, + 391 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 379, + 505, + 388 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 361, + 505, + 391 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 406, + 451, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 452, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 452, + 421 + ], + "score": 1.0, + "content": "E GRADIENT DESCENT: SMOOTH HOMOGENEOUS MODELS WITH", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 125, + 420, + 234, + 434 + ], + "spans": [ + { + "bbox": [ + 125, + 420, + 234, + 434 + ], + "score": 1.0, + "content": "EXPONENTIAL LOSS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 503, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "In this section, we discretize our proof to prove similar results for gradient descent on smooth ho-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 458, + 420, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 420, + 470 + ], + "score": 1.0, + "content": "mogeneous models. As usual, the update rule of gradient descent is defined as", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 447, + 505, + 470 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 476, + 367, + 491 + ], + "lines": [ + { + "bbox": [ + 244, + 476, + 367, + 491 + ], + "spans": [ + { + "bbox": [ + 244, + 476, + 367, + 491 + ], + "score": 0.91, + "content": "\\pmb \\theta ( t + 1 ) = \\pmb \\theta ( t ) - \\eta ( t ) \\nabla \\mathcal L ( t )", + "type": "interline_equation", + "image_path": "6fb50094d1e04ee2654af4670b3920e27bf088cd4cd48b37572f80637967a50e.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 244, + 476, + 367, + 491 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 438, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 439, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 128, + 512 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 497, + 146, + 510 + ], + "score": 0.92, + "content": "\\eta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 495, + 243, + 512 + ], + "score": 1.0, + "content": "is the learning rate, and", + "type": "text" + }, + { + "bbox": [ + 244, + 497, + 327, + 510 + ], + "score": 0.93, + "content": "\\nabla \\mathcal { L } ( t ) : = \\nabla \\mathcal { L } ( \\pmb { \\theta } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 495, + 398, + 512 + ], + "score": 1.0, + "content": "is the gradient of", + "type": "text" + }, + { + "bbox": [ + 398, + 498, + 406, + 507 + ], + "score": 0.82, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 495, + 417, + 512 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 417, + 497, + 435, + 510 + ], + "score": 0.92, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 495, + 439, + 512 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 495, + 439, + 512 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "We first focus on the exponential loss. At the end of this section (Appendix F), we discuss how to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 525, + 413, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 413, + 538 + ], + "score": 1.0, + "content": "extend the proof to general loss functions with a similar assumption as (B3).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 514, + 505, + 538 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 620 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "The main difficulty for discretizing our previous analysis comes from the fact that the original ver-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 271, + 567 + ], + "score": 1.0, + "content": "sion of the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 271, + 552, + 349, + 565 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\tilde { \\gamma } ( \\pmb { \\theta } ) : = \\rho ^ { - \\tilde { L } } \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 552, + 461, + 567 + ], + "score": 1.0, + "content": "becomes less smooth when", + "type": "text" + }, + { + "bbox": [ + 461, + 555, + 501, + 565 + ], + "score": 0.9, + "content": "\\rho \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 552, + 506, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 271, + 577 + ], + "score": 1.0, + "content": "Thus, if we take a Taylor expansion for", + "type": "text" + }, + { + "bbox": [ + 272, + 565, + 321, + 576 + ], + "score": 0.9, + "content": "\\tilde { \\gamma } ( \\pmb { \\theta } ( t + 1 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 564, + 385, + 577 + ], + "score": 1.0, + "content": "from the point", + "type": "text" + }, + { + "bbox": [ + 386, + 564, + 404, + 576 + ], + "score": 0.9, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 564, + 505, + 577 + ], + "score": 1.0, + "content": ", although one can show", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "that the first-order term is positive as in the gradient flow analysis, the second-order term is unlikely", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "score": 1.0, + "content": "to be bounded during gradient descent with a constant step size. To get a smoothed version of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 597, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 610 + ], + "score": 1.0, + "content": "normalized margin that is monotone increasing, we need to define another one that is even smoother", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 606, + 138, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 126, + 621 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 126, + 609, + 133, + 620 + ], + "score": 0.84, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 606, + 138, + 621 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 542, + 506, + 621 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 504, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 201, + 640 + ], + "score": 1.0, + "content": "Technically, recall that", + "type": "text" + }, + { + "bbox": [ + 201, + 624, + 265, + 637 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { d \\mathcal { L } } { d t } = - \\| \\nabla \\mathcal { L } \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 622, + 506, + 640 + ], + "score": 1.0, + "content": "does not hold exactly for gradient descent. However, if the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 636, + 358, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 233, + 648 + ], + "score": 1.0, + "content": "smoothness can be bounded by", + "type": "text" + }, + { + "bbox": [ + 233, + 637, + 250, + 648 + ], + "score": 0.89, + "content": "s ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 636, + 358, + 648 + ], + "score": 1.0, + "content": ", then it is well-known that", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 622, + 506, + 648 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 655, + 403, + 670 + ], + "lines": [ + { + "bbox": [ + 207, + 655, + 403, + 670 + ], + "spans": [ + { + "bbox": [ + 207, + 655, + 403, + 670 + ], + "score": 0.87, + "content": "\\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\leq - \\eta ( t ) ( 1 - s ( t ) \\eta ( t ) ) \\| \\nabla \\mathcal { L } \\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "aaacdcd760519b27b1e1ddb258d530f907b63adb2ed34eb23e35cdba79a065d1.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 207, + 655, + 403, + 670 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 235, + 690 + ], + "score": 1.0, + "content": "By analyzing the landscape of", + "type": "text" + }, + { + "bbox": [ + 235, + 677, + 243, + 687 + ], + "score": 0.8, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 676, + 505, + 690 + ], + "score": 1.0, + "content": ", one can easily find that the smoothness is bounded locally by", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 185, + 701 + ], + "score": 0.91, + "content": "\\dot { O ( } \\mathcal { L } \\cdot \\mathrm { p o l y l o g } ( \\frac { 1 } { \\mathcal { L } } ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 687, + 254, + 701 + ], + "score": 1.0, + "content": ". Thus, if we set", + "type": "text" + }, + { + "bbox": [ + 254, + 688, + 271, + 700 + ], + "score": 0.91, + "content": "\\eta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "to a constant or set it appropriately according to the loss,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "then this discretization error becomes negligible. Using this insight, we define the new smoothed", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 185, + 722 + ], + "score": 1.0, + "content": "normalized margin", + "type": "text" + }, + { + "bbox": [ + 185, + 710, + 193, + 721 + ], + "score": 0.86, + "content": "\\hat { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 710, + 379, + 722 + ], + "score": 1.0, + "content": "in a way that it increases slightly slower than", + "type": "text" + }, + { + "bbox": [ + 379, + 711, + 387, + 721 + ], + "score": 0.85, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "during training to cancel the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 222, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 222, + 733 + ], + "score": 1.0, + "content": "effect of discretization error.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 676, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 196, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 196, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 196, + 95 + ], + "score": 1.0, + "content": "E.1 ASSUMPTIONS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "As stated in Section 4.1, we assume (A2), (A3), (A4) similarly as for gradient flow, and two addi-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 242, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 242, + 127 + ], + "score": 1.0, + "content": "tional assumptions (S1) and (S5).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 112, + 133, + 466, + 207 + ], + "lines": [ + { + "bbox": [ + 117, + 132, + 414, + 147 + ], + "spans": [ + { + "bbox": [ + 117, + 132, + 258, + 147 + ], + "score": 1.0, + "content": "(S1). (Smoothness). For any fixed", + "type": "text" + }, + { + "bbox": [ + 258, + 136, + 266, + 144 + ], + "score": 0.6, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 132, + 269, + 147 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 270, + 134, + 302, + 146 + ], + "score": 0.9, + "content": "\\Phi ( \\cdot ; x )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 132, + 313, + 147 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 313, + 134, + 324, + 144 + ], + "score": 0.87, + "content": "\\mathcal { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 132, + 371, + 147 + ], + "score": 1.0, + "content": "-smooth on", + "type": "text" + }, + { + "bbox": [ + 371, + 133, + 409, + 147 + ], + "score": 0.93, + "content": "\\mathbb { R } ^ { d } \\backslash \\{ \\mathbf { 0 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 132, + 414, + 147 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 115, + 147, + 465, + 162 + ], + "spans": [ + { + "bbox": [ + 115, + 147, + 259, + 162 + ], + "score": 1.0, + "content": "(A2). (Homogeneity). There exists", + "type": "text" + }, + { + "bbox": [ + 260, + 149, + 286, + 159 + ], + "score": 0.91, + "content": "L > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 147, + 325, + 162 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 325, + 148, + 462, + 161 + ], + "score": 0.89, + "content": "\\forall \\alpha > 0 : \\Phi ( \\alpha \\pmb { \\theta } ; \\pmb { x } ) = \\alpha ^ { L } \\Phi ( \\pmb { \\theta } ; \\pmb { x } ) ;", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 147, + 465, + 162 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 115, + 163, + 276, + 176 + ], + "spans": [ + { + "bbox": [ + 115, + 163, + 223, + 176 + ], + "score": 1.0, + "content": "(A3). (Exponential Loss).", + "type": "text" + }, + { + "bbox": [ + 224, + 163, + 270, + 176 + ], + "score": 0.91, + "content": "\\ell ( q ) = e ^ { - q }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 163, + 276, + 176 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 115, + 178, + 389, + 191 + ], + "spans": [ + { + "bbox": [ + 115, + 178, + 280, + 191 + ], + "score": 1.0, + "content": "(A4). (Separability). There exists a time", + "type": "text" + }, + { + "bbox": [ + 280, + 180, + 290, + 190 + ], + "score": 0.87, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 178, + 329, + 191 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 330, + 178, + 385, + 191 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb { \\theta } ( t _ { 0 } ) ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 178, + 389, + 191 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 117, + 192, + 420, + 208 + ], + "spans": [ + { + "bbox": [ + 117, + 192, + 243, + 208 + ], + "score": 1.0, + "content": "(S5). (Learing rate condition).", + "type": "text" + }, + { + "bbox": [ + 243, + 192, + 318, + 207 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { t \\geq 0 } \\eta ( t ) = + \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 192, + 336, + 208 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 336, + 193, + 416, + 205 + ], + "score": 0.9, + "content": "\\eta ( t ) \\leq H ( \\mathcal { L } ( \\pmb { \\theta } ( t ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 192, + 420, + 208 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 215, + 500, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 501, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 128, + 229 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 216, + 153, + 228 + ], + "score": 0.94, + "content": "H ( \\mathcal { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 214, + 412, + 229 + ], + "score": 1.0, + "content": "is a function of the current training loss. The explicit formula of", + "type": "text" + }, + { + "bbox": [ + 412, + 216, + 437, + 228 + ], + "score": 0.93, + "content": "H ( \\mathcal { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 214, + 501, + 229 + ], + "score": 1.0, + "content": "is given below:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 231, + 394, + 261 + ], + "lines": [ + { + "bbox": [ + 216, + 231, + 394, + 261 + ], + "spans": [ + { + "bbox": [ + 216, + 231, + 394, + 261 + ], + "score": 0.94, + "content": "H ( x ) : = \\frac { \\mu ( x ) } { C _ { \\eta } \\kappa ( x ) } = \\Theta \\left( \\frac { 1 } { x ( \\log \\frac { 1 } { x } ) ^ { 3 - 2 / L } } \\right) ,", + "type": "interline_equation", + "image_path": "f5fae6732bf21d556254fcdd78830af65d7096820d74ad81d547f0c3d3db5fc9.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 216, + 231, + 394, + 261 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 502, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 504, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 133, + 278 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 265, + 146, + 276 + ], + "score": 0.85, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 263, + 218, + 278 + ], + "score": 1.0, + "content": "is a constant, and", + "type": "text" + }, + { + "bbox": [ + 218, + 264, + 262, + 277 + ], + "score": 0.93, + "content": "\\kappa ( \\boldsymbol { x } ) , \\mu ( \\boldsymbol { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 263, + 504, + 278 + ], + "score": 1.0, + "content": "are two non-decreasing functions. For constant learning rate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 275, + 333, + 288 + ], + "spans": [ + { + "bbox": [ + 107, + 275, + 147, + 288 + ], + "score": 0.9, + "content": "\\eta ( t ) = \\eta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 275, + 239, + 287 + ], + "score": 1.0, + "content": ", (S5) is satisfied when", + "type": "text" + }, + { + "bbox": [ + 239, + 278, + 249, + 287 + ], + "score": 0.86, + "content": "\\eta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 275, + 333, + 287 + ], + "score": 1.0, + "content": "if sufficiently small.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 292, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 183, + 306 + ], + "score": 1.0, + "content": "Roughly speaking,", + "type": "text" + }, + { + "bbox": [ + 184, + 292, + 216, + 305 + ], + "score": 0.92, + "content": "C _ { \\eta } \\kappa ( \\boldsymbol { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 291, + 378, + 306 + ], + "score": 1.0, + "content": "is an upper bound for the smoothness of", + "type": "text" + }, + { + "bbox": [ + 378, + 293, + 387, + 302 + ], + "score": 0.82, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 291, + 473, + 306 + ], + "score": 1.0, + "content": "in a neighborhood of", + "type": "text" + }, + { + "bbox": [ + 473, + 293, + 480, + 302 + ], + "score": 0.8, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 291, + 506, + 306 + ], + "score": 1.0, + "content": "when", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 302, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 107, + 304, + 149, + 315 + ], + "score": 0.91, + "content": "x = { \\mathcal { L } } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 302, + 274, + 317 + ], + "score": 1.0, + "content": ". And we set the learning rate", + "type": "text" + }, + { + "bbox": [ + 275, + 303, + 292, + 316 + ], + "score": 0.9, + "content": "\\eta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 302, + 506, + 317 + ], + "score": 1.0, + "content": "to be the inverse of the smoothness multiplied by a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 313, + 504, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 132, + 327 + ], + "score": 1.0, + "content": "factor", + "type": "text" + }, + { + "bbox": [ + 133, + 315, + 185, + 326 + ], + "score": 0.92, + "content": "\\dot { \\mu ( x ) } = o ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 313, + 255, + 327 + ], + "score": 1.0, + "content": ". In our analysis,", + "type": "text" + }, + { + "bbox": [ + 255, + 315, + 275, + 326 + ], + "score": 0.9, + "content": "\\mu ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 313, + 464, + 327 + ], + "score": 1.0, + "content": "can be any non-decreasing function that maps", + "type": "text" + }, + { + "bbox": [ + 465, + 314, + 504, + 326 + ], + "score": 0.93, + "content": "\\left[ 0 , \\mathcal { L } ( \\dot { t } _ { 0 } ) \\right]", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 326, + 471, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 117, + 343 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 329, + 149, + 341 + ], + "score": 0.92, + "content": "( 0 , 1 / 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 326, + 243, + 343 + ], + "score": 1.0, + "content": "and makes the integral", + "type": "text" + }, + { + "bbox": [ + 243, + 327, + 295, + 342 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\int _ { 0 } ^ { 1 / 2 } \\mu ( x ) d x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 326, + 436, + 343 + ], + "score": 1.0, + "content": "exist. 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And then we set", + "type": "text" + }, + { + "bbox": [ + 281, + 659, + 302, + 672 + ], + "score": 0.92, + "content": "\\phi ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 658, + 326, + 673 + ], + "score": 1.0, + "content": "to be", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 658, + 326, + 673 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 676, + 407, + 705 + ], + "lines": [ + { + "bbox": [ + 204, + 676, + 407, + 705 + ], + "spans": [ + { + "bbox": [ + 204, + 676, + 407, + 705 + ], + "score": 0.94, + "content": "\\phi ( x ) = \\log \\log { \\frac { 1 } { x } } + \\int _ { 0 } ^ { x } \\left( \\phi ^ { \\prime } ( w ) + { \\frac { 1 } { w \\log { \\frac { 1 } { w } } } } \\right) d w .", + "type": "interline_equation", + "image_path": "d18ef29e6d3088fac2ea801684d9cc811c727b2d319e2aa63dae973925cd0ed2.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 204, + 676, + 407, + 705 + ], + 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723 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 720, + 366, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 320, + 734 + ], + "score": 1.0, + "content": "Moreover, we have the following relationship among", + "type": "text" + }, + { + "bbox": [ + 320, + 721, + 337, + 732 + ], + "score": 0.9, + "content": "\\hat { \\gamma } , \\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 720, + 355, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 355, + 722, + 362, + 732 + ], + "score": 0.83, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 720, + 366, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 707, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 452, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 454, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 162, + 97 + ], + "score": 1.0, + "content": "Lemma E.1.", + "type": "text" + }, + { + "bbox": [ + 163, + 82, + 183, + 95 + ], + "score": 0.9, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 80, + 259, + 97 + ], + "score": 1.0, + "content": "is well-defined for", + "type": "text" + }, + { + "bbox": [ + 259, + 82, + 317, + 95 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb { \\theta } ) \\leq \\mathcal { L } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 80, + 454, + 97 + ], + "score": 1.0, + "content": "and has the following properties:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 126, + 103, + 321, + 117 + ], + "lines": [ + { + "bbox": [ + 124, 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138, + 135 + ], + "score": 0.32, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 122, + 158, + 138 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 159, + 123, + 242, + 136 + ], + "score": 0.91, + "content": "\\{ \\pmb \\theta _ { m } \\in \\mathbb { R } ^ { d } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 122, + 371, + 138 + ], + "score": 1.0, + "content": "be a sequence of parameters. If", + "type": "text" + }, + { + "bbox": [ + 372, + 124, + 421, + 136 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb { \\theta } _ { m } ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 122, + 445, + 138 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 142, + 414, + 176 + ], + "lines": [ + { + "bbox": [ + 233, + 142, + 414, + 176 + ], + "spans": [ + { + "bbox": [ + 233, + 142, + 414, + 176 + ], + "score": 0.94, + "content": "\\frac { \\hat { \\gamma } ( \\pmb \\theta _ { m } ) } { \\bar { \\gamma } ( \\pmb \\theta _ { m } ) } = 1 - O ( ( \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta _ { m } ) } ) ^ { - 1 } ) 1 .", + "type": "interline_equation", + "image_path": "efbe5835d504517b82c20b63fc0db86b460a672e6b397378b2e09133e22844d2.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 233, + 142, + 414, + 159.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 233, + 159.0, + 414, + 176.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 190, + 444, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 445, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 214, + 205 + ], + "score": 1.0, + "content": "Proof. First we verify that", + "type": "text" + }, + { + "bbox": [ + 214, + 192, + 222, + 203 + ], + "score": 0.87, + "content": "\\hat { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 190, + 445, + 205 + ], + "score": 1.0, + "content": "is well-defined. To see this, we only need to verify that", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 209, + 383, + 238 + ], + "lines": [ + { + "bbox": [ + 228, + 209, + 383, + 238 + ], + "spans": [ + { + "bbox": [ + 228, + 209, + 383, + 238 + ], + "score": 0.95, + "content": "I ( x ) : = \\int _ { 0 } ^ { x } \\left( \\phi ^ { \\prime } ( w ) + { \\frac { 1 } { w \\log { \\frac { 1 } { w } } } } \\right) d w", + "type": "interline_equation", + "image_path": "51700aaf316ed1b0d60ffdefe3113f1f9e3896b13706f525123e6c3dda575d9f.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 228, + 209, + 383, + 238 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 244, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 159, + 259 + ], + "score": 1.0, + "content": "exists for all", + "type": "text" + }, + { + "bbox": [ + 160, + 245, + 219, + 257 + ], + "score": 0.89, + "content": "x \\in ( 0 , \\mathcal { L } ( t _ { 0 } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 243, + 334, + 259 + ], + "score": 1.0, + "content": ", then it is trivial to see that", + "type": "text" + }, + { + "bbox": [ + 334, + 245, + 359, + 257 + ], + "score": 0.93, + "content": "\\phi ^ { \\prime } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 243, + 468, + 259 + ], + "score": 1.0, + "content": "is indeed the derivative of", + "type": "text" + }, + { + "bbox": [ + 469, + 245, + 491, + 257 + ], + "score": 0.92, + "content": "\\phi ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 243, + 505, + 259 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 254, + 207, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 207, + 273 + ], + "score": 1.0, + "content": "I 0 ( x ) = φ 0 ( x ) + 1x log 1 .", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 504, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 147, + 290 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 147, + 277, + 167, + 289 + ], + "score": 0.92, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 276, + 223, + 290 + ], + "score": 1.0, + "content": "exists for all", + "type": "text" + }, + { + "bbox": [ + 223, + 277, + 283, + 289 + ], + "score": 0.93, + "content": "x \\in ( 0 , \\mathcal { L } ( t _ { 0 } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 276, + 329, + 290 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + }, + { + "bbox": [ + 329, + 277, + 349, + 289 + ], + "score": 0.9, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 276, + 456, + 290 + ], + "score": 1.0, + "content": "exists for a small enough", + "type": "text" + }, + { + "bbox": [ + 456, + 278, + 484, + 288 + ], + "score": 0.89, + "content": "x > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 276, + 504, + 290 + ], + "score": 1.0, + "content": ". By", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 102, + 283, + 507, + 309 + ], + "spans": [ + { + "bbox": [ + 102, + 283, + 238, + 309 + ], + "score": 1.0, + "content": "definition, it is easy to verify that", + "type": "text" + }, + { + "bbox": [ + 238, + 289, + 351, + 307 + ], + "score": 0.93, + "content": "\\begin{array} { r } { r ( w ) : = \\frac { 1 + 2 ( 1 + \\lambda ( w ) / L ) \\mu ( w ) } { w \\log \\frac { 1 } { w } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 283, + 429, + 309 + ], + "score": 1.0, + "content": "is decreasing when", + "type": "text" + }, + { + "bbox": [ + 429, + 293, + 438, + 300 + ], + "score": 0.77, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 283, + 507, + 309 + ], + "score": 1.0, + "content": "is small enough.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 303, + 275, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 208, + 318 + ], + "score": 1.0, + "content": "Thus, for a small enough", + "type": "text" + }, + { + "bbox": [ + 208, + 306, + 235, + 315 + ], + "score": 0.89, + "content": "w > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 303, + 275, + 318 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 323, + 498, + 361 + ], + "lines": [ + { + "bbox": [ + 110, + 323, + 498, + 361 + ], + "spans": [ + { + "bbox": [ + 110, + 323, + 498, + 361 + ], + "score": 0.93, + "content": "- \\phi ^ { \\prime } ( w ) = r ( w ) = \\frac { 1 + 2 ( 1 + \\frac { 1 } { L } ( \\log \\frac { 1 } { w } ) ^ { - 1 } ) \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } { 2 \\log \\frac { 1 } { w } } } { w \\log \\frac { 1 } { w } } = \\frac { 1 } { w \\log \\frac { 1 } { w } } + \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( 1 + \\frac { 1 } { L } ( \\log \\frac { 1 } { w } ) ^ { - 1 } ) } { w ( \\log \\frac { 1 } { w } ) ^ { 2 } } .", + "type": "interline_equation", + "image_path": "16a00bf4775b9b516dda8090436b581751f73528e435f67849132bc1958c1cca.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 110, + 323, + 498, + 335.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 110, + 335.6666666666667, + 498, + 348.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 110, + 348.33333333333337, + 498, + 361.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 290, + 378 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 291, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 280, + 380 + ], + "score": 1.0, + "content": "So we have the following for small enough", + "type": "text" + }, + { + "bbox": [ + 281, + 369, + 287, + 377 + ], + "score": 0.8, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 365, + 291, + 380 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 384, + 480, + 448 + ], + "lines": [ + { + "bbox": [ + 120, + 384, + 480, + 448 + ], + "spans": [ + { + "bbox": [ + 120, + 384, + 480, + 448 + ], + "score": 0.94, + "content": "\\begin{array} { r } { I ( x ) = \\displaystyle \\int _ { 0 } ^ { x } - \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( 1 + \\frac { 1 } { L } ( \\log \\frac { 1 } { w } ) ^ { - 1 } ) } { w ( \\log \\frac { 1 } { w } ) ^ { 2 } } d w = - \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( \\frac { 1 } { \\log \\frac { 1 } { w } } + \\frac { 1 } { 2 L ( \\log \\frac { 1 } { w } ) ^ { 2 } } ) _ { 0 } ^ { x } } \\\\ { \\displaystyle = - \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( \\frac { 1 } { \\log \\frac { 1 } { x } } + \\frac { 1 } { 2 L ( \\log \\frac { 1 } { x } ) ^ { 2 } } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "0d76a71d36a3071f2c2b07b6ffdce8bad5b562c86abaf274d8e5a8fe04497cc2.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 120, + 384, + 480, + 405.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 120, + 405.3333333333333, + 480, + 426.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 120, + 426.66666666666663, + 480, + 447.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 243, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 244, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 220, + 467 + ], + "score": 1.0, + "content": "This proves the existence of", + "type": "text" + }, + { + "bbox": [ + 221, + 453, + 240, + 465 + ], + "score": 0.92, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 451, + 244, + 467 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 258, + 484 + ], + "score": 1.0, + "content": "Now we prove (a). By Lemma A.5,", + "type": "text" + }, + { + "bbox": [ + 258, + 470, + 314, + 482 + ], + "score": 0.87, + "content": "\\tilde { \\gamma } ( \\pmb \\theta ) \\leq \\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 468, + 445, + 484 + ], + "score": 1.0, + "content": ", so we only need to prove that", + "type": "text" + }, + { + "bbox": [ + 445, + 470, + 501, + 482 + ], + "score": 0.9, + "content": "\\hat { \\gamma } _ { \\perp } ( \\pmb { \\theta } ) < \\tilde { \\gamma } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 468, + 506, + 484 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 478, + 509, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 220, + 502 + ], + "score": 1.0, + "content": "To see this, note that for all", + "type": "text" + }, + { + "bbox": [ + 221, + 481, + 335, + 497 + ], + "score": 0.83, + "content": "w \\leq \\mathcal { L } ( t _ { 0 } ) , r ( w ) > \\frac { 1 } { w \\log \\frac { 1 } { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 478, + 390, + 502 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + }, + { + "bbox": [ + 390, + 481, + 466, + 497 + ], + "score": 0.92, + "content": "- \\bar { \\phi } ^ { \\prime } ( w ) > \\frac { 1 } { w \\log \\frac { 1 } { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 478, + 509, + 499 + ], + "score": 1.0, + "content": "1g 1w , and this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 494, + 440, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 138, + 510 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 138, + 496, + 177, + 508 + ], + "score": 0.93, + "content": "I ( x ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 494, + 204, + 510 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 204, + 496, + 247, + 508 + ], + "score": 0.92, + "content": "x \\le \\mathcal { L } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 494, + 333, + 510 + ], + "score": 1.0, + "content": ". Then it holds for all", + "type": "text" + }, + { + "bbox": [ + 333, + 497, + 340, + 506 + ], + "score": 0.79, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 494, + 362, + 510 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 362, + 496, + 419, + 508 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb { \\theta } ) \\leq \\mathcal { L } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 494, + 440, + 510 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 514, + 422, + 548 + ], + "lines": [ + { + "bbox": [ + 189, + 514, + 422, + 548 + ], + "spans": [ + { + "bbox": [ + 189, + 514, + 422, + 548 + ], + "score": 0.95, + "content": "\\frac { \\widehat { \\gamma } ( \\pmb \\theta ) } { \\widetilde { \\gamma } ( \\pmb \\theta ) } = \\frac { e ^ { \\phi ( \\mathcal { L } ( \\pmb \\theta ) ) } } { \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } } = \\frac { e ^ { \\log \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } + I ( \\mathcal { L } ( \\pmb \\theta ) ) } } { e ^ { \\log \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } } } = e ^ { I ( \\mathcal { L } ( \\pmb \\theta ) ) } < 0 .", + "type": "interline_equation", + "image_path": "550b1be6b6722356e10c92dc0dcbf0ab17c7d6770f47bcbb052da087eff9c243.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 514, + 422, + 531.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 189, + 531.0, + 422, + 548.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 560, + 426, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 427, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 359, + 574 + ], + "score": 1.0, + "content": "To prove (b), we combine (19) and (20), then for small enough", + "type": "text" + }, + { + "bbox": [ + 359, + 561, + 388, + 573 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb { \\theta } _ { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 560, + 427, + 574 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 579, + 459, + 709 + ], + "lines": [ + { + "bbox": [ + 152, + 579, + 459, + 709 + ], + "spans": [ + { + "bbox": [ + 152, + 579, + 459, + 709 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\frac { \\hat { \\gamma } ( \\theta ) } { \\hat { \\gamma } ( \\theta ) } = \\frac { \\hat { \\gamma } ( \\theta ) } { \\hat { \\gamma } ( \\theta ) } \\cdot \\frac { \\hat { \\gamma } ( \\theta ) } { \\hat { \\gamma } ( \\theta ) } } \\\\ & { \\quad \\quad \\quad = \\exp \\left( - \\log \\frac { 1 } { \\mathcal { L } ( { t _ { 0 } } ) } \\left( \\frac { 1 } { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } + \\frac { 1 } { 2 L ( \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } ) ^ { 2 } } \\right) \\right) \\cdot \\frac { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } { q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) } } \\\\ & { \\quad \\quad \\quad = \\exp \\left( - \\mathcal { O } \\left( \\frac { 1 } { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } \\right) \\right) \\cdot \\frac { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } + \\mathcal { O } ( 1 ) } } \\\\ & { \\quad \\quad \\quad = 1 - \\mathcal { O } \\left( \\left( \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } \\right) ^ { - 1 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "31d99d28e06520ea4f5c5dece7d2b24466be3c45b9daa6030f2bb33d54d290e1.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 152, + 579, + 459, + 622.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 152, + 622.3333333333334, + 459, + 665.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 152, + 665.6666666666667, + 459, + 709.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "page_idx": 30, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 719, + 505, + 730 + ], + "lines": [ + { + "bbox": [ + 496, + 721, + 504, + 729 + ], + "spans": [ + { + "bbox": [ + 496, + 721, + 504, + 729 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "31", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 452, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 454, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 162, + 97 + ], + "score": 1.0, + "content": "Lemma E.1.", + "type": "text" + }, + { + "bbox": [ + 163, + 82, + 183, + 95 + ], + "score": 0.9, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 80, + 259, + 97 + ], + "score": 1.0, + "content": "is well-defined for", + "type": "text" + }, + { + "bbox": [ + 259, + 82, + 317, + 95 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb { \\theta } ) \\leq \\mathcal { L } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 80, + 454, + 97 + ], + "score": 1.0, + "content": "and has the following properties:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 454, + 97 + ] + }, + { + "type": "text", + "bbox": [ + 126, + 103, + 321, + 117 + ], + "lines": [ + { + "bbox": [ + 124, + 102, + 321, + 118 + ], + "spans": [ + { + "bbox": [ + 124, + 102, + 151, + 118 + ], + "score": 1.0, + "content": "(a) If", + "type": "text" + }, + { + "bbox": [ + 151, + 104, + 208, + 116 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\pmb { \\theta } ) \\leq \\mathcal { L } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 102, + 232, + 118 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 232, + 104, + 318, + 117 + ], + "score": 0.88, + "content": "\\hat { \\gamma } ( \\pmb \\theta ) < \\tilde { \\gamma } ( \\pmb \\theta ) \\leq \\bar { \\gamma } ( \\pmb \\theta ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 102, + 321, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 124, + 102, + 321, + 118 + ] + }, + { + "type": "text", + "bbox": [ + 126, + 122, + 444, + 136 + ], + "lines": [ + { + "bbox": [ + 126, + 122, + 445, + 138 + ], + "spans": [ + { + "bbox": [ + 126, + 124, + 138, + 135 + ], + "score": 0.32, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 122, + 158, + 138 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 159, + 123, + 242, + 136 + ], + "score": 0.91, + "content": "\\{ \\pmb \\theta _ { m } \\in \\mathbb { R } ^ { d } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 122, + 371, + 138 + ], + "score": 1.0, + "content": "be a sequence of parameters. If", + "type": "text" + }, + { + "bbox": [ + 372, + 124, + 421, + 136 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb { \\theta } _ { m } ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 122, + 445, + 138 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 126, + 122, + 445, + 138 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 142, + 414, + 176 + ], + "lines": [ + { + "bbox": [ + 233, + 142, + 414, + 176 + ], + "spans": [ + { + "bbox": [ + 233, + 142, + 414, + 176 + ], + "score": 0.94, + "content": "\\frac { \\hat { \\gamma } ( \\pmb \\theta _ { m } ) } { \\bar { \\gamma } ( \\pmb \\theta _ { m } ) } = 1 - O ( ( \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta _ { m } ) } ) ^ { - 1 } ) 1 .", + "type": "interline_equation", + "image_path": "efbe5835d504517b82c20b63fc0db86b460a672e6b397378b2e09133e22844d2.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 233, + 142, + 414, + 159.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 233, + 159.0, + 414, + 176.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 190, + 444, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 445, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 214, + 205 + ], + "score": 1.0, + "content": "Proof. First we verify that", + "type": "text" + }, + { + "bbox": [ + 214, + 192, + 222, + 203 + ], + "score": 0.87, + "content": "\\hat { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 190, + 445, + 205 + ], + "score": 1.0, + "content": "is well-defined. To see this, we only need to verify that", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 190, + 445, + 205 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 209, + 383, + 238 + ], + "lines": [ + { + "bbox": [ + 228, + 209, + 383, + 238 + ], + "spans": [ + { + "bbox": [ + 228, + 209, + 383, + 238 + ], + "score": 0.95, + "content": "I ( x ) : = \\int _ { 0 } ^ { x } \\left( \\phi ^ { \\prime } ( w ) + { \\frac { 1 } { w \\log { \\frac { 1 } { w } } } } \\right) d w", + "type": "interline_equation", + "image_path": "51700aaf316ed1b0d60ffdefe3113f1f9e3896b13706f525123e6c3dda575d9f.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 228, + 209, + 383, + 238 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 244, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 159, + 259 + ], + "score": 1.0, + "content": "exists for all", + "type": "text" + }, + { + "bbox": [ + 160, + 245, + 219, + 257 + ], + "score": 0.89, + "content": "x \\in ( 0 , \\mathcal { L } ( t _ { 0 } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 243, + 334, + 259 + ], + "score": 1.0, + "content": ", then it is trivial to see that", + "type": "text" + }, + { + "bbox": [ + 334, + 245, + 359, + 257 + ], + "score": 0.93, + "content": "\\phi ^ { \\prime } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 243, + 468, + 259 + ], + "score": 1.0, + "content": "is indeed the derivative of", + "type": "text" + }, + { + "bbox": [ + 469, + 245, + 491, + 257 + ], + "score": 0.92, + "content": "\\phi ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 243, + 505, + 259 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 254, + 207, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 207, + 273 + ], + "score": 1.0, + "content": "I 0 ( x ) = φ 0 ( x ) + 1x log 1 .", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 243, + 505, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 504, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 147, + 290 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 147, + 277, + 167, + 289 + ], + "score": 0.92, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 276, + 223, + 290 + ], + "score": 1.0, + "content": "exists for all", + "type": "text" + }, + { + "bbox": [ + 223, + 277, + 283, + 289 + ], + "score": 0.93, + "content": "x \\in ( 0 , \\mathcal { L } ( t _ { 0 } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 276, + 329, + 290 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + }, + { + "bbox": [ + 329, + 277, + 349, + 289 + ], + "score": 0.9, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 276, + 456, + 290 + ], + "score": 1.0, + "content": "exists for a small enough", + "type": "text" + }, + { + "bbox": [ + 456, + 278, + 484, + 288 + ], + "score": 0.89, + "content": "x > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 276, + 504, + 290 + ], + "score": 1.0, + "content": ". By", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 102, + 283, + 507, + 309 + ], + "spans": [ + { + "bbox": [ + 102, + 283, + 238, + 309 + ], + "score": 1.0, + "content": "definition, it is easy to verify that", + "type": "text" + }, + { + "bbox": [ + 238, + 289, + 351, + 307 + ], + "score": 0.93, + "content": "\\begin{array} { r } { r ( w ) : = \\frac { 1 + 2 ( 1 + \\lambda ( w ) / L ) \\mu ( w ) } { w \\log \\frac { 1 } { w } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 283, + 429, + 309 + ], + "score": 1.0, + "content": "is decreasing when", + "type": "text" + }, + { + "bbox": [ + 429, + 293, + 438, + 300 + ], + "score": 0.77, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 283, + 507, + 309 + ], + "score": 1.0, + "content": "is small enough.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 303, + 275, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 208, + 318 + ], + "score": 1.0, + "content": "Thus, for a small enough", + "type": "text" + }, + { + "bbox": [ + 208, + 306, + 235, + 315 + ], + "score": 0.89, + "content": "w > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 303, + 275, + 318 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 102, + 276, + 507, + 318 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 323, + 498, + 361 + ], + "lines": [ + { + "bbox": [ + 110, + 323, + 498, + 361 + ], + "spans": [ + { + "bbox": [ + 110, + 323, + 498, + 361 + ], + "score": 0.93, + "content": "- \\phi ^ { \\prime } ( w ) = r ( w ) = \\frac { 1 + 2 ( 1 + \\frac { 1 } { L } ( \\log \\frac { 1 } { w } ) ^ { - 1 } ) \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } { 2 \\log \\frac { 1 } { w } } } { w \\log \\frac { 1 } { w } } = \\frac { 1 } { w \\log \\frac { 1 } { w } } + \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( 1 + \\frac { 1 } { L } ( \\log \\frac { 1 } { w } ) ^ { - 1 } ) } { w ( \\log \\frac { 1 } { w } ) ^ { 2 } } .", + "type": "interline_equation", + "image_path": "16a00bf4775b9b516dda8090436b581751f73528e435f67849132bc1958c1cca.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 110, + 323, + 498, + 335.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 110, + 335.6666666666667, + 498, + 348.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 110, + 348.33333333333337, + 498, + 361.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 290, + 378 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 291, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 280, + 380 + ], + "score": 1.0, + "content": "So we have the following for small enough", + "type": "text" + }, + { + "bbox": [ + 281, + 369, + 287, + 377 + ], + "score": 0.8, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 365, + 291, + 380 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 365, + 291, + 380 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 384, + 480, + 448 + ], + "lines": [ + { + "bbox": [ + 120, + 384, + 480, + 448 + ], + "spans": [ + { + "bbox": [ + 120, + 384, + 480, + 448 + ], + "score": 0.94, + "content": "\\begin{array} { r } { I ( x ) = \\displaystyle \\int _ { 0 } ^ { x } - \\frac { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( 1 + \\frac { 1 } { L } ( \\log \\frac { 1 } { w } ) ^ { - 1 } ) } { w ( \\log \\frac { 1 } { w } ) ^ { 2 } } d w = - \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( \\frac { 1 } { \\log \\frac { 1 } { w } } + \\frac { 1 } { 2 L ( \\log \\frac { 1 } { w } ) ^ { 2 } } ) _ { 0 } ^ { x } } \\\\ { \\displaystyle = - \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ( \\frac { 1 } { \\log \\frac { 1 } { x } } + \\frac { 1 } { 2 L ( \\log \\frac { 1 } { x } ) ^ { 2 } } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "0d76a71d36a3071f2c2b07b6ffdce8bad5b562c86abaf274d8e5a8fe04497cc2.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 120, + 384, + 480, + 405.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 120, + 405.3333333333333, + 480, + 426.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 120, + 426.66666666666663, + 480, + 447.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 243, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 244, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 220, + 467 + ], + "score": 1.0, + "content": "This proves the existence of", + "type": "text" + }, + { + "bbox": [ + 221, + 453, + 240, + 465 + ], + "score": 0.92, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 451, + 244, + 467 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 451, + 244, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 258, + 484 + ], + "score": 1.0, + "content": "Now we prove (a). By Lemma A.5,", + "type": "text" + }, + { + "bbox": [ + 258, + 470, + 314, + 482 + ], + "score": 0.87, + "content": "\\tilde { \\gamma } ( \\pmb \\theta ) \\leq \\tilde { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 468, + 445, + 484 + ], + "score": 1.0, + "content": ", so we only need to prove that", + "type": "text" + }, + { + "bbox": [ + 445, + 470, + 501, + 482 + ], + "score": 0.9, + "content": "\\hat { \\gamma } _ { \\perp } ( \\pmb { \\theta } ) < \\tilde { \\gamma } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 468, + 506, + 484 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 478, + 509, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 220, + 502 + ], + "score": 1.0, + "content": "To see this, note that for all", + "type": "text" + }, + { + "bbox": [ + 221, + 481, + 335, + 497 + ], + "score": 0.83, + "content": "w \\leq \\mathcal { L } ( t _ { 0 } ) , r ( w ) > \\frac { 1 } { w \\log \\frac { 1 } { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 478, + 390, + 502 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + }, + { + "bbox": [ + 390, + 481, + 466, + 497 + ], + "score": 0.92, + "content": "- \\bar { \\phi } ^ { \\prime } ( w ) > \\frac { 1 } { w \\log \\frac { 1 } { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 478, + 509, + 499 + ], + "score": 1.0, + "content": "1g 1w , and this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 494, + 440, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 138, + 510 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 138, + 496, + 177, + 508 + ], + "score": 0.93, + "content": "I ( x ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 494, + 204, + 510 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 204, + 496, + 247, + 508 + ], + "score": 0.92, + "content": "x \\le \\mathcal { L } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 494, + 333, + 510 + ], + "score": 1.0, + "content": ". Then it holds for all", + "type": "text" + }, + { + "bbox": [ + 333, + 497, + 340, + 506 + ], + "score": 0.79, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 494, + 362, + 510 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 362, + 496, + 419, + 508 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb { \\theta } ) \\leq \\mathcal { L } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 494, + 440, + 510 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 468, + 509, + 510 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 514, + 422, + 548 + ], + "lines": [ + { + "bbox": [ + 189, + 514, + 422, + 548 + ], + "spans": [ + { + "bbox": [ + 189, + 514, + 422, + 548 + ], + "score": 0.95, + "content": "\\frac { \\widehat { \\gamma } ( \\pmb \\theta ) } { \\widetilde { \\gamma } ( \\pmb \\theta ) } = \\frac { e ^ { \\phi ( \\mathcal { L } ( \\pmb \\theta ) ) } } { \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } } = \\frac { e ^ { \\log \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } + I ( \\mathcal { L } ( \\pmb \\theta ) ) } } { e ^ { \\log \\log \\frac { 1 } { \\mathcal { L } ( \\pmb \\theta ) } } } = e ^ { I ( \\mathcal { L } ( \\pmb \\theta ) ) } < 0 .", + "type": "interline_equation", + "image_path": "550b1be6b6722356e10c92dc0dcbf0ab17c7d6770f47bcbb052da087eff9c243.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 514, + 422, + 531.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 189, + 531.0, + 422, + 548.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 560, + 426, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 427, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 359, + 574 + ], + "score": 1.0, + "content": "To prove (b), we combine (19) and (20), then for small enough", + "type": "text" + }, + { + "bbox": [ + 359, + 561, + 388, + 573 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\pmb { \\theta } _ { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 560, + 427, + 574 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 560, + 427, + 574 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 579, + 459, + 709 + ], + "lines": [ + { + "bbox": [ + 152, + 579, + 459, + 709 + ], + "spans": [ + { + "bbox": [ + 152, + 579, + 459, + 709 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\frac { \\hat { \\gamma } ( \\theta ) } { \\hat { \\gamma } ( \\theta ) } = \\frac { \\hat { \\gamma } ( \\theta ) } { \\hat { \\gamma } ( \\theta ) } \\cdot \\frac { \\hat { \\gamma } ( \\theta ) } { \\hat { \\gamma } ( \\theta ) } } \\\\ & { \\quad \\quad \\quad = \\exp \\left( - \\log \\frac { 1 } { \\mathcal { L } ( { t _ { 0 } } ) } \\left( \\frac { 1 } { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } + \\frac { 1 } { 2 L ( \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } ) ^ { 2 } } \\right) \\right) \\cdot \\frac { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } { q _ { \\operatorname* { m i n } } ( \\theta _ { m } ) } } \\\\ & { \\quad \\quad \\quad = \\exp \\left( - \\mathcal { O } \\left( \\frac { 1 } { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } \\right) \\right) \\cdot \\frac { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } } { \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } + \\mathcal { O } ( 1 ) } } \\\\ & { \\quad \\quad \\quad = 1 - \\mathcal { O } \\left( \\left( \\log \\frac { 1 } { \\mathcal { L } ( \\theta _ { m } ) } \\right) ^ { - 1 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "31d99d28e06520ea4f5c5dece7d2b24466be3c45b9daa6030f2bb33d54d290e1.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 152, + 579, + 459, + 622.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 152, + 622.3333333333334, + 459, + 665.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 152, + 665.6666666666667, + 459, + 709.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 463, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 464, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 223, + 96 + ], + "score": 1.0, + "content": "Now we specify the value of", + "type": "text" + }, + { + "bbox": [ + 223, + 83, + 236, + 95 + ], + "score": 0.89, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 81, + 370, + 96 + ], + "score": 1.0, + "content": ". By (S1) and (S2), we can define", + "type": "text" + }, + { + "bbox": [ + 370, + 83, + 416, + 94 + ], + "score": 0.93, + "content": "B _ { 0 } , B _ { 1 } , B _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 81, + 464, + 96 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 104, + 399, + 156 + ], + "lines": [ + { + "bbox": [ + 211, + 104, + 399, + 156 + ], + "spans": [ + { + "bbox": [ + 211, + 104, + 399, + 156 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { B _ { 0 } : = \\operatorname* { s u p } \\left\\{ q _ { n } : \\pmb \\theta \\in S ^ { d - 1 } , n \\in [ N ] \\right\\} , } \\\\ & { B _ { 1 } : = \\operatorname* { s u p } \\left\\{ \\left\\| \\nabla q _ { n } \\right\\| _ { 2 } : \\pmb \\theta \\in S ^ { d - 1 } , n \\in [ N ] \\right\\} , } \\\\ & { B _ { 2 } : = \\operatorname* { s u p } \\left\\{ \\left\\| \\nabla ^ { 2 } q _ { n } \\right\\| _ { 2 } : \\pmb \\theta \\in S ^ { d - 1 } , n \\in [ N ] \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "f7c7b94b2919472564c0f5c3e77b005b1ea3ea6243bb887a17bc8a4676f3f94b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 211, + 104, + 399, + 121.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 211, + 121.33333333333333, + 399, + 138.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 211, + 138.66666666666666, + 399, + 156.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 405, + 186 + ], + "lines": [ + { + "bbox": [ + 101, + 156, + 405, + 190 + ], + "spans": [ + { + "bbox": [ + 101, + 156, + 156, + 190 + ], + "score": 1.0, + "content": "Then we set", + "type": "text" + }, + { + "bbox": [ + 157, + 164, + 405, + 186 + ], + "score": 0.74, + "content": "\\begin{array} { r l } & { C _ { \\eta } : = \\frac { 1 } { 2 } \\left( B _ { 1 } ^ { 2 } + \\rho ( t _ { 0 } ) ^ { - L } B _ { 2 } \\right) \\operatorname* { m i n } \\Big \\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\Big \\} . } \\end{array}", + "type": "inline_equation", + "image_path": "24f2335a827180f6b97fe05428bdaf53f7a4f89ac58564c5e4ff72cf7009c98c.jpg" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 106, + 202, + 181, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 182, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 182, + 215 + ], + "score": 1.0, + "content": "E.3 THEOREMS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 506, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "score": 1.0, + "content": "Now we state our main theorems for the monotonicity of the normalized margin and the convergence", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "to KKT points. We will prove Theorem E.2 in Appendix E.4, and prove Theorem E.3 and E.4 in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 248, + 168, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 168, + 261 + ], + "score": 1.0, + "content": "Appendix E.5.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 105, + 265, + 503, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 279 + ], + "score": 1.0, + "content": "Theorem E.2. Under assumptions (S1), (A2) - (A4), (S5), the following are true for gradient de-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 276, + 133, + 289 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 133, + 289 + ], + "score": 1.0, + "content": "scent:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 131, + 299, + 272, + 313 + ], + "lines": [ + { + "bbox": [ + 129, + 298, + 272, + 315 + ], + "spans": [ + { + "bbox": [ + 129, + 298, + 171, + 315 + ], + "score": 1.0, + "content": "1. For all", + "type": "text" + }, + { + "bbox": [ + 172, + 300, + 198, + 312 + ], + "score": 0.7, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 298, + 201, + 315 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 202, + 300, + 267, + 313 + ], + "score": 0.77, + "content": "\\hat { \\gamma } ( t + 1 ) \\geq \\hat { \\gamma } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 298, + 272, + 315 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 129, + 322, + 377, + 337 + ], + "lines": [ + { + "bbox": [ + 128, + 321, + 377, + 338 + ], + "spans": [ + { + "bbox": [ + 128, + 321, + 171, + 338 + ], + "score": 1.0, + "content": "2. For all", + "type": "text" + }, + { + "bbox": [ + 172, + 324, + 198, + 335 + ], + "score": 0.9, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 321, + 227, + 338 + ], + "score": 1.0, + "content": ", either", + "type": "text" + }, + { + "bbox": [ + 228, + 323, + 293, + 336 + ], + "score": 0.91, + "content": "\\hat { \\gamma } ( t + 1 ) > \\hat { \\gamma } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 321, + 306, + 338 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 306, + 322, + 372, + 336 + ], + "score": 0.91, + "content": "\\hat { \\pmb { \\theta } } ( t + 1 ) = \\hat { \\pmb { \\theta } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 321, + 377, + 338 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 129, + 344, + 416, + 358 + ], + "lines": [ + { + "bbox": [ + 129, + 344, + 417, + 359 + ], + "spans": [ + { + "bbox": [ + 129, + 344, + 142, + 359 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 142, + 345, + 182, + 357 + ], + "score": 0.91, + "content": "\\mathcal { L } ( t ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 344, + 201, + 359 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 345, + 244, + 357 + ], + "score": 0.88, + "content": "\\rho ( t ) \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 344, + 257, + 359 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 257, + 346, + 295, + 356 + ], + "score": 0.83, + "content": "t \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 344, + 340, + 359 + ], + "score": 1.0, + "content": "; therefore,", + "type": "text" + }, + { + "bbox": [ + 340, + 345, + 413, + 357 + ], + "score": 0.91, + "content": "| \\bar { \\gamma } ( t ) - \\tilde { \\gamma } ( t ) | 0", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 344, + 417, + 359 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 504, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 376 + ], + "score": 1.0, + "content": "Theorem E.3. Consider gradient flow under assumptions (S1), (A2) - (A4), (S5). For every limit", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 374, + 363, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 129, + 393 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 376, + 137, + 388 + ], + "score": 0.55, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 374, + 148, + 393 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 149, + 374, + 209, + 394 + ], + "score": 0.85, + "content": "\\left\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 374, + 214, + 393 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 214, + 375, + 272, + 392 + ], + "score": 0.77, + "content": "\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 374, + 344, + 393 + ], + "score": 1.0, + "content": "is a KKT point of", + "type": "text" + }, + { + "bbox": [ + 344, + 378, + 358, + 389 + ], + "score": 0.74, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 374, + 363, + 393 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 108, + 398, + 504, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 479, + 412 + ], + "score": 1.0, + "content": "Theorem E.4. Consider gradient descent under assumptions (S1), (A2) - (A4), (S5). For any", + "type": "text" + }, + { + "bbox": [ + 479, + 399, + 505, + 410 + ], + "score": 0.87, + "content": "\\epsilon , \\delta >", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 407, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 163, + 423 + ], + "score": 1.0, + "content": "0, there exists", + "type": "text" + }, + { + "bbox": [ + 164, + 409, + 231, + 421 + ], + "score": 0.91, + "content": "r : = \\Theta ( \\log { \\delta ^ { - 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 407, + 250, + 423 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 250, + 409, + 305, + 421 + ], + "score": 0.91, + "content": "\\Delta : = \\Theta ( \\epsilon ^ { - 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 407, + 345, + 423 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 346, + 409, + 403, + 421 + ], + "score": 0.95, + "content": "\\pmb { \\theta } / q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 407, + 426, + 423 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 427, + 410, + 448, + 421 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 407, + 506, + 423 + ], + "score": 1.0, + "content": "-KKT point at", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 420, + 294, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 149, + 433 + ], + "score": 1.0, + "content": "some time", + "type": "text" + }, + { + "bbox": [ + 149, + 421, + 158, + 431 + ], + "score": 0.86, + "content": "t _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 420, + 200, + 433 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 201, + 420, + 290, + 432 + ], + "score": 0.91, + "content": "\\log \\rho ( t _ { * } ) \\in ( r , r + \\Delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 420, + 294, + 433 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 504, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "With a refined analysis, we can also derive tight rates for loss convergence and weight growth. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 454, + 238, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 238, + 467 + ], + "score": 1.0, + "content": "defer the proof to Appendix E.6.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 504, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 486 + ], + "score": 1.0, + "content": "Theorem E.5. Under assumptions (S1), (A2) - (A4), (S5), we have the following tight rates for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 483, + 231, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 231, + 495 + ], + "score": 1.0, + "content": "training loss and weight norm:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 505, + 429, + 533 + ], + "lines": [ + { + "bbox": [ + 180, + 505, + 429, + 533 + ], + "spans": [ + { + "bbox": [ + 180, + 505, + 429, + 533 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t ) = \\Theta \\left( \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } \\right) \\quad a n d \\quad \\rho ( t ) = \\Theta ( ( \\log T ) ^ { 1 / L } ) ,", + "type": "interline_equation", + "image_path": "4b2e0c5ff6fe08becd78235a33e1ba14845baf8dc9f226e051b65cbfefdd8b65.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 180, + 505, + 429, + 533 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 206, + 559 + ], + "lines": [ + { + "bbox": [ + 102, + 538, + 203, + 564 + ], + "spans": [ + { + "bbox": [ + 102, + 538, + 133, + 564 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 543, + 203, + 559 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 575, + 245, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 245, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 245, + 588 + ], + "score": 1.0, + "content": "E.4 PROOF FOR THEOREM E.2", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 506, + 633 + ], + "lines": [ + { + "bbox": [ + 100, + 588, + 511, + 624 + ], + "spans": [ + { + "bbox": [ + 100, + 588, + 122, + 624 + ], + "score": 1.0, + "content": "Wefor", + "type": "text" + }, + { + "bbox": [ + 150, + 597, + 263, + 612 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nu ( t ) : = \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( t ) } q _ { n } ( t ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 588, + 460, + 624 + ], + "score": 1.0, + "content": "as we do for gradient flow. Then we can get a clary B.3. Also, we can get a lower bound for", + "type": "text" + }, + { + "bbox": [ + 478, + 588, + 511, + 624 + ], + "score": 1.0, + "content": "d formusing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 123, + 610, + 478, + 622 + ], + "spans": [ + { + "bbox": [ + 123, + 610, + 187, + 622 + ], + "score": 0.91, + "content": "\\langle \\pmb { \\theta } ( t ) , - \\nabla \\mathcal { L } ( t ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 610, + 478, + 622 + ], + "score": 0.91, + "content": "\\nu ( t )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 620, + 271, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 271, + 634 + ], + "score": 1.0, + "content": "Lemma B.5 for exponential loss directly.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 410, + 656 + ], + "lines": [ + { + "bbox": [ + 103, + 635, + 414, + 658 + ], + "spans": [ + { + "bbox": [ + 103, + 635, + 414, + 658 + ], + "score": 1.0, + "content": "Corollary E.6. hθ(t), −∇L(t)i = Lν(t). If L(t) < 1, then ν(t) ≥ L(t)λ(L(t)) .", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 667, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 312, + 680 + ], + "score": 1.0, + "content": "As we are analyzing gradient descent, the norm of", + "type": "text" + }, + { + "bbox": [ + 313, + 668, + 329, + 678 + ], + "score": 0.83, + "content": "\\nabla \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 667, + 348, + 680 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 667, + 369, + 678 + ], + "score": 0.91, + "content": "\\nabla ^ { 2 } \\boldsymbol { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "appear naturally in discretization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 365, + 691 + ], + "score": 1.0, + "content": "error terms. To bound them, we have the following lemma when", + "type": "text" + }, + { + "bbox": [ + 365, + 679, + 385, + 691 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 678, + 404, + 691 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 404, + 679, + 424, + 691 + ], + "score": 0.92, + "content": "\\rho ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "have lower bounds:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 105, + 695, + 352, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 695, + 352, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 196, + 710 + ], + "score": 1.0, + "content": "Lemma E.7. For any", + "type": "text" + }, + { + "bbox": [ + 197, + 696, + 328, + 708 + ], + "score": 0.91, + "content": "\\pmb \\theta , i f \\tilde { \\gamma } ( \\pmb \\theta ) \\geq \\hat { \\gamma } ( t _ { 0 } ) , \\rho ( \\pmb \\theta ) \\geq \\rho ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 695, + 352, + 710 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 718, + 447, + 734 + ], + "lines": [ + { + "bbox": [ + 163, + 718, + 447, + 734 + ], + "spans": [ + { + "bbox": [ + 163, + 718, + 447, + 734 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\| \\nabla \\mathcal { L } ( \\pmb { \\theta } ) \\| _ { 2 } ^ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) \\mathcal { L } ( \\pmb { \\theta } ) \\quad a n d \\quad \\left\\| \\nabla ^ { 2 } \\mathcal { L } ( \\pmb { \\theta } ) \\right\\| _ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6b9909f7f2e24d69959da3441023ee1ddf18b5d8bfa804e9ac3b3cb0565ddf4c.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 163, + 718, + 447, + 734 + ], + "spans": [], + "index": 33 + } + ] + } + ], + "page_idx": 31, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 16, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 463, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 464, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 223, + 96 + ], + "score": 1.0, + "content": "Now we specify the value of", + "type": "text" + }, + { + "bbox": [ + 223, + 83, + 236, + 95 + ], + "score": 0.89, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 81, + 370, + 96 + ], + "score": 1.0, + "content": ". By (S1) and (S2), we can define", + "type": "text" + }, + { + "bbox": [ + 370, + 83, + 416, + 94 + ], + "score": 0.93, + "content": "B _ { 0 } , B _ { 1 } , B _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 81, + 464, + 96 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 464, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 104, + 399, + 156 + ], + "lines": [ + { + "bbox": [ + 211, + 104, + 399, + 156 + ], + "spans": [ + { + "bbox": [ + 211, + 104, + 399, + 156 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { B _ { 0 } : = \\operatorname* { s u p } \\left\\{ q _ { n } : \\pmb \\theta \\in S ^ { d - 1 } , n \\in [ N ] \\right\\} , } \\\\ & { B _ { 1 } : = \\operatorname* { s u p } \\left\\{ \\left\\| \\nabla q _ { n } \\right\\| _ { 2 } : \\pmb \\theta \\in S ^ { d - 1 } , n \\in [ N ] \\right\\} , } \\\\ & { B _ { 2 } : = \\operatorname* { s u p } \\left\\{ \\left\\| \\nabla ^ { 2 } q _ { n } \\right\\| _ { 2 } : \\pmb \\theta \\in S ^ { d - 1 } , n \\in [ N ] \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "f7c7b94b2919472564c0f5c3e77b005b1ea3ea6243bb887a17bc8a4676f3f94b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 211, + 104, + 399, + 121.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 211, + 121.33333333333333, + 399, + 138.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 211, + 138.66666666666666, + 399, + 156.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 405, + 186 + ], + "lines": [ + { + "bbox": [ + 101, + 156, + 405, + 190 + ], + "spans": [ + { + "bbox": [ + 101, + 156, + 156, + 190 + ], + "score": 1.0, + "content": "Then we set", + "type": "text" + }, + { + "bbox": [ + 157, + 164, + 405, + 186 + ], + "score": 0.74, + "content": "\\begin{array} { r l } & { C _ { \\eta } : = \\frac { 1 } { 2 } \\left( B _ { 1 } ^ { 2 } + \\rho ( t _ { 0 } ) ^ { - L } B _ { 2 } \\right) \\operatorname* { m i n } \\Big \\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\Big \\} . } \\end{array}", + "type": "inline_equation", + "image_path": "24f2335a827180f6b97fe05428bdaf53f7a4f89ac58564c5e4ff72cf7009c98c.jpg" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 101, + 156, + 405, + 190 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 202, + 181, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 182, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 182, + 215 + ], + "score": 1.0, + "content": "E.3 THEOREMS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 506, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "score": 1.0, + "content": "Now we state our main theorems for the monotonicity of the normalized margin and the convergence", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "to KKT points. We will prove Theorem E.2 in Appendix E.4, and prove Theorem E.3 and E.4 in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 248, + 168, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 168, + 261 + ], + "score": 1.0, + "content": "Appendix E.5.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 224, + 505, + 261 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 265, + 503, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 279 + ], + "score": 1.0, + "content": "Theorem E.2. Under assumptions (S1), (A2) - (A4), (S5), the following are true for gradient de-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 276, + 133, + 289 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 133, + 289 + ], + "score": 1.0, + "content": "scent:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 264, + 505, + 289 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 299, + 272, + 313 + ], + "lines": [ + { + "bbox": [ + 129, + 298, + 272, + 315 + ], + "spans": [ + { + "bbox": [ + 129, + 298, + 171, + 315 + ], + "score": 1.0, + "content": "1. For all", + "type": "text" + }, + { + "bbox": [ + 172, + 300, + 198, + 312 + ], + "score": 0.7, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 298, + 201, + 315 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 202, + 300, + 267, + 313 + ], + "score": 0.77, + "content": "\\hat { \\gamma } ( t + 1 ) \\geq \\hat { \\gamma } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 298, + 272, + 315 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 129, + 298, + 272, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 322, + 377, + 337 + ], + "lines": [ + { + "bbox": [ + 128, + 321, + 377, + 338 + ], + "spans": [ + { + "bbox": [ + 128, + 321, + 171, + 338 + ], + "score": 1.0, + "content": "2. 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Consider gradient flow under assumptions (S1), (A2) - (A4), (S5). For every limit", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 374, + 363, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 129, + 393 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 376, + 137, + 388 + ], + "score": 0.55, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 374, + 148, + 393 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 149, + 374, + 209, + 394 + ], + "score": 0.85, + "content": "\\left\\{ { \\hat { \\pmb { \\theta } } } ( t ) : t \\geq 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 374, + 214, + 393 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 214, + 375, + 272, + 392 + ], + "score": 0.77, + "content": "\\bar { \\theta } / q _ { \\mathrm { m i n } } ( \\bar { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 374, + 344, + 393 + ], + "score": 1.0, + "content": "is a KKT point of", + "type": "text" + }, + { + "bbox": [ + 344, + 378, + 358, + 389 + ], + "score": 0.74, + "content": "( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 374, + 363, + 393 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 361, + 505, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 398, + 504, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 479, + 412 + ], + "score": 1.0, + "content": "Theorem E.4. Consider gradient descent under assumptions (S1), (A2) - (A4), (S5). For any", + "type": "text" + }, + { + "bbox": [ + 479, + 399, + 505, + 410 + ], + "score": 0.87, + "content": "\\epsilon , \\delta >", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 407, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 163, + 423 + ], + "score": 1.0, + "content": "0, there exists", + "type": "text" + }, + { + "bbox": [ + 164, + 409, + 231, + 421 + ], + "score": 0.91, + "content": "r : = \\Theta ( \\log { \\delta ^ { - 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 407, + 250, + 423 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 250, + 409, + 305, + 421 + ], + "score": 0.91, + "content": "\\Delta : = \\Theta ( \\epsilon ^ { - 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 407, + 345, + 423 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 346, + 409, + 403, + 421 + ], + "score": 0.95, + "content": "\\pmb { \\theta } / q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) ^ { 1 / L }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 407, + 426, + 423 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 427, + 410, + 448, + 421 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 407, + 506, + 423 + ], + "score": 1.0, + "content": "-KKT point at", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 420, + 294, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 149, + 433 + ], + "score": 1.0, + "content": "some time", + "type": "text" + }, + { + "bbox": [ + 149, + 421, + 158, + 431 + ], + "score": 0.86, + "content": "t _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 420, + 200, + 433 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 201, + 420, + 290, + 432 + ], + "score": 0.91, + "content": "\\log \\rho ( t _ { * } ) \\in ( r , r + \\Delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 420, + 294, + 433 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 397, + 506, + 433 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 504, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "With a refined analysis, we can also derive tight rates for loss convergence and weight growth. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 454, + 238, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 238, + 467 + ], + "score": 1.0, + "content": "defer the proof to Appendix E.6.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 443, + 506, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 504, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 486 + ], + "score": 1.0, + "content": "Theorem E.5. Under assumptions (S1), (A2) - (A4), (S5), we have the following tight rates for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 483, + 231, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 231, + 495 + ], + "score": 1.0, + "content": "training loss and weight norm:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 106, + 471, + 505, + 495 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 505, + 429, + 533 + ], + "lines": [ + { + "bbox": [ + 180, + 505, + 429, + 533 + ], + "spans": [ + { + "bbox": [ + 180, + 505, + 429, + 533 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t ) = \\Theta \\left( \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } \\right) \\quad a n d \\quad \\rho ( t ) = \\Theta ( ( \\log T ) ^ { 1 / L } ) ,", + "type": "interline_equation", + "image_path": "4b2e0c5ff6fe08becd78235a33e1ba14845baf8dc9f226e051b65cbfefdd8b65.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 180, + 505, + 429, + 533 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 206, + 559 + ], + "lines": [ + { + "bbox": [ + 102, + 538, + 203, + 564 + ], + "spans": [ + { + "bbox": [ + 102, + 538, + 133, + 564 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 543, + 203, + 559 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 102, + 538, + 203, + 564 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 575, + 245, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 245, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 245, + 588 + ], + "score": 1.0, + "content": "E.4 PROOF FOR THEOREM E.2", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 506, + 633 + ], + "lines": [ + { + "bbox": [ + 100, + 588, + 511, + 624 + ], + "spans": [ + { + "bbox": [ + 100, + 588, + 122, + 624 + ], + "score": 1.0, + "content": "Wefor", + "type": "text" + }, + { + "bbox": [ + 150, + 597, + 263, + 612 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nu ( t ) : = \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( t ) } q _ { n } ( t ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 588, + 460, + 624 + ], + "score": 1.0, + "content": "as we do for gradient flow. Then we can get a clary B.3. Also, we can get a lower bound for", + "type": "text" + }, + { + "bbox": [ + 478, + 588, + 511, + 624 + ], + "score": 1.0, + "content": "d formusing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 123, + 610, + 478, + 622 + ], + "spans": [ + { + "bbox": [ + 123, + 610, + 187, + 622 + ], + "score": 0.91, + "content": "\\langle \\pmb { \\theta } ( t ) , - \\nabla \\mathcal { L } ( t ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 610, + 478, + 622 + ], + "score": 0.91, + "content": "\\nu ( t )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 620, + 271, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 271, + 634 + ], + "score": 1.0, + "content": "Lemma B.5 for exponential loss directly.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 100, + 588, + 511, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 410, + 656 + ], + "lines": [ + { + "bbox": [ + 103, + 635, + 414, + 658 + ], + "spans": [ + { + "bbox": [ + 103, + 635, + 414, + 658 + ], + "score": 1.0, + "content": "Corollary E.6. hθ(t), −∇L(t)i = Lν(t). If L(t) < 1, then ν(t) ≥ L(t)λ(L(t)) .", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 103, + 635, + 414, + 658 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 667, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 312, + 680 + ], + "score": 1.0, + "content": "As we are analyzing gradient descent, the norm of", + "type": "text" + }, + { + "bbox": [ + 313, + 668, + 329, + 678 + ], + "score": 0.83, + "content": "\\nabla \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 667, + 348, + 680 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 667, + 369, + 678 + ], + "score": 0.91, + "content": "\\nabla ^ { 2 } \\boldsymbol { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "appear naturally in discretization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 365, + 691 + ], + "score": 1.0, + "content": "error terms. To bound them, we have the following lemma when", + "type": "text" + }, + { + "bbox": [ + 365, + 679, + 385, + 691 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 678, + 404, + 691 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 404, + 679, + 424, + 691 + ], + "score": 0.92, + "content": "\\rho ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "have lower bounds:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 106, + 667, + 505, + 691 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 695, + 352, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 695, + 352, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 196, + 710 + ], + "score": 1.0, + "content": "Lemma E.7. For any", + "type": "text" + }, + { + "bbox": [ + 197, + 696, + 328, + 708 + ], + "score": 0.91, + "content": "\\pmb \\theta , i f \\tilde { \\gamma } ( \\pmb \\theta ) \\geq \\hat { \\gamma } ( t _ { 0 } ) , \\rho ( \\pmb \\theta ) \\geq \\rho ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 695, + 352, + 710 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 695, + 352, + 710 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 718, + 447, + 734 + ], + "lines": [ + { + "bbox": [ + 163, + 718, + 447, + 734 + ], + "spans": [ + { + "bbox": [ + 163, + 718, + 447, + 734 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\| \\nabla \\mathcal { L } ( \\pmb { \\theta } ) \\| _ { 2 } ^ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) \\mathcal { L } ( \\pmb { \\theta } ) \\quad a n d \\quad \\left\\| \\nabla ^ { 2 } \\mathcal { L } ( \\pmb { \\theta } ) \\right\\| _ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6b9909f7f2e24d69959da3441023ee1ddf18b5d8bfa804e9ac3b3cb0565ddf4c.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 163, + 718, + 447, + 734 + ], + "spans": [], + "index": 33 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 363, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 362, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 294, + 95 + ], + "score": 1.0, + "content": "Proof. By the chain rule and the definitions of", + "type": "text" + }, + { + "bbox": [ + 294, + 83, + 323, + 94 + ], + "score": 0.92, + "content": "B _ { 1 } , B _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 82, + 362, + 95 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 99, + 454, + 206 + ], + "lines": [ + { + "bbox": [ + 156, + 99, + 454, + 206 + ], + "spans": [ + { + "bbox": [ + 156, + 99, + 454, + 206 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\left\\| \\nabla \\mathcal { L } \\right\\| _ { 2 } = \\left\\| - \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\nabla q _ { n } \\right\\| _ { 2 } \\leq \\mathcal { L } \\rho ^ { L - 1 } B _ { 1 } } \\\\ & { \\displaystyle \\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\big ( \\nabla q _ { n } \\nabla q _ { n } ^ { \\top } - \\nabla ^ { 2 } q _ { n } \\big ) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\left( B _ { 1 } ^ { 2 } \\rho ^ { 2 L - 2 } + B _ { 2 } \\rho ^ { L - 2 } \\right) \\leq \\mathcal { L } \\rho ^ { 2 L - 2 } \\left( B _ { 1 } ^ { 2 } + \\rho ^ { - L } B _ { 2 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "3b2d6ecf8165136c6bb28b50d62034c9846f50e4ff3e1d5fe8b793823622b626.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 156, + 99, + 454, + 134.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 156, + 134.66666666666666, + 454, + 170.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 156, + 170.33333333333331, + 454, + 205.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 104, + 209, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 146, + 226 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 211, + 284, + 225 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\hat { \\gamma } ( t _ { 0 } ) \\rho ^ { L } \\leq \\tilde { \\gamma } \\rho ^ { L } \\leq \\log \\frac { 1 } { \\mathcal { L } } \\leq B _ { 0 } \\rho ^ { L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 209, + 300, + 226 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 301, + 212, + 440, + 226 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\gamma } ( t _ { 0 } ) ^ { - 1 } \\log \\frac { 1 } { \\mathcal { L } } \\leq \\rho ^ { L } \\leq B _ { 0 } ^ { - 1 } \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 209, + 505, + 226 + ], + "score": 1.0, + "content": ". Combining all", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 222, + 226, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 226, + 236 + ], + "score": 1.0, + "content": "these formulas together gives", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 241, + 493, + 303 + ], + "lines": [ + { + "bbox": [ + 118, + 241, + 493, + 303 + ], + "spans": [ + { + "bbox": [ + 118, + 241, + 493, + 303 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\left\\| \\nabla { \\mathcal { L } } \\right\\| _ { 2 } ^ { 2 } \\leq B _ { 1 } ^ { 2 } \\operatorname* { m i n } \\left\\{ \\widehat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\} { \\mathcal { L } } ^ { 2 } \\cdot \\left( \\log \\displaystyle \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta } \\kappa ( { \\mathcal { L } } ) \\mathcal { L } } \\\\ & { \\left\\| \\nabla ^ { 2 } { \\mathcal { L } } \\right\\| _ { 2 } \\leq \\left( B _ { 1 } ^ { 2 } + \\rho ^ { - L } B _ { 2 } \\right) \\operatorname* { m i n } \\left\\{ \\widehat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\} { \\mathcal { L } } \\cdot \\left( \\log \\displaystyle \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta } \\kappa ( { \\mathcal { L } } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "39183ed1cf78814025b802e05f6085544b47d48f0fea52af899f6bea1e7ddb56.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 118, + 241, + 493, + 261.6666666666667 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 118, + 261.6666666666667, + 493, + 282.33333333333337 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 118, + 282.33333333333337, + 493, + 303.00000000000006 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 217, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 217, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 217, + 320 + ], + "score": 1.0, + "content": "which completes the proof.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "For proving the first two propositions in Theorem E.2, we only need to prove Lemma E.8. (P1) gives", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 179, + 357 + ], + "score": 1.0, + "content": "a lower bound for", + "type": "text" + }, + { + "bbox": [ + 180, + 345, + 186, + 356 + ], + "score": 0.84, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 344, + 463, + 357 + ], + "score": 1.0, + "content": ". (P2) gives both lower and upper bounds for the weight growth using", + "type": "text" + }, + { + "bbox": [ + 463, + 345, + 481, + 357 + ], + "score": 0.9, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 344, + 506, + 357 + ], + "score": 1.0, + "content": ". (P3)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 494, + 369 + ], + "score": 1.0, + "content": "gives a lower bound for the decrement of training loss. Finally, (P4) shows the monotonicity of", + "type": "text" + }, + { + "bbox": [ + 494, + 356, + 501, + 367 + ], + "score": 0.84, + "content": "\\hat { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 355, + 506, + 369 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 366, + 418, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 418, + 378 + ], + "score": 1.0, + "content": "and it is trivial to deduce the first two propositions in Theorem E.2 from (P4).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 504, + 416 + ], + "lines": [ + { + "bbox": [ + 104, + 379, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 379, + 196, + 396 + ], + "score": 1.0, + "content": "Lemma E.8. For all", + "type": "text" + }, + { + "bbox": [ + 197, + 382, + 276, + 393 + ], + "score": 0.88, + "content": "t = t _ { 0 } , t _ { 0 } + 1 , \\ldots", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 379, + 378, + 396 + ], + "score": 1.0, + "content": ", we interpolate between", + "type": "text" + }, + { + "bbox": [ + 379, + 381, + 397, + 393 + ], + "score": 0.9, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 379, + 418, + 396 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 418, + 381, + 455, + 393 + ], + "score": 0.9, + "content": "\\pmb \\theta ( t + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 379, + 506, + 396 + ], + "score": 1.0, + "content": "by defining", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 391, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 392, + 241, + 405 + ], + "score": 0.92, + "content": "\\pmb \\theta ( t + \\alpha ) = \\pmb \\theta ( t ) - \\alpha \\eta ( t ) \\nabla \\mathcal L ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 391, + 258, + 406 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 258, + 393, + 303, + 405 + ], + "score": 0.92, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 391, + 393, + 406 + ], + "score": 1.0, + "content": ". Then for all integer", + "type": "text" + }, + { + "bbox": [ + 393, + 393, + 423, + 404 + ], + "score": 0.78, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 391, + 427, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 428, + 393, + 467, + 405 + ], + "score": 0.76, + "content": "\\nu ( t ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 391, + 506, + 406 + ], + "score": 1.0, + "content": ", and the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 403, + 243, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 198, + 417 + ], + "score": 1.0, + "content": "following holds for all", + "type": "text" + }, + { + "bbox": [ + 198, + 405, + 237, + 416 + ], + "score": 0.91, + "content": "\\alpha \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 403, + 243, + 417 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 421, + 445, + 517 + ], + "lines": [ + { + "bbox": [ + 115, + 421, + 445, + 517 + ], + "spans": [ + { + "bbox": [ + 115, + 421, + 445, + 517 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { ( P I ) . \\ \\tilde { \\gamma } ( t + \\alpha ) > \\hat { \\gamma } ( t _ { 0 } ) . } \\\\ & { ( P 2 ) . \\ 2 L \\alpha \\eta ( t ) \\nu ( t ) \\leq \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } \\leq 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) } { L } \\right) . } \\\\ & { ( P 3 ) . \\ \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) \\leq - \\alpha \\eta ( t ) ( 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } . } \\\\ & { ( P 4 ) . \\ \\log \\hat { \\gamma } ( t + \\alpha ) - \\log \\hat { \\gamma } ( t ) \\geq \\frac { \\rho ( t ) ^ { 2 } } { L \\nu ( t ) ^ { 2 } } \\left. \\left( I - \\hat { \\theta } ( t ) \\hat { \\theta } ( t ) ^ { \\top } \\right) \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } . \\log \\frac { \\rho ( t + \\alpha ) } { \\rho ( t ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "316658965977837af2fb156a3fb6bcbdbde6a08040c00be4cb1e023d533b0f84.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 115, + 421, + 445, + 453.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 115, + 453.0, + 445, + 485.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 115, + 485.0, + 445, + 517.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 478, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 479, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 479, + 538 + ], + "score": 1.0, + "content": "To prove Lemma E.8, we only need to prove the following lemma and then use an induction:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 222, + 553 + ], + "score": 1.0, + "content": "Lemma E.9. Fix an integer", + "type": "text" + }, + { + "bbox": [ + 223, + 540, + 253, + 551 + ], + "score": 0.89, + "content": "T \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 538, + 314, + 553 + ], + "score": 1.0, + "content": ". Suppose that", + "type": "text" + }, + { + "bbox": [ + 315, + 540, + 402, + 551 + ], + "score": 0.36, + "content": "( P I ) , ( P 2 ) , ( P 3 ) , ( P 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 538, + 456, + 553 + ], + "score": 1.0, + "content": "hold for any", + "type": "text" + }, + { + "bbox": [ + 456, + 540, + 501, + 551 + ], + "score": 0.9, + "content": "t + \\alpha \\leq T", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 538, + 505, + 553 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 550, + 504, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 129, + 564 + ], + "score": 1.0, + "content": "Then", + "type": "text" + }, + { + "bbox": [ + 129, + 551, + 156, + 562 + ], + "score": 0.42, + "content": "i f ( P l )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 550, + 197, + 564 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + }, + { + "bbox": [ + 197, + 551, + 286, + 563 + ], + "score": 0.92, + "content": "( t , \\alpha ) \\in \\{ T \\} \\times [ 0 , A )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 550, + 325, + 564 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 326, + 551, + 369, + 563 + ], + "score": 0.92, + "content": "A \\in ( 0 , 1 ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 550, + 416, + 564 + ], + "score": 1.0, + "content": ", then all of", + "type": "text" + }, + { + "bbox": [ + 416, + 551, + 504, + 563 + ], + "score": 0.34, + "content": "( P I ) , ( P 2 ) , ( P 3 ) , ( P 4 )", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 561, + 231, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 141, + 574 + ], + "score": 1.0, + "content": "hold for", + "type": "text" + }, + { + "bbox": [ + 141, + 562, + 228, + 574 + ], + "score": 0.93, + "content": "( t , \\alpha ) \\in \\{ T \\} \\times [ 0 , A ]", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 561, + 231, + 574 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 358, + 600 + ], + "score": 1.0, + "content": "Proof for Lemma E.8. We prove this lemma by induction. For", + "type": "text" + }, + { + "bbox": [ + 358, + 588, + 414, + 599 + ], + "score": 0.67, + "content": "t = t _ { 0 } , \\alpha = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 587, + 417, + 600 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 418, + 588, + 455, + 599 + ], + "score": 0.87, + "content": "\\nu ( t ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "by (S4) and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 311, + 612 + ], + "score": 1.0, + "content": "Corollary E.6. (P2), (P3), (P4) hold trivially since", + "type": "text" + }, + { + "bbox": [ + 312, + 599, + 410, + 611 + ], + "score": 0.69, + "content": "\\log \\hat { \\gamma } ( t + \\alpha ) = \\log \\hat { \\gamma } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 598, + 415, + 612 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 415, + 599, + 486, + 611 + ], + "score": 0.74, + "content": "\\textstyle { \\mathcal { L } } ( t + \\alpha ) = { \\mathcal { L } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 609, + 371, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 203, + 622 + ], + "score": 0.9, + "content": "\\log \\hat { \\gamma } ( t + \\alpha ) = \\log \\hat { \\gamma } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 609, + 371, + 623 + ], + "score": 1.0, + "content": ". By Lemma E.1, (P1) also holds trivially.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 504, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 196, + 640 + ], + "score": 1.0, + "content": "Now we fix an integer", + "type": "text" + }, + { + "bbox": [ + 197, + 627, + 226, + 638 + ], + "score": 0.91, + "content": "T \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 626, + 430, + 640 + ], + "score": 1.0, + "content": "and assume that (P1), (P2), (P3), (P4) hold for any", + "type": "text" + }, + { + "bbox": [ + 430, + 627, + 474, + 638 + ], + "score": 0.91, + "content": "t + \\alpha \\leq T", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "(where", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 637, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 132, + 649 + ], + "score": 0.9, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 637, + 201, + 650 + ], + "score": 1.0, + "content": "is an integer and", + "type": "text" + }, + { + "bbox": [ + 202, + 638, + 243, + 650 + ], + "score": 0.89, + "content": "\\alpha \\in [ 0 , 1 ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 637, + 285, + 650 + ], + "score": 1.0, + "content": "). By (P3),", + "type": "text" + }, + { + "bbox": [ + 285, + 638, + 359, + 650 + ], + "score": 0.9, + "content": "\\mathcal { L } ( t ) \\leq \\mathcal { L } ( t _ { 0 } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 637, + 374, + 650 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 374, + 638, + 410, + 650 + ], + "score": 0.92, + "content": "\\nu ( t ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 637, + 504, + 650 + ], + "score": 1.0, + "content": ". We only need to show", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 648, + 332, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 244, + 662 + ], + "score": 1.0, + "content": "that (P1), (P2), (P3), (P4) hold for", + "type": "text" + }, + { + "bbox": [ + 245, + 649, + 270, + 659 + ], + "score": 0.91, + "content": "t = T", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 648, + 288, + 662 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 288, + 650, + 327, + 661 + ], + "score": 0.9, + "content": "\\alpha \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 648, + 332, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 123, + 678 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 666, + 244, + 677 + ], + "score": 0.91, + "content": "A : = \\operatorname* { i n f } \\{ \\alpha \\in [ 0 , 1 ] : \\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 664, + 346, + 678 + ], + "score": 1.0, + "content": "or (P1) does not hold for", + "type": "text" + }, + { + "bbox": [ + 346, + 666, + 377, + 678 + ], + "score": 0.93, + "content": "( T , \\alpha ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 664, + 393, + 678 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 393, + 666, + 421, + 676 + ], + "score": 0.89, + "content": "A = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 664, + 506, + 678 + ], + "score": 1.0, + "content": ", then (P1) holds for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 675, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 134, + 689 + ], + "score": 0.92, + "content": "( T , A )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 675, + 219, + 691 + ], + "score": 1.0, + "content": "since (P1) holds for", + "type": "text" + }, + { + "bbox": [ + 220, + 677, + 262, + 689 + ], + "score": 0.87, + "content": "( T - 1 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 675, + 276, + 691 + ], + "score": 1.0, + "content": "; if", + "type": "text" + }, + { + "bbox": [ + 277, + 677, + 306, + 687 + ], + "score": 0.9, + "content": "A > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 675, + 464, + 691 + ], + "score": 1.0, + "content": ", we can also know that (P1) holds for", + "type": "text" + }, + { + "bbox": [ + 464, + 677, + 491, + 689 + ], + "score": 0.92, + "content": "( T , A )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 675, + 506, + 691 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 212, + 700 + ], + "score": 1.0, + "content": "Lemma E.9. Suppose that", + "type": "text" + }, + { + "bbox": [ + 212, + 688, + 239, + 698 + ], + "score": 0.89, + "content": "A < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 686, + 347, + 700 + ], + "score": 1.0, + "content": ". Then by the continuity of", + "type": "text" + }, + { + "bbox": [ + 347, + 687, + 386, + 699 + ], + "score": 0.93, + "content": "\\tilde { \\gamma } ( T + \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 686, + 452, + 700 + ], + "score": 1.0, + "content": "(with respect to", + "type": "text" + }, + { + "bbox": [ + 452, + 690, + 460, + 698 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 686, + 505, + 700 + ], + "score": 1.0, + "content": "), we know", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 172, + 711 + ], + "score": 1.0, + "content": "that there exists", + "type": "text" + }, + { + "bbox": [ + 173, + 699, + 206, + 709 + ], + "score": 0.91, + "content": "A ^ { \\prime } > A", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 698, + 247, + 711 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 247, + 699, + 324, + 711 + ], + "score": 0.91, + "content": "\\tilde { \\gamma } ( T + \\alpha ) > \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 698, + 353, + 711 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 353, + 699, + 401, + 711 + ], + "score": 0.92, + "content": "\\alpha \\in [ A , A ^ { \\prime } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 698, + 505, + 711 + ], + "score": 1.0, + "content": ", which contradicts to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 158, + 722 + ], + "score": 1.0, + "content": "definition of", + "type": "text" + }, + { + "bbox": [ + 159, + 710, + 167, + 720 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 708, + 216, + 722 + ], + "score": 1.0, + "content": ". 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Using Lemma E.9 again, we can conclude that (P1), (P2), (P3),", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 160, + 733 + ], + "score": 1.0, + "content": "(P4) hold for", + "type": "text" + }, + { + "bbox": [ + 160, + 721, + 186, + 731 + ], + "score": 0.9, + "content": "t = T", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 720, + 204, + 733 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 204, + 721, + 243, + 732 + ], + "score": 0.91, + "content": "\\alpha \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 720, + 248, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 506, + 732 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + } + ], + "page_idx": 32, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "33", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 307, + 504, + 318 + ], + "lines": [ + { + "bbox": [ + 496, + 309, + 504, + 317 + ], + "spans": [ + { + "bbox": [ + 496, + 309, + 504, + 317 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 363, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 362, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 294, + 95 + ], + "score": 1.0, + "content": "Proof. By the chain rule and the definitions of", + "type": "text" + }, + { + "bbox": [ + 294, + 83, + 323, + 94 + ], + "score": 0.92, + "content": "B _ { 1 } , B _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 82, + 362, + 95 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 362, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 99, + 454, + 206 + ], + "lines": [ + { + "bbox": [ + 156, + 99, + 454, + 206 + ], + "spans": [ + { + "bbox": [ + 156, + 99, + 454, + 206 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\left\\| \\nabla \\mathcal { L } \\right\\| _ { 2 } = \\left\\| - \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\nabla q _ { n } \\right\\| _ { 2 } \\leq \\mathcal { L } \\rho ^ { L - 1 } B _ { 1 } } \\\\ & { \\displaystyle \\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\big ( \\nabla q _ { n } \\nabla q _ { n } ^ { \\top } - \\nabla ^ { 2 } q _ { n } \\big ) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } } \\left( B _ { 1 } ^ { 2 } \\rho ^ { 2 L - 2 } + B _ { 2 } \\rho ^ { L - 2 } \\right) \\leq \\mathcal { L } \\rho ^ { 2 L - 2 } \\left( B _ { 1 } ^ { 2 } + \\rho ^ { - L } B _ { 2 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "3b2d6ecf8165136c6bb28b50d62034c9846f50e4ff3e1d5fe8b793823622b626.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 156, + 99, + 454, + 134.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 156, + 134.66666666666666, + 454, + 170.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 156, + 170.33333333333331, + 454, + 205.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 104, + 209, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 146, + 226 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 211, + 284, + 225 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\hat { \\gamma } ( t _ { 0 } ) \\rho ^ { L } \\leq \\tilde { \\gamma } \\rho ^ { L } \\leq \\log \\frac { 1 } { \\mathcal { L } } \\leq B _ { 0 } \\rho ^ { L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 209, + 300, + 226 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 301, + 212, + 440, + 226 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\gamma } ( t _ { 0 } ) ^ { - 1 } \\log \\frac { 1 } { \\mathcal { L } } \\leq \\rho ^ { L } \\leq B _ { 0 } ^ { - 1 } \\log \\frac { 1 } { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 209, + 505, + 226 + ], + "score": 1.0, + "content": ". Combining all", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 222, + 226, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 226, + 236 + ], + "score": 1.0, + "content": "these formulas together gives", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 209, + 505, + 236 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 241, + 493, + 303 + ], + "lines": [ + { + "bbox": [ + 118, + 241, + 493, + 303 + ], + "spans": [ + { + "bbox": [ + 118, + 241, + 493, + 303 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\left\\| \\nabla { \\mathcal { L } } \\right\\| _ { 2 } ^ { 2 } \\leq B _ { 1 } ^ { 2 } \\operatorname* { m i n } \\left\\{ \\widehat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\} { \\mathcal { L } } ^ { 2 } \\cdot \\left( \\log \\displaystyle \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta } \\kappa ( { \\mathcal { L } } ) \\mathcal { L } } \\\\ & { \\left\\| \\nabla ^ { 2 } { \\mathcal { L } } \\right\\| _ { 2 } \\leq \\left( B _ { 1 } ^ { 2 } + \\rho ^ { - L } B _ { 2 } \\right) \\operatorname* { m i n } \\left\\{ \\widehat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\} { \\mathcal { L } } \\cdot \\left( \\log \\displaystyle \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta } \\kappa ( { \\mathcal { L } } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "39183ed1cf78814025b802e05f6085544b47d48f0fea52af899f6bea1e7ddb56.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 118, + 241, + 493, + 261.6666666666667 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 118, + 261.6666666666667, + 493, + 282.33333333333337 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 118, + 282.33333333333337, + 493, + 303.00000000000006 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 217, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 217, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 217, + 320 + ], + "score": 1.0, + "content": "which completes the proof.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 306, + 217, + 320 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "For proving the first two propositions in Theorem E.2, we only need to prove Lemma E.8. (P1) gives", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 179, + 357 + ], + "score": 1.0, + "content": "a lower bound for", + "type": "text" + }, + { + "bbox": [ + 180, + 345, + 186, + 356 + ], + "score": 0.84, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 344, + 463, + 357 + ], + "score": 1.0, + "content": ". (P2) gives both lower and upper bounds for the weight growth using", + "type": "text" + }, + { + "bbox": [ + 463, + 345, + 481, + 357 + ], + "score": 0.9, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 344, + 506, + 357 + ], + "score": 1.0, + "content": ". (P3)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 494, + 369 + ], + "score": 1.0, + "content": "gives a lower bound for the decrement of training loss. Finally, (P4) shows the monotonicity of", + "type": "text" + }, + { + "bbox": [ + 494, + 356, + 501, + 367 + ], + "score": 0.84, + "content": "\\hat { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 355, + 506, + 369 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 366, + 418, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 418, + 378 + ], + "score": 1.0, + "content": "and it is trivial to deduce the first two propositions in Theorem E.2 from (P4).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 333, + 506, + 378 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 504, + 416 + ], + "lines": [ + { + "bbox": [ + 104, + 379, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 379, + 196, + 396 + ], + "score": 1.0, + "content": "Lemma E.8. For all", + "type": "text" + }, + { + "bbox": [ + 197, + 382, + 276, + 393 + ], + "score": 0.88, + "content": "t = t _ { 0 } , t _ { 0 } + 1 , \\ldots", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 379, + 378, + 396 + ], + "score": 1.0, + "content": ", we interpolate between", + "type": "text" + }, + { + "bbox": [ + 379, + 381, + 397, + 393 + ], + "score": 0.9, + "content": "\\pmb \\theta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 379, + 418, + 396 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 418, + 381, + 455, + 393 + ], + "score": 0.9, + "content": "\\pmb \\theta ( t + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 379, + 506, + 396 + ], + "score": 1.0, + "content": "by defining", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 391, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 392, + 241, + 405 + ], + "score": 0.92, + "content": "\\pmb \\theta ( t + \\alpha ) = \\pmb \\theta ( t ) - \\alpha \\eta ( t ) \\nabla \\mathcal L ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 391, + 258, + 406 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 258, + 393, + 303, + 405 + ], + "score": 0.92, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 391, + 393, + 406 + ], + "score": 1.0, + "content": ". Then for all integer", + "type": "text" + }, + { + "bbox": [ + 393, + 393, + 423, + 404 + ], + "score": 0.78, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 391, + 427, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 428, + 393, + 467, + 405 + ], + "score": 0.76, + "content": "\\nu ( t ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 391, + 506, + 406 + ], + "score": 1.0, + "content": ", and the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 403, + 243, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 198, + 417 + ], + "score": 1.0, + "content": "following holds for all", + "type": "text" + }, + { + "bbox": [ + 198, + 405, + 237, + 416 + ], + "score": 0.91, + "content": "\\alpha \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 403, + 243, + 417 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 379, + 506, + 417 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 421, + 445, + 517 + ], + "lines": [ + { + "bbox": [ + 115, + 421, + 445, + 517 + ], + "spans": [ + { + "bbox": [ + 115, + 421, + 445, + 517 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { ( P I ) . \\ \\tilde { \\gamma } ( t + \\alpha ) > \\hat { \\gamma } ( t _ { 0 } ) . } \\\\ & { ( P 2 ) . \\ 2 L \\alpha \\eta ( t ) \\nu ( t ) \\leq \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } \\leq 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) } { L } \\right) . } \\\\ & { ( P 3 ) . \\ \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) \\leq - \\alpha \\eta ( t ) ( 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } . } \\\\ & { ( P 4 ) . \\ \\log \\hat { \\gamma } ( t + \\alpha ) - \\log \\hat { \\gamma } ( t ) \\geq \\frac { \\rho ( t ) ^ { 2 } } { L \\nu ( t ) ^ { 2 } } \\left. \\left( I - \\hat { \\theta } ( t ) \\hat { \\theta } ( t ) ^ { \\top } \\right) \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } . \\log \\frac { \\rho ( t + \\alpha ) } { \\rho ( t ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "316658965977837af2fb156a3fb6bcbdbde6a08040c00be4cb1e023d533b0f84.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 115, + 421, + 445, + 453.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 115, + 453.0, + 445, + 485.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 115, + 485.0, + 445, + 517.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 478, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 479, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 479, + 538 + ], + "score": 1.0, + "content": "To prove Lemma E.8, we only need to prove the following lemma and then use an induction:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 524, + 479, + 538 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 222, + 553 + ], + "score": 1.0, + "content": "Lemma E.9. Fix an integer", + "type": "text" + }, + { + "bbox": [ + 223, + 540, + 253, + 551 + ], + "score": 0.89, + "content": "T \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 538, + 314, + 553 + ], + "score": 1.0, + "content": ". Suppose that", + "type": "text" + }, + { + "bbox": [ + 315, + 540, + 402, + 551 + ], + "score": 0.36, + "content": "( P I ) , ( P 2 ) , ( P 3 ) , ( P 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 538, + 456, + 553 + ], + "score": 1.0, + "content": "hold for any", + "type": "text" + }, + { + "bbox": [ + 456, + 540, + 501, + 551 + ], + "score": 0.9, + "content": "t + \\alpha \\leq T", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 538, + 505, + 553 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 550, + 504, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 129, + 564 + ], + "score": 1.0, + "content": "Then", + "type": "text" + }, + { + "bbox": [ + 129, + 551, + 156, + 562 + ], + "score": 0.42, + "content": "i f ( P l )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 550, + 197, + 564 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + }, + { + "bbox": [ + 197, + 551, + 286, + 563 + ], + "score": 0.92, + "content": "( t , \\alpha ) \\in \\{ T \\} \\times [ 0 , A )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 550, + 325, + 564 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 326, + 551, + 369, + 563 + ], + "score": 0.92, + "content": "A \\in ( 0 , 1 ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 550, + 416, + 564 + ], + "score": 1.0, + "content": ", then all of", + "type": "text" + }, + { + "bbox": [ + 416, + 551, + 504, + 563 + ], + "score": 0.34, + "content": "( P I ) , ( P 2 ) , ( P 3 ) , ( P 4 )", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 561, + 231, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 141, + 574 + ], + "score": 1.0, + "content": "hold for", + "type": "text" + }, + { + "bbox": [ + 141, + 562, + 228, + 574 + ], + "score": 0.93, + "content": "( t , \\alpha ) \\in \\{ T \\} \\times [ 0 , A ]", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 561, + 231, + 574 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 538, + 505, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 358, + 600 + ], + "score": 1.0, + "content": "Proof for Lemma E.8. We prove this lemma by induction. For", + "type": "text" + }, + { + "bbox": [ + 358, + 588, + 414, + 599 + ], + "score": 0.67, + "content": "t = t _ { 0 } , \\alpha = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 587, + 417, + 600 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 418, + 588, + 455, + 599 + ], + "score": 0.87, + "content": "\\nu ( t ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "by (S4) and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 311, + 612 + ], + "score": 1.0, + "content": "Corollary E.6. (P2), (P3), (P4) hold trivially since", + "type": "text" + }, + { + "bbox": [ + 312, + 599, + 410, + 611 + ], + "score": 0.69, + "content": "\\log \\hat { \\gamma } ( t + \\alpha ) = \\log \\hat { \\gamma } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 598, + 415, + 612 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 415, + 599, + 486, + 611 + ], + "score": 0.74, + "content": "\\textstyle { \\mathcal { L } } ( t + \\alpha ) = { \\mathcal { L } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 609, + 371, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 203, + 622 + ], + "score": 0.9, + "content": "\\log \\hat { \\gamma } ( t + \\alpha ) = \\log \\hat { \\gamma } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 609, + 371, + 623 + ], + "score": 1.0, + "content": ". By Lemma E.1, (P1) also holds trivially.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 587, + 505, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 504, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 196, + 640 + ], + "score": 1.0, + "content": "Now we fix an integer", + "type": "text" + }, + { + "bbox": [ + 197, + 627, + 226, + 638 + ], + "score": 0.91, + "content": "T \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 626, + 430, + 640 + ], + "score": 1.0, + "content": "and assume that (P1), (P2), (P3), (P4) hold for any", + "type": "text" + }, + { + "bbox": [ + 430, + 627, + 474, + 638 + ], + "score": 0.91, + "content": "t + \\alpha \\leq T", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "(where", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 637, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 132, + 649 + ], + "score": 0.9, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 637, + 201, + 650 + ], + "score": 1.0, + "content": "is an integer and", + "type": "text" + }, + { + "bbox": [ + 202, + 638, + 243, + 650 + ], + "score": 0.89, + "content": "\\alpha \\in [ 0 , 1 ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 637, + 285, + 650 + ], + "score": 1.0, + "content": "). By (P3),", + "type": "text" + }, + { + "bbox": [ + 285, + 638, + 359, + 650 + ], + "score": 0.9, + "content": "\\mathcal { L } ( t ) \\leq \\mathcal { L } ( t _ { 0 } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 637, + 374, + 650 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 374, + 638, + 410, + 650 + ], + "score": 0.92, + "content": "\\nu ( t ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 637, + 504, + 650 + ], + "score": 1.0, + "content": ". We only need to show", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 648, + 332, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 244, + 662 + ], + "score": 1.0, + "content": "that (P1), (P2), (P3), (P4) hold for", + "type": "text" + }, + { + "bbox": [ + 245, + 649, + 270, + 659 + ], + "score": 0.91, + "content": "t = T", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 648, + 288, + 662 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 288, + 650, + 327, + 661 + ], + "score": 0.9, + "content": "\\alpha \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 648, + 332, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 626, + 505, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 123, + 678 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 666, + 244, + 677 + ], + "score": 0.91, + "content": "A : = \\operatorname* { i n f } \\{ \\alpha \\in [ 0 , 1 ] : \\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 664, + 346, + 678 + ], + "score": 1.0, + "content": "or (P1) does not hold for", + "type": "text" + }, + { + "bbox": [ + 346, + 666, + 377, + 678 + ], + "score": 0.93, + "content": "( T , \\alpha ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 664, + 393, + 678 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 393, + 666, + 421, + 676 + ], + "score": 0.89, + "content": "A = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 664, + 506, + 678 + ], + "score": 1.0, + "content": ", then (P1) holds for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 675, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 134, + 689 + ], + "score": 0.92, + "content": "( T , A )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 675, + 219, + 691 + ], + "score": 1.0, + "content": "since (P1) holds for", + "type": "text" + }, + { + "bbox": [ + 220, + 677, + 262, + 689 + ], + "score": 0.87, + "content": "( T - 1 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 675, + 276, + 691 + ], + "score": 1.0, + "content": "; if", + "type": "text" + }, + { + "bbox": [ + 277, + 677, + 306, + 687 + ], + "score": 0.9, + "content": "A > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 675, + 464, + 691 + ], + "score": 1.0, + "content": ", we can also know that (P1) holds for", + "type": "text" + }, + { + "bbox": [ + 464, + 677, + 491, + 689 + ], + "score": 0.92, + "content": "( T , A )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 675, + 506, + 691 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 212, + 700 + ], + "score": 1.0, + "content": "Lemma E.9. Suppose that", + "type": "text" + }, + { + "bbox": [ + 212, + 688, + 239, + 698 + ], + "score": 0.89, + "content": "A < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 686, + 347, + 700 + ], + "score": 1.0, + "content": ". Then by the continuity of", + "type": "text" + }, + { + "bbox": [ + 347, + 687, + 386, + 699 + ], + "score": 0.93, + "content": "\\tilde { \\gamma } ( T + \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 686, + 452, + 700 + ], + "score": 1.0, + "content": "(with respect to", + "type": "text" + }, + { + "bbox": [ + 452, + 690, + 460, + 698 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 686, + 505, + 700 + ], + "score": 1.0, + "content": "), we know", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 172, + 711 + ], + "score": 1.0, + "content": "that there exists", + "type": "text" + }, + { + "bbox": [ + 173, + 699, + 206, + 709 + ], + "score": 0.91, + "content": "A ^ { \\prime } > A", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 698, + 247, + 711 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 247, + 699, + 324, + 711 + ], + "score": 0.91, + "content": "\\tilde { \\gamma } ( T + \\alpha ) > \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 698, + 353, + 711 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 353, + 699, + 401, + 711 + ], + "score": 0.92, + "content": "\\alpha \\in [ A , A ^ { \\prime } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 698, + 505, + 711 + ], + "score": 1.0, + "content": ", which contradicts to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 158, + 722 + ], + "score": 1.0, + "content": "definition of", + "type": "text" + }, + { + "bbox": [ + 159, + 710, + 167, + 720 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 708, + 216, + 722 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 217, + 710, + 244, + 720 + ], + "score": 0.89, + "content": "A = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 708, + 505, + 722 + ], + "score": 1.0, + "content": ". Using Lemma E.9 again, we can conclude that (P1), (P2), (P3),", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 160, + 733 + ], + "score": 1.0, + "content": "(P4) hold for", + "type": "text" + }, + { + "bbox": [ + 160, + 721, + 186, + 731 + ], + "score": 0.9, + "content": "t = T", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 720, + 204, + 733 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 204, + 721, + 243, + 732 + ], + "score": 0.91, + "content": "\\alpha \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 720, + 248, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 506, + 732 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 664, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 246, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 247, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 247, + 95 + ], + "score": 1.0, + "content": "Now we turn to prove Lemma E.9.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 141 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 271, + 119 + ], + "score": 1.0, + "content": "Proof for Lemma E.9. Applying (P3) on", + "type": "text" + }, + { + "bbox": [ + 271, + 106, + 388, + 118 + ], + "score": 0.92, + "content": "( t , \\alpha ) \\in \\{ t _ { 0 } , \\ldots , T - 1 \\} \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 104, + 427, + 119 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 428, + 106, + 501, + 118 + ], + "score": 0.91, + "content": "{ \\mathcal { L } } ( t ) \\leq { \\mathcal { L } } ( t _ { 0 } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 104, + 506, + 119 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 237, + 131 + ], + "score": 1.0, + "content": "Then by Corollary E.6, we have", + "type": "text" + }, + { + "bbox": [ + 237, + 118, + 274, + 129 + ], + "score": 0.92, + "content": "\\nu ( t ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 116, + 352, + 131 + ], + "score": 1.0, + "content": ". Applying (P2) on", + "type": "text" + }, + { + "bbox": [ + 353, + 118, + 470, + 129 + ], + "score": 0.92, + "content": "( t , \\alpha ) \\in \\{ t _ { 0 } , \\ldots , T - 1 \\} \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 116, + 505, + 131 + ], + "score": 1.0, + "content": ", we can", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 178, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 121, + 142 + ], + "score": 1.0, + "content": "get", + "type": "text" + }, + { + "bbox": [ + 121, + 128, + 173, + 141 + ], + "score": 0.92, + "content": "\\rho ( t ) \\geq \\rho ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 127, + 178, + 142 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 504, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 122, + 158 + ], + "score": 1.0, + "content": "Fix", + "type": "text" + }, + { + "bbox": [ + 123, + 145, + 151, + 155 + ], + "score": 0.88, + "content": "t = T", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 144, + 213, + 158 + ], + "score": 1.0, + "content": ". By (P2) with", + "type": "text" + }, + { + "bbox": [ + 214, + 145, + 259, + 157 + ], + "score": 0.91, + "content": "\\alpha \\in [ 0 , A )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 144, + 349, + 158 + ], + "score": 1.0, + "content": "and the continuity of", + "type": "text" + }, + { + "bbox": [ + 349, + 146, + 356, + 157 + ], + "score": 0.84, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 144, + 397, + 158 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 397, + 145, + 474, + 157 + ], + "score": 0.94, + "content": "\\tilde { \\gamma } ( t + A ) \\geq \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 144, + 505, + 158 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 156, + 253, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 178, + 168 + ], + "score": 0.93, + "content": "\\tilde { \\gamma } ( t + \\alpha ) \\geq \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 156, + 206, + 169 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 206, + 157, + 248, + 168 + ], + "score": 0.92, + "content": "\\alpha \\in [ 0 , A ]", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 156, + 253, + 169 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 105, + 179, + 504, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 234, + 196 + ], + "score": 1.0, + "content": "Proof for (P2). By definition,", + "type": "text" + }, + { + "bbox": [ + 234, + 180, + 505, + 195 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\rho ( t + \\alpha ) ^ { 2 } = \\| \\pmb { \\theta } ( t ) - \\alpha \\eta ( t ) \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } = \\rho ( t ) ^ { 2 } + \\alpha ^ { 2 } \\eta ( t ) ^ { 2 } \\left\\| \\nabla \\mathcal { L } ( t ) \\right\\| _ { 2 } ^ { 2 } - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 191, + 307, + 207 + ], + "spans": [ + { + "bbox": [ + 107, + 193, + 193, + 205 + ], + "score": 0.92, + "content": "2 \\alpha \\eta ( t ) \\left. \\pmb { \\theta } ( t ) , \\nabla \\mathcal { L } ( t ) \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 191, + 307, + 207 + ], + "score": 1.0, + "content": ". By Corollary E.6, we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 209, + 420, + 226 + ], + "lines": [ + { + "bbox": [ + 190, + 209, + 420, + 226 + ], + "spans": [ + { + "bbox": [ + 190, + 209, + 420, + 226 + ], + "score": 0.86, + "content": "\\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } = \\alpha ^ { 2 } \\eta ( t ) ^ { 2 } \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } + 2 L \\alpha \\eta ( t ) \\nu ( t ) .", + "type": "interline_equation", + "image_path": "8f168583b603c09c8053f0866cd04c928047b67272355679e44287b3fe622000.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 190, + 209, + 420, + 226 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 231, + 504, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 120, + 245 + ], + "score": 1.0, + "content": "So", + "type": "text" + }, + { + "bbox": [ + 120, + 231, + 260, + 244 + ], + "score": 0.92, + "content": "\\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } \\geq 2 L \\alpha \\eta ( t ) \\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 230, + 505, + 245 + ], + "score": 1.0, + "content": ". For the other direction, we have the following using Corol-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 242, + 210, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 210, + 255 + ], + "score": 1.0, + "content": "lary E.6 and Lemma E.7,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 259, + 478, + 343 + ], + "lines": [ + { + "bbox": [ + 131, + 259, + 478, + 343 + ], + "spans": [ + { + "bbox": [ + 131, + 259, + 478, + 343 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } = 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { \\alpha \\eta ( t ) \\left\\| \\nabla \\mathcal { L } ( t ) \\right\\| _ { 2 } ^ { 2 } } { 2 L \\nu ( t ) } \\right) } \\\\ & { \\qquad \\le 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { C _ { \\eta } ^ { - 1 } \\kappa ( \\mathcal { L } ( t ) ) ^ { - 1 } \\mu ( \\mathcal { L } ( t ) ) \\cdot 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( t ) ) \\mathcal { L } ( t ) } { 2 L \\mathcal { L } ( t ) \\lambda ( \\mathcal { L } ( t ) ) ^ { - 1 } } \\right) } \\\\ & { \\qquad = 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) / L \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "532123aadf62da03b786440165d395dfb0505ea4db3bcf295f923b1b5c1b0346.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 131, + 259, + 478, + 287.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 131, + 287.0, + 478, + 315.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 131, + 315.0, + 478, + 343.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 507, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 271, + 365 + ], + "score": 1.0, + "content": "Proof for (P3). (P3) holds trivially for", + "type": "text" + }, + { + "bbox": [ + 271, + 353, + 298, + 363 + ], + "score": 0.9, + "content": "\\alpha = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 351, + 310, + 365 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 311, + 352, + 358, + 365 + ], + "score": 0.92, + "content": "\\nabla \\mathcal { L } ( t ) = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 351, + 460, + 365 + ], + "score": 1.0, + "content": ". So now we assume that", + "type": "text" + }, + { + "bbox": [ + 460, + 353, + 487, + 364 + ], + "score": 0.94, + "content": "\\alpha \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 363, + 473, + 376 + ], + "spans": [ + { + "bbox": [ + 107, + 363, + 153, + 375 + ], + "score": 0.92, + "content": "\\nabla \\mathcal { L } ( t ) \\neq \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 363, + 390, + 376 + ], + "score": 1.0, + "content": ". By the update rule (18) and Taylor expansion, there exists", + "type": "text" + }, + { + "bbox": [ + 390, + 363, + 432, + 375 + ], + "score": 0.93, + "content": "\\xi \\in ( 0 , \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 363, + 473, + 376 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 379, + 514, + 456 + ], + "lines": [ + { + "bbox": [ + 111, + 379, + 514, + 456 + ], + "spans": [ + { + "bbox": [ + 111, + 379, + 514, + 456 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\mathrel { \\mathop : } ( t + \\alpha ) = \\mathcal { L } ( t ) + ( \\theta ( t + \\alpha ) - \\theta ( t ) ) ^ { \\top } \\nabla \\mathcal { L } ( t ) + \\frac { 1 } { 2 } ( \\theta ( t + \\alpha ) - \\theta ( t ) ) ^ { \\top } \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) ( \\theta ( t + \\alpha ) - \\theta ( t ) ) } \\\\ & { \\qquad \\le \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } + \\frac { 1 } { 2 } \\alpha ^ { 2 } \\eta ( t ) ^ { 2 } \\| \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) \\| _ { 2 } \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } } \\\\ & { \\qquad = \\mathcal { L } ( t ) - \\alpha \\eta ( t ) ( 1 - \\frac { 1 } { 2 } \\alpha \\eta ( t ) ) \\| \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) \\| _ { 2 } ) \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "e61b2ec6f0513fe0fb54efdea65daa4166a3ac97997812cbba20047c9df69747.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 111, + 379, + 514, + 404.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 111, + 404.6666666666667, + 514, + 430.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 111, + 430.33333333333337, + 514, + 456.00000000000006 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 373, + 475 + ], + "lines": [ + { + "bbox": [ + 104, + 459, + 374, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 172, + 477 + ], + "score": 1.0, + "content": "By Lemma E.7,", + "type": "text" + }, + { + "bbox": [ + 172, + 461, + 323, + 475 + ], + "score": 0.9, + "content": "\\left\\| \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) \\right\\| _ { 2 } \\leq 2 C _ { \\eta } \\cdot \\kappa ( \\mathcal { L } ( t + \\xi ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 459, + 374, + 477 + ], + "score": 1.0, + "content": ", so we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 480, + 438, + 497 + ], + "lines": [ + { + "bbox": [ + 174, + 480, + 438, + 497 + ], + "spans": [ + { + "bbox": [ + 174, + 480, + 438, + 497 + ], + "score": 0.82, + "content": "\\mathcal { L } ( t + \\alpha ) \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\alpha C _ { \\eta } \\eta ( t ) \\kappa ( \\mathcal { L } ( t + \\xi ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "83f4dc1988319f0e135ea298c68c1db54722cebfd2ccc210bc1930640b4d19ac.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 174, + 480, + 438, + 497 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 506, + 504, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 238, + 521 + ], + "score": 1.0, + "content": "Now we only need to show that", + "type": "text" + }, + { + "bbox": [ + 239, + 507, + 309, + 519 + ], + "score": 0.93, + "content": "\\mathscr { L } ( t + \\alpha ) < \\mathscr { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 505, + 339, + 521 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 339, + 507, + 384, + 519 + ], + "score": 0.92, + "content": "\\alpha \\in ( 0 , A ]", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 505, + 505, + 521 + ], + "score": 1.0, + "content": ". Assuming this, we can have", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 518, + 343, + 531 + ], + "spans": [ + { + "bbox": [ + 107, + 518, + 201, + 531 + ], + "score": 0.91, + "content": "\\kappa ( \\mathcal { L } ( t + \\xi ) ) \\leq \\kappa ( \\mathcal { L } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 518, + 296, + 531 + ], + "score": 1.0, + "content": "by the monotonicity of", + "type": "text" + }, + { + "bbox": [ + 297, + 522, + 303, + 528 + ], + "score": 0.8, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 518, + 343, + 531 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 535, + 428, + 570 + ], + "lines": [ + { + "bbox": [ + 183, + 535, + 428, + 570 + ], + "spans": [ + { + "bbox": [ + 183, + 535, + 428, + 570 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\mathcal { L } ( t + \\alpha ) \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\alpha C _ { \\eta } \\eta ( t ) \\kappa ( \\mathcal { L } ( t ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } } \\\\ & { \\qquad \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\mu ( \\mathcal { L } ( t ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "90fb9d163295e531359ad0271e4b8aee6b96a0f7b1345e45296eb29ca8d4eaa5.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 183, + 535, + 428, + 546.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 183, + 546.6666666666666, + 428, + 558.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 183, + 558.3333333333333, + 428, + 569.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 506, + 627 + ], + "lines": [ + { + "bbox": [ + 104, + 576, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 189, + 592 + ], + "score": 1.0, + "content": "Now we show that", + "type": "text" + }, + { + "bbox": [ + 189, + 579, + 266, + 590 + ], + "score": 0.9, + "content": "\\mathcal { L } ( t + \\alpha ) ~ < ~ \\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 576, + 298, + 592 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 298, + 579, + 347, + 591 + ], + "score": 0.92, + "content": "\\alpha \\in \\mathsf { \\Gamma } ( 0 , A ]", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 576, + 475, + 592 + ], + "score": 1.0, + "content": ". Assume to the contrary that", + "type": "text" + }, + { + "bbox": [ + 476, + 580, + 505, + 590 + ], + "score": 0.83, + "content": "\\alpha _ { 0 } : =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 262, + 602 + ], + "score": 0.85, + "content": "\\operatorname* { i n f } \\{ \\alpha ^ { \\prime } \\in ( 0 , A ] : { \\mathcal { L } } ( t + \\alpha ^ { \\prime } ) \\geq { \\mathcal { L } } ( t ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 589, + 303, + 603 + ], + "score": 1.0, + "content": "exists. If", + "type": "text" + }, + { + "bbox": [ + 303, + 591, + 335, + 601 + ], + "score": 0.89, + "content": "\\alpha _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 589, + 352, + 603 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 353, + 590, + 481, + 602 + ], + "score": 0.89, + "content": "p : [ 0 , 1 ] \\to \\mathbb { R } , \\alpha \\mapsto \\mathcal { L } ( t + \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 589, + 506, + 603 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 599, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 243, + 617 + ], + "score": 0.88, + "content": "\\begin{array} { r } { p ^ { \\prime } ( 0 ) = \\operatorname* { l i m } _ { \\alpha \\downarrow 0 } \\frac { \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) } { \\alpha } \\geq 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 599, + 507, + 617 + ], + "score": 1.0, + "content": "L(t+α)−L(t) ≥ 0, but it contradicts to p0(0) = −η(t) k∇L(t)k22 < 0. If α0 > 0,", + "type": "text" + }, + { + "bbox": [ + 220, + 617, + 227, + 624 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 602, + 263, + 627 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 263, + 614, + 358, + 627 + ], + "score": 0.92, + "content": "\\kappa ( \\mathcal { L } ( t + \\xi ) ) < \\kappa ( \\dot { \\mathcal { L } } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 615, + 432, + 627 + ], + "score": 0.91, + "content": "\\xi \\in ( 0 , \\alpha _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 602, + 469, + 627 + ], + "score": 1.0, + "content": "thus", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 631, + 425, + 648 + ], + "lines": [ + { + "bbox": [ + 185, + 631, + 425, + 648 + ], + "spans": [ + { + "bbox": [ + 185, + 631, + 425, + 648 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\mathcal { L } ( t + \\alpha ) \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\mu ( \\mathcal { L } ( t ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } < \\mathcal { L } ( t ) , } \\end{array}", + "type": "interline_equation", + "image_path": "78a72fa74328b662e81fda8ae4cfee9030f3e404a05693a1d60124b6373fcde4.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 185, + 631, + 425, + 648 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 652, + 231, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 651, + 231, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 231, + 664 + ], + "score": 1.0, + "content": "which leads to a contradiction.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 675, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 102, + 669, + 504, + 695 + ], + "spans": [ + { + "bbox": [ + 102, + 669, + 147, + 695 + ], + "score": 1.0, + "content": "Proof for", + "type": "text" + }, + { + "bbox": [ + 147, + 678, + 165, + 689 + ], + "score": 0.42, + "content": "( \\mathbf { P 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 669, + 219, + 695 + ], + "score": 1.0, + "content": ". We define", + "type": "text" + }, + { + "bbox": [ + 219, + 676, + 336, + 690 + ], + "score": 0.92, + "content": "\\pmb { v } ( t ) : = \\hat { \\pmb { \\theta } } ( t ) \\hat { \\pmb { \\theta } } ( t ) ^ { \\top } ( - \\nabla \\mathcal { L } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 669, + 353, + 695 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 673, + 504, + 694 + ], + "score": 0.9, + "content": "\\pmb { u } ( t ) : = \\left( \\pmb { I } - \\hat { \\pmb { \\theta } } ( t ) \\hat { \\pmb { \\theta } } ( t ) ^ { \\top } \\right) ( - \\nabla \\mathcal { L } ( t ) )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 691, + 361, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 303, + 705 + ], + "score": 1.0, + "content": "similarly as in the analysis for gradient flow. For", + "type": "text" + }, + { + "bbox": [ + 303, + 693, + 321, + 705 + ], + "score": 0.91, + "content": "{ \\pmb v } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 691, + 361, + 705 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 709, + 399, + 736 + ], + "lines": [ + { + "bbox": [ + 210, + 709, + 399, + 736 + ], + "spans": [ + { + "bbox": [ + 210, + 709, + 399, + 736 + ], + "score": 0.94, + "content": "\\| \\pmb { v } ( t ) \\| _ { 2 } = \\frac { 1 } { \\rho ( t ) } \\left. \\pmb { \\theta } ( t ) , - \\pmb { \\mathcal { L } } ( t ) \\right. = \\frac { 1 } { \\rho ( t ) } \\cdot L \\nu ( t ) .", + "type": "interline_equation", + "image_path": "b92fc95da92cc467c3f2cd46a13526f3347b1c4bf9e75014ac3007e2b217ccea.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 210, + 709, + 399, + 736 + ], + "spans": [], + "index": 33 + } + ] + } + ], + "page_idx": 33, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 246, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 247, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 247, + 95 + ], + "score": 1.0, + "content": "Now we turn to prove Lemma E.9.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 82, + 247, + 95 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 141 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 271, + 119 + ], + "score": 1.0, + "content": "Proof for Lemma E.9. Applying (P3) on", + "type": "text" + }, + { + "bbox": [ + 271, + 106, + 388, + 118 + ], + "score": 0.92, + "content": "( t , \\alpha ) \\in \\{ t _ { 0 } , \\ldots , T - 1 \\} \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 104, + 427, + 119 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 428, + 106, + 501, + 118 + ], + "score": 0.91, + "content": "{ \\mathcal { L } } ( t ) \\leq { \\mathcal { L } } ( t _ { 0 } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 104, + 506, + 119 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 237, + 131 + ], + "score": 1.0, + "content": "Then by Corollary E.6, we have", + "type": "text" + }, + { + "bbox": [ + 237, + 118, + 274, + 129 + ], + "score": 0.92, + "content": "\\nu ( t ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 116, + 352, + 131 + ], + "score": 1.0, + "content": ". Applying (P2) on", + "type": "text" + }, + { + "bbox": [ + 353, + 118, + 470, + 129 + ], + "score": 0.92, + "content": "( t , \\alpha ) \\in \\{ t _ { 0 } , \\ldots , T - 1 \\} \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 116, + 505, + 131 + ], + "score": 1.0, + "content": ", we can", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 178, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 121, + 142 + ], + "score": 1.0, + "content": "get", + "type": "text" + }, + { + "bbox": [ + 121, + 128, + 173, + 141 + ], + "score": 0.92, + "content": "\\rho ( t ) \\geq \\rho ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 127, + 178, + 142 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 104, + 506, + 142 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 504, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 122, + 158 + ], + "score": 1.0, + "content": "Fix", + "type": "text" + }, + { + "bbox": [ + 123, + 145, + 151, + 155 + ], + "score": 0.88, + "content": "t = T", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 144, + 213, + 158 + ], + "score": 1.0, + "content": ". By (P2) with", + "type": "text" + }, + { + "bbox": [ + 214, + 145, + 259, + 157 + ], + "score": 0.91, + "content": "\\alpha \\in [ 0 , A )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 144, + 349, + 158 + ], + "score": 1.0, + "content": "and the continuity of", + "type": "text" + }, + { + "bbox": [ + 349, + 146, + 356, + 157 + ], + "score": 0.84, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 144, + 397, + 158 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 397, + 145, + 474, + 157 + ], + "score": 0.94, + "content": "\\tilde { \\gamma } ( t + A ) \\geq \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 144, + 505, + 158 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 156, + 253, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 178, + 168 + ], + "score": 0.93, + "content": "\\tilde { \\gamma } ( t + \\alpha ) \\geq \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 156, + 206, + 169 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 206, + 157, + 248, + 168 + ], + "score": 0.92, + "content": "\\alpha \\in [ 0 , A ]", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 156, + 253, + 169 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 144, + 505, + 169 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 179, + 504, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 234, + 196 + ], + "score": 1.0, + "content": "Proof for (P2). By definition,", + "type": "text" + }, + { + "bbox": [ + 234, + 180, + 505, + 195 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\rho ( t + \\alpha ) ^ { 2 } = \\| \\pmb { \\theta } ( t ) - \\alpha \\eta ( t ) \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } = \\rho ( t ) ^ { 2 } + \\alpha ^ { 2 } \\eta ( t ) ^ { 2 } \\left\\| \\nabla \\mathcal { L } ( t ) \\right\\| _ { 2 } ^ { 2 } - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 191, + 307, + 207 + ], + "spans": [ + { + "bbox": [ + 107, + 193, + 193, + 205 + ], + "score": 0.92, + "content": "2 \\alpha \\eta ( t ) \\left. \\pmb { \\theta } ( t ) , \\nabla \\mathcal { L } ( t ) \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 191, + 307, + 207 + ], + "score": 1.0, + "content": ". By Corollary E.6, we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 177, + 505, + 207 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 209, + 420, + 226 + ], + "lines": [ + { + "bbox": [ + 190, + 209, + 420, + 226 + ], + "spans": [ + { + "bbox": [ + 190, + 209, + 420, + 226 + ], + "score": 0.86, + "content": "\\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } = \\alpha ^ { 2 } \\eta ( t ) ^ { 2 } \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } + 2 L \\alpha \\eta ( t ) \\nu ( t ) .", + "type": "interline_equation", + "image_path": "8f168583b603c09c8053f0866cd04c928047b67272355679e44287b3fe622000.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 190, + 209, + 420, + 226 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 231, + 504, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 120, + 245 + ], + "score": 1.0, + "content": "So", + "type": "text" + }, + { + "bbox": [ + 120, + 231, + 260, + 244 + ], + "score": 0.92, + "content": "\\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } \\geq 2 L \\alpha \\eta ( t ) \\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 230, + 505, + 245 + ], + "score": 1.0, + "content": ". For the other direction, we have the following using Corol-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 242, + 210, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 210, + 255 + ], + "score": 1.0, + "content": "lary E.6 and Lemma E.7,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 230, + 505, + 255 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 259, + 478, + 343 + ], + "lines": [ + { + "bbox": [ + 131, + 259, + 478, + 343 + ], + "spans": [ + { + "bbox": [ + 131, + 259, + 478, + 343 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } = 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { \\alpha \\eta ( t ) \\left\\| \\nabla \\mathcal { L } ( t ) \\right\\| _ { 2 } ^ { 2 } } { 2 L \\nu ( t ) } \\right) } \\\\ & { \\qquad \\le 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { C _ { \\eta } ^ { - 1 } \\kappa ( \\mathcal { L } ( t ) ) ^ { - 1 } \\mu ( \\mathcal { L } ( t ) ) \\cdot 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( t ) ) \\mathcal { L } ( t ) } { 2 L \\mathcal { L } ( t ) \\lambda ( \\mathcal { L } ( t ) ) ^ { - 1 } } \\right) } \\\\ & { \\qquad = 2 L \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) / L \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "532123aadf62da03b786440165d395dfb0505ea4db3bcf295f923b1b5c1b0346.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 131, + 259, + 478, + 287.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 131, + 287.0, + 478, + 315.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 131, + 315.0, + 478, + 343.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 507, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 271, + 365 + ], + "score": 1.0, + "content": "Proof for (P3). (P3) holds trivially for", + "type": "text" + }, + { + "bbox": [ + 271, + 353, + 298, + 363 + ], + "score": 0.9, + "content": "\\alpha = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 351, + 310, + 365 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 311, + 352, + 358, + 365 + ], + "score": 0.92, + "content": "\\nabla \\mathcal { L } ( t ) = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 351, + 460, + 365 + ], + "score": 1.0, + "content": ". So now we assume that", + "type": "text" + }, + { + "bbox": [ + 460, + 353, + 487, + 364 + ], + "score": 0.94, + "content": "\\alpha \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 363, + 473, + 376 + ], + "spans": [ + { + "bbox": [ + 107, + 363, + 153, + 375 + ], + "score": 0.92, + "content": "\\nabla \\mathcal { L } ( t ) \\neq \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 363, + 390, + 376 + ], + "score": 1.0, + "content": ". By the update rule (18) and Taylor expansion, there exists", + "type": "text" + }, + { + "bbox": [ + 390, + 363, + 432, + 375 + ], + "score": 0.93, + "content": "\\xi \\in ( 0 , \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 363, + 473, + 376 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 351, + 505, + 376 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 379, + 514, + 456 + ], + "lines": [ + { + "bbox": [ + 111, + 379, + 514, + 456 + ], + "spans": [ + { + "bbox": [ + 111, + 379, + 514, + 456 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\mathrel { \\mathop : } ( t + \\alpha ) = \\mathcal { L } ( t ) + ( \\theta ( t + \\alpha ) - \\theta ( t ) ) ^ { \\top } \\nabla \\mathcal { L } ( t ) + \\frac { 1 } { 2 } ( \\theta ( t + \\alpha ) - \\theta ( t ) ) ^ { \\top } \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) ( \\theta ( t + \\alpha ) - \\theta ( t ) ) } \\\\ & { \\qquad \\le \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } + \\frac { 1 } { 2 } \\alpha ^ { 2 } \\eta ( t ) ^ { 2 } \\| \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) \\| _ { 2 } \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } } \\\\ & { \\qquad = \\mathcal { L } ( t ) - \\alpha \\eta ( t ) ( 1 - \\frac { 1 } { 2 } \\alpha \\eta ( t ) ) \\| \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) \\| _ { 2 } ) \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "e61b2ec6f0513fe0fb54efdea65daa4166a3ac97997812cbba20047c9df69747.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 111, + 379, + 514, + 404.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 111, + 404.6666666666667, + 514, + 430.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 111, + 430.33333333333337, + 514, + 456.00000000000006 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 373, + 475 + ], + "lines": [ + { + "bbox": [ + 104, + 459, + 374, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 172, + 477 + ], + "score": 1.0, + "content": "By Lemma E.7,", + "type": "text" + }, + { + "bbox": [ + 172, + 461, + 323, + 475 + ], + "score": 0.9, + "content": "\\left\\| \\nabla ^ { 2 } \\mathcal { L } ( t + \\xi ) \\right\\| _ { 2 } \\leq 2 C _ { \\eta } \\cdot \\kappa ( \\mathcal { L } ( t + \\xi ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 459, + 374, + 477 + ], + "score": 1.0, + "content": ", so we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 459, + 374, + 477 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 480, + 438, + 497 + ], + "lines": [ + { + "bbox": [ + 174, + 480, + 438, + 497 + ], + "spans": [ + { + "bbox": [ + 174, + 480, + 438, + 497 + ], + "score": 0.82, + "content": "\\mathcal { L } ( t + \\alpha ) \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\alpha C _ { \\eta } \\eta ( t ) \\kappa ( \\mathcal { L } ( t + \\xi ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "83f4dc1988319f0e135ea298c68c1db54722cebfd2ccc210bc1930640b4d19ac.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 174, + 480, + 438, + 497 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 506, + 504, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 238, + 521 + ], + "score": 1.0, + "content": "Now we only need to show that", + "type": "text" + }, + { + "bbox": [ + 239, + 507, + 309, + 519 + ], + "score": 0.93, + "content": "\\mathscr { L } ( t + \\alpha ) < \\mathscr { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 505, + 339, + 521 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 339, + 507, + 384, + 519 + ], + "score": 0.92, + "content": "\\alpha \\in ( 0 , A ]", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 505, + 505, + 521 + ], + "score": 1.0, + "content": ". Assuming this, we can have", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 518, + 343, + 531 + ], + "spans": [ + { + "bbox": [ + 107, + 518, + 201, + 531 + ], + "score": 0.91, + "content": "\\kappa ( \\mathcal { L } ( t + \\xi ) ) \\leq \\kappa ( \\mathcal { L } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 518, + 296, + 531 + ], + "score": 1.0, + "content": "by the monotonicity of", + "type": "text" + }, + { + "bbox": [ + 297, + 522, + 303, + 528 + ], + "score": 0.8, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 518, + 343, + 531 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 505, + 505, + 531 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 535, + 428, + 570 + ], + "lines": [ + { + "bbox": [ + 183, + 535, + 428, + 570 + ], + "spans": [ + { + "bbox": [ + 183, + 535, + 428, + 570 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\mathcal { L } ( t + \\alpha ) \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\alpha C _ { \\eta } \\eta ( t ) \\kappa ( \\mathcal { L } ( t ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } } \\\\ & { \\qquad \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\mu ( \\mathcal { L } ( t ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "90fb9d163295e531359ad0271e4b8aee6b96a0f7b1345e45296eb29ca8d4eaa5.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 183, + 535, + 428, + 546.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 183, + 546.6666666666666, + 428, + 558.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 183, + 558.3333333333333, + 428, + 569.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 506, + 627 + ], + "lines": [ + { + "bbox": [ + 104, + 576, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 189, + 592 + ], + "score": 1.0, + "content": "Now we show that", + "type": "text" + }, + { + "bbox": [ + 189, + 579, + 266, + 590 + ], + "score": 0.9, + "content": "\\mathcal { L } ( t + \\alpha ) ~ < ~ \\mathcal { L } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 576, + 298, + 592 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 298, + 579, + 347, + 591 + ], + "score": 0.92, + "content": "\\alpha \\in \\mathsf { \\Gamma } ( 0 , A ]", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 576, + 475, + 592 + ], + "score": 1.0, + "content": ". Assume to the contrary that", + "type": "text" + }, + { + "bbox": [ + 476, + 580, + 505, + 590 + ], + "score": 0.83, + "content": "\\alpha _ { 0 } : =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 262, + 602 + ], + "score": 0.85, + "content": "\\operatorname* { i n f } \\{ \\alpha ^ { \\prime } \\in ( 0 , A ] : { \\mathcal { L } } ( t + \\alpha ^ { \\prime } ) \\geq { \\mathcal { L } } ( t ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 589, + 303, + 603 + ], + "score": 1.0, + "content": "exists. If", + "type": "text" + }, + { + "bbox": [ + 303, + 591, + 335, + 601 + ], + "score": 0.89, + "content": "\\alpha _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 589, + 352, + 603 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 353, + 590, + 481, + 602 + ], + "score": 0.89, + "content": "p : [ 0 , 1 ] \\to \\mathbb { R } , \\alpha \\mapsto \\mathcal { L } ( t + \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 589, + 506, + 603 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 599, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 243, + 617 + ], + "score": 0.88, + "content": "\\begin{array} { r } { p ^ { \\prime } ( 0 ) = \\operatorname* { l i m } _ { \\alpha \\downarrow 0 } \\frac { \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) } { \\alpha } \\geq 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 599, + 507, + 617 + ], + "score": 1.0, + "content": "L(t+α)−L(t) ≥ 0, but it contradicts to p0(0) = −η(t) k∇L(t)k22 < 0. If α0 > 0,", + "type": "text" + }, + { + "bbox": [ + 220, + 617, + 227, + 624 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 602, + 263, + 627 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 263, + 614, + 358, + 627 + ], + "score": 0.92, + "content": "\\kappa ( \\mathcal { L } ( t + \\xi ) ) < \\kappa ( \\dot { \\mathcal { L } } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 615, + 432, + 627 + ], + "score": 0.91, + "content": "\\xi \\in ( 0 , \\alpha _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 602, + 469, + 627 + ], + "score": 1.0, + "content": "thus", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 576, + 507, + 627 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 631, + 425, + 648 + ], + "lines": [ + { + "bbox": [ + 185, + 631, + 425, + 648 + ], + "spans": [ + { + "bbox": [ + 185, + 631, + 425, + 648 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\mathcal { L } ( t + \\alpha ) \\leq \\mathcal { L } ( t ) - \\alpha \\eta ( t ) \\left( 1 - \\mu ( \\mathcal { L } ( t ) ) \\right) \\left. \\nabla \\mathcal { L } ( t ) \\right. _ { 2 } ^ { 2 } < \\mathcal { L } ( t ) , } \\end{array}", + "type": "interline_equation", + "image_path": "78a72fa74328b662e81fda8ae4cfee9030f3e404a05693a1d60124b6373fcde4.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 185, + 631, + 425, + 648 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 652, + 231, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 651, + 231, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 231, + 664 + ], + "score": 1.0, + "content": "which leads to a contradiction.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 651, + 231, + 664 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 675, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 102, + 669, + 504, + 695 + ], + "spans": [ + { + "bbox": [ + 102, + 669, + 147, + 695 + ], + "score": 1.0, + "content": "Proof for", + "type": "text" + }, + { + "bbox": [ + 147, + 678, + 165, + 689 + ], + "score": 0.42, + "content": "( \\mathbf { P 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 669, + 219, + 695 + ], + "score": 1.0, + "content": ". We define", + "type": "text" + }, + { + "bbox": [ + 219, + 676, + 336, + 690 + ], + "score": 0.92, + "content": "\\pmb { v } ( t ) : = \\hat { \\pmb { \\theta } } ( t ) \\hat { \\pmb { \\theta } } ( t ) ^ { \\top } ( - \\nabla \\mathcal { L } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 669, + 353, + 695 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 673, + 504, + 694 + ], + "score": 0.9, + "content": "\\pmb { u } ( t ) : = \\left( \\pmb { I } - \\hat { \\pmb { \\theta } } ( t ) \\hat { \\pmb { \\theta } } ( t ) ^ { \\top } \\right) ( - \\nabla \\mathcal { L } ( t ) )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 691, + 361, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 303, + 705 + ], + "score": 1.0, + "content": "similarly as in the analysis for gradient flow. For", + "type": "text" + }, + { + "bbox": [ + 303, + 693, + 321, + 705 + ], + "score": 0.91, + "content": "{ \\pmb v } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 691, + 361, + 705 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 102, + 669, + 504, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 709, + 399, + 736 + ], + "lines": [ + { + "bbox": [ + 210, + 709, + 399, + 736 + ], + "spans": [ + { + "bbox": [ + 210, + 709, + 399, + 736 + ], + "score": 0.94, + "content": "\\| \\pmb { v } ( t ) \\| _ { 2 } = \\frac { 1 } { \\rho ( t ) } \\left. \\pmb { \\theta } ( t ) , - \\pmb { \\mathcal { L } } ( t ) \\right. = \\frac { 1 } { \\rho ( t ) } \\cdot L \\nu ( t ) .", + "type": "interline_equation", + "image_path": "b92fc95da92cc467c3f2cd46a13526f3347b1c4bf9e75014ac3007e2b217ccea.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 210, + 709, + 399, + 736 + ], + "spans": [], + "index": 33 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 79, + 385, + 96 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 387, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 156, + 96 + ], + "score": 1.0, + "content": "Decompose", + "type": "text" + }, + { + "bbox": [ + 156, + 80, + 288, + 95 + ], + "score": 0.94, + "content": "\\left\\| \\nabla \\mathcal { L } ( t ) \\right\\| _ { 2 } ^ { 2 } = \\left\\| \\pmb { v } ( t ) \\right\\| _ { 2 } ^ { 2 } + \\left\\| \\pmb { u } ( t ) \\right\\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 79, + 387, + 96 + ], + "score": 1.0, + "content": ". Then by (P3), we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 99, + 450, + 128 + ], + "lines": [ + { + "bbox": [ + 160, + 99, + 450, + 128 + ], + "spans": [ + { + "bbox": [ + 160, + 99, + 450, + 128 + ], + "score": 0.94, + "content": "\\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) \\leq - \\alpha \\eta ( t ) ( 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\left( \\frac { 1 } { \\rho ( t ) ^ { 2 } } \\cdot L ^ { 2 } \\nu ( t ) ^ { 2 } + \\| \\pmb { u } \\| _ { 2 } ^ { 2 } \\right) .", + "type": "interline_equation", + "image_path": "7c9aaf127b18144c4b24a4eec566428ff4e5f9a23c1464022b4b997b660a7913.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 160, + 99, + 450, + 128 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 137, + 325, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 136, + 327, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 158, + 153 + ], + "score": 1.0, + "content": "Multiplying", + "type": "text" + }, + { + "bbox": [ + 155, + 136, + 327, + 153 + ], + "score": 1.0, + "content": "1+λ(L(t))µ(L(t))/L on both sides, we have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 160, + 504, + 193 + ], + "lines": [ + { + "bbox": [ + 111, + 160, + 504, + 193 + ], + "spans": [ + { + "bbox": [ + 111, + 160, + 504, + 193 + ], + "score": 0.89, + "content": "\\frac { \\iota + \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) / L } { ( 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\nu ( t ) } ( \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) ) \\leq - \\alpha \\eta ( t ) \\left( 1 + \\frac { \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) } { L } \\right) \\left( \\frac { L ^ { 2 } \\nu ( t ) } { \\rho ( t ) ^ { 2 } } + \\frac { \\| u \\| _ { 2 } ^ { 2 } } { \\nu ( t ) } \\right)", + "type": "interline_equation", + "image_path": "73a889ce67684e9f3bf0117eb1719cac86d2189e2b6f7bdaa87052072288c303.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 160, + 504, + 171.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 111, + 171.0, + 504, + 182.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 182.0, + 504, + 193.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 243, + 212 + ], + "score": 1.0, + "content": "By Corollary E.6, we can bound", + "type": "text" + }, + { + "bbox": [ + 243, + 199, + 261, + 210 + ], + "score": 0.91, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 198, + 276, + 212 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 276, + 198, + 366, + 211 + ], + "score": 0.91, + "content": "\\nu ( t ) \\geq \\mathcal { L } ( t ) / \\lambda ( \\mathcal { L } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 198, + 505, + 212 + ], + "score": 1.0, + "content": ". By (P2), we have the inequality", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 207, + 387, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 300, + 230 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) } { L } \\right) \\ge \\frac { \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } } { 2 L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 207, + 387, + 229 + ], + "score": 1.0, + "content": ". So we further have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 234, + 506, + 262 + ], + "lines": [ + { + "bbox": [ + 111, + 234, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 111, + 234, + 506, + 262 + ], + "score": 0.89, + "content": "\\frac { 1 + \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) / L } { 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\mathcal { L } ( t ) / \\lambda ( \\mathcal { L } ( t ) ) } ( \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) ) \\leq - \\frac { 1 } { \\rho ( t ) ^ { 2 } } ( \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } ) \\left( \\frac { L } { 2 } + \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\| u \\| _ { 2 } ^ { 2 } \\right)", + "type": "interline_equation", + "image_path": "e87b8b099422fa54774c2f4f7bd6994ec44dd74a88bb7dd95d2c9cc7e7a57135.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 111, + 234, + 506, + 262 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 103, + 267, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 103, + 267, + 421, + 286 + ], + "score": 1.0, + "content": "From the definition φ, it is easy to see that −φ0(L(t)) ≥ 1+λ(L(t))µ(L(t))/L(1−µ(L(t)))L(t)/λ(L(t)) .", + "type": "text" + }, + { + "bbox": [ + 418, + 268, + 437, + 284 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 437, + 269, + 502, + 282 + ], + "score": 0.75, + "content": "\\psi ( x ) = - \\log x .", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 268, + 506, + 284 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 281, + 312, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 126, + 298 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 126, + 283, + 178, + 298 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\psi ^ { \\prime } ( x ) = - \\frac { 1 } { x } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 281, + 312, + 298 + ], + "score": 1.0, + "content": ". Combining these together gives", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 302, + 485, + 330 + ], + "lines": [ + { + "bbox": [ + 126, + 302, + 485, + 330 + ], + "spans": [ + { + "bbox": [ + 126, + 302, + 485, + 330 + ], + "score": 0.92, + "content": "\\phi ^ { \\prime } ( \\mathcal { L } ( t ) ) ( \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) ) + \\psi ^ { \\prime } ( \\rho ( t ) ^ { 2 } ) ( \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } ) \\left( \\frac { L } { 2 } + \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\left. \\boldsymbol { u } \\right. _ { 2 } ^ { 2 } \\right) \\geq 0", + "type": "interline_equation", + "image_path": "a22eef9844716928d49ae2e19d2d41a45d1003cbd5cd08d7e4ebf219c1f34d4c.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 126, + 302, + 485, + 330 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 265, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 266, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 194, + 348 + ], + "score": 1.0, + "content": "Then by convexity of", + "type": "text" + }, + { + "bbox": [ + 194, + 335, + 201, + 346 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 333, + 219, + 348 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 335, + 227, + 346 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 333, + 266, + 348 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 351, + 482, + 379 + ], + "lines": [ + { + "bbox": [ + 129, + 351, + 482, + 379 + ], + "spans": [ + { + "bbox": [ + 129, + 351, + 482, + 379 + ], + "score": 0.91, + "content": "\\left( \\phi ( \\mathcal { L } ( t + \\alpha ) ) - \\phi ( \\mathcal { L } ( t ) ) \\right) + \\left( \\log \\frac { 1 } { \\rho ( t + \\alpha ) ^ { 2 } } - \\log \\frac { 1 } { \\rho ( t ) ^ { 2 } } \\right) \\left( \\frac { L } { 2 } + \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\left\\| \\boldsymbol { u } \\right\\| _ { 2 } ^ { 2 } \\right) \\ge 0", + "type": "interline_equation", + "image_path": "7b9c33f7365b54d05d135e8e630b77fe5470f1fc6f203b3993633338da462ee0.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 129, + 351, + 482, + 379 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 298, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 299, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 190, + 396 + ], + "score": 1.0, + "content": "And by definition of", + "type": "text" + }, + { + "bbox": [ + 190, + 384, + 197, + 395 + ], + "score": 0.86, + "content": "\\hat { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 383, + 299, + 396 + ], + "score": 1.0, + "content": ", this can be re-written as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 399, + 479, + 486 + ], + "lines": [ + { + "bbox": [ + 132, + 399, + 479, + 486 + ], + "spans": [ + { + "bbox": [ + 132, + 399, + 479, + 486 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\log \\widehat { \\gamma } ( t + \\alpha ) - \\log \\widehat { \\gamma } ( t ) = \\left( \\phi ( \\mathcal { L } ( t + \\alpha ) ) - \\phi ( \\mathcal { L } ( t ) ) \\right) + L \\left( \\log \\displaystyle \\frac { 1 } { \\rho ( t + \\alpha ) } - \\log \\displaystyle \\frac { 1 } { \\rho ( t ) } \\right) } \\\\ { \\qquad \\geq - \\left( \\log \\displaystyle \\frac { 1 } { \\rho ( t + \\alpha ) ^ { 2 } } - \\log \\displaystyle \\frac { 1 } { \\rho ( t ) ^ { 2 } } \\right) \\displaystyle \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\left. u \\right. _ { 2 } ^ { 2 } } \\\\ { \\qquad = \\displaystyle \\frac { \\rho ( t ) ^ { 2 } } { L \\nu ( t ) ^ { 2 } } \\left. u \\right. _ { 2 } ^ { 2 } \\cdot \\log \\displaystyle \\frac { \\rho ( t + \\alpha ) } { \\rho ( t ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "5d4ab5a58ba9a8fa3b99b4af7a49c29d8d7e017e4cbff4001677d3a3bdab56dd.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 132, + 399, + 479, + 428.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 132, + 428.0, + 479, + 457.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 132, + 457.0, + 479, + 486.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 495, + 504, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 148, + 510 + ], + "score": 1.0, + "content": "Proof for", + "type": "text" + }, + { + "bbox": [ + 148, + 496, + 166, + 507 + ], + "score": 0.31, + "content": "( \\mathbf { P 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 493, + 213, + 510 + ], + "score": 1.0, + "content": ". By (P4),", + "type": "text" + }, + { + "bbox": [ + 214, + 495, + 360, + 508 + ], + "score": 0.9, + "content": "\\log \\hat { \\gamma } ( t + \\alpha ) \\geq \\log \\hat { \\gamma } ( t ) \\geq \\log \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 493, + 403, + 510 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 404, + 495, + 474, + 508 + ], + "score": 0.94, + "content": "\\phi ( x ) \\geq \\log \\log \\frac { 1 } { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 493, + 505, + 510 + ], + "score": 1.0, + "content": ". So we", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 516, + 367, + 530 + ], + "lines": [ + { + "bbox": [ + 243, + 516, + 367, + 530 + ], + "spans": [ + { + "bbox": [ + 243, + 516, + 367, + 530 + ], + "score": 0.87, + "content": "\\tilde { \\gamma } ( t + \\alpha ) > \\hat { \\gamma } ( t + \\alpha ) \\geq \\hat { \\gamma } ( t _ { 0 } ) ,", + "type": "interline_equation", + "image_path": "0e4a0d183ce8236e8e7a64f10fbcc37b1193e05ea26e02f7e53e2a0c859d3311.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 243, + 516, + 367, + 530 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 216, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 217, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 217, + 545 + ], + "score": 1.0, + "content": "which completes the proof.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 459, + 569 + ], + "score": 1.0, + "content": "For showing the third proposition in Theorem E.2, we use (P1) to give a lower bound for", + "type": "text" + }, + { + "bbox": [ + 460, + 556, + 501, + 568 + ], + "score": 0.9, + "content": "\\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 554, + 505, + 569 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 412, + 580 + ], + "score": 1.0, + "content": "and use (P3) to show the speed of loss decreasing. Then it can be seen that", + "type": "text" + }, + { + "bbox": [ + 413, + 567, + 454, + 579 + ], + "score": 0.93, + "content": "\\mathcal { L } ( t ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 567, + 473, + 580 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 473, + 568, + 505, + 579 + ], + "score": 0.9, + "content": "\\rho ( t ) ", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 578, + 304, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 126, + 589 + ], + "score": 0.83, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 578, + 250, + 591 + ], + "score": 1.0, + "content": ". By Lemma E.1, we then have", + "type": "text" + }, + { + "bbox": [ + 250, + 578, + 301, + 590 + ], + "score": 0.91, + "content": "| \\bar { \\gamma } - \\hat { \\gamma } | \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 578, + 304, + 591 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 592, + 390, + 609 + ], + "lines": [ + { + "bbox": [ + 101, + 588, + 387, + 613 + ], + "spans": [ + { + "bbox": [ + 101, + 588, + 183, + 613 + ], + "score": 1.0, + "content": "Lemma E.10. Let", + "type": "text" + }, + { + "bbox": [ + 184, + 593, + 307, + 609 + ], + "score": 0.92, + "content": "\\begin{array} { r } { E _ { 0 } : = \\mathcal { L } ( t _ { 0 } ) ^ { 2 } ( \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ) ^ { 2 - 2 / L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 588, + 361, + 613 + ], + "score": 1.0, + "content": ". Then for all", + "type": "text" + }, + { + "bbox": [ + 361, + 595, + 387, + 606 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 614, + 490, + 648 + ], + "lines": [ + { + "bbox": [ + 120, + 614, + 490, + 648 + ], + "spans": [ + { + "bbox": [ + 120, + 614, + 490, + 648 + ], + "score": 0.94, + "content": "E ( \\mathcal { L } ( t ) ) \\geq \\frac { 1 } { 2 } L ^ { 2 } \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) ~ f o r ~ E ( x ) : = \\int _ { x } ^ { \\mathcal { L } ( t _ { 0 } ) } \\frac { 1 } { \\operatorname* { m i n } \\{ u ^ { 2 } ( \\log \\frac { 1 } { u } ) ^ { 2 - 2 / L } , E _ { 0 } \\} } d u .", + "type": "interline_equation", + "image_path": "f645396e0220c7fed40bcf10891ebe26c16f7ea1e08868316f7f4d3af3b0e824.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 120, + 614, + 490, + 625.3333333333334 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 120, + 625.3333333333334, + 490, + 636.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 120, + 636.6666666666667, + 490, + 648.0000000000001 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 305, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 652, + 306, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 150, + 668 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 150, + 654, + 190, + 666 + ], + "score": 0.93, + "content": "\\mathcal { L } ( t ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 652, + 208, + 668 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 209, + 653, + 259, + 666 + ], + "score": 0.89, + "content": "\\rho ( t ) \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 652, + 272, + 668 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 272, + 655, + 302, + 664 + ], + "score": 0.86, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 652, + 306, + 668 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 199, + 692 + ], + "score": 1.0, + "content": "Proof. For any integer", + "type": "text" + }, + { + "bbox": [ + 199, + 678, + 225, + 689 + ], + "score": 0.86, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 676, + 230, + 692 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 230, + 676, + 282, + 691 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mu ( \\mathcal { L } ( t ) ) \\leq \\frac { 1 } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 676, + 301, + 692 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 301, + 677, + 388, + 690 + ], + "score": 0.93, + "content": "\\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } \\geq \\| \\pmb { v } ( t ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 676, + 505, + 692 + ], + "score": 1.0, + "content": ". Combining these with (P3),", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 687, + 143, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 143, + 702 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 703, + 420, + 730 + ], + "lines": [ + { + "bbox": [ + 190, + 703, + 420, + 730 + ], + "spans": [ + { + "bbox": [ + 190, + 703, + 420, + 730 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\leq - \\frac { 1 } { 2 } \\eta ( t ) \\left. \\pmb { v } ( t ) \\right. _ { 2 } ^ { 2 } = - \\frac { 1 } { 2 } \\eta ( t ) \\frac { L ^ { 2 } \\nu ( t ) ^ { 2 } } { \\rho ( t ) ^ { 2 } } .", + "type": "interline_equation", + "image_path": "9eb663b924ad5ab9003e51e9429420374a61fc6e3b2d8f00ae0a6717681e52c0.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 190, + 703, + 420, + 730 + ], + "spans": [], + "index": 31 + } + ] + } + ], + "page_idx": 34, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "35", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 532, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 495, + 534, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 495, + 534, + 505, + 544 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 79, + 385, + 96 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 387, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 156, + 96 + ], + "score": 1.0, + "content": "Decompose", + "type": "text" + }, + { + "bbox": [ + 156, + 80, + 288, + 95 + ], + "score": 0.94, + "content": "\\left\\| \\nabla \\mathcal { L } ( t ) \\right\\| _ { 2 } ^ { 2 } = \\left\\| \\pmb { v } ( t ) \\right\\| _ { 2 } ^ { 2 } + \\left\\| \\pmb { u } ( t ) \\right\\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 79, + 387, + 96 + ], + "score": 1.0, + "content": ". Then by (P3), we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 104, + 79, + 387, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 99, + 450, + 128 + ], + "lines": [ + { + "bbox": [ + 160, + 99, + 450, + 128 + ], + "spans": [ + { + "bbox": [ + 160, + 99, + 450, + 128 + ], + "score": 0.94, + "content": "\\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) \\leq - \\alpha \\eta ( t ) ( 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\left( \\frac { 1 } { \\rho ( t ) ^ { 2 } } \\cdot L ^ { 2 } \\nu ( t ) ^ { 2 } + \\| \\pmb { u } \\| _ { 2 } ^ { 2 } \\right) .", + "type": "interline_equation", + "image_path": "7c9aaf127b18144c4b24a4eec566428ff4e5f9a23c1464022b4b997b660a7913.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 160, + 99, + 450, + 128 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 137, + 325, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 136, + 327, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 158, + 153 + ], + "score": 1.0, + "content": "Multiplying", + "type": "text" + }, + { + "bbox": [ + 155, + 136, + 327, + 153 + ], + "score": 1.0, + "content": "1+λ(L(t))µ(L(t))/L on both sides, we have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 136, + 327, + 153 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 160, + 504, + 193 + ], + "lines": [ + { + "bbox": [ + 111, + 160, + 504, + 193 + ], + "spans": [ + { + "bbox": [ + 111, + 160, + 504, + 193 + ], + "score": 0.89, + "content": "\\frac { \\iota + \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) / L } { ( 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\nu ( t ) } ( \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) ) \\leq - \\alpha \\eta ( t ) \\left( 1 + \\frac { \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) } { L } \\right) \\left( \\frac { L ^ { 2 } \\nu ( t ) } { \\rho ( t ) ^ { 2 } } + \\frac { \\| u \\| _ { 2 } ^ { 2 } } { \\nu ( t ) } \\right)", + "type": "interline_equation", + "image_path": "73a889ce67684e9f3bf0117eb1719cac86d2189e2b6f7bdaa87052072288c303.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 160, + 504, + 171.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 111, + 171.0, + 504, + 182.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 182.0, + 504, + 193.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 243, + 212 + ], + "score": 1.0, + "content": "By Corollary E.6, we can bound", + "type": "text" + }, + { + "bbox": [ + 243, + 199, + 261, + 210 + ], + "score": 0.91, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 198, + 276, + 212 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 276, + 198, + 366, + 211 + ], + "score": 0.91, + "content": "\\nu ( t ) \\geq \\mathcal { L } ( t ) / \\lambda ( \\mathcal { L } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 198, + 505, + 212 + ], + "score": 1.0, + "content": ". By (P2), we have the inequality", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 207, + 387, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 300, + 230 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\alpha \\eta ( t ) \\nu ( t ) \\left( 1 + \\frac { \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) } { L } \\right) \\ge \\frac { \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } } { 2 L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 207, + 387, + 229 + ], + "score": 1.0, + "content": ". So we further have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 198, + 505, + 230 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 234, + 506, + 262 + ], + "lines": [ + { + "bbox": [ + 111, + 234, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 111, + 234, + 506, + 262 + ], + "score": 0.89, + "content": "\\frac { 1 + \\lambda ( \\mathcal { L } ( t ) ) \\mu ( \\mathcal { L } ( t ) ) / L } { 1 - \\mu ( \\mathcal { L } ( t ) ) ) \\mathcal { L } ( t ) / \\lambda ( \\mathcal { L } ( t ) ) } ( \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) ) \\leq - \\frac { 1 } { \\rho ( t ) ^ { 2 } } ( \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } ) \\left( \\frac { L } { 2 } + \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\| u \\| _ { 2 } ^ { 2 } \\right)", + "type": "interline_equation", + "image_path": "e87b8b099422fa54774c2f4f7bd6994ec44dd74a88bb7dd95d2c9cc7e7a57135.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 111, + 234, + 506, + 262 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 103, + 267, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 103, + 267, + 421, + 286 + ], + "score": 1.0, + "content": "From the definition φ, it is easy to see that −φ0(L(t)) ≥ 1+λ(L(t))µ(L(t))/L(1−µ(L(t)))L(t)/λ(L(t)) .", + "type": "text" + }, + { + "bbox": [ + 418, + 268, + 437, + 284 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 437, + 269, + 502, + 282 + ], + "score": 0.75, + "content": "\\psi ( x ) = - \\log x .", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 268, + 506, + 284 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 281, + 312, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 126, + 298 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 126, + 283, + 178, + 298 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\psi ^ { \\prime } ( x ) = - \\frac { 1 } { x } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 281, + 312, + 298 + ], + "score": 1.0, + "content": ". Combining these together gives", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 103, + 267, + 506, + 298 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 302, + 485, + 330 + ], + "lines": [ + { + "bbox": [ + 126, + 302, + 485, + 330 + ], + "spans": [ + { + "bbox": [ + 126, + 302, + 485, + 330 + ], + "score": 0.92, + "content": "\\phi ^ { \\prime } ( \\mathcal { L } ( t ) ) ( \\mathcal { L } ( t + \\alpha ) - \\mathcal { L } ( t ) ) + \\psi ^ { \\prime } ( \\rho ( t ) ^ { 2 } ) ( \\rho ( t + \\alpha ) ^ { 2 } - \\rho ( t ) ^ { 2 } ) \\left( \\frac { L } { 2 } + \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\left. \\boldsymbol { u } \\right. _ { 2 } ^ { 2 } \\right) \\geq 0", + "type": "interline_equation", + "image_path": "a22eef9844716928d49ae2e19d2d41a45d1003cbd5cd08d7e4ebf219c1f34d4c.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 126, + 302, + 485, + 330 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 265, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 266, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 194, + 348 + ], + "score": 1.0, + "content": "Then by convexity of", + "type": "text" + }, + { + "bbox": [ + 194, + 335, + 201, + 346 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 333, + 219, + 348 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 335, + 227, + 346 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 333, + 266, + 348 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 333, + 266, + 348 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 351, + 482, + 379 + ], + "lines": [ + { + "bbox": [ + 129, + 351, + 482, + 379 + ], + "spans": [ + { + "bbox": [ + 129, + 351, + 482, + 379 + ], + "score": 0.91, + "content": "\\left( \\phi ( \\mathcal { L } ( t + \\alpha ) ) - \\phi ( \\mathcal { L } ( t ) ) \\right) + \\left( \\log \\frac { 1 } { \\rho ( t + \\alpha ) ^ { 2 } } - \\log \\frac { 1 } { \\rho ( t ) ^ { 2 } } \\right) \\left( \\frac { L } { 2 } + \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\left\\| \\boldsymbol { u } \\right\\| _ { 2 } ^ { 2 } \\right) \\ge 0", + "type": "interline_equation", + "image_path": "7b9c33f7365b54d05d135e8e630b77fe5470f1fc6f203b3993633338da462ee0.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 129, + 351, + 482, + 379 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 298, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 299, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 190, + 396 + ], + "score": 1.0, + "content": "And by definition of", + "type": "text" + }, + { + "bbox": [ + 190, + 384, + 197, + 395 + ], + "score": 0.86, + "content": "\\hat { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 383, + 299, + 396 + ], + "score": 1.0, + "content": ", this can be re-written as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 383, + 299, + 396 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 399, + 479, + 486 + ], + "lines": [ + { + "bbox": [ + 132, + 399, + 479, + 486 + ], + "spans": [ + { + "bbox": [ + 132, + 399, + 479, + 486 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\log \\widehat { \\gamma } ( t + \\alpha ) - \\log \\widehat { \\gamma } ( t ) = \\left( \\phi ( \\mathcal { L } ( t + \\alpha ) ) - \\phi ( \\mathcal { L } ( t ) ) \\right) + L \\left( \\log \\displaystyle \\frac { 1 } { \\rho ( t + \\alpha ) } - \\log \\displaystyle \\frac { 1 } { \\rho ( t ) } \\right) } \\\\ { \\qquad \\geq - \\left( \\log \\displaystyle \\frac { 1 } { \\rho ( t + \\alpha ) ^ { 2 } } - \\log \\displaystyle \\frac { 1 } { \\rho ( t ) ^ { 2 } } \\right) \\displaystyle \\frac { \\rho ( t ) ^ { 2 } } { 2 L \\nu ( t ) ^ { 2 } } \\left. u \\right. _ { 2 } ^ { 2 } } \\\\ { \\qquad = \\displaystyle \\frac { \\rho ( t ) ^ { 2 } } { L \\nu ( t ) ^ { 2 } } \\left. u \\right. _ { 2 } ^ { 2 } \\cdot \\log \\displaystyle \\frac { \\rho ( t + \\alpha ) } { \\rho ( t ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "5d4ab5a58ba9a8fa3b99b4af7a49c29d8d7e017e4cbff4001677d3a3bdab56dd.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 132, + 399, + 479, + 428.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 132, + 428.0, + 479, + 457.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 132, + 457.0, + 479, + 486.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 495, + 504, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 148, + 510 + ], + "score": 1.0, + "content": "Proof for", + "type": "text" + }, + { + "bbox": [ + 148, + 496, + 166, + 507 + ], + "score": 0.31, + "content": "( \\mathbf { P 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 493, + 213, + 510 + ], + "score": 1.0, + "content": ". By (P4),", + "type": "text" + }, + { + "bbox": [ + 214, + 495, + 360, + 508 + ], + "score": 0.9, + "content": "\\log \\hat { \\gamma } ( t + \\alpha ) \\geq \\log \\hat { \\gamma } ( t ) \\geq \\log \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 493, + 403, + 510 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 404, + 495, + 474, + 508 + ], + "score": 0.94, + "content": "\\phi ( x ) \\geq \\log \\log \\frac { 1 } { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 493, + 505, + 510 + ], + "score": 1.0, + "content": ". So we", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 493, + 505, + 510 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 516, + 367, + 530 + ], + "lines": [ + { + "bbox": [ + 243, + 516, + 367, + 530 + ], + "spans": [ + { + "bbox": [ + 243, + 516, + 367, + 530 + ], + "score": 0.87, + "content": "\\tilde { \\gamma } ( t + \\alpha ) > \\hat { \\gamma } ( t + \\alpha ) \\geq \\hat { \\gamma } ( t _ { 0 } ) ,", + "type": "interline_equation", + "image_path": "0e4a0d183ce8236e8e7a64f10fbcc37b1193e05ea26e02f7e53e2a0c859d3311.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 243, + 516, + 367, + 530 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 216, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 217, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 217, + 545 + ], + "score": 1.0, + "content": "which completes the proof.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 531, + 217, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 459, + 569 + ], + "score": 1.0, + "content": "For showing the third proposition in Theorem E.2, we use (P1) to give a lower bound for", + "type": "text" + }, + { + "bbox": [ + 460, + 556, + 501, + 568 + ], + "score": 0.9, + "content": "\\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 554, + 505, + 569 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 412, + 580 + ], + "score": 1.0, + "content": "and use (P3) to show the speed of loss decreasing. Then it can be seen that", + "type": "text" + }, + { + "bbox": [ + 413, + 567, + 454, + 579 + ], + "score": 0.93, + "content": "\\mathcal { L } ( t ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 567, + 473, + 580 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 473, + 568, + 505, + 579 + ], + "score": 0.9, + "content": "\\rho ( t ) ", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 578, + 304, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 126, + 589 + ], + "score": 0.83, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 578, + 250, + 591 + ], + "score": 1.0, + "content": ". By Lemma E.1, we then have", + "type": "text" + }, + { + "bbox": [ + 250, + 578, + 301, + 590 + ], + "score": 0.91, + "content": "| \\bar { \\gamma } - \\hat { \\gamma } | \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 578, + 304, + 591 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 554, + 505, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 592, + 390, + 609 + ], + "lines": [ + { + "bbox": [ + 101, + 588, + 387, + 613 + ], + "spans": [ + { + "bbox": [ + 101, + 588, + 183, + 613 + ], + "score": 1.0, + "content": "Lemma E.10. Let", + "type": "text" + }, + { + "bbox": [ + 184, + 593, + 307, + 609 + ], + "score": 0.92, + "content": "\\begin{array} { r } { E _ { 0 } : = \\mathcal { L } ( t _ { 0 } ) ^ { 2 } ( \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ) ^ { 2 - 2 / L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 588, + 361, + 613 + ], + "score": 1.0, + "content": ". Then for all", + "type": "text" + }, + { + "bbox": [ + 361, + 595, + 387, + 606 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 101, + 588, + 387, + 613 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 614, + 490, + 648 + ], + "lines": [ + { + "bbox": [ + 120, + 614, + 490, + 648 + ], + "spans": [ + { + "bbox": [ + 120, + 614, + 490, + 648 + ], + "score": 0.94, + "content": "E ( \\mathcal { L } ( t ) ) \\geq \\frac { 1 } { 2 } L ^ { 2 } \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) ~ f o r ~ E ( x ) : = \\int _ { x } ^ { \\mathcal { L } ( t _ { 0 } ) } \\frac { 1 } { \\operatorname* { m i n } \\{ u ^ { 2 } ( \\log \\frac { 1 } { u } ) ^ { 2 - 2 / L } , E _ { 0 } \\} } d u .", + "type": "interline_equation", + "image_path": "f645396e0220c7fed40bcf10891ebe26c16f7ea1e08868316f7f4d3af3b0e824.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 120, + 614, + 490, + 625.3333333333334 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 120, + 625.3333333333334, + 490, + 636.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 120, + 636.6666666666667, + 490, + 648.0000000000001 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 305, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 652, + 306, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 150, + 668 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 150, + 654, + 190, + 666 + ], + "score": 0.93, + "content": "\\mathcal { L } ( t ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 652, + 208, + 668 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 209, + 653, + 259, + 666 + ], + "score": 0.89, + "content": "\\rho ( t ) \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 652, + 272, + 668 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 272, + 655, + 302, + 664 + ], + "score": 0.86, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 652, + 306, + 668 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 652, + 306, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 199, + 692 + ], + "score": 1.0, + "content": "Proof. For any integer", + "type": "text" + }, + { + "bbox": [ + 199, + 678, + 225, + 689 + ], + "score": 0.86, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 676, + 230, + 692 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 230, + 676, + 282, + 691 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mu ( \\mathcal { L } ( t ) ) \\leq \\frac { 1 } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 676, + 301, + 692 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 301, + 677, + 388, + 690 + ], + "score": 0.93, + "content": "\\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } \\geq \\| \\pmb { v } ( t ) \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 676, + 505, + 692 + ], + "score": 1.0, + "content": ". Combining these with (P3),", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 687, + 143, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 143, + 702 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 676, + 505, + 702 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 703, + 420, + 730 + ], + "lines": [ + { + "bbox": [ + 190, + 703, + 420, + 730 + ], + "spans": [ + { + "bbox": [ + 190, + 703, + 420, + 730 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\leq - \\frac { 1 } { 2 } \\eta ( t ) \\left. \\pmb { v } ( t ) \\right. _ { 2 } ^ { 2 } = - \\frac { 1 } { 2 } \\eta ( t ) \\frac { L ^ { 2 } \\nu ( t ) ^ { 2 } } { \\rho ( t ) ^ { 2 } } .", + "type": "interline_equation", + "image_path": "9eb663b924ad5ab9003e51e9429420374a61fc6e3b2d8f00ae0a6717681e52c0.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 190, + 703, + 420, + 730 + ], + "spans": [], + "index": 31 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 506, + 112 + ], + "lines": [ + { + "bbox": [ + 102, + 76, + 511, + 109 + ], + "spans": [ + { + "bbox": [ + 102, + 79, + 143, + 106 + ], + "score": 1.0, + "content": "By (P1),", + "type": "text" + }, + { + "bbox": [ + 143, + 80, + 287, + 103 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\rho ( t ) ^ { - 2 } \\leq \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { - 2 / L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 76, + 511, + 109 + ], + "score": 1.0, + "content": "/L. By Corollary E.6, ν(t)2 ≥ L(t)2 \u0010log 1L(t) \u00112. Thus", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 144, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 144, + 114 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 110, + 440, + 140 + ], + "lines": [ + { + "bbox": [ + 170, + 110, + 440, + 140 + ], + "spans": [ + { + "bbox": [ + 170, + 110, + 440, + 140 + ], + "score": 0.93, + "content": "\\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\leq - \\frac { 1 } { 2 } \\eta ( t ) \\cdot L ^ { 2 } \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\mathcal { L } ( t ) ^ { 2 } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { 2 - 2 / L } .", + "type": "interline_equation", + "image_path": "80583424360afeb17edd7cddcc971c3dac68aaa4ad5b9e81271adce99d4081d5.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 170, + 110, + 440, + 120.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 170, + 120.0, + 440, + 130.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 170, + 130.0, + 440, + 140.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 504, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 192, + 158 + ], + "score": 1.0, + "content": "It is easy to see that", + "type": "text" + }, + { + "bbox": [ + 189, + 140, + 507, + 162 + ], + "score": 1.0, + "content": "1u2(log 1 )2−2/L is unimodal in (0, 1), so E0(x) is non-decreasing and E(x) is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 159, + 189, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 189, + 171 + ], + "score": 1.0, + "content": "convex. So we have", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 174, + 471, + 257 + ], + "lines": [ + { + "bbox": [ + 139, + 174, + 471, + 257 + ], + "spans": [ + { + "bbox": [ + 139, + 174, + 471, + 257 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { E ( \\mathcal { L } ( t + 1 ) ) - E ( \\mathcal { L } ( t ) ) \\geq E ^ { \\prime } ( \\mathcal { L } ( t ) ) \\left( \\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\right) } \\\\ & { \\qquad \\quad \\geq \\frac { 1 } { 2 } \\eta ( t ) \\cdot L ^ { 2 } \\widehat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\cdot \\frac { \\mathcal { L } ( t ) ^ { 2 } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { 2 - 2 / L } } { \\operatorname* { m i n } \\{ \\mathcal { L } ( t ) ^ { 2 } ( \\log \\frac { 1 } { \\mathcal { L } ( t ) } ) ^ { 2 - 2 / L } , E _ { 0 } \\} } } \\\\ & { \\qquad \\quad \\geq \\frac { 1 } { 2 } \\eta ( t ) \\cdot L ^ { 2 } \\widehat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } , } \\end{array}", + "type": "interline_equation", + "image_path": "2f2970618c2bfe5c8bfd0294796f29cb3f4cf9996088db9691cd6a187a70588a.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 139, + 174, + 471, + 201.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 139, + 201.66666666666666, + 471, + 229.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 139, + 229.33333333333331, + 471, + 257.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 504, + 300 + ], + "lines": [ + { + "bbox": [ + 103, + 262, + 507, + 279 + ], + "spans": [ + { + "bbox": [ + 103, + 262, + 163, + 279 + ], + "score": 1.0, + "content": "which proves", + "type": "text" + }, + { + "bbox": [ + 163, + 263, + 316, + 278 + ], + "score": 0.93, + "content": "\\begin{array} { r } { E ( \\mathcal { L } ( t ) ) \\ge \\frac { 1 } { 2 } L ^ { 2 } \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 262, + 362, + 279 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 362, + 265, + 370, + 274 + ], + "score": 0.83, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 262, + 459, + 279 + ], + "score": 1.0, + "content": "is non-decreasing. If", + "type": "text" + }, + { + "bbox": [ + 459, + 265, + 467, + 274 + ], + "score": 0.79, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 262, + 507, + 279 + ], + "score": 1.0, + "content": "does not", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 238, + 289 + ], + "score": 1.0, + "content": "decreases to 0, then neither does", + "type": "text" + }, + { + "bbox": [ + 239, + 277, + 262, + 289 + ], + "score": 0.89, + "content": "E ( \\mathcal { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 276, + 357, + 289 + ], + "score": 1.0, + "content": ". But the RHS grows to", + "type": "text" + }, + { + "bbox": [ + 358, + 278, + 376, + 288 + ], + "score": 0.88, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 276, + 505, + 289 + ], + "score": 1.0, + "content": ", which leads to a contradiction.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 119, + 300 + ], + "score": 1.0, + "content": "So", + "type": "text" + }, + { + "bbox": [ + 119, + 288, + 148, + 298 + ], + "score": 0.87, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 287, + 189, + 300 + ], + "score": 1.0, + "content": ". To make", + "type": "text" + }, + { + "bbox": [ + 189, + 288, + 218, + 298 + ], + "score": 0.88, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 287, + 221, + 300 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 222, + 289, + 241, + 299 + ], + "score": 0.83, + "content": "q _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 287, + 312, + 300 + ], + "score": 1.0, + "content": "must converge to", + "type": "text" + }, + { + "bbox": [ + 313, + 289, + 331, + 298 + ], + "score": 0.86, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 287, + 348, + 300 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 349, + 289, + 388, + 299 + ], + "score": 0.85, + "content": "\\rho \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 287, + 393, + 300 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 288, + 506, + 299 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 313, + 284, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 285, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 285, + 326 + ], + "score": 1.0, + "content": "E.5 PROOF FOR THEOREM E.3 AND E.4", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 104, + 334, + 500, + 347 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 500, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 500, + 348 + ], + "score": 1.0, + "content": "The proofs for Theorem E.3 and E.4 are similar as those for Theorem A.8 and A.9 in Appendix C.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 506, + 404 + ], + "lines": [ + { + "bbox": [ + 136, + 350, + 507, + 371 + ], + "spans": [ + { + "bbox": [ + 136, + 351, + 267, + 371 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\beta ( t ) : = \\frac { 1 } { \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } } \\left. \\hat { \\pmb { \\theta } } , - \\nabla \\mathcal { L } ( t ) \\right. } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 350, + 507, + 370 + ], + "score": 1.0, + "content": "as we do in Appendix C. It is easy to see that Lemma C.8", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 368, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 203, + 384 + ], + "score": 1.0, + "content": "still holds if we replace", + "type": "text" + }, + { + "bbox": [ + 203, + 371, + 226, + 382 + ], + "score": 0.92, + "content": "\\widetilde { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 368, + 248, + 384 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 248, + 371, + 270, + 382 + ], + "score": 0.88, + "content": "\\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 368, + 379, + 384 + ], + "score": 1.0, + "content": ". So we only need to show", + "type": "text" + }, + { + "bbox": [ + 379, + 370, + 409, + 380 + ], + "score": 0.9, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 368, + 427, + 384 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 427, + 370, + 456, + 381 + ], + "score": 0.92, + "content": "\\beta 1", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 368, + 506, + 384 + ], + "score": 1.0, + "content": "for proving", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 225, + 393 + ], + "score": 1.0, + "content": "convergence to KKT points.", + "type": "text" + }, + { + "bbox": [ + 225, + 382, + 256, + 392 + ], + "score": 0.9, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "can be followed from Theorem E.2. Similar as the proof for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 391, + 405, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 349, + 405 + ], + "score": 1.0, + "content": "Lemma C.9, it follows from Lemma E.8 and (15) that for all", + "type": "text" + }, + { + "bbox": [ + 349, + 393, + 401, + 403 + ], + "score": 0.91, + "content": "t _ { 2 } > t _ { 1 } \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 391, + 405, + 405 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 409, + 406, + 444 + ], + "lines": [ + { + "bbox": [ + 205, + 409, + 406, + 444 + ], + "spans": [ + { + "bbox": [ + 205, + 409, + 406, + 444 + ], + "score": 0.93, + "content": "\\sum _ { t = t _ { 1 } } ^ { t _ { 2 } - 1 } \\left( \\beta ( t ) ^ { - 2 } - 1 \\right) \\cdot \\log \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } \\leq \\frac { 1 } { L } \\log \\frac { \\hat { \\gamma } ( t _ { 2 } ) } { \\hat { \\gamma } ( t _ { 1 } ) } .", + "type": "interline_equation", + "image_path": "987d22e74afb6b535b7b0cb33eeb316a1b20c2bc66a8045bcfa0d45f18e51b63.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 409, + 406, + 426.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 205, + 426.5, + 406, + 444.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 223, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 224, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 224, + 469 + ], + "score": 1.0, + "content": "Now we prove Theorem E.4.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "Proof for Theorem E.4. We make the following changes in the proof for Theorem A.9. First, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 137, + 506 + ], + "score": 1.0, + "content": "replace", + "type": "text" + }, + { + "bbox": [ + 138, + 494, + 160, + 505 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 493, + 182, + 506 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 183, + 493, + 205, + 505 + ], + "score": 0.92, + "content": "\\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 493, + 232, + 506 + ], + "score": 1.0, + "content": ", since", + "type": "text" + }, + { + "bbox": [ + 233, + 493, + 251, + 505 + ], + "score": 0.63, + "content": "\\tilde { \\gamma } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 494, + 286, + 505 + ], + "score": 0.5, + "content": "\\left( t \\geq t _ { 0 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 493, + 372, + 506 + ], + "score": 1.0, + "content": "is lower bounded by", + "type": "text" + }, + { + "bbox": [ + 372, + 494, + 395, + 505 + ], + "score": 0.92, + "content": "\\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 493, + 442, + 506 + ], + "score": 1.0, + "content": "rather than", + "type": "text" + }, + { + "bbox": [ + 443, + 493, + 465, + 506 + ], + "score": 0.91, + "content": "\\tilde { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 493, + 505, + 506 + ], + "score": 1.0, + "content": ". Second,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 168, + 518 + ], + "score": 1.0, + "content": "when choosing", + "type": "text" + }, + { + "bbox": [ + 169, + 506, + 178, + 515 + ], + "score": 0.86, + "content": "t _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 504, + 196, + 518 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 196, + 506, + 205, + 515 + ], + "score": 0.85, + "content": "t _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 504, + 246, + 518 + ], + "score": 1.0, + "content": ", we make", + "type": "text" + }, + { + "bbox": [ + 246, + 505, + 283, + 516 + ], + "score": 0.91, + "content": "\\log \\rho ( t _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 504, + 301, + 518 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 301, + 505, + 338, + 516 + ], + "score": 0.92, + "content": "\\log \\rho ( t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "equal to the chosen values approximately", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 195, + 528 + ], + "score": 1.0, + "content": "with an additive error", + "type": "text" + }, + { + "bbox": [ + 196, + 516, + 214, + 527 + ], + "score": 0.9, + "content": "o ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 514, + 505, + 528 + ], + "score": 1.0, + "content": ", rather than make them equal exactly. This is possible because it can be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 335, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 335, + 540 + ], + "score": 1.0, + "content": "shown from (P2) in Lemma E.8 that the following holds:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 542, + 485, + 593 + ], + "lines": [ + { + "bbox": [ + 126, + 542, + 485, + 593 + ], + "spans": [ + { + "bbox": [ + 126, + 542, + 485, + 593 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\rho ( t + 1 ) ^ { 2 } - \\rho ( t ) ^ { 2 } = O ( \\eta ( t ) \\nu ( t ) ) } \\\\ & { \\qquad = o ( \\kappa ( \\mathcal { L } ( t ) ) ^ { - 1 } \\cdot \\mathcal { L } ( t ) \\rho ( t ) ^ { L } ) = o \\left( \\frac { \\rho ( t ) ^ { L } } { ( \\log \\frac { 1 } { \\mathcal { L } ( t ) } ) ^ { 2 - 2 / L } } \\right) = o ( \\rho ( t ) ^ { - L + 2 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "19d084ef2bf0ce91197061c49b69fba8700147822442605eaa552017e226493a.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 126, + 542, + 485, + 559.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 126, + 559.0, + 485, + 576.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 126, + 576.0, + 485, + 593.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 348, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 348, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 144, + 613 + ], + "score": 1.0, + "content": "Dividing", + "type": "text" + }, + { + "bbox": [ + 144, + 599, + 166, + 612 + ], + "score": 0.92, + "content": "\\rho ( t ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 597, + 348, + 613 + ], + "score": 1.0, + "content": "on the leftmost and rightmost sides, we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 617, + 394, + 632 + ], + "lines": [ + { + "bbox": [ + 215, + 617, + 394, + 632 + ], + "spans": [ + { + "bbox": [ + 215, + 617, + 394, + 632 + ], + "score": 0.91, + "content": "\\rho ( t + 1 ) / \\rho ( t ) = 1 + o ( \\rho ( t ) ^ { - L } ) = 1 + o ( 1 ) ,", + "type": "interline_equation", + "image_path": "ef385e89d8ac940acf65c1c02a95badb997b78c3eb502f771e11f641d3d670ef.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 215, + 617, + 394, + 632 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 637, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 184, + 651 + ], + "score": 1.0, + "content": "which implies that", + "type": "text" + }, + { + "bbox": [ + 185, + 638, + 312, + 650 + ], + "score": 0.9, + "content": "\\log \\rho ( t + 1 ) - \\log \\rho ( t ) = o ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 638, + 396, + 651 + ], + "score": 1.0, + "content": ". Therefore, for any", + "type": "text" + }, + { + "bbox": [ + 396, + 639, + 405, + 648 + ], + "score": 0.84, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 638, + 505, + 651 + ], + "score": 1.0, + "content": ", we can always find the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 167, + 662 + ], + "score": 1.0, + "content": "minimum time", + "type": "text" + }, + { + "bbox": [ + 167, + 650, + 172, + 659 + ], + "score": 0.79, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 649, + 210, + 662 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 210, + 650, + 262, + 661 + ], + "score": 0.9, + "content": "\\log \\rho ( t ) \\geq R", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 649, + 362, + 662 + ], + "score": 1.0, + "content": ", and it holds for sure that", + "type": "text" + }, + { + "bbox": [ + 363, + 649, + 430, + 661 + ], + "score": 0.92, + "content": "\\log \\rho ( t ) - R \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 649, + 442, + 662 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 442, + 650, + 483, + 660 + ], + "score": 0.91, + "content": "R \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 649, + 506, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 105, + 675, + 466, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 468, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 468, + 689 + ], + "score": 1.0, + "content": "For proving Theorem E.3, we also need the following lemma as a variant of Lemma C.11.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 691, + 226, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 223, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 198, + 705 + ], + "score": 1.0, + "content": "Lemma E.11. For all", + "type": "text" + }, + { + "bbox": [ + 198, + 692, + 223, + 703 + ], + "score": 0.89, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 708, + 429, + 736 + ], + "lines": [ + { + "bbox": [ + 181, + 708, + 429, + 736 + ], + "spans": [ + { + "bbox": [ + 181, + 708, + 429, + 736 + ], + "score": 0.92, + "content": "\\left\\| \\hat { \\pmb \\theta } ( t + 1 ) - \\hat { \\pmb \\theta } ( t ) \\right\\| _ { 2 } \\leq \\left( \\frac { B _ { 1 } } { L \\tilde { \\gamma } ( t ) } \\cdot \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } + 1 \\right) \\log \\frac { \\rho ( t + 1 ) } { \\rho ( t ) }", + "type": "interline_equation", + "image_path": "04e07b017ded711eec7404460b58453680292024ccf2b986bd1ee2f80cf13b60.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 181, + 708, + 429, + 736 + ], + "spans": [], + "index": 36 + } + ] + } + ], + "page_idx": 35, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "36", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 506, + 112 + ], + "lines": [ + { + "bbox": [ + 102, + 76, + 511, + 109 + ], + "spans": [ + { + "bbox": [ + 102, + 79, + 143, + 106 + ], + "score": 1.0, + "content": "By (P1),", + "type": "text" + }, + { + "bbox": [ + 143, + 80, + 287, + 103 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\rho ( t ) ^ { - 2 } \\leq \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { - 2 / L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 76, + 511, + 109 + ], + "score": 1.0, + "content": "/L. By Corollary E.6, ν(t)2 ≥ L(t)2 \u0010log 1L(t) \u00112. Thus", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 144, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 144, + 114 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 102, + 76, + 511, + 114 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 110, + 440, + 140 + ], + "lines": [ + { + "bbox": [ + 170, + 110, + 440, + 140 + ], + "spans": [ + { + "bbox": [ + 170, + 110, + 440, + 140 + ], + "score": 0.93, + "content": "\\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\leq - \\frac { 1 } { 2 } \\eta ( t ) \\cdot L ^ { 2 } \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\mathcal { L } ( t ) ^ { 2 } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { 2 - 2 / L } .", + "type": "interline_equation", + "image_path": "80583424360afeb17edd7cddcc971c3dac68aaa4ad5b9e81271adce99d4081d5.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 170, + 110, + 440, + 120.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 170, + 120.0, + 440, + 130.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 170, + 130.0, + 440, + 140.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 504, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 192, + 158 + ], + "score": 1.0, + "content": "It is easy to see that", + "type": "text" + }, + { + "bbox": [ + 189, + 140, + 507, + 162 + ], + "score": 1.0, + "content": "1u2(log 1 )2−2/L is unimodal in (0, 1), so E0(x) is non-decreasing and E(x) is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 159, + 189, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 189, + 171 + ], + "score": 1.0, + "content": "convex. So we have", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 140, + 507, + 171 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 174, + 471, + 257 + ], + "lines": [ + { + "bbox": [ + 139, + 174, + 471, + 257 + ], + "spans": [ + { + "bbox": [ + 139, + 174, + 471, + 257 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { E ( \\mathcal { L } ( t + 1 ) ) - E ( \\mathcal { L } ( t ) ) \\geq E ^ { \\prime } ( \\mathcal { L } ( t ) ) \\left( \\mathcal { L } ( t + 1 ) - \\mathcal { L } ( t ) \\right) } \\\\ & { \\qquad \\quad \\geq \\frac { 1 } { 2 } \\eta ( t ) \\cdot L ^ { 2 } \\widehat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\cdot \\frac { \\mathcal { L } ( t ) ^ { 2 } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { 2 - 2 / L } } { \\operatorname* { m i n } \\{ \\mathcal { L } ( t ) ^ { 2 } ( \\log \\frac { 1 } { \\mathcal { L } ( t ) } ) ^ { 2 - 2 / L } , E _ { 0 } \\} } } \\\\ & { \\qquad \\quad \\geq \\frac { 1 } { 2 } \\eta ( t ) \\cdot L ^ { 2 } \\widehat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } , } \\end{array}", + "type": "interline_equation", + "image_path": "2f2970618c2bfe5c8bfd0294796f29cb3f4cf9996088db9691cd6a187a70588a.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 139, + 174, + 471, + 201.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 139, + 201.66666666666666, + 471, + 229.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 139, + 229.33333333333331, + 471, + 257.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 504, + 300 + ], + "lines": [ + { + "bbox": [ + 103, + 262, + 507, + 279 + ], + "spans": [ + { + "bbox": [ + 103, + 262, + 163, + 279 + ], + "score": 1.0, + "content": "which proves", + "type": "text" + }, + { + "bbox": [ + 163, + 263, + 316, + 278 + ], + "score": 0.93, + "content": "\\begin{array} { r } { E ( \\mathcal { L } ( t ) ) \\ge \\frac { 1 } { 2 } L ^ { 2 } \\hat { \\gamma } ( t _ { 0 } ) ^ { 2 / L } \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 262, + 362, + 279 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 362, + 265, + 370, + 274 + ], + "score": 0.83, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 262, + 459, + 279 + ], + "score": 1.0, + "content": "is non-decreasing. If", + "type": "text" + }, + { + "bbox": [ + 459, + 265, + 467, + 274 + ], + "score": 0.79, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 262, + 507, + 279 + ], + "score": 1.0, + "content": "does not", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 238, + 289 + ], + "score": 1.0, + "content": "decreases to 0, then neither does", + "type": "text" + }, + { + "bbox": [ + 239, + 277, + 262, + 289 + ], + "score": 0.89, + "content": "E ( \\mathcal { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 276, + 357, + 289 + ], + "score": 1.0, + "content": ". But the RHS grows to", + "type": "text" + }, + { + "bbox": [ + 358, + 278, + 376, + 288 + ], + "score": 0.88, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 276, + 505, + 289 + ], + "score": 1.0, + "content": ", which leads to a contradiction.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 119, + 300 + ], + "score": 1.0, + "content": "So", + "type": "text" + }, + { + "bbox": [ + 119, + 288, + 148, + 298 + ], + "score": 0.87, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 287, + 189, + 300 + ], + "score": 1.0, + "content": ". To make", + "type": "text" + }, + { + "bbox": [ + 189, + 288, + 218, + 298 + ], + "score": 0.88, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 287, + 221, + 300 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 222, + 289, + 241, + 299 + ], + "score": 0.83, + "content": "q _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 287, + 312, + 300 + ], + "score": 1.0, + "content": "must converge to", + "type": "text" + }, + { + "bbox": [ + 313, + 289, + 331, + 298 + ], + "score": 0.86, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 287, + 348, + 300 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 349, + 289, + 388, + 299 + ], + "score": 0.85, + "content": "\\rho \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 287, + 393, + 300 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 288, + 506, + 299 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 103, + 262, + 507, + 300 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 313, + 284, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 285, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 285, + 326 + ], + "score": 1.0, + "content": "E.5 PROOF FOR THEOREM E.3 AND E.4", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 104, + 334, + 500, + 347 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 500, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 500, + 348 + ], + "score": 1.0, + "content": "The proofs for Theorem E.3 and E.4 are similar as those for Theorem A.8 and A.9 in Appendix C.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 334, + 500, + 348 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 506, + 404 + ], + "lines": [ + { + "bbox": [ + 136, + 350, + 507, + 371 + ], + "spans": [ + { + "bbox": [ + 136, + 351, + 267, + 371 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\beta ( t ) : = \\frac { 1 } { \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } } \\left. \\hat { \\pmb { \\theta } } , - \\nabla \\mathcal { L } ( t ) \\right. } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 350, + 507, + 370 + ], + "score": 1.0, + "content": "as we do in Appendix C. It is easy to see that Lemma C.8", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 368, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 203, + 384 + ], + "score": 1.0, + "content": "still holds if we replace", + "type": "text" + }, + { + "bbox": [ + 203, + 371, + 226, + 382 + ], + "score": 0.92, + "content": "\\widetilde { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 368, + 248, + 384 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 248, + 371, + 270, + 382 + ], + "score": 0.88, + "content": "\\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 368, + 379, + 384 + ], + "score": 1.0, + "content": ". So we only need to show", + "type": "text" + }, + { + "bbox": [ + 379, + 370, + 409, + 380 + ], + "score": 0.9, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 368, + 427, + 384 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 427, + 370, + 456, + 381 + ], + "score": 0.92, + "content": "\\beta 1", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 368, + 506, + 384 + ], + "score": 1.0, + "content": "for proving", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 225, + 393 + ], + "score": 1.0, + "content": "convergence to KKT points.", + "type": "text" + }, + { + "bbox": [ + 225, + 382, + 256, + 392 + ], + "score": 0.9, + "content": "\\mathcal { L } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "can be followed from Theorem E.2. Similar as the proof for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 391, + 405, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 349, + 405 + ], + "score": 1.0, + "content": "Lemma C.9, it follows from Lemma E.8 and (15) that for all", + "type": "text" + }, + { + "bbox": [ + 349, + 393, + 401, + 403 + ], + "score": 0.91, + "content": "t _ { 2 } > t _ { 1 } \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 391, + 405, + 405 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 350, + 507, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 409, + 406, + 444 + ], + "lines": [ + { + "bbox": [ + 205, + 409, + 406, + 444 + ], + "spans": [ + { + "bbox": [ + 205, + 409, + 406, + 444 + ], + "score": 0.93, + "content": "\\sum _ { t = t _ { 1 } } ^ { t _ { 2 } - 1 } \\left( \\beta ( t ) ^ { - 2 } - 1 \\right) \\cdot \\log \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } \\leq \\frac { 1 } { L } \\log \\frac { \\hat { \\gamma } ( t _ { 2 } ) } { \\hat { \\gamma } ( t _ { 1 } ) } .", + "type": "interline_equation", + "image_path": "987d22e74afb6b535b7b0cb33eeb316a1b20c2bc66a8045bcfa0d45f18e51b63.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 409, + 406, + 426.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 205, + 426.5, + 406, + 444.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 223, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 224, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 224, + 469 + ], + "score": 1.0, + "content": "Now we prove Theorem E.4.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 455, + 224, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "Proof for Theorem E.4. We make the following changes in the proof for Theorem A.9. First, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 137, + 506 + ], + "score": 1.0, + "content": "replace", + "type": "text" + }, + { + "bbox": [ + 138, + 494, + 160, + 505 + ], + "score": 0.92, + "content": "\\tilde { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 493, + 182, + 506 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 183, + 493, + 205, + 505 + ], + "score": 0.92, + "content": "\\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 493, + 232, + 506 + ], + "score": 1.0, + "content": ", since", + "type": "text" + }, + { + "bbox": [ + 233, + 493, + 251, + 505 + ], + "score": 0.63, + "content": "\\tilde { \\gamma } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 494, + 286, + 505 + ], + "score": 0.5, + "content": "\\left( t \\geq t _ { 0 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 493, + 372, + 506 + ], + "score": 1.0, + "content": "is lower bounded by", + "type": "text" + }, + { + "bbox": [ + 372, + 494, + 395, + 505 + ], + "score": 0.92, + "content": "\\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 493, + 442, + 506 + ], + "score": 1.0, + "content": "rather than", + "type": "text" + }, + { + "bbox": [ + 443, + 493, + 465, + 506 + ], + "score": 0.91, + "content": "\\tilde { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 493, + 505, + 506 + ], + "score": 1.0, + "content": ". Second,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 168, + 518 + ], + "score": 1.0, + "content": "when choosing", + "type": "text" + }, + { + "bbox": [ + 169, + 506, + 178, + 515 + ], + "score": 0.86, + "content": "t _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 504, + 196, + 518 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 196, + 506, + 205, + 515 + ], + "score": 0.85, + "content": "t _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 504, + 246, + 518 + ], + "score": 1.0, + "content": ", we make", + "type": "text" + }, + { + "bbox": [ + 246, + 505, + 283, + 516 + ], + "score": 0.91, + "content": "\\log \\rho ( t _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 504, + 301, + 518 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 301, + 505, + 338, + 516 + ], + "score": 0.92, + "content": "\\log \\rho ( t _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "equal to the chosen values approximately", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 195, + 528 + ], + "score": 1.0, + "content": "with an additive error", + "type": "text" + }, + { + "bbox": [ + 196, + 516, + 214, + 527 + ], + "score": 0.9, + "content": "o ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 514, + 505, + 528 + ], + "score": 1.0, + "content": ", rather than make them equal exactly. This is possible because it can be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 335, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 335, + 540 + ], + "score": 1.0, + "content": "shown from (P2) in Lemma E.8 that the following holds:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 482, + 506, + 540 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 542, + 485, + 593 + ], + "lines": [ + { + "bbox": [ + 126, + 542, + 485, + 593 + ], + "spans": [ + { + "bbox": [ + 126, + 542, + 485, + 593 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\rho ( t + 1 ) ^ { 2 } - \\rho ( t ) ^ { 2 } = O ( \\eta ( t ) \\nu ( t ) ) } \\\\ & { \\qquad = o ( \\kappa ( \\mathcal { L } ( t ) ) ^ { - 1 } \\cdot \\mathcal { L } ( t ) \\rho ( t ) ^ { L } ) = o \\left( \\frac { \\rho ( t ) ^ { L } } { ( \\log \\frac { 1 } { \\mathcal { L } ( t ) } ) ^ { 2 - 2 / L } } \\right) = o ( \\rho ( t ) ^ { - L + 2 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "19d084ef2bf0ce91197061c49b69fba8700147822442605eaa552017e226493a.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 126, + 542, + 485, + 559.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 126, + 559.0, + 485, + 576.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 126, + 576.0, + 485, + 593.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 348, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 348, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 144, + 613 + ], + "score": 1.0, + "content": "Dividing", + "type": "text" + }, + { + "bbox": [ + 144, + 599, + 166, + 612 + ], + "score": 0.92, + "content": "\\rho ( t ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 597, + 348, + 613 + ], + "score": 1.0, + "content": "on the leftmost and rightmost sides, we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 597, + 348, + 613 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 617, + 394, + 632 + ], + "lines": [ + { + "bbox": [ + 215, + 617, + 394, + 632 + ], + "spans": [ + { + "bbox": [ + 215, + 617, + 394, + 632 + ], + "score": 0.91, + "content": "\\rho ( t + 1 ) / \\rho ( t ) = 1 + o ( \\rho ( t ) ^ { - L } ) = 1 + o ( 1 ) ,", + "type": "interline_equation", + "image_path": "ef385e89d8ac940acf65c1c02a95badb997b78c3eb502f771e11f641d3d670ef.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 215, + 617, + 394, + 632 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 637, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 184, + 651 + ], + "score": 1.0, + "content": "which implies that", + "type": "text" + }, + { + "bbox": [ + 185, + 638, + 312, + 650 + ], + "score": 0.9, + "content": "\\log \\rho ( t + 1 ) - \\log \\rho ( t ) = o ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 638, + 396, + 651 + ], + "score": 1.0, + "content": ". Therefore, for any", + "type": "text" + }, + { + "bbox": [ + 396, + 639, + 405, + 648 + ], + "score": 0.84, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 638, + 505, + 651 + ], + "score": 1.0, + "content": ", we can always find the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 167, + 662 + ], + "score": 1.0, + "content": "minimum time", + "type": "text" + }, + { + "bbox": [ + 167, + 650, + 172, + 659 + ], + "score": 0.79, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 649, + 210, + 662 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 210, + 650, + 262, + 661 + ], + "score": 0.9, + "content": "\\log \\rho ( t ) \\geq R", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 649, + 362, + 662 + ], + "score": 1.0, + "content": ", and it holds for sure that", + "type": "text" + }, + { + "bbox": [ + 363, + 649, + 430, + 661 + ], + "score": 0.92, + "content": "\\log \\rho ( t ) - R \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 649, + 442, + 662 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 442, + 650, + 483, + 660 + ], + "score": 0.91, + "content": "R \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 649, + 506, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 638, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 675, + 466, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 468, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 468, + 689 + ], + "score": 1.0, + "content": "For proving Theorem E.3, we also need the following lemma as a variant of Lemma C.11.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 675, + 468, + 689 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 691, + 226, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 223, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 198, + 705 + ], + "score": 1.0, + "content": "Lemma E.11. For all", + "type": "text" + }, + { + "bbox": [ + 198, + 692, + 223, + 703 + ], + "score": 0.89, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 689, + 223, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 708, + 429, + 736 + ], + "lines": [ + { + "bbox": [ + 181, + 708, + 429, + 736 + ], + "spans": [ + { + "bbox": [ + 181, + 708, + 429, + 736 + ], + "score": 0.92, + "content": "\\left\\| \\hat { \\pmb \\theta } ( t + 1 ) - \\hat { \\pmb \\theta } ( t ) \\right\\| _ { 2 } \\leq \\left( \\frac { B _ { 1 } } { L \\tilde { \\gamma } ( t ) } \\cdot \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } + 1 \\right) \\log \\frac { \\rho ( t + 1 ) } { \\rho ( t ) }", + "type": "interline_equation", + "image_path": "04e07b017ded711eec7404460b58453680292024ccf2b986bd1ee2f80cf13b60.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 181, + 708, + 429, + 736 + ], + "spans": [], + "index": 36 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 290, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 290, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 290, + 96 + ], + "score": 1.0, + "content": "Proof. According to the update rule, we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 98, + 477, + 185 + ], + "lines": [ + { + "bbox": [ + 133, + 98, + 477, + 185 + ], + "spans": [ + { + "bbox": [ + 133, + 98, + 477, + 185 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\left\\| \\hat { \\theta } ( t + 1 ) - \\hat { \\theta } ( t ) \\right\\| _ { 2 } = \\displaystyle \\frac { 1 } { \\rho ( t + 1 ) } \\left\\| \\theta ( t + 1 ) - \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } \\pmb { \\theta } ( t ) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { 1 } { \\rho ( t + 1 ) } \\left( \\| \\theta ( t + 1 ) - \\theta ( t ) \\| _ { 2 } + \\left\\| \\left( \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } - 1 \\right) \\pmb { \\theta } ( t ) \\right\\| _ { 2 } \\right) } \\\\ & { \\qquad = \\displaystyle \\frac { \\eta ( t ) } { \\rho ( t + 1 ) } \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } + \\left( 1 - \\frac { \\rho ( t ) } { \\rho ( t + 1 ) } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "1e83536a68da147fbf43e0db7553f754c84329ecf0f6347db38b29ccecdce645.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 133, + 98, + 477, + 127.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 133, + 127.0, + 477, + 156.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 133, + 156.0, + 477, + 185.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 187, + 367, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 367, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 142, + 206 + ], + "score": 1.0, + "content": "By (16),", + "type": "text" + }, + { + "bbox": [ + 142, + 187, + 227, + 205 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } \\le \\frac { B _ { 1 } \\nu ( t ) } { \\widetilde { \\gamma } ( t ) \\rho ( t ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 185, + 367, + 206 + ], + "score": 1.0, + "content": ". So we can bound the first term as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 208, + 478, + 294 + ], + "lines": [ + { + "bbox": [ + 133, + 208, + 478, + 294 + ], + "spans": [ + { + "bbox": [ + 133, + 208, + 478, + 294 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\displaystyle \\frac { \\eta ( t ) } { \\rho ( t + 1 ) } \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } \\le \\frac { B _ { 1 } } { \\widetilde { \\gamma } ( t ) } \\cdot \\frac { \\eta ( t ) \\nu ( t ) } { \\rho ( t + 1 ) \\rho ( t ) } \\le \\frac { B _ { 1 } } { \\widetilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) ^ { 2 } - \\rho ( t ) ^ { 2 } } { 2 L \\rho ( t + 1 ) \\rho ( t ) } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad = \\frac { B _ { 1 } } { 2 L \\widetilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) ^ { 2 } - \\rho ( t ) ^ { 2 } } { \\rho ( t + 1 ) ^ { 2 } } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\le \\frac { B _ { 1 } } { L \\widetilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } \\log \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } } \\end{array}", + "type": "interline_equation", + "image_path": "b22a2de5f367e3d9f03863120c7ac46c7b27eeba4eb65e33116274a3f759edf5.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 133, + 208, + 478, + 236.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 133, + 236.66666666666666, + 478, + 265.3333333333333 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 133, + 265.3333333333333, + 478, + 294.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 103, + 293, + 508, + 322 + ], + "spans": [ + { + "bbox": [ + 103, + 293, + 287, + 322 + ], + "score": 1.0, + "content": "where the last inequality uses the inequality bound the second term by", + "type": "text" + }, + { + "bbox": [ + 288, + 297, + 354, + 311 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { a - b } { a } \\leq \\log ( a / b ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 293, + 508, + 322 + ], + "score": 1.0, + "content": ". Using this inequality again, we can", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 317, + 459, + 370 + ], + "lines": [ + { + "bbox": [ + 237, + 317, + 459, + 370 + ], + "spans": [ + { + "bbox": [ + 237, + 317, + 459, + 370 + ], + "score": 0.55, + "content": "\\begin{array} { r l } & { \\quad 1 - \\displaystyle \\frac { \\rho ( t ) } { \\rho ( t + 1 ) } \\leq \\log \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } . } \\\\ & { \\quad \\left\\| \\hat { \\pmb \\theta } ( t + 1 ) - \\hat { \\pmb \\theta } ( t ) \\right\\| _ { 2 } \\leq \\left( \\frac { B _ { 1 } } { L \\tilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } + 1 \\right) \\log \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "810d60710ce103b6ac97f702d5f73a7fd659607aed9bc0686c52e8f17301147f.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 237, + 317, + 459, + 334.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 237, + 334.6666666666667, + 459, + 352.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 237, + 352.33333333333337, + 459, + 370.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 347, + 458, + 369 + ], + "lines": [ + { + "bbox": [ + 100, + 343, + 235, + 375 + ], + "spans": [ + { + "bbox": [ + 100, + 343, + 235, + 375 + ], + "score": 1.0, + "content": "Combining these together gives", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 379, + 272, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 274, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 274, + 392 + ], + "score": 1.0, + "content": "Now we are ready to prove Theorem E.3.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "Proof for Theorem E.3. As discussed above, we only need to show a variant of Lemma C.12 for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 414, + 504, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 258, + 432 + ], + "score": 1.0, + "content": "gradient descent: for every limit point", + "type": "text" + }, + { + "bbox": [ + 259, + 416, + 266, + 428 + ], + "score": 0.77, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 415, + 277, + 432 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 278, + 414, + 339, + 434 + ], + "score": 0.94, + "content": "\\left\\{ { \\hat { \\theta } } ( t ) : t \\geq 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 415, + 445, + 432 + ], + "score": 1.0, + "content": ", there exists a sequence of", + "type": "text" + }, + { + "bbox": [ + 446, + 417, + 504, + 430 + ], + "score": 0.92, + "content": "\\{ t _ { m } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 433, + 308, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 145, + 448 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 145, + 433, + 236, + 447 + ], + "score": 0.92, + "content": "t _ { m } \\uparrow + \\infty , \\hat { \\pmb \\theta } ( t _ { m } ) \\bar { \\pmb \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 434, + 257, + 448 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 257, + 435, + 303, + 447 + ], + "score": 0.92, + "content": "\\beta ( t _ { m } ) 1", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 434, + 308, + 448 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 503, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 264, + 465 + ], + "score": 1.0, + "content": "We only need to change the choices of", + "type": "text" + }, + { + "bbox": [ + 264, + 452, + 309, + 464 + ], + "score": 0.94, + "content": "s _ { m } , s _ { m } ^ { \\prime } , t _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 450, + 479, + 465 + ], + "score": 1.0, + "content": "in the proof for Lemma C.12. We choose", + "type": "text" + }, + { + "bbox": [ + 479, + 453, + 505, + 463 + ], + "score": 0.86, + "content": "s _ { m } >", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 463, + 218, + 475 + ], + "spans": [ + { + "bbox": [ + 107, + 464, + 129, + 475 + ], + "score": 0.9, + "content": "t _ { m - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 463, + 218, + 474 + ], + "score": 1.0, + "content": "to be a time such that", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 478, + 425, + 519 + ], + "lines": [ + { + "bbox": [ + 185, + 478, + 425, + 519 + ], + "spans": [ + { + "bbox": [ + 185, + 478, + 425, + 519 + ], + "score": 0.93, + "content": "\\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\| _ { 2 } \\leq \\epsilon _ { m } \\quad \\mathrm { a n d } \\quad \\frac { 1 } { L } \\log ( \\frac { \\operatorname* { l i m } _ { t + \\infty } \\hat { \\gamma } ( t ) } { \\hat { \\gamma } ( s _ { m } ) } ) \\leq \\epsilon _ { m } ^ { 3 } .", + "type": "interline_equation", + "image_path": "629597a695ebe3b0c26369724e6c0c4851c5e766e141044aa84c43276bcf56a7.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 478, + 425, + 498.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 185, + 498.5, + 425, + 519.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 157, + 536 + ], + "score": 1.0, + "content": "Then we let", + "type": "text" + }, + { + "bbox": [ + 157, + 523, + 196, + 534 + ], + "score": 0.92, + "content": "s _ { m } ^ { \\prime } > s _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 520, + 326, + 536 + ], + "score": 1.0, + "content": "be the minimum time such that", + "type": "text" + }, + { + "bbox": [ + 326, + 522, + 445, + 534 + ], + "score": 0.89, + "content": "\\log { \\rho ( s _ { m } ^ { \\prime } ) } \\ge \\log { \\rho ( s _ { m } ) } + \\epsilon _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 520, + 506, + 536 + ], + "score": 1.0, + "content": ". According to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 533, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 163, + 548 + ], + "score": 1.0, + "content": "Theorem E.2,", + "type": "text" + }, + { + "bbox": [ + 163, + 536, + 177, + 545 + ], + "score": 0.82, + "content": "s _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 533, + 194, + 548 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 534, + 208, + 546 + ], + "score": 0.9, + "content": "s _ { m } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 533, + 340, + 548 + ], + "score": 1.0, + "content": "must exist. Finally, we construct", + "type": "text" + }, + { + "bbox": [ + 341, + 535, + 438, + 546 + ], + "score": 0.88, + "content": "t _ { m } \\in \\{ s _ { m } , \\ldots , s _ { m } ^ { \\prime } - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 533, + 506, + 548 + ], + "score": 1.0, + "content": "to be a time that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 542, + 362, + 559 + ], + "spans": [ + { + "bbox": [ + 107, + 544, + 186, + 557 + ], + "score": 0.91, + "content": "\\beta ( t _ { m } ) ^ { - 2 } - 1 \\leq \\epsilon _ { m } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 542, + 362, + 559 + ], + "score": 1.0, + "content": ", where the existence can be shown by (22).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 470, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 470, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 352, + 575 + ], + "score": 1.0, + "content": "To see that this construction meets our requirement, note that", + "type": "text" + }, + { + "bbox": [ + 352, + 560, + 452, + 574 + ], + "score": 0.93, + "content": "\\beta ( t _ { m } ) ^ { - 2 } - 1 \\le \\epsilon _ { m } ^ { 2 } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 560, + 470, + 575 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 577, + 495, + 605 + ], + "lines": [ + { + "bbox": [ + 114, + 577, + 495, + 605 + ], + "spans": [ + { + "bbox": [ + 114, + 577, + 495, + 605 + ], + "score": 0.92, + "content": "\\Big \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\bar { \\pmb \\theta } \\Big \\| _ { 2 } \\leq \\Big \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\hat { \\pmb \\theta } ( s _ { m } ) \\Big \\| _ { 2 } + \\Big \\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\Big \\| _ { 2 } \\leq \\bigg ( \\frac { B _ { 1 } } { L \\hat { \\gamma } ( t _ { 0 } ) } e ^ { \\epsilon _ { m } } + 1 \\bigg ) \\cdot \\epsilon _ { m } + \\epsilon _ { m } 0 ,", + "type": "interline_equation", + "image_path": "3f93d9b25c0dde3e6b9639bf089f54f4edc0df6cc8f9aaaa925fb3b9b1bf4139.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 114, + 577, + 495, + 605 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 609, + 283, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 285, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 285, + 622 + ], + "score": 1.0, + "content": "where the last inequality is by Lemma E.11.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 107, + 633, + 245, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 245, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 245, + 646 + ], + "score": 1.0, + "content": "E.6 PROOF FOR THEOREM E.5", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 321, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 321, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 321, + 668 + ], + "score": 1.0, + "content": "Proof. By a similar analysis as Lemma D.4, we have", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 670, + 461, + 732 + ], + "lines": [ + { + "bbox": [ + 150, + 670, + 461, + 732 + ], + "spans": [ + { + "bbox": [ + 150, + 670, + 461, + 732 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } { { E ( x ) = \\int _ { x } ^ { \\mathcal { L } ( t _ { 0 } ) } \\Theta ( \\frac { 1 } { u ^ { 2 } ( \\log \\frac { 1 } { u } ) ^ { 2 - 2 / L } } ) d u = \\int _ { 1 / \\mathcal { L } ( t _ { 0 } ) } ^ { 1 / x } \\Theta ( \\frac { 1 } { ( \\log u ) ^ { 2 - 2 / L } } ) d u } } \\\\ & { } & { = \\Theta ( \\frac { 1 } { x ( \\log \\frac { 1 } { x } ) ^ { 2 - 2 / L } } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6bac32ce62dd4860ece8d882769d3d52b411a87de66fc87b777fcf0ff1975e0c.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 150, + 670, + 461, + 690.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 150, + 690.6666666666666, + 461, + 711.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 150, + 711.3333333333333, + 461, + 731.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + } + ], + "page_idx": 36, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "37", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 352, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 495, + 353, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 495, + 353, + 505, + 363 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 610, + 505, + 620 + ], + "lines": [ + { + "bbox": [ + 496, + 611, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 496, + 611, + 505, + 621 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 290, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 290, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 290, + 96 + ], + "score": 1.0, + "content": "Proof. According to the update rule, we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 290, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 98, + 477, + 185 + ], + "lines": [ + { + "bbox": [ + 133, + 98, + 477, + 185 + ], + "spans": [ + { + "bbox": [ + 133, + 98, + 477, + 185 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\left\\| \\hat { \\theta } ( t + 1 ) - \\hat { \\theta } ( t ) \\right\\| _ { 2 } = \\displaystyle \\frac { 1 } { \\rho ( t + 1 ) } \\left\\| \\theta ( t + 1 ) - \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } \\pmb { \\theta } ( t ) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { 1 } { \\rho ( t + 1 ) } \\left( \\| \\theta ( t + 1 ) - \\theta ( t ) \\| _ { 2 } + \\left\\| \\left( \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } - 1 \\right) \\pmb { \\theta } ( t ) \\right\\| _ { 2 } \\right) } \\\\ & { \\qquad = \\displaystyle \\frac { \\eta ( t ) } { \\rho ( t + 1 ) } \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } + \\left( 1 - \\frac { \\rho ( t ) } { \\rho ( t + 1 ) } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "1e83536a68da147fbf43e0db7553f754c84329ecf0f6347db38b29ccecdce645.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 133, + 98, + 477, + 127.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 133, + 127.0, + 477, + 156.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 133, + 156.0, + 477, + 185.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 187, + 367, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 367, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 142, + 206 + ], + "score": 1.0, + "content": "By (16),", + "type": "text" + }, + { + "bbox": [ + 142, + 187, + 227, + 205 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } \\le \\frac { B _ { 1 } \\nu ( t ) } { \\widetilde { \\gamma } ( t ) \\rho ( t ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 185, + 367, + 206 + ], + "score": 1.0, + "content": ". So we can bound the first term as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 185, + 367, + 206 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 208, + 478, + 294 + ], + "lines": [ + { + "bbox": [ + 133, + 208, + 478, + 294 + ], + "spans": [ + { + "bbox": [ + 133, + 208, + 478, + 294 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\displaystyle \\frac { \\eta ( t ) } { \\rho ( t + 1 ) } \\| \\nabla \\mathcal { L } ( t ) \\| _ { 2 } \\le \\frac { B _ { 1 } } { \\widetilde { \\gamma } ( t ) } \\cdot \\frac { \\eta ( t ) \\nu ( t ) } { \\rho ( t + 1 ) \\rho ( t ) } \\le \\frac { B _ { 1 } } { \\widetilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) ^ { 2 } - \\rho ( t ) ^ { 2 } } { 2 L \\rho ( t + 1 ) \\rho ( t ) } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad = \\frac { B _ { 1 } } { 2 L \\widetilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) ^ { 2 } - \\rho ( t ) ^ { 2 } } { \\rho ( t + 1 ) ^ { 2 } } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\le \\frac { B _ { 1 } } { L \\widetilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } \\log \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } } \\end{array}", + "type": "interline_equation", + "image_path": "b22a2de5f367e3d9f03863120c7ac46c7b27eeba4eb65e33116274a3f759edf5.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 133, + 208, + 478, + 236.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 133, + 236.66666666666666, + 478, + 265.3333333333333 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 133, + 265.3333333333333, + 478, + 294.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 103, + 293, + 508, + 322 + ], + "spans": [ + { + "bbox": [ + 103, + 293, + 287, + 322 + ], + "score": 1.0, + "content": "where the last inequality uses the inequality bound the second term by", + "type": "text" + }, + { + "bbox": [ + 288, + 297, + 354, + 311 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { a - b } { a } \\leq \\log ( a / b ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 293, + 508, + 322 + ], + "score": 1.0, + "content": ". Using this inequality again, we can", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 103, + 293, + 508, + 322 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 317, + 459, + 370 + ], + "lines": [ + { + "bbox": [ + 237, + 317, + 459, + 370 + ], + "spans": [ + { + "bbox": [ + 237, + 317, + 459, + 370 + ], + "score": 0.55, + "content": "\\begin{array} { r l } & { \\quad 1 - \\displaystyle \\frac { \\rho ( t ) } { \\rho ( t + 1 ) } \\leq \\log \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } . } \\\\ & { \\quad \\left\\| \\hat { \\pmb \\theta } ( t + 1 ) - \\hat { \\pmb \\theta } ( t ) \\right\\| _ { 2 } \\leq \\left( \\frac { B _ { 1 } } { L \\tilde { \\gamma } ( t ) } \\cdot \\displaystyle \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } + 1 \\right) \\log \\frac { \\rho ( t + 1 ) } { \\rho ( t ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "810d60710ce103b6ac97f702d5f73a7fd659607aed9bc0686c52e8f17301147f.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 237, + 317, + 459, + 334.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 237, + 334.6666666666667, + 459, + 352.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 237, + 352.33333333333337, + 459, + 370.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 347, + 458, + 369 + ], + "lines": [ + { + "bbox": [ + 100, + 343, + 235, + 375 + ], + "spans": [ + { + "bbox": [ + 100, + 343, + 235, + 375 + ], + "score": 1.0, + "content": "Combining these together gives", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 100, + 343, + 235, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 379, + 272, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 274, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 274, + 392 + ], + "score": 1.0, + "content": "Now we are ready to prove Theorem E.3.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 378, + 274, + 392 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "Proof for Theorem E.3. As discussed above, we only need to show a variant of Lemma C.12 for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 414, + 504, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 258, + 432 + ], + "score": 1.0, + "content": "gradient descent: for every limit point", + "type": "text" + }, + { + "bbox": [ + 259, + 416, + 266, + 428 + ], + "score": 0.77, + "content": "\\bar { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 415, + 277, + 432 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 278, + 414, + 339, + 434 + ], + "score": 0.94, + "content": "\\left\\{ { \\hat { \\theta } } ( t ) : t \\geq 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 415, + 445, + 432 + ], + "score": 1.0, + "content": ", there exists a sequence of", + "type": "text" + }, + { + "bbox": [ + 446, + 417, + 504, + 430 + ], + "score": 0.92, + "content": "\\{ t _ { m } : m \\in \\mathbb { N } \\}", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 433, + 308, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 145, + 448 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 145, + 433, + 236, + 447 + ], + "score": 0.92, + "content": "t _ { m } \\uparrow + \\infty , \\hat { \\pmb \\theta } ( t _ { m } ) \\bar { \\pmb \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 434, + 257, + 448 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 257, + 435, + 303, + 447 + ], + "score": 0.92, + "content": "\\beta ( t _ { m } ) 1", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 434, + 308, + 448 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 402, + 505, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 503, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 264, + 465 + ], + "score": 1.0, + "content": "We only need to change the choices of", + "type": "text" + }, + { + "bbox": [ + 264, + 452, + 309, + 464 + ], + "score": 0.94, + "content": "s _ { m } , s _ { m } ^ { \\prime } , t _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 450, + 479, + 465 + ], + "score": 1.0, + "content": "in the proof for Lemma C.12. We choose", + "type": "text" + }, + { + "bbox": [ + 479, + 453, + 505, + 463 + ], + "score": 0.86, + "content": "s _ { m } >", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 463, + 218, + 475 + ], + "spans": [ + { + "bbox": [ + 107, + 464, + 129, + 475 + ], + "score": 0.9, + "content": "t _ { m - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 463, + 218, + 474 + ], + "score": 1.0, + "content": "to be a time such that", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 450, + 505, + 475 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 478, + 425, + 519 + ], + "lines": [ + { + "bbox": [ + 185, + 478, + 425, + 519 + ], + "spans": [ + { + "bbox": [ + 185, + 478, + 425, + 519 + ], + "score": 0.93, + "content": "\\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\| _ { 2 } \\leq \\epsilon _ { m } \\quad \\mathrm { a n d } \\quad \\frac { 1 } { L } \\log ( \\frac { \\operatorname* { l i m } _ { t + \\infty } \\hat { \\gamma } ( t ) } { \\hat { \\gamma } ( s _ { m } ) } ) \\leq \\epsilon _ { m } ^ { 3 } .", + "type": "interline_equation", + "image_path": "629597a695ebe3b0c26369724e6c0c4851c5e766e141044aa84c43276bcf56a7.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 478, + 425, + 498.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 185, + 498.5, + 425, + 519.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 157, + 536 + ], + "score": 1.0, + "content": "Then we let", + "type": "text" + }, + { + "bbox": [ + 157, + 523, + 196, + 534 + ], + "score": 0.92, + "content": "s _ { m } ^ { \\prime } > s _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 520, + 326, + 536 + ], + "score": 1.0, + "content": "be the minimum time such that", + "type": "text" + }, + { + "bbox": [ + 326, + 522, + 445, + 534 + ], + "score": 0.89, + "content": "\\log { \\rho ( s _ { m } ^ { \\prime } ) } \\ge \\log { \\rho ( s _ { m } ) } + \\epsilon _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 520, + 506, + 536 + ], + "score": 1.0, + "content": ". According to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 533, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 163, + 548 + ], + "score": 1.0, + "content": "Theorem E.2,", + "type": "text" + }, + { + "bbox": [ + 163, + 536, + 177, + 545 + ], + "score": 0.82, + "content": "s _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 533, + 194, + 548 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 534, + 208, + 546 + ], + "score": 0.9, + "content": "s _ { m } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 533, + 340, + 548 + ], + "score": 1.0, + "content": "must exist. Finally, we construct", + "type": "text" + }, + { + "bbox": [ + 341, + 535, + 438, + 546 + ], + "score": 0.88, + "content": "t _ { m } \\in \\{ s _ { m } , \\ldots , s _ { m } ^ { \\prime } - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 533, + 506, + 548 + ], + "score": 1.0, + "content": "to be a time that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 542, + 362, + 559 + ], + "spans": [ + { + "bbox": [ + 107, + 544, + 186, + 557 + ], + "score": 0.91, + "content": "\\beta ( t _ { m } ) ^ { - 2 } - 1 \\leq \\epsilon _ { m } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 542, + 362, + 559 + ], + "score": 1.0, + "content": ", where the existence can be shown by (22).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 520, + 506, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 470, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 470, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 352, + 575 + ], + "score": 1.0, + "content": "To see that this construction meets our requirement, note that", + "type": "text" + }, + { + "bbox": [ + 352, + 560, + 452, + 574 + ], + "score": 0.93, + "content": "\\beta ( t _ { m } ) ^ { - 2 } - 1 \\le \\epsilon _ { m } ^ { 2 } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 560, + 470, + 575 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 560, + 470, + 575 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 577, + 495, + 605 + ], + "lines": [ + { + "bbox": [ + 114, + 577, + 495, + 605 + ], + "spans": [ + { + "bbox": [ + 114, + 577, + 495, + 605 + ], + "score": 0.92, + "content": "\\Big \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\bar { \\pmb \\theta } \\Big \\| _ { 2 } \\leq \\Big \\| \\hat { \\pmb \\theta } ( t _ { m } ) - \\hat { \\pmb \\theta } ( s _ { m } ) \\Big \\| _ { 2 } + \\Big \\| \\hat { \\pmb \\theta } ( s _ { m } ) - \\bar { \\pmb \\theta } \\Big \\| _ { 2 } \\leq \\bigg ( \\frac { B _ { 1 } } { L \\hat { \\gamma } ( t _ { 0 } ) } e ^ { \\epsilon _ { m } } + 1 \\bigg ) \\cdot \\epsilon _ { m } + \\epsilon _ { m } 0 ,", + "type": "interline_equation", + "image_path": "3f93d9b25c0dde3e6b9639bf089f54f4edc0df6cc8f9aaaa925fb3b9b1bf4139.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 114, + 577, + 495, + 605 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 609, + 283, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 285, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 285, + 622 + ], + "score": 1.0, + "content": "where the last inequality is by Lemma E.11.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 609, + 285, + 622 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 633, + 245, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 245, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 245, + 646 + ], + "score": 1.0, + "content": "E.6 PROOF FOR THEOREM E.5", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 321, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 321, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 321, + 668 + ], + "score": 1.0, + "content": "Proof. By a similar analysis as Lemma D.4, we have", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 654, + 321, + 668 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 670, + 461, + 732 + ], + "lines": [ + { + "bbox": [ + 150, + 670, + 461, + 732 + ], + "spans": [ + { + "bbox": [ + 150, + 670, + 461, + 732 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } { { E ( x ) = \\int _ { x } ^ { \\mathcal { L } ( t _ { 0 } ) } \\Theta ( \\frac { 1 } { u ^ { 2 } ( \\log \\frac { 1 } { u } ) ^ { 2 - 2 / L } } ) d u = \\int _ { 1 / \\mathcal { L } ( t _ { 0 } ) } ^ { 1 / x } \\Theta ( \\frac { 1 } { ( \\log u ) ^ { 2 - 2 / L } } ) d u } } \\\\ & { } & { = \\Theta ( \\frac { 1 } { x ( \\log \\frac { 1 } { x } ) ^ { 2 - 2 / L } } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6bac32ce62dd4860ece8d882769d3d52b411a87de66fc87b777fcf0ff1975e0c.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 150, + 670, + 461, + 690.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 150, + 690.6666666666666, + 461, + 711.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 150, + 711.3333333333333, + 461, + 731.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 80, + 505, + 110 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 509, + 100 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 271, + 99 + ], + "score": 1.0, + "content": "We can also bound the inverse function", + "type": "text" + }, + { + "bbox": [ + 272, + 83, + 304, + 96 + ], + "score": 0.93, + "content": "E ^ { - 1 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 81, + 320, + 99 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 320, + 80, + 392, + 100 + ], + "score": 0.93, + "content": "\\Theta \\left( { \\frac { 1 } { y ( \\log y ) ^ { 2 - 2 / L } } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 81, + 509, + 99 + ], + "score": 1.0, + "content": ". With these, we can use a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 97, + 335, + 110 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 335, + 110 + ], + "score": 1.0, + "content": "similar analysis as Theorem A.10 to prove Theorem E.5.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 114, + 505, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 299, + 128 + ], + "score": 1.0, + "content": "First, using a similar proof as for (17), we have", + "type": "text" + }, + { + "bbox": [ + 300, + 114, + 364, + 128 + ], + "score": 0.93, + "content": "\\rho ^ { L } = \\Theta ( \\log \\frac { 1 } { \\mathcal { L } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 114, + 473, + 128 + ], + "score": 1.0, + "content": ". So we only need to show", + "type": "text" + }, + { + "bbox": [ + 474, + 115, + 505, + 127 + ], + "score": 0.9, + "content": "{ \\mathcal { L } } ( t ) =", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 123, + 509, + 148 + ], + "spans": [ + { + "bbox": [ + 107, + 127, + 176, + 142 + ], + "score": 0.92, + "content": "\\Theta \\big ( { \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 123, + 509, + 148 + ], + "score": 1.0, + "content": ". With a similar analysis as for (P3) in Lemma E.9, we have the following bound", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 141, + 195, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 120, + 156 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 142, + 191, + 154 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\tau + 1 ) - \\mathcal { L } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 141, + 195, + 156 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 160, + 415, + 175 + ], + "lines": [ + { + "bbox": [ + 195, + 160, + 415, + 175 + ], + "spans": [ + { + "bbox": [ + 195, + 160, + 415, + 175 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\tau + 1 ) - \\mathcal { L } ( \\tau ) \\geq - \\eta ( \\tau ) ( 1 + \\mu ( \\mathcal { L } ( \\tau ) ) ) \\left. \\nabla \\mathcal { L } ( \\tau ) \\right. _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "4833b3bb9de80ed41571b6663bedda4d9a852b5d2d7e435f5dedb5905c0e7093.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 195, + 160, + 415, + 175 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 181, + 506, + 231 + ], + "lines": [ + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 185, + 196 + ], + "score": 1.0, + "content": "Using the fact that", + "type": "text" + }, + { + "bbox": [ + 185, + 183, + 223, + 194 + ], + "score": 0.91, + "content": "\\mu \\leq 1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 181, + 265, + 196 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 265, + 181, + 431, + 196 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\tau + 1 ) - \\mathcal { L } ( \\tau ) \\geq - \\frac { 3 } { 2 } \\eta ( \\tau ) \\left. \\nabla \\mathcal { L } ( \\tau ) \\right. _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 181, + 506, + 196 + ], + "score": 1.0, + "content": ". By Lemma E.7,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 192, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 240, + 206 + ], + "score": 0.88, + "content": "\\| \\nabla \\mathcal { L } ( \\tau ) \\| _ { 2 } \\le 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\tau ) ) \\mathcal { L } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 192, + 506, + 208 + ], + "score": 1.0, + "content": ". Using a similar proof as for Lemma E.10, we can show that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 203, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 107, + 205, + 182, + 216 + ], + "score": 0.85, + "content": "E ( \\mathcal { L } ( t ) ) \\ \\stackrel { } { \\leq } \\ O ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 203, + 376, + 219 + ], + "score": 1.0, + "content": ". Combining this with Lemma E.10, we have", + "type": "text" + }, + { + "bbox": [ + 377, + 205, + 452, + 217 + ], + "score": 0.92, + "content": "E ( \\mathcal { L } ( t ) ) ~ = ~ \\Theta ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 203, + 506, + 219 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 213, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 207, + 231 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { L } ( t ) = \\Theta \\big ( \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 213, + 214, + 235 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 216, + 505, + 227 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 243, + 386, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 387, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 387, + 259 + ], + "score": 1.0, + "content": "F GRADIENT DESCENT: GENERAL LOSS FUNCTIONS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 506, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "It is worth to note that the above analysis can be extended to other loss functions. For this, we need", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "to replace (B3) with a strong assumption (S3), which takes into account the second-order derivatives", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 288, + 131, + 305 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 117, + 305 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 291, + 124, + 303 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 288, + 131, + 305 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 117, + 311, + 411, + 324 + ], + "lines": [ + { + "bbox": [ + 116, + 310, + 412, + 325 + ], + "spans": [ + { + "bbox": [ + 116, + 310, + 214, + 325 + ], + "score": 1.0, + "content": "(S3). The loss function", + "type": "text" + }, + { + "bbox": [ + 214, + 312, + 232, + 324 + ], + "score": 0.9, + "content": "\\ell ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 310, + 313, + 325 + ], + "score": 1.0, + "content": "can be expressed as", + "type": "text" + }, + { + "bbox": [ + 314, + 311, + 371, + 324 + ], + "score": 0.93, + "content": "\\ell ( q ) = e ^ { - f ( q ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 310, + 412, + 325 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 144, + 326, + 277, + 338 + ], + "lines": [ + { + "bbox": [ + 141, + 325, + 278, + 340 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 174, + 340 + ], + "score": 1.0, + "content": "(S3.1).", + "type": "text" + }, + { + "bbox": [ + 174, + 327, + 220, + 339 + ], + "score": 0.89, + "content": "f : \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 325, + 230, + 340 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 230, + 326, + 241, + 337 + ], + "score": 0.84, + "content": "\\mathcal { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 325, + 278, + 340 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 143, + 340, + 272, + 352 + ], + "lines": [ + { + "bbox": [ + 142, + 339, + 272, + 353 + ], + "spans": [ + { + "bbox": [ + 142, + 339, + 174, + 353 + ], + "score": 1.0, + "content": "(S3.2).", + "type": "text" + }, + { + "bbox": [ + 174, + 340, + 216, + 352 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 339, + 243, + 353 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 243, + 340, + 268, + 352 + ], + "score": 0.9, + "content": "q \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 339, + 272, + 353 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 140, + 353, + 504, + 376 + ], + "lines": [ + { + "bbox": [ + 141, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 351, + 227, + 366 + ], + "score": 1.0, + "content": "(S3.3). There exists", + "type": "text" + }, + { + "bbox": [ + 227, + 353, + 260, + 365 + ], + "score": 0.92, + "content": "b _ { f } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 351, + 302, + 366 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 303, + 353, + 330, + 365 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q ) q", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 351, + 422, + 366 + ], + "score": 1.0, + "content": "is non-decreasing for", + "type": "text" + }, + { + "bbox": [ + 422, + 353, + 483, + 366 + ], + "score": 0.92, + "content": "q \\in ( b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 351, + 506, + 366 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 174, + 363, + 290, + 377 + ], + "spans": [ + { + "bbox": [ + 174, + 364, + 235, + 376 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q ) q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 363, + 247, + 377 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 247, + 366, + 286, + 376 + ], + "score": 0.88, + "content": "q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 363, + 290, + 377 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 147, + 378, + 504, + 401 + ], + "lines": [ + { + "bbox": [ + 145, + 375, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 145, + 375, + 191, + 390 + ], + "score": 1.0, + "content": "(S3.4). Let", + "type": "text" + }, + { + "bbox": [ + 192, + 377, + 321, + 390 + ], + "score": 0.9, + "content": "g \\ : \\ [ f ( b _ { f } ) , + \\infty ) \\ \\ [ b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 375, + 433, + 390 + ], + "score": 1.0, + "content": "be the inverse function of", + "type": "text" + }, + { + "bbox": [ + 433, + 378, + 441, + 389 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 375, + 505, + 390 + ], + "score": 1.0, + "content": "on the domain", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 175, + 387, + 440, + 402 + ], + "spans": [ + { + "bbox": [ + 175, + 389, + 213, + 401 + ], + "score": 0.93, + "content": "[ b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 387, + 269, + 402 + ], + "score": 1.0, + "content": ". There exists", + "type": "text" + }, + { + "bbox": [ + 269, + 389, + 294, + 400 + ], + "score": 0.89, + "content": "p \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 387, + 360, + 402 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 360, + 388, + 435, + 401 + ], + "score": 0.92, + "content": "x > f ( b _ { f } ) , y > b _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 387, + 440, + 402 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 405, + 414, + 434 + ], + "lines": [ + { + "bbox": [ + 264, + 405, + 414, + 434 + ], + "spans": [ + { + "bbox": [ + 264, + 405, + 414, + 434 + ], + "score": 0.93, + "content": "\\left| { \\frac { g ^ { \\prime \\prime } ( x ) } { g ^ { \\prime } ( x ) } } \\right| \\leq { \\frac { p } { x } } \\quad { \\mathrm { a n d } } \\quad \\left| { \\frac { f ^ { \\prime \\prime } ( y ) } { f ^ { \\prime } ( y ) } } \\right| \\leq { \\frac { p } { y } } .", + "type": "interline_equation", + "image_path": "19221d8c5f4383da742c9294e37b08def7e22fe213443ac5a840b50ba551ac1d.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 264, + 405, + 414, + 434 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 441, + 506, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "It can be verified that (S3) is satisfied by exponential loss and logistic loss. Now we explain each of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "the assumptions in (S3). (S3.2) and (S3.3) are essentially the same as (B3.2) and (B3.3). (S3.1) and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 462, + 452, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 295, + 475 + ], + "score": 1.0, + "content": "(B3.1) are the same except that (S3.1) assumes", + "type": "text" + }, + { + "bbox": [ + 296, + 464, + 303, + 475 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 463, + 313, + 475 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 313, + 462, + 324, + 473 + ], + "score": 0.87, + "content": "\\mathcal { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 463, + 404, + 475 + ], + "score": 1.0, + "content": "-smooth rather than", + "type": "text" + }, + { + "bbox": [ + 404, + 463, + 415, + 473 + ], + "score": 0.87, + "content": "{ \\mathcal { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 463, + 452, + 475 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 301, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 302, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 302, + 493 + ], + "score": 1.0, + "content": "(S3.4) can also be written in the following form:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 496, + 445, + 525 + ], + "lines": [ + { + "bbox": [ + 167, + 496, + 445, + 525 + ], + "spans": [ + { + "bbox": [ + 167, + 496, + 445, + 525 + ], + "score": 0.92, + "content": "\\left| { \\frac { d } { d x } } \\log g ^ { \\prime } ( x ) \\right| \\leq p \\cdot { \\frac { d } { d x } } \\log x \\quad { \\mathrm { a n d } } \\quad \\left| { \\frac { d } { d y } } \\log f ^ { \\prime } ( y ) \\right| \\leq p \\cdot { \\frac { d } { d y } } \\log y .", + "type": "interline_equation", + "image_path": "9c68847be24d355830261446fe2dd4337cfe6393449001bb700e8de41317834c.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 167, + 496, + 445, + 525 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 506, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 140, + 543 + ], + "score": 1.0, + "content": "That is,", + "type": "text" + }, + { + "bbox": [ + 141, + 530, + 178, + 542 + ], + "score": 0.92, + "content": "\\log g ^ { \\prime } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 529, + 197, + 543 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 198, + 529, + 235, + 542 + ], + "score": 0.92, + "content": "\\log f ^ { \\prime } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 529, + 320, + 543 + ], + "score": 1.0, + "content": "grow no faster than", + "type": "text" + }, + { + "bbox": [ + 320, + 530, + 357, + 542 + ], + "score": 0.92, + "content": "O ( \\log x )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 529, + 377, + 543 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 377, + 530, + 413, + 542 + ], + "score": 0.91, + "content": "O ( \\log y )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 529, + 505, + 543 + ], + "score": 1.0, + "content": ", respectively. In fact,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 393, + 554 + ], + "score": 1.0, + "content": "(B3.4) can be deduced from (S3.4). Recall that (B3.4) ensures that", + "type": "text" + }, + { + "bbox": [ + 393, + 541, + 485, + 553 + ], + "score": 0.91, + "content": "{ \\Theta } \\bar { ( g ^ { \\prime } ( x ) ) \\ } \\stackrel { - } { = } \\ g ^ { \\prime } ( \\stackrel { . } { \\Theta } ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 551, + 484, + 565 + ], + "spans": [ + { + "bbox": [ + 107, + 552, + 195, + 564 + ], + "score": 0.92, + "content": "\\Theta ( f ^ { \\prime } ( y ) ) = f ^ { \\prime } ( \\Theta ( y ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 551, + 429, + 565 + ], + "score": 1.0, + "content": ". Thus, (S3.4) also gives us the interchangeability between", + "type": "text" + }, + { + "bbox": [ + 430, + 553, + 452, + 564 + ], + "score": 0.89, + "content": "f ^ { \\prime } , g ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 551, + 470, + 565 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 553, + 479, + 562 + ], + "score": 0.81, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 551, + 484, + 565 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 435, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 435, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 267, + 581 + ], + "score": 1.0, + "content": "Lemma F.1. (S3.4) implies (B3.4) with", + "type": "text" + }, + { + "bbox": [ + 267, + 566, + 379, + 579 + ], + "score": 0.92, + "content": "b _ { g } = \\operatorname* { m a x } \\{ 2 f ( b _ { f } ) , f ( 2 b _ { f } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 566, + 398, + 581 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 398, + 567, + 431, + 577 + ], + "score": 0.86, + "content": "K = 2 ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 566, + 435, + 581 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 504, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 152, + 606 + ], + "score": 1.0, + "content": "Proof. Fix", + "type": "text" + }, + { + "bbox": [ + 152, + 592, + 285, + 604 + ], + "score": 0.9, + "content": "x \\in ( b _ { g } , + \\infty ) , y \\in ( g ( b _ { g } ) , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 590, + 303, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 304, + 592, + 353, + 604 + ], + "score": 0.91, + "content": "\\theta \\in [ 1 / 2 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 590, + 506, + 606 + ], + "score": 1.0, + "content": ". Integrating (23) on both sides of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 602, + 294, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 177, + 615 + ], + "score": 1.0, + "content": "inequalities from", + "type": "text" + }, + { + "bbox": [ + 177, + 604, + 189, + 613 + ], + "score": 0.84, + "content": "\\theta x", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 602, + 200, + 615 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 200, + 605, + 207, + 613 + ], + "score": 0.8, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 602, + 225, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 604, + 237, + 614 + ], + "score": 0.86, + "content": "\\theta y", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 602, + 248, + 615 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 248, + 605, + 254, + 614 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 602, + 294, + 615 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 619, + 469, + 643 + ], + "lines": [ + { + "bbox": [ + 143, + 619, + 469, + 643 + ], + "spans": [ + { + "bbox": [ + 143, + 619, + 469, + 643 + ], + "score": 0.89, + "content": "| \\log g ^ { \\prime } ( x ) - \\log g ^ { \\prime } ( \\theta x ) | \\leq p \\cdot \\log { \\frac { 1 } { \\theta } } \\quad { \\mathrm { a n d } } \\quad | \\log f ^ { \\prime } ( y ) - \\log f ^ { \\prime } ( \\theta y ) | \\leq p \\cdot \\log { \\frac { 1 } { \\theta } } .", + "type": "interline_equation", + "image_path": "60f4ae8b4640b186ededf3922e9f8301286e8b556d8c88cf7798cd093936b682.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 143, + 619, + 469, + 643 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 648, + 462, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 463, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 186, + 663 + ], + "score": 1.0, + "content": "Therefore, we have", + "type": "text" + }, + { + "bbox": [ + 186, + 648, + 313, + 662 + ], + "score": 0.93, + "content": "g ^ { \\prime } ( x ) \\leq \\theta ^ { - p } g ^ { \\prime } ( \\theta x ) \\leq K g ^ { \\prime } ( \\theta x )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 648, + 331, + 663 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 331, + 648, + 460, + 662 + ], + "score": 0.89, + "content": "f ^ { \\prime } ( y ) \\leq \\theta ^ { - p } f ^ { \\prime } ( \\theta y ) \\leq K f ^ { \\prime } ( \\theta x ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 648, + 463, + 663 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 508, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 672, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 381, + 687 + ], + "score": 1.0, + "content": "To extend our results to general loss functions, we need to redefine", + "type": "text" + }, + { + "bbox": [ + 382, + 673, + 493, + 686 + ], + "score": 0.93, + "content": "\\kappa ( x ) , \\lambda ( x ) , \\phi ( x ) , C _ { \\eta } , H ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 672, + 505, + 687 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 684, + 328, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 149, + 697 + ], + "score": 1.0, + "content": "order. For", + "type": "text" + }, + { + "bbox": [ + 149, + 685, + 169, + 697 + ], + "score": 0.92, + "content": "\\kappa ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 684, + 187, + 697 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 685, + 207, + 697 + ], + "score": 0.92, + "content": "\\lambda ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 684, + 328, + 697 + ], + "score": 1.0, + "content": ", we redefine them as follows:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 702, + 452, + 736 + ], + "lines": [ + { + "bbox": [ + 158, + 702, + 452, + 736 + ], + "spans": [ + { + "bbox": [ + 158, + 702, + 452, + 736 + ], + "score": 0.93, + "content": "\\kappa ( x ) : = \\operatorname* { s u p } \\left\\{ \\frac { w ( \\log \\frac { 1 } { w } ) ^ { 2 - 2 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { w } ) ^ { 2 } } : w \\in ( 0 , x ] \\right\\} \\qquad \\lambda ( x ) : = \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { x } ) } { g ( \\log \\frac { 1 } { x } ) } .", + "type": "interline_equation", + "image_path": "b21d2113bed7f553675a58b92a5cc488a5ff98be53adf0938cf3b05f219cf04d.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 158, + 702, + 452, + 713.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 158, + 713.3333333333334, + 452, + 724.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 158, + 724.6666666666667, + 452, + 736.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + } + ], + "page_idx": 37, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "38", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 649, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 496, + 651, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 496, + 651, + 505, + 660 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 80, + 505, + 110 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 509, + 100 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 271, + 99 + ], + "score": 1.0, + "content": "We can also bound the inverse function", + "type": "text" + }, + { + "bbox": [ + 272, + 83, + 304, + 96 + ], + "score": 0.93, + "content": "E ^ { - 1 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 81, + 320, + 99 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 320, + 80, + 392, + 100 + ], + "score": 0.93, + "content": "\\Theta \\left( { \\frac { 1 } { y ( \\log y ) ^ { 2 - 2 / L } } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 81, + 509, + 99 + ], + "score": 1.0, + "content": ". With these, we can use a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 97, + 335, + 110 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 335, + 110 + ], + "score": 1.0, + "content": "similar analysis as Theorem A.10 to prove Theorem E.5.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 80, + 509, + 110 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 114, + 505, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 299, + 128 + ], + "score": 1.0, + "content": "First, using a similar proof as for (17), we have", + "type": "text" + }, + { + "bbox": [ + 300, + 114, + 364, + 128 + ], + "score": 0.93, + "content": "\\rho ^ { L } = \\Theta ( \\log \\frac { 1 } { \\mathcal { L } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 114, + 473, + 128 + ], + "score": 1.0, + "content": ". So we only need to show", + "type": "text" + }, + { + "bbox": [ + 474, + 115, + 505, + 127 + ], + "score": 0.9, + "content": "{ \\mathcal { L } } ( t ) =", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 123, + 509, + 148 + ], + "spans": [ + { + "bbox": [ + 107, + 127, + 176, + 142 + ], + "score": 0.92, + "content": "\\Theta \\big ( { \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 123, + 509, + 148 + ], + "score": 1.0, + "content": ". With a similar analysis as for (P3) in Lemma E.9, we have the following bound", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 141, + 195, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 120, + 156 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 142, + 191, + 154 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\tau + 1 ) - \\mathcal { L } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 141, + 195, + 156 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 114, + 509, + 156 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 160, + 415, + 175 + ], + "lines": [ + { + "bbox": [ + 195, + 160, + 415, + 175 + ], + "spans": [ + { + "bbox": [ + 195, + 160, + 415, + 175 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\tau + 1 ) - \\mathcal { L } ( \\tau ) \\geq - \\eta ( \\tau ) ( 1 + \\mu ( \\mathcal { L } ( \\tau ) ) ) \\left. \\nabla \\mathcal { L } ( \\tau ) \\right. _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "4833b3bb9de80ed41571b6663bedda4d9a852b5d2d7e435f5dedb5905c0e7093.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 195, + 160, + 415, + 175 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 181, + 506, + 231 + ], + "lines": [ + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 185, + 196 + ], + "score": 1.0, + "content": "Using the fact that", + "type": "text" + }, + { + "bbox": [ + 185, + 183, + 223, + 194 + ], + "score": 0.91, + "content": "\\mu \\leq 1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 181, + 265, + 196 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 265, + 181, + 431, + 196 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\tau + 1 ) - \\mathcal { L } ( \\tau ) \\geq - \\frac { 3 } { 2 } \\eta ( \\tau ) \\left. \\nabla \\mathcal { L } ( \\tau ) \\right. _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 181, + 506, + 196 + ], + "score": 1.0, + "content": ". By Lemma E.7,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 192, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 240, + 206 + ], + "score": 0.88, + "content": "\\| \\nabla \\mathcal { L } ( \\tau ) \\| _ { 2 } \\le 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\tau ) ) \\mathcal { L } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 192, + 506, + 208 + ], + "score": 1.0, + "content": ". Using a similar proof as for Lemma E.10, we can show that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 203, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 107, + 205, + 182, + 216 + ], + "score": 0.85, + "content": "E ( \\mathcal { L } ( t ) ) \\ \\stackrel { } { \\leq } \\ O ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 203, + 376, + 219 + ], + "score": 1.0, + "content": ". Combining this with Lemma E.10, we have", + "type": "text" + }, + { + "bbox": [ + 377, + 205, + 452, + 217 + ], + "score": 0.92, + "content": "E ( \\mathcal { L } ( t ) ) ~ = ~ \\Theta ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 203, + 506, + 219 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 213, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 207, + 231 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { L } ( t ) = \\Theta \\big ( \\frac { 1 } { T ( \\log T ) ^ { 2 - 2 / L } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 213, + 214, + 235 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 216, + 505, + 227 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 181, + 506, + 235 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 243, + 386, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 387, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 387, + 259 + ], + "score": 1.0, + "content": "F GRADIENT DESCENT: GENERAL LOSS FUNCTIONS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 506, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "It is worth to note that the above analysis can be extended to other loss functions. For this, we need", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "to replace (B3) with a strong assumption (S3), which takes into account the second-order derivatives", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 288, + 131, + 305 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 117, + 305 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 291, + 124, + 303 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 288, + 131, + 305 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 268, + 505, + 305 + ] + }, + { + "type": "text", + "bbox": [ + 117, + 311, + 411, + 324 + ], + "lines": [ + { + "bbox": [ + 116, + 310, + 412, + 325 + ], + "spans": [ + { + "bbox": [ + 116, + 310, + 214, + 325 + ], + "score": 1.0, + "content": "(S3). The loss function", + "type": "text" + }, + { + "bbox": [ + 214, + 312, + 232, + 324 + ], + "score": 0.9, + "content": "\\ell ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 310, + 313, + 325 + ], + "score": 1.0, + "content": "can be expressed as", + "type": "text" + }, + { + "bbox": [ + 314, + 311, + 371, + 324 + ], + "score": 0.93, + "content": "\\ell ( q ) = e ^ { - f ( q ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 310, + 412, + 325 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 116, + 310, + 412, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 144, + 326, + 277, + 338 + ], + "lines": [ + { + "bbox": [ + 141, + 325, + 278, + 340 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 174, + 340 + ], + "score": 1.0, + "content": "(S3.1).", + "type": "text" + }, + { + "bbox": [ + 174, + 327, + 220, + 339 + ], + "score": 0.89, + "content": "f : \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 325, + 230, + 340 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 230, + 326, + 241, + 337 + ], + "score": 0.84, + "content": "\\mathcal { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 325, + 278, + 340 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 141, + 325, + 278, + 340 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 340, + 272, + 352 + ], + "lines": [ + { + "bbox": [ + 142, + 339, + 272, + 353 + ], + "spans": [ + { + "bbox": [ + 142, + 339, + 174, + 353 + ], + "score": 1.0, + "content": "(S3.2).", + "type": "text" + }, + { + "bbox": [ + 174, + 340, + 216, + 352 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 339, + 243, + 353 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 243, + 340, + 268, + 352 + ], + "score": 0.9, + "content": "q \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 339, + 272, + 353 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 142, + 339, + 272, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 140, + 353, + 504, + 376 + ], + "lines": [ + { + "bbox": [ + 141, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 351, + 227, + 366 + ], + "score": 1.0, + "content": "(S3.3). There exists", + "type": "text" + }, + { + "bbox": [ + 227, + 353, + 260, + 365 + ], + "score": 0.92, + "content": "b _ { f } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 351, + 302, + 366 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 303, + 353, + 330, + 365 + ], + "score": 0.92, + "content": "f ^ { \\prime } ( q ) q", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 351, + 422, + 366 + ], + "score": 1.0, + "content": "is non-decreasing for", + "type": "text" + }, + { + "bbox": [ + 422, + 353, + 483, + 366 + ], + "score": 0.92, + "content": "q \\in ( b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 351, + 506, + 366 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 174, + 363, + 290, + 377 + ], + "spans": [ + { + "bbox": [ + 174, + 364, + 235, + 376 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q ) q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 363, + 247, + 377 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 247, + 366, + 286, + 376 + ], + "score": 0.88, + "content": "q \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 363, + 290, + 377 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 141, + 351, + 506, + 377 + ] + }, + { + "type": "text", + "bbox": [ + 147, + 378, + 504, + 401 + ], + "lines": [ + { + "bbox": [ + 145, + 375, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 145, + 375, + 191, + 390 + ], + "score": 1.0, + "content": "(S3.4). Let", + "type": "text" + }, + { + "bbox": [ + 192, + 377, + 321, + 390 + ], + "score": 0.9, + "content": "g \\ : \\ [ f ( b _ { f } ) , + \\infty ) \\ \\ [ b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 375, + 433, + 390 + ], + "score": 1.0, + "content": "be the inverse function of", + "type": "text" + }, + { + "bbox": [ + 433, + 378, + 441, + 389 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 375, + 505, + 390 + ], + "score": 1.0, + "content": "on the domain", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 175, + 387, + 440, + 402 + ], + "spans": [ + { + "bbox": [ + 175, + 389, + 213, + 401 + ], + "score": 0.93, + "content": "[ b _ { f } , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 387, + 269, + 402 + ], + "score": 1.0, + "content": ". There exists", + "type": "text" + }, + { + "bbox": [ + 269, + 389, + 294, + 400 + ], + "score": 0.89, + "content": "p \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 387, + 360, + 402 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 360, + 388, + 435, + 401 + ], + "score": 0.92, + "content": "x > f ( b _ { f } ) , y > b _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 387, + 440, + 402 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 145, + 375, + 505, + 402 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 405, + 414, + 434 + ], + "lines": [ + { + "bbox": [ + 264, + 405, + 414, + 434 + ], + "spans": [ + { + "bbox": [ + 264, + 405, + 414, + 434 + ], + "score": 0.93, + "content": "\\left| { \\frac { g ^ { \\prime \\prime } ( x ) } { g ^ { \\prime } ( x ) } } \\right| \\leq { \\frac { p } { x } } \\quad { \\mathrm { a n d } } \\quad \\left| { \\frac { f ^ { \\prime \\prime } ( y ) } { f ^ { \\prime } ( y ) } } \\right| \\leq { \\frac { p } { y } } .", + "type": "interline_equation", + "image_path": "19221d8c5f4383da742c9294e37b08def7e22fe213443ac5a840b50ba551ac1d.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 264, + 405, + 414, + 434 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 441, + 506, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "It can be verified that (S3) is satisfied by exponential loss and logistic loss. Now we explain each of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "the assumptions in (S3). (S3.2) and (S3.3) are essentially the same as (B3.2) and (B3.3). (S3.1) and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 462, + 452, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 295, + 475 + ], + "score": 1.0, + "content": "(B3.1) are the same except that (S3.1) assumes", + "type": "text" + }, + { + "bbox": [ + 296, + 464, + 303, + 475 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 463, + 313, + 475 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 313, + 462, + 324, + 473 + ], + "score": 0.87, + "content": "\\mathcal { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 463, + 404, + 475 + ], + "score": 1.0, + "content": "-smooth rather than", + "type": "text" + }, + { + "bbox": [ + 404, + 463, + 415, + 473 + ], + "score": 0.87, + "content": "{ \\mathcal { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 463, + 452, + 475 + ], + "score": 1.0, + "content": "-smooth.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 440, + 506, + 475 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 301, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 302, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 302, + 493 + ], + "score": 1.0, + "content": "(S3.4) can also be written in the following form:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 479, + 302, + 493 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 496, + 445, + 525 + ], + "lines": [ + { + "bbox": [ + 167, + 496, + 445, + 525 + ], + "spans": [ + { + "bbox": [ + 167, + 496, + 445, + 525 + ], + "score": 0.92, + "content": "\\left| { \\frac { d } { d x } } \\log g ^ { \\prime } ( x ) \\right| \\leq p \\cdot { \\frac { d } { d x } } \\log x \\quad { \\mathrm { a n d } } \\quad \\left| { \\frac { d } { d y } } \\log f ^ { \\prime } ( y ) \\right| \\leq p \\cdot { \\frac { d } { d y } } \\log y .", + "type": "interline_equation", + "image_path": "9c68847be24d355830261446fe2dd4337cfe6393449001bb700e8de41317834c.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 167, + 496, + 445, + 525 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 506, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 140, + 543 + ], + "score": 1.0, + "content": "That is,", + "type": "text" + }, + { + "bbox": [ + 141, + 530, + 178, + 542 + ], + "score": 0.92, + "content": "\\log g ^ { \\prime } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 529, + 197, + 543 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 198, + 529, + 235, + 542 + ], + "score": 0.92, + "content": "\\log f ^ { \\prime } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 529, + 320, + 543 + ], + "score": 1.0, + "content": "grow no faster than", + "type": "text" + }, + { + "bbox": [ + 320, + 530, + 357, + 542 + ], + "score": 0.92, + "content": "O ( \\log x )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 529, + 377, + 543 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 377, + 530, + 413, + 542 + ], + "score": 0.91, + "content": "O ( \\log y )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 529, + 505, + 543 + ], + "score": 1.0, + "content": ", respectively. In fact,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 393, + 554 + ], + "score": 1.0, + "content": "(B3.4) can be deduced from (S3.4). Recall that (B3.4) ensures that", + "type": "text" + }, + { + "bbox": [ + 393, + 541, + 485, + 553 + ], + "score": 0.91, + "content": "{ \\Theta } \\bar { ( g ^ { \\prime } ( x ) ) \\ } \\stackrel { - } { = } \\ g ^ { \\prime } ( \\stackrel { . } { \\Theta } ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 551, + 484, + 565 + ], + "spans": [ + { + "bbox": [ + 107, + 552, + 195, + 564 + ], + "score": 0.92, + "content": "\\Theta ( f ^ { \\prime } ( y ) ) = f ^ { \\prime } ( \\Theta ( y ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 551, + 429, + 565 + ], + "score": 1.0, + "content": ". Thus, (S3.4) also gives us the interchangeability between", + "type": "text" + }, + { + "bbox": [ + 430, + 553, + 452, + 564 + ], + "score": 0.89, + "content": "f ^ { \\prime } , g ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 551, + 470, + 565 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 553, + 479, + 562 + ], + "score": 0.81, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 551, + 484, + 565 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 529, + 506, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 435, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 435, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 267, + 581 + ], + "score": 1.0, + "content": "Lemma F.1. (S3.4) implies (B3.4) with", + "type": "text" + }, + { + "bbox": [ + 267, + 566, + 379, + 579 + ], + "score": 0.92, + "content": "b _ { g } = \\operatorname* { m a x } \\{ 2 f ( b _ { f } ) , f ( 2 b _ { f } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 566, + 398, + 581 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 398, + 567, + 431, + 577 + ], + "score": 0.86, + "content": "K = 2 ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 566, + 435, + 581 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 566, + 435, + 581 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 504, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 152, + 606 + ], + "score": 1.0, + "content": "Proof. Fix", + "type": "text" + }, + { + "bbox": [ + 152, + 592, + 285, + 604 + ], + "score": 0.9, + "content": "x \\in ( b _ { g } , + \\infty ) , y \\in ( g ( b _ { g } ) , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 590, + 303, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 304, + 592, + 353, + 604 + ], + "score": 0.91, + "content": "\\theta \\in [ 1 / 2 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 590, + 506, + 606 + ], + "score": 1.0, + "content": ". Integrating (23) on both sides of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 602, + 294, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 177, + 615 + ], + "score": 1.0, + "content": "inequalities from", + "type": "text" + }, + { + "bbox": [ + 177, + 604, + 189, + 613 + ], + "score": 0.84, + "content": "\\theta x", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 602, + 200, + 615 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 200, + 605, + 207, + 613 + ], + "score": 0.8, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 602, + 225, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 604, + 237, + 614 + ], + "score": 0.86, + "content": "\\theta y", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 602, + 248, + 615 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 248, + 605, + 254, + 614 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 602, + 294, + 615 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 590, + 506, + 615 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 619, + 469, + 643 + ], + "lines": [ + { + "bbox": [ + 143, + 619, + 469, + 643 + ], + "spans": [ + { + "bbox": [ + 143, + 619, + 469, + 643 + ], + "score": 0.89, + "content": "| \\log g ^ { \\prime } ( x ) - \\log g ^ { \\prime } ( \\theta x ) | \\leq p \\cdot \\log { \\frac { 1 } { \\theta } } \\quad { \\mathrm { a n d } } \\quad | \\log f ^ { \\prime } ( y ) - \\log f ^ { \\prime } ( \\theta y ) | \\leq p \\cdot \\log { \\frac { 1 } { \\theta } } .", + "type": "interline_equation", + "image_path": "60f4ae8b4640b186ededf3922e9f8301286e8b556d8c88cf7798cd093936b682.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 143, + 619, + 469, + 643 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 648, + 462, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 463, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 186, + 663 + ], + "score": 1.0, + "content": "Therefore, we have", + "type": "text" + }, + { + "bbox": [ + 186, + 648, + 313, + 662 + ], + "score": 0.93, + "content": "g ^ { \\prime } ( x ) \\leq \\theta ^ { - p } g ^ { \\prime } ( \\theta x ) \\leq K g ^ { \\prime } ( \\theta x )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 648, + 331, + 663 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 331, + 648, + 460, + 662 + ], + "score": 0.89, + "content": "f ^ { \\prime } ( y ) \\leq \\theta ^ { - p } f ^ { \\prime } ( \\theta y ) \\leq K f ^ { \\prime } ( \\theta x ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 648, + 463, + 663 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 648, + 463, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 508, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 672, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 381, + 687 + ], + "score": 1.0, + "content": "To extend our results to general loss functions, we need to redefine", + "type": "text" + }, + { + "bbox": [ + 382, + 673, + 493, + 686 + ], + "score": 0.93, + "content": "\\kappa ( x ) , \\lambda ( x ) , \\phi ( x ) , C _ { \\eta } , H ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 672, + 505, + 687 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 684, + 328, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 149, + 697 + ], + "score": 1.0, + "content": "order. For", + "type": "text" + }, + { + "bbox": [ + 149, + 685, + 169, + 697 + ], + "score": 0.92, + "content": "\\kappa ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 684, + 187, + 697 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 685, + 207, + 697 + ], + "score": 0.92, + "content": "\\lambda ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 684, + 328, + 697 + ], + "score": 1.0, + "content": ", we redefine them as follows:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 672, + 505, + 697 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 702, + 452, + 736 + ], + "lines": [ + { + "bbox": [ + 158, + 702, + 452, + 736 + ], + "spans": [ + { + "bbox": [ + 158, + 702, + 452, + 736 + ], + "score": 0.93, + "content": "\\kappa ( x ) : = \\operatorname* { s u p } \\left\\{ \\frac { w ( \\log \\frac { 1 } { w } ) ^ { 2 - 2 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { w } ) ^ { 2 } } : w \\in ( 0 , x ] \\right\\} \\qquad \\lambda ( x ) : = \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { x } ) } { g ( \\log \\frac { 1 } { x } ) } .", + "type": "interline_equation", + "image_path": "b21d2113bed7f553675a58b92a5cc488a5ff98be53adf0938cf3b05f219cf04d.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 158, + 702, + 452, + 713.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 158, + 713.3333333333334, + 452, + 724.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 158, + 724.6666666666667, + 452, + 736.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 79, + 505, + 111 + ], + "lines": [ + { + "bbox": [ + 104, + 81, + 505, + 100 + ], + "spans": [ + { + "bbox": [ + 104, + 84, + 176, + 96 + ], + "score": 1.0, + "content": "By Lemma D.1,", + "type": "text" + }, + { + "bbox": [ + 176, + 81, + 398, + 100 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { w ( \\log \\frac { 1 } { w } ) ^ { 2 - 2 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { w } ) ^ { 2 } } = O \\left( w ( \\log \\frac { 1 } { w } ) ^ { 4 - 2 / L } / g ( \\log \\frac { 1 } { w } ) ^ { 2 } \\right) \\to 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 82, + 417, + 97 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 418, + 84, + 438, + 97 + ], + "score": 0.91, + "content": "\\kappa ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 82, + 505, + 97 + ], + "score": 1.0, + "content": "is well-defined.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 321, + 112 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 133, + 112 + ], + "score": 1.0, + "content": "Using", + "type": "text" + }, + { + "bbox": [ + 133, + 100, + 153, + 111 + ], + "score": 0.91, + "content": "\\lambda ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 99, + 214, + 112 + ], + "score": 1.0, + "content": ", we can define", + "type": "text" + }, + { + "bbox": [ + 214, + 100, + 235, + 112 + ], + "score": 0.91, + "content": "\\phi ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 99, + 253, + 112 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 100, + 273, + 112 + ], + "score": 0.91, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 99, + 321, + 112 + ], + "score": 1.0, + "content": "as follows.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 115, + 447, + 200 + ], + "lines": [ + { + "bbox": [ + 164, + 115, + 447, + 200 + ], + "spans": [ + { + "bbox": [ + 164, + 115, + 447, + 200 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\phi ^ { \\prime } ( x ) : = - \\operatorname* { s u p } \\left\\{ \\displaystyle \\frac { \\lambda ( w ) } { w } \\left( 1 + 2 ( 1 + \\lambda ( w ) / L ) \\mu ( w ) \\right) : w \\in [ x , \\mathcal { L } ( t _ { 0 } ) ] \\right\\} } \\\\ & { \\displaystyle \\phi ( x ) : = \\log \\left( g ( \\log \\frac { 1 } { x } ) \\right) + \\int _ { 0 } ^ { x } \\left( \\phi ^ { \\prime } ( w ) + \\frac { \\lambda ( w ) } { w } \\right) d w } \\\\ & { \\displaystyle \\hat { \\gamma } ( \\pmb { \\theta } ) : = \\frac { e ^ { \\phi ( \\mathcal { L } ( \\pmb { \\theta } ) ) } } { \\rho ^ { L } } . } \\end{array}", + "type": "interline_equation", + "image_path": "b2233ecccbab3a94fb199d3489c179e9d15a45110c22cf1582d9e7db4f46f7cc.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 164, + 115, + 447, + 143.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 164, + 143.33333333333334, + 447, + 171.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 164, + 171.66666666666669, + 447, + 200.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 201, + 504, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 366, + 216 + ], + "score": 1.0, + "content": "Using a similar argument as in Lemma E.1, we can show that", + "type": "text" + }, + { + "bbox": [ + 366, + 203, + 387, + 214 + ], + "score": 0.9, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 200, + 470, + 216 + ], + "score": 1.0, + "content": "is well-defined and", + "type": "text" + }, + { + "bbox": [ + 471, + 202, + 505, + 215 + ], + "score": 0.9, + "content": "\\hat { \\gamma } ( \\pmb \\theta ) ~ <", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 212, + 427, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 227, + 227 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\tilde { \\gamma } ( \\pmb { \\theta } ) : = g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\breve { / } \\rho ^ { L } \\leq \\bar { \\gamma } ( \\pmb { \\theta } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 213, + 258, + 226 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 258, + 213, + 300, + 225 + ], + "score": 0.92, + "content": "{ \\mathcal { L } } ( \\pmb { \\theta } ) \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 213, + 357, + 226 + ], + "score": 1.0, + "content": ", we also have", + "type": "text" + }, + { + "bbox": [ + 358, + 213, + 423, + 225 + ], + "score": 0.92, + "content": "\\hat { \\gamma } ( \\pmb { \\theta } ) / \\bar { \\gamma } ( \\pmb { \\theta } ) 1", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 213, + 427, + 226 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 230, + 268, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 269, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 122, + 244 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 230, + 136, + 243 + ], + "score": 0.88, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 228, + 269, + 244 + ], + "score": 1.0, + "content": ", we define it to be the following.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 245, + 507, + 279 + ], + "lines": [ + { + "bbox": [ + 111, + 245, + 507, + 279 + ], + "spans": [ + { + "bbox": [ + 111, + 245, + 507, + 279 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\gamma _ { \\eta } : = \\frac { 1 } { 2 } B _ { 1 } \\left( \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } B _ { 1 } + \\frac { 2 ^ { p + 1 } } { \\log \\frac { 1 } { C ( t _ { 0 } ) } } ( p B _ { 1 } + B _ { 2 } ) \\right) \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } \\operatorname* { m i n } \\left\\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "ec4bb9e70cec3464c55ef0cd792ecb8d9a5a2e02759e0e0dfed7ae19882259bc.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 111, + 245, + 507, + 256.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 256.3333333333333, + 507, + 267.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 267.66666666666663, + 507, + 278.99999999999994 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 504, + 310 + ], + "lines": [ + { + "bbox": [ + 104, + 280, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 218, + 301 + ], + "score": 1.0, + "content": "Finally, the definitions for", + "type": "text" + }, + { + "bbox": [ + 218, + 283, + 343, + 299 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu ( x ) : = ( \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ) / ( 2 \\log \\frac { 1 } { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 280, + 363, + 301 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 364, + 284, + 471, + 297 + ], + "score": 0.93, + "content": "H ( x ) : = \\mu ( x ) / ( C _ { \\eta } \\kappa ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 280, + 506, + 301 + ], + "score": 1.0, + "content": "in (S5)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 296, + 397, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 228, + 311 + ], + "score": 1.0, + "content": "remain unchanged except that", + "type": "text" + }, + { + "bbox": [ + 228, + 298, + 248, + 309 + ], + "score": 0.91, + "content": "\\kappa ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 296, + 266, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 267, + 298, + 280, + 310 + ], + "score": 0.88, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 296, + 397, + 311 + ], + "score": 1.0, + "content": "now use the new definitions.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 506, + 352 + ], + "lines": [ + { + "bbox": [ + 101, + 308, + 510, + 337 + ], + "spans": [ + { + "bbox": [ + 101, + 308, + 251, + 337 + ], + "score": 1.0, + "content": "Similar as gradient flow, we define", + "type": "text" + }, + { + "bbox": [ + 251, + 315, + 414, + 330 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nu ( t ) : = \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ( t ) ) } f ^ { \\prime } ( q _ { n } ( t ) ) q _ { n } ( t ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 308, + 510, + 337 + ], + "score": 1.0, + "content": ". The key idea behind", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 328, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 346, + 343 + ], + "score": 1.0, + "content": "the above definitions is that we can prove similar bounds for", + "type": "text" + }, + { + "bbox": [ + 347, + 330, + 437, + 342 + ], + "score": 0.91, + "content": "\\nu ( t ) , \\| \\nabla \\mathcal { L } \\| _ { 2 } , \\| \\nabla ^ { 2 } \\mathcal { L } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 328, + 506, + 343 + ], + "score": 1.0, + "content": "as Corollary E.6", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 340, + 175, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 175, + 351 + ], + "score": 1.0, + "content": "and Lemma E.7.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 505, + 391 + ], + "lines": [ + { + "bbox": [ + 103, + 347, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 103, + 347, + 162, + 374 + ], + "score": 1.0, + "content": "Lemma F.2.", + "type": "text" + }, + { + "bbox": [ + 162, + 354, + 263, + 367 + ], + "score": 0.9, + "content": "\\langle \\pmb \\theta ( t ) , - \\nabla \\mathcal L ( t ) \\rangle = L \\nu ( t ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 347, + 276, + 374 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 276, + 353, + 338, + 367 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t ) < e ^ { - f ( b _ { f } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 347, + 362, + 374 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 362, + 352, + 421, + 370 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\nu ( t ) \\geq \\frac { \\mathcal { L } ( t ) } { \\lambda ( \\mathcal { L } ( t ) ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 347, + 440, + 374 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 440, + 354, + 473, + 367 + ], + "score": 0.91, + "content": "\\lambda ( \\mathcal { L } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 347, + 506, + 374 + ], + "score": 1.0, + "content": "has the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 102, + 369, + 283, + 392 + ], + "spans": [ + { + "bbox": [ + 102, + 370, + 159, + 392 + ], + "score": 1.0, + "content": "lower bound", + "type": "text" + }, + { + "bbox": [ + 160, + 369, + 280, + 390 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\lambda ( \\mathcal { L } ( t ) ) \\leq 2 ^ { p + 1 } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 370, + 283, + 392 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 400, + 504, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 414 + ], + "score": 1.0, + "content": "Proof. It can be easily proved by combining Theorem B.4, Lemma B.5 and Lemma D.1 together.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 496, + 414, + 504, + 421 + ], + "spans": [ + { + "bbox": [ + 496, + 414, + 504, + 421 + ], + "score": 0.991, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 105, + 429, + 355, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 353, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 195, + 443 + ], + "score": 1.0, + "content": "Lemma F.3. For any", + "type": "text" + }, + { + "bbox": [ + 195, + 431, + 201, + 440 + ], + "score": 0.63, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 429, + 214, + 443 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 214, + 430, + 330, + 443 + ], + "score": 0.89, + "content": "\\mathcal { L } ( \\pmb \\theta ) \\leq \\mathcal { L } ( t _ { 0 } ) , \\tilde { \\gamma } ( \\pmb \\theta ) \\geq \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 429, + 353, + 443 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 445, + 447, + 461 + ], + "lines": [ + { + "bbox": [ + 163, + 445, + 447, + 461 + ], + "spans": [ + { + "bbox": [ + 163, + 445, + 447, + 461 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\| \\nabla \\mathcal { L } ( \\pmb { \\theta } ) \\| _ { 2 } ^ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) \\mathcal { L } ( \\pmb { \\theta } ) \\quad a n d \\quad \\left\\| \\nabla ^ { 2 } \\mathcal { L } ( \\pmb { \\theta } ) \\right\\| _ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "5e3aa293e04ba12492267f2503a2228a2c9280b89b0018630b475b8731c12dca.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 163, + 445, + 447, + 461 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 305, + 487 + ], + "score": 1.0, + "content": "Proof. Note that a direct corollary of (23) is that", + "type": "text" + }, + { + "bbox": [ + 305, + 472, + 411, + 485 + ], + "score": 0.93, + "content": "f ^ { \\prime } ( x _ { 1 } ) \\leq f ^ { \\prime } ( x _ { 2 } ) ( x _ { 1 } / x _ { 2 } ) ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 471, + 427, + 487 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 428, + 473, + 487, + 485 + ], + "score": 0.92, + "content": "x _ { 1 } \\geq x _ { 2 } > b _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 471, + 506, + 487 + ], + "score": 1.0, + "content": ". So", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 143, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 143, + 497 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 496, + 466, + 527 + ], + "lines": [ + { + "bbox": [ + 144, + 496, + 466, + 527 + ], + "spans": [ + { + "bbox": [ + 144, + 496, + 466, + 527 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q _ { n } ) \\leq f ^ { \\prime } \\left( g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) \\right) \\left( { \\frac { q _ { n } } { g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } \\right) ^ { p } = { \\frac { 1 } { g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } \\left( { \\frac { q _ { n } / \\rho ^ { L } } { \\widetilde { \\gamma } } } \\right) ^ { p } \\leq { \\frac { R } { g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } ,", + "type": "interline_equation", + "image_path": "73613c208a21c367087550ae5413da2b7d66550b4522a0baeea4c6ac3e6c12b2.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 144, + 496, + 466, + 506.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 144, + 506.3333333333333, + 466, + 516.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 144, + 516.6666666666666, + 466, + 527.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 381, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 381, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 133, + 544 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 530, + 209, + 542 + ], + "score": 0.93, + "content": "R : = ( B _ { 0 } / \\hat { \\gamma } ( t _ { 0 } ) ) ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 529, + 381, + 544 + ], + "score": 1.0, + "content": ". Applying (S3.4), we can also deduce that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 545, + 399, + 574 + ], + "lines": [ + { + "bbox": [ + 211, + 545, + 399, + 574 + ], + "spans": [ + { + "bbox": [ + 211, + 545, + 399, + 574 + ], + "score": 0.93, + "content": "| f ^ { \\prime \\prime } ( q _ { n } ) | \\leq { \\frac { p } { q _ { n } } } \\cdot f ^ { \\prime } ( q _ { n } ) \\leq { \\frac { p R } { g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } .", + "type": "interline_equation", + "image_path": "25293948d1d5ce46b2034a1543169fe865fb0827e643f8636917f4c19324091e.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 211, + 545, + 399, + 574 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 369, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 368, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 170, + 592 + ], + "score": 1.0, + "content": "Now we bound", + "type": "text" + }, + { + "bbox": [ + 170, + 578, + 200, + 591 + ], + "score": 0.93, + "content": "\\lVert \\nabla \\mathcal { L } \\rVert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 577, + 219, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 577, + 255, + 592 + ], + "score": 0.93, + "content": "\\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 577, + 368, + 592 + ], + "score": 1.0, + "content": ". By the chain rule, we have", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 595, + 500, + 733 + ], + "lines": [ + { + "bbox": [ + 112, + 595, + 500, + 733 + ], + "spans": [ + { + "bbox": [ + 112, + 595, + 500, + 733 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\left\\| \\nabla \\mathcal { L } \\right\\| _ { 2 } = \\left\\| - \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } f ^ { \\prime } ( \\boldsymbol { q } _ { n } ) \\nabla \\boldsymbol { q } _ { n } \\right\\| _ { 2 } \\leq B _ { 1 } R \\cdot \\frac { \\mathcal { L } \\rho ^ { L - 1 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } } \\\\ & { \\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } \\left( \\big ( f ^ { \\prime } ( \\boldsymbol { q } _ { n } ) ^ { 2 } - f ^ { \\prime \\prime } ( \\boldsymbol { q } _ { n } ) \\big ) \\nabla q _ { n } \\nabla q _ { n } ^ { \\top } - f ^ { \\prime } ( q _ { n } ) \\nabla ^ { 2 } q _ { n } \\right) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } \\left( \\left( \\frac { R ^ { 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } + \\frac { p R } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\right) B _ { 1 } ^ { 2 } \\rho ^ { 2 L - 2 } + \\frac { R } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } B _ { 2 } \\rho ^ { L - 2 } \\right) } \\\\ & { \\qquad \\leq \\frac { \\mathcal { L } \\rho ^ { 2 L - 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\left( \\left( R + p \\lambda ( \\mathcal { L } ) \\right) B _ { 1 } ^ { 2 } + \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\rho ^ { L } } B _ { 2 } \\right) R . } \\end{array}", + "type": "interline_equation", + "image_path": "63f9e0fdb6110c12727f7233454a668cd4590f7bc7d3ef7a807c85fdcca98eb4.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 112, + 595, + 500, + 641.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 112, + 641.0, + 500, + 687.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 112, + 687.0, + 500, + 733.0 + ], + "spans": [], + "index": 32 + } + ] + } + ], + "page_idx": 38, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "39", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 107, + 79, + 505, + 111 + ], + "lines": [ + { + "bbox": [ + 104, + 81, + 505, + 100 + ], + "spans": [ + { + "bbox": [ + 104, + 84, + 176, + 96 + ], + "score": 1.0, + "content": "By Lemma D.1,", + "type": "text" + }, + { + "bbox": [ + 176, + 81, + 398, + 100 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { w ( \\log \\frac { 1 } { w } ) ^ { 2 - 2 / L } } { g ^ { \\prime } ( \\log \\frac { 1 } { w } ) ^ { 2 } } = O \\left( w ( \\log \\frac { 1 } { w } ) ^ { 4 - 2 / L } / g ( \\log \\frac { 1 } { w } ) ^ { 2 } \\right) \\to 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 82, + 417, + 97 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 418, + 84, + 438, + 97 + ], + "score": 0.91, + "content": "\\kappa ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 82, + 505, + 97 + ], + "score": 1.0, + "content": "is well-defined.", + "type": "text" + } + ], + "index": 0, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 99, + 321, + 112 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 133, + 112 + ], + "score": 1.0, + "content": "Using", + "type": "text" + }, + { + "bbox": [ + 133, + 100, + 153, + 111 + ], + "score": 0.91, + "content": "\\lambda ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 99, + 214, + 112 + ], + "score": 1.0, + "content": ", we can define", + "type": "text" + }, + { + "bbox": [ + 214, + 100, + 235, + 112 + ], + "score": 0.91, + "content": "\\phi ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 99, + 253, + 112 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 100, + 273, + 112 + ], + "score": 0.91, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 99, + 321, + 112 + ], + "score": 1.0, + "content": "as follows.", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 81, + 505, + 112 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 115, + 447, + 200 + ], + "lines": [ + { + "bbox": [ + 164, + 115, + 447, + 200 + ], + "spans": [ + { + "bbox": [ + 164, + 115, + 447, + 200 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\phi ^ { \\prime } ( x ) : = - \\operatorname* { s u p } \\left\\{ \\displaystyle \\frac { \\lambda ( w ) } { w } \\left( 1 + 2 ( 1 + \\lambda ( w ) / L ) \\mu ( w ) \\right) : w \\in [ x , \\mathcal { L } ( t _ { 0 } ) ] \\right\\} } \\\\ & { \\displaystyle \\phi ( x ) : = \\log \\left( g ( \\log \\frac { 1 } { x } ) \\right) + \\int _ { 0 } ^ { x } \\left( \\phi ^ { \\prime } ( w ) + \\frac { \\lambda ( w ) } { w } \\right) d w } \\\\ & { \\displaystyle \\hat { \\gamma } ( \\pmb { \\theta } ) : = \\frac { e ^ { \\phi ( \\mathcal { L } ( \\pmb { \\theta } ) ) } } { \\rho ^ { L } } . } \\end{array}", + "type": "interline_equation", + "image_path": "b2233ecccbab3a94fb199d3489c179e9d15a45110c22cf1582d9e7db4f46f7cc.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 164, + 115, + 447, + 143.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 164, + 143.33333333333334, + 447, + 171.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 164, + 171.66666666666669, + 447, + 200.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 201, + 504, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 366, + 216 + ], + "score": 1.0, + "content": "Using a similar argument as in Lemma E.1, we can show that", + "type": "text" + }, + { + "bbox": [ + 366, + 203, + 387, + 214 + ], + "score": 0.9, + "content": "\\hat { \\gamma } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 200, + 470, + 216 + ], + "score": 1.0, + "content": "is well-defined and", + "type": "text" + }, + { + "bbox": [ + 471, + 202, + 505, + 215 + ], + "score": 0.9, + "content": "\\hat { \\gamma } ( \\pmb \\theta ) ~ <", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 212, + 427, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 227, + 227 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\tilde { \\gamma } ( \\pmb { \\theta } ) : = g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\breve { / } \\rho ^ { L } \\leq \\bar { \\gamma } ( \\pmb { \\theta } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 213, + 258, + 226 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 258, + 213, + 300, + 225 + ], + "score": 0.92, + "content": "{ \\mathcal { L } } ( \\pmb { \\theta } ) \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 213, + 357, + 226 + ], + "score": 1.0, + "content": ", we also have", + "type": "text" + }, + { + "bbox": [ + 358, + 213, + 423, + 225 + ], + "score": 0.92, + "content": "\\hat { \\gamma } ( \\pmb { \\theta } ) / \\bar { \\gamma } ( \\pmb { \\theta } ) 1", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 213, + 427, + 226 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 200, + 505, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 230, + 268, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 269, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 122, + 244 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 230, + 136, + 243 + ], + "score": 0.88, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 228, + 269, + 244 + ], + "score": 1.0, + "content": ", we define it to be the following.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 228, + 269, + 244 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 245, + 507, + 279 + ], + "lines": [ + { + "bbox": [ + 111, + 245, + 507, + 279 + ], + "spans": [ + { + "bbox": [ + 111, + 245, + 507, + 279 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\gamma _ { \\eta } : = \\frac { 1 } { 2 } B _ { 1 } \\left( \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } B _ { 1 } + \\frac { 2 ^ { p + 1 } } { \\log \\frac { 1 } { C ( t _ { 0 } ) } } ( p B _ { 1 } + B _ { 2 } ) \\right) \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } \\operatorname* { m i n } \\left\\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "ec4bb9e70cec3464c55ef0cd792ecb8d9a5a2e02759e0e0dfed7ae19882259bc.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 111, + 245, + 507, + 256.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 256.3333333333333, + 507, + 267.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 267.66666666666663, + 507, + 278.99999999999994 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 504, + 310 + ], + "lines": [ + { + "bbox": [ + 104, + 280, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 218, + 301 + ], + "score": 1.0, + "content": "Finally, the definitions for", + "type": "text" + }, + { + "bbox": [ + 218, + 283, + 343, + 299 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu ( x ) : = ( \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } ) / ( 2 \\log \\frac { 1 } { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 280, + 363, + 301 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 364, + 284, + 471, + 297 + ], + "score": 0.93, + "content": "H ( x ) : = \\mu ( x ) / ( C _ { \\eta } \\kappa ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 280, + 506, + 301 + ], + "score": 1.0, + "content": "in (S5)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 296, + 397, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 228, + 311 + ], + "score": 1.0, + "content": "remain unchanged except that", + "type": "text" + }, + { + "bbox": [ + 228, + 298, + 248, + 309 + ], + "score": 0.91, + "content": "\\kappa ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 296, + 266, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 267, + 298, + 280, + 310 + ], + "score": 0.88, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 296, + 397, + 311 + ], + "score": 1.0, + "content": "now use the new definitions.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 280, + 506, + 311 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 506, + 352 + ], + "lines": [ + { + "bbox": [ + 101, + 308, + 510, + 337 + ], + "spans": [ + { + "bbox": [ + 101, + 308, + 251, + 337 + ], + "score": 1.0, + "content": "Similar as gradient flow, we define", + "type": "text" + }, + { + "bbox": [ + 251, + 315, + 414, + 330 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nu ( t ) : = \\sum _ { n = 1 } ^ { N } e ^ { - f ( q _ { n } ( t ) ) } f ^ { \\prime } ( q _ { n } ( t ) ) q _ { n } ( t ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 308, + 510, + 337 + ], + "score": 1.0, + "content": ". The key idea behind", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 328, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 346, + 343 + ], + "score": 1.0, + "content": "the above definitions is that we can prove similar bounds for", + "type": "text" + }, + { + "bbox": [ + 347, + 330, + 437, + 342 + ], + "score": 0.91, + "content": "\\nu ( t ) , \\| \\nabla \\mathcal { L } \\| _ { 2 } , \\| \\nabla ^ { 2 } \\mathcal { L } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 328, + 506, + 343 + ], + "score": 1.0, + "content": "as Corollary E.6", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 340, + 175, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 175, + 351 + ], + "score": 1.0, + "content": "and Lemma E.7.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 101, + 308, + 510, + 351 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 505, + 391 + ], + "lines": [ + { + "bbox": [ + 103, + 347, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 103, + 347, + 162, + 374 + ], + "score": 1.0, + "content": "Lemma F.2.", + "type": "text" + }, + { + "bbox": [ + 162, + 354, + 263, + 367 + ], + "score": 0.9, + "content": "\\langle \\pmb \\theta ( t ) , - \\nabla \\mathcal L ( t ) \\rangle = L \\nu ( t ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 347, + 276, + 374 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 276, + 353, + 338, + 367 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t ) < e ^ { - f ( b _ { f } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 347, + 362, + 374 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 362, + 352, + 421, + 370 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\nu ( t ) \\geq \\frac { \\mathcal { L } ( t ) } { \\lambda ( \\mathcal { L } ( t ) ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 347, + 440, + 374 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 440, + 354, + 473, + 367 + ], + "score": 0.91, + "content": "\\lambda ( \\mathcal { L } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 347, + 506, + 374 + ], + "score": 1.0, + "content": "has the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 102, + 369, + 283, + 392 + ], + "spans": [ + { + "bbox": [ + 102, + 370, + 159, + 392 + ], + "score": 1.0, + "content": "lower bound", + "type": "text" + }, + { + "bbox": [ + 160, + 369, + 280, + 390 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\lambda ( \\mathcal { L } ( t ) ) \\leq 2 ^ { p + 1 } \\left( \\log \\frac { 1 } { \\mathcal { L } ( t ) } \\right) ^ { - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 370, + 283, + 392 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 102, + 347, + 506, + 392 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 400, + 504, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 414 + ], + "score": 1.0, + "content": "Proof. It can be easily proved by combining Theorem B.4, Lemma B.5 and Lemma D.1 together.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 496, + 414, + 504, + 421 + ], + "spans": [ + { + "bbox": [ + 496, + 414, + 504, + 421 + ], + "score": 0.991, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 399, + 505, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 429, + 355, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 353, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 195, + 443 + ], + "score": 1.0, + "content": "Lemma F.3. For any", + "type": "text" + }, + { + "bbox": [ + 195, + 431, + 201, + 440 + ], + "score": 0.63, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 429, + 214, + 443 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 214, + 430, + 330, + 443 + ], + "score": 0.89, + "content": "\\mathcal { L } ( \\pmb \\theta ) \\leq \\mathcal { L } ( t _ { 0 } ) , \\tilde { \\gamma } ( \\pmb \\theta ) \\geq \\hat { \\gamma } ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 429, + 353, + 443 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 429, + 353, + 443 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 445, + 447, + 461 + ], + "lines": [ + { + "bbox": [ + 163, + 445, + 447, + 461 + ], + "spans": [ + { + "bbox": [ + 163, + 445, + 447, + 461 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\| \\nabla \\mathcal { L } ( \\pmb { \\theta } ) \\| _ { 2 } ^ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) \\mathcal { L } ( \\pmb { \\theta } ) \\quad a n d \\quad \\left\\| \\nabla ^ { 2 } \\mathcal { L } ( \\pmb { \\theta } ) \\right\\| _ { 2 } \\leq 2 C _ { \\eta } \\kappa ( \\mathcal { L } ( \\pmb { \\theta } ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "5e3aa293e04ba12492267f2503a2228a2c9280b89b0018630b475b8731c12dca.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 163, + 445, + 447, + 461 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 305, + 487 + ], + "score": 1.0, + "content": "Proof. Note that a direct corollary of (23) is that", + "type": "text" + }, + { + "bbox": [ + 305, + 472, + 411, + 485 + ], + "score": 0.93, + "content": "f ^ { \\prime } ( x _ { 1 } ) \\leq f ^ { \\prime } ( x _ { 2 } ) ( x _ { 1 } / x _ { 2 } ) ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 471, + 427, + 487 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 428, + 473, + 487, + 485 + ], + "score": 0.92, + "content": "x _ { 1 } \\geq x _ { 2 } > b _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 471, + 506, + 487 + ], + "score": 1.0, + "content": ". So", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 143, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 143, + 497 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 471, + 506, + 497 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 496, + 466, + 527 + ], + "lines": [ + { + "bbox": [ + 144, + 496, + 466, + 527 + ], + "spans": [ + { + "bbox": [ + 144, + 496, + 466, + 527 + ], + "score": 0.91, + "content": "f ^ { \\prime } ( q _ { n } ) \\leq f ^ { \\prime } \\left( g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) \\right) \\left( { \\frac { q _ { n } } { g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } \\right) ^ { p } = { \\frac { 1 } { g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } \\left( { \\frac { q _ { n } / \\rho ^ { L } } { \\widetilde { \\gamma } } } \\right) ^ { p } \\leq { \\frac { R } { g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } ,", + "type": "interline_equation", + "image_path": "73613c208a21c367087550ae5413da2b7d66550b4522a0baeea4c6ac3e6c12b2.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 144, + 496, + 466, + 506.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 144, + 506.3333333333333, + 466, + 516.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 144, + 516.6666666666666, + 466, + 527.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 381, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 381, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 133, + 544 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 530, + 209, + 542 + ], + "score": 0.93, + "content": "R : = ( B _ { 0 } / \\hat { \\gamma } ( t _ { 0 } ) ) ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 529, + 381, + 544 + ], + "score": 1.0, + "content": ". Applying (S3.4), we can also deduce that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 529, + 381, + 544 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 545, + 399, + 574 + ], + "lines": [ + { + "bbox": [ + 211, + 545, + 399, + 574 + ], + "spans": [ + { + "bbox": [ + 211, + 545, + 399, + 574 + ], + "score": 0.93, + "content": "| f ^ { \\prime \\prime } ( q _ { n } ) | \\leq { \\frac { p } { q _ { n } } } \\cdot f ^ { \\prime } ( q _ { n } ) \\leq { \\frac { p R } { g ( \\log { \\frac { 1 } { \\mathcal { L } } } ) g ^ { \\prime } ( \\log { \\frac { 1 } { \\mathcal { L } } } ) } } .", + "type": "interline_equation", + "image_path": "25293948d1d5ce46b2034a1543169fe865fb0827e643f8636917f4c19324091e.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 211, + 545, + 399, + 574 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 369, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 368, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 170, + 592 + ], + "score": 1.0, + "content": "Now we bound", + "type": "text" + }, + { + "bbox": [ + 170, + 578, + 200, + 591 + ], + "score": 0.93, + "content": "\\lVert \\nabla \\mathcal { L } \\rVert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 577, + 219, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 577, + 255, + 592 + ], + "score": 0.93, + "content": "\\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 577, + 368, + 592 + ], + "score": 1.0, + "content": ". By the chain rule, we have", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 577, + 368, + 592 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 595, + 500, + 733 + ], + "lines": [ + { + "bbox": [ + 112, + 595, + 500, + 733 + ], + "spans": [ + { + "bbox": [ + 112, + 595, + 500, + 733 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\left\\| \\nabla \\mathcal { L } \\right\\| _ { 2 } = \\left\\| - \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } f ^ { \\prime } ( \\boldsymbol { q } _ { n } ) \\nabla \\boldsymbol { q } _ { n } \\right\\| _ { 2 } \\leq B _ { 1 } R \\cdot \\frac { \\mathcal { L } \\rho ^ { L - 1 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } } \\\\ & { \\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } \\left( \\big ( f ^ { \\prime } ( \\boldsymbol { q } _ { n } ) ^ { 2 } - f ^ { \\prime \\prime } ( \\boldsymbol { q } _ { n } ) \\big ) \\nabla q _ { n } \\nabla q _ { n } ^ { \\top } - f ^ { \\prime } ( q _ { n } ) \\nabla ^ { 2 } q _ { n } \\right) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } \\left( \\left( \\frac { R ^ { 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } + \\frac { p R } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\right) B _ { 1 } ^ { 2 } \\rho ^ { 2 L - 2 } + \\frac { R } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } B _ { 2 } \\rho ^ { L - 2 } \\right) } \\\\ & { \\qquad \\leq \\frac { \\mathcal { L } \\rho ^ { 2 L - 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\left( \\left( R + p \\lambda ( \\mathcal { L } ) \\right) B _ { 1 } ^ { 2 } + \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\rho ^ { L } } B _ { 2 } \\right) R . } \\end{array}", + "type": "interline_equation", + "image_path": "63f9e0fdb6110c12727f7233454a668cd4590f7bc7d3ef7a807c85fdcca98eb4.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 112, + 595, + 500, + 641.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 112, + 641.0, + 500, + 687.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 112, + 687.0, + 500, + 733.0 + ], + "spans": [], + "index": 32 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 506, + 135 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 123, + 96 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 84, + 130, + 94 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 79, + 168, + 96 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 168, + 81, + 280, + 95 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\hat { \\gamma } ( t _ { 0 } ) \\rho ^ { L } \\le \\log \\frac { 1 } { \\mathcal L } \\le B _ { 0 } \\rho ^ { L } , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 79, + 356, + 96 + ], + "score": 1.0, + "content": ", so we can bound", + "type": "text" + }, + { + "bbox": [ + 356, + 81, + 383, + 94 + ], + "score": 0.91, + "content": "\\rho ^ { 2 L - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 79, + 398, + 96 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 398, + 81, + 501, + 96 + ], + "score": 0.92, + "content": "\\rho ^ { 2 L - 2 } \\leq M ( \\log { \\frac { 1 } { \\mathcal { L } } } ) ^ { 2 L - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 79, + 506, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 134, + 115 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 95, + 293, + 114 + ], + "score": 0.93, + "content": "{ \\cal M } : = \\operatorname * { m i n } \\left\\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 91, + 409, + 115 + ], + "score": 1.0, + "content": ". By Lemma F.2, we have", + "type": "text" + }, + { + "bbox": [ + 410, + 97, + 505, + 115 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\lambda ( \\mathcal { L } ) \\ \\leq \\ 2 ^ { p + 1 } \\frac { 1 } { \\log \\frac { 1 } { \\mathcal { L } } } \\ \\leq \\ } \\end{array}", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 114, + 492, + 135 + ], + "spans": [ + { + "bbox": [ + 107, + 114, + 141, + 135 + ], + "score": 0.92, + "content": "\\frac { 2 ^ { p + 1 } } { \\log { \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 115, + 181, + 129 + ], + "score": 1.0, + "content": ", and also", + "type": "text" + }, + { + "bbox": [ + 181, + 114, + 309, + 135 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\rho ^ { L } } = \\lambda ( \\mathcal { L } ) \\cdot \\tilde { \\gamma } \\leq \\frac { 2 ^ { p + 1 } B _ { 1 } } { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 114, + 492, + 132 + ], + "score": 1.0, + "content": ". Combining all these formulas together gives", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 138, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 111, + 138, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 111, + 138, + 505, + 206 + ], + "score": 0.91, + "content": "\\begin{array} { r l r } & { \\| \\nabla \\mathcal { L } \\| _ { 2 } ^ { 2 } \\leq B _ { 1 } ^ { 2 } R ^ { 2 } M \\displaystyle \\frac { \\mathcal { L } ^ { 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\left( \\log \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta ^ { \\kappa } } ( \\mathcal { L } ) \\mathcal { L } } & \\\\ & { \\big | \\nabla ^ { 2 } \\mathcal { L } \\big \\| _ { 2 } \\leq \\left( \\left( R + \\frac { p 2 ^ { p + 1 } } { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } \\right) B _ { 1 } ^ { 2 } + \\frac { 2 ^ { p + 1 } B _ { 1 } B _ { 2 } } { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } \\right) R M \\displaystyle \\frac { \\mathcal { L } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\left( \\log \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta ^ { \\kappa } } ( \\mathcal { L } ) , } & \\end{array}", + "type": "interline_equation", + "image_path": "94c47cfd0cd7aac7ab543d51c14820eb758303a09fcd837db2e3d5684acfa512.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 138, + 505, + 160.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 111, + 160.66666666666666, + 505, + 183.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 183.33333333333331, + 505, + 205.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 207, + 217, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 217, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 217, + 221 + ], + "score": 1.0, + "content": "which completes the proof.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 504, + 244 + ], + "score": 1.0, + "content": "With Corollary E.6 and Lemma E.7, we can prove Lemma E.8 with exactly the same argument.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "Then Theorem E.2, E.3, E.4 can also be proved similarly. For Theorem E.5, we can follow the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 253, + 443, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 443, + 265 + ], + "score": 1.0, + "content": "argument for gradient flow to show that it holds with slightly different tight bounds:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 504, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 504, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 504, + 280 + ], + "score": 1.0, + "content": "Theorem F.4. Under assumptions (S1), (A2), (S3), (A4), (S5), we have the following tight rates for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 279, + 231, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 231, + 291 + ], + "score": 1.0, + "content": "training loss and weight norm:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 294, + 426, + 322 + ], + "lines": [ + { + "bbox": [ + 184, + 294, + 426, + 322 + ], + "spans": [ + { + "bbox": [ + 184, + 294, + 426, + 322 + ], + "score": 0.91, + "content": "\\mathcal { L } ( t ) = \\Theta \\left( \\frac { g ( \\log T ) ^ { 2 / L } } { T ( \\log T ) ^ { 2 } } \\right) \\quad a n d \\quad \\rho ( t ) = \\Theta ( g ( \\log T ) ^ { 1 / L } ) ,", + "type": "interline_equation", + "image_path": "19d3c481f8f5f29d6dbf2aaa60834dec89dc6a726bf5949e087976f458eabef6.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 184, + 294, + 426, + 322 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 206, + 342 + ], + "lines": [ + { + "bbox": [ + 104, + 322, + 203, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 133, + 347 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 326, + 203, + 342 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 106, + 356, + 358, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 359, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 359, + 371 + ], + "score": 1.0, + "content": "G EXTENSION: MULTI-CLASS CLASSIFICATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "In this section, we generalize our results to multi-class classification with cross-entropy loss. This", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 392, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 392, + 506, + 404 + ], + "score": 1.0, + "content": "part of analysis is inspired by Theorem 1 in (Zhang et al., 2019), which gives a lower bound for the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 403, + 232, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 220, + 415 + ], + "score": 1.0, + "content": "gradient in terms of the loss", + "type": "text" + }, + { + "bbox": [ + 220, + 403, + 228, + 413 + ], + "score": 0.81, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 403, + 232, + 415 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 481, + 432 + ], + "score": 1.0, + "content": "Since now a neural network has multiple outputs, we need to redefine our notations. Let", + "type": "text" + }, + { + "bbox": [ + 482, + 420, + 491, + 430 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 340, + 443 + ], + "score": 1.0, + "content": "the number of classes. The output of a neural network", + "type": "text" + }, + { + "bbox": [ + 340, + 431, + 349, + 441 + ], + "score": 0.85, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 428, + 399, + 443 + ], + "score": 1.0, + "content": "is a vector", + "type": "text" + }, + { + "bbox": [ + 399, + 430, + 464, + 443 + ], + "score": 0.93, + "content": "\\Phi ( \\pmb { \\theta } ; \\pmb { x } ) \\in \\mathbb { R } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 428, + 506, + 443 + ], + "score": 1.0, + "content": ". 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Combining all these formulas together gives", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 104, + 79, + 506, + 135 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 138, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 111, + 138, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 111, + 138, + 505, + 206 + ], + "score": 0.91, + "content": "\\begin{array} { r l r } & { \\| \\nabla \\mathcal { L } \\| _ { 2 } ^ { 2 } \\leq B _ { 1 } ^ { 2 } R ^ { 2 } M \\displaystyle \\frac { \\mathcal { L } ^ { 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\left( \\log \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta ^ { \\kappa } } ( \\mathcal { L } ) \\mathcal { L } } & \\\\ & { \\big | \\nabla ^ { 2 } \\mathcal { L } \\big \\| _ { 2 } \\leq \\left( \\left( R + \\frac { p 2 ^ { p + 1 } } { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } \\right) B _ { 1 } ^ { 2 } + \\frac { 2 ^ { p + 1 } B _ { 1 } B _ { 2 } } { \\log \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } \\right) R M \\displaystyle \\frac { \\mathcal { L } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } \\left( \\log \\frac { 1 } { \\mathcal { L } } \\right) ^ { 2 - 2 / L } \\leq 2 C _ { \\eta ^ { \\kappa } } ( \\mathcal { L } ) , } & \\end{array}", + "type": "interline_equation", + "image_path": "94c47cfd0cd7aac7ab543d51c14820eb758303a09fcd837db2e3d5684acfa512.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 138, + 505, + 160.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 111, + 160.66666666666666, + 505, + 183.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 183.33333333333331, + 505, + 205.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 207, + 217, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 217, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 217, + 221 + ], + "score": 1.0, + "content": "which completes the proof.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 206, + 217, + 221 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 504, + 244 + ], + "score": 1.0, + "content": "With Corollary E.6 and Lemma E.7, we can prove Lemma E.8 with exactly the same argument.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "Then Theorem E.2, E.3, E.4 can also be proved similarly. For Theorem E.5, we can follow the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 253, + 443, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 443, + 265 + ], + "score": 1.0, + "content": "argument for gradient flow to show that it holds with slightly different tight bounds:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 230, + 505, + 265 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 504, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 504, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 504, + 280 + ], + "score": 1.0, + "content": "Theorem F.4. Under assumptions (S1), (A2), (S3), (A4), (S5), we have the following tight rates for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 279, + 231, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 231, + 291 + ], + "score": 1.0, + "content": "training loss and weight norm:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 106, + 266, + 504, + 291 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 294, + 426, + 322 + ], + "lines": [ + { + "bbox": [ + 184, + 294, + 426, + 322 + ], + "spans": [ + { + "bbox": [ + 184, + 294, + 426, + 322 + ], + "score": 0.91, + "content": "\\mathcal { L } ( t ) = \\Theta \\left( \\frac { g ( \\log T ) ^ { 2 / L } } { T ( \\log T ) ^ { 2 } } \\right) \\quad a n d \\quad \\rho ( t ) = \\Theta ( g ( \\log T ) ^ { 1 / L } ) ,", + "type": "interline_equation", + "image_path": "19d3c481f8f5f29d6dbf2aaa60834dec89dc6a726bf5949e087976f458eabef6.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 184, + 294, + 426, + 322 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 206, + 342 + ], + "lines": [ + { + "bbox": [ + 104, + 322, + 203, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 133, + 347 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 326, + 203, + 342 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\sum _ { \\tau = t _ { 0 } } ^ { t - 1 } \\eta ( \\tau ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 322, + 203, + 347 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 356, + 358, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 359, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 359, + 371 + ], + "score": 1.0, + "content": "G EXTENSION: MULTI-CLASS CLASSIFICATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "In this section, we generalize our results to multi-class classification with cross-entropy loss. This", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 392, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 392, + 506, + 404 + ], + "score": 1.0, + "content": "part of analysis is inspired by Theorem 1 in (Zhang et al., 2019), which gives a lower bound for the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 403, + 232, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 220, + 415 + ], + "score": 1.0, + "content": "gradient in terms of the loss", + "type": "text" + }, + { + "bbox": [ + 220, + 403, + 228, + 413 + ], + "score": 0.81, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 403, + 232, + 415 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 380, + 506, + 415 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 481, + 432 + ], + "score": 1.0, + "content": "Since now a neural network has multiple outputs, we need to redefine our notations. Let", + "type": "text" + }, + { + "bbox": [ + 482, + 420, + 491, + 430 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 340, + 443 + ], + "score": 1.0, + "content": "the number of classes. The output of a neural network", + "type": "text" + }, + { + "bbox": [ + 340, + 431, + 349, + 441 + ], + "score": 0.85, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 428, + 399, + 443 + ], + "score": 1.0, + "content": "is a vector", + "type": "text" + }, + { + "bbox": [ + 399, + 430, + 464, + 443 + ], + "score": 0.93, + "content": "\\Phi ( \\pmb { \\theta } ; \\pmb { x } ) \\in \\mathbb { R } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 428, + 506, + 443 + ], + "score": 1.0, + "content": ". We use", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 439, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 107, + 442, + 168, + 454 + ], + "score": 0.89, + "content": "\\Phi _ { j } ( \\pmb { \\theta } ; \\pmb { x } ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 439, + 228, + 456 + ], + "score": 1.0, + "content": "to denote the", + "type": "text" + }, + { + "bbox": [ + 228, + 442, + 235, + 453 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 439, + 290, + 456 + ], + "score": 1.0, + "content": "-th output of", + "type": "text" + }, + { + "bbox": [ + 290, + 442, + 299, + 452 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 439, + 354, + 456 + ], + "score": 1.0, + "content": "on the input", + "type": "text" + }, + { + "bbox": [ + 355, + 441, + 395, + 452 + ], + "score": 0.92, + "content": "\\textbf { \\em x } \\in \\mathbb { R } ^ { d _ { \\mathrm { { x } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 439, + 506, + 456 + ], + "score": 1.0, + "content": ". A dataset is denoted by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 447, + 509, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 286, + 466 + ], + "score": 0.93, + "content": "\\bar { \\mathcal { D } } = \\{ \\pmb { x } _ { n } , y _ { n } \\} _ { n = 1 } ^ { N } = \\{ ( \\pmb { x } _ { n } , y _ { n } ) : n \\in [ N ] \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 447, + 318, + 477 + ], + "score": 1.0, + "content": ", where on the d", + "type": "text" + }, + { + "bbox": [ + 318, + 453, + 361, + 465 + ], + "score": 0.93, + "content": "\\pmb { x } _ { n } \\in \\mathbb { R } ^ { d _ { \\mathbf { x } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 447, + 440, + 477 + ], + "score": 1.0, + "content": "is a data input and defined as", + "type": "text" + }, + { + "bbox": [ + 441, + 453, + 479, + 466 + ], + "score": 0.93, + "content": "y _ { n } \\in [ C ]", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 447, + 509, + 477 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 274, + 466, + 349, + 475 + ], + "spans": [ + { + "bbox": [ + 274, + 466, + 282, + 475 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 466, + 349, + 474 + ], + "score": 0.84, + "content": "\\mathcal { D }", + "type": "inline_equation" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 419, + 509, + 477 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 480, + 380, + 515 + ], + "lines": [ + { + "bbox": [ + 231, + 480, + 380, + 515 + ], + "spans": [ + { + "bbox": [ + 231, + 480, + 380, + 515 + ], + "score": 0.95, + "content": "\\mathcal { L } ( \\pmb \\theta ) : = \\sum _ { n = 1 } ^ { N } - \\log \\frac { e ^ { \\Phi _ { y _ { n } } ( \\pmb \\theta ; \\pmb x _ { n } ) } } { \\sum _ { j = 1 } ^ { C } e ^ { \\Phi _ { j } ( \\pmb \\theta ; \\pmb x _ { n } ) } } .", + "type": "interline_equation", + "image_path": "767e11d63b44c6dae3d317aeda636a07489924812f02e176285a4a6d6a4b60d3.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 231, + 480, + 380, + 497.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 231, + 497.5, + 380, + 515.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 271, + 533 + ], + "score": 1.0, + "content": "The margin for a single data point", + "type": "text" + }, + { + "bbox": [ + 272, + 519, + 307, + 532 + ], + "score": 0.93, + "content": "( { \\pmb x } _ { n } , y _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 517, + 389, + 533 + ], + "score": 1.0, + "content": "is defined to be", + "type": "text" + }, + { + "bbox": [ + 390, + 519, + 504, + 532 + ], + "score": 0.9, + "content": "\\begin{array} { r c l } { q _ { n } ( \\pmb { \\theta } ) } & { : = } & { \\Phi _ { y _ { n } } ( \\pmb { \\theta } ; \\pmb { x } _ { n } ) - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 529, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 107, + 530, + 196, + 543 + ], + "score": 0.91, + "content": "\\operatorname* { m a x } _ { j \\neq y _ { n } } \\bar { \\{ \\Phi _ { j } ( \\pmb { \\theta } ; \\pmb { x } _ { n } ) \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 529, + 453, + 543 + ], + "score": 1.0, + "content": ", and the margin for the entire dataset is defined to be", + "type": "text" + }, + { + "bbox": [ + 453, + 531, + 504, + 542 + ], + "score": 0.84, + "content": "q _ { \\mathrm { m i n } } ( \\pmb { \\theta } ) =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 542, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 107, + 544, + 171, + 556 + ], + "score": 0.9, + "content": "\\mathrm { m i n } _ { n \\in [ N ] } q _ { n } ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 542, + 339, + 557 + ], + "score": 1.0, + "content": ". 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So", + "type": "text" + }, + { + "bbox": [ + 341, + 586, + 438, + 601 + ], + "score": 0.91, + "content": "g ( q ) = - \\log ( e ^ { e ^ { - q } } - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 586, + 443, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 573, + 506, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 348, + 620 + ], + "score": 1.0, + "content": "The cross-entropy loss can be rewritten in other ways. Let", + "type": "text" + }, + { + "bbox": [ + 348, + 605, + 483, + 618 + ], + "score": 0.91, + "content": "s _ { n j } : = \\Phi _ { y _ { n } } ( \\pmb { \\theta } ; \\pmb { x } _ { n } ) - \\Phi _ { j } ( \\pmb { \\theta } ; \\pmb { x } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 602, + 506, + 620 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 617, + 371, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 340, + 637 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\widetilde { q } _ { n } : = - \\mathrm { L S E } ( \\{ - s _ { n j } : j \\neq y _ { n } \\} ) = - \\log \\left( \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 617, + 371, + 639 + ], + "score": 1.0, + "content": ". 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(Homogeneity). There exists", + "type": "text" + }, + { + "bbox": [ + 269, + 83, + 302, + 93 + ], + "score": 0.91, + "content": "L \\ > \\ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 80, + 346, + 96 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 347, + 82, + 505, + 95 + ], + "score": 0.87, + "content": "\\forall j ~ \\in ~ [ C ] , \\forall \\alpha ~ > ~ 0 ~ : ~ \\Phi _ { j } ( \\alpha \\pmb \\theta ; \\pmb x ) ~ =", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 91, + 198, + 109 + ], + "spans": [ + { + "bbox": [ + 142, + 94, + 192, + 107 + ], + "score": 0.91, + "content": "\\alpha ^ { L } \\Phi _ { j } ( \\pmb { \\theta } ; \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 91, + 198, + 109 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 114, + 110, + 474, + 123 + ], + "lines": [ + { + "bbox": [ + 113, + 109, + 474, + 125 + ], + "spans": [ + { + "bbox": [ + 113, + 109, + 232, + 125 + ], + "score": 1.0, + "content": "(M3). (Cross-entropy Loss).", + "type": "text" + }, + { + "bbox": [ + 232, + 110, + 254, + 123 + ], + "score": 0.9, + "content": "\\mathcal { L } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 109, + 474, + 125 + ], + "score": 1.0, + "content": "is defined as the cross-entropy loss on the training set;", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 114, + 126, + 390, + 139 + ], + "lines": [ + { + "bbox": [ + 113, + 125, + 390, + 141 + ], + "spans": [ + { + "bbox": [ + 113, + 125, + 280, + 141 + ], + "score": 1.0, + "content": "(M4). (Separability). There exists a time", + "type": "text" + }, + { + "bbox": [ + 280, + 128, + 290, + 138 + ], + "score": 0.87, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 125, + 329, + 141 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 330, + 126, + 386, + 139 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t _ { 0 } ) < \\log 2", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 125, + 390, + 141 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 109, + 146, + 504, + 172 + ], + "lines": [ + { + "bbox": [ + 116, + 142, + 508, + 167 + ], + "spans": [ + { + "bbox": [ + 116, + 148, + 157, + 159 + ], + "score": 0.89, + "content": "\\mathcal { L } < \\log 2", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 142, + 180, + 167 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 181, + 148, + 254, + 162 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } < 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 142, + 282, + 167 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 283, + 148, + 317, + 160 + ], + "score": 0.91, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 142, + 357, + 167 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + }, + { + "bbox": [ + 357, + 149, + 390, + 161 + ], + "score": 0.92, + "content": "s _ { n j } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 142, + 418, + 167 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 419, + 148, + 487, + 160 + ], + "score": 0.92, + "content": "n \\in [ N ] , j \\in [ C ]", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 142, + 508, + 167 + ], + "score": 1.0, + "content": ". So", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 160, + 291, + 173 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 291, + 173 + ], + "score": 1.0, + "content": "(M4) ensures the separability of training data.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 175, + 496, + 188 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 496, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 403, + 189 + ], + "score": 1.0, + "content": "Definition G.1. For cross-entropy loss, the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 403, + 176, + 424, + 188 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 174, + 436, + 189 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 436, + 176, + 443, + 186 + ], + "score": 0.82, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 174, + 496, + 189 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 193, + 378, + 222 + ], + "lines": [ + { + "bbox": [ + 232, + 193, + 378, + 222 + ], + "spans": [ + { + "bbox": [ + 232, + 193, + 378, + 222 + ], + "score": 0.94, + "content": "\\widetilde { \\gamma } ( \\pmb \\theta ) : = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\rho ^ { L } } = \\frac { - \\log ( e ^ { \\mathcal { L } } - 1 ) } { \\rho ^ { L } } ,", + "type": "interline_equation", + "image_path": "c4580bf722dd3b82530e20221db120f440274e72452dae9c0e1287f8c45c5c55.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 232, + 193, + 378, + 222 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 346, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 346, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 133, + 242 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 228, + 160, + 240 + ], + "score": 0.92, + "content": "\\ell ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 226, + 326, + 242 + ], + "score": 1.0, + "content": "is the inverse function of the logistic loss", + "type": "text" + }, + { + "bbox": [ + 327, + 228, + 342, + 240 + ], + "score": 0.9, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 226, + 346, + 242 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 443, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 443, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 443, + 263 + ], + "score": 1.0, + "content": "Theorem 4.1 and 4.4 still hold. Here we redefine the optimization problem (P) to be", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 266, + 407, + 307 + ], + "lines": [ + { + "bbox": [ + 203, + 266, + 407, + 307 + ], + "spans": [ + { + "bbox": [ + 203, + 266, + 407, + 307 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\mathrm { s . t . } } & { s _ { n j } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ] , \\forall j \\in [ C ] \\setminus \\{ y _ { n } \\} } \\end{array}", + "type": "interline_equation", + "image_path": "4eea707e36a5a57fd8bc00c5d254c4a50c40eedf8a156a0885388a87eb0183a0.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 266, + 407, + 286.5 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 203, + 286.5, + 407, + 307.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 504, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "Most of our proofs are very similar as before. Here we only show the proof for the generalized", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 322, + 200, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 200, + 334 + ], + "score": 1.0, + "content": "version of Lemma 5.1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 104, + 337, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 420, + 351 + ], + "score": 1.0, + "content": "Lemma G.2. Lemma 5.1 is also true for the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 421, + 339, + 428, + 350 + ], + "score": 0.83, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 336, + 505, + 351 + ], + "score": 1.0, + "content": "defined in Defini-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 145, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 145, + 361 + ], + "score": 1.0, + "content": "tion G.1.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 289, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 289, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 165, + 390 + ], + "score": 1.0, + "content": "Proof. Define", + "type": "text" + }, + { + "bbox": [ + 165, + 376, + 183, + 388 + ], + "score": 0.92, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 375, + 289, + 390 + ], + "score": 1.0, + "content": "by the following formula:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 394, + 370, + 429 + ], + "lines": [ + { + "bbox": [ + 240, + 394, + 370, + 429 + ], + "spans": [ + { + "bbox": [ + 240, + 394, + 370, + 429 + ], + "score": 0.94, + "content": "\\nu ( t ) : = \\sum _ { n = 1 } ^ { N } \\frac { \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } s _ { n j } } { 1 + \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } } .", + "type": "interline_equation", + "image_path": "a3a6ef5aea6f1853718c2f6cae40809c10d680847a84088d87e6016c2c48144c.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 394, + 370, + 411.5 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 240, + 411.5, + 370, + 429.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 435, + 489, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 490, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 372, + 454 + ], + "score": 1.0, + "content": "Using a similar argument as in Theorem B.4, it can be proved that", + "type": "text" + }, + { + "bbox": [ + 372, + 435, + 431, + 451 + ], + "score": 0.93, + "content": "\\textstyle { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } = L \\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 437, + 462, + 451 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 463, + 439, + 486, + 449 + ], + "score": 0.88, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 437, + 490, + 451 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 104, + 455, + 507, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 326, + 477 + ], + "score": 1.0, + "content": "It can be shown that Lemma B.5, which asserts that", + "type": "text" + }, + { + "bbox": [ + 327, + 456, + 403, + 476 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nu ( t ) \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 455, + 507, + 477 + ], + "score": 1.0, + "content": ", still holds for this new", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 471, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 158, + 491 + ], + "score": 1.0, + "content": "definition of", + "type": "text" + }, + { + "bbox": [ + 158, + 475, + 176, + 488 + ], + "score": 0.91, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 471, + 237, + 491 + ], + "score": 1.0, + "content": ". By definition,", + "type": "text" + }, + { + "bbox": [ + 238, + 476, + 276, + 488 + ], + "score": 0.92, + "content": "s _ { n j } \\geq q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 471, + 291, + 491 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 291, + 476, + 320, + 487 + ], + "score": 0.9, + "content": "j \\neq y _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 471, + 383, + 491 + ], + "score": 1.0, + "content": ". Also note that", + "type": "text" + }, + { + "bbox": [ + 384, + 475, + 437, + 487 + ], + "score": 0.91, + "content": "\\tilde { e } ^ { - \\tilde { q } _ { n } } \\geq e ^ { - q _ { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 471, + 455, + 491 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 455, + 476, + 505, + 488 + ], + "score": 0.89, + "content": "s _ { n j } \\geq q _ { n } \\geq", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 487, + 180, + 498 + ], + "spans": [ + { + "bbox": [ + 107, + 487, + 117, + 498 + ], + "score": 0.86, + "content": "\\tilde { q } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 487, + 180, + 498 + ], + "score": 1.0, + "content": ". Then we have", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 504, + 468, + 539 + ], + "lines": [ + { + "bbox": [ + 142, + 504, + 468, + 539 + ], + "spans": [ + { + "bbox": [ + 142, + 504, + 468, + 539 + ], + "score": 0.94, + "content": "\\nu ( t ) \\geq \\sum _ { n = 1 } ^ { N } \\frac { \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } } { 1 + \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } } \\cdot \\tilde { q } _ { n } = \\sum _ { n = 1 } ^ { N } \\frac { e ^ { - \\tilde { q } _ { n } } } { 1 + e ^ { - \\tilde { q } _ { n } } } \\cdot \\tilde { q } _ { n } = \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\tilde { q } _ { n } ) } f ^ { \\prime } ( \\tilde { q } _ { n } ) \\tilde { q } _ { n } .", + "type": "interline_equation", + "image_path": "894d2b9a3b10d6d26717c4b6e7efa95bda05ec6943f4ea768d45a6c7cd5321bc.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 142, + 504, + 468, + 515.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 142, + 515.6666666666666, + 468, + 527.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 142, + 527.3333333333333, + 468, + 538.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 504, + 578 + ], + "lines": [ + { + "bbox": [ + 102, + 543, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 102, + 543, + 147, + 562 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 147, + 545, + 228, + 560 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { L } = \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\tilde { q } _ { n } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 543, + 474, + 562 + ], + "score": 1.0, + "content": ". Then using Lemma B.5 for logistic loss can conclude that", + "type": "text" + }, + { + "bbox": [ + 474, + 547, + 505, + 560 + ], + "score": 0.9, + "content": "\\nu ( t ) \\geq", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 108, + 559, + 149, + 579 + ], + "spans": [ + { + "bbox": [ + 108, + 559, + 149, + 579 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 422, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 422, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 422, + 596 + ], + "score": 1.0, + "content": "The rest of the proof for this lemma is exactly the same as that for Lemma 5.1.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 506, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "Gradient Descent. 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(Learing rate condition).", + "type": "text" + }, + { + "bbox": [ + 243, + 658, + 318, + 672 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\sum _ { t \\geq 0 } \\eta ( t ) = + \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 657, + 336, + 673 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 336, + 658, + 416, + 671 + ], + "score": 0.9, + "content": "\\eta ( t ) \\leq H ( \\mathcal { L } ( \\pmb { \\theta } ( t ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 657, + 420, + 673 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 108, + 680, + 504, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 129, + 695 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 130, + 681, + 237, + 694 + ], + "score": 0.93, + "content": "H ( x ) : = \\mu ( x ) / ( C _ { \\eta } \\kappa ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 680, + 413, + 695 + ], + "score": 1.0, + "content": "is defined to be the same as in Appendix", + "type": "text" + }, + { + "bbox": [ + 413, + 682, + 421, + 691 + ], + "score": 0.51, + "content": "\\mathrm { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 680, + 451, + 695 + ], + "score": 1.0, + "content": "(when", + "type": "text" + }, + { + "bbox": [ + 451, + 681, + 467, + 694 + ], + "score": 0.88, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 680, + 506, + 695 + ], + "score": 1.0, + "content": "is set to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 692, + 399, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 268, + 705 + ], + "score": 1.0, + "content": "logistic loss) except that we use another", + "type": "text" + }, + { + "bbox": [ + 268, + 693, + 281, + 705 + ], + "score": 0.9, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 692, + 399, + 705 + ], + "score": 1.0, + "content": "which will be specified later.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "We only need to show that Lemma F.2 and Lemma F.3 continue to hold. 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There exists", + "type": "text" + }, + { + "bbox": [ + 269, + 83, + 302, + 93 + ], + "score": 0.91, + "content": "L \\ > \\ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 80, + 346, + 96 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 347, + 82, + 505, + 95 + ], + "score": 0.87, + "content": "\\forall j ~ \\in ~ [ C ] , \\forall \\alpha ~ > ~ 0 ~ : ~ \\Phi _ { j } ( \\alpha \\pmb \\theta ; \\pmb x ) ~ =", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 91, + 198, + 109 + ], + "spans": [ + { + "bbox": [ + 142, + 94, + 192, + 107 + ], + "score": 0.91, + "content": "\\alpha ^ { L } \\Phi _ { j } ( \\pmb { \\theta } ; \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 91, + 198, + 109 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 112, + 80, + 505, + 109 + ] + }, + { + "type": "text", + "bbox": [ + 114, + 110, + 474, + 123 + ], + "lines": [ + { + "bbox": [ + 113, + 109, + 474, + 125 + ], + "spans": [ + { + "bbox": [ + 113, + 109, + 232, + 125 + ], + "score": 1.0, + "content": "(M3). (Cross-entropy Loss).", + "type": "text" + }, + { + "bbox": [ + 232, + 110, + 254, + 123 + ], + "score": 0.9, + "content": "\\mathcal { L } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 109, + 474, + 125 + ], + "score": 1.0, + "content": "is defined as the cross-entropy loss on the training set;", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 113, + 109, + 474, + 125 + ] + }, + { + "type": "text", + "bbox": [ + 114, + 126, + 390, + 139 + ], + "lines": [ + { + "bbox": [ + 113, + 125, + 390, + 141 + ], + "spans": [ + { + "bbox": [ + 113, + 125, + 280, + 141 + ], + "score": 1.0, + "content": "(M4). (Separability). There exists a time", + "type": "text" + }, + { + "bbox": [ + 280, + 128, + 290, + 138 + ], + "score": 0.87, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 125, + 329, + 141 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 330, + 126, + 386, + 139 + ], + "score": 0.92, + "content": "\\mathcal { L } ( t _ { 0 } ) < \\log 2", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 125, + 390, + 141 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 113, + 125, + 390, + 141 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 146, + 504, + 172 + ], + "lines": [ + { + "bbox": [ + 116, + 142, + 508, + 167 + ], + "spans": [ + { + "bbox": [ + 116, + 148, + 157, + 159 + ], + "score": 0.89, + "content": "\\mathcal { L } < \\log 2", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 142, + 180, + 167 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 181, + 148, + 254, + 162 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } < 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 142, + 282, + 167 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 283, + 148, + 317, + 160 + ], + "score": 0.91, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 142, + 357, + 167 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + }, + { + "bbox": [ + 357, + 149, + 390, + 161 + ], + "score": 0.92, + "content": "s _ { n j } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 142, + 418, + 167 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 419, + 148, + 487, + 160 + ], + "score": 0.92, + "content": "n \\in [ N ] , j \\in [ C ]", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 142, + 508, + 167 + ], + "score": 1.0, + "content": ". So", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 160, + 291, + 173 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 291, + 173 + ], + "score": 1.0, + "content": "(M4) ensures the separability of training data.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 107, + 142, + 508, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 175, + 496, + 188 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 496, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 403, + 189 + ], + "score": 1.0, + "content": "Definition G.1. For cross-entropy loss, the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 403, + 176, + 424, + 188 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 174, + 436, + 189 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 436, + 176, + 443, + 186 + ], + "score": 0.82, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 174, + 496, + 189 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 174, + 496, + 189 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 193, + 378, + 222 + ], + "lines": [ + { + "bbox": [ + 232, + 193, + 378, + 222 + ], + "spans": [ + { + "bbox": [ + 232, + 193, + 378, + 222 + ], + "score": 0.94, + "content": "\\widetilde { \\gamma } ( \\pmb \\theta ) : = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\rho ^ { L } } = \\frac { - \\log ( e ^ { \\mathcal { L } } - 1 ) } { \\rho ^ { L } } ,", + "type": "interline_equation", + "image_path": "c4580bf722dd3b82530e20221db120f440274e72452dae9c0e1287f8c45c5c55.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 232, + 193, + 378, + 222 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 346, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 346, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 133, + 242 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 228, + 160, + 240 + ], + "score": 0.92, + "content": "\\ell ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 226, + 326, + 242 + ], + "score": 1.0, + "content": "is the inverse function of the logistic loss", + "type": "text" + }, + { + "bbox": [ + 327, + 228, + 342, + 240 + ], + "score": 0.9, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 226, + 346, + 242 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 226, + 346, + 242 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 443, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 443, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 443, + 263 + ], + "score": 1.0, + "content": "Theorem 4.1 and 4.4 still hold. Here we redefine the optimization problem (P) to be", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 248, + 443, + 263 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 266, + 407, + 307 + ], + "lines": [ + { + "bbox": [ + 203, + 266, + 407, + 307 + ], + "spans": [ + { + "bbox": [ + 203, + 266, + 407, + 307 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\operatorname* { m i n } } & { \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } \\right\\| _ { 2 } ^ { 2 } } \\\\ { \\mathrm { s . t . } } & { s _ { n j } ( \\pmb { \\theta } ) \\geq 1 \\qquad \\forall n \\in [ N ] , \\forall j \\in [ C ] \\setminus \\{ y _ { n } \\} } \\end{array}", + "type": "interline_equation", + "image_path": "4eea707e36a5a57fd8bc00c5d254c4a50c40eedf8a156a0885388a87eb0183a0.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 266, + 407, + 286.5 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 203, + 286.5, + 407, + 307.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 504, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "Most of our proofs are very similar as before. Here we only show the proof for the generalized", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 322, + 200, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 200, + 334 + ], + "score": 1.0, + "content": "version of Lemma 5.1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 311, + 505, + 334 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 337, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 420, + 351 + ], + "score": 1.0, + "content": "Lemma G.2. Lemma 5.1 is also true for the smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 421, + 339, + 428, + 350 + ], + "score": 0.83, + "content": "\\tilde { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 336, + 505, + 351 + ], + "score": 1.0, + "content": "defined in Defini-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 145, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 145, + 361 + ], + "score": 1.0, + "content": "tion G.1.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 336, + 505, + 361 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 289, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 289, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 165, + 390 + ], + "score": 1.0, + "content": "Proof. Define", + "type": "text" + }, + { + "bbox": [ + 165, + 376, + 183, + 388 + ], + "score": 0.92, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 375, + 289, + 390 + ], + "score": 1.0, + "content": "by the following formula:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 375, + 289, + 390 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 394, + 370, + 429 + ], + "lines": [ + { + "bbox": [ + 240, + 394, + 370, + 429 + ], + "spans": [ + { + "bbox": [ + 240, + 394, + 370, + 429 + ], + "score": 0.94, + "content": "\\nu ( t ) : = \\sum _ { n = 1 } ^ { N } \\frac { \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } s _ { n j } } { 1 + \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } } .", + "type": "interline_equation", + "image_path": "a3a6ef5aea6f1853718c2f6cae40809c10d680847a84088d87e6016c2c48144c.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 394, + 370, + 411.5 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 240, + 411.5, + 370, + 429.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 435, + 489, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 490, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 372, + 454 + ], + "score": 1.0, + "content": "Using a similar argument as in Theorem B.4, it can be proved that", + "type": "text" + }, + { + "bbox": [ + 372, + 435, + 431, + 451 + ], + "score": 0.93, + "content": "\\textstyle { \\frac { 1 } { 2 } } { \\frac { d \\rho ^ { 2 } } { d t } } = L \\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 437, + 462, + 451 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 463, + 439, + 486, + 449 + ], + "score": 0.88, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 437, + 490, + 451 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 435, + 490, + 454 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 104, + 455, + 507, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 326, + 477 + ], + "score": 1.0, + "content": "It can be shown that Lemma B.5, which asserts that", + "type": "text" + }, + { + "bbox": [ + 327, + 456, + 403, + 476 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nu ( t ) \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 455, + 507, + 477 + ], + "score": 1.0, + "content": ", still holds for this new", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 471, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 158, + 491 + ], + "score": 1.0, + "content": "definition of", + "type": "text" + }, + { + "bbox": [ + 158, + 475, + 176, + 488 + ], + "score": 0.91, + "content": "\\nu ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 471, + 237, + 491 + ], + "score": 1.0, + "content": ". By definition,", + "type": "text" + }, + { + "bbox": [ + 238, + 476, + 276, + 488 + ], + "score": 0.92, + "content": "s _ { n j } \\geq q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 471, + 291, + 491 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 291, + 476, + 320, + 487 + ], + "score": 0.9, + "content": "j \\neq y _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 471, + 383, + 491 + ], + "score": 1.0, + "content": ". Also note that", + "type": "text" + }, + { + "bbox": [ + 384, + 475, + 437, + 487 + ], + "score": 0.91, + "content": "\\tilde { e } ^ { - \\tilde { q } _ { n } } \\geq e ^ { - q _ { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 471, + 455, + 491 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 455, + 476, + 505, + 488 + ], + "score": 0.89, + "content": "s _ { n j } \\geq q _ { n } \\geq", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 487, + 180, + 498 + ], + "spans": [ + { + "bbox": [ + 107, + 487, + 117, + 498 + ], + "score": 0.86, + "content": "\\tilde { q } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 487, + 180, + 498 + ], + "score": 1.0, + "content": ". Then we have", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 455, + 507, + 498 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 504, + 468, + 539 + ], + "lines": [ + { + "bbox": [ + 142, + 504, + 468, + 539 + ], + "spans": [ + { + "bbox": [ + 142, + 504, + 468, + 539 + ], + "score": 0.94, + "content": "\\nu ( t ) \\geq \\sum _ { n = 1 } ^ { N } \\frac { \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } } { 1 + \\sum _ { j \\neq y _ { n } } e ^ { - s _ { n j } } } \\cdot \\tilde { q } _ { n } = \\sum _ { n = 1 } ^ { N } \\frac { e ^ { - \\tilde { q } _ { n } } } { 1 + e ^ { - \\tilde { q } _ { n } } } \\cdot \\tilde { q } _ { n } = \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\tilde { q } _ { n } ) } f ^ { \\prime } ( \\tilde { q } _ { n } ) \\tilde { q } _ { n } .", + "type": "interline_equation", + "image_path": "894d2b9a3b10d6d26717c4b6e7efa95bda05ec6943f4ea768d45a6c7cd5321bc.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 142, + 504, + 468, + 515.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 142, + 515.6666666666666, + 468, + 527.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 142, + 527.3333333333333, + 468, + 538.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 504, + 578 + ], + "lines": [ + { + "bbox": [ + 102, + 543, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 102, + 543, + 147, + 562 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 147, + 545, + 228, + 560 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { L } = \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\tilde { q } _ { n } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 543, + 474, + 562 + ], + "score": 1.0, + "content": ". Then using Lemma B.5 for logistic loss can conclude that", + "type": "text" + }, + { + "bbox": [ + 474, + 547, + 505, + 560 + ], + "score": 0.9, + "content": "\\nu ( t ) \\geq", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 108, + 559, + 149, + 579 + ], + "spans": [ + { + "bbox": [ + 108, + 559, + 149, + 579 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 102, + 543, + 505, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 422, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 422, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 422, + 596 + ], + "score": 1.0, + "content": "The rest of the proof for this lemma is exactly the same as that for Lemma 5.1.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 582, + 422, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 506, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "Gradient Descent. For gradient descent, we only need to replace (M1) with (S1) and make the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 622, + 256, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 256, + 633 + ], + "score": 1.0, + "content": "assumption (S5) on the learning rate.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 106, + 610, + 505, + 633 + ] + }, + { + "type": "list", + "bbox": [ + 115, + 641, + 419, + 672 + ], + "lines": [ + { + "bbox": [ + 117, + 641, + 416, + 655 + ], + "spans": [ + { + "bbox": [ + 117, + 641, + 258, + 655 + ], + "score": 1.0, + "content": "(S1). (Smoothness). For any fixed", + "type": "text" + }, + { + "bbox": [ + 258, + 645, + 266, + 653 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 641, + 284, + 655 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 284, + 642, + 356, + 655 + ], + "score": 0.6, + "content": "j \\in [ C ] , \\Phi _ { j } ( \\cdot ; x )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 641, + 367, + 655 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 367, + 642, + 378, + 653 + ], + "score": 0.86, + "content": "\\mathcal { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 641, + 416, + 655 + ], + "score": 1.0, + "content": "-smooth;", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + }, + { + "bbox": [ + 116, + 657, + 420, + 673 + ], + "spans": [ + { + "bbox": [ + 116, + 657, + 243, + 673 + ], + "score": 1.0, + "content": "(S5). 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So now we redefine them as", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 112, + 418, + 164 + ], + "lines": [ + { + "bbox": [ + 192, + 112, + 418, + 164 + ], + "spans": [ + { + "bbox": [ + 192, + 112, + 418, + 164 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { B _ { 0 } : = \\operatorname* { s u p } \\left. s _ { n j } : \\pmb \\theta \\in { \\cal S } ^ { d - 1 } , n \\in [ N ] , j \\in [ C ] \\right. , } \\\\ & { B _ { 1 } : = \\operatorname* { s u p } \\left. \\left\\| \\nabla s _ { n j } \\right\\| _ { 2 } : \\pmb \\theta \\in { \\cal S } ^ { d - 1 } , n \\in [ N ] , j \\in [ C ] \\right. , } \\\\ & { B _ { 2 } : = \\operatorname* { s u p } \\left. \\left\\| \\nabla ^ { 2 } s _ { n j } \\right\\| _ { 2 } : \\pmb \\theta \\in { \\cal S } ^ { d - 1 } , n \\in [ N ] , j \\in [ C ] \\right. . } \\end{array}", + "type": "interline_equation", + "image_path": "b799d068e64b7aa07d6cb0056e98a70266e4fc15361e1caaef279a10ffd5cd44.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 192, + 112, + 418, + 129.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 192, + 129.33333333333334, + 418, + 146.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 192, + 146.66666666666669, + 418, + 164.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 401, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 402, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 252, + 186 + ], + "score": 1.0, + "content": "By the property of LSE, we can use", + "type": "text" + }, + { + "bbox": [ + 252, + 172, + 298, + 183 + ], + "score": 0.93, + "content": "B _ { 0 } , B _ { 1 } , B _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 170, + 337, + 186 + ], + "score": 1.0, + "content": "to bound", + "type": "text" + }, + { + "bbox": [ + 337, + 171, + 398, + 183 + ], + "score": 0.91, + "content": "\\tilde { q } _ { n } , \\nabla \\tilde { q } _ { n } , \\nabla ^ { 2 } \\tilde { q } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 170, + 402, + 186 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 190, + 493, + 311 + ], + "lines": [ + { + "bbox": [ + 119, + 190, + 493, + 311 + ], + "spans": [ + { + "bbox": [ + 119, + 190, + 493, + 311 + ], + "score": 0.95, + "content": "\\begin{array} { c } { \\displaystyle \\tilde { q } _ { n } \\leq q _ { n } \\leq B _ { 0 } \\rho ^ { L } , } \\\\ { \\| \\nabla \\tilde { q } _ { n } \\| _ { 2 } = \\left\\| \\sum _ { j = 1 } ^ { C } \\frac { e ^ { - s _ { n j } } } { \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } \\nabla s _ { n j } \\right\\| _ { 2 } \\leq \\sum _ { j = 1 } ^ { C } \\frac { e ^ { - s _ { n j } } } { \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } \\| \\nabla s _ { n j } \\| _ { 2 } \\leq B _ { 1 } \\rho ^ { L - 1 } , } \\\\ { \\| \\nabla ^ { 2 } \\tilde { q } _ { n } \\| _ { 2 } = \\left\\| \\sum _ { j = 1 } ^ { C } \\left( \\frac { e ^ { - s _ { n j } } } { \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } ( \\nabla ^ { 2 } s _ { n j } - \\nabla s _ { n j } \\nabla s _ { n j } ^ { \\top } ) + \\frac { e ^ { - 2 s _ { n j } } } { \\left( \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } \\right) ^ { 2 } } \\nabla s _ { n j } \\nabla s _ { n j } ^ { \\top } \\right) \\right\\| _ { 2 } } \\\\ { \\leq B _ { 2 } \\rho ^ { L - 2 } + 2 B _ { 1 } ^ { 2 } \\rho ^ { 2 L - 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "f28750a72d31d71ed4c06b244441f500142fa238d4f842c59ebcd7ab4ade0c1e.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 119, + 190, + 493, + 230.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 119, + 230.33333333333334, + 493, + 270.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 119, + 270.6666666666667, + 493, + 311.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 407, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 408, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 315, + 331 + ], + "score": 1.0, + "content": "Using these bounds, we can prove with the constant", + "type": "text" + }, + { + "bbox": [ + 316, + 319, + 329, + 330 + ], + "score": 0.88, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 317, + 408, + 331 + ], + "score": 1.0, + "content": "defined as follows:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 337, + 485, + 403 + ], + "lines": [ + { + "bbox": [ + 126, + 337, + 485, + 403 + ], + "spans": [ + { + "bbox": [ + 126, + 337, + 485, + 403 + ], + "score": 0.65, + "content": "\\begin{array} { r l } & { C _ { \\eta } : = \\frac { 1 } { 2 } \\left( \\left( \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } + \\frac { p 2 ^ { p + 1 } } { \\log { \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } } \\right) B _ { 1 } ^ { 2 } + ( 2 \\log { 2 } ) \\left( \\frac { B _ { 2 } } { \\rho ( t _ { 0 } ) ^ { L } } + 2 B _ { 1 } ^ { 2 } \\right) \\right) \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } M , } \\\\ & { \\textsf { r e } M : = \\operatorname* { m i n } \\Big \\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\Big \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "f2df2ed0c09873a866782ce039c4fd60775ef759223ef50e9af242ee12758847.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 106, + 337, + 485, + 359.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 106, + 359.0, + 485, + 381.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 106, + 381.0, + 485, + 403.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 417, + 365, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 366, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 366, + 432 + ], + "score": 1.0, + "content": "H EXTENSION: MULTI-HOMOGENEOUS MODELS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "In this section, we extend our results to multi-homogeneous models. For this, the main difference", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 456, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 107, + 456, + 504, + 466 + ], + "score": 1.0, + "content": "from the proof for homogeneous models is that now we have to separate the norm of each homo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "geneous parts of the parameter, rather than consider them as a whole. So only a small part to proof", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "needs to be changed. We focus on gradient flow, but it is worth to note that following the same", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 489, + 363, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 363, + 501 + ], + "score": 1.0, + "content": "argument, it is not hard to extend the results to gradient descent.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 503, + 531 + ], + "lines": [ + { + "bbox": [ + 102, + 501, + 509, + 532 + ], + "spans": [ + { + "bbox": [ + 102, + 501, + 123, + 532 + ], + "score": 1.0, + "content": "Let smo", + "type": "text" + }, + { + "bbox": [ + 124, + 505, + 201, + 518 + ], + "score": 0.91, + "content": "\\Phi ( { \\pmb w } _ { 1 } , \\ldots , { \\pmb w } _ { m } ; { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 501, + 217, + 532 + ], + "score": 1.0, + "content": "be arg", + "type": "text" + }, + { + "bbox": [ + 217, + 505, + 270, + 518 + ], + "score": 0.92, + "content": "\\left( k _ { 1 } , \\ldots , k _ { m } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 501, + 352, + 532 + ], + "score": 1.0, + "content": "-homogeneous. Let (5) can be rewritten a", + "type": "text" + }, + { + "bbox": [ + 353, + 505, + 406, + 518 + ], + "score": 0.93, + "content": "\\rho _ { i } = \\| \\pmb { w } _ { i } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 501, + 425, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 505, + 479, + 521 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { \\pmb { w } } _ { i } = \\frac { { \\pmb w } _ { i } } { \\| { \\pmb w } _ { i } \\| _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 501, + 509, + 532 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 105, + 534, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 401, + 549 + ], + "score": 1.0, + "content": "Definition H.1. For a multi-homogeneous model with loss function", + "type": "text" + }, + { + "bbox": [ + 401, + 535, + 417, + 547 + ], + "score": 0.89, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 533, + 505, + 549 + ], + "score": 1.0, + "content": "satisfying (B3), the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 546, + 319, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 225, + 559 + ], + "score": 1.0, + "content": "smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 226, + 546, + 246, + 559 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 546, + 258, + 559 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 258, + 547, + 266, + 556 + ], + "score": 0.81, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 546, + 319, + 559 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 566, + 372, + 598 + ], + "lines": [ + { + "bbox": [ + 237, + 566, + 372, + 598 + ], + "spans": [ + { + "bbox": [ + 237, + 566, + 372, + 598 + ], + "score": 0.94, + "content": "{ \\tilde { \\gamma } } ( \\pmb { \\theta } ) : = \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } .", + "type": "interline_equation", + "image_path": "3ae3606dfb09b2840c65f58b06091b37967bf8963d073a84af9b3575f8217724.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 566, + 372, + 582.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 237, + 582.0, + 372, + 598.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 610, + 494, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 495, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 495, + 624 + ], + "score": 1.0, + "content": "We only prove the generalized version of Lemma 5.1 here. The other proofs are almost the same.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 222, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 223, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 194, + 641 + ], + "score": 1.0, + "content": "Lemma H.2. For all", + "type": "text" + }, + { + "bbox": [ + 194, + 628, + 219, + 639 + ], + "score": 0.91, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 626, + 223, + 641 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 646, + 464, + 679 + ], + "lines": [ + { + "bbox": [ + 129, + 646, + 464, + 679 + ], + "spans": [ + { + "bbox": [ + 129, + 646, + 464, + 679 + ], + "score": 0.92, + "content": "{ \\frac { d } { d t } } \\log \\rho _ { i } > 0 f o r a l l i \\in [ m ] \\quad a n d \\quad { \\frac { d } { d t } } \\log { \\tilde { \\gamma } } \\geq \\sum _ { i = 1 } ^ { m } k _ { i } \\left( { \\frac { d } { d t } } \\log \\rho _ { i } \\right) ^ { - 1 } \\left\\| { \\frac { d { \\hat { w } } _ { i } } { d t } } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "dcfeeec80720f64ad1fbe77ac3c3da3e230fc876b8d8a3344fc5780ef1a15457.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 129, + 646, + 464, + 657.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 129, + 657.0, + 464, + 668.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 129, + 668.0, + 464, + 679.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 701, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 256, + 721 + ], + "score": 1.0, + "content": "Proof. Note that ddt log ρi = 12ρ2 dρidt", + "type": "text" + }, + { + "bbox": [ + 177, + 701, + 290, + 720 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho _ { i } = \\frac { 1 } { 2 \\rho _ { i } ^ { 2 } } \\frac { d \\rho _ { i } ^ { 2 } } { d t } = \\frac { k _ { i } \\nu ( t ) } { \\rho _ { i } ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 700, + 506, + 721 + ], + "score": 1.0, + "content": "by Theorem B.4. It simply follows from Lemma B.5", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 717, + 507, + 736 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 124, + 736 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 719, + 176, + 734 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho > 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 717, + 207, + 736 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 208, + 721, + 234, + 732 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 717, + 341, + 736 + ], + "score": 1.0, + "content": ". And it is easy to see that", + "type": "text" + }, + { + "bbox": [ + 341, + 719, + 451, + 734 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\log { \\tilde { \\gamma } } = \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\rho ^ { L } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 717, + 507, + 736 + ], + "score": 1.0, + "content": "exists for all", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + } + ], + "page_idx": 41, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 322, + 96 + ], + "score": 1.0, + "content": "For Lemma F.3, we first note the original definition of", + "type": "text" + }, + { + "bbox": [ + 323, + 83, + 336, + 95 + ], + "score": 0.89, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 81, + 372, + 96 + ], + "score": 1.0, + "content": "involves", + "type": "text" + }, + { + "bbox": [ + 372, + 83, + 418, + 94 + ], + "score": 0.92, + "content": "B _ { 0 } , B _ { 1 } , B _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 81, + 506, + 96 + ], + "score": 1.0, + "content": ", which are undefined", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 324, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 324, + 105 + ], + "score": 1.0, + "content": "in the multi-class setting. So now we redefine them as", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 105 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 112, + 418, + 164 + ], + "lines": [ + { + "bbox": [ + 192, + 112, + 418, + 164 + ], + "spans": [ + { + "bbox": [ + 192, + 112, + 418, + 164 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { B _ { 0 } : = \\operatorname* { s u p } \\left. s _ { n j } : \\pmb \\theta \\in { \\cal S } ^ { d - 1 } , n \\in [ N ] , j \\in [ C ] \\right. , } \\\\ & { B _ { 1 } : = \\operatorname* { s u p } \\left. \\left\\| \\nabla s _ { n j } \\right\\| _ { 2 } : \\pmb \\theta \\in { \\cal S } ^ { d - 1 } , n \\in [ N ] , j \\in [ C ] \\right. , } \\\\ & { B _ { 2 } : = \\operatorname* { s u p } \\left. \\left\\| \\nabla ^ { 2 } s _ { n j } \\right\\| _ { 2 } : \\pmb \\theta \\in { \\cal S } ^ { d - 1 } , n \\in [ N ] , j \\in [ C ] \\right. . } \\end{array}", + "type": "interline_equation", + "image_path": "b799d068e64b7aa07d6cb0056e98a70266e4fc15361e1caaef279a10ffd5cd44.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 192, + 112, + 418, + 129.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 192, + 129.33333333333334, + 418, + 146.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 192, + 146.66666666666669, + 418, + 164.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 401, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 402, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 252, + 186 + ], + "score": 1.0, + "content": "By the property of LSE, we can use", + "type": "text" + }, + { + "bbox": [ + 252, + 172, + 298, + 183 + ], + "score": 0.93, + "content": "B _ { 0 } , B _ { 1 } , B _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 170, + 337, + 186 + ], + "score": 1.0, + "content": "to bound", + "type": "text" + }, + { + "bbox": [ + 337, + 171, + 398, + 183 + ], + "score": 0.91, + "content": "\\tilde { q } _ { n } , \\nabla \\tilde { q } _ { n } , \\nabla ^ { 2 } \\tilde { q } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 170, + 402, + 186 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 170, + 402, + 186 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 190, + 493, + 311 + ], + "lines": [ + { + "bbox": [ + 119, + 190, + 493, + 311 + ], + "spans": [ + { + "bbox": [ + 119, + 190, + 493, + 311 + ], + "score": 0.95, + "content": "\\begin{array} { c } { \\displaystyle \\tilde { q } _ { n } \\leq q _ { n } \\leq B _ { 0 } \\rho ^ { L } , } \\\\ { \\| \\nabla \\tilde { q } _ { n } \\| _ { 2 } = \\left\\| \\sum _ { j = 1 } ^ { C } \\frac { e ^ { - s _ { n j } } } { \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } \\nabla s _ { n j } \\right\\| _ { 2 } \\leq \\sum _ { j = 1 } ^ { C } \\frac { e ^ { - s _ { n j } } } { \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } \\| \\nabla s _ { n j } \\| _ { 2 } \\leq B _ { 1 } \\rho ^ { L - 1 } , } \\\\ { \\| \\nabla ^ { 2 } \\tilde { q } _ { n } \\| _ { 2 } = \\left\\| \\sum _ { j = 1 } ^ { C } \\left( \\frac { e ^ { - s _ { n j } } } { \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } } ( \\nabla ^ { 2 } s _ { n j } - \\nabla s _ { n j } \\nabla s _ { n j } ^ { \\top } ) + \\frac { e ^ { - 2 s _ { n j } } } { \\left( \\sum _ { k = 1 } ^ { C } e ^ { - s _ { n k } } \\right) ^ { 2 } } \\nabla s _ { n j } \\nabla s _ { n j } ^ { \\top } \\right) \\right\\| _ { 2 } } \\\\ { \\leq B _ { 2 } \\rho ^ { L - 2 } + 2 B _ { 1 } ^ { 2 } \\rho ^ { 2 L - 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "f28750a72d31d71ed4c06b244441f500142fa238d4f842c59ebcd7ab4ade0c1e.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 119, + 190, + 493, + 230.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 119, + 230.33333333333334, + 493, + 270.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 119, + 270.6666666666667, + 493, + 311.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 407, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 408, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 315, + 331 + ], + "score": 1.0, + "content": "Using these bounds, we can prove with the constant", + "type": "text" + }, + { + "bbox": [ + 316, + 319, + 329, + 330 + ], + "score": 0.88, + "content": "C _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 317, + 408, + 331 + ], + "score": 1.0, + "content": "defined as follows:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 317, + 408, + 331 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 337, + 485, + 403 + ], + "lines": [ + { + "bbox": [ + 126, + 337, + 485, + 403 + ], + "spans": [ + { + "bbox": [ + 126, + 337, + 485, + 403 + ], + "score": 0.65, + "content": "\\begin{array} { r l } & { C _ { \\eta } : = \\frac { 1 } { 2 } \\left( \\left( \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } + \\frac { p 2 ^ { p + 1 } } { \\log { \\frac { 1 } { \\mathcal { L } ( t _ { 0 } ) } } } \\right) B _ { 1 } ^ { 2 } + ( 2 \\log { 2 } ) \\left( \\frac { B _ { 2 } } { \\rho ( t _ { 0 } ) ^ { L } } + 2 B _ { 1 } ^ { 2 } \\right) \\right) \\left( \\frac { B _ { 0 } } { \\hat { \\gamma } ( t _ { 0 } ) } \\right) ^ { p } M , } \\\\ & { \\textsf { r e } M : = \\operatorname* { m i n } \\Big \\{ \\hat { \\gamma } ( t _ { 0 } ) ^ { - 2 + 2 / L } , B _ { 0 } ^ { - 2 + 2 / L } \\Big \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "f2df2ed0c09873a866782ce039c4fd60775ef759223ef50e9af242ee12758847.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 106, + 337, + 485, + 359.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 106, + 359.0, + 485, + 381.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 106, + 381.0, + 485, + 403.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 417, + 365, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 366, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 366, + 432 + ], + "score": 1.0, + "content": "H EXTENSION: MULTI-HOMOGENEOUS MODELS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "In this section, we extend our results to multi-homogeneous models. For this, the main difference", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 456, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 107, + 456, + 504, + 466 + ], + "score": 1.0, + "content": "from the proof for homogeneous models is that now we have to separate the norm of each homo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "geneous parts of the parameter, rather than consider them as a whole. So only a small part to proof", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "needs to be changed. We focus on gradient flow, but it is worth to note that following the same", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 489, + 363, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 363, + 501 + ], + "score": 1.0, + "content": "argument, it is not hard to extend the results to gradient descent.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 444, + 506, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 503, + 531 + ], + "lines": [ + { + "bbox": [ + 102, + 501, + 509, + 532 + ], + "spans": [ + { + "bbox": [ + 102, + 501, + 123, + 532 + ], + "score": 1.0, + "content": "Let smo", + "type": "text" + }, + { + "bbox": [ + 124, + 505, + 201, + 518 + ], + "score": 0.91, + "content": "\\Phi ( { \\pmb w } _ { 1 } , \\ldots , { \\pmb w } _ { m } ; { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 501, + 217, + 532 + ], + "score": 1.0, + "content": "be arg", + "type": "text" + }, + { + "bbox": [ + 217, + 505, + 270, + 518 + ], + "score": 0.92, + "content": "\\left( k _ { 1 } , \\ldots , k _ { m } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 501, + 352, + 532 + ], + "score": 1.0, + "content": "-homogeneous. Let (5) can be rewritten a", + "type": "text" + }, + { + "bbox": [ + 353, + 505, + 406, + 518 + ], + "score": 0.93, + "content": "\\rho _ { i } = \\| \\pmb { w } _ { i } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 501, + 425, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 505, + 479, + 521 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { \\pmb { w } } _ { i } = \\frac { { \\pmb w } _ { i } } { \\| { \\pmb w } _ { i } \\| _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 501, + 509, + 532 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 102, + 501, + 509, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 534, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 401, + 549 + ], + "score": 1.0, + "content": "Definition H.1. For a multi-homogeneous model with loss function", + "type": "text" + }, + { + "bbox": [ + 401, + 535, + 417, + 547 + ], + "score": 0.89, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 533, + 505, + 549 + ], + "score": 1.0, + "content": "satisfying (B3), the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 546, + 319, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 225, + 559 + ], + "score": 1.0, + "content": "smoothed normalized margin", + "type": "text" + }, + { + "bbox": [ + 226, + 546, + 246, + 559 + ], + "score": 0.92, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 546, + 258, + 559 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 258, + 547, + 266, + 556 + ], + "score": 0.81, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 546, + 319, + 559 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 533, + 505, + 559 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 566, + 372, + 598 + ], + "lines": [ + { + "bbox": [ + 237, + 566, + 372, + 598 + ], + "spans": [ + { + "bbox": [ + 237, + 566, + 372, + 598 + ], + "score": 0.94, + "content": "{ \\tilde { \\gamma } } ( \\pmb { \\theta } ) : = \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } .", + "type": "interline_equation", + "image_path": "3ae3606dfb09b2840c65f58b06091b37967bf8963d073a84af9b3575f8217724.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 566, + 372, + 582.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 237, + 582.0, + 372, + 598.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 610, + 494, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 495, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 495, + 624 + ], + "score": 1.0, + "content": "We only prove the generalized version of Lemma 5.1 here. The other proofs are almost the same.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 609, + 495, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 222, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 223, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 194, + 641 + ], + "score": 1.0, + "content": "Lemma H.2. For all", + "type": "text" + }, + { + "bbox": [ + 194, + 628, + 219, + 639 + ], + "score": 0.91, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 626, + 223, + 641 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 626, + 223, + 641 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 646, + 464, + 679 + ], + "lines": [ + { + "bbox": [ + 129, + 646, + 464, + 679 + ], + "spans": [ + { + "bbox": [ + 129, + 646, + 464, + 679 + ], + "score": 0.92, + "content": "{ \\frac { d } { d t } } \\log \\rho _ { i } > 0 f o r a l l i \\in [ m ] \\quad a n d \\quad { \\frac { d } { d t } } \\log { \\tilde { \\gamma } } \\geq \\sum _ { i = 1 } ^ { m } k _ { i } \\left( { \\frac { d } { d t } } \\log \\rho _ { i } \\right) ^ { - 1 } \\left\\| { \\frac { d { \\hat { w } } _ { i } } { d t } } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "dcfeeec80720f64ad1fbe77ac3c3da3e230fc876b8d8a3344fc5780ef1a15457.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 129, + 646, + 464, + 657.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 129, + 657.0, + 464, + 668.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 129, + 668.0, + 464, + 679.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 701, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 256, + 721 + ], + "score": 1.0, + "content": "Proof. Note that ddt log ρi = 12ρ2 dρidt", + "type": "text" + }, + { + "bbox": [ + 177, + 701, + 290, + 720 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho _ { i } = \\frac { 1 } { 2 \\rho _ { i } ^ { 2 } } \\frac { d \\rho _ { i } ^ { 2 } } { d t } = \\frac { k _ { i } \\nu ( t ) } { \\rho _ { i } ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 700, + 506, + 721 + ], + "score": 1.0, + "content": "by Theorem B.4. It simply follows from Lemma B.5", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 717, + 507, + 736 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 124, + 736 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 719, + 176, + 734 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\rho > 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 717, + 207, + 736 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 208, + 721, + 234, + 732 + ], + "score": 0.9, + "content": "t > t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 717, + 341, + 736 + ], + "score": 1.0, + "content": ". And it is easy to see that", + "type": "text" + }, + { + "bbox": [ + 341, + 719, + 451, + 734 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\log { \\tilde { \\gamma } } = \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) / \\rho ^ { L } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 717, + 507, + 736 + ], + "score": 1.0, + "content": "exists for all", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 700, + 507, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 311, + 94 + ], + "lines": [ + { + "bbox": [ + 107, + 81, + 312, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 132, + 94 + ], + "score": 0.89, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 81, + 312, + 95 + ], + "score": 1.0, + "content": ". By the chain rule and Lemma B.5, we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 96, + 494, + 198 + ], + "lines": [ + { + "bbox": [ + 118, + 96, + 494, + 198 + ], + "spans": [ + { + "bbox": [ + 118, + 96, + 494, + 198 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\tilde { \\gamma } = \\frac { d } { d t } \\left( \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\right) - \\displaystyle \\sum _ { i = 1 } ^ { m } k _ { i } \\log \\rho _ { i } \\right) = \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\cdot \\frac { 1 } { \\mathcal { L } } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - \\displaystyle \\sum _ { i = 1 } ^ { m } \\frac { k _ { i } ^ { 2 } \\nu ( t ) } { \\rho _ { i } ^ { 2 } } } \\\\ { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - \\displaystyle \\sum _ { i = 1 } ^ { m } \\frac { k _ { i } ^ { 2 } \\nu ( t ) } { \\rho _ { i } ^ { 2 } } } \\\\ { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } - \\displaystyle \\sum _ { i = 1 } ^ { m } \\frac { k _ { i } ^ { 2 } \\nu ( t ) ^ { 2 } } { \\rho _ { i } ^ { 2 } } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "15b8d64da4d0d441df55c40893b3efafb4a77051c8c500dd022e79b71d155104.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 118, + 96, + 494, + 130.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 118, + 130.0, + 494, + 164.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 118, + 164.0, + 494, + 198.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 506, + 229 + ], + "lines": [ + { + "bbox": [ + 104, + 197, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 178, + 219 + ], + "score": 1.0, + "content": "On the one hand,", + "type": "text" + }, + { + "bbox": [ + 178, + 199, + 271, + 217 + ], + "score": 0.93, + "content": "\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } = \\sum _ { i = 1 } ^ { m } \\left\\| \\frac { d \\pmb { w } _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 200, + 302, + 216 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 303, + 203, + 327, + 214 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 200, + 466, + 216 + ], + "score": 1.0, + "content": "by Lemma I.3; on the other hand,", + "type": "text" + }, + { + "bbox": [ + 466, + 202, + 505, + 215 + ], + "score": 0.93, + "content": "k _ { i } \\nu ( t ) =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 208, + 357, + 237 + ], + "spans": [ + { + "bbox": [ + 107, + 215, + 149, + 230 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\left. w _ { i } , \\frac { d w _ { i } } { d t } \\right. } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 208, + 357, + 237 + ], + "score": 1.0, + "content": "dt i=1 \r dt \r2 by Theorem B.4. Combining these together yields", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 231, + 470, + 265 + ], + "lines": [ + { + "bbox": [ + 141, + 231, + 470, + 265 + ], + "spans": [ + { + "bbox": [ + 141, + 231, + 470, + 265 + ], + "score": 0.92, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq \\frac { 1 } { \\nu ( t ) } \\sum _ { i = 1 } ^ { m } \\left( \\left\\| \\frac { d w _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } - \\left. \\hat { w } _ { i } , \\frac { d w _ { i } } { d t } \\right. ^ { 2 } \\right) = \\frac { 1 } { \\nu ( t ) } \\left\\| ( I - \\hat { w } _ { i } \\hat { w } _ { i } ^ { \\top } ) \\frac { d w _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "47f7987bb5760861c83e4842fce0b8ecdec07f1d936900cac1ee117fb7476335.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 141, + 231, + 470, + 242.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 141, + 242.33333333333334, + 470, + 253.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 141, + 253.66666666666669, + 470, + 265.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 400, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 401, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 180, + 287 + ], + "score": 1.0, + "content": "By the chain rule,", + "type": "text" + }, + { + "bbox": [ + 181, + 273, + 292, + 290 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d \\hat { \\pmb { w } } _ { i } } { d t } = \\frac { 1 } { \\rho _ { i } } ( \\pmb { I } - \\hat { \\pmb { w } _ { i } } \\hat { \\pmb { w } _ { i } } ^ { \\top } ) \\frac { d \\pmb { w } _ { i } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 272, + 401, + 289 + ], + "score": 1.0, + "content": "i>) dwidt for a.e. t > 0. So we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 291, + 437, + 324 + ], + "lines": [ + { + "bbox": [ + 174, + 291, + 437, + 324 + ], + "spans": [ + { + "bbox": [ + 174, + 291, + 437, + 324 + ], + "score": 0.93, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq \\sum _ { i = 1 } ^ { m } \\frac { \\rho _ { i } ^ { 2 } } { \\nu ( t ) } \\left\\| \\frac { d \\hat { w } _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } = \\sum _ { i = 1 } ^ { m } k _ { i } \\left( \\frac { d } { d t } \\log \\rho _ { i } \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { w } _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "41c70e65772082b7eff9d9af185f6da7de896636485fd823cf76e8e7a892004d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 174, + 291, + 437, + 302.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 174, + 302.0, + 437, + 313.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 174, + 313.0, + 437, + 324.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 504, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 361, + 361 + ], + "score": 1.0, + "content": "For cross-entropy loss, we can combine the proofs in Appendix", + "type": "text" + }, + { + "bbox": [ + 362, + 350, + 370, + 360 + ], + "score": 0.46, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 349, + 506, + 361 + ], + "score": 1.0, + "content": "to show that Lemma H.2 holds if", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 359, + 380, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 380, + 373 + ], + "score": 1.0, + "content": "we use the following definition of the smoothed normalized margin:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 502, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 504, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 504, + 387 + ], + "score": 1.0, + "content": "Definition H.3. For a multi-homogeneous model with cross-entropy, the smoothed normalized mar-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 385, + 216, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 121, + 397 + ], + "score": 1.0, + "content": "gin", + "type": "text" + }, + { + "bbox": [ + 122, + 385, + 142, + 397 + ], + "score": 0.91, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 385, + 154, + 397 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 155, + 385, + 162, + 395 + ], + "score": 0.8, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 385, + 216, + 397 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 395, + 374, + 425 + ], + "lines": [ + { + "bbox": [ + 236, + 395, + 374, + 425 + ], + "spans": [ + { + "bbox": [ + 236, + 395, + 374, + 425 + ], + "score": 0.94, + "content": "\\widetilde { \\gamma } = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } = \\frac { - \\log ( e ^ { \\mathcal { L } } - 1 ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } .", + "type": "interline_equation", + "image_path": "c5a3f3fe44de61c42ae4a8bc947e189166dd41676dd27f985a02670e35472171.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 236, + 395, + 374, + 410.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 236, + 410.0, + 374, + 425.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 346, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 346, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 133, + 440 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 425, + 160, + 438 + ], + "score": 0.92, + "content": "\\ell ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 423, + 326, + 440 + ], + "score": 1.0, + "content": "is the inverse function of the logistic loss", + "type": "text" + }, + { + "bbox": [ + 327, + 426, + 342, + 438 + ], + "score": 0.87, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 423, + 346, + 440 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 506, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "The only place we need to change in the proof for Lemma H.2 is that instead of using Lemma B.5,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 455, + 507, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 177, + 474 + ], + "score": 1.0, + "content": "we need to prove", + "type": "text" + }, + { + "bbox": [ + 178, + 456, + 250, + 476 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\nu ( t ) \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 455, + 507, + 475 + ], + "score": 1.0, + "content": "g(log 1L )g0(log 1 ) L in a similar way as in Lemma G.2. The other parts of the proof", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 474, + 231, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 231, + 486 + ], + "score": 1.0, + "content": "are exactly the same as before.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 108, + 501, + 403, + 514 + ], + "lines": [ + { + "bbox": [ + 104, + 500, + 403, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 403, + 516 + ], + "score": 1.0, + "content": "I CHAIN RULES FOR NON-DIFFERENTIABLE FUNCTIONS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 506, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "In this section, we provide some background on the chain rule for non-differentiable functions. The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "ordinary chain rule for differentiable functions is a very useful formula for computing derivatives", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "in calculus. However, for non-differentiable functions, it is difficult to find a natural definition of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "subdifferential so that the chain rule equation holds exactly. To solve this issue, Clarke proposed", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "score": 1.0, + "content": "Clarke’s subdifferential (Clarke, 1975; 1990; Clarke et al., 2008) for locally Lipschitz functions, for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 580, + 372, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 372, + 593 + ], + "score": 1.0, + "content": "which the chain rule holds as an inclusion rather than an equation:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 506, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 375, + 607 + ], + "score": 1.0, + "content": "Theorem I.1 (Theorem 2.3.9 and 2.3.10 of (Clarke, 1990)). Let", + "type": "text" + }, + { + "bbox": [ + 375, + 593, + 468, + 605 + ], + "score": 0.89, + "content": "z _ { 1 } , \\dots , z _ { n } : \\mathbb { R } ^ { d } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 592, + 489, + 607 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 489, + 594, + 505, + 606 + ], + "score": 0.71, + "content": "f :", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 144, + 615 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { n } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 603, + 284, + 619 + ], + "score": 1.0, + "content": "be locally Lipschitz functions. Let", + "type": "text" + }, + { + "bbox": [ + 284, + 605, + 425, + 617 + ], + "score": 0.91, + "content": "( f \\circ z ) ( { \\pmb x } ) = f ( z _ { 1 } ( { \\pmb x } ) , \\ldots , z _ { n } ( { \\pmb x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 603, + 506, + 619 + ], + "score": 1.0, + "content": "be the composition", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 180, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 117, + 629 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 617, + 124, + 628 + ], + "score": 0.81, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 615, + 142, + 629 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 143, + 618, + 149, + 626 + ], + "score": 0.65, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 615, + 180, + 629 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 629, + 470, + 663 + ], + "lines": [ + { + "bbox": [ + 141, + 629, + 470, + 663 + ], + "spans": [ + { + "bbox": [ + 141, + 629, + 470, + 663 + ], + "score": 0.92, + "content": "\\partial ^ { \\circ } ( f \\circ z ) ( \\pmb { x } ) \\subseteq \\mathrm { c o n v } \\left\\{ \\sum _ { i = 1 } ^ { n } \\alpha _ { i } \\pmb { h } _ { i } : \\pmb { \\alpha } \\in \\partial ^ { \\circ } f ( z _ { 1 } ( \\pmb { x } ) , \\ldots , z _ { n } ( \\pmb { x } ) ) , \\pmb { h } _ { i } \\in \\partial ^ { \\circ } z _ { i } ( \\pmb { x } ) \\right\\} .", + "type": "interline_equation", + "image_path": "525ea9af94868c51dd6fd748596ad09fb133baa862186595e71700a703df6b97.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 141, + 629, + 470, + 640.3333333333334 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 141, + 640.3333333333334, + 470, + 651.6666666666667 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 141, + 651.6666666666667, + 470, + 663.0000000000001 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 670, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 464, + 684 + ], + "score": 1.0, + "content": "For analyzing gradient flow, the chain rule is crucial. For a differentiable loss function", + "type": "text" + }, + { + "bbox": [ + 465, + 671, + 486, + 683 + ], + "score": 0.9, + "content": "\\mathcal { L } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 670, + 506, + 684 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 157, + 681, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 157, + 681, + 472, + 709 + ], + "score": 1.0, + "content": "the chain rule that the function value keeps decreasing along the gradient flow :", + "type": "text" + }, + { + "bbox": [ + 473, + 683, + 505, + 698 + ], + "score": 0.91, + "content": "\\begin{array} { r l r } { \\mathrm { ~ } } & { { } } & { \\frac { d \\pmb \\theta ( t ) } { d t } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 695, + 156, + 708 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 156, + 708 + ], + "score": 0.89, + "content": "- \\nabla \\mathcal { L } ( \\pmb \\theta ( t ) )", + "type": "inline_equation" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 705, + 409, + 736 + ], + "lines": [ + { + "bbox": [ + 202, + 705, + 409, + 736 + ], + "spans": [ + { + "bbox": [ + 202, + 705, + 409, + 736 + ], + "score": 0.93, + "content": "\\frac { d \\mathcal { L } ( \\pmb { \\theta } ( t ) ) } { d t } = \\left. \\nabla \\mathcal { L } ( \\pmb { \\theta } ( t ) ) , \\frac { d \\pmb { \\theta } ( t ) } { d t } \\right. = - \\left\\| \\frac { d \\pmb { \\theta } ( t ) } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "58b648788b9f4aa77f760b8e37d496e4fd0939dfd021f3ff35649e1fcd6be781.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 202, + 705, + 409, + 720.5 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 202, + 720.5, + 409, + 736.0 + ], + "spans": [], + "index": 40 + } + ] + } + ], + "page_idx": 42, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "43", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 326, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 496, + 327, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 496, + 327, + 505, + 336 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 311, + 94 + ], + "lines": [ + { + "bbox": [ + 107, + 81, + 312, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 132, + 94 + ], + "score": 0.89, + "content": "t \\geq t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 81, + 312, + 95 + ], + "score": 1.0, + "content": ". By the chain rule and Lemma B.5, we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 107, + 81, + 312, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 96, + 494, + 198 + ], + "lines": [ + { + "bbox": [ + 118, + 96, + 494, + 198 + ], + "spans": [ + { + "bbox": [ + 118, + 96, + 494, + 198 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d } { d t } \\log \\tilde { \\gamma } = \\frac { d } { d t } \\left( \\log \\left( g ( \\log \\frac { 1 } { \\mathcal { L } } ) \\right) - \\displaystyle \\sum _ { i = 1 } ^ { m } k _ { i } \\log \\rho _ { i } \\right) = \\frac { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\cdot \\frac { 1 } { \\mathcal { L } } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - \\displaystyle \\sum _ { i = 1 } ^ { m } \\frac { k _ { i } ^ { 2 } \\nu ( t ) } { \\rho _ { i } ^ { 2 } } } \\\\ { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } \\right) - \\displaystyle \\sum _ { i = 1 } ^ { m } \\frac { k _ { i } ^ { 2 } \\nu ( t ) } { \\rho _ { i } ^ { 2 } } } \\\\ { \\geq \\frac { 1 } { \\nu ( t ) } \\cdot \\left( - \\frac { d \\mathcal { L } } { d t } - \\displaystyle \\sum _ { i = 1 } ^ { m } \\frac { k _ { i } ^ { 2 } \\nu ( t ) ^ { 2 } } { \\rho _ { i } ^ { 2 } } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "15b8d64da4d0d441df55c40893b3efafb4a77051c8c500dd022e79b71d155104.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 118, + 96, + 494, + 130.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 118, + 130.0, + 494, + 164.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 118, + 164.0, + 494, + 198.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 506, + 229 + ], + "lines": [ + { + "bbox": [ + 104, + 197, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 178, + 219 + ], + "score": 1.0, + "content": "On the one hand,", + "type": "text" + }, + { + "bbox": [ + 178, + 199, + 271, + 217 + ], + "score": 0.93, + "content": "\\begin{array} { r } { - \\frac { d \\mathcal { L } } { d t } = \\sum _ { i = 1 } ^ { m } \\left\\| \\frac { d \\pmb { w } _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 200, + 302, + 216 + ], + "score": 1.0, + "content": "for a.e.", + "type": "text" + }, + { + "bbox": [ + 303, + 203, + 327, + 214 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 200, + 466, + 216 + ], + "score": 1.0, + "content": "by Lemma I.3; on the other hand,", + "type": "text" + }, + { + "bbox": [ + 466, + 202, + 505, + 215 + ], + "score": 0.93, + "content": "k _ { i } \\nu ( t ) =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 208, + 357, + 237 + ], + "spans": [ + { + "bbox": [ + 107, + 215, + 149, + 230 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\left. w _ { i } , \\frac { d w _ { i } } { d t } \\right. } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 208, + 357, + 237 + ], + "score": 1.0, + "content": "dt i=1 \r dt \r2 by Theorem B.4. Combining these together yields", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 197, + 505, + 237 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 231, + 470, + 265 + ], + "lines": [ + { + "bbox": [ + 141, + 231, + 470, + 265 + ], + "spans": [ + { + "bbox": [ + 141, + 231, + 470, + 265 + ], + "score": 0.92, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq \\frac { 1 } { \\nu ( t ) } \\sum _ { i = 1 } ^ { m } \\left( \\left\\| \\frac { d w _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } - \\left. \\hat { w } _ { i } , \\frac { d w _ { i } } { d t } \\right. ^ { 2 } \\right) = \\frac { 1 } { \\nu ( t ) } \\left\\| ( I - \\hat { w } _ { i } \\hat { w } _ { i } ^ { \\top } ) \\frac { d w _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "47f7987bb5760861c83e4842fce0b8ecdec07f1d936900cac1ee117fb7476335.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 141, + 231, + 470, + 242.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 141, + 242.33333333333334, + 470, + 253.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 141, + 253.66666666666669, + 470, + 265.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 400, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 401, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 180, + 287 + ], + "score": 1.0, + "content": "By the chain rule,", + "type": "text" + }, + { + "bbox": [ + 181, + 273, + 292, + 290 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { d \\hat { \\pmb { w } } _ { i } } { d t } = \\frac { 1 } { \\rho _ { i } } ( \\pmb { I } - \\hat { \\pmb { w } _ { i } } \\hat { \\pmb { w } _ { i } } ^ { \\top } ) \\frac { d \\pmb { w } _ { i } } { d t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 272, + 401, + 289 + ], + "score": 1.0, + "content": "i>) dwidt for a.e. t > 0. So we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 272, + 401, + 290 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 291, + 437, + 324 + ], + "lines": [ + { + "bbox": [ + 174, + 291, + 437, + 324 + ], + "spans": [ + { + "bbox": [ + 174, + 291, + 437, + 324 + ], + "score": 0.93, + "content": "\\frac { d } { d t } \\log \\tilde { \\gamma } \\geq \\sum _ { i = 1 } ^ { m } \\frac { \\rho _ { i } ^ { 2 } } { \\nu ( t ) } \\left\\| \\frac { d \\hat { w } _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } = \\sum _ { i = 1 } ^ { m } k _ { i } \\left( \\frac { d } { d t } \\log \\rho _ { i } \\right) ^ { - 1 } \\left\\| \\frac { d \\hat { w } _ { i } } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "41c70e65772082b7eff9d9af185f6da7de896636485fd823cf76e8e7a892004d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 174, + 291, + 437, + 302.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 174, + 302.0, + 437, + 313.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 174, + 313.0, + 437, + 324.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 504, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 361, + 361 + ], + "score": 1.0, + "content": "For cross-entropy loss, we can combine the proofs in Appendix", + "type": "text" + }, + { + "bbox": [ + 362, + 350, + 370, + 360 + ], + "score": 0.46, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 349, + 506, + 361 + ], + "score": 1.0, + "content": "to show that Lemma H.2 holds if", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 359, + 380, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 380, + 373 + ], + "score": 1.0, + "content": "we use the following definition of the smoothed normalized margin:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 349, + 506, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 502, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 504, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 504, + 387 + ], + "score": 1.0, + "content": "Definition H.3. For a multi-homogeneous model with cross-entropy, the smoothed normalized mar-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 385, + 216, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 121, + 397 + ], + "score": 1.0, + "content": "gin", + "type": "text" + }, + { + "bbox": [ + 122, + 385, + 142, + 397 + ], + "score": 0.91, + "content": "\\widetilde \\gamma ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 385, + 154, + 397 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 155, + 385, + 162, + 395 + ], + "score": 0.8, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 385, + 216, + 397 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 373, + 504, + 397 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 395, + 374, + 425 + ], + "lines": [ + { + "bbox": [ + 236, + 395, + 374, + 425 + ], + "spans": [ + { + "bbox": [ + 236, + 395, + 374, + 425 + ], + "score": 0.94, + "content": "\\widetilde { \\gamma } = \\frac { \\ell ^ { - 1 } ( \\mathcal { L } ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } = \\frac { - \\log ( e ^ { \\mathcal { L } } - 1 ) } { \\prod _ { i = 1 } ^ { m } \\rho _ { i } ^ { k _ { i } } } .", + "type": "interline_equation", + "image_path": "c5a3f3fe44de61c42ae4a8bc947e189166dd41676dd27f985a02670e35472171.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 236, + 395, + 374, + 410.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 236, + 410.0, + 374, + 425.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 346, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 346, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 133, + 440 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 425, + 160, + 438 + ], + "score": 0.92, + "content": "\\ell ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 423, + 326, + 440 + ], + "score": 1.0, + "content": "is the inverse function of the logistic loss", + "type": "text" + }, + { + "bbox": [ + 327, + 426, + 342, + 438 + ], + "score": 0.87, + "content": "\\ell ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 423, + 346, + 440 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 423, + 346, + 440 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 506, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "The only place we need to change in the proof for Lemma H.2 is that instead of using Lemma B.5,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 455, + 507, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 177, + 474 + ], + "score": 1.0, + "content": "we need to prove", + "type": "text" + }, + { + "bbox": [ + 178, + 456, + 250, + 476 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\nu ( t ) \\geq \\frac { g ( \\log \\frac { 1 } { \\mathcal { L } } ) } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\mathcal { L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 455, + 507, + 475 + ], + "score": 1.0, + "content": "g(log 1L )g0(log 1 ) L in a similar way as in Lemma G.2. The other parts of the proof", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 474, + 231, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 231, + 486 + ], + "score": 1.0, + "content": "are exactly the same as before.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 445, + 507, + 486 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 501, + 403, + 514 + ], + "lines": [ + { + "bbox": [ + 104, + 500, + 403, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 403, + 516 + ], + "score": 1.0, + "content": "I CHAIN RULES FOR NON-DIFFERENTIABLE FUNCTIONS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 506, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "In this section, we provide some background on the chain rule for non-differentiable functions. The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "ordinary chain rule for differentiable functions is a very useful formula for computing derivatives", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "in calculus. However, for non-differentiable functions, it is difficult to find a natural definition of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "subdifferential so that the chain rule equation holds exactly. To solve this issue, Clarke proposed", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "score": 1.0, + "content": "Clarke’s subdifferential (Clarke, 1975; 1990; Clarke et al., 2008) for locally Lipschitz functions, for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 580, + 372, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 372, + 593 + ], + "score": 1.0, + "content": "which the chain rule holds as an inclusion rather than an equation:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 525, + 506, + 593 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 506, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 375, + 607 + ], + "score": 1.0, + "content": "Theorem I.1 (Theorem 2.3.9 and 2.3.10 of (Clarke, 1990)). Let", + "type": "text" + }, + { + "bbox": [ + 375, + 593, + 468, + 605 + ], + "score": 0.89, + "content": "z _ { 1 } , \\dots , z _ { n } : \\mathbb { R } ^ { d } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 592, + 489, + 607 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 489, + 594, + 505, + 606 + ], + "score": 0.71, + "content": "f :", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 144, + 615 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { n } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 603, + 284, + 619 + ], + "score": 1.0, + "content": "be locally Lipschitz functions. Let", + "type": "text" + }, + { + "bbox": [ + 284, + 605, + 425, + 617 + ], + "score": 0.91, + "content": "( f \\circ z ) ( { \\pmb x } ) = f ( z _ { 1 } ( { \\pmb x } ) , \\ldots , z _ { n } ( { \\pmb x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 603, + 506, + 619 + ], + "score": 1.0, + "content": "be the composition", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 180, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 117, + 629 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 617, + 124, + 628 + ], + "score": 0.81, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 615, + 142, + 629 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 143, + 618, + 149, + 626 + ], + "score": 0.65, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 615, + 180, + 629 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 592, + 506, + 629 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 629, + 470, + 663 + ], + "lines": [ + { + "bbox": [ + 141, + 629, + 470, + 663 + ], + "spans": [ + { + "bbox": [ + 141, + 629, + 470, + 663 + ], + "score": 0.92, + "content": "\\partial ^ { \\circ } ( f \\circ z ) ( \\pmb { x } ) \\subseteq \\mathrm { c o n v } \\left\\{ \\sum _ { i = 1 } ^ { n } \\alpha _ { i } \\pmb { h } _ { i } : \\pmb { \\alpha } \\in \\partial ^ { \\circ } f ( z _ { 1 } ( \\pmb { x } ) , \\ldots , z _ { n } ( \\pmb { x } ) ) , \\pmb { h } _ { i } \\in \\partial ^ { \\circ } z _ { i } ( \\pmb { x } ) \\right\\} .", + "type": "interline_equation", + "image_path": "525ea9af94868c51dd6fd748596ad09fb133baa862186595e71700a703df6b97.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 141, + 629, + 470, + 640.3333333333334 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 141, + 640.3333333333334, + 470, + 651.6666666666667 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 141, + 651.6666666666667, + 470, + 663.0000000000001 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 670, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 464, + 684 + ], + "score": 1.0, + "content": "For analyzing gradient flow, the chain rule is crucial. For a differentiable loss function", + "type": "text" + }, + { + "bbox": [ + 465, + 671, + 486, + 683 + ], + "score": 0.9, + "content": "\\mathcal { L } ( \\pmb { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 670, + 506, + 684 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 157, + 681, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 157, + 681, + 472, + 709 + ], + "score": 1.0, + "content": "the chain rule that the function value keeps decreasing along the gradient flow :", + "type": "text" + }, + { + "bbox": [ + 473, + 683, + 505, + 698 + ], + "score": 0.91, + "content": "\\begin{array} { r l r } { \\mathrm { ~ } } & { { } } & { \\frac { d \\pmb \\theta ( t ) } { d t } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 695, + 156, + 708 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 156, + 708 + ], + "score": 0.89, + "content": "- \\nabla \\mathcal { L } ( \\pmb \\theta ( t ) )", + "type": "inline_equation" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 670, + 506, + 709 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 705, + 409, + 736 + ], + "lines": [ + { + "bbox": [ + 202, + 705, + 409, + 736 + ], + "spans": [ + { + "bbox": [ + 202, + 705, + 409, + 736 + ], + "score": 0.93, + "content": "\\frac { d \\mathcal { L } ( \\pmb { \\theta } ( t ) ) } { d t } = \\left. \\nabla \\mathcal { L } ( \\pmb { \\theta } ( t ) ) , \\frac { d \\pmb { \\theta } ( t ) } { d t } \\right. = - \\left\\| \\frac { d \\pmb { \\theta } ( t ) } { d t } \\right\\| _ { 2 } ^ { 2 } .", + "type": "interline_equation", + "image_path": "58b648788b9f4aa77f760b8e37d496e4fd0939dfd021f3ff35649e1fcd6be781.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 202, + 705, + 409, + 720.5 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 202, + 720.5, + 409, + 736.0 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "But for locally Lipschitz functions which could be non-differentiable, (25) may not hold in general", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 293, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 293, + 106 + ], + "score": 1.0, + "content": "since Theorem I.1 only holds for an inclusion.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 105, + 110, + 504, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "score": 1.0, + "content": "Following (Davis et al., 2020; Drusvyatskiy et al., 2015), we consider the functions that admit a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 121, + 197, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 197, + 134 + ], + "score": 1.0, + "content": "chain rule for any arc.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 136, + 504, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 134, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 337, + 151 + ], + "score": 1.0, + "content": "Definition I.2 (Chain Rule). A locally Lipschitz function", + "type": "text" + }, + { + "bbox": [ + 338, + 137, + 388, + 149 + ], + "score": 0.93, + "content": "f : { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 134, + 506, + 151 + ], + "score": 1.0, + "content": "admits a chain rule if for any", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 203, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 121, + 162 + ], + "score": 1.0, + "content": "arc", + "type": "text" + }, + { + "bbox": [ + 121, + 148, + 198, + 160 + ], + "score": 0.92, + "content": "z : [ 0 , + \\infty ) \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 146, + 203, + 162 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 161, + 392, + 176 + ], + "lines": [ + { + "bbox": [ + 218, + 161, + 392, + 176 + ], + "spans": [ + { + "bbox": [ + 218, + 161, + 392, + 176 + ], + "score": 0.91, + "content": "\\forall h \\in \\partial ^ { \\circ } f ( z ( t ) ) : ( f \\circ z ) ^ { \\prime } ( t ) = \\langle h , z ^ { \\prime } ( t ) \\rangle", + "type": "interline_equation", + "image_path": "15165a0145e9b766471106cd7edd1e4318a24b5556a8fb6e0abc1338004f8c1b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 218, + 161, + 392, + 176 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 188, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 188, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 161, + 192 + ], + "score": 1.0, + "content": "holds for a.e.", + "type": "text" + }, + { + "bbox": [ + 162, + 181, + 185, + 191 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 181, + 188, + 192 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 105, + 202, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "score": 1.0, + "content": "It is shown in (Davis et al., 2020; Drusvyatskiy et al., 2015) that a generalized version of (25) holds", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 213, + 183, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 183, + 226 + ], + "score": 1.0, + "content": "for such functions:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 105, + 228, + 504, + 253 + ], + "lines": [ + { + "bbox": [ + 104, + 227, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 104, + 227, + 311, + 243 + ], + "score": 1.0, + "content": "Lemma I.3 (Lemma 5.2 (Davis et al., 2020)). Let", + "type": "text" + }, + { + "bbox": [ + 311, + 229, + 363, + 240 + ], + "score": 0.92, + "content": "\\mathcal { L } : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 227, + 506, + 243 + ], + "score": 1.0, + "content": "be a locally Lipschitz function that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 239, + 389, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 204, + 253 + ], + "score": 1.0, + "content": "admits a chain rule. Let", + "type": "text" + }, + { + "bbox": [ + 204, + 241, + 281, + 253 + ], + "score": 0.93, + "content": "\\pmb \\theta : [ 0 , + \\infty ) \\mathbb R ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 239, + 376, + 253 + ], + "score": 1.0, + "content": "be the gradient flow on", + "type": "text" + }, + { + "bbox": [ + 376, + 241, + 384, + 251 + ], + "score": 0.78, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 239, + 389, + 253 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 260, + 387, + 285 + ], + "lines": [ + { + "bbox": [ + 223, + 260, + 387, + 285 + ], + "spans": [ + { + "bbox": [ + 223, + 260, + 387, + 285 + ], + "score": 0.92, + "content": "\\frac { d \\pmb { \\theta } ( t ) } { d t } \\in - \\partial ^ { \\circ } \\mathcal { L } ( \\pmb { \\theta } ( t ) ) \\qquad f o r a . e . \\ t \\geq 0 ,", + "type": "interline_equation", + "image_path": "5d8820c1aac7a673b6fdad51023cc575b991266002b1818644a2261fd1c7cbfa.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 223, + 260, + 387, + 285 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 128, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 130, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 130, + 304 + ], + "score": 1.0, + "content": "Then", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 302, + 428, + 333 + ], + "lines": [ + { + "bbox": [ + 183, + 302, + 428, + 333 + ], + "spans": [ + { + "bbox": [ + 183, + 302, + 428, + 333 + ], + "score": 0.94, + "content": "{ \\frac { d { \\mathcal { L } } ( \\pmb { \\theta } ( t ) ) } { d t } } = - \\left\\| { \\frac { d \\pmb { \\theta } ( t ) } { d t } } \\right\\| _ { 2 } ^ { 2 } = - \\operatorname* { m i n } \\{ \\| \\pmb { h } \\| _ { 2 } ^ { 2 } : \\pmb { h } \\in \\partial ^ { \\circ } { \\mathcal { L } } ( \\pmb { \\theta } ( t ) ) \\}", + "type": "interline_equation", + "image_path": "0bc592a7d66b49bc959f304957e4f98563f200683921f779299506d906db8a28.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 183, + 302, + 428, + 312.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 183, + 312.3333333333333, + 428, + 322.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 183, + 322.66666666666663, + 428, + 332.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 187, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 188, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 161, + 350 + ], + "score": 1.0, + "content": "holds for a.e.", + "type": "text" + }, + { + "bbox": [ + 161, + 338, + 184, + 348 + ], + "score": 0.87, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 337, + 188, + 350 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 358, + 506, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 176, + 371 + ], + "score": 1.0, + "content": "We can see that", + "type": "text" + }, + { + "bbox": [ + 177, + 359, + 187, + 370 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "-smooth functions admit chain rules. As shown in (Davis et al., 2020), if a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 377, + 383 + ], + "score": 1.0, + "content": "locally Lipschitz function is subdifferentiablly regular or Whitney", + "type": "text" + }, + { + "bbox": [ + 378, + 370, + 389, + 381 + ], + "score": 0.87, + "content": "\\mathcal { C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "-stratifiable, then it admits a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "chain rule. The latter one includes a large family of functions, e.g., semi-algebraic functions, semi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "analytic functions, and definable functions in an o-minimal structure (Coste, 2002; van den Dries &", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 402, + 164, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 164, + 416 + ], + "score": 1.0, + "content": "Miller, 1996).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 503, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "It is worth noting that the class of functions that admits chain rules is closed under composition.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 432, + 304, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 304, + 443 + ], + "score": 1.0, + "content": "This is indeed a simple corollary of Theorem I.1.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 181, + 460 + ], + "score": 1.0, + "content": "Theorem I.4. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 446, + 267, + 459 + ], + "score": 0.81, + "content": "z _ { 1 } , \\dots , z _ { n } : \\mathbb { R } ^ { d } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 445, + 285, + 460 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 447, + 336, + 458 + ], + "score": 0.87, + "content": "f : \\mathbb { R } ^ { n } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 445, + 505, + 460 + ], + "score": 1.0, + "content": "be locally Lipschitz functions and assume", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 245, + 471 + ], + "score": 1.0, + "content": "all of them admit chain rules. Let", + "type": "text" + }, + { + "bbox": [ + 245, + 458, + 388, + 470 + ], + "score": 0.9, + "content": "( f \\circ z ) ( \\pmb { x } ) = f ( z _ { 1 } ( \\pmb { x } ) , \\dots , z _ { n } ( \\pmb { x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 457, + 479, + 471 + ], + "score": 1.0, + "content": "be the composition of", + "type": "text" + }, + { + "bbox": [ + 479, + 459, + 486, + 470 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 469, + 265, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 114, + 479 + ], + "score": 0.46, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 469, + 140, + 481 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 141, + 470, + 163, + 481 + ], + "score": 0.9, + "content": "f \\circ z", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 469, + 265, + 481 + ], + "score": 1.0, + "content": "also admits a chain rule.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 104, + 498, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 204, + 513 + ], + "score": 1.0, + "content": "Proof. We can see that", + "type": "text" + }, + { + "bbox": [ + 204, + 501, + 228, + 511 + ], + "score": 0.9, + "content": "f \\circ z", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 498, + 332, + 513 + ], + "score": 1.0, + "content": "is locally Lipschitz. Let", + "type": "text" + }, + { + "bbox": [ + 333, + 499, + 460, + 512 + ], + "score": 0.93, + "content": "\\pmb { x } : [ 0 , + \\infty ) \\mathbb { R } ^ { d } , t \\mapsto \\pmb { x } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 498, + 505, + 513 + ], + "score": 1.0, + "content": "be an arc.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 185, + 524 + ], + "score": 1.0, + "content": "First, we show that", + "type": "text" + }, + { + "bbox": [ + 185, + 511, + 334, + 523 + ], + "score": 0.9, + "content": "z \\circ \\dot { \\pmb x } : [ 0 , + \\infty ) \\mathbb { R } ^ { d } , t \\mapsto z ( { \\pmb x } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "is also an arc. For any closed sub-interval", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 113, + 532 + ], + "score": 0.26, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 522, + 117, + 534 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 117, + 522, + 151, + 534 + ], + "score": 0.87, + "content": "z ( \\pmb { x } ( I ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 522, + 298, + 534 + ], + "score": 1.0, + "content": "must be contained in a compact set", + "type": "text" + }, + { + "bbox": [ + 298, + 523, + 307, + 532 + ], + "score": 0.79, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 522, + 505, + 534 + ], + "score": 1.0, + "content": ". Then it can be shown that the locally Lipschitz", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 190, + 545 + ], + "score": 1.0, + "content": "continuous function", + "type": "text" + }, + { + "bbox": [ + 190, + 535, + 198, + 543 + ], + "score": 0.74, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 534, + 353, + 545 + ], + "score": 1.0, + "content": "is (globally) Lipschitz continuous on", + "type": "text" + }, + { + "bbox": [ + 354, + 534, + 363, + 543 + ], + "score": 0.8, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 534, + 505, + 545 + ], + "score": 1.0, + "content": ". By the fact that the composition", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 470, + 557 + ], + "score": 1.0, + "content": "of a Lipschitz continuous and an absolutely continuous function is absolutely continuous,", + "type": "text" + }, + { + "bbox": [ + 470, + 546, + 494, + 554 + ], + "score": 0.87, + "content": "z \\circ x", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 543, + 506, + 557 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 301, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 208, + 567 + ], + "score": 1.0, + "content": "absolutely continuous on", + "type": "text" + }, + { + "bbox": [ + 209, + 556, + 215, + 565 + ], + "score": 0.74, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 555, + 301, + 567 + ], + "score": 1.0, + "content": ", and thus it is an arc.", + 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} ( t ) \\rangle , } \\\\ { \\forall \\pmb { h } _ { i } \\in \\partial ^ { \\circ } z _ { i } ( \\pmb { x } ( t ) ) : } & { ( z _ { i } \\circ \\pmb { x } ) ^ { \\prime } ( t ) = \\langle \\pmb { h } _ { i } , \\pmb { x } ^ { \\prime } ( t ) \\rangle . } \\end{array}", + "type": "interline_equation", + "image_path": "7ab99dc6f7bef7a94477202573d304224d964786573688b98bd1c2a53ba4ed72.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 181, + 590, + 429, + 600.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 181, + 600.6666666666666, + 429, + 611.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 181, + 611.3333333333333, + 429, + 621.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 293, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 293, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 266, + 640 + ], + "score": 1.0, + "content": "Combining these we obtain that for a.e.", + "type": "text" + }, + { + "bbox": [ + 266, + 628, + 289, + 638 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 627, + 293, + 640 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 646, + 380, + 679 + ], + "lines": [ + { + "bbox": [ + 230, + 646, + 380, + 679 + ], + "spans": [ + { + "bbox": [ + 230, + 646, + 380, + 679 + ], + "score": 0.94, + "content": "( f \\circ z \\circ x ) ^ { \\prime } ( t ) = \\sum _ { i = 1 } ^ { n } \\alpha _ { i } \\left. h _ { i } , x ^ { \\prime } ( t ) \\right. ,", + "type": "interline_equation", + "image_path": "1ff7032281da7fde9bc23e878b2c698dd5d49689f0be7f0d2170438550c190ad.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 230, + 646, + 380, + 662.5 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 230, + 662.5, + 380, + 679.0 + 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The RHS can be rewritten as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 691, + 509, + 716 + ], + "spans": [ + { + "bbox": [ + 107, + 698, + 185, + 711 + ], + "score": 0.94, + "content": "\\textstyle \\langle \\sum _ { i = 1 } ^ { n } \\alpha _ { i } \\pmb { h } _ { i } , \\pmb { x } ^ { \\prime } ( t ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 691, + 282, + 716 + ], + "score": 1.0, + "content": ". By Theorem I.1, every", + "type": "text" + }, + { + "bbox": [ + 283, + 698, + 366, + 709 + ], + "score": 0.88, + "content": "k \\in \\partial ^ { \\circ } ( f \\circ z ) ( { \\pmb x } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 691, + 509, + 716 + ], + "score": 1.0, + "content": "can be written as a convex combi-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 286, + 724 + ], + "score": 1.0, + "content": "nation of a finite set of points in the form of", + "type": "text" + }, + { + "bbox": [ + 286, + 709, + 332, + 722 + ], + "score": 0.9, + "content": "\\scriptstyle \\sum _ { i = 1 } ^ { n } \\alpha _ { i } h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 708, + 351, + 724 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 351, + 709, + 465, + 722 + ], + "score": 0.92, + "content": "( f \\circ z \\circ x ) ^ { \\prime } ( t ) = \\langle k , x ^ { \\prime } ( t ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 104, + 720, + 122, + 732 + ], + "score": 1.0, + "content": "a.e.", + "type": "text" + }, + { + "bbox": [ + 123, + 721, + 146, + 731 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 720, + 150, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 505, + 731 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + } + ], + "page_idx": 43, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "44", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "But for locally Lipschitz functions which could be non-differentiable, (25) may not hold in general", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 293, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 293, + 106 + ], + "score": 1.0, + "content": "since Theorem I.1 only holds for an inclusion.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 110, + 504, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "score": 1.0, + "content": "Following (Davis et al., 2020; Drusvyatskiy et al., 2015), we consider the functions that admit a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 121, + 197, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 197, + 134 + ], + "score": 1.0, + "content": "chain rule for any arc.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 109, + 506, + 134 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 136, + 504, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 134, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 337, + 151 + ], + "score": 1.0, + "content": "Definition I.2 (Chain Rule). A locally Lipschitz function", + "type": "text" + }, + { + "bbox": [ + 338, + 137, + 388, + 149 + ], + "score": 0.93, + "content": "f : { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 134, + 506, + 151 + ], + "score": 1.0, + "content": "admits a chain rule if for any", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 203, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 121, + 162 + ], + "score": 1.0, + "content": "arc", + "type": "text" + }, + { + "bbox": [ + 121, + 148, + 198, + 160 + ], + "score": 0.92, + "content": "z : [ 0 , + \\infty ) \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 146, + 203, + 162 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 134, + 506, + 162 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 161, + 392, + 176 + ], + "lines": [ + { + "bbox": [ + 218, + 161, + 392, + 176 + ], + "spans": [ + { + "bbox": [ + 218, + 161, + 392, + 176 + ], + "score": 0.91, + "content": "\\forall h \\in \\partial ^ { \\circ } f ( z ( t ) ) : ( f \\circ z ) ^ { \\prime } ( t ) = \\langle h , z ^ { \\prime } ( t ) \\rangle", + "type": "interline_equation", + "image_path": "15165a0145e9b766471106cd7edd1e4318a24b5556a8fb6e0abc1338004f8c1b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 218, + 161, + 392, + 176 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 188, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 188, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 161, + 192 + ], + "score": 1.0, + "content": "holds for a.e.", + "type": "text" + }, + { + "bbox": [ + 162, + 181, + 185, + 191 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 181, + 188, + 192 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 181, + 188, + 192 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 202, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "score": 1.0, + "content": "It is shown in (Davis et al., 2020; Drusvyatskiy et al., 2015) that a generalized version of (25) holds", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 213, + 183, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 183, + 226 + ], + "score": 1.0, + "content": "for such functions:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 201, + 505, + 226 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 228, + 504, + 253 + ], + "lines": [ + { + "bbox": [ + 104, + 227, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 104, + 227, + 311, + 243 + ], + "score": 1.0, + "content": "Lemma I.3 (Lemma 5.2 (Davis et al., 2020)). Let", + "type": "text" + }, + { + "bbox": [ + 311, + 229, + 363, + 240 + ], + "score": 0.92, + "content": "\\mathcal { L } : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 227, + 506, + 243 + ], + "score": 1.0, + "content": "be a locally Lipschitz function that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 239, + 389, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 204, + 253 + ], + "score": 1.0, + "content": "admits a chain rule. Let", + "type": "text" + }, + { + "bbox": [ + 204, + 241, + 281, + 253 + ], + "score": 0.93, + "content": "\\pmb \\theta : [ 0 , + \\infty ) \\mathbb R ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 239, + 376, + 253 + ], + "score": 1.0, + "content": "be the gradient flow on", + "type": "text" + }, + { + "bbox": [ + 376, + 241, + 384, + 251 + ], + "score": 0.78, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 239, + 389, + 253 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 227, + 506, + 253 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 260, + 387, + 285 + ], + "lines": [ + { + "bbox": [ + 223, + 260, + 387, + 285 + ], + "spans": [ + { + "bbox": [ + 223, + 260, + 387, + 285 + ], + "score": 0.92, + "content": "\\frac { d \\pmb { \\theta } ( t ) } { d t } \\in - \\partial ^ { \\circ } \\mathcal { L } ( \\pmb { \\theta } ( t ) ) \\qquad f o r a . e . \\ t \\geq 0 ,", + "type": "interline_equation", + "image_path": "5d8820c1aac7a673b6fdad51023cc575b991266002b1818644a2261fd1c7cbfa.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 223, + 260, + 387, + 285 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 128, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 130, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 130, + 304 + ], + "score": 1.0, + "content": "Then", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 290, + 130, + 304 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 302, + 428, + 333 + ], + "lines": [ + { + "bbox": [ + 183, + 302, + 428, + 333 + ], + "spans": [ + { + "bbox": [ + 183, + 302, + 428, + 333 + ], + "score": 0.94, + "content": "{ \\frac { d { \\mathcal { L } } ( \\pmb { \\theta } ( t ) ) } { d t } } = - \\left\\| { \\frac { d \\pmb { \\theta } ( t ) } { d t } } \\right\\| _ { 2 } ^ { 2 } = - \\operatorname* { m i n } \\{ \\| \\pmb { h } \\| _ { 2 } ^ { 2 } : \\pmb { h } \\in \\partial ^ { \\circ } { \\mathcal { L } } ( \\pmb { \\theta } ( t ) ) \\}", + "type": "interline_equation", + "image_path": "0bc592a7d66b49bc959f304957e4f98563f200683921f779299506d906db8a28.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 183, + 302, + 428, + 312.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 183, + 312.3333333333333, + 428, + 322.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 183, + 322.66666666666663, + 428, + 332.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 187, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 188, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 161, + 350 + ], + "score": 1.0, + "content": "holds for a.e.", + "type": "text" + }, + { + "bbox": [ + 161, + 338, + 184, + 348 + ], + "score": 0.87, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 337, + 188, + 350 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 337, + 188, + 350 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 358, + 506, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 176, + 371 + ], + "score": 1.0, + "content": "We can see that", + "type": "text" + }, + { + "bbox": [ + 177, + 359, + 187, + 370 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "-smooth functions admit chain rules. As shown in (Davis et al., 2020), if a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 377, + 383 + ], + "score": 1.0, + "content": "locally Lipschitz function is subdifferentiablly regular or Whitney", + "type": "text" + }, + { + "bbox": [ + 378, + 370, + 389, + 381 + ], + "score": 0.87, + "content": "\\mathcal { C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "-stratifiable, then it admits a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "chain rule. The latter one includes a large family of functions, e.g., semi-algebraic functions, semi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "analytic functions, and definable functions in an o-minimal structure (Coste, 2002; van den Dries &", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 402, + 164, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 164, + 416 + ], + "score": 1.0, + "content": "Miller, 1996).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 358, + 506, + 416 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 420, + 503, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "It is worth noting that the class of functions that admits chain rules is closed under composition.", + "type": "text" + } + ], + "index": 23, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 432, + 304, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 304, + 443 + ], + "score": 1.0, + "content": "This is indeed a simple corollary of Theorem I.1.", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 420, + 505, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 181, + 460 + ], + "score": 1.0, + "content": "Theorem I.4. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 446, + 267, + 459 + ], + "score": 0.81, + "content": "z _ { 1 } , \\dots , z _ { n } : \\mathbb { R } ^ { d } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 445, + 285, + 460 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 447, + 336, + 458 + ], + "score": 0.87, + "content": "f : \\mathbb { R } ^ { n } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 445, + 505, + 460 + ], + "score": 1.0, + "content": "be locally Lipschitz functions and assume", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 245, + 471 + ], + "score": 1.0, + "content": "all of them admit chain rules. Let", + "type": "text" + }, + { + "bbox": [ + 245, + 458, + 388, + 470 + ], + "score": 0.9, + "content": "( f \\circ z ) ( \\pmb { x } ) = f ( z _ { 1 } ( \\pmb { x } ) , \\dots , z _ { n } ( \\pmb { x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 457, + 479, + 471 + ], + "score": 1.0, + "content": "be the composition of", + "type": "text" + }, + { + "bbox": [ + 479, + 459, + 486, + 470 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 469, + 265, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 114, + 479 + ], + "score": 0.46, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 469, + 140, + 481 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 141, + 470, + 163, + 481 + ], + "score": 0.9, + "content": "f \\circ z", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 469, + 265, + 481 + ], + "score": 1.0, + "content": "also admits a chain rule.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 445, + 506, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 104, + 498, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 204, + 513 + ], + "score": 1.0, + "content": "Proof. We can see that", + "type": "text" + }, + { + "bbox": [ + 204, + 501, + 228, + 511 + ], + "score": 0.9, + "content": "f \\circ z", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 498, + 332, + 513 + ], + "score": 1.0, + "content": "is locally Lipschitz. Let", + "type": "text" + }, + { + "bbox": [ + 333, + 499, + 460, + 512 + ], + "score": 0.93, + "content": "\\pmb { x } : [ 0 , + \\infty ) \\mathbb { R } ^ { d } , t \\mapsto \\pmb { x } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 498, + 505, + 513 + ], + "score": 1.0, + "content": "be an arc.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 185, + 524 + ], + "score": 1.0, + "content": "First, we show that", + "type": "text" + }, + { + "bbox": [ + 185, + 511, + 334, + 523 + ], + "score": 0.9, + "content": "z \\circ \\dot { \\pmb x } : [ 0 , + \\infty ) \\mathbb { R } ^ { d } , t \\mapsto z ( { \\pmb x } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "is also an arc. For any closed sub-interval", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 113, + 532 + ], + "score": 0.26, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 522, + 117, + 534 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 117, + 522, + 151, + 534 + ], + "score": 0.87, + "content": "z ( \\pmb { x } ( I ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 522, + 298, + 534 + ], + "score": 1.0, + "content": "must be contained in a compact set", + "type": "text" + }, + { + "bbox": [ + 298, + 523, + 307, + 532 + ], + "score": 0.79, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 522, + 505, + 534 + ], + "score": 1.0, + "content": ". Then it can be shown that the locally Lipschitz", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 190, + 545 + ], + "score": 1.0, + "content": "continuous function", + "type": "text" + }, + { + "bbox": [ + 190, + 535, + 198, + 543 + ], + "score": 0.74, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 534, + 353, + 545 + ], + "score": 1.0, + "content": "is (globally) Lipschitz continuous on", + "type": "text" + }, + { + "bbox": [ + 354, + 534, + 363, + 543 + ], + "score": 0.8, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 534, + 505, + 545 + ], + "score": 1.0, + "content": ". By the fact that the composition", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 470, + 557 + ], + "score": 1.0, + "content": "of a Lipschitz continuous and an absolutely continuous function is absolutely continuous,", + "type": "text" + }, + { + "bbox": [ + 470, + 546, + 494, + 554 + ], + "score": 0.87, + "content": "z \\circ x", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 543, + 506, + 557 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 301, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 208, + 567 + ], + "score": 1.0, + "content": "absolutely continuous on", + "type": "text" + }, + { + "bbox": [ + 209, + 556, + 215, + 565 + ], + "score": 0.74, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 555, + 301, + 567 + ], + "score": 1.0, + "content": ", and thus it is an arc.", + 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\\prime } ( t ) = \\langle \\pmb { \\alpha } , ( \\pmb { z } \\circ \\pmb { x } ) ^ { \\prime } ( t ) \\rangle , } \\\\ { \\forall \\pmb { h } _ { i } \\in \\partial ^ { \\circ } z _ { i } ( \\pmb { x } ( t ) ) : } & { ( z _ { i } \\circ \\pmb { x } ) ^ { \\prime } ( t ) = \\langle \\pmb { h } _ { i } , \\pmb { x } ^ { \\prime } ( t ) \\rangle . } \\end{array}", + "type": "interline_equation", + "image_path": "7ab99dc6f7bef7a94477202573d304224d964786573688b98bd1c2a53ba4ed72.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 181, + 590, + 429, + 600.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 181, + 600.6666666666666, + 429, + 611.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 181, + 611.3333333333333, + 429, + 621.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 293, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 293, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 266, + 640 + ], + "score": 1.0, + "content": "Combining these we obtain that for a.e.", + "type": "text" + }, + { + "bbox": [ + 266, + 628, + 289, + 638 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 627, + 293, + 640 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 627, + 293, + 640 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 646, + 380, + 679 + ], + "lines": [ + { + "bbox": [ + 230, + 646, + 380, + 679 + ], + "spans": [ + { + "bbox": [ + 230, + 646, + 380, + 679 + ], + "score": 0.94, + "content": "( f \\circ z \\circ x ) ^ { \\prime } ( t ) = \\sum _ { i = 1 } ^ { n } \\alpha _ { i } \\left. h _ { i } , x ^ { \\prime } ( t ) \\right. ,", + "type": "interline_equation", + "image_path": "1ff7032281da7fde9bc23e878b2c698dd5d49689f0be7f0d2170438550c190ad.jpg" + } + ] + } + ], + "index": 39.5, + 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The RHS can be rewritten as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 691, + 509, + 716 + ], + "spans": [ + { + "bbox": [ + 107, + 698, + 185, + 711 + ], + "score": 0.94, + "content": "\\textstyle \\langle \\sum _ { i = 1 } ^ { n } \\alpha _ { i } \\pmb { h } _ { i } , \\pmb { x } ^ { \\prime } ( t ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 691, + 282, + 716 + ], + "score": 1.0, + "content": ". By Theorem I.1, every", + "type": "text" + }, + { + "bbox": [ + 283, + 698, + 366, + 709 + ], + "score": 0.88, + "content": "k \\in \\partial ^ { \\circ } ( f \\circ z ) ( { \\pmb x } ( t ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 691, + 509, + 716 + ], + "score": 1.0, + "content": "can be written as a convex combi-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 286, + 724 + ], + "score": 1.0, + "content": "nation of a finite set of points in the form of", + "type": "text" + }, + { + "bbox": [ + 286, + 709, + 332, + 722 + ], + "score": 0.9, + "content": "\\scriptstyle \\sum _ { i = 1 } ^ { n } \\alpha _ { i } h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 708, + 351, + 724 + ], + "score": 1.0, + "content": ". So", + "type": "text" + }, + { + "bbox": [ + 351, + 709, + 465, + 722 + ], + "score": 0.92, + "content": "( f \\circ z \\circ x ) ^ { \\prime } ( t ) = \\langle k , x ^ { \\prime } ( t ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 104, + 720, + 122, + 732 + ], + "score": 1.0, + "content": "a.e.", + "type": "text" + }, + { + "bbox": [ + 123, + 721, + 146, + 731 + ], + "score": 0.89, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 720, + 150, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 505, + 731 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 104, + 685, + 509, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 201, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 203, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 203, + 96 + ], + "score": 1.0, + "content": "J MEXICAN HAT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "In this section, we give an example to illustrate that gradient flow does not necessarily converge in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 325, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 182, + 129 + ], + "score": 1.0, + "content": "direction, even for", + "type": "text" + }, + { + "bbox": [ + 182, + 117, + 199, + 126 + ], + "score": 0.86, + "content": "C ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 114, + 325, + 129 + ], + "score": 1.0, + "content": "-smooth homogeneous models.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 506, + 167 + ], + "lines": [ + { + "bbox": [ + 104, + 132, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 104, + 132, + 505, + 146 + ], + "score": 1.0, + "content": "It is known that gradient flow (or gradient descent) may not converge to any point even when opti-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 148, + 157 + ], + "score": 1.0, + "content": "mizing an", + "type": "text" + }, + { + "bbox": [ + 149, + 145, + 165, + 154 + ], + "score": 0.83, + "content": "C ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 144, + 505, + 157 + ], + "score": 1.0, + "content": "function (Curry, 1944; Zoutendijk, 1976; Palis & De Melo, 2012; Absil et al., 2005).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 477, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 477, + 168 + ], + "score": 1.0, + "content": "One famous counterexample is the “Mexican Hat” function described in (Absil et al., 2005):", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 167, + 474, + 201 + ], + "lines": [ + { + "bbox": [ + 137, + 167, + 474, + 201 + ], + "spans": [ + { + "bbox": [ + 137, + 167, + 474, + 201 + ], + "score": 0.93, + "content": "f ( u , v ) : = f ( r \\cos \\varphi , r \\sin \\varphi ) : = \\left\\{ { \\begin{array} { l l } { e ^ { - { \\frac { 1 } { 1 - r ^ { 2 } } } } \\left( 1 - C ( r ) \\sin \\left( \\varphi - { \\frac { 1 } { 1 - r ^ { 2 } } } \\right) \\right) \\quad } & { r < 1 } \\\\ { 0 \\quad } & { r \\geq 1 } \\end{array} } \\right.", + "type": "interline_equation", + "image_path": "3b091b743632542554d9606ad91468856cfbd26acbf370ee6eb9ce8e13409653.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 137, + 167, + 474, + 178.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 137, + 178.33333333333334, + 474, + 189.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 137, + 189.66666666666669, + 474, + 201.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 102, + 201, + 507, + 223 + ], + "spans": [ + { + "bbox": [ + 102, + 201, + 133, + 223 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 203, + 252, + 220 + ], + "score": 0.92, + "content": "\\begin{array} { r } { C ( r ) = \\frac { 4 r ^ { 4 } } { 4 r ^ { 4 } + ( 1 - r ^ { 2 } ) ^ { 4 } } \\in [ 0 , 1 ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 201, + 342, + 223 + ], + "score": 1.0, + "content": ". It can be shown that", + "type": "text" + }, + { + "bbox": [ + 342, + 205, + 349, + 217 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 201, + 360, + 223 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 360, + 205, + 375, + 216 + ], + "score": 0.88, + "content": "{ \\mathcal { C } } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 201, + 423, + 223 + ], + "score": 1.0, + "content": "-smooth on", + "type": "text" + }, + { + "bbox": [ + 423, + 205, + 436, + 215 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 201, + 507, + 223 + ], + "score": 1.0, + "content": "but not analytic.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 247, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 213, + 232 + ], + "score": 1.0, + "content": "See Figure 2 for a plot for", + "type": "text" + }, + { + "bbox": [ + 213, + 219, + 242, + 231 + ], + "score": 0.91, + "content": "f ( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 217, + 247, + 232 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "image", + "bbox": [ + 213, + 249, + 388, + 429 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 213, + 249, + 388, + 429 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 213, + 249, + 388, + 429 + ], + "spans": [ + { + "bbox": [ + 213, + 249, + 388, + 429 + ], + "score": 0.967, + "type": "image", + "image_path": "fdd8b960a9c32bef3e12b8b8208114e0c2be68507e93e7f18e6e76881a178ab2.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 213, + 249, + 388, + 264.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 213, + 264.0, + 388, + 279.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 213, + 279.0, + 388, + 294.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 213, + 294.0, + 388, + 309.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 213, + 309.0, + 388, + 324.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 213, + 324.0, + 388, + 339.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 213, + 339.0, + 388, + 354.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 213, + 354.0, + 388, + 369.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 213, + 369.0, + 388, + 384.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 213, + 384.0, + 388, + 399.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 213, + 399.0, + 388, + 414.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 213, + 414.0, + 388, + 429.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 195, + 439, + 415, + 452 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 195, + 439, + 415, + 453 + ], + "spans": [ + { + "bbox": [ + 195, + 439, + 381, + 453 + ], + "score": 1.0, + "content": "Figure 2: A plot for the Mexican Hat function", + "type": "text" + }, + { + "bbox": [ + 382, + 440, + 411, + 452 + ], + "score": 0.92, + "content": "f ( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 439, + 415, + 453 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + } + ], + "index": 19.75 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "However, the Maxican Hat function is not homogeneous, and Absil et al. (2005) did not consider", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 478, + 484 + ], + "score": 1.0, + "content": "the directional convergence, either. To make it homogeneous, we introduce an extra variable", + "type": "text" + }, + { + "bbox": [ + 478, + 474, + 484, + 482 + ], + "score": 0.75, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 472, + 505, + 484 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 483, + 432, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 269, + 496 + ], + "score": 1.0, + "content": "normalize the parameter before evaluate", + "type": "text" + }, + { + "bbox": [ + 269, + 484, + 276, + 495 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 483, + 361, + 496 + ], + "score": 1.0, + "content": ". In particular, we fix", + "type": "text" + }, + { + "bbox": [ + 361, + 483, + 388, + 493 + ], + "score": 0.9, + "content": "L > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 483, + 432, + 496 + ], + "score": 1.0, + "content": "and define", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 107, + 497, + 499, + 511 + ], + "lines": [ + { + "bbox": [ + 107, + 497, + 499, + 511 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 499, + 511 + ], + "score": 0.8, + "content": "h ( \\theta ) = h ( x , y , z ) = \\rho ^ { L } ( 1 - f ( u , v ) ) \\qquad \\mathrm { w h e r e } \\quad u = x / \\rho , \\quad v = y / \\rho , \\quad \\rho = \\sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } .", + "type": "interline_equation", + "image_path": "4ba558e383b22d3ba772f532746acd869b33691841b2fb0b2b8e76532b201413.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 107, + 497, + 499, + 511 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 512, + 254, + 524 + ], + "lines": [ + { + "bbox": [ + 108, + 512, + 254, + 524 + ], + "spans": [], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 506, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 511, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 281, + 545 + ], + "score": 1.0, + "content": "Theorem J.1. Consider gradient flow on", + "type": "text" + }, + { + "bbox": [ + 281, + 525, + 378, + 540 + ], + "score": 0.91, + "content": "\\begin{array} { r } { { \\mathcal L } ( \\pmb { \\theta } ) = \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( \\pmb { \\theta } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 515, + 412, + 545 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 412, + 527, + 474, + 539 + ], + "score": 0.91, + "content": "q _ { n } ( \\pmb \\theta ) \\ = \\ h ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 515, + 511, + 545 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 145, + 550 + ], + "score": 0.9, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 538, + 307, + 551 + ], + "score": 1.0, + "content": ". Suppose the polar representation of", + "type": "text" + }, + { + "bbox": [ + 307, + 539, + 331, + 550 + ], + "score": 0.8, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 538, + 346, + 551 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 346, + 539, + 411, + 550 + ], + "score": 0.82, + "content": "( r \\cos \\varphi , r \\sin \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 538, + 431, + 551 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 431, + 539, + 485, + 550 + ], + "score": 0.87, + "content": "0 ~ < ~ r ~ < ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 507, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 149, + 565 + ], + "score": 0.93, + "content": "\\textstyle \\varphi = { \\frac { 1 } { 1 - r ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 549, + 205, + 569 + ], + "score": 1.0, + "content": "holds at time", + "type": "text" + }, + { + "bbox": [ + 205, + 552, + 230, + 562 + ], + "score": 0.87, + "content": "t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 549, + 254, + 569 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 255, + 549, + 283, + 567 + ], + "score": 0.94, + "content": "\\frac { \\pmb { \\theta } ( t ) } { \\| \\pmb { \\theta } ( t ) \\| _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 549, + 507, + 569 + ], + "score": 1.0, + "content": "does not converge to any point, and the limit points of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 564, + 388, + 584 + ], + "spans": [ + { + "bbox": [ + 107, + 566, + 176, + 583 + ], + "score": 0.89, + "content": "\\big \\{ \\frac { \\pmb \\theta ( t ) } { \\lVert \\pmb \\theta ( t ) \\rVert _ { 2 } } : t > 0 \\big \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 564, + 231, + 584 + ], + "score": 1.0, + "content": "form a circle", + "type": "text" + }, + { + "bbox": [ + 231, + 567, + 384, + 580 + ], + "score": 0.86, + "content": "\\left\\{ \\left( x , y , z \\right) \\in S ^ { 2 } : x ^ { 2 } + y ^ { 2 } = 1 , z = 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 564, + 388, + 584 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 103, + 592, + 507, + 612 + ], + "spans": [ + { + "bbox": [ + 103, + 592, + 166, + 612 + ], + "score": 1.0, + "content": "Proof. Define", + "type": "text" + }, + { + "bbox": [ + 166, + 594, + 229, + 609 + ], + "score": 0.94, + "content": "\\textstyle \\psi = \\varphi - { \\frac { 1 } { 1 - r ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 592, + 507, + 612 + ], + "score": 1.0, + "content": ". Our proof consists of two parts, following from the idea in (Absil", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 242, + 623 + ], + "score": 1.0, + "content": "et al., 2005). First, we show that", + "type": "text" + }, + { + "bbox": [ + 243, + 608, + 276, + 622 + ], + "score": 0.92, + "content": "\\begin{array} { r } { { \\frac { d \\psi } { d t } } = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 608, + 320, + 623 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + }, + { + "bbox": [ + 320, + 610, + 348, + 621 + ], + "score": 0.9, + "content": "\\psi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 608, + 448, + 623 + ], + "score": 1.0, + "content": ". Then we can infer that", + "type": "text" + }, + { + "bbox": [ + 448, + 610, + 477, + 621 + ], + "score": 0.91, + "content": "\\psi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 608, + 506, + 623 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 620, + 504, + 632 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 129, + 631 + ], + "score": 0.87, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 620, + 211, + 632 + ], + "score": 1.0, + "content": ". Next, we show that", + "type": "text" + }, + { + "bbox": [ + 212, + 621, + 238, + 630 + ], + "score": 0.88, + "content": "r 1", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 620, + 249, + 632 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 249, + 622, + 287, + 631 + ], + "score": 0.88, + "content": "t \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 620, + 317, + 632 + ], + "score": 1.0, + "content": ". Using", + "type": "text" + }, + { + "bbox": [ + 318, + 621, + 343, + 631 + ], + "score": 0.89, + "content": "\\psi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 620, + 462, + 632 + ], + "score": 1.0, + "content": ", we know that the polar angle", + "type": "text" + }, + { + "bbox": [ + 463, + 622, + 504, + 632 + ], + "score": 0.91, + "content": "\\varphi + \\infty", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 630, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 630, + 117, + 645 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 632, + 155, + 642 + ], + "score": 0.9, + "content": "t \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 630, + 204, + 645 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 204, + 632, + 228, + 644 + ], + "score": 0.92, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 630, + 288, + 645 + ], + "score": 1.0, + "content": "circles around", + "type": "text" + }, + { + "bbox": [ + 289, + 631, + 380, + 644 + ], + "score": 0.91, + "content": "\\{ ( u , v ) : u ^ { 2 } + v ^ { 2 } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 630, + 505, + 645 + ], + "score": 1.0, + "content": ", and thus it does not converge.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 649, + 504, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 148, + 666 + ], + "score": 1.0, + "content": "Proof for", + "type": "text" + }, + { + "bbox": [ + 149, + 649, + 180, + 664 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { d \\psi } { d t } = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 645, + 291, + 666 + ], + "score": 1.0, + "content": ". For convenience, we use", + "type": "text" + }, + { + "bbox": [ + 291, + 653, + 300, + 660 + ], + "score": 0.79, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 645, + 341, + 666 + ], + "score": 1.0, + "content": "to denote", + "type": "text" + }, + { + "bbox": [ + 341, + 650, + 357, + 662 + ], + "score": 0.9, + "content": "z / \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 645, + 506, + 666 + ], + "score": 1.0, + "content": ". By simple calculation, we have the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 660, + 276, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 276, + 674 + ], + "score": 1.0, + "content": "following formulas for partial derivatives:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 673, + 473, + 731 + ], + "lines": [ + { + "bbox": [ + 129, + 673, + 473, + 731 + ], + "spans": [ + { + "bbox": [ + 129, + 673, + 473, + 731 + ], + "score": 0.94, + "content": "\\left\\{ \\begin{array} { l l } { \\frac { \\partial h } { \\partial x } = \\rho ^ { L - 1 } \\left( L u ( 1 - f ) - ( 1 - u ^ { 2 } ) \\frac { \\partial f } { \\partial u } + u v \\frac { \\partial f } { \\partial v } \\right) } \\\\ { \\frac { \\partial h } { \\partial y } = \\rho ^ { L - 1 } \\left( L v ( 1 - f ) + u v \\frac { \\partial f } { \\partial u } - ( 1 - v ^ { 2 } ) \\frac { \\partial f } { \\partial v } \\right) } \\\\ { \\frac { \\partial h } { \\partial z } = \\rho ^ { L - 1 } \\left( L w ( 1 - f ) + u w \\frac { \\partial f } { \\partial u } + v w \\frac { \\partial f } { \\partial v } \\right) } \\end{array} \\right. \\qquad \\left\\{ \\begin{array} { l l } { \\frac { \\partial f } { \\partial u } = \\frac { u } { r } \\cdot \\frac { \\partial f } { \\partial r } - \\frac { v } { r ^ { 2 } } \\cdot \\frac { \\partial f } { \\partial \\varphi } } \\\\ { \\frac { \\partial f } { \\partial v } = \\frac { v } { r } \\cdot \\frac { \\partial f } { \\partial r } + \\frac { u } { r ^ { 2 } } \\cdot \\frac { \\partial f } { \\partial \\varphi } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "7256c675819920b017f2ef7258f04927dbaa92c7af1376f5458ea52079a674e2.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 129, + 673, + 473, + 692.3333333333334 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 129, + 692.3333333333334, + 473, + 711.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 129, + 711.6666666666667, + 473, + 731.0000000000001 + ], + "spans": [], + "index": 41 + } + ] + } + ], + "page_idx": 44, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "45", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 201, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 203, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 203, + 96 + ], + "score": 1.0, + "content": "J MEXICAN HAT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "In this section, we give an example to illustrate that gradient flow does not necessarily converge in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 325, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 182, + 129 + ], + "score": 1.0, + "content": "direction, even for", + "type": "text" + }, + { + "bbox": [ + 182, + 117, + 199, + 126 + ], + "score": 0.86, + "content": "C ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 114, + 325, + 129 + ], + "score": 1.0, + "content": "-smooth homogeneous models.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 105, + 505, + 129 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 506, + 167 + ], + "lines": [ + { + "bbox": [ + 104, + 132, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 104, + 132, + 505, + 146 + ], + "score": 1.0, + "content": "It is known that gradient flow (or gradient descent) may not converge to any point even when opti-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 148, + 157 + ], + "score": 1.0, + "content": "mizing an", + "type": "text" + }, + { + "bbox": [ + 149, + 145, + 165, + 154 + ], + "score": 0.83, + "content": "C ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 144, + 505, + 157 + ], + "score": 1.0, + "content": "function (Curry, 1944; Zoutendijk, 1976; Palis & De Melo, 2012; Absil et al., 2005).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 477, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 477, + 168 + ], + "score": 1.0, + "content": "One famous counterexample is the “Mexican Hat” function described in (Absil et al., 2005):", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 132, + 505, + 168 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 167, + 474, + 201 + ], + "lines": [ + { + "bbox": [ + 137, + 167, + 474, + 201 + ], + "spans": [ + { + "bbox": [ + 137, + 167, + 474, + 201 + ], + "score": 0.93, + "content": "f ( u , v ) : = f ( r \\cos \\varphi , r \\sin \\varphi ) : = \\left\\{ { \\begin{array} { l l } { e ^ { - { \\frac { 1 } { 1 - r ^ { 2 } } } } \\left( 1 - C ( r ) \\sin \\left( \\varphi - { \\frac { 1 } { 1 - r ^ { 2 } } } \\right) \\right) \\quad } & { r < 1 } \\\\ { 0 \\quad } & { r \\geq 1 } \\end{array} } \\right.", + "type": "interline_equation", + "image_path": "3b091b743632542554d9606ad91468856cfbd26acbf370ee6eb9ce8e13409653.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 137, + 167, + 474, + 178.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 137, + 178.33333333333334, + 474, + 189.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 137, + 189.66666666666669, + 474, + 201.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "list", + "bbox": [ + 106, + 203, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 102, + 201, + 507, + 223 + ], + "spans": [ + { + "bbox": [ + 102, + 201, + 133, + 223 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 203, + 252, + 220 + ], + "score": 0.92, + "content": "\\begin{array} { r } { C ( r ) = \\frac { 4 r ^ { 4 } } { 4 r ^ { 4 } + ( 1 - r ^ { 2 } ) ^ { 4 } } \\in [ 0 , 1 ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 201, + 342, + 223 + ], + "score": 1.0, + "content": ". 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(2005) did not consider", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 478, + 484 + ], + "score": 1.0, + "content": "the directional convergence, either. To make it homogeneous, we introduce an extra variable", + "type": "text" + }, + { + "bbox": [ + 478, + 474, + 484, + 482 + ], + "score": 0.75, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 472, + 505, + 484 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 483, + 432, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 269, + 496 + ], + "score": 1.0, + "content": "normalize the parameter before evaluate", + "type": "text" + }, + { + "bbox": [ + 269, + 484, + 276, + 495 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 483, + 361, + 496 + ], + "score": 1.0, + "content": ". 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Consider gradient flow on", + "type": "text" + }, + { + "bbox": [ + 281, + 525, + 378, + 540 + ], + "score": 0.91, + "content": "\\begin{array} { r } { { \\mathcal L } ( \\pmb { \\theta } ) = \\sum _ { n = 1 } ^ { N } e ^ { - q _ { n } ( \\pmb { \\theta } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 515, + 412, + 545 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 412, + 527, + 474, + 539 + ], + "score": 0.91, + "content": "q _ { n } ( \\pmb \\theta ) \\ = \\ h ( \\pmb \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 515, + 511, + 545 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 145, + 550 + ], + "score": 0.9, + "content": "n \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 538, + 307, + 551 + ], + "score": 1.0, + "content": ". Suppose the polar representation of", + "type": "text" + }, + { + "bbox": [ + 307, + 539, + 331, + 550 + ], + "score": 0.8, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 538, + 346, + 551 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 346, + 539, + 411, + 550 + ], + "score": 0.82, + "content": "( r \\cos \\varphi , r \\sin \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 538, + 431, + 551 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 431, + 539, + 485, + 550 + ], + "score": 0.87, + "content": "0 ~ < ~ r ~ < ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 507, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 149, + 565 + ], + "score": 0.93, + "content": "\\textstyle \\varphi = { \\frac { 1 } { 1 - r ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 549, + 205, + 569 + ], + "score": 1.0, + "content": "holds at time", + "type": "text" + }, + { + "bbox": [ + 205, + 552, + 230, + 562 + ], + "score": 0.87, + "content": "t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 549, + 254, + 569 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 255, + 549, + 283, + 567 + ], + "score": 0.94, + "content": "\\frac { \\pmb { \\theta } ( t ) } { \\| \\pmb { \\theta } ( t ) \\| _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 549, + 507, + 569 + ], + "score": 1.0, + "content": "does not converge to any point, and the limit points of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 564, + 388, + 584 + ], + "spans": [ + { + "bbox": [ + 107, + 566, + 176, + 583 + ], + "score": 0.89, + "content": "\\big \\{ \\frac { \\pmb \\theta ( t ) } { \\lVert \\pmb \\theta ( t ) \\rVert _ { 2 } } : t > 0 \\big \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 564, + 231, + 584 + ], + "score": 1.0, + "content": "form a circle", + "type": "text" + }, + { + "bbox": [ + 231, + 567, + 384, + 580 + ], + "score": 0.86, + "content": "\\left\\{ \\left( x , y , z \\right) \\in S ^ { 2 } : x ^ { 2 } + y ^ { 2 } = 1 , z = 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 564, + 388, + 584 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 106, + 515, + 511, + 584 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 103, + 592, + 507, + 612 + ], + "spans": [ + { + "bbox": [ + 103, + 592, + 166, + 612 + ], + "score": 1.0, + "content": "Proof. Define", + "type": "text" + }, + { + "bbox": [ + 166, + 594, + 229, + 609 + ], + "score": 0.94, + "content": "\\textstyle \\psi = \\varphi - { \\frac { 1 } { 1 - r ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 592, + 507, + 612 + ], + "score": 1.0, + "content": ". Our proof consists of two parts, following from the idea in (Absil", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 242, + 623 + ], + "score": 1.0, + "content": "et al., 2005). First, we show that", + "type": "text" + }, + { + "bbox": [ + 243, + 608, + 276, + 622 + ], + "score": 0.92, + "content": "\\begin{array} { r } { { \\frac { d \\psi } { d t } } = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 608, + 320, + 623 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + }, + { + "bbox": [ + 320, + 610, + 348, + 621 + ], + "score": 0.9, + "content": "\\psi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 608, + 448, + 623 + ], + "score": 1.0, + "content": ". Then we can infer that", + "type": "text" + }, + { + "bbox": [ + 448, + 610, + 477, + 621 + ], + "score": 0.91, + "content": "\\psi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 608, + 506, + 623 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 620, + 504, + 632 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 129, + 631 + ], + "score": 0.87, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 620, + 211, + 632 + ], + "score": 1.0, + "content": ". Next, we show that", + "type": "text" + }, + { + "bbox": [ + 212, + 621, + 238, + 630 + ], + "score": 0.88, + "content": "r 1", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 620, + 249, + 632 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 249, + 622, + 287, + 631 + ], + "score": 0.88, + "content": "t \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 620, + 317, + 632 + ], + "score": 1.0, + "content": ". Using", + "type": "text" + }, + { + "bbox": [ + 318, + 621, + 343, + 631 + ], + "score": 0.89, + "content": "\\psi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 620, + 462, + 632 + ], + "score": 1.0, + "content": ", we know that the polar angle", + "type": "text" + }, + { + "bbox": [ + 463, + 622, + 504, + 632 + ], + "score": 0.91, + "content": "\\varphi + \\infty", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 630, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 630, + 117, + 645 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 632, + 155, + 642 + ], + "score": 0.9, + "content": "t \\to + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 630, + 204, + 645 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 204, + 632, + 228, + 644 + ], + "score": 0.92, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 630, + 288, + 645 + ], + "score": 1.0, + "content": "circles around", + "type": "text" + }, + { + "bbox": [ + 289, + 631, + 380, + 644 + ], + "score": 0.91, + "content": "\\{ ( u , v ) : u ^ { 2 } + v ^ { 2 } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 630, + 505, + 645 + ], + "score": 1.0, + "content": ", and thus it does not converge.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 103, + 592, + 507, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 649, + 504, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 148, + 666 + ], + "score": 1.0, + "content": "Proof for", + "type": "text" + }, + { + "bbox": [ + 149, + 649, + 180, + 664 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { d \\psi } { d t } = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 645, + 291, + 666 + ], + "score": 1.0, + "content": ". For convenience, we use", + "type": "text" + }, + { + "bbox": [ + 291, + 653, + 300, + 660 + ], + "score": 0.79, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 645, + 341, + 666 + ], + "score": 1.0, + "content": "to denote", + "type": "text" + }, + { + "bbox": [ + 341, + 650, + 357, + 662 + ], + "score": 0.9, + "content": "z / \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 645, + 506, + 666 + ], + "score": 1.0, + "content": ". By simple calculation, we have the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 660, + 276, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 276, + 674 + ], + "score": 1.0, + "content": "following formulas for partial derivatives:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 645, + 506, + 674 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 673, + 473, + 731 + ], + "lines": [ + { + "bbox": [ + 129, + 673, + 473, + 731 + ], + "spans": [ + { + "bbox": [ + 129, + 673, + 473, + 731 + ], + "score": 0.94, + "content": "\\left\\{ \\begin{array} { l l } { \\frac { \\partial h } { \\partial x } = \\rho ^ { L - 1 } \\left( L u ( 1 - f ) - ( 1 - u ^ { 2 } ) \\frac { \\partial f } { \\partial u } + u v \\frac { \\partial f } { \\partial v } \\right) } \\\\ { \\frac { \\partial h } { \\partial y } = \\rho ^ { L - 1 } \\left( L v ( 1 - f ) + u v \\frac { \\partial f } { \\partial u } - ( 1 - v ^ { 2 } ) \\frac { \\partial f } { \\partial v } \\right) } \\\\ { \\frac { \\partial h } { \\partial z } = \\rho ^ { L - 1 } \\left( L w ( 1 - f ) + u w \\frac { \\partial f } { \\partial u } + v w \\frac { \\partial f } { \\partial v } \\right) } \\end{array} \\right. \\qquad \\left\\{ \\begin{array} { l l } { \\frac { \\partial f } { \\partial u } = \\frac { u } { r } \\cdot \\frac { \\partial f } { \\partial r } - \\frac { v } { r ^ { 2 } } \\cdot \\frac { \\partial f } { \\partial \\varphi } } \\\\ { \\frac { \\partial f } { \\partial v } = \\frac { v } { r } \\cdot \\frac { \\partial f } { \\partial r } + \\frac { u } { r ^ { 2 } } \\cdot \\frac { \\partial f } { \\partial \\varphi } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "7256c675819920b017f2ef7258f04927dbaa92c7af1376f5458ea52079a674e2.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 129, + 673, + 473, + 692.3333333333334 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 129, + 692.3333333333334, + 473, + 711.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 129, + 711.6666666666667, + 473, + 731.0000000000001 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 214, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 215, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 215, + 96 + ], + "score": 1.0, + "content": "For gradient flow, we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 96, + 480, + 175 + ], + "lines": [ + { + "bbox": [ + 132, + 96, + 480, + 175 + ], + "spans": [ + { + "bbox": [ + 132, + 96, + 480, + 175 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\frac { d \\theta } { d t } = N e ^ { - h } \\nabla h } \\\\ & { \\frac { d u } { d t } = \\frac { 1 } { \\rho } \\left( ( 1 - u ^ { 2 } ) \\frac { d x } { d t } - u v \\frac { d y } { d t } - u w \\frac { d z } { d t } \\right) = N e ^ { - h } \\rho ^ { L - 2 } \\left( - ( 1 - u ^ { 2 } ) \\frac { \\partial f } { \\partial u } + u v \\frac { \\partial f } { \\partial v } \\right) } \\\\ & { \\frac { d v } { d t } = \\frac { 1 } { \\rho } \\left( ( 1 - v ^ { 2 } ) \\frac { d y } { d t } - u v \\frac { d x } { d t } - v w \\frac { d z } { d t } \\right) = N e ^ { - h } \\rho ^ { L - 2 } \\left( - ( 1 - v ^ { 2 } ) \\frac { \\partial f } { \\partial v } + u v \\frac { \\partial f } { \\partial u } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "1316172d676f8122217c655e87f03702f8b0fb555e4ff8bbdf9e28a84b2146f6.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 132, + 96, + 480, + 122.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 122.33333333333333, + 480, + 148.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 132, + 148.66666666666666, + 480, + 175.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 175, + 432, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 173, + 433, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 246, + 190 + ], + "score": 1.0, + "content": "By writing down the movement of", + "type": "text" + }, + { + "bbox": [ + 246, + 175, + 270, + 187 + ], + "score": 0.91, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 173, + 433, + 190 + ], + "score": 1.0, + "content": "in the polar coordinate system, we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 189, + 416, + 344 + ], + "lines": [ + { + "bbox": [ + 193, + 189, + 416, + 344 + ], + "spans": [ + { + "bbox": [ + 193, + 189, + 416, + 344 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\frac { d r } { d t } = \\frac { u } { r } \\cdot \\frac { d u } { d t } + \\frac { v } { r } \\cdot \\frac { d v } { d t } } \\\\ & { \\quad = - 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\\frac { 1 } { 1 - r ^ { 2 } } } \\right) - e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ^ { \\prime } ( r ) \\sin \\psi - e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ( r ) \\cos \\psi \\cdot \\frac { \\partial \\psi } { \\partial r } } \\\\ { = \\frac { d } { d r } \\left( e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } \\right) + e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ( r ) \\frac { d } { d r } \\left( \\frac { 1 } { 1 - r ^ { 2 } } \\right) } \\\\ { = - \\frac { 2 r } { ( 1 - r ^ { 2 } ) ^ { 2 } } ( 1 - C ( r ) ) e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } . } \\\\ { \\displaystyle \\frac { \\partial f } { \\partial \\varphi } = - e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ( r ) \\cos \\psi \\cdot \\frac { \\partial \\psi } { \\partial \\varphi } } \\\\ { = - e ^ { - \\frac { 1 } { 1 - r ^ { 2 } } } C ( r ) . } \\end{array}", + "type": "interline_equation", + "image_path": "bd526aaa270ec74261c2e9754b774c77b6fa1033556ac90dd5d54c026d698207.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 163, + 357, + 447, + 397.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 163, + 397.6666666666667, + 447, + 438.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 163, + 438.33333333333337, + 447, + 479.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 481, + 338, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 340, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 128, + 496 + ], + "score": 1.0, + "content": "So if", + "type": "text" + }, + { + "bbox": [ + 128, + 482, + 154, + 494 + ], + "score": 0.92, + "content": "\\psi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 481, + 178, + 496 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 175, + 478, + 340, + 497 + ], + "score": 1.0, + "content": "dψdt = 0 by the direct calculation below:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 496, + 455, + 581 + ], + "lines": [ + { + "bbox": [ + 156, + 496, + 455, + 581 + ], + "spans": [ + { + "bbox": [ + 156, + 496, + 455, + 581 + ], + "score": 0.96, + "content": "{ \\begin{array} { r l } & { { \\frac { d \\psi } { d t } } = { \\frac { d \\varphi } { d t } } - 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The layers of this model can be described as conv-32 with filter size", + "type": "text" + }, + { + "bbox": [ + 429, + 83, + 449, + 93 + ], + "score": 0.86, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 81, + 505, + 95 + ], + "score": 1.0, + "content": ", max-pool,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 207, + 105 + ], + "score": 1.0, + "content": "conv-64 with filter size", + "type": "text" + }, + { + "bbox": [ + 208, + 94, + 232, + 104 + ], + "score": 0.87, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 94, + 505, + 105 + ], + "score": 1.0, + "content": ", max-pool, fc-1024, fc-10 in order. Notice that this model has", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "bias terms in each layer, and thus does not satisfy homogeneity. To make its outputs homogeneous", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "to its parameters, we also trained this model after removing all the bias terms except those in the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "first layer (the modified model is called the CNN without bias). Note that keeping the bias terms in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "the first layer prevents the model to be homogeneous in the input data while retains the homogeneity", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "in parameters. We initialize all layer weights by He normal initializer (He et al., 2015) and all bias", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "terms by zero. In training the models, we use SGD with batch size 100 without momentum. We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 168, + 388, + 184 + ], + "spans": [ + { + "bbox": [ + 104, + 168, + 217, + 184 + ], + "score": 1.0, + "content": "normalize all the images to", + "type": "text" + }, + { + "bbox": [ + 217, + 169, + 261, + 182 + ], + "score": 0.93, + "content": "[ 0 , 1 ] ^ { 3 2 \\times 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 168, + 388, + 184 + ], + "score": 1.0, + "content": "by dividing 255 for each pixel.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 196, + 310, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 195, + 312, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 312, + 209 + ], + "score": 1.0, + "content": "K.1 EVALUATION FOR NORMALIZED MARGIN", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 217, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "In the first part of our experiments, we evaluate the normalized margin every few epochs to see how", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "it changes over time. From now on, we view the bias term in the first layer as a part of the weight", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "in the first layer for convenience. Observe that the CNN without bias is multi-homogeneous in layer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "weights (see (4) in Section 4.4). So for the CNN without bias, we define the normalized margin", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 114, + 272 + ], + "score": 0.82, + "content": "\\bar { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 261, + 293, + 273 + ], + "score": 1.0, + "content": "as the margin divided by the product of the", + "type": "text" + }, + { + "bbox": [ + 293, + 261, + 305, + 271 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "-norm of all layer weights. Here we compute the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 119, + 282 + ], + "score": 0.87, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "-norm of a layer weight parameter after flattening it into a one-dimensional vector. For the CNN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 487, + 295 + ], + "score": 1.0, + "content": "with bias, we still compute the smoothed normalized margin in this way. When computing the", + "type": "text" + }, + { + "bbox": [ + 488, + 282, + 500, + 293 + ], + "score": 0.86, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "norm of every layer weight, we simply ignore the bias terms if they are not in the first layer. 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The layers of this model can be described as conv-32 with filter size", + "type": "text" + }, + { + "bbox": [ + 429, + 83, + 449, + 93 + ], + "score": 0.86, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 81, + 505, + 95 + ], + "score": 1.0, + "content": ", max-pool,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 207, + 105 + ], + "score": 1.0, + "content": "conv-64 with filter size", + "type": "text" + }, + { + "bbox": [ + 208, + 94, + 232, + 104 + ], + "score": 0.87, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 94, + 505, + 105 + ], + "score": 1.0, + "content": ", max-pool, fc-1024, fc-10 in order. Notice that this model has", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "bias terms in each layer, and thus does not satisfy homogeneity. To make its outputs homogeneous", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "to its parameters, we also trained this model after removing all the bias terms except those in the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "first layer (the modified model is called the CNN without bias). Note that keeping the bias terms in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "the first layer prevents the model to be homogeneous in the input data while retains the homogeneity", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "in parameters. We initialize all layer weights by He normal initializer (He et al., 2015) and all bias", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "terms by zero. In training the models, we use SGD with batch size 100 without momentum. We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 168, + 388, + 184 + ], + "spans": [ + { + "bbox": [ + 104, + 168, + 217, + 184 + ], + "score": 1.0, + "content": "normalize all the images to", + "type": "text" + }, + { + "bbox": [ + 217, + 169, + 261, + 182 + ], + "score": 0.93, + "content": "[ 0 , 1 ] ^ { 3 2 \\times 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 168, + 388, + 184 + ], + "score": 1.0, + "content": "by dividing 255 for each pixel.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 81, + 506, + 184 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 196, + 310, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 195, + 312, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 312, + 209 + ], + "score": 1.0, + "content": "K.1 EVALUATION FOR NORMALIZED MARGIN", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 217, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "In the first part of our experiments, we evaluate the normalized margin every few epochs to see how", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "it changes over time. 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Here we compute the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 119, + 282 + ], + "score": 0.87, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "-norm of a layer weight parameter after flattening it into a one-dimensional vector. For the CNN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 487, + 295 + ], + "score": 1.0, + "content": "with bias, we still compute the smoothed normalized margin in this way. When computing the", + "type": "text" + }, + { + "bbox": [ + 488, + 282, + 500, + 293 + ], + "score": 0.86, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "norm of every layer weight, we simply ignore the bias terms if they are not in the first layer. For", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "completeness, we include the plots for the normalized margin using the original definition (2) in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 316, + 170, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 170, + 329 + ], + "score": 1.0, + "content": "Figure 3 and 4.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 216, + 506, + 329 + ] + }, + { + "type": "image", + "bbox": [ + 166, + 342, + 438, + 465 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 166, + 342, + 438, + 465 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 342, + 438, + 465 + ], + "spans": [ + { + "bbox": [ + 166, + 342, + 438, + 465 + ], + "score": 0.967, + "type": "image", + "image_path": "537667231ed5f10711f6cfa0e044280666fd050879ce6aa6d81b43708667e02e.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 166, + 342, + 438, + 383.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 166, + 383.0, + 438, + 424.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 166, + 424.0, + 438, + 465.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 480, + 505, + 514 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "Figure 3: Training CNNs with and without bias on MNIST, using SGD with learning rate 0.01. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 251, + 504 + ], + "score": 1.0, + "content": "training accuracy (left) increases to", + "type": "text" + }, + { + "bbox": [ + 251, + 491, + 276, + 502 + ], + "score": 0.9, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "after about 100 epochs, and the normalized margin with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 502, + 389, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 389, + 514 + ], + "score": 1.0, + "content": "the original definition (right) keeps increasing after the model is fitted.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "image", + "bbox": [ + 126, + 533, + 491, + 642 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 533, + 491, + 642 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 126, + 533, + 491, + 642 + ], + "spans": [ + { + "bbox": [ + 126, + 533, + 491, + 642 + ], + "score": 0.968, + "type": "image", + "image_path": "6a977df3714686b1cdc63609ca27ce2478cbf361e5fd64ab3db15c2a4a1067fc.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 126, + 533, + 491, + 569.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 126, + 569.3333333333334, + 491, + 605.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 126, + 605.6666666666667, + 491, + 642.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 655, + 505, + 711 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 669 + ], + "score": 1.0, + "content": "Figure 4: Training CNNs with and without bias on MNIST, using SGD with the loss-based learning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 334, + 679 + ], + "score": 1.0, + "content": "rate scheduler. The training accuracy (left) increases to", + "type": "text" + }, + { + "bbox": [ + 334, + 667, + 359, + 677 + ], + "score": 0.89, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "after about 20 epochs, and the nor-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "malized margin with the original definition (middle) increases rapidly after the model is fitted. The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 687, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 344, + 703 + ], + "score": 1.0, + "content": "right figure shows the change of the relative learning rate", + "type": "text" + }, + { + "bbox": [ + 344, + 689, + 363, + 701 + ], + "score": 0.91, + "content": "\\alpha ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 687, + 506, + 703 + ], + "score": 1.0, + "content": "(see (27) for its definition) during", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 698, + 143, + 714 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 143, + 714 + ], + "score": 1.0, + "content": "training.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + } + ], + "index": 29.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "SGD with Constant Learning Rate. We first train the CNNs using SGD with constant learning", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "rate 0.01. After about 100 epochs, both CNNs have fitted the training set. After that, we can see that", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "the normalized margins of both CNNs increase. However, the growth rate of the normalized margin", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "is rather slow. The results are shown in Figure 1 in Section 1. We also tried other learning rates", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 335, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 335, + 138 + ], + "score": 1.0, + "content": "other than 0.01, and similar phenomena can be observed.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "SGD with Loss-based Learning Rate. Indeed, we can speed up the training by using a proper", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "scheduling of learning rates for SGD. We propose a heuristic learning rate scheduling method, called", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 179 + ], + "score": 1.0, + "content": "the loss-based learning rate scheduling. The basic idea is to find the maximum possible learning", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 504, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 504, + 188 + ], + "score": 1.0, + "content": "rate at each epoch based on the current training loss (in a similar way as the line search method).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "See Appendix L.1 for the details. As shown in Figure 1, SGD with loss-based learning rate schedul-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "ing decreases the training loss exponentially faster than SGD with constant learning rate. Also, a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "rapid growth of normalized margin is observed for both CNNs. Note that with this scheduling the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 218, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 237, + 233 + ], + "score": 1.0, + "content": "training loss can be as small as", + "type": "text" + }, + { + "bbox": [ + 237, + 219, + 266, + 231 + ], + "score": 0.9, + "content": "\\overline { { 1 0 ^ { - 8 0 0 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 218, + 506, + 233 + ], + "score": 1.0, + "content": ", which may lead to numerical issues. To address such is-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "sues, we applied some re-parameterization tricks and numerical tricks in our implementation. See", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 223, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 223, + 253 + ], + "score": 1.0, + "content": "Appendix L.2 for the details.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "Experiments on CIFAR-10. To verify whether the normalized margin is increasing in practice, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "also conduct experiments on CIFAR-10. We use a modified version of VGGNet-16. 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We train two networks: one is exactly the same as the VGGNet", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "we described, and the other one is the VGGNet without any bias terms except those in the first layer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "(similar as in the experiments on MNIST). The experiment results are shown in Figure 5 and 6. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 338, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 338, + 348 + ], + "score": 1.0, + "content": "can see that the normalize margin is increasing over time.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "image", + "bbox": [ + 163, + 360, + 446, + 487 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 163, + 360, + 446, + 487 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 163, + 360, + 446, + 487 + ], + "spans": [ + { + "bbox": [ + 163, + 360, + 446, + 487 + ], + "score": 0.969, + "type": "image", + "image_path": "58ff1562f78c324281e18d2aa99b831e549164308703f02119c458659d4d2e3c.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 163, + 360, + 446, + 402.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 163, + 402.3333333333333, + 446, + 444.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 163, + 444.66666666666663, + 446, + 486.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 502, + 504, + 515 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "score": 1.0, + "content": "Figure 5: Training VGGNet with and without bias on CIFAR-10, using SGD with learning rate 0.1.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + } + ], + "index": 25.0 + }, + { + "type": "image", + "bbox": [ + 155, + 533, + 448, + 660 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 155, + 533, + 448, + 660 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 155, + 533, + 448, + 660 + ], + "spans": [ + { + "bbox": [ + 155, + 533, + 448, + 660 + ], + "score": 0.968, + "type": "image", + "image_path": "495a9fd0459b12209dbeccc75dc0894e2a64a5daed8bfc4e279791f66c40feb6.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 155, + 533, + 448, + 575.3333333333334 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 155, + 575.3333333333334, + 448, + 617.6666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 155, + 617.6666666666667, + 448, + 660.0000000000001 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 675, + 505, + 698 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "Figure 6: Training VGGNet with and without bias on CIFAR-10, using SGD with the loss-based", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 686, + 201, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 201, + 699 + ], + "score": 1.0, + "content": "learning rate scheduler.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + } + ], + "index": 29.25 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Test Accuracy. 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We first train the CNNs using SGD with constant learning", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "rate 0.01. After about 100 epochs, both CNNs have fitted the training set. After that, we can see that", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "the normalized margins of both CNNs increase. However, the growth rate of the normalized margin", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "is rather slow. The results are shown in Figure 1 in Section 1. We also tried other learning rates", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 335, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 335, + 138 + ], + "score": 1.0, + "content": "other than 0.01, and similar phenomena can be observed.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 81, + 506, + 138 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "SGD with Loss-based Learning Rate. Indeed, we can speed up the training by using a proper", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "scheduling of learning rates for SGD. We propose a heuristic learning rate scheduling method, called", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 179 + ], + "score": 1.0, + "content": "the loss-based learning rate scheduling. The basic idea is to find the maximum possible learning", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 504, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 504, + 188 + ], + "score": 1.0, + "content": "rate at each epoch based on the current training loss (in a similar way as the line search method).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "See Appendix L.1 for the details. As shown in Figure 1, SGD with loss-based learning rate schedul-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "ing decreases the training loss exponentially faster than SGD with constant learning rate. Also, a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "rapid growth of normalized margin is observed for both CNNs. Note that with this scheduling the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 218, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 237, + 233 + ], + "score": 1.0, + "content": "training loss can be as small as", + "type": "text" + }, + { + "bbox": [ + 237, + 219, + 266, + 231 + ], + "score": 0.9, + "content": "\\overline { { 1 0 ^ { - 8 0 0 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 218, + 506, + 233 + ], + "score": 1.0, + "content": ", which may lead to numerical issues. To address such is-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "sues, we applied some re-parameterization tricks and numerical tricks in our implementation. See", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 223, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 223, + 253 + ], + "score": 1.0, + "content": "Appendix L.2 for the details.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 142, + 506, + 253 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "Experiments on CIFAR-10. To verify whether the normalized margin is increasing in practice, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "also conduct experiments on CIFAR-10. We use a modified version of VGGNet-16. 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We train two networks: one is exactly the same as the VGGNet", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "we described, and the other one is the VGGNet without any bias terms except those in the first layer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "(similar as in the experiments on MNIST). The experiment results are shown in Figure 5 and 6. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 338, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 338, + 348 + ], + "score": 1.0, + "content": "can see that the normalize margin is increasing over time.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 258, + 506, + 348 + ] + }, + { + "type": "image", + "bbox": [ + 163, + 360, + 446, + 487 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 163, + 360, + 446, + 487 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 163, + 360, + 446, + 487 + ], + "spans": [ + { + "bbox": [ + 163, + 360, + 446, + 487 + ], + "score": 0.969, + "type": "image", + "image_path": "58ff1562f78c324281e18d2aa99b831e549164308703f02119c458659d4d2e3c.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 163, + 360, + 446, + 402.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 163, + 402.3333333333333, + 446, + 444.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 163, + 444.66666666666663, + 446, + 486.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 502, + 504, + 515 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "score": 1.0, + "content": "Figure 5: Training VGGNet with and without bias on CIFAR-10, using SGD with learning rate 0.1.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + } + ], + "index": 25.0 + }, + { + "type": "image", + "bbox": [ + 155, + 533, + 448, + 660 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 155, + 533, + 448, + 660 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 155, + 533, + 448, + 660 + ], + "spans": [ + { + "bbox": [ + 155, + 533, + 448, + 660 + ], + "score": 0.968, + "type": "image", + "image_path": "495a9fd0459b12209dbeccc75dc0894e2a64a5daed8bfc4e279791f66c40feb6.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 155, + 533, + 448, + 575.3333333333334 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 155, + 575.3333333333334, + 448, + 617.6666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 155, + 617.6666666666667, + 448, + 660.0000000000001 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 675, + 505, + 698 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "Figure 6: Training VGGNet with and without bias on CIFAR-10, using SGD with the loss-based", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 686, + 201, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 201, + 699 + ], + "score": 1.0, + "content": "learning rate scheduler.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + } + ], + "index": 29.25 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Test Accuracy. Previous works on margin-based generalization bounds (Neyshabur et al., 2018;", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "Bartlett et al., 2017; Golowich et al., 2018; Li et al., 2018a; Wei et al., 2019; Banburski et al.,", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 709, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "2019) usually suggest that a larger margin implies a better generalization bound. To see whether the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "generalization error also gets smaller in practice, we plot train and test accuracy for both MNIST and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "CIFAR-10. As shown in Figure 7, the test accuracy changes only slightly after training with loss-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "based learning rate scheduling for 10000 epochs, although the normalized margin does increase a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "lot. We leave it as a future work to study this interesting gap between margin-based generalization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "bound and generalization error. Concurrent to this work, Wei & Ma (2020) proposed a generalization", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "bound based on a new notion of margin called all-layer margin, and showed via experiments that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 504, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 504, + 172 + ], + "score": 1.0, + "content": "enlarging all-layer margin can indeed improve generalization. It would be an interesting research", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 502, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 502, + 182 + ], + "score": 1.0, + "content": "direction to study how different definitions of margin may lead to different generalization abilities.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "image", + "bbox": [ + 166, + 196, + 444, + 320 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 166, + 196, + 444, + 320 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 196, + 444, + 320 + ], + "spans": [ + { + "bbox": [ + 166, + 196, + 444, + 320 + ], + "score": 0.967, + "type": "image", + "image_path": "30c454b39b58b3eb361944408c107dca51e6a4f80df7e6e0667b0ce844855b9b.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 166, + 196, + 444, + 237.33333333333334 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 166, + 237.33333333333334, + 444, + 278.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 166, + 278.6666666666667, + 444, + 320.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 334, + 506, + 379 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "score": 1.0, + "content": "Figure 7: (Left). Training and test accuracy during training CNNs without bias on MNIST, using", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "SGD with the loss-based learning rate scheduler. Every number is averaged over 10 runs. (Right).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "Training and test accuracy during training VGGNet without bias on CIFAR-10, using SGD with the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 367, + 402, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 402, + 380 + ], + "score": 1.0, + "content": "loss-based learning rate scheduler. Every number is averaged over 3 runs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + } + ], + "index": 11.75 + }, + { + "type": "title", + "bbox": [ + 108, + 400, + 268, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 400, + 269, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 269, + 413 + ], + "score": 1.0, + "content": "K.2 EVALUATION FOR ROBUSTNESS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "score": 1.0, + "content": "Recently, robustness of deep learning has received considerable attention (Szegedy et al., 2013; Big-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "gio et al., 2013; Athalye et al., 2018), since most state-of-the-arts deep neural networks are found", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "to be very vulnerable against small but adversarial perturbations of the input points. In our experi-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "ments, we found that enlarging the normalized margin can improve the robustness. In particular, by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "simply training the neural network for a longer time with our loss-based learning rate, we observe", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 476, + 423, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 220, + 489 + ], + "score": 1.0, + "content": "noticeable improvements of", + "type": "text" + }, + { + "bbox": [ + 220, + 476, + 232, + 486 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 476, + 423, + 489 + ], + "score": 1.0, + "content": "-robustness on both the training set and test set.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "We first elaborate the relationship between the normalized margin and the robustness from a theo-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 250, + 516 + ], + "score": 1.0, + "content": "retical perspective. For a data point", + "type": "text" + }, + { + "bbox": [ + 251, + 505, + 295, + 516 + ], + "score": 0.93, + "content": "\\boldsymbol { z } = ( \\boldsymbol { x } , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 504, + 505, + 516 + ], + "score": 1.0, + "content": ", we can define the robustness (with respect to some", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 515, + 288, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 130, + 527 + ], + "score": 1.0, + "content": "norm", + "type": "text" + }, + { + "bbox": [ + 131, + 515, + 151, + 527 + ], + "score": 0.88, + "content": "\\| \\cdot \\| )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 516, + 233, + 527 + ], + "score": 1.0, + "content": "of a neural network", + "type": "text" + }, + { + "bbox": [ + 233, + 516, + 242, + 525 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 516, + 257, + 527 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 257, + 518, + 264, + 525 + ], + "score": 0.8, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 516, + 288, + 527 + ], + "score": 1.0, + "content": "to be", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 533, + 413, + 552 + ], + "lines": [ + { + "bbox": [ + 196, + 533, + 413, + 552 + ], + "spans": [ + { + "bbox": [ + 196, + 533, + 413, + 552 + ], + "score": 0.9, + "content": "R _ { \\theta } ( z ) : = \\operatorname* { i n f } _ { x ^ { \\prime } \\in X } \\left\\{ \\| x - x ^ { \\prime } \\| : ( x ^ { \\prime } , y ) { \\mathrm { ~ i s ~ m i s c l a s s i f i e d } } \\right\\} .", + "type": "interline_equation", + "image_path": "d0f4d23a57fce2b2b32f43b75f804e39c37bf29b1e16d9831407e64ddd9245cb.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 196, + 533, + 413, + 552 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 505, + 661 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 133, + 572 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 560, + 144, + 569 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 558, + 260, + 572 + ], + "score": 1.0, + "content": "is the data domain (which is", + "type": "text" + }, + { + "bbox": [ + 260, + 558, + 303, + 571 + ], + "score": 0.92, + "content": "[ 0 , 1 ] ^ { 3 2 \\times 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "for MNIST). It is well-known that the normalized", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 222, + 582 + ], + "score": 1.0, + "content": "margin is a lower bound of", + "type": "text" + }, + { + "bbox": [ + 222, + 570, + 234, + 580 + ], + "score": 0.89, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "-robustness for fully-connected networks (See, e.g., Theorem 4 in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "(Sokolic et al., 2017)). Indeed, a general relationship between those two quantities can be easily", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 233, + 605 + ], + "score": 1.0, + "content": "shown. Note that a data point", + "type": "text" + }, + { + "bbox": [ + 234, + 594, + 241, + 602 + ], + "score": 0.74, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 592, + 407, + 605 + ], + "score": 1.0, + "content": "is correctly classified iff the margin for", + "type": "text" + }, + { + "bbox": [ + 407, + 594, + 415, + 602 + ], + "score": 0.71, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 592, + 466, + 605 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 466, + 592, + 491, + 604 + ], + "score": 0.9, + "content": "q _ { \\pmb { \\theta } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 592, + 505, + 605 + ], + "score": 1.0, + "content": ", is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 324, + 617 + ], + "score": 1.0, + "content": "larger than 0. For homogeneous models, the margin", + "type": "text" + }, + { + "bbox": [ + 325, + 603, + 349, + 615 + ], + "score": 0.92, + "content": "q _ { \\pmb { \\theta } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 603, + 464, + 617 + ], + "score": 1.0, + "content": "and the normalized margin", + "type": "text" + }, + { + "bbox": [ + 464, + 604, + 488, + 616 + ], + "score": 0.91, + "content": "q _ { \\hat { \\theta } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 614, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 114, + 626 + ], + "score": 0.68, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 614, + 210, + 629 + ], + "score": 1.0, + "content": "have the same sign. If", + "type": "text" + }, + { + "bbox": [ + 210, + 615, + 282, + 628 + ], + "score": 0.87, + "content": "q _ { \\hat { \\theta } } ( \\cdot ) : \\mathbb { R } ^ { d _ { \\mathrm { x } } } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 614, + 294, + 629 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 294, + 616, + 302, + 627 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 614, + 458, + 629 + ], + "score": 1.0, + "content": "-Lipschitz (with respect to some norm", + "type": "text" + }, + { + "bbox": [ + 459, + 616, + 479, + 628 + ], + "score": 0.88, + "content": "\\| \\cdot \\| )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 614, + 506, + 629 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 193, + 641 + ], + "score": 1.0, + "content": "it is easy to see that", + "type": "text" + }, + { + "bbox": [ + 194, + 628, + 272, + 639 + ], + "score": 0.89, + "content": "R _ { \\pmb \\theta } ( z ) \\ge q _ { \\hat { \\pmb \\theta } } ( z ) / \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 626, + 506, + 641 + ], + "score": 1.0, + "content": ". This suggests that improving the normalize margin on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "the training set can improve the robustness on the training set. Therefore, our theoretical analysis", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 462, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 462, + 662 + ], + "score": 1.0, + "content": "suggests that training longer can improve the robustness of the model on the training set.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 488, + 678 + ], + "score": 1.0, + "content": "This observation does match with our experiment results. In the experiments, we measure the", + "type": "text" + }, + { + "bbox": [ + 488, + 665, + 500, + 676 + ], + "score": 0.86, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 675, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 416, + 689 + ], + "score": 1.0, + "content": "robustness of the CNN without bias for the first time its loss decreases below", + "type": "text" + }, + { + "bbox": [ + 416, + 676, + 442, + 687 + ], + "score": 0.82, + "content": "1 0 ^ { - 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 675, + 445, + 689 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 446, + 676, + 471, + 687 + ], + "score": 0.78, + "content": "1 0 ^ { - 1 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 675, + 475, + 689 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 475, + 676, + 501, + 687 + ], + "score": 0.83, + "content": "1 0 ^ { - 2 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 675, + 505, + 689 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 685, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 136, + 698 + ], + "score": 0.9, + "content": "1 0 ^ { - 1 2 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 685, + 186, + 702 + ], + "score": 1.0, + "content": "(labelled as", + "type": "text" + }, + { + "bbox": [ + 187, + 689, + 230, + 698 + ], + "score": 0.5, + "content": "\\mathtt { m o d e l - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 685, + 242, + 702 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 243, + 689, + 285, + 698 + ], + "score": 0.38, + "content": "\\mathtt { m o d e l - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 685, + 431, + 702 + ], + "score": 1.0, + "content": "respectively). We also measure the", + "type": "text" + }, + { + "bbox": [ + 432, + 687, + 444, + 698 + ], + "score": 0.87, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 685, + 507, + 702 + ], + "score": 1.0, + "content": "-robustness for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 366, + 711 + ], + "score": 1.0, + "content": "the final model after training for 10000 epochs (labelled as mode", + "type": "text" + }, + { + "bbox": [ + 367, + 699, + 385, + 709 + ], + "score": 0.26, + "content": "\\boldsymbol { \\perp } - \\boldsymbol { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "), whose training loss is about", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 706, + 507, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 136, + 720 + ], + "score": 0.89, + "content": "1 0 ^ { - 8 8 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 706, + 507, + 724 + ], + "score": 1.0, + "content": ". The normalized margin of each model is monotone increasing with respect to the number", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 235, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 235, + 733 + ], + "score": 1.0, + "content": "of epochs, as shown in Table 1.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5 + } + ], + "page_idx": 48, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "49", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "2019) usually suggest that a larger margin implies a better generalization bound. To see whether the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "generalization error also gets smaller in practice, we plot train and test accuracy for both MNIST and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "CIFAR-10. As shown in Figure 7, the test accuracy changes only slightly after training with loss-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "based learning rate scheduling for 10000 epochs, although the normalized margin does increase a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "lot. We leave it as a future work to study this interesting gap between margin-based generalization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "bound and generalization error. Concurrent to this work, Wei & Ma (2020) proposed a generalization", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "bound based on a new notion of margin called all-layer margin, and showed via experiments that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 504, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 504, + 172 + ], + "score": 1.0, + "content": "enlarging all-layer margin can indeed improve generalization. It would be an interesting research", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 502, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 502, + 182 + ], + "score": 1.0, + "content": "direction to study how different definitions of margin may lead to different generalization abilities.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 506, + 182 + ] + }, + { + "type": "image", + "bbox": [ + 166, + 196, + 444, + 320 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 166, + 196, + 444, + 320 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 196, + 444, + 320 + ], + "spans": [ + { + "bbox": [ + 166, + 196, + 444, + 320 + ], + "score": 0.967, + "type": "image", + "image_path": "30c454b39b58b3eb361944408c107dca51e6a4f80df7e6e0667b0ce844855b9b.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 166, + 196, + 444, + 237.33333333333334 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 166, + 237.33333333333334, + 444, + 278.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 166, + 278.6666666666667, + 444, + 320.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 334, + 506, + 379 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "score": 1.0, + "content": "Figure 7: (Left). Training and test accuracy during training CNNs without bias on MNIST, using", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "SGD with the loss-based learning rate scheduler. Every number is averaged over 10 runs. (Right).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "Training and test accuracy during training VGGNet without bias on CIFAR-10, using SGD with the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 367, + 402, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 402, + 380 + ], + "score": 1.0, + "content": "loss-based learning rate scheduler. Every number is averaged over 3 runs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + } + ], + "index": 11.75 + }, + { + "type": "title", + "bbox": [ + 108, + 400, + 268, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 400, + 269, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 269, + 413 + ], + "score": 1.0, + "content": "K.2 EVALUATION FOR ROBUSTNESS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "score": 1.0, + "content": "Recently, robustness of deep learning has received considerable attention (Szegedy et al., 2013; Big-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "gio et al., 2013; Athalye et al., 2018), since most state-of-the-arts deep neural networks are found", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "to be very vulnerable against small but adversarial perturbations of the input points. In our experi-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "ments, we found that enlarging the normalized margin can improve the robustness. In particular, by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "simply training the neural network for a longer time with our loss-based learning rate, we observe", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 476, + 423, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 220, + 489 + ], + "score": 1.0, + "content": "noticeable improvements of", + "type": "text" + }, + { + "bbox": [ + 220, + 476, + 232, + 486 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 476, + 423, + 489 + ], + "score": 1.0, + "content": "-robustness on both the training set and test set.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 420, + 506, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "We first elaborate the relationship between the normalized margin and the robustness from a theo-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 250, + 516 + ], + "score": 1.0, + "content": "retical perspective. 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It is well-known that the normalized", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 222, + 582 + ], + "score": 1.0, + "content": "margin is a lower bound of", + "type": "text" + }, + { + "bbox": [ + 222, + 570, + 234, + 580 + ], + "score": 0.89, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "-robustness for fully-connected networks (See, e.g., Theorem 4 in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "(Sokolic et al., 2017)). Indeed, a general relationship between those two quantities can be easily", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 233, + 605 + ], + "score": 1.0, + "content": "shown. Note that a data point", + "type": "text" + }, + { + "bbox": [ + 234, + 594, + 241, + 602 + ], + "score": 0.74, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 592, + 407, + 605 + ], + "score": 1.0, + "content": "is correctly classified iff the margin for", + "type": "text" + }, + { + "bbox": [ + 407, + 594, + 415, + 602 + ], + "score": 0.71, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 592, + 466, + 605 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 466, + 592, + 491, + 604 + ], + "score": 0.9, + "content": "q _ { \\pmb { \\theta } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 592, + 505, + 605 + ], + "score": 1.0, + "content": ", is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 324, + 617 + ], + "score": 1.0, + "content": "larger than 0. 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If", + "type": "text" + }, + { + "bbox": [ + 210, + 615, + 282, + 628 + ], + "score": 0.87, + "content": "q _ { \\hat { \\theta } } ( \\cdot ) : \\mathbb { R } ^ { d _ { \\mathrm { x } } } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 614, + 294, + 629 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 294, + 616, + 302, + 627 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 614, + 458, + 629 + ], + "score": 1.0, + "content": "-Lipschitz (with respect to some norm", + "type": "text" + }, + { + "bbox": [ + 459, + 616, + 479, + 628 + ], + "score": 0.88, + "content": "\\| \\cdot \\| )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 614, + 506, + 629 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 193, + 641 + ], + "score": 1.0, + "content": "it is easy to see that", + "type": "text" + }, + { + "bbox": [ + 194, + 628, + 272, + 639 + ], + "score": 0.89, + "content": "R _ { \\pmb \\theta } ( z ) \\ge q _ { \\hat { \\pmb \\theta } } ( z ) / \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 626, + 506, + 641 + ], + "score": 1.0, + "content": ". 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Figures on the first row show the robust accuracy on the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 359, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 505, + 370 + ], + "score": 1.0, + "content": "training set, and figures on the second row show that on the test set. On every row, the left figure and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 369, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 506, + 381 + ], + "score": 1.0, + "content": "the right figure plot the same curves but they are in different scales. From model-1 to model-4,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "noticeable robust accuracy improvements can be observed. The improvement of model-5 upon", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "model-4 is marginal or nonexistent for some \u000f, but the improvement upon model-1 is always", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 402, + 153, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 153, + 415 + ], + "score": 1.0, + "content": "significant.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "table", + "bbox": [ + 118, + 447, + 491, + 527 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 125, + 435, + 483, + 446 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 433, + 483, + 448 + ], + "spans": [ + { + "bbox": [ + 127, + 433, + 483, + 448 + ], + "score": 1.0, + "content": "Table 1: Statistics of the CNN without bias after training for different number of epochs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "table_body", + "bbox": [ + 118, + 447, + 491, + 527 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 447, + 491, + 527 + ], + "spans": [ + { + "bbox": [ + 118, + 447, + 491, + 527 + ], + "score": 0.983, + "html": "
model namenumber of epochstrain lossnormalized margintrain acctest acc
model-13810-10.045.65× 10-5100%99.3%
model-27510-15.129.50 ×10-5100%99.3%
model-310710-20.071.30 × 10-4100%99.3%
model-493510- -120.014.61 × 10-4100%99.2%
model-51000010 -881.511.18 × 10-3100%99.1%
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model namenumber of epochstrain lossnormalized margintrain acctest acc
model-13810-10.045.65× 10-5100%99.3%
model-27510-15.129.50 ×10-5100%99.3%
model-310710-20.071.30 × 10-4100%99.3%
model-493510- -120.014.61 × 10-4100%99.2%
model-51000010 -881.511.18 × 10-3100%99.1%
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This shows that training longer can also help to", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 295, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 156, + 106 + ], + "score": 1.0, + "content": "improve the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 157, + 93, + 169, + 104 + ], + "score": 0.88, + "content": "\\bar { L } ^ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 169, + 93, + 295, + 106 + ], + "score": 1.0, + "content": "-robust accuracy on the test set.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 654, + 506, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "model-5 upon model-1 is always significant. This shows that training longer can also help to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 295, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 156, + 106 + ], + "score": 1.0, + "content": "improve the", + "type": "text" + }, + { + "bbox": [ + 157, + 93, + 169, + 104 + ], + "score": 0.88, + "content": "\\bar { L } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 93, + 295, + 106 + ], + "score": 1.0, + "content": "-robust accuracy on the test set.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 504, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "score": 1.0, + "content": "We tried various different settings of hyperparameters for the evaluation method (including different", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "learning rates, different binary search steps, etc.) and we observed that the shapes and relative", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 450, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 450, + 146 + ], + "score": 1.0, + "content": "positions of the curves in Figure 8 are stable across different hyperparameter settings.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 149, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "It is worth to note that the normalized margin and robustness do not grow in the same speed in our", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 279, + 174 + ], + "score": 1.0, + "content": "experiments, although the theory suggests", + "type": "text" + }, + { + "bbox": [ + 279, + 160, + 356, + 173 + ], + "score": 0.93, + "content": "R _ { \\pmb \\theta } ( \\pmb z ) \\geq q _ { \\hat { \\pmb \\theta } } ( \\pmb z ) / \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 160, + 506, + 174 + ], + "score": 1.0, + "content": ". This may be because the Lipschitz", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 143, + 184 + ], + "score": 1.0, + "content": "constant", + "type": "text" + }, + { + "bbox": [ + 143, + 172, + 151, + 182 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "(if defined locally) is also changing during training. Combining training longer with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "existing techniques for constraining Lipschitz number (Anil et al., 2019; Cisse et al., 2017) could", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 361, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 361, + 205 + ], + "score": 1.0, + "content": "potentially alleviate this issue, and we leave it as a future work.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 220, + 327, + 233 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 327, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 327, + 234 + ], + "score": 1.0, + "content": "L ADDITIONAL EXPERIMENTAL DETAILS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 244, + 364, + 256 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 365, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 365, + 259 + ], + "score": 1.0, + "content": "In this section, we provide additional details of our experiments.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 106, + 268, + 322, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 322, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 322, + 282 + ], + "score": 1.0, + "content": "L.1 LOSS-BASED LEARNING RATE SCHEDULING", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 492, + 302 + ], + "score": 1.0, + "content": "The intuition of the loss-based learning rate scheduling is as follows. If the training loss is", + "type": "text" + }, + { + "bbox": [ + 493, + 292, + 500, + 300 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 466, + 313 + ], + "score": 1.0, + "content": "smooth, then optimization theory suggests that we should set the learning rate to roughly", + "type": "text" + }, + { + "bbox": [ + 466, + 301, + 484, + 313 + ], + "score": 0.9, + "content": "1 / \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 300, + 506, + 313 + ], + "score": 1.0, + "content": ". 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In particular, we initialize", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 423, + 342, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 127, + 435 + ], + "score": 0.9, + "content": "\\alpha ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 423, + 332, + 435 + ], + "score": 1.0, + "content": "by some value, and do the following at each epoch", + "type": "text" + }, + { + "bbox": [ + 332, + 424, + 337, + 433 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 423, + 342, + 435 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 442, + 504, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 491, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 173, + 456 + ], + "score": 1.0, + "content": "Step 1. Initially", + "type": "text" + }, + { + "bbox": [ + 173, + 443, + 242, + 456 + ], + "score": 0.93, + "content": "\\alpha ( t ) \\alpha ( t - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 442, + 262, + 456 + ], + "score": 1.0, + "content": "; Let", + "type": "text" + }, + { + "bbox": [ + 263, + 443, + 299, + 456 + ], + "score": 0.93, + "content": "\\bar { \\mathcal { L } } ( t - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 442, + 491, + 456 + ], + "score": 1.0, + "content": "be the training loss at the end of the last epoch;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 457, + 472, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 375, + 471 + ], + "score": 1.0, + "content": "Step 2. Run SGD through the whole training set with learning rate", + "type": "text" + }, + { + "bbox": [ + 375, + 457, + 467, + 470 + ], + "score": 0.92, + "content": "\\eta ( t ) : = \\alpha ( t ) / \\mathcal { L } ( t - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 457, + 472, + 471 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 471, + 367, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 242, + 485 + ], + "score": 1.0, + "content": "Step 3. Evaluate the training loss", + "type": "text" + }, + { + "bbox": [ + 243, + 471, + 262, + 484 + ], + "score": 0.91, + "content": "\\bar { \\mathcal { L } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 471, + 367, + 485 + ], + "score": 1.0, + "content": "on the whole training set;", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 149, + 499 + ], + "score": 1.0, + "content": "Step 4. If", + "type": "text" + }, + { + "bbox": [ + 149, + 486, + 218, + 499 + ], + "score": 0.89, + "content": "\\bar { \\mathcal { L } } ( t ) < \\bar { \\mathcal { L } } ( t - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 486, + 222, + 499 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 222, + 486, + 293, + 499 + ], + "score": 0.91, + "content": "\\alpha ( t ) \\gets \\alpha ( t ) \\cdot r _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 486, + 419, + 499 + ], + "score": 1.0, + "content": "and end this epoch; otherwise,", + "type": "text" + }, + { + "bbox": [ + 419, + 486, + 487, + 499 + ], + "score": 0.93, + "content": "\\alpha ( t ) \\gets \\alpha ( t ) / r _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 137, + 497, + 193, + 512 + ], + "spans": [ + { + "bbox": [ + 137, + 497, + 193, + 512 + ], + "score": 1.0, + "content": "go to Step 2.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 518, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 104, + 517, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 517, + 227, + 533 + ], + "score": 1.0, + "content": "In all our experiments, we set", + "type": "text" + }, + { + "bbox": [ + 227, + 518, + 448, + 532 + ], + "score": 0.88, + "content": "\\alpha ( 0 ) : = 0 . 1 , r _ { u } : = 2 ^ { 1 / 5 } \\approx 1 . 1 4 9 , r _ { d } : = 2 ^ { 1 / 1 0 } \\approx 1 . 0 7 2 .", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 517, + 506, + 533 + ], + "score": 1.0, + "content": "This specific", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "score": 1.0, + "content": "choice of those hyperparameters is not important; other choices can only affact the computational", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 541, + 353, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 353, + 554 + ], + "score": 1.0, + "content": "efficiency, but not the overall tendency of normalized margin.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 565, + 278, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 279, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 279, + 578 + ], + "score": 1.0, + "content": "L.2 ADDRESSING NUMERICAL ISSUES", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 586, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 347, + 599 + ], + "score": 1.0, + "content": "Since we are dealing with extremely small loss (as small as", + "type": "text" + }, + { + "bbox": [ + 347, + 586, + 376, + 597 + ], + "score": 0.89, + "content": "1 0 ^ { - 8 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 585, + 505, + 599 + ], + "score": 1.0, + "content": "), the current Tensorflow imple-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 597, + 504, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 475, + 610 + ], + "score": 1.0, + "content": "mentation would run into numerical issues. To address the issues, we work as follows. Let", + "type": "text" + }, + { + "bbox": [ + 475, + 597, + 504, + 609 + ], + "score": 0.92, + "content": "\\bar { \\mathcal { L } } _ { B } ( \\pmb { \\theta } )", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 290, + 622 + ], + "score": 1.0, + "content": "be the (average) training loss within a batch", + "type": "text" + }, + { + "bbox": [ + 290, + 608, + 330, + 621 + ], + "score": 0.93, + "content": "B \\subseteq [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 608, + 426, + 622 + ], + "score": 1.0, + "content": ". We use the notations", + "type": "text" + }, + { + "bbox": [ + 426, + 609, + 481, + 620 + ], + "score": 0.9, + "content": "C , s _ { n j } , \\tilde { q } _ { n } , q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 620, + 481, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 449, + 633 + ], + "score": 1.0, + "content": "Appendix G. 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We use the notations", + "type": "text" + }, + { + "bbox": [ + 426, + 609, + 481, + 620 + ], + "score": 0.9, + "content": "C , s _ { n j } , \\tilde { q } _ { n } , q _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 620, + 481, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 449, + 633 + ], + "score": 1.0, + "content": "Appendix G. 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In our experiments, we set", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 526, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 107, + 526, + 198, + 540 + ], + "score": 0.92, + "content": "\\widetilde { \\mathcal { F } } ( t ) : = \\log \\bar { \\mathcal { L } } ( t - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 526, + 505, + 541 + ], + "score": 1.0, + "content": "to be the training loss at the end of the last epoch, since the training loss", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 405, + 552 + ], + "score": 1.0, + "content": "cannot change a lot within one single epoch. 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\\nabla \\boldsymbol { q } _ { n } \\right\\| _ { 2 } \\leq B _ { 1 } R \\cdot \\frac { \\mathcal { L } \\rho ^ { L - 1 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } } \\\\ & { \\left\\| \\nabla ^ { 2 } \\mathcal { L } \\right\\| _ { 2 } = \\left\\| \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } \\left( \\big ( f ^ { \\prime } ( \\boldsymbol { q } _ { n } ) ^ { 2 } - f ^ { \\prime \\prime } ( \\boldsymbol { q } _ { n } ) \\big ) \\nabla q _ { n } \\nabla q _ { n } ^ { \\top } - f ^ { \\prime } ( q _ { n } ) \\nabla ^ { 2 } q _ { n } \\right) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\sum _ { n = 1 } ^ { N } e ^ { - f ( \\boldsymbol { q } _ { n } ) } \\left( \\left( \\frac { R ^ { 2 } } { g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) ^ { 2 } } + \\frac { p R } { g ( \\log \\frac { 1 } { \\mathcal { L } } ) g ^ { \\prime } ( \\log \\frac { 1 } { \\mathcal { L } } ) } \\right) B _ { 1 } ^ { 2 } \\rho 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model namenumber of epochstrain lossnormalized margintrain acctest acc
model-13810-10.045.65× 10-5100%99.3%
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model-493510- -120.014.61 × 10-4100%99.2%
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LAPLACE APPROXIMATION FOR NEURAL NETWORKS + +Hippolyt Ritter1∗, Aleksandar Botev1, David Barber1 2 1University College London 2Alan Turing Institute + +# ABSTRACT + +We leverage recent insights from second-order optimisation for neural networks to construct a Kronecker factored Laplace approximation to the posterior over the weights of a trained network. Our approximation requires no modification of the training procedure, enabling practitioners to estimate the uncertainty of their models currently used in production without having to retrain them. We extensively compare our method to using Dropout and a diagonal Laplace approximation for estimating the uncertainty of a network. We demonstrate that our Kronecker factored method leads to better uncertainty estimates on out-of-distribution data and is more robust to simple adversarial attacks. Our approach only requires calculating two square curvature factor matrices for each layer. Their size is equal to the respective square of the input and output size of the layer, making the method efficient both computationally and in terms of memory usage. We illustrate its scalability by applying it to a state-of-the-art convolutional network architecture. + +# 1 INTRODUCTION + +Neural networks are most commonly trained in a maximum a posteriori (MAP) setting, which only yields point estimates of the parameters, ignoring any uncertainty about them. This often leads to overconfident predictions, especially in regimes that are weakly covered by training data or far away from the data manifold. While the confidence of wrong predictions is usually irrelevant in a research context, it is essential that a Machine Learning algorithm knows when it does not know in the real world, as the consequences of mistakes can be fatal, be it when driving a car or diagnosing a disease. + +The Bayesian framework of statistics provides a principled way for avoiding overconfidence in the parameters by treating them as unknown quantities and integrating over all possible values. Specifically, for the prediction of new data under a model, it fits a posterior distribution over the parameters given the training data and weighs the contribution of each setting of the parameters to the prediction by the probability of the data under those parameters times their prior probability. However, the posterior of neural networks is usually intractable due to their size and nonlinearity. + +There has been previous interest in integrating neural networks into the Bayesian framework (MacKay, 1992; Hinton & Van Camp, 1993; Neal, 1993; Barber & Bishop, 1998), however these approaches were designed for small networks by current standards. Recent adaptations to architectures of modern scale rely on crude approximations of the posterior to become tractable. All of (Graves, 2011; Hernandez-Lobato & Adams, 2015; Blundell et al., 2015) assume independence between the ´ individual weights. While they achieve good results on small datasets, this strong restriction of the posterior is susceptible to underestimating the uncertainty, in particular when optimising the variational bound. The approach in (Gal & Ghahramani, 2016) requires the use of certain stochastic regularisers which are not commonly present in most recent architectures. Furthermore, it is not clear if the approximate posterior defined by these regularisers is a good fit to the true posterior. + +Recent work on second-order optimisation of neural networks (Martens & Grosse, 2015; Botev et al., 2017) has demonstrated that the diagonal blocks of the curvature can be well approximated by a Kronecker product. We combine this insight with the idea of modelling the posterior over the weights as a Gaussian, using a Laplace approximation (MacKay, 1992) with Kronecker factored covariance matrices. This leads to a computationally efficient matrix normal posterior distribution (Gupta & Nagar, 1999) over the weights of every layer. Since the Laplace approximation is applied after training, our approach can be used to obtain uncertainty estimates from existing networks. + +# 2 THE CURVATURE OF NEURAL NETWORKS + +Our method is inspired by recent Kronecker factored approximations of the curvature of a neural network (Martens & Grosse, 2015; Botev et al., 2017) for optimisation and we give a high-level review of these in the following. While the two methods approximate the Gauss-Newton and Fisher matrix respectively, as they are guaranteed to be positive semi-definite (p.s.d.), we base all of our discussion on the Hessian in order to be as general as possible. + +# 2.1 NEURAL NETWORK NOTATION + +We denote a feedforward network as taking an input $a _ { 0 } = x$ and producing an output $h _ { L }$ . The intermediate representations for layers $\lambda = 1 , . . . , L$ are denoted as $h _ { \lambda } = W _ { \lambda } a _ { \lambda - 1 }$ and $a _ { \lambda } = f _ { \lambda } ( h _ { \lambda } )$ . We refer to $a _ { \lambda }$ as the activations, and $h _ { \lambda }$ as the (linear) pre-activations. The bias terms are absorbed into the $W _ { \lambda }$ by appending a 1 to each $a _ { \lambda }$ . The network parameters are optimised w.r.t. an error function $E ( y , h _ { L } )$ for targets $y$ . Most commonly used error functions, such as squared error and categorical cross-entropy, can be interpreted as exponential family negative log likelihoods $- \log p ( y | h _ { L } )$ . + +# 2.2 KRONECKER FACTORED SECOND-ORDER OPTIMISATION + +Traditional second-order methods use either the Hessian matrix or a positive semi-definite approximation thereof to generate parameter updates of the form $\Delta = C ^ { - 1 } g$ , where $C$ is the chosen curvature matrix and $g$ the gradient of the error function parameterised by the network. However, this curvature matrix is infeasbile to compute for modern neural networks as their number of parameters is often in the millions, rendering the size of $C$ of the order of several terabytes. + +Recent work (Martens & Grosse, 2015; Botev et al., 2017) exploits that, for a single data point, the diagonal blocks of these curvature matrices are Kronecker factored: + +$$ +H _ { \lambda } = \frac { \partial ^ { 2 } E } { \partial \operatorname { v e c } ( W _ { \lambda } ) \partial \operatorname { v e c } ( W _ { \lambda } ) } = \mathcal { Q } _ { \lambda } \otimes \mathcal { H } _ { \lambda } +$$ + +where $H _ { \lambda }$ is the Hessian w.r.t. the weights in layer $\lambda$ . $\mathcal { Q } _ { \lambda } = a _ { \lambda - 1 } a _ { \lambda - 1 } ^ { \top }$ denotes the covariance of the incoming activations aλ−1 and Hλ = ∂ E∂h ∂h the pre-activation Hessian, i.e. the Hessian of the error w.r.t. the linear pre-activations $h _ { \lambda }$ in a layer. We provide the derivation for this result as well as the recursion for calculating $\mathcal { H }$ in Appendix A. + +The Kronecker factorisation holds two key advantages: the matrices that need be computed and stored are much smaller — if we assume all layers to be of dimensionality $D$ , the two factors are each of size $D ^ { 2 }$ , whereas the full Hessian for the weights of only one layer would have $D ^ { 4 }$ elements. Furthermore, the inverse of a Kronecker product is equal to the Kronecker product of the inverses, so it is only necessary to invert those two moderately sized matrices. + +In order to maintain this structure over a minibatch of data, all Kronecker factored second-order methods make two core approximations: First, they only model the diagonal blocks corresponding to the weights of a layer, such that the curvature decomposes into $L$ independent matrices. Second, they assume $\mathcal { Q } _ { \lambda }$ and $\mathcal { H } _ { \lambda }$ to be independent. This is in order to maintain the Kronecker factorisation in expectation, i.e. $\mathbb { E } \left[ \mathcal { Q } _ { \lambda } \otimes \mathcal { H } _ { \lambda } \right] \approx \mathbb { E } \left[ \mathcal { Q } _ { \lambda } \right] \otimes \mathbb { E } \left[ \mathcal { H } _ { \lambda } \right]$ , since the expectation of a Kronecker product is not guaranteed to be Kronecker factored itself. + +The main difference between the Kronecker factored second-order optimisers lies in how they efficiently approximate $\mathbb { E } \left[ \mathcal { H } _ { \lambda } \right]$ . For exact calculation, it would be necessary to pass back an entire matrix per data point in a minibatch, which imposes infeasible memory and computational requirements. KFRA (Botev et al., 2017) simply passes back the expectation at every layer, while KFAC (Martens & Grosse, 2015) utilises the Fisher identity to only propagate a vector rather than a matrix, approximating the Kronecker factors with a stochastic rank-one matrix for each data point. + +The diagonal blocks of the Hessian and Gauss-Newton matrix are equal for neural networks with piecewise linear activation functions (Botev et al., 2017), thus both methods can be used to directly approximate the diagonal blocks of the Hessian of such networks, as the Gauss-Newton and Fisher are equivalent for networks that parameterise an exponential family log likelihood. + +# 3 A SCALABLE LAPLACE APPROXIMATION FOR NEURAL NETWORKS + +# 3.1 THE LAPLACE APPROXIMATION + +The standard Laplace approximation is obtained by taking the second-order Taylor expansion around a mode of a distribution. For a neural network, such a mode can be found using standard gradientbased methods. Specifically, if we approximate the log posterior over the weights of a network given some data $\mathcal { D }$ around a MAP estimate $\boldsymbol { \theta } ^ { * }$ , we obtain: + +$$ +\log p ( \boldsymbol { \theta } | \mathcal { D } ) \approx \log p ( \boldsymbol { \theta } ^ { * } | \mathcal { D } ) - \frac { 1 } { 2 } \big ( \boldsymbol { \theta } - \boldsymbol { \theta } ^ { * } \big ) ^ { \top } \bar { H } ( \boldsymbol { \theta } - \boldsymbol { \theta } ^ { * } ) +$$ + +where $\theta = [ \mathrm { v e c } ( W _ { 1 } ) , . . . , \mathrm { v e c } ( W _ { L } ) ]$ is the stacked vector of weights and $\bar { H } = \mathbb { E } \left[ H \right]$ the average Hessian of the negative log posterior1. The first order term is missing because we expand the function around a maximum $\boldsymbol { \theta } ^ { * }$ , where the gradient is zero. If we exponentiate this equation, it is easy to notice that the right-hand side is of Gaussian functional form for $\theta$ , thus we obtain a normal distribution by integrating over it. The posterior over the weights is then approximated as Gaussian: + +$$ +\theta \sim \mathcal { N } ( \theta ^ { * } , \bar { H } ^ { - 1 } ) +$$ + +assuming $\bar { H }$ is p.s.d. We can then approximate the posterior mean when predicting on unseen data $D ^ { * }$ by averaging the predictions of $T$ Monte Carlo samples ${ \boldsymbol { \theta } } ^ { ( t ) }$ from the approximate posterior: + +$$ +p ( \mathcal { D } ^ { * } | \mathcal { D } ) = \int p ( \mathcal { D } ^ { * } | \boldsymbol { \theta } ) p ( \boldsymbol { \theta } | \mathcal { D } ) d \boldsymbol { \theta } \approx \frac { 1 } { T } \sum _ { t = 1 } ^ { T } p ( \mathcal { D } ^ { * } | \boldsymbol { \theta } ^ { ( t ) } ) +$$ + +# 3.2 DIAGONAL LAPLACE APPROXIMATION + +Unfortunately, it is not feasible to compute or invert the Hessian matrix w.r.t. all of the weights jointly. An approximation that is easy to compute in modern automatic differentiation frameworks is the diagonal of the Fisher matrix $F$ , which is simply the expectation of the squared gradients: + +$$ +H \approx \mathrm { d i a g } ( F ) = \mathrm { d i a g } ( \mathbb { E } \left[ \nabla _ { \theta } \log p ( y | x ) \nabla _ { \theta } \log p ( y | x ) ^ { \top } \right] ) = \mathrm { d i a g } ( \mathbb { E } \left[ ( \nabla _ { \theta } \log p ( y | x ) ) ^ { 2 } \right] ) +$$ + +where diag extracts the diagonal of a matrix or turns a vector into a diagonal matrix. Such diagonal approximations to the curvature of a neural network have been used successfully for pruning the weights (LeCun et al., 1990) and, more recently, for transfer learning (Kirkpatrick et al., 2017). + +This corresponds to modelling the weights with a Normal distribution with diagonal covariance: + +$$ +\operatorname { v e c } ( W _ { \lambda } ) \sim { \mathcal { N } } ( \operatorname { v e c } ( W _ { \lambda } ^ { * } ) , \operatorname { d i a g } ( F _ { \lambda } ) ^ { - 1 } ) \quad { \mathrm { f o r ~ } } \lambda = 1 , \dots , L +$$ + +Unfortunately, even if the Taylor approximation is accurate, this will place significant probability mass in low probability areas of the true posterior if some weights exhibit high covariance. + +# 3.3 KRONECKER FACTORED LAPLACE APPROXIMATION + +So while it is desirable to model the covariance between the weights, some approximations are needed in order to remain computationally efficient. First, we assume the weights of the different layers to be independent. This corresponds to the block-diagonal approximation in KFAC and KFRA, which empirically preserves sufficient information about the curvature to obtain competitive optimisation performance. For our purposes this means that our posterior factorises over the layers. + +As discussed above, the Hessian of the log-likelihood for a single datapoint is Kronecker factored, and we denote the two factor matrices as ${ \cal H } _ { \lambda } = { \mathcal Q } _ { \lambda } \otimes { \mathcal H } _ { \lambda }$ .2 By further assuming independence between $\mathcal { Q }$ and $\mathcal { H }$ in all layers, we can approximate the expected Hessian of each layer as: + +$$ +\mathbb { E } \left[ H _ { \lambda } \right] = \mathbb { E } \left[ \mathcal { Q } _ { \lambda } \otimes \mathcal { H } _ { \lambda } \right] \approx \mathbb { E } \left[ \mathcal { Q } _ { \lambda } \right] \otimes \mathbb { E } \left[ \mathcal { H } _ { \lambda } \right] +$$ + +Hence, the Hessian of every layer is Kronecker factored over an entire dataset and the Laplace approximation can be approximated by a product of Gaussians. Each Gaussian has a Kronecker factored covariance, corresponding to a matrix normal distribution (Gupta & Nagar, 1999), which considers the two Kronecker factors of the covariance to be the covariances of the rows and columns of a matrix. The two factors are much smaller than the full covariance and allow for significantly more efficient inversion and sampling (we review the matrix normal distribution in Appendix B). + +Our resulting posterior for the weights in layer $\lambda$ is then: + +$$ +W _ { \lambda } \sim \mathcal { M N } ( W _ { \lambda } ^ { \ast } , \bar { \mathcal { Q } } _ { \lambda } ^ { - 1 } , \bar { \mathcal { H } } _ { \lambda } ^ { - 1 } ) +$$ + +In contrast to optimisation methods, we do not need to approximate $\mathbb { E } \left[ \mathcal { H } _ { \lambda } \right]$ as it is only calculated once. However, when it is possible to augment the data (e.g. randomised cropping of images), it may be advantageous. We provide a more detailed discussion of this in Appendix C. + +# 3.4 INCORPORATING THE PRIOR AND REGULARISING THE CURVATURE FACTORS + +Just as the log posterior, the Hessian decomposes into a term depending on the data log likelihood and one on the prior. For the commonly used $L _ { 2 }$ -regularisation, corresponding to a Gaussian prior, the Hessian is equal to the precision of the prior times the identity matrix. We approximate this by adding a multiple of the identity to each of the Kronecker factors from the log likelihood: + +$$ +H _ { \lambda } = N \mathbb { E } \left[ - \frac { \partial ^ { 2 } \log p ( \mathcal { D } | \theta ) } { \partial \theta ^ { 2 } } \right] + \tau I \approx ( \sqrt { N } \mathbb { E } \left[ \mathcal { Q } _ { \lambda } \right] + \sqrt { \tau } I ) \otimes ( \sqrt { N } \mathbb { E } \left[ \mathcal { H } _ { \lambda } \right] + \sqrt { \tau } I ) +$$ + +where $\tau$ is the precision of the Gaussian prior on the weights and $N$ the size of the dataset. However, we can also treat them as hyperparameters and optimise them w.r.t. the predictive performance on a validation set. We emphasise that this can be done without retraining the network, so it does not impose a large computational overhead and is trivial to parallelise. + +Setting $N$ to a larger value than the size of the dataset can be interpreted as including duplicates of the data points as pseudo-observations. Adding a multiple of the uncertainty to the precision matrix decreases the uncertainty about each parameter. This has a regularising effect both on our approximation to the true Laplace, which may be overestimating the variance in certain directions due to ignoring the covariances between the layers, as well as the Laplace approximation itself, which may be placing probability mass in low probability areas of the true posterior. + +# 4 RELATED WORK + +Most recent attempts to approximating the posterior of a neural network are based on formulating an approximate distribution to the posterior and optimising the variational lower bound w.r.t. its parameters. (Graves, 2011; Blundell et al., 2015; Kingma et al., 2015) as well as the expectation propagation based approaches of (Hernandez-Lobato & Adams, 2015) and (Ghosh et al., 2016) ´ assume independence between the individual weights which, particularly when optimising the KL divergence, often lets the model underestimate the uncertainty about the weights. Gal & Ghahramani (2016) interpret Dropout to approximate the posterior with a mixture of delta functions, assuming independence between the columns. (Lakshminarayanan et al., 2016) suggest using an ensemble of networks for estimating the uncertainty. + +Our work is a scalable approximation of (MacKay, 1992). Since the per-layer Hessian of a neural network is infeasible to compute, we suggest a factorisation of the covariance into a Kronecker product, leading to a more efficient matrix normal distribution. The posterior that we obtain is reminiscent of (Louizos & Welling, 2016) and (Sun et al., 2017), who optimise the parameters of a matrix normal distribution as their weights, which requires a modification of the training procedure. + +# 5 EXPERIMENTS + +Since the Laplace approximation is a method for predicting in a Bayesian manner and not for training, we focus on comparing to uncertainty estimates obtained from Dropout (Gal & Ghahramani, 2016). The trained networks will be identical, but the prediction methods will differ. We also compare to a diagonal Laplace approximation to highlight the benefit from modelling the covariances between the weights. All experiments are implemented using Theano (Theano Development Team, 2016) and Lasagne (Dieleman et al., 2015).3 + +# 5.1 TOY REGRESSION DATASET + +As a first experiment, we visualise the uncertainty obtained from the Laplace approximations on a toy regression dataset, similar to (Hernandez-Lobato´ $\&$ Adams, 2015). We create a dataset of 20 uniformly distributed points $x \sim \mathcal { U } ( - 4 , 4 )$ and sample $y \sim \mathcal { N } ( x ^ { 3 } , 3 ^ { 2 } )$ . In contrast to (Hernandez- ´ Lobato & Adams, 2015), we use a two-layer network with seven units per layer rather than one layer with 100 units. This is because both the input and output are one-dimensional, hence the weight matrices are vectors and the matrix normal distribution reduces to a multivariate normal distribution. Furthermore, the Laplace approximation is sensitive to the ratio of the number of data points to parameters, and we want to visualise it both with and without hyperparameter tuning. + +![](images/debac792cdab601dfb06184753b4dcbf42158c1124917b3079299c753416bc64.jpg) +Figure 1: Toy regression uncertainty. Black dots are data points, the black line shows the noiseless function. The red line shows the deterministic prediction of the network, the blue line the mean output. Each shade of blue visualises one additional standard deviation. Best viewed on screen. + +Fig. 1 shows the uncertainty obtained from the Kronecker factored and diagonal Laplace approximation applied to the same network, as well as from a full Laplace approximation and $5 0 , 0 0 0 \mathrm { H M C }$ (Neal, 1993) samples. The latter two methods are feasible only for such a small model and dataset. For the diagonal and full Laplace approximation we use the Fisher identity and draw one sample per data point. We set the hyperparameters of the Laplace approximations (see Section 3.4) using a grid search over the likelihood of 20 validation points that are sampled the same way as the training set. + +The regularised Laplace approximations all give an overall good fit to the HMC predictive posterior. Their uncertainty is slightly higher close to the training data and increases more slowly away from the data than that of the HMC posterior. The diagonal and full Laplace approximation require stronger regularisation than our Kronecker factored one, as they have higher uncertainty when not regularised. In particular the full Laplace approximation vastly overestimates the uncertainty without additional regularisation, leading to a bad predictive mean (see Appendix E for the corresponding figures), as the Hessian of the log likelihood is underdetermined. This is commonly the case in deep learning, as the number of parameters is typically much larger than the number of data points. Hence restricting the structure of the covariance is not only a computational necessity for most architectures, but also allows for more precise estimation of the approximate covariance. + +# 5.2 OUT-OF-DISTRIBUTION UNCERTAINTY + +For a more realistic test, similar to (Louizos & Welling, 2017), we assess the uncertainty of the predictions when classifying data from a different distribution than the training data. For this we train a network with two layers of 1024 hidden units and ReLU transfer functions to classify MNIST digits. We use a learning rate of $1 0 ^ { - 2 }$ and momentum of 0.9 for 250 epochs. We apply Dropout with $p { = } 0 . 5$ after each inner layer, as our chief interest is to compare against its uncertainty estimates. We further use $L _ { 2 }$ -regularisation with a factor of $1 0 ^ { - 2 }$ and randomly binarise the images during training according to their pixel intensities and draw $1 , 0 0 0$ such samples per datapoint for estimating the curvature factors. We use this network to classify the images in the notMNIST dataset4, which contains $2 8 \times 2 8$ grey-scale images of the letters $\mathbf { \dot { A } } ^ { \prime }$ to $\mathbf { \hat { J } } ^ { \star }$ from various computer fonts, i.e. not digits. An ideal classifier would make uniform predictions over its classes. + +We compare the uncertainty obtained by predicting the digit class of the notMNIST images using 1. a deterministic forward pass through the Dropout trained network, 2. by sampling different Dropout masks and averaging the predictions, and by sampling different weight matrices from 3. the matrix normal distribution obtained from our Kronecker factored Laplace approximation as well as 4. the diagonal one. As an additional baseline similar to (Blundell et al., 2015; Graves, 2011), we compare to a network with identical architecture with a fully factorised Gaussian (FFG) approximate posterior on the weights and a standard normal prior. We train the model on the variational lower bound using the reparametrisation trick (Kingma & Welling, 2013). We use 100 samples for the stochastic forward passes and optimise the hyperparameters of the Laplace approximations w.r.t. the cross-entropy on the validation set of MNIST. + +![](images/d894a7edf56530a450cd12d773fa589c25cc119d5b25ad9f5c7d7640b554c41f.jpg) +Figure 2: Predictive entropy on notMNIST obtained from different methods for the forward pass on a network trained on MNIST. + +We measure the uncertainty of the different methods as the entropy of the predictive distribution, which has a minimal value of 0 when a single class is predicted with certainty and a maximum of about 2.3 for uniform predictions. Fig. 2 shows the inverse empirical cumulative distribution of the entropy values obtained from the four methods. Consistent with the results in (Gal & Ghahramani, 2016), averaging the probabilities of multiple passes through the network yields predictions with higher uncertainty than a deterministic pass that approximates the geometric average (Srivastava et al., 2014). However, there still are some images that are predicted to be a digit with certainty. Our Kronecker factored Laplace approximation makes hardly any predictions with absolute certainty and assigns high uncertainty to most of the letters as desired. The diagonal Laplace approximation required stronger regularisation towards predicting deterministically, yet it performs similarly to Dropout. As shown in Table 1, however, the network makes predictions on the test set of MNIST with similar accuracy to the deterministic forward pass and MC Dropout when using our approximation. The variational factorised Gaussian posterior has low uncertainty as expected. + +# 5.3 ADVERSARIAL EXAMPLES + +![](images/ed41abe45c3623949c3b8185e71c83cf08c0f3240991f92ab6369cada44c3cfe.jpg) +Figure 3: Untargeted adversarial attack. + +![](images/bd7abf0110987c858acf90a823a00cb267f5508f27375bc544386b20b41304ac.jpg) +Figure 4: Targeted adversarial attack. + +To further test the robustness of our prediction method close to the data distribution, we perform an adversarial attack on a neural network. As first demonstrated in (Szegedy et al., 2013), neural networks are prone to being fooled by gradient-based changes to their inputs. Li & Gal (2017) suggest, and provide empirical support, that Bayesian models may be more robust to such attacks, since they implicitly form an infinitely large ensemble by integrating over the model parameters. For our experiments, we use the fully connected net trained on MNIST from the previous section and compare the sensitivity of the different prediction methods for two kinds of adversarial attacks. + +First, we use the untargeted Fast Gradient Sign method $\begin{array} { r } { x _ { a d v } = x - \eta \mathrm { s g n } ( \nabla _ { x } \operatorname* { m a x } _ { y } \log p ^ { ( M ) } ( y | x ) ) } \end{array}$ suggested in (Goodfellow et al., 2014), which takes the gradient of the class predicted with maximal probability by method $M$ w.r.t. the input $x$ and reduces this probability with varying step size $\eta$ This step size is rescaled by the difference between the maximal and minimal value per dimension in the dataset. It is to be expected that this method generates examples away from the data manifold, as there is no clear subset of the data that corresponds to e.g. ”not ones”. + +Fig. 3 shows the average predictive uncertainty and the accuracy on the original class on the MNIST test set as the step size $\eta$ increases. The Kronecker factored Laplace approximation achieves significantly higher uncertainty than any other prediction method as the images move away from the data. Both the diagonal and the Kronecker factored Laplace maintain higher accuracy than MC Dropout on their original predictions. Interestingly, the deterministic forward pass appears to be most robust in terms of accuracy, however it has much smaller uncertainty on the predictions it makes and will confidently predict a false class for most images, whereas the other methods are more uncertain. + +Furthermore, we perform a targeted attack that attempts to force the network to predict a specific class, in our case $\cdot _ { 0 } \cdot \mathrm { \ }$ following (Li & Gal, 2017). Hence, for each method, we exclude all data points in the test set that are already predicted as $\cdot _ { 0 } \cdot \mathrm { \ }$ . The updates are of similar form to the untargeted attack, however they increase the probability of the pre-specified class $y$ rather than decreasing the current maximum as x(ty $x _ { y } ^ { ( t + 1 ) } = x _ { y } ^ { ( \bar { t } ) } + \eta \mathrm { s g n } \big ( \bar { \nabla } _ { x } \log \bar { p ^ { ( M ) } } ( y | x _ { y } ^ { ( t ) } ) \big )$ , where $x _ { y } ^ { ( 0 ) } = x$ . + +We use a step size of $\eta { = } 1 0 ^ { - 2 }$ for the targeted attack. The uncertainty and accuracy on the original and target class are shown in Fig. 4. Here, the Kronecker factored Laplace approximation has slightly smaller uncertainty at its peak in comparison to the other methods, however it appears to be much more robust. It only misclassifies over $5 0 \%$ of the images after about 20 steps, whereas for the other methods this is the case after roughly 10 steps and reaches $1 0 0 \%$ accuracy on the target class after almost 50 updates, whereas the other methods are fooled on all images after about 25 steps. + +In conjunction with the experiment on notMNIST, it appears that the Laplace approximation achieves higher uncertainty than Dropout away from the data, as in the untargeted attack. In the targeted attack it exhibits smaller uncertainty than Dropout, yet it is more robust to having its prediction changed. The diagonal Laplace approximation again performs similarly to Dropout. + +# 5.4 UNCERTAINTY ON MISCLASSIFICATIONS + +To highlight the scalability of our method, we apply it to a state-of-the-art convolutional network architecture. Recently, deep residual networks (He et al., 2016a;b) have been the most successful ones among those. As demonstrated in (Grosse & Martens, 2016), Kronecker factored curvature methods are applicable to convolutional layers by interpreting them as matrix-matrix multiplications. + +![](images/5ed41153d9002457c121cab0e2d0510ad1bedc3708935fa9b5bfa70ad80fe8a9.jpg) +Figure 5: Inverse ecdf of the predictive entropy from Wide Residual Networks trained with and without Dropout on CIFAR100. For misclassifications, curves on top corresponding to higher uncertainty are desirable, and curves on the bottom for correct classifications. + +We compare our uncertainty estimates on wide residual networks (Zagoruyko & Komodakis, 2016), a recent variation that achieved competitive performance on CIFAR100 (Krizhevsky & Hinton, 2009) while, in contrast to most other residual architectures, including Dropout at specific points. While this does not correspond to using Dropout in the Bayesian sense (Gal & Ghahramani, 2015), it allows us to at least compare our method to the uncertainty estimates obtained from Dropout. + +We note that it is straightforward to incorporate batch normalisation (Ioffe & Szegedy, 2015) into the curvature backpropagation algorithms, so we apply a standard Laplace approximation to its parameters as well. We are not aware of any interpretation of Dropout as performing Bayesian inference on the parameters of batch normalisation. Further implementation details are in Appendix G. + +Again, the accuracy of the prediction methods is comparable (see Table 2 in Appendix F). For calculating the curvature factors, we draw 5, 000 samples per image using the same data augmentation as during training, effectively increasing the dataset size to $2 . 5 \times 1 0 ^ { 8 }$ . The diagonal approximation had to be regularised to the extent of becoming deterministic, so we omit it from the results. + +In Fig. 5 we compare the distribution of the predictive uncertainty on the test set.5 We distinguish between the uncertainty on correct and incorrect classifications, as the mistakes of a system used in practice may be less severe if the network can at least indicate that it is uncertain. Thus, high uncertainty on misclassifications and low uncertainty on correct ones would be desirable, such that a system could return control to a human expert when it can not make a confident decision. In general, the network tends to be more uncertain on its misclassifcations than its correct ones regardless of whether it was trained with or without Dropout and of the method used for prediction. Both Dropout and the Laplace approximation similarly increase the uncertainty in the predictions, however this is irrespective of the correctness of the classification. Yet, our experiments show that the Kronecker factored Laplace approximation can be scaled to modern convolutional networks and maintain good classification accuracy while having similar uncertainty about the predictions as Dropout. + +We had to use much stronger regularisation for the Laplace approximation on the wide residual network, possibly because the block-diagonal approximation becomes more inaccurate on deep networks, possibly because the number of parameters is much higher relative to the number of data. It would be interesting to see how the Laplace approximations behaves on a much larger dataset like ImageNet for similarly sized networks, where we have a better ratio of data to parameters and curvature directions. However, even on a relatively small dataset like CIFAR we did not have to regularise the Laplace approximation to the degree of the posterior becoming deterministic. + +# 6 CONCLUSION + +We presented a scalable approximation to the Laplace approximation for the posterior of a neural network and provided experimental results suggesting that the uncertainty estimates are on par with current alternatives like Dropout, if not better. It enables practitioners to obtain principled uncertainty estimates from their models, even if they were trained in a maximum likelihood/MAP setting. + +There are many possible extensions to this work. One would be to automatically determine the scale and regularisation hyperparameters of the Kronecker factored Laplace approximation using the model evidence similar to how (MacKay, 1992) interpolates between the data log likelihood and the width of the prior. The model evidence could further be used to perform Bayesian model averaging on ensembles of neural networks, potentially improving their generalisation ability and uncertainty estimates. A challenging application would be active learning, where only little data is available relative to the number of curvature directions that need to be estimated. + +# ACKNOWLEDGEMENTS + +This work was supported by the Alan Turing Institute under the EPSRC grant EP/N510129/1. We thank the anonymous reviewers for their feedback and Harshil Shah for his comments on an earlier draft of this paper. + +# REFERENCES + +Anoop Korattikara Balan, Vivek Rathod, Kevin P Murphy, and Max Welling. Bayesian Dark Knowledge. In Advances in Neural Information Processing Systems, pp. 3438–3446, 2015. +David Barber and Christopher M Bishop. Ensemble Learning for Multi-layer Networks. 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Proceedings of the National Academy of Sciences, pp. 201611835, 2017. + +Alex Krizhevsky and Geoffrey Hinton. Learning Multiple Layers of Features from Tiny Images. 2009. + +Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles. arXiv preprint arXiv:1612.01474, 2016. + +Yann LeCun, John S. Denker, and Sara A. Solla. Optimal Brain Damage. In Advances in Neural Information Processing Systems, pp. 598–605, 1990. + +Yingzhen Li and Yarin Gal. Dropout Inference in Bayesian Neural Networks with Alphadivergences. arXiv preprint arXiv:1703.02914, 2017. + +Christos Louizos and Max Welling. Structured and Efficient Variational Deep Learning with Matrix Gaussian Posteriors. In ICML, pp. 1708–1716, 2016. + +Christos Louizos and Max Welling. Multiplicative Normalizing Flows for Variational Bayesian Neural Networks. In ICML, pp. 2218–2227, 2017. + +David J. C. MacKay. A Practical Bayesian Framework for Backpropagation Networks. Neural Computation, 4(3):448–472, 1992. + +James Martens and Roger Grosse. Optimizing Neural Networks with Kronecker-factored Approximate Curvature. In ICML, pp. 2408–2417, 2015. + +Radford M Neal. Bayesian Learning via Stochastic Dynamics. In Advances in Neural Information Processing Systems, pp. 475–482, 1993. + +Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A Simple Way to Prevent Neural Networks from Overfitting. Journal of Machine Learning Research, 15(1):1929–1958, 2014. + +Shengyang Sun, Changyou Chen, and Lawrence Carin. Learning Structured Weight Uncertainty in Bayesian Neural Networks. In Artificial Intelligence and Statistics, pp. 1283–1292, 2017. + +Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing Properties of Neural Networks. arXiv preprint arXiv:1312.6199, 2013. + +Theano Development Team. Theano: A Python Framework for Fast Computation of Mathematical Expressions. arXiv e-prints, abs/1605.02688, May 2016. + +Sergey Zagoruyko and Nikos Komodakis. Wide Residual Networks. arXiv preprint arXiv:1605.07146, 2016. + +# Appendices + +# A DERIVATION OF THE ACTIVATION HESSIAN RECURSION + +Here, we provide the basic derivation of the factorisation of the diagonal blocks of the Hessian in Eq. 1 and the recursive formula for calculating $\mathcal { H }$ as presented in (Botev et al., 2017). + +The Hessian of a neural network with parameters $\theta$ as defined in the main text has elements: + +$$ +[ H ] _ { i j } = \frac { \partial ^ { 2 } } { \partial \theta _ { i } \partial \theta _ { j } } E ( \theta ) +$$ + +For a given layer $\lambda$ , the gradient w.r.t. a weight $W _ { a , b } ^ { \lambda }$ is: + +$$ +{ \frac { \partial E } { \partial W _ { a , b } ^ { \lambda } } } = \sum _ { i } { \frac { \partial h _ { i } ^ { \lambda } } { \partial W _ { a , b } ^ { \lambda } } } { \frac { \partial E } { \partial h _ { i } ^ { \lambda } } } = a _ { b } ^ { \lambda - 1 } { \frac { \partial E } { \partial h _ { a } ^ { \lambda } } } +$$ + +Keeping $\lambda$ fixed and differentiating again, we find that the per-sample Hessian of that layer is: + +$$ +\left[ H _ { \lambda } \right] _ { ( a , b ) , ( c , d ) } \equiv \frac { \partial ^ { 2 } E } { \partial W _ { a , b } ^ { \lambda } \partial W _ { c , d } ^ { \lambda } } = a _ { b } ^ { \lambda - 1 } a _ { d } ^ { \lambda - 1 } \left[ \mathcal { H } _ { \lambda } \right] _ { a , c } +$$ + +where + +$$ +[ \mathcal { H } _ { \lambda } ] _ { a , b } = \frac { \partial ^ { 2 } E } { \partial h _ { a } ^ { \lambda } \partial h _ { b } ^ { \lambda } } +$$ + +is the pre-activation Hessian. + +We can reexpress this in matrix notation as a Kronecker product as in Eq. 1: + +$$ +H _ { \lambda } = \frac { \partial ^ { 2 } E } { \partial \operatorname { v e c } \left( W _ { \lambda } \right) \partial \operatorname { v e c } \left( W _ { \lambda } \right) } = \left( a _ { \lambda - 1 } a _ { \lambda - 1 } ^ { \mathsf { T } } \right) \otimes \mathcal { H } _ { \lambda } +$$ + +The pre-activation Hessian can be calculated recursively as: + +$$ +\mathcal { H } _ { \lambda } = B _ { \lambda } W _ { \lambda + 1 } ^ { \mathsf { T } } \mathcal { H } _ { \lambda + 1 } W _ { \lambda + 1 } B _ { \lambda } + D _ { \lambda } +$$ + +where the diagonal matrices $B$ and $D$ are defined as: + +$$ +\begin{array} { l } { { B _ { \lambda } = \mathrm { d i a g } \left( \mathbf f _ { \lambda } ^ { \prime } ( h _ { \lambda } ) \right) } } \\ { { D _ { \lambda } = \mathrm { d i a g } \left( \mathbf f _ { \lambda } ^ { \prime \prime } ( h _ { \lambda } ) { \frac { \partial E } { \partial a _ { \lambda } } } \right) } } \end{array} +$$ + +$\mathbf { f ^ { \prime } }$ and $\mathbf { f } ^ { \prime \prime }$ denote the first and second derivative of the transfer function. The recursion is initialised with the Hessian of the error w.r.t. the linear network outputs. + +For further details and on how to calculate the diagonal blocks of the Gauss-Newton and Fisher matrix, we refer the reader to (Botev et al., 2017) and (Martens & Grosse, 2015). + +# B MATRIX NORMAL DISTRIBUTION + +The matrix normal distribution (Gupta & Nagar, 1999) is a multivariate distribution over an entire matrix of shape $n \times p$ rather than just a vector. In contrast to the multivariate normal distribution, it is parameterised by two p.s.d. covariance matrices, $U : n \times n$ and $V : p \times p$ , which indicate the covariance of the rows and columns respectively. In addition it has a mean matrix $M : n \times p$ . + +A vectorised sample from a matrix normal distribution $X \sim { \mathcal { M N } } ( M , U , V )$ corresponds to a sample from a normal distribution $\operatorname { v e c } ( X ) \sim { \mathcal { N } } ( \operatorname { v e c } ( M ) , U \otimes V )$ . However, samples can be drawn more efficiently as $X = M + A Z B$ with $Z \sim \mathcal { M N } ( 0 , I , I )$ , and $A A ^ { \mathsf { T } } = U$ and $B ^ { \intercal } B = V$ . The sample $Z$ corresponds to a sample from a normal distribution of length $n p$ that has been reshaped to a $n \times p$ matrix. This is more efficient in the sense that we only need to calculate two matrix-matrix products of small matrices, rather than a matrix-vector product with one big one. + +# C APPROXIMATION OF THE EXPECTED ACTIVATION HESSIAN + +While the square root of $\mathcal { Q } _ { \lambda }$ is calculated during the forward pass on all layers, $\mathcal { H }$ requires an additional backward pass. Strictly speaking, it is not essential to approximate $\mathbb { E } \left[ \mathcal { H } \right]$ for the Kronecker factored Laplace approximation, as in contrast to optimisation procedures the curvature only needs to be calculated once and is thus not time critical. For datasets of the scale of ImageNet and the networks used for such datasets, it would still be impractically slow to perform the calculation for every data point individually. Furthermore, as most datasets are augmented during training, e.g. random cropping or reflections of images, the curvature of the network can be estimated using the same augmentations, effectively increasing the size of the dataset by orders of magnitude. Thus, we make use of the minibatch approximation in our experiments — as we make use of data augmentation — in order to demonstrate its practical applicability. + +We note that E [H] can be calculated exactly by running KFRA (Botev et al., 2017) with a minibatchsize of one, and then averaging the results. KFAC (Martens & Grosse, 2015), in contrast, stochastically approximates the Fisher matrix, so even when run for every datapoint separately, it cannot calculate the curvature factor exactly. + +In the following, we also show figures for the adversarial experiments in which we calculate the curvature per datapoint and without data augmentation: + +![](images/bd401ad0b8e9d0a6e372927c134409d5b015ccf19eb9e8745ca37ac78fb7f7a9.jpg) +Figure 6: Untargeted adversarial attack for Kronecker factored Laplace approximation with the curvature calculated with and without data augmentation/approximating the activation Hessian. + +Fig. 6 and Fig. 7 show how the Laplace approximation with the curvature estimated from 1000 randomly sampled binary MNIST images and the activation Hessian calculated with a minibatch size of 100 performs in comparison to the curvature factor being calculated without any data augmentation with a batch size of 100 or exactly. We note that without data augmentation we had to use much stronger regularisation of the curvature factors, in particular we had to add a non-negligible multiple of the identity to the factors, whereas with data augmentation it was only needed to ensure that the matrices are invertible. The Kronecker factored Laplace approximation reaches particularly high uncertainty on the untargeted adversarial attack and is most robust on the targeted attack when using data augmentation, suggesting that it is particularly well suited for large datasets and ones where some form of data augmentation can be applied. The difference between approximating the activation Hessian over a minibatch and calculating it exactly appears to be negligible. + +![](images/e31a6b6b0d0ab2399091e0aa6f1e8ffc0b21f98d8546841f2ad9096b7c2ec352.jpg) +Figure 7: Targeted adversarial attack for Kronecker factored Laplace approximation with the curvature calculated with and without data augmentation/approximating the activation Hessian. + +# D MEMORY AND COMPUTATIONAL REQUIREMENTS + +If we denote the dimensionality of the input to layer $\lambda$ as $D _ { \lambda - 1 }$ and its output as $D _ { \lambda }$ , the curvature factors correspond to the two precision matrices with Dλ−1(Dλ−1+1)2 and $\frac { D _ { \lambda } ( D _ { \lambda } + 1 ) } { 2 }$ ‘parameters’ to estimate, since they are symmetric. So across a network, the number of curvature directions that we are estimating grows linearly in the number of layers and quadratically in the dimension of the layers, i.e. the number of columns of the weight matrices. The size of the full Hessian, on the other hand, grows quadratically in the number of layers and with the fourth power in the dimensionality of the layers (assuming they are all the same size). + +Once the curvature factors are calculated, which only needs to be done once, we use their Cholesky decomposition to solve two triangular linear systems when sampling weights from the matrix normal distribution. We use the same weight samples for each minibatch, i.e. we do not sample a weight matrix per datapoint. This is for computational efficiency and does not change the expectation. + +One possibility to save computation time would be to sample a fixed set of weight matrices from the approximate posterior — in order to avoid solving the linear system on every forward pass — and treat the networks that they define as an ensemble. The individual ensemble members can be evaluated in parallel and their outputs averaged, which can be done with a small overhead over evaluating a single network given sufficient compute resources. A further speed up can be achieved by distilling the predictive distributions of the Laplace network into a smaller, deterministic feedforward network as successfully demonstrated in (Balan et al., 2015) for posterior samples using HMC. + +# E COMPLEMENTARY FIGURES FOR THE TOY DATASET + +Fig. 8 shows the different Laplace approximations (Kronecker factored, diagonal, full) from the main text without any hyperparameter tuning. The figure of the uncertainty obtained from samples using HMC is repeated. Note that the scale is larger than in the main text due to the high uncertainty of the Laplace approximations. + +The Laplace approximations are increasingly uncertain away from the data, as the true posterior estimated from HMC samples, however they all overestimate the uncertainty without regularisation. This is easy to fix by optimising the hyperparameters on a validation set as discussed in the main text, resulting in posterior uncertainty much more similar to the true posterior. As previously discussed in (Botev et al., 2017), the Hessian of a neural network is usually underdetermined as the number of data points is much smaller than the number of parameters — in our case we have 20 data points to estimate a $7 8 \times 7 8$ precision matrix. This leads to the full Laplace approximation vastly overestimating the uncertainty and a bad predictive mean. Both the Kronecker factored and the diagonal approximation exhibit smaller variance than the full Laplace approximation as they restrict the structure of the precision matrix. Consistently with the other experiments, we find the diagonal + +![](images/8728ab113d8def68159f046cb816ff9c7101a0cf17ad7545a887f6a29fd79c4f.jpg) +Figure 8: Toy regression uncertainty. Black dots are data points, the black line shows the underlying noiseless function. The red line shows the deterministic prediction of the trained network, the blue line the mean output. Each shade of blue visualises one additional standard deviation. + +Laplace approximation to place more mass in low probability areas of the posterior than the Kronecker factored approximation, resulting in higher variance on the regression problem. This leads to a need for greater regularisation of the diagonal approximation to obtain acceptable predictive performance, and underestimating the uncertainty. + +# F PREDICTION ACCURACY + +This section shows the accuracy values obtained from the different predictions methods on the feedforward networks for MNIST and the wide residual network for CIFAR100. The results for MNIST are shown in Table 1 and the results for CIFAR in Table 2. + +In all cases, neither MC Dropout nor the Laplace approximation significantly change the classification accuracy of the network in comparison to a deterministic forward pass. + +Table 1: Test accuracy of the feedforward network trained on MNIST + +
Prediction MethodAccuracy
FFG Deterministic98.88% 98.86%
MC Dropout Diagonal Laplace98.85% 98.85%
KF Laplace98.80%
+ +Table 2: Accuracy on the final 5, 000 CIFAR100 test images for a wide residual network trained with and without Dropout. + +
Accuracy
Prediction MethodDropoutDeterministic
Deterministic79.12%79.18%
MC Dropout79.20%1
KF Laplace79.10%79.36%
+ +# G IMPLEMENTATION DETAILS FOR RESIDUAL NETWORKS + +Our wide residual network has $n { = } 3$ block repetitions and a width factor of $k { = } 8$ on CIFAR100 with and without Dropout using hyperparameters taken from (Zagoruyko & Komodakis, 2016): the network parameters are trained on a cross-entropy loss using Nesterov momentum with an initial learning rate of 0.1 and momentum of 0.9 for 200 epochs with a minibatch size of 128. We decay the learning rate every 50 epochs by a factor of 0.2, which is slightly different to the schedule used in (Zagoruyko & Komodakis, 2016) (they decay after 60, 120 and 160 epochs). As the original authors, we use $L _ { 2 }$ -regularisation with a factor of $5 \times 1 0 ^ { - 4 }$ . + +We make one small modification to the architecture: instead of downsampling with $1 \times 1$ convolutions with stride 2, we use $2 \times 2$ convolutions. This is due to Theano not supporting the transformation of images into the patches extracted by a convolution for $1 \times 1$ convolutions with stride greater than 1, which we require for our curvature backpropagation through convolutions. + +We apply a standard Laplace approximation to the batch normalisation parameters — a Kronecker factorisation is not needed, since the parameters are one-dimensional. When calculating the curvature factors, we use the moving averages for the per-layer means and standard deviations obtained after training, in order to maintain independence between the data points in a minibatch. + +We need to make a further approximation to the ones discussed in Section 2.2 when backpropagating the curvature for residual networks. The residual blocks compute a function of the form $r e s ( x ) =$ $x + f _ { \phi } ( x )$ , where $f _ { \phi }$ typically is a sequence of convolutions, batch normalisation and elementwise nonlinearities. This means that we would need to pass back two curvature matrices, one for each summand. However, this would double the number of backpropagated matrices for each residual connection, hence the computation time/memory requirements would grow exponentially in the number of residual blocks. Therefore, we simply add the curvature matrices after each residual connection. \ No newline at end of file diff --git a/parse/train/Skdvd2xAZ/Skdvd2xAZ_content_list.json b/parse/train/Skdvd2xAZ/Skdvd2xAZ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6af7c8c9db51a432415da57e00fe94f2385a27bc --- /dev/null +++ b/parse/train/Skdvd2xAZ/Skdvd2xAZ_content_list.json @@ -0,0 +1,1919 @@ +[ + { + "type": "text", + "text": "A SCALABLE LAPLACE APPROXIMATION FOR NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 99, + 666, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hippolyt Ritter1∗, Aleksandar Botev1, David Barber1 2 1University College London 2Alan Turing Institute ", + "bbox": [ + 184, + 166, + 562, + 200 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 237, + 544, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We leverage recent insights from second-order optimisation for neural networks to construct a Kronecker factored Laplace approximation to the posterior over the weights of a trained network. Our approximation requires no modification of the training procedure, enabling practitioners to estimate the uncertainty of their models currently used in production without having to retrain them. We extensively compare our method to using Dropout and a diagonal Laplace approximation for estimating the uncertainty of a network. We demonstrate that our Kronecker factored method leads to better uncertainty estimates on out-of-distribution data and is more robust to simple adversarial attacks. Our approach only requires calculating two square curvature factor matrices for each layer. Their size is equal to the respective square of the input and output size of the layer, making the method efficient both computationally and in terms of memory usage. We illustrate its scalability by applying it to a state-of-the-art convolutional network architecture. ", + "bbox": [ + 233, + 267, + 764, + 448 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 473, + 336, + 489 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neural networks are most commonly trained in a maximum a posteriori (MAP) setting, which only yields point estimates of the parameters, ignoring any uncertainty about them. This often leads to overconfident predictions, especially in regimes that are weakly covered by training data or far away from the data manifold. While the confidence of wrong predictions is usually irrelevant in a research context, it is essential that a Machine Learning algorithm knows when it does not know in the real world, as the consequences of mistakes can be fatal, be it when driving a car or diagnosing a disease. ", + "bbox": [ + 174, + 505, + 823, + 588 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The Bayesian framework of statistics provides a principled way for avoiding overconfidence in the parameters by treating them as unknown quantities and integrating over all possible values. Specifically, for the prediction of new data under a model, it fits a posterior distribution over the parameters given the training data and weighs the contribution of each setting of the parameters to the prediction by the probability of the data under those parameters times their prior probability. However, the posterior of neural networks is usually intractable due to their size and nonlinearity. ", + "bbox": [ + 174, + 594, + 823, + 678 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There has been previous interest in integrating neural networks into the Bayesian framework (MacKay, 1992; Hinton & Van Camp, 1993; Neal, 1993; Barber & Bishop, 1998), however these approaches were designed for small networks by current standards. Recent adaptations to architectures of modern scale rely on crude approximations of the posterior to become tractable. All of (Graves, 2011; Hernandez-Lobato & Adams, 2015; Blundell et al., 2015) assume independence between the ´ individual weights. While they achieve good results on small datasets, this strong restriction of the posterior is susceptible to underestimating the uncertainty, in particular when optimising the variational bound. The approach in (Gal & Ghahramani, 2016) requires the use of certain stochastic regularisers which are not commonly present in most recent architectures. Furthermore, it is not clear if the approximate posterior defined by these regularisers is a good fit to the true posterior. ", + "bbox": [ + 174, + 685, + 825, + 824 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent work on second-order optimisation of neural networks (Martens & Grosse, 2015; Botev et al., 2017) has demonstrated that the diagonal blocks of the curvature can be well approximated by a Kronecker product. We combine this insight with the idea of modelling the posterior over the weights as a Gaussian, using a Laplace approximation (MacKay, 1992) with Kronecker factored covariance matrices. This leads to a computationally efficient matrix normal posterior distribution (Gupta & Nagar, 1999) over the weights of every layer. Since the Laplace approximation is applied after training, our approach can be used to obtain uncertainty estimates from existing networks. ", + "bbox": [ + 174, + 832, + 823, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 THE CURVATURE OF NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 152, + 555, + 169 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our method is inspired by recent Kronecker factored approximations of the curvature of a neural network (Martens & Grosse, 2015; Botev et al., 2017) for optimisation and we give a high-level review of these in the following. While the two methods approximate the Gauss-Newton and Fisher matrix respectively, as they are guaranteed to be positive semi-definite (p.s.d.), we base all of our discussion on the Hessian in order to be as general as possible. ", + "bbox": [ + 174, + 184, + 825, + 255 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 NEURAL NETWORK NOTATION ", + "text_level": 1, + "bbox": [ + 176, + 272, + 429, + 286 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We denote a feedforward network as taking an input $a _ { 0 } = x$ and producing an output $h _ { L }$ . The intermediate representations for layers $\\lambda = 1 , . . . , L$ are denoted as $h _ { \\lambda } = W _ { \\lambda } a _ { \\lambda - 1 }$ and $a _ { \\lambda } = f _ { \\lambda } ( h _ { \\lambda } )$ . We refer to $a _ { \\lambda }$ as the activations, and $h _ { \\lambda }$ as the (linear) pre-activations. The bias terms are absorbed into the $W _ { \\lambda }$ by appending a 1 to each $a _ { \\lambda }$ . The network parameters are optimised w.r.t. an error function $E ( y , h _ { L } )$ for targets $y$ . Most commonly used error functions, such as squared error and categorical cross-entropy, can be interpreted as exponential family negative log likelihoods $- \\log p ( y | h _ { L } )$ . ", + "bbox": [ + 173, + 297, + 825, + 382 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 KRONECKER FACTORED SECOND-ORDER OPTIMISATION ", + "text_level": 1, + "bbox": [ + 174, + 398, + 609, + 414 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Traditional second-order methods use either the Hessian matrix or a positive semi-definite approximation thereof to generate parameter updates of the form $\\Delta = C ^ { - 1 } g$ , where $C$ is the chosen curvature matrix and $g$ the gradient of the error function parameterised by the network. However, this curvature matrix is infeasbile to compute for modern neural networks as their number of parameters is often in the millions, rendering the size of $C$ of the order of several terabytes. ", + "bbox": [ + 173, + 424, + 825, + 497 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Recent work (Martens & Grosse, 2015; Botev et al., 2017) exploits that, for a single data point, the diagonal blocks of these curvature matrices are Kronecker factored: ", + "bbox": [ + 171, + 503, + 823, + 532 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/b8521042e25f111bdaac593dbe6b67fc61fa70b6a3348105b00e1d36391546d9.jpg", + "text": "$$\nH _ { \\lambda } = \\frac { \\partial ^ { 2 } E } { \\partial \\operatorname { v e c } ( W _ { \\lambda } ) \\partial \\operatorname { v e c } ( W _ { \\lambda } ) } = \\mathcal { Q } _ { \\lambda } \\otimes \\mathcal { H } _ { \\lambda }\n$$", + "text_format": "latex", + "bbox": [ + 357, + 549, + 638, + 587 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $H _ { \\lambda }$ is the Hessian w.r.t. the weights in layer $\\lambda$ . $\\mathcal { Q } _ { \\lambda } = a _ { \\lambda - 1 } a _ { \\lambda - 1 } ^ { \\top }$ denotes the covariance of the incoming activations aλ−1 and Hλ = ∂ E∂h ∂h the pre-activation Hessian, i.e. the Hessian of the error w.r.t. the linear pre-activations $h _ { \\lambda }$ in a layer. We provide the derivation for this result as well as the recursion for calculating $\\mathcal { H }$ in Appendix A. ", + "bbox": [ + 174, + 598, + 825, + 665 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The Kronecker factorisation holds two key advantages: the matrices that need be computed and stored are much smaller — if we assume all layers to be of dimensionality $D$ , the two factors are each of size $D ^ { 2 }$ , whereas the full Hessian for the weights of only one layer would have $D ^ { 4 }$ elements. Furthermore, the inverse of a Kronecker product is equal to the Kronecker product of the inverses, so it is only necessary to invert those two moderately sized matrices. ", + "bbox": [ + 174, + 670, + 825, + 742 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In order to maintain this structure over a minibatch of data, all Kronecker factored second-order methods make two core approximations: First, they only model the diagonal blocks corresponding to the weights of a layer, such that the curvature decomposes into $L$ independent matrices. Second, they assume $\\mathcal { Q } _ { \\lambda }$ and $\\mathcal { H } _ { \\lambda }$ to be independent. This is in order to maintain the Kronecker factorisation in expectation, i.e. $\\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\otimes \\mathcal { H } _ { \\lambda } \\right] \\approx \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\right] \\otimes \\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]$ , since the expectation of a Kronecker product is not guaranteed to be Kronecker factored itself. ", + "bbox": [ + 173, + 750, + 825, + 833 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main difference between the Kronecker factored second-order optimisers lies in how they efficiently approximate $\\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]$ . For exact calculation, it would be necessary to pass back an entire matrix per data point in a minibatch, which imposes infeasible memory and computational requirements. KFRA (Botev et al., 2017) simply passes back the expectation at every layer, while KFAC (Martens & Grosse, 2015) utilises the Fisher identity to only propagate a vector rather than a matrix, approximating the Kronecker factors with a stochastic rank-one matrix for each data point. ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The diagonal blocks of the Hessian and Gauss-Newton matrix are equal for neural networks with piecewise linear activation functions (Botev et al., 2017), thus both methods can be used to directly approximate the diagonal blocks of the Hessian of such networks, as the Gauss-Newton and Fisher are equivalent for networks that parameterise an exponential family log likelihood. ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 A SCALABLE LAPLACE APPROXIMATION FOR NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 171, + 183, + 764, + 199 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 THE LAPLACE APPROXIMATION ", + "text_level": 1, + "bbox": [ + 176, + 214, + 436, + 229 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The standard Laplace approximation is obtained by taking the second-order Taylor expansion around a mode of a distribution. For a neural network, such a mode can be found using standard gradientbased methods. Specifically, if we approximate the log posterior over the weights of a network given some data $\\mathcal { D }$ around a MAP estimate $\\boldsymbol { \\theta } ^ { * }$ , we obtain: ", + "bbox": [ + 173, + 241, + 825, + 299 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/333314683c0abf6a5a733994f08d5666621f38a95390596a723f4505fce14fb6.jpg", + "text": "$$\n\\log p ( \\boldsymbol { \\theta } | \\mathcal { D } ) \\approx \\log p ( \\boldsymbol { \\theta } ^ { * } | \\mathcal { D } ) - \\frac { 1 } { 2 } \\big ( \\boldsymbol { \\theta } - \\boldsymbol { \\theta } ^ { * } \\big ) ^ { \\top } \\bar { H } ( \\boldsymbol { \\theta } - \\boldsymbol { \\theta } ^ { * } )\n$$", + "text_format": "latex", + "bbox": [ + 328, + 318, + 671, + 348 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\theta = [ \\mathrm { v e c } ( W _ { 1 } ) , . . . , \\mathrm { v e c } ( W _ { L } ) ]$ is the stacked vector of weights and $\\bar { H } = \\mathbb { E } \\left[ H \\right]$ the average Hessian of the negative log posterior1. The first order term is missing because we expand the function around a maximum $\\boldsymbol { \\theta } ^ { * }$ , where the gradient is zero. If we exponentiate this equation, it is easy to notice that the right-hand side is of Gaussian functional form for $\\theta$ , thus we obtain a normal distribution by integrating over it. The posterior over the weights is then approximated as Gaussian: ", + "bbox": [ + 173, + 359, + 825, + 435 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/cf1e95882914704d05c830b48d2f12d37440a3d90aa302355aba74889230764b.jpg", + "text": "$$\n\\theta \\sim \\mathcal { N } ( \\theta ^ { * } , \\bar { H } ^ { - 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 441, + 454, + 558, + 474 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "assuming $\\bar { H }$ is p.s.d. We can then approximate the posterior mean when predicting on unseen data $D ^ { * }$ by averaging the predictions of $T$ Monte Carlo samples ${ \\boldsymbol { \\theta } } ^ { ( t ) }$ from the approximate posterior: ", + "bbox": [ + 173, + 486, + 828, + 520 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/7f13d48c5e36c255d6f912a3e1f350bef6320aee46d47d9383bec0c07e6aafda.jpg", + "text": "$$\np ( \\mathcal { D } ^ { * } | \\mathcal { D } ) = \\int p ( \\mathcal { D } ^ { * } | \\boldsymbol { \\theta } ) p ( \\boldsymbol { \\theta } | \\mathcal { D } ) d \\boldsymbol { \\theta } \\approx \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } p ( \\mathcal { D } ^ { * } | \\boldsymbol { \\theta } ^ { ( t ) } )\n$$", + "text_format": "latex", + "bbox": [ + 320, + 537, + 678, + 582 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 DIAGONAL LAPLACE APPROXIMATION ", + "text_level": 1, + "bbox": [ + 176, + 598, + 483, + 613 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Unfortunately, it is not feasible to compute or invert the Hessian matrix w.r.t. all of the weights jointly. An approximation that is easy to compute in modern automatic differentiation frameworks is the diagonal of the Fisher matrix $F$ , which is simply the expectation of the squared gradients: ", + "bbox": [ + 174, + 625, + 826, + 667 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/67d188bafccb6d7e7a78a805a8871adc852fc39c49b2f49bd9a5d47743c3ee5d.jpg", + "text": "$$\nH \\approx \\mathrm { d i a g } ( F ) = \\mathrm { d i a g } ( \\mathbb { E } \\left[ \\nabla _ { \\theta } \\log p ( y | x ) \\nabla _ { \\theta } \\log p ( y | x ) ^ { \\top } \\right] ) = \\mathrm { d i a g } ( \\mathbb { E } \\left[ ( \\nabla _ { \\theta } \\log p ( y | x ) ) ^ { 2 } \\right] )\n$$", + "text_format": "latex", + "bbox": [ + 196, + 694, + 781, + 722 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where diag extracts the diagonal of a matrix or turns a vector into a diagonal matrix. Such diagonal approximations to the curvature of a neural network have been used successfully for pruning the weights (LeCun et al., 1990) and, more recently, for transfer learning (Kirkpatrick et al., 2017). ", + "bbox": [ + 174, + 737, + 825, + 780 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This corresponds to modelling the weights with a Normal distribution with diagonal covariance: ", + "bbox": [ + 171, + 785, + 803, + 801 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d202f8c5ff2b2edab4d5ba7d43596938a2eb555b65c7068640ed3f9320b4c52f.jpg", + "text": "$$\n\\operatorname { v e c } ( W _ { \\lambda } ) \\sim { \\mathcal { N } } ( \\operatorname { v e c } ( W _ { \\lambda } ^ { * } ) , \\operatorname { d i a g } ( F _ { \\lambda } ) ^ { - 1 } ) \\quad { \\mathrm { f o r ~ } } \\lambda = 1 , \\dots , L\n$$", + "text_format": "latex", + "bbox": [ + 305, + 820, + 694, + 840 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Unfortunately, even if the Taylor approximation is accurate, this will place significant probability mass in low probability areas of the true posterior if some weights exhibit high covariance. ", + "bbox": [ + 171, + 853, + 823, + 883 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.3 KRONECKER FACTORED LAPLACE APPROXIMATION ", + "text_level": 1, + "bbox": [ + 173, + 103, + 575, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "So while it is desirable to model the covariance between the weights, some approximations are needed in order to remain computationally efficient. First, we assume the weights of the different layers to be independent. This corresponds to the block-diagonal approximation in KFAC and KFRA, which empirically preserves sufficient information about the curvature to obtain competitive optimisation performance. For our purposes this means that our posterior factorises over the layers. ", + "bbox": [ + 174, + 128, + 825, + 200 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As discussed above, the Hessian of the log-likelihood for a single datapoint is Kronecker factored, and we denote the two factor matrices as ${ \\cal H } _ { \\lambda } = { \\mathcal Q } _ { \\lambda } \\otimes { \\mathcal H } _ { \\lambda }$ .2 By further assuming independence between $\\mathcal { Q }$ and $\\mathcal { H }$ in all layers, we can approximate the expected Hessian of each layer as: ", + "bbox": [ + 174, + 205, + 825, + 252 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d4e47d88af0576da23b9a0a7ce485ed4397c8ae1f2380f8a4bc9dfa977f43fd8.jpg", + "text": "$$\n\\mathbb { E } \\left[ H _ { \\lambda } \\right] = \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\otimes \\mathcal { H } _ { \\lambda } \\right] \\approx \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\right] \\otimes \\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 354, + 270, + 645, + 287 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Hence, the Hessian of every layer is Kronecker factored over an entire dataset and the Laplace approximation can be approximated by a product of Gaussians. Each Gaussian has a Kronecker factored covariance, corresponding to a matrix normal distribution (Gupta & Nagar, 1999), which considers the two Kronecker factors of the covariance to be the covariances of the rows and columns of a matrix. The two factors are much smaller than the full covariance and allow for significantly more efficient inversion and sampling (we review the matrix normal distribution in Appendix B). ", + "bbox": [ + 174, + 296, + 825, + 382 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our resulting posterior for the weights in layer $\\lambda$ is then: ", + "bbox": [ + 173, + 388, + 545, + 404 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f8e7ca58ef6b39545d43d71a592e83d9d0fa209d6d79cfde44af829dd7be389d.jpg", + "text": "$$\nW _ { \\lambda } \\sim \\mathcal { M N } ( W _ { \\lambda } ^ { \\ast } , \\bar { \\mathcal { Q } } _ { \\lambda } ^ { - 1 } , \\bar { \\mathcal { H } } _ { \\lambda } ^ { - 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 398, + 420, + 599, + 440 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In contrast to optimisation methods, we do not need to approximate $\\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]$ as it is only calculated once. However, when it is possible to augment the data (e.g. randomised cropping of images), it may be advantageous. We provide a more detailed discussion of this in Appendix C. ", + "bbox": [ + 174, + 449, + 826, + 492 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.4 INCORPORATING THE PRIOR AND REGULARISING THE CURVATURE FACTORS ", + "text_level": 1, + "bbox": [ + 171, + 508, + 748, + 523 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Just as the log posterior, the Hessian decomposes into a term depending on the data log likelihood and one on the prior. For the commonly used $L _ { 2 }$ -regularisation, corresponding to a Gaussian prior, the Hessian is equal to the precision of the prior times the identity matrix. We approximate this by adding a multiple of the identity to each of the Kronecker factors from the log likelihood: ", + "bbox": [ + 174, + 534, + 825, + 592 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6f831ef21c86d130330cc83051752a1f5eecb53501d281a26f3ec861b7202903.jpg", + "text": "$$\nH _ { \\lambda } = N \\mathbb { E } \\left[ - \\frac { \\partial ^ { 2 } \\log p ( \\mathcal { D } | \\theta ) } { \\partial \\theta ^ { 2 } } \\right] + \\tau I \\approx ( \\sqrt { N } \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\right] + \\sqrt { \\tau } I ) \\otimes ( \\sqrt { N } \\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right] + \\sqrt { \\tau } I )\n$$", + "text_format": "latex", + "bbox": [ + 217, + 607, + 781, + 650 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\tau$ is the precision of the Gaussian prior on the weights and $N$ the size of the dataset. However, we can also treat them as hyperparameters and optimise them w.r.t. the predictive performance on a validation set. We emphasise that this can be done without retraining the network, so it does not impose a large computational overhead and is trivial to parallelise. ", + "bbox": [ + 176, + 659, + 823, + 715 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Setting $N$ to a larger value than the size of the dataset can be interpreted as including duplicates of the data points as pseudo-observations. Adding a multiple of the uncertainty to the precision matrix decreases the uncertainty about each parameter. This has a regularising effect both on our approximation to the true Laplace, which may be overestimating the variance in certain directions due to ignoring the covariances between the layers, as well as the Laplace approximation itself, which may be placing probability mass in low probability areas of the true posterior. ", + "bbox": [ + 174, + 722, + 825, + 806 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 827, + 344, + 842 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Most recent attempts to approximating the posterior of a neural network are based on formulating an approximate distribution to the posterior and optimising the variational lower bound w.r.t. its parameters. (Graves, 2011; Blundell et al., 2015; Kingma et al., 2015) as well as the expectation propagation based approaches of (Hernandez-Lobato & Adams, 2015) and (Ghosh et al., 2016) ´ assume independence between the individual weights which, particularly when optimising the KL divergence, often lets the model underestimate the uncertainty about the weights. Gal & Ghahramani (2016) interpret Dropout to approximate the posterior with a mixture of delta functions, assuming independence between the columns. (Lakshminarayanan et al., 2016) suggest using an ensemble of networks for estimating the uncertainty. ", + "bbox": [ + 176, + 858, + 821, + 887 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our work is a scalable approximation of (MacKay, 1992). Since the per-layer Hessian of a neural network is infeasible to compute, we suggest a factorisation of the covariance into a Kronecker product, leading to a more efficient matrix normal distribution. The posterior that we obtain is reminiscent of (Louizos & Welling, 2016) and (Sun et al., 2017), who optimise the parameters of a matrix normal distribution as their weights, which requires a modification of the training procedure. ", + "bbox": [ + 174, + 208, + 825, + 279 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 301, + 326, + 316 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Since the Laplace approximation is a method for predicting in a Bayesian manner and not for training, we focus on comparing to uncertainty estimates obtained from Dropout (Gal & Ghahramani, 2016). The trained networks will be identical, but the prediction methods will differ. We also compare to a diagonal Laplace approximation to highlight the benefit from modelling the covariances between the weights. All experiments are implemented using Theano (Theano Development Team, 2016) and Lasagne (Dieleman et al., 2015).3 ", + "bbox": [ + 174, + 334, + 825, + 420 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.1 TOY REGRESSION DATASET ", + "text_level": 1, + "bbox": [ + 176, + 440, + 406, + 454 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As a first experiment, we visualise the uncertainty obtained from the Laplace approximations on a toy regression dataset, similar to (Hernandez-Lobato´ $\\&$ Adams, 2015). We create a dataset of 20 uniformly distributed points $x \\sim \\mathcal { U } ( - 4 , 4 )$ and sample $y \\sim \\mathcal { N } ( x ^ { 3 } , 3 ^ { 2 } )$ . In contrast to (Hernandez- ´ Lobato & Adams, 2015), we use a two-layer network with seven units per layer rather than one layer with 100 units. This is because both the input and output are one-dimensional, hence the weight matrices are vectors and the matrix normal distribution reduces to a multivariate normal distribution. Furthermore, the Laplace approximation is sensitive to the ratio of the number of data points to parameters, and we want to visualise it both with and without hyperparameter tuning. ", + "bbox": [ + 173, + 467, + 825, + 582 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/debac792cdab601dfb06184753b4dcbf42158c1124917b3079299c753416bc64.jpg", + "image_caption": [ + "Figure 1: Toy regression uncertainty. Black dots are data points, the black line shows the noiseless function. The red line shows the deterministic prediction of the network, the blue line the mean output. Each shade of blue visualises one additional standard deviation. Best viewed on screen. " + ], + "image_footnote": [], + "bbox": [ + 184, + 608, + 807, + 739 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Fig. 1 shows the uncertainty obtained from the Kronecker factored and diagonal Laplace approximation applied to the same network, as well as from a full Laplace approximation and $5 0 , 0 0 0 \\mathrm { H M C }$ (Neal, 1993) samples. The latter two methods are feasible only for such a small model and dataset. For the diagonal and full Laplace approximation we use the Fisher identity and draw one sample per data point. We set the hyperparameters of the Laplace approximations (see Section 3.4) using a grid search over the likelihood of 20 validation points that are sampled the same way as the training set. ", + "bbox": [ + 173, + 809, + 825, + 893 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The regularised Laplace approximations all give an overall good fit to the HMC predictive posterior. Their uncertainty is slightly higher close to the training data and increases more slowly away from the data than that of the HMC posterior. The diagonal and full Laplace approximation require stronger regularisation than our Kronecker factored one, as they have higher uncertainty when not regularised. In particular the full Laplace approximation vastly overestimates the uncertainty without additional regularisation, leading to a bad predictive mean (see Appendix E for the corresponding figures), as the Hessian of the log likelihood is underdetermined. This is commonly the case in deep learning, as the number of parameters is typically much larger than the number of data points. Hence restricting the structure of the covariance is not only a computational necessity for most architectures, but also allows for more precise estimation of the approximate covariance. ", + "bbox": [ + 173, + 103, + 825, + 242 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 OUT-OF-DISTRIBUTION UNCERTAINTY ", + "text_level": 1, + "bbox": [ + 176, + 267, + 482, + 281 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For a more realistic test, similar to (Louizos & Welling, 2017), we assess the uncertainty of the predictions when classifying data from a different distribution than the training data. For this we train a network with two layers of 1024 hidden units and ReLU transfer functions to classify MNIST digits. We use a learning rate of $1 0 ^ { - 2 }$ and momentum of 0.9 for 250 epochs. We apply Dropout with $p { = } 0 . 5$ after each inner layer, as our chief interest is to compare against its uncertainty estimates. We further use $L _ { 2 }$ -regularisation with a factor of $1 0 ^ { - 2 }$ and randomly binarise the images during training according to their pixel intensities and draw $1 , 0 0 0$ such samples per datapoint for estimating the curvature factors. We use this network to classify the images in the notMNIST dataset4, which contains $2 8 \\times 2 8$ grey-scale images of the letters $\\mathbf { \\dot { A } } ^ { \\prime }$ to $\\mathbf { \\hat { J } } ^ { \\star }$ from various computer fonts, i.e. not digits. An ideal classifier would make uniform predictions over its classes. ", + "bbox": [ + 174, + 296, + 825, + 440 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We compare the uncertainty obtained by predicting the digit class of the notMNIST images using 1. a deterministic forward pass through the Dropout trained network, 2. by sampling different Dropout masks and averaging the predictions, and by sampling different weight matrices from 3. the matrix normal distribution obtained from our Kronecker factored Laplace approximation as well as 4. the diagonal one. As an additional baseline similar to (Blundell et al., 2015; Graves, 2011), we compare to a network with identical architecture with a fully factorised Gaussian (FFG) approximate posterior on the weights and a standard normal prior. We train the model on the variational lower bound using the reparametrisation trick (Kingma & Welling, 2013). We use 100 samples for the stochastic forward passes and optimise the hyperparameters of the Laplace approximations w.r.t. the cross-entropy on the validation set of MNIST. ", + "bbox": [ + 174, + 449, + 483, + 712 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/d894a7edf56530a450cd12d773fa589c25cc119d5b25ad9f5c7d7640b554c41f.jpg", + "image_caption": [ + "Figure 2: Predictive entropy on notMNIST obtained from different methods for the forward pass on a network trained on MNIST. " + ], + "image_footnote": [], + "bbox": [ + 511, + 483, + 792, + 631 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 712, + 625, + 726 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We measure the uncertainty of the different methods as the entropy of the predictive distribution, which has a minimal value of 0 when a single class is predicted with certainty and a maximum of about 2.3 for uniform predictions. Fig. 2 shows the inverse empirical cumulative distribution of the entropy values obtained from the four methods. Consistent with the results in (Gal & Ghahramani, 2016), averaging the probabilities of multiple passes through the network yields predictions with higher uncertainty than a deterministic pass that approximates the geometric average (Srivastava et al., 2014). However, there still are some images that are predicted to be a digit with certainty. Our Kronecker factored Laplace approximation makes hardly any predictions with absolute certainty and assigns high uncertainty to most of the letters as desired. The diagonal Laplace approximation required stronger regularisation towards predicting deterministically, yet it performs similarly to Dropout. As shown in Table 1, however, the network makes predictions on the test set of MNIST with similar accuracy to the deterministic forward pass and MC Dropout when using our approximation. The variational factorised Gaussian posterior has low uncertainty as expected. ", + "bbox": [ + 174, + 733, + 825, + 885 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 821, + 132 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 ADVERSARIAL EXAMPLES ", + "text_level": 1, + "bbox": [ + 174, + 148, + 398, + 161 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/ed41abe45c3623949c3b8185e71c83cf08c0f3240991f92ab6369cada44c3cfe.jpg", + "image_caption": [ + "Figure 3: Untargeted adversarial attack. " + ], + "image_footnote": [], + "bbox": [ + 204, + 176, + 495, + 279 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/bd7abf0110987c858acf90a823a00cb267f5508f27375bc544386b20b41304ac.jpg", + "image_caption": [ + "Figure 4: Targeted adversarial attack. " + ], + "image_footnote": [], + "bbox": [ + 501, + 176, + 792, + 279 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To further test the robustness of our prediction method close to the data distribution, we perform an adversarial attack on a neural network. As first demonstrated in (Szegedy et al., 2013), neural networks are prone to being fooled by gradient-based changes to their inputs. Li & Gal (2017) suggest, and provide empirical support, that Bayesian models may be more robust to such attacks, since they implicitly form an infinitely large ensemble by integrating over the model parameters. For our experiments, we use the fully connected net trained on MNIST from the previous section and compare the sensitivity of the different prediction methods for two kinds of adversarial attacks. ", + "bbox": [ + 174, + 321, + 825, + 419 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "First, we use the untargeted Fast Gradient Sign method $\\begin{array} { r } { x _ { a d v } = x - \\eta \\mathrm { s g n } ( \\nabla _ { x } \\operatorname* { m a x } _ { y } \\log p ^ { ( M ) } ( y | x ) ) } \\end{array}$ suggested in (Goodfellow et al., 2014), which takes the gradient of the class predicted with maximal probability by method $M$ w.r.t. the input $x$ and reduces this probability with varying step size $\\eta$ This step size is rescaled by the difference between the maximal and minimal value per dimension in the dataset. It is to be expected that this method generates examples away from the data manifold, as there is no clear subset of the data that corresponds to e.g. ”not ones”. ", + "bbox": [ + 174, + 428, + 823, + 512 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Fig. 3 shows the average predictive uncertainty and the accuracy on the original class on the MNIST test set as the step size $\\eta$ increases. The Kronecker factored Laplace approximation achieves significantly higher uncertainty than any other prediction method as the images move away from the data. Both the diagonal and the Kronecker factored Laplace maintain higher accuracy than MC Dropout on their original predictions. Interestingly, the deterministic forward pass appears to be most robust in terms of accuracy, however it has much smaller uncertainty on the predictions it makes and will confidently predict a false class for most images, whereas the other methods are more uncertain. ", + "bbox": [ + 173, + 518, + 825, + 617 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Furthermore, we perform a targeted attack that attempts to force the network to predict a specific class, in our case $\\cdot _ { 0 } \\cdot \\mathrm { \\ }$ following (Li & Gal, 2017). Hence, for each method, we exclude all data points in the test set that are already predicted as $\\cdot _ { 0 } \\cdot \\mathrm { \\ }$ . The updates are of similar form to the untargeted attack, however they increase the probability of the pre-specified class $y$ rather than decreasing the current maximum as x(ty $x _ { y } ^ { ( t + 1 ) } = x _ { y } ^ { ( \\bar { t } ) } + \\eta \\mathrm { s g n } \\big ( \\bar { \\nabla } _ { x } \\log \\bar { p ^ { ( M ) } } ( y | x _ { y } ^ { ( t ) } ) \\big )$ , where $x _ { y } ^ { ( 0 ) } = x$ . ", + "bbox": [ + 173, + 623, + 825, + 698 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use a step size of $\\eta { = } 1 0 ^ { - 2 }$ for the targeted attack. The uncertainty and accuracy on the original and target class are shown in Fig. 4. Here, the Kronecker factored Laplace approximation has slightly smaller uncertainty at its peak in comparison to the other methods, however it appears to be much more robust. It only misclassifies over $5 0 \\%$ of the images after about 20 steps, whereas for the other methods this is the case after roughly 10 steps and reaches $1 0 0 \\%$ accuracy on the target class after almost 50 updates, whereas the other methods are fooled on all images after about 25 steps. ", + "bbox": [ + 174, + 705, + 825, + 790 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In conjunction with the experiment on notMNIST, it appears that the Laplace approximation achieves higher uncertainty than Dropout away from the data, as in the untargeted attack. In the targeted attack it exhibits smaller uncertainty than Dropout, yet it is more robust to having its prediction changed. The diagonal Laplace approximation again performs similarly to Dropout. ", + "bbox": [ + 174, + 797, + 825, + 854 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.4 UNCERTAINTY ON MISCLASSIFICATIONS ", + "text_level": 1, + "bbox": [ + 176, + 869, + 495, + 883 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To highlight the scalability of our method, we apply it to a state-of-the-art convolutional network architecture. Recently, deep residual networks (He et al., 2016a;b) have been the most successful ones among those. As demonstrated in (Grosse & Martens, 2016), Kronecker factored curvature methods are applicable to convolutional layers by interpreting them as matrix-matrix multiplications. ", + "bbox": [ + 176, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/5ed41153d9002457c121cab0e2d0510ad1bedc3708935fa9b5bfa70ad80fe8a9.jpg", + "image_caption": [ + "Figure 5: Inverse ecdf of the predictive entropy from Wide Residual Networks trained with and without Dropout on CIFAR100. For misclassifications, curves on top corresponding to higher uncertainty are desirable, and curves on the bottom for correct classifications. " + ], + "image_footnote": [], + "bbox": [ + 204, + 99, + 794, + 241 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 343, + 823, + 372 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare our uncertainty estimates on wide residual networks (Zagoruyko & Komodakis, 2016), a recent variation that achieved competitive performance on CIFAR100 (Krizhevsky & Hinton, 2009) while, in contrast to most other residual architectures, including Dropout at specific points. While this does not correspond to using Dropout in the Bayesian sense (Gal & Ghahramani, 2015), it allows us to at least compare our method to the uncertainty estimates obtained from Dropout. ", + "bbox": [ + 174, + 378, + 825, + 449 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We note that it is straightforward to incorporate batch normalisation (Ioffe & Szegedy, 2015) into the curvature backpropagation algorithms, so we apply a standard Laplace approximation to its parameters as well. We are not aware of any interpretation of Dropout as performing Bayesian inference on the parameters of batch normalisation. Further implementation details are in Appendix G. ", + "bbox": [ + 174, + 455, + 823, + 511 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Again, the accuracy of the prediction methods is comparable (see Table 2 in Appendix F). For calculating the curvature factors, we draw 5, 000 samples per image using the same data augmentation as during training, effectively increasing the dataset size to $2 . 5 \\times 1 0 ^ { 8 }$ . The diagonal approximation had to be regularised to the extent of becoming deterministic, so we omit it from the results. ", + "bbox": [ + 174, + 518, + 825, + 575 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Fig. 5 we compare the distribution of the predictive uncertainty on the test set.5 We distinguish between the uncertainty on correct and incorrect classifications, as the mistakes of a system used in practice may be less severe if the network can at least indicate that it is uncertain. Thus, high uncertainty on misclassifications and low uncertainty on correct ones would be desirable, such that a system could return control to a human expert when it can not make a confident decision. In general, the network tends to be more uncertain on its misclassifcations than its correct ones regardless of whether it was trained with or without Dropout and of the method used for prediction. Both Dropout and the Laplace approximation similarly increase the uncertainty in the predictions, however this is irrespective of the correctness of the classification. Yet, our experiments show that the Kronecker factored Laplace approximation can be scaled to modern convolutional networks and maintain good classification accuracy while having similar uncertainty about the predictions as Dropout. ", + "bbox": [ + 174, + 584, + 825, + 738 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We had to use much stronger regularisation for the Laplace approximation on the wide residual network, possibly because the block-diagonal approximation becomes more inaccurate on deep networks, possibly because the number of parameters is much higher relative to the number of data. It would be interesting to see how the Laplace approximations behaves on a much larger dataset like ImageNet for similarly sized networks, where we have a better ratio of data to parameters and curvature directions. However, even on a relatively small dataset like CIFAR we did not have to regularise the Laplace approximation to the degree of the posterior becoming deterministic. ", + "bbox": [ + 174, + 744, + 825, + 843 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 318, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We presented a scalable approximation to the Laplace approximation for the posterior of a neural network and provided experimental results suggesting that the uncertainty estimates are on par with current alternatives like Dropout, if not better. It enables practitioners to obtain principled uncertainty estimates from their models, even if they were trained in a maximum likelihood/MAP setting. ", + "bbox": [ + 174, + 133, + 823, + 189 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "There are many possible extensions to this work. One would be to automatically determine the scale and regularisation hyperparameters of the Kronecker factored Laplace approximation using the model evidence similar to how (MacKay, 1992) interpolates between the data log likelihood and the width of the prior. The model evidence could further be used to perform Bayesian model averaging on ensembles of neural networks, potentially improving their generalisation ability and uncertainty estimates. A challenging application would be active learning, where only little data is available relative to the number of curvature directions that need to be estimated. ", + "bbox": [ + 174, + 196, + 825, + 294 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 316, + 367, + 330 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work was supported by the Alan Turing Institute under the EPSRC grant EP/N510129/1. 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", + "bbox": [ + 174, + 857, + 821, + 887 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Sergey Zagoruyko and Nikos Komodakis. Wide Residual Networks. arXiv preprint arXiv:1605.07146, 2016. ", + "bbox": [ + 173, + 893, + 823, + 922 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Appendices ", + "text_level": 1, + "bbox": [ + 176, + 99, + 334, + 127 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A DERIVATION OF THE ACTIVATION HESSIAN RECURSION ", + "text_level": 1, + "bbox": [ + 176, + 145, + 676, + 162 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Here, we provide the basic derivation of the factorisation of the diagonal blocks of the Hessian in Eq. 1 and the recursive formula for calculating $\\mathcal { H }$ as presented in (Botev et al., 2017). ", + "bbox": [ + 171, + 179, + 825, + 208 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The Hessian of a neural network with parameters $\\theta$ as defined in the main text has elements: ", + "bbox": [ + 176, + 214, + 777, + 231 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/0f296d35cb406ef6eb9012ee3b6a4216e5267cab6413a299488245fd078eadb1.jpg", + "text": "$$\n[ H ] _ { i j } = \\frac { \\partial ^ { 2 } } { \\partial \\theta _ { i } \\partial \\theta _ { j } } E ( \\theta )\n$$", + "text_format": "latex", + "bbox": [ + 426, + 250, + 573, + 287 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "For a given layer $\\lambda$ , the gradient w.r.t. a weight $W _ { a , b } ^ { \\lambda }$ is: ", + "bbox": [ + 173, + 303, + 539, + 320 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/83dba684467e560f40061ae37edaec35ad0a2115e940c900c683886e03489863.jpg", + "text": "$$\n{ \\frac { \\partial E } { \\partial W _ { a , b } ^ { \\lambda } } } = \\sum _ { i } { \\frac { \\partial h _ { i } ^ { \\lambda } } { \\partial W _ { a , b } ^ { \\lambda } } } { \\frac { \\partial E } { \\partial h _ { i } ^ { \\lambda } } } = a _ { b } ^ { \\lambda - 1 } { \\frac { \\partial E } { \\partial h _ { a } ^ { \\lambda } } }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 339, + 630, + 381 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Keeping $\\lambda$ fixed and differentiating again, we find that the per-sample Hessian of that layer is: ", + "bbox": [ + 171, + 393, + 787, + 410 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/e4a81dbe7d6040535c7d848c28f2c79f66c521dabc27e35ecc127cf4e545e4a9.jpg", + "text": "$$\n\\left[ H _ { \\lambda } \\right] _ { ( a , b ) , ( c , d ) } \\equiv \\frac { \\partial ^ { 2 } E } { \\partial W _ { a , b } ^ { \\lambda } \\partial W _ { c , d } ^ { \\lambda } } = a _ { b } ^ { \\lambda - 1 } a _ { d } ^ { \\lambda - 1 } \\left[ \\mathcal { H } _ { \\lambda } \\right] _ { a , c }\n$$", + "text_format": "latex", + "bbox": [ + 326, + 429, + 671, + 469 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 483, + 217, + 497 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/19269707e999e2664986cce1d5888ad8b2f6e20b86f3dc7bbadf68aee2228634.jpg", + "text": "$$\n[ \\mathcal { H } _ { \\lambda } ] _ { a , b } = \\frac { \\partial ^ { 2 } E } { \\partial h _ { a } ^ { \\lambda } \\partial h _ { b } ^ { \\lambda } }\n$$", + "text_format": "latex", + "bbox": [ + 433, + 517, + 566, + 556 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "is the pre-activation Hessian. ", + "bbox": [ + 174, + 569, + 364, + 583 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We can reexpress this in matrix notation as a Kronecker product as in Eq. 1: ", + "bbox": [ + 171, + 590, + 671, + 606 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/36187593be22e70d34cda56a97545b1794bc291950b32db101df9d3a58cf5d5e.jpg", + "text": "$$\nH _ { \\lambda } = \\frac { \\partial ^ { 2 } E } { \\partial \\operatorname { v e c } \\left( W _ { \\lambda } \\right) \\partial \\operatorname { v e c } \\left( W _ { \\lambda } \\right) } = \\left( a _ { \\lambda - 1 } a _ { \\lambda - 1 } ^ { \\mathsf { T } } \\right) \\otimes \\mathcal { H } _ { \\lambda }\n$$", + "text_format": "latex", + "bbox": [ + 323, + 625, + 674, + 662 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The pre-activation Hessian can be calculated recursively as: ", + "bbox": [ + 173, + 675, + 565, + 690 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/066d8f5662937414b7ff787c091a3dc30e6371e80531770315d17352185108d7.jpg", + "text": "$$\n\\mathcal { H } _ { \\lambda } = B _ { \\lambda } W _ { \\lambda + 1 } ^ { \\mathsf { T } } \\mathcal { H } _ { \\lambda + 1 } W _ { \\lambda + 1 } B _ { \\lambda } + D _ { \\lambda }\n$$", + "text_format": "latex", + "bbox": [ + 372, + 709, + 625, + 732 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "where the diagonal matrices $B$ and $D$ are defined as: ", + "bbox": [ + 173, + 744, + 521, + 760 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/2d283728938d16ef2672bdb5f416bcd470208dcd734b67d9666da1668ae8810e.jpg", + "text": "$$\n\\begin{array} { l } { { B _ { \\lambda } = \\mathrm { d i a g } \\left( \\mathbf f _ { \\lambda } ^ { \\prime } ( h _ { \\lambda } ) \\right) } } \\\\ { { D _ { \\lambda } = \\mathrm { d i a g } \\left( \\mathbf f _ { \\lambda } ^ { \\prime \\prime } ( h _ { \\lambda } ) { \\frac { \\partial E } { \\partial a _ { \\lambda } } } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 413, + 787, + 584, + 843 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "$\\mathbf { f ^ { \\prime } }$ and $\\mathbf { f } ^ { \\prime \\prime }$ denote the first and second derivative of the transfer function. The recursion is initialised with the Hessian of the error w.r.t. the linear network outputs. ", + "bbox": [ + 174, + 859, + 823, + 888 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "For further details and on how to calculate the diagonal blocks of the Gauss-Newton and Fisher matrix, we refer the reader to (Botev et al., 2017) and (Martens & Grosse, 2015). ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B MATRIX NORMAL DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 488, + 118 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The matrix normal distribution (Gupta & Nagar, 1999) is a multivariate distribution over an entire matrix of shape $n \\times p$ rather than just a vector. In contrast to the multivariate normal distribution, it is parameterised by two p.s.d. covariance matrices, $U : n \\times n$ and $V : p \\times p$ , which indicate the covariance of the rows and columns respectively. In addition it has a mean matrix $M : n \\times p$ . ", + "bbox": [ + 174, + 133, + 825, + 190 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A vectorised sample from a matrix normal distribution $X \\sim { \\mathcal { M N } } ( M , U , V )$ corresponds to a sample from a normal distribution $\\operatorname { v e c } ( X ) \\sim { \\mathcal { N } } ( \\operatorname { v e c } ( M ) , U \\otimes V )$ . However, samples can be drawn more efficiently as $X = M + A Z B$ with $Z \\sim \\mathcal { M N } ( 0 , I , I )$ , and $A A ^ { \\mathsf { T } } = U$ and $B ^ { \\intercal } B = V$ . The sample $Z$ corresponds to a sample from a normal distribution of length $n p$ that has been reshaped to a $n \\times p$ matrix. This is more efficient in the sense that we only need to calculate two matrix-matrix products of small matrices, rather than a matrix-vector product with one big one. ", + "bbox": [ + 173, + 196, + 825, + 284 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C APPROXIMATION OF THE EXPECTED ACTIVATION HESSIAN ", + "text_level": 1, + "bbox": [ + 176, + 304, + 699, + 320 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "While the square root of $\\mathcal { Q } _ { \\lambda }$ is calculated during the forward pass on all layers, $\\mathcal { H }$ requires an additional backward pass. Strictly speaking, it is not essential to approximate $\\mathbb { E } \\left[ \\mathcal { H } \\right]$ for the Kronecker factored Laplace approximation, as in contrast to optimisation procedures the curvature only needs to be calculated once and is thus not time critical. For datasets of the scale of ImageNet and the networks used for such datasets, it would still be impractically slow to perform the calculation for every data point individually. Furthermore, as most datasets are augmented during training, e.g. random cropping or reflections of images, the curvature of the network can be estimated using the same augmentations, effectively increasing the size of the dataset by orders of magnitude. Thus, we make use of the minibatch approximation in our experiments — as we make use of data augmentation — in order to demonstrate its practical applicability. ", + "bbox": [ + 173, + 335, + 825, + 476 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We note that E [H] can be calculated exactly by running KFRA (Botev et al., 2017) with a minibatchsize of one, and then averaging the results. KFAC (Martens & Grosse, 2015), in contrast, stochastically approximates the Fisher matrix, so even when run for every datapoint separately, it cannot calculate the curvature factor exactly. ", + "bbox": [ + 174, + 482, + 825, + 537 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In the following, we also show figures for the adversarial experiments in which we calculate the curvature per datapoint and without data augmentation: ", + "bbox": [ + 174, + 545, + 823, + 573 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/bd401ad0b8e9d0a6e372927c134409d5b015ccf19eb9e8745ca37ac78fb7f7a9.jpg", + "image_caption": [ + "Figure 6: Untargeted adversarial attack for Kronecker factored Laplace approximation with the curvature calculated with and without data augmentation/approximating the activation Hessian. " + ], + "image_footnote": [], + "bbox": [ + 178, + 587, + 821, + 741 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Fig. 6 and Fig. 7 show how the Laplace approximation with the curvature estimated from 1000 randomly sampled binary MNIST images and the activation Hessian calculated with a minibatch size of 100 performs in comparison to the curvature factor being calculated without any data augmentation with a batch size of 100 or exactly. We note that without data augmentation we had to use much stronger regularisation of the curvature factors, in particular we had to add a non-negligible multiple of the identity to the factors, whereas with data augmentation it was only needed to ensure that the matrices are invertible. The Kronecker factored Laplace approximation reaches particularly high uncertainty on the untargeted adversarial attack and is most robust on the targeted attack when using data augmentation, suggesting that it is particularly well suited for large datasets and ones where some form of data augmentation can be applied. The difference between approximating the activation Hessian over a minibatch and calculating it exactly appears to be negligible. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/e31a6b6b0d0ab2399091e0aa6f1e8ffc0b21f98d8546841f2ad9096b7c2ec352.jpg", + "image_caption": [ + "Figure 7: Targeted adversarial attack for Kronecker factored Laplace approximation with the curvature calculated with and without data augmentation/approximating the activation Hessian. " + ], + "image_footnote": [], + "bbox": [ + 176, + 101, + 823, + 256 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 321, + 823, + 351 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D MEMORY AND COMPUTATIONAL REQUIREMENTS ", + "text_level": 1, + "bbox": [ + 174, + 371, + 620, + 386 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "If we denote the dimensionality of the input to layer $\\lambda$ as $D _ { \\lambda - 1 }$ and its output as $D _ { \\lambda }$ , the curvature factors correspond to the two precision matrices with Dλ−1(Dλ−1+1)2 and $\\frac { D _ { \\lambda } ( D _ { \\lambda } + 1 ) } { 2 }$ ‘parameters’ to estimate, since they are symmetric. So across a network, the number of curvature directions that we are estimating grows linearly in the number of layers and quadratically in the dimension of the layers, i.e. the number of columns of the weight matrices. The size of the full Hessian, on the other hand, grows quadratically in the number of layers and with the fourth power in the dimensionality of the layers (assuming they are all the same size). ", + "bbox": [ + 173, + 400, + 825, + 503 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Once the curvature factors are calculated, which only needs to be done once, we use their Cholesky decomposition to solve two triangular linear systems when sampling weights from the matrix normal distribution. We use the same weight samples for each minibatch, i.e. we do not sample a weight matrix per datapoint. This is for computational efficiency and does not change the expectation. ", + "bbox": [ + 174, + 511, + 825, + 566 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "One possibility to save computation time would be to sample a fixed set of weight matrices from the approximate posterior — in order to avoid solving the linear system on every forward pass — and treat the networks that they define as an ensemble. The individual ensemble members can be evaluated in parallel and their outputs averaged, which can be done with a small overhead over evaluating a single network given sufficient compute resources. A further speed up can be achieved by distilling the predictive distributions of the Laplace network into a smaller, deterministic feedforward network as successfully demonstrated in (Balan et al., 2015) for posterior samples using HMC. ", + "bbox": [ + 173, + 573, + 825, + 671 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "E COMPLEMENTARY FIGURES FOR THE TOY DATASET ", + "text_level": 1, + "bbox": [ + 174, + 690, + 640, + 707 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Fig. 8 shows the different Laplace approximations (Kronecker factored, diagonal, full) from the main text without any hyperparameter tuning. The figure of the uncertainty obtained from samples using HMC is repeated. Note that the scale is larger than in the main text due to the high uncertainty of the Laplace approximations. ", + "bbox": [ + 173, + 722, + 823, + 777 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The Laplace approximations are increasingly uncertain away from the data, as the true posterior estimated from HMC samples, however they all overestimate the uncertainty without regularisation. This is easy to fix by optimising the hyperparameters on a validation set as discussed in the main text, resulting in posterior uncertainty much more similar to the true posterior. As previously discussed in (Botev et al., 2017), the Hessian of a neural network is usually underdetermined as the number of data points is much smaller than the number of parameters — in our case we have 20 data points to estimate a $7 8 \\times 7 8$ precision matrix. This leads to the full Laplace approximation vastly overestimating the uncertainty and a bad predictive mean. Both the Kronecker factored and the diagonal approximation exhibit smaller variance than the full Laplace approximation as they restrict the structure of the precision matrix. Consistently with the other experiments, we find the diagonal ", + "bbox": [ + 173, + 785, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/8728ab113d8def68159f046cb816ff9c7101a0cf17ad7545a887f6a29fd79c4f.jpg", + "image_caption": [ + "Figure 8: Toy regression uncertainty. Black dots are data points, the black line shows the underlying noiseless function. The red line shows the deterministic prediction of the trained network, the blue line the mean output. Each shade of blue visualises one additional standard deviation. " + ], + "image_footnote": [], + "bbox": [ + 210, + 121, + 769, + 593 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Laplace approximation to place more mass in low probability areas of the posterior than the Kronecker factored approximation, resulting in higher variance on the regression problem. This leads to a need for greater regularisation of the diagonal approximation to obtain acceptable predictive performance, and underestimating the uncertainty. ", + "bbox": [ + 173, + 696, + 825, + 753 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "F PREDICTION ACCURACY ", + "text_level": 1, + "bbox": [ + 176, + 799, + 411, + 815 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This section shows the accuracy values obtained from the different predictions methods on the feedforward networks for MNIST and the wide residual network for CIFAR100. The results for MNIST are shown in Table 1 and the results for CIFAR in Table 2. ", + "bbox": [ + 174, + 847, + 825, + 888 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In all cases, neither MC Dropout nor the Laplace approximation significantly change the classification accuracy of the network in comparison to a deterministic forward pass. ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/2917d1b2ef29894240396badcccee0730023770a1a2a6738ee3bf4a150af869d.jpg", + "table_caption": [ + "Table 1: Test accuracy of the feedforward network trained on MNIST " + ], + "table_footnote": [], + "table_body": "
Prediction MethodAccuracy
FFG Deterministic98.88% 98.86%
MC Dropout Diagonal Laplace98.85% 98.85%
KF Laplace98.80%
", + "bbox": [ + 382, + 126, + 611, + 215 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/52092b0b4adc367a00e011aebda2cedf442164045a787b26e985b415a1e76a5f.jpg", + "table_caption": [ + "Table 2: Accuracy on the final 5, 000 CIFAR100 test images for a wide residual network trained with and without Dropout. " + ], + "table_footnote": [], + "table_body": "
Accuracy
Prediction MethodDropoutDeterministic
Deterministic79.12%79.18%
MC Dropout79.20%1
KF Laplace79.10%79.36%
", + "bbox": [ + 333, + 270, + 660, + 343 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "G IMPLEMENTATION DETAILS FOR RESIDUAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 367, + 673, + 385 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Our wide residual network has $n { = } 3$ block repetitions and a width factor of $k { = } 8$ on CIFAR100 with and without Dropout using hyperparameters taken from (Zagoruyko & Komodakis, 2016): the network parameters are trained on a cross-entropy loss using Nesterov momentum with an initial learning rate of 0.1 and momentum of 0.9 for 200 epochs with a minibatch size of 128. We decay the learning rate every 50 epochs by a factor of 0.2, which is slightly different to the schedule used in (Zagoruyko & Komodakis, 2016) (they decay after 60, 120 and 160 epochs). As the original authors, we use $L _ { 2 }$ -regularisation with a factor of $5 \\times 1 0 ^ { - 4 }$ . ", + "bbox": [ + 173, + 398, + 825, + 500 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We make one small modification to the architecture: instead of downsampling with $1 \\times 1$ convolutions with stride 2, we use $2 \\times 2$ convolutions. This is due to Theano not supporting the transformation of images into the patches extracted by a convolution for $1 \\times 1$ convolutions with stride greater than 1, which we require for our curvature backpropagation through convolutions. ", + "bbox": [ + 174, + 506, + 825, + 563 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We apply a standard Laplace approximation to the batch normalisation parameters — a Kronecker factorisation is not needed, since the parameters are one-dimensional. When calculating the curvature factors, we use the moving averages for the per-layer means and standard deviations obtained after training, in order to maintain independence between the data points in a minibatch. ", + "bbox": [ + 174, + 569, + 825, + 625 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We need to make a further approximation to the ones discussed in Section 2.2 when backpropagating the curvature for residual networks. The residual blocks compute a function of the form $r e s ( x ) =$ $x + f _ { \\phi } ( x )$ , where $f _ { \\phi }$ typically is a sequence of convolutions, batch normalisation and elementwise nonlinearities. This means that we would need to pass back two curvature matrices, one for each summand. However, this would double the number of backpropagated matrices for each residual connection, hence the computation time/memory requirements would grow exponentially in the number of residual blocks. Therefore, we simply add the curvature matrices after each residual connection. 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We extensively", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 268, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 470, + 279 + ], + "score": 1.0, + "content": "compare our method to using Dropout and a diagonal Laplace approximation for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "score": 1.0, + "content": "estimating the uncertainty of a network. We demonstrate that our Kronecker fac-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 470, + 301 + ], + "score": 1.0, + "content": "tored method leads to better uncertainty estimates on out-of-distribution data and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 469, + 312 + ], + "score": 1.0, + "content": "is more robust to simple adversarial attacks. Our approach only requires calcu-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 312, + 469, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 469, + 324 + ], + "score": 1.0, + "content": "lating two square curvature factor matrices for each layer. Their size is equal to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "score": 1.0, + "content": "the respective square of the input and output size of the layer, making the method", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 469, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 346 + ], + "score": 1.0, + "content": "efficient both computationally and in terms of memory usage. We illustrate its", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 467, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 467, + 356 + ], + "score": 1.0, + "content": "scalability by applying it to a state-of-the-art convolutional network architecture.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 375, + 206, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 208, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 208, + 391 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 504, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "Neural networks are most commonly trained in a maximum a posteriori (MAP) setting, which only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "yields point estimates of the parameters, ignoring any uncertainty about them. This often leads to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 104, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "overconfident predictions, especially in regimes that are weakly covered by training data or far away", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "from the data manifold. While the confidence of wrong predictions is usually irrelevant in a research", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "context, it is essential that a Machine Learning algorithm knows when it does not know in the real", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "world, as the consequences of mistakes can be fatal, be it when driving a car or diagnosing a disease.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 504, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "The Bayesian framework of statistics provides a principled way for avoiding overconfidence in the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "parameters by treating them as unknown quantities and integrating over all possible values. Specifi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "cally, for the prediction of new data under a model, it fits a posterior distribution over the parameters", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "given the training data and weighs the contribution of each setting of the parameters to the predic-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "tion by the probability of the data under those parameters times their prior probability. However, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 527, + 442, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 442, + 539 + ], + "score": 1.0, + "content": "posterior of neural networks is usually intractable due to their size and nonlinearity.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "There has been previous interest in integrating neural networks into the Bayesian framework", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 553, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 569 + ], + "score": 1.0, + "content": "(MacKay, 1992; Hinton & Van Camp, 1993; Neal, 1993; Barber & Bishop, 1998), however these ap-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "proaches were designed for small networks by current standards. Recent adaptations to architectures", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 575, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 506, + 590 + ], + "score": 1.0, + "content": "of modern scale rely on crude approximations of the posterior to become tractable. All of (Graves,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "score": 1.0, + "content": "2011; Hernandez-Lobato & Adams, 2015; Blundell et al., 2015) assume independence between the ´", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "individual weights. While they achieve good results on small datasets, this strong restriction of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "posterior is susceptible to underestimating the uncertainty, in particular when optimising the vari-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "ational bound. The approach in (Gal & Ghahramani, 2016) requires the use of certain stochastic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "regularisers which are not commonly present in most recent architectures. Furthermore, it is not", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 642, + 489, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 489, + 655 + ], + "score": 1.0, + "content": "clear if the approximate posterior defined by these regularisers is a good fit to the true posterior.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "Recent work on second-order optimisation of neural networks (Martens & Grosse, 2015; Botev", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "score": 1.0, + "content": "et al., 2017) has demonstrated that the diagonal blocks of the curvature can be well approximated", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "by a Kronecker product. We combine this insight with the idea of modelling the posterior over the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "score": 1.0, + "content": "weights as a Gaussian, using a Laplace approximation (MacKay, 1992) with Kronecker factored", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "score": 1.0, + "content": "covariance matrices. This leads to a computationally efficient matrix normal posterior distribution", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 119, + 722, + 321, + 731 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 322, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 322, + 734 + ], + "score": 1.0, + "content": "∗Corresponding author: j.ritter@cs.ucl.ac.uk", + "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": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 79, + 408, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 410, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 410, + 97 + ], + "score": 1.0, + "content": "A SCALABLE LAPLACE APPROXIMATION", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 297, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 297, + 118 + ], + "score": 1.0, + "content": "FOR NEURAL NETWORKS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 132, + 344, + 159 + ], + "lines": [ + { + "bbox": [ + 110, + 133, + 346, + 148 + ], + "spans": [ + { + "bbox": [ + 110, + 133, + 346, + 148 + ], + "score": 1.0, + "content": "Hippolyt Ritter1∗, Aleksandar Botev1, David Barber1 2", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 113, + 145, + 319, + 161 + ], + "spans": [ + { + "bbox": [ + 113, + 145, + 319, + 161 + ], + "score": 1.0, + "content": "1University College London 2Alan Turing Institute", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 110, + 133, + 346, + 161 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 188, + 333, + 200 + ], + "lines": [ + { + "bbox": [ + 277, + 187, + 335, + 201 + ], + "spans": [ + { + "bbox": [ + 277, + 187, + 335, + 201 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 212, + 468, + 355 + ], + "lines": [ + { + "bbox": [ + 142, + 213, + 469, + 225 + ], + "spans": [ + { + "bbox": [ + 142, + 213, + 469, + 225 + ], + "score": 1.0, + "content": "We leverage recent insights from second-order optimisation for neural networks", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 223, + 470, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 223, + 470, + 236 + ], + "score": 1.0, + "content": "to construct a Kronecker factored Laplace approximation to the posterior over the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 470, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 470, + 247 + ], + "score": 1.0, + "content": "weights of a trained network. Our approximation requires no modification of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 246, + 469, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 246, + 469, + 257 + ], + "score": 1.0, + "content": "training procedure, enabling practitioners to estimate the uncertainty of their mod-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 257, + 469, + 269 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 469, + 269 + ], + "score": 1.0, + "content": "els currently used in production without having to retrain them. We extensively", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 268, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 470, + 279 + ], + "score": 1.0, + "content": "compare our method to using Dropout and a diagonal Laplace approximation for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "score": 1.0, + "content": "estimating the uncertainty of a network. We demonstrate that our Kronecker fac-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 470, + 301 + ], + "score": 1.0, + "content": "tored method leads to better uncertainty estimates on out-of-distribution data and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 469, + 312 + ], + "score": 1.0, + "content": "is more robust to simple adversarial attacks. Our approach only requires calcu-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 312, + 469, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 469, + 324 + ], + "score": 1.0, + "content": "lating two square curvature factor matrices for each layer. Their size is equal to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "score": 1.0, + "content": "the respective square of the input and output size of the layer, making the method", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 469, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 346 + ], + "score": 1.0, + "content": "efficient both computationally and in terms of memory usage. We illustrate its", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 467, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 467, + 356 + ], + "score": 1.0, + "content": "scalability by applying it to a state-of-the-art convolutional network architecture.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 141, + 213, + 470, + 356 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 375, + 206, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 208, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 208, + 391 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 504, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "Neural networks are most commonly trained in a maximum a posteriori (MAP) setting, which only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "yields point estimates of the parameters, ignoring any uncertainty about them. This often leads to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 104, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "overconfident predictions, especially in regimes that are weakly covered by training data or far away", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "from the data manifold. While the confidence of wrong predictions is usually irrelevant in a research", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "context, it is essential that a Machine Learning algorithm knows when it does not know in the real", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "world, as the consequences of mistakes can be fatal, be it when driving a car or diagnosing a disease.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 399, + 506, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 504, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "The Bayesian framework of statistics provides a principled way for avoiding overconfidence in the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "parameters by treating them as unknown quantities and integrating over all possible values. Specifi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "cally, for the prediction of new data under a model, it fits a posterior distribution over the parameters", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "given the training data and weighs the contribution of each setting of the parameters to the predic-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "tion by the probability of the data under those parameters times their prior probability. However, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 527, + 442, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 442, + 539 + ], + "score": 1.0, + "content": "posterior of neural networks is usually intractable due to their size and nonlinearity.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 471, + 506, + 539 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "There has been previous interest in integrating neural networks into the Bayesian framework", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 553, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 569 + ], + "score": 1.0, + "content": "(MacKay, 1992; Hinton & Van Camp, 1993; Neal, 1993; Barber & Bishop, 1998), however these ap-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "proaches were designed for small networks by current standards. Recent adaptations to architectures", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 575, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 506, + 590 + ], + "score": 1.0, + "content": "of modern scale rely on crude approximations of the posterior to become tractable. All of (Graves,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "score": 1.0, + "content": "2011; Hernandez-Lobato & Adams, 2015; Blundell et al., 2015) assume independence between the ´", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "individual weights. While they achieve good results on small datasets, this strong restriction of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "posterior is susceptible to underestimating the uncertainty, in particular when optimising the vari-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "ational bound. The approach in (Gal & Ghahramani, 2016) requires the use of certain stochastic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "regularisers which are not commonly present in most recent architectures. Furthermore, it is not", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 642, + 489, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 489, + 655 + ], + "score": 1.0, + "content": "clear if the approximate posterior defined by these regularisers is a good fit to the true posterior.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 542, + 506, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "Recent work on second-order optimisation of neural networks (Martens & Grosse, 2015; Botev", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "score": 1.0, + "content": "et al., 2017) has demonstrated that the diagonal blocks of the curvature can be well approximated", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "by a Kronecker product. We combine this insight with the idea of modelling the posterior over the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "score": 1.0, + "content": "weights as a Gaussian, using a Laplace approximation (MacKay, 1992) with Kronecker factored", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "score": 1.0, + "content": "covariance matrices. This leads to a computationally efficient matrix normal posterior distribution", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "(Gupta & Nagar, 1999) over the weights of every layer. Since the Laplace approximation is applied", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 489, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 489, + 106 + ], + "score": 1.0, + "content": "after training, our approach can be used to obtain uncertainty estimates from existing networks.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 659, + 506, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "(Gupta & Nagar, 1999) over the weights of every layer. Since the Laplace approximation is applied", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 489, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 489, + 106 + ], + "score": 1.0, + "content": "after training, our approach can be used to obtain uncertainty estimates from existing networks.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 107, + 121, + 340, + 134 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 340, + 136 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 340, + 136 + ], + "score": 1.0, + "content": "2 THE CURVATURE OF NEURAL NETWORKS", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 146, + 505, + 202 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 504, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 504, + 158 + ], + "score": 1.0, + "content": "Our method is inspired by recent Kronecker factored approximations of the curvature of a neural", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "score": 1.0, + "content": "network (Martens & Grosse, 2015; Botev et al., 2017) for optimisation and we give a high-level", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 169, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 505, + 181 + ], + "score": 1.0, + "content": "review of these in the following. 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The inter-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 243, + 261 + ], + "score": 1.0, + "content": "mediate representations for layers", + "type": "text" + }, + { + "bbox": [ + 244, + 248, + 293, + 259 + ], + "score": 0.92, + "content": "\\lambda = 1 , . . . , L", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 248, + 352, + 261 + ], + "score": 1.0, + "content": "are denoted as", + "type": "text" + }, + { + "bbox": [ + 353, + 248, + 413, + 259 + ], + "score": 0.93, + "content": "h _ { \\lambda } = W _ { \\lambda } a _ { \\lambda - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 248, + 431, + 261 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 432, + 248, + 485, + 260 + ], + "score": 0.92, + "content": "a _ { \\lambda } = f _ { \\lambda } ( h _ { \\lambda } )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 248, + 505, + 261 + ], + "score": 1.0, + "content": ". 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Most commonly used error functions, such as squared error and categorical", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 291, + 486, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 424, + 304 + ], + "score": 1.0, + "content": "cross-entropy, can be interpreted as exponential family negative log likelihoods", + "type": "text" + }, + { + "bbox": [ + 425, + 291, + 482, + 303 + ], + "score": 0.92, + "content": "- \\log p ( y | h _ { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 291, + 486, + 304 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 316, + 373, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 375, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 375, + 330 + ], + "score": 1.0, + "content": "2.2 KRONECKER FACTORED SECOND-ORDER OPTIMISATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 336, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "Traditional second-order methods use either the Hessian matrix or a positive semi-definite approx-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 348, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 342, + 363 + ], + "score": 1.0, + "content": "imation thereof to generate parameter updates of the form", + "type": "text" + }, + { + "bbox": [ + 343, + 348, + 390, + 362 + ], + "score": 0.93, + "content": "\\Delta = C ^ { - 1 } g", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 348, + 421, + 363 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 421, + 350, + 430, + 360 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 348, + 506, + 363 + ], + "score": 1.0, + "content": "is the chosen cur-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 180, + 374 + ], + "score": 1.0, + "content": "vature matrix and", + "type": "text" + }, + { + "bbox": [ + 180, + 363, + 186, + 372 + ], + "score": 0.77, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "the gradient of the error function parameterised by the network. However, this", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "curvature matrix is infeasbile to compute for modern neural networks as their number of parameters", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 381, + 427, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 381, + 285, + 396 + ], + "score": 1.0, + "content": "is often in the millions, rendering the size of", + "type": "text" + }, + { + "bbox": [ + 286, + 384, + 294, + 393 + ], + "score": 0.85, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 381, + 427, + 396 + ], + "score": 1.0, + "content": "of the order of several terabytes.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 105, + 399, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "score": 1.0, + "content": "Recent work (Martens & Grosse, 2015; Botev et al., 2017) exploits that, for a single data point, the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 410, + 378, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 378, + 424 + ], + "score": 1.0, + "content": "diagonal blocks of these curvature matrices are Kronecker factored:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 435, + 391, + 465 + ], + "lines": [ + { + "bbox": [ + 219, + 435, + 391, + 465 + ], + "spans": [ + { + "bbox": [ + 219, + 435, + 391, + 465 + ], + "score": 0.94, + "content": "H _ { \\lambda } = \\frac { \\partial ^ { 2 } E } { \\partial \\operatorname { v e c } ( W _ { \\lambda } ) \\partial \\operatorname { v e c } ( W _ { \\lambda } ) } = \\mathcal { Q } _ { \\lambda } \\otimes \\mathcal { H } _ { \\lambda }", + "type": "interline_equation", + "image_path": "b8521042e25f111bdaac593dbe6b67fc61fa70b6a3348105b00e1d36391546d9.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 435, + 391, + 450.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 219, + 450.0, + 391, + 465.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 133, + 489 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 476, + 148, + 488 + ], + "score": 0.89, + "content": "H _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 475, + 315, + 489 + ], + "score": 1.0, + "content": "is the Hessian w.r.t. the weights in layer", + "type": "text" + }, + { + "bbox": [ + 315, + 476, + 322, + 486 + ], + "score": 0.62, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 475, + 328, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 328, + 474, + 398, + 489 + ], + "score": 0.93, + "content": "\\mathcal { Q } _ { \\lambda } = a _ { \\lambda - 1 } a _ { \\lambda - 1 } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "denotes the covariance of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 485, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 488, + 307, + 507 + ], + "score": 1.0, + "content": "the incoming activations aλ−1 and Hλ = ∂ E∂h ∂h", + "type": "text" + }, + { + "bbox": [ + 306, + 485, + 505, + 503 + ], + "score": 1.0, + "content": "the pre-activation Hessian, i.e. the Hessian of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 254, + 516 + ], + "score": 1.0, + "content": "error w.r.t. the linear pre-activations", + "type": "text" + }, + { + "bbox": [ + 254, + 505, + 266, + 515 + ], + "score": 0.86, + "content": "h _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "in a layer. We provide the derivation for this result as well", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 514, + 307, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 232, + 527 + ], + "score": 1.0, + "content": "as the recursion for calculating", + "type": "text" + }, + { + "bbox": [ + 232, + 515, + 241, + 525 + ], + "score": 0.78, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 514, + 307, + 527 + ], + "score": 1.0, + "content": "in Appendix A.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 530, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 505, + 544 + ], + "score": 1.0, + "content": "The Kronecker factorisation holds two key advantages: the matrices that need be computed and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 541, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 413, + 556 + ], + "score": 1.0, + "content": "stored are much smaller — if we assume all layers to be of dimensionality", + "type": "text" + }, + { + "bbox": [ + 413, + 543, + 423, + 553 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 541, + 505, + 556 + ], + "score": 1.0, + "content": ", the two factors are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 552, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 155, + 569 + ], + "score": 1.0, + "content": "each of size", + "type": "text" + }, + { + "bbox": [ + 155, + 554, + 169, + 565 + ], + "score": 0.89, + "content": "D ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 552, + 449, + 569 + ], + "score": 1.0, + "content": ", whereas the full Hessian for the weights of only one layer would have", + "type": "text" + }, + { + "bbox": [ + 450, + 554, + 464, + 565 + ], + "score": 0.89, + "content": "D ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 552, + 506, + 569 + ], + "score": 1.0, + "content": "elements.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 580 + ], + "score": 1.0, + "content": "Furthermore, the inverse of a Kronecker product is equal to the Kronecker product of the inverses,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 577, + 382, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 382, + 590 + ], + "score": 1.0, + "content": "so it is only necessary to invert those two moderately sized matrices.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "In order to maintain this structure over a minibatch of data, all Kronecker factored second-order", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "score": 1.0, + "content": "methods make two core approximations: First, they only model the diagonal blocks corresponding", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 370, + 628 + ], + "score": 1.0, + "content": "to the weights of a layer, such that the curvature decomposes into", + "type": "text" + }, + { + "bbox": [ + 370, + 617, + 379, + 626 + ], + "score": 0.74, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "independent matrices. Second,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 158, + 639 + ], + "score": 1.0, + "content": "they assume", + "type": "text" + }, + { + "bbox": [ + 158, + 627, + 172, + 639 + ], + "score": 0.9, + "content": "\\mathcal { Q } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 627, + 190, + 639 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 627, + 205, + 638 + ], + "score": 0.89, + "content": "\\mathcal { H } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "to be independent. This is in order to maintain the Kronecker factorisation", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 186, + 651 + ], + "score": 1.0, + "content": "in expectation, i.e.", + "type": "text" + }, + { + "bbox": [ + 186, + 638, + 319, + 650 + ], + "score": 0.89, + "content": "\\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\otimes \\mathcal { H } _ { \\lambda } \\right] \\approx \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\right] \\otimes \\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 637, + 506, + 651 + ], + "score": 1.0, + "content": ", since the expectation of a Kronecker product", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 649, + 303, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 303, + 660 + ], + "score": 1.0, + "content": "is not guaranteed to be Kronecker factored itself.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "The main difference between the Kronecker factored second-order optimisers lies in how they ef-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 195, + 689 + ], + "score": 1.0, + "content": "ficiently approximate", + "type": "text" + }, + { + "bbox": [ + 195, + 677, + 224, + 689 + ], + "score": 0.88, + "content": "\\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 676, + 505, + 689 + ], + "score": 1.0, + "content": ". For exact calculation, it would be necessary to pass back an entire", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "matrix per data point in a minibatch, which imposes infeasible memory and computational require-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "ments. KFRA (Botev et al., 2017) simply passes back the expectation at every layer, while KFAC", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "(Martens & Grosse, 2015) utilises the Fisher identity to only propagate a vector rather than a ma-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "trix, approximating the Kronecker factors with a stochastic rank-one matrix for each data point.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 506, + 106 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 121, + 340, + 134 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 340, + 136 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 340, + 136 + ], + "score": 1.0, + "content": "2 THE CURVATURE OF NEURAL NETWORKS", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 146, + 505, + 202 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 504, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 504, + 158 + ], + "score": 1.0, + "content": "Our method is inspired by recent Kronecker factored approximations of the curvature of a neural", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "score": 1.0, + "content": "network (Martens & Grosse, 2015; Botev et al., 2017) for optimisation and we give a high-level", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 169, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 505, + 181 + ], + "score": 1.0, + "content": "review of these in the following. While the two methods approximate the Gauss-Newton and Fisher", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "matrix respectively, as they are guaranteed to be positive semi-definite (p.s.d.), we base all of our", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 357, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 357, + 203 + ], + "score": 1.0, + "content": "discussion on the Hessian in order to be as general as possible.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 146, + 505, + 203 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 216, + 263, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 264, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 264, + 228 + ], + "score": 1.0, + "content": "2.1 NEURAL NETWORK NOTATION", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 316, + 249 + ], + "score": 1.0, + "content": "We denote a feedforward network as taking an input", + "type": "text" + }, + { + "bbox": [ + 316, + 239, + 346, + 248 + ], + "score": 0.89, + "content": "a _ { 0 } = x", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 236, + 446, + 249 + ], + "score": 1.0, + "content": "and producing an output", + "type": "text" + }, + { + "bbox": [ + 446, + 237, + 459, + 248 + ], + "score": 0.87, + "content": "h _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 236, + 505, + 249 + ], + "score": 1.0, + "content": ". The inter-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 243, + 261 + ], + "score": 1.0, + "content": "mediate representations for layers", + "type": "text" + }, + { + "bbox": [ + 244, + 248, + 293, + 259 + ], + "score": 0.92, + "content": "\\lambda = 1 , . . . , L", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 248, + 352, + 261 + ], + "score": 1.0, + "content": "are denoted as", + "type": "text" + }, + { + "bbox": [ + 353, + 248, + 413, + 259 + ], + "score": 0.93, + "content": "h _ { \\lambda } = W _ { \\lambda } a _ { \\lambda - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 248, + 431, + 261 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 432, + 248, + 485, + 260 + ], + "score": 0.92, + "content": "a _ { \\lambda } = f _ { \\lambda } ( h _ { \\lambda } )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 248, + 505, + 261 + ], + "score": 1.0, + "content": ". 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The network parameters are optimised w.r.t. an error function", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 279, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 107, + 281, + 145, + 293 + ], + "score": 0.93, + "content": "E ( y , h _ { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 279, + 189, + 294 + ], + "score": 1.0, + "content": "for targets", + "type": "text" + }, + { + "bbox": [ + 190, + 282, + 196, + 292 + ], + "score": 0.72, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 279, + 506, + 294 + ], + "score": 1.0, + "content": ". Most commonly used error functions, such as squared error and categorical", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 291, + 486, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 424, + 304 + ], + "score": 1.0, + "content": "cross-entropy, can be interpreted as exponential family negative log likelihoods", + "type": "text" + }, + { + "bbox": [ + 425, + 291, + 482, + 303 + ], + "score": 0.92, + "content": "- \\log p ( y | h _ { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 291, + 486, + 304 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 236, + 506, + 304 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 316, + 373, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 375, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 375, + 330 + ], + "score": 1.0, + "content": "2.2 KRONECKER FACTORED SECOND-ORDER OPTIMISATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 336, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "Traditional second-order methods use either the Hessian matrix or a positive semi-definite approx-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 348, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 342, + 363 + ], + "score": 1.0, + "content": "imation thereof to generate parameter updates of the form", + "type": "text" + }, + { + "bbox": [ + 343, + 348, + 390, + 362 + ], + "score": 0.93, + "content": "\\Delta = C ^ { - 1 } g", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 348, + 421, + 363 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 421, + 350, + 430, + 360 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 348, + 506, + 363 + ], + "score": 1.0, + "content": "is the chosen cur-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 180, + 374 + ], + "score": 1.0, + "content": "vature matrix and", + "type": "text" + }, + { + "bbox": [ + 180, + 363, + 186, + 372 + ], + "score": 0.77, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "the gradient of the error function parameterised by the network. However, this", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "curvature matrix is infeasbile to compute for modern neural networks as their number of parameters", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 381, + 427, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 381, + 285, + 396 + ], + "score": 1.0, + "content": "is often in the millions, rendering the size of", + "type": "text" + }, + { + "bbox": [ + 286, + 384, + 294, + 393 + ], + "score": 0.85, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 381, + 427, + 396 + ], + "score": 1.0, + "content": "of the order of several terabytes.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 336, + 506, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 399, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "score": 1.0, + "content": "Recent work (Martens & Grosse, 2015; Botev et al., 2017) exploits that, for a single data point, the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 410, + 378, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 378, + 424 + ], + "score": 1.0, + "content": "diagonal blocks of these curvature matrices are Kronecker factored:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 398, + 505, + 424 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 435, + 391, + 465 + ], + "lines": [ + { + "bbox": [ + 219, + 435, + 391, + 465 + ], + "spans": [ + { + "bbox": [ + 219, + 435, + 391, + 465 + ], + "score": 0.94, + "content": "H _ { \\lambda } = \\frac { \\partial ^ { 2 } E } { \\partial \\operatorname { v e c } ( W _ { \\lambda } ) \\partial \\operatorname { v e c } ( W _ { \\lambda } ) } = \\mathcal { Q } _ { \\lambda } \\otimes \\mathcal { H } _ { \\lambda }", + "type": "interline_equation", + "image_path": "b8521042e25f111bdaac593dbe6b67fc61fa70b6a3348105b00e1d36391546d9.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 435, + 391, + 450.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 219, + 450.0, + 391, + 465.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 133, + 489 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 476, + 148, + 488 + ], + "score": 0.89, + "content": "H _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 475, + 315, + 489 + ], + "score": 1.0, + "content": "is the Hessian w.r.t. the weights in layer", + "type": "text" + }, + { + "bbox": [ + 315, + 476, + 322, + 486 + ], + "score": 0.62, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 475, + 328, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 328, + 474, + 398, + 489 + ], + "score": 0.93, + "content": "\\mathcal { Q } _ { \\lambda } = a _ { \\lambda - 1 } a _ { \\lambda - 1 } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "denotes the covariance of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 485, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 488, + 307, + 507 + ], + "score": 1.0, + "content": "the incoming activations aλ−1 and Hλ = ∂ E∂h ∂h", + "type": "text" + }, + { + "bbox": [ + 306, + 485, + 505, + 503 + ], + "score": 1.0, + "content": "the pre-activation Hessian, i.e. the Hessian of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 254, + 516 + ], + "score": 1.0, + "content": "error w.r.t. the linear pre-activations", + "type": "text" + }, + { + "bbox": [ + 254, + 505, + 266, + 515 + ], + "score": 0.86, + "content": "h _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "in a layer. We provide the derivation for this result as well", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 514, + 307, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 232, + 527 + ], + "score": 1.0, + "content": "as the recursion for calculating", + "type": "text" + }, + { + "bbox": [ + 232, + 515, + 241, + 525 + ], + "score": 0.78, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 514, + 307, + 527 + ], + "score": 1.0, + "content": "in Appendix A.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 474, + 506, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 530, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 505, + 544 + ], + "score": 1.0, + "content": "The Kronecker factorisation holds two key advantages: the matrices that need be computed and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 541, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 413, + 556 + ], + "score": 1.0, + "content": "stored are much smaller — if we assume all layers to be of dimensionality", + "type": "text" + }, + { + "bbox": [ + 413, + 543, + 423, + 553 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 541, + 505, + 556 + ], + "score": 1.0, + "content": ", the two factors are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 552, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 155, + 569 + ], + "score": 1.0, + "content": "each of size", + "type": "text" + }, + { + "bbox": [ + 155, + 554, + 169, + 565 + ], + "score": 0.89, + "content": "D ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 552, + 449, + 569 + ], + "score": 1.0, + "content": ", whereas the full Hessian for the weights of only one layer would have", + "type": "text" + }, + { + "bbox": [ + 450, + 554, + 464, + 565 + ], + "score": 0.89, + "content": "D ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 552, + 506, + 569 + ], + "score": 1.0, + "content": "elements.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 580 + ], + "score": 1.0, + "content": "Furthermore, the inverse of a Kronecker product is equal to the Kronecker product of the inverses,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 577, + 382, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 382, + 590 + ], + "score": 1.0, + "content": "so it is only necessary to invert those two moderately sized matrices.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 530, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "In order to maintain this structure over a minibatch of data, all Kronecker factored second-order", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "score": 1.0, + "content": "methods make two core approximations: First, they only model the diagonal blocks corresponding", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 370, + 628 + ], + "score": 1.0, + "content": "to the weights of a layer, such that the curvature decomposes into", + "type": "text" + }, + { + "bbox": [ + 370, + 617, + 379, + 626 + ], + "score": 0.74, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "independent matrices. Second,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 158, + 639 + ], + "score": 1.0, + "content": "they assume", + "type": "text" + }, + { + "bbox": [ + 158, + 627, + 172, + 639 + ], + "score": 0.9, + "content": "\\mathcal { Q } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 627, + 190, + 639 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 627, + 205, + 638 + ], + "score": 0.89, + "content": "\\mathcal { H } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "to be independent. This is in order to maintain the Kronecker factorisation", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 186, + 651 + ], + "score": 1.0, + "content": "in expectation, i.e.", + "type": "text" + }, + { + "bbox": [ + 186, + 638, + 319, + 650 + ], + "score": 0.89, + "content": "\\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\otimes \\mathcal { H } _ { \\lambda } \\right] \\approx \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\right] \\otimes \\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 637, + 506, + 651 + ], + "score": 1.0, + "content": ", since the expectation of a Kronecker product", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 649, + 303, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 303, + 660 + ], + "score": 1.0, + "content": "is not guaranteed to be Kronecker factored itself.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 594, + 506, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "The main difference between the Kronecker factored second-order optimisers lies in how they ef-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 195, + 689 + ], + "score": 1.0, + "content": "ficiently approximate", + "type": "text" + }, + { + "bbox": [ + 195, + 677, + 224, + 689 + ], + "score": 0.88, + "content": "\\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 676, + 505, + 689 + ], + "score": 1.0, + "content": ". For exact calculation, it would be necessary to pass back an entire", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "matrix per data point in a minibatch, which imposes infeasible memory and computational require-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "ments. 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An approximation that is easy to compute in modern automatic differentiation frameworks", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 518, + 490, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 248, + 530 + ], + "score": 1.0, + "content": "is the diagonal of the Fisher matrix", + "type": "text" + }, + { + "bbox": [ + 249, + 518, + 257, + 527 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 518, + 490, + 530 + ], + "score": 1.0, + "content": ", which is simply the expectation of the squared gradients:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 550, + 478, + 572 + ], + "lines": [ + { + "bbox": [ + 120, + 550, + 478, + 572 + ], + "spans": [ + { + "bbox": [ + 120, + 550, + 478, + 572 + ], + "score": 0.92, + "content": "H \\approx \\mathrm { d i a g } ( F ) = \\mathrm { d i a g } ( \\mathbb { E } \\left[ \\nabla _ { \\theta } \\log p ( y | x ) \\nabla _ { \\theta } \\log p ( y | x ) ^ { \\top } \\right] ) = \\mathrm { d i a g } ( \\mathbb { E } \\left[ ( \\nabla _ { \\theta } \\log p ( y | x ) ) ^ { 2 } \\right] )", + "type": "interline_equation", + "image_path": "67d188bafccb6d7e7a78a805a8871adc852fc39c49b2f49bd9a5d47743c3ee5d.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 120, + 550, + 478, + 572 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "where diag extracts the diagonal of a matrix or turns a vector into a diagonal matrix. 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Specifically, if we approximate the log posterior over the weights of a network given", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 225, + 316, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 149, + 237 + ], + "score": 1.0, + "content": "some data", + "type": "text" + }, + { + "bbox": [ + 149, + 226, + 159, + 235 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 225, + 256, + 237 + ], + "score": 1.0, + "content": "around a MAP estimate", + "type": "text" + }, + { + "bbox": [ + 257, + 225, + 267, + 235 + ], + "score": 0.85, + "content": "\\boldsymbol { \\theta } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 225, + 316, + 237 + ], + "score": 1.0, + "content": ", we obtain:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 191, + 505, + 237 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 252, + 411, + 276 + ], + "lines": [ + { + "bbox": [ + 201, + 252, + 411, + 276 + ], + "spans": [ + { + "bbox": [ + 201, + 252, + 411, + 276 + ], + "score": 0.92, + "content": "\\log p ( \\boldsymbol { \\theta } | \\mathcal { D } ) \\approx \\log p ( \\boldsymbol { \\theta } ^ { * } | \\mathcal { D } ) - \\frac { 1 } { 2 } \\big ( \\boldsymbol { \\theta } - \\boldsymbol { \\theta } ^ { * } \\big ) ^ { \\top } \\bar { H } ( \\boldsymbol { \\theta } - \\boldsymbol { \\theta } ^ { * } )", + "type": "interline_equation", + "image_path": "333314683c0abf6a5a733994f08d5666621f38a95390596a723f4505fce14fb6.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 201, + 252, + 411, + 276 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 134, + 300 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 286, + 252, + 299 + ], + "score": 0.92, + "content": "\\theta = [ \\mathrm { v e c } ( W _ { 1 } ) , . . . , \\mathrm { v e c } ( W _ { L } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 285, + 405, + 300 + ], + "score": 1.0, + "content": "is the stacked vector of weights and", + "type": "text" + }, + { + "bbox": [ + 405, + 286, + 454, + 299 + ], + "score": 0.93, + "content": "\\bar { H } = \\mathbb { E } \\left[ H \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 285, + 505, + 300 + ], + "score": 1.0, + "content": "the average", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 300, + 504, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 504, + 312 + ], + "score": 1.0, + "content": "Hessian of the negative log posterior1. The first order term is missing because we expand the func-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 206, + 324 + ], + "score": 1.0, + "content": "tion around a maximum", + "type": "text" + }, + { + "bbox": [ + 206, + 310, + 216, + 321 + ], + "score": 0.84, + "content": "\\boldsymbol { \\theta } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 310, + 505, + 324 + ], + "score": 1.0, + "content": ", where the gradient is zero. If we exponentiate this equation, it is easy", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 390, + 334 + ], + "score": 1.0, + "content": "to notice that the right-hand side is of Gaussian functional form for", + "type": "text" + }, + { + "bbox": [ + 390, + 322, + 397, + 332 + ], + "score": 0.76, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 322, + 505, + 334 + ], + "score": 1.0, + "content": ", thus we obtain a normal", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "distribution by integrating over it. The posterior over the weights is then approximated as Gaussian:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 285, + 505, + 345 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 270, + 360, + 342, + 376 + ], + "lines": [ + { + "bbox": [ + 270, + 360, + 342, + 376 + ], + "spans": [ + { + "bbox": [ + 270, + 360, + 342, + 376 + ], + "score": 0.91, + "content": "\\theta \\sim \\mathcal { N } ( \\theta ^ { * } , \\bar { H } ^ { - 1 } )", + "type": "interline_equation", + "image_path": "cf1e95882914704d05c830b48d2f12d37440a3d90aa302355aba74889230764b.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 270, + 360, + 342, + 376 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 507, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 147, + 399 + ], + "score": 1.0, + "content": "assuming", + "type": "text" + }, + { + "bbox": [ + 147, + 385, + 157, + 396 + ], + "score": 0.84, + "content": "\\bar { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "is p.s.d. We can then approximate the posterior mean when predicting on unseen data", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 396, + 493, + 414 + ], + "spans": [ + { + "bbox": [ + 107, + 399, + 120, + 410 + ], + "score": 0.86, + "content": "D ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 396, + 248, + 414 + ], + "score": 1.0, + "content": "by averaging the predictions of", + "type": "text" + }, + { + "bbox": [ + 248, + 400, + 257, + 410 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 396, + 345, + 414 + ], + "score": 1.0, + "content": "Monte Carlo samples", + "type": "text" + }, + { + "bbox": [ + 346, + 397, + 361, + 410 + ], + "score": 0.9, + "content": "{ \\boldsymbol { \\theta } } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 396, + 493, + 414 + ], + "score": 1.0, + "content": "from the approximate posterior:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 106, + 385, + 506, + 414 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 426, + 415, + 461 + ], + "lines": [ + { + "bbox": [ + 196, + 426, + 415, + 461 + ], + "spans": [ + { + "bbox": [ + 196, + 426, + 415, + 461 + ], + "score": 0.93, + "content": "p ( \\mathcal { D } ^ { * } | \\mathcal { D } ) = \\int p ( \\mathcal { D } ^ { * } | \\boldsymbol { \\theta } ) p ( \\boldsymbol { \\theta } | \\mathcal { D } ) d \\boldsymbol { \\theta } \\approx \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } p ( \\mathcal { D } ^ { * } | \\boldsymbol { \\theta } ^ { ( t ) } )", + "type": "interline_equation", + "image_path": "7f13d48c5e36c255d6f912a3e1f350bef6320aee46d47d9383bec0c07e6aafda.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 426, + 415, + 443.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 196, + 443.5, + 415, + 461.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 474, + 296, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 298, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 298, + 486 + ], + "score": 1.0, + "content": "3.2 DIAGONAL LAPLACE APPROXIMATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 506, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "Unfortunately, it is not feasible to compute or invert the Hessian matrix w.r.t. all of the weights", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "jointly. An approximation that is easy to compute in modern automatic differentiation frameworks", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 518, + 490, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 248, + 530 + ], + "score": 1.0, + "content": "is the diagonal of the Fisher matrix", + "type": "text" + }, + { + "bbox": [ + 249, + 518, + 257, + 527 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 518, + 490, + 530 + ], + "score": 1.0, + "content": ", which is simply the expectation of the squared gradients:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 495, + 505, + 530 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 550, + 478, + 572 + ], + "lines": [ + { + "bbox": [ + 120, + 550, + 478, + 572 + ], + "spans": [ + { + "bbox": [ + 120, + 550, + 478, + 572 + ], + "score": 0.92, + "content": "H \\approx \\mathrm { d i a g } ( F ) = \\mathrm { d i a g } ( \\mathbb { E } \\left[ \\nabla _ { \\theta } \\log p ( y | x ) \\nabla _ { \\theta } \\log p ( y | x ) ^ { \\top } \\right] ) = \\mathrm { d i a g } ( \\mathbb { E } \\left[ ( \\nabla _ { \\theta } \\log p ( y | x ) ) ^ { 2 } \\right] )", + "type": "interline_equation", + "image_path": "67d188bafccb6d7e7a78a805a8871adc852fc39c49b2f49bd9a5d47743c3ee5d.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 120, + 550, + 478, + 572 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "where diag extracts the diagonal of a matrix or turns a vector into a diagonal matrix. Such diagonal", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "approximations to the curvature of a neural network have been used successfully for pruning the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 606, + 489, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 489, + 619 + ], + "score": 1.0, + "content": "weights (LeCun et al., 1990) and, more recently, for transfer learning (Kirkpatrick et al., 2017).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 584, + 505, + 619 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 622, + 492, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 492, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 492, + 637 + ], + "score": 1.0, + "content": "This corresponds to modelling the weights with a Normal distribution with diagonal covariance:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 621, + 492, + 637 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 650, + 425, + 666 + ], + "lines": [ + { + "bbox": [ + 187, + 650, + 425, + 666 + ], + "spans": [ + { + "bbox": [ + 187, + 650, + 425, + 666 + ], + "score": 0.9, + "content": "\\operatorname { v e c } ( W _ { \\lambda } ) \\sim { \\mathcal { N } } ( \\operatorname { v e c } ( W _ { \\lambda } ^ { * } ) , \\operatorname { d i a g } ( F _ { \\lambda } ) ^ { - 1 } ) \\quad { \\mathrm { f o r ~ } } \\lambda = 1 , \\dots , L", + "type": "interline_equation", + "image_path": "d202f8c5ff2b2edab4d5ba7d43596938a2eb555b65c7068640ed3f9320b4c52f.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 187, + 650, + 425, + 666 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 676, + 504, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Unfortunately, even if the Taylor approximation is accurate, this will place significant probability", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 687, + 470, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 470, + 700 + ], + "score": 1.0, + "content": "mass in low probability areas of the true posterior if some weights exhibit high covariance.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 676, + 505, + 700 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 352, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 353, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 353, + 95 + ], + "score": 1.0, + "content": "3.3 KRONECKER FACTORED LAPLACE APPROXIMATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 117 + ], + "score": 1.0, + "content": "So while it is desirable to model the covariance between the weights, some approximations are", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 504, + 127 + ], + "score": 1.0, + "content": "needed in order to remain computationally efficient. First, we assume the weights of the differ-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "ent layers to be independent. This corresponds to the block-diagonal approximation in KFAC and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "score": 1.0, + "content": "KFRA, which empirically preserves sufficient information about the curvature to obtain competitive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 504, + 160 + ], + "score": 1.0, + "content": "optimisation performance. For our purposes this means that our posterior factorises over the layers.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 177 + ], + "score": 1.0, + "content": "As discussed above, the Hessian of the log-likelihood for a single datapoint is Kronecker factored,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 279, + 190 + ], + "score": 1.0, + "content": "and we denote the two factor matrices as", + "type": "text" + }, + { + "bbox": [ + 280, + 175, + 352, + 189 + ], + "score": 0.92, + "content": "{ \\cal H } _ { \\lambda } = { \\mathcal Q } _ { \\lambda } \\otimes { \\mathcal H } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 176, + 505, + 190 + ], + "score": 1.0, + "content": ".2 By further assuming independence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 469, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 142, + 201 + ], + "score": 1.0, + "content": "between", + "type": "text" + }, + { + "bbox": [ + 142, + 188, + 152, + 199 + ], + "score": 0.82, + "content": "\\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 187, + 169, + 201 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 170, + 188, + 180, + 198 + ], + "score": 0.81, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 187, + 469, + 201 + ], + "score": 1.0, + "content": "in all layers, we can approximate the expected Hessian of each layer as:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 214, + 395, + 228 + ], + "lines": [ + { + "bbox": [ + 217, + 214, + 395, + 228 + ], + "spans": [ + { + "bbox": [ + 217, + 214, + 395, + 228 + ], + "score": 0.91, + "content": "\\mathbb { E } \\left[ H _ { \\lambda } \\right] = \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\otimes \\mathcal { H } _ { \\lambda } \\right] \\approx \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\right] \\otimes \\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]", + "type": "interline_equation", + "image_path": "d4e47d88af0576da23b9a0a7ce485ed4397c8ae1f2380f8a4bc9dfa977f43fd8.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 217, + 214, + 395, + 228 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "Hence, the Hessian of every layer is Kronecker factored over an entire dataset and the Laplace", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 248, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 259 + ], + "score": 1.0, + "content": "approximation can be approximated by a product of Gaussians. Each Gaussian has a Kronecker", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "score": 1.0, + "content": "factored covariance, corresponding to a matrix normal distribution (Gupta & Nagar, 1999), which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "considers the two Kronecker factors of the covariance to be the covariances of the rows and columns", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "of a matrix. The two factors are much smaller than the full covariance and allow for significantly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 292, + 495, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 495, + 303 + ], + "score": 1.0, + "content": "more efficient inversion and sampling (we review the matrix normal distribution in Appendix B).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 334, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 334, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 294, + 321 + ], + "score": 1.0, + "content": "Our resulting posterior for the weights in layer", + "type": "text" + }, + { + "bbox": [ + 295, + 309, + 302, + 318 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 307, + 334, + 321 + ], + "score": 1.0, + "content": "is then:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 333, + 367, + 349 + ], + "lines": [ + { + "bbox": [ + 244, + 333, + 367, + 349 + ], + "spans": [ + { + "bbox": [ + 244, + 333, + 367, + 349 + ], + "score": 0.91, + "content": "W _ { \\lambda } \\sim \\mathcal { M N } ( W _ { \\lambda } ^ { \\ast } , \\bar { \\mathcal { Q } } _ { \\lambda } ^ { - 1 } , \\bar { \\mathcal { H } } _ { \\lambda } ^ { - 1 } )", + "type": "interline_equation", + "image_path": "f8e7ca58ef6b39545d43d71a592e83d9d0fa209d6d79cfde44af829dd7be389d.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 244, + 333, + 367, + 349 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 506, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 382, + 370 + ], + "score": 1.0, + "content": "In contrast to optimisation methods, we do not need to approximate", + "type": "text" + }, + { + "bbox": [ + 382, + 357, + 411, + 369 + ], + "score": 0.92, + "content": "\\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "as it is only calculated", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "once. However, when it is possible to augment the data (e.g. randomised cropping of images), it", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 379, + 445, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 445, + 391 + ], + "score": 1.0, + "content": "may be advantageous. We provide a more detailed discussion of this in Appendix C.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 105, + 403, + 458, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 458, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 458, + 415 + ], + "score": 1.0, + "content": "3.4 INCORPORATING THE PRIOR AND REGULARISING THE CURVATURE FACTORS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "Just as the log posterior, the Hessian decomposes into a term depending on the data log likelihood", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 290, + 448 + ], + "score": 1.0, + "content": "and one on the prior. For the commonly used", + "type": "text" + }, + { + "bbox": [ + 290, + 435, + 303, + 447 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "-regularisation, corresponding to a Gaussian prior,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "the Hessian is equal to the precision of the prior times the identity matrix. We approximate this by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 457, + 465, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 465, + 470 + ], + "score": 1.0, + "content": "adding a multiple of the identity to each of the Kronecker factors from the log likelihood:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 481, + 478, + 515 + ], + "lines": [ + { + "bbox": [ + 133, + 481, + 478, + 515 + ], + "spans": [ + { + "bbox": [ + 133, + 481, + 478, + 515 + ], + "score": 0.93, + "content": "H _ { \\lambda } = N \\mathbb { E } \\left[ - \\frac { \\partial ^ { 2 } \\log p ( \\mathcal { D } | \\theta ) } { \\partial \\theta ^ { 2 } } \\right] + \\tau I \\approx ( \\sqrt { N } \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\right] + \\sqrt { \\tau } I ) \\otimes ( \\sqrt { N } \\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right] + \\sqrt { \\tau } I )", + "type": "interline_equation", + "image_path": "6f831ef21c86d130330cc83051752a1f5eecb53501d281a26f3ec861b7202903.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 133, + 481, + 478, + 492.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 133, + 492.3333333333333, + 478, + 503.66666666666663 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 133, + 503.66666666666663, + 478, + 515.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 522, + 504, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 133, + 536 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 525, + 140, + 533 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 522, + 363, + 536 + ], + "score": 1.0, + "content": "is the precision of the Gaussian prior on the weights and", + "type": "text" + }, + { + "bbox": [ + 363, + 523, + 374, + 533 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "the size of the dataset. However,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "we can also treat them as hyperparameters and optimise them w.r.t. the predictive performance on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "a validation set. We emphasise that this can be done without retraining the network, so it does not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 556, + 373, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 373, + 568 + ], + "score": 1.0, + "content": "impose a large computational overhead and is trivial to parallelise.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 138, + 585 + ], + "score": 1.0, + "content": "Setting", + "type": "text" + }, + { + "bbox": [ + 138, + 573, + 149, + 582 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "to a larger value than the size of the dataset can be interpreted as including duplicates", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "of the data points as pseudo-observations. Adding a multiple of the uncertainty to the precision", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "matrix decreases the uncertainty about each parameter. This has a regularising effect both on our", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "approximation to the true Laplace, which may be overestimating the variance in certain directions", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "due to ignoring the covariances between the layers, as well as the Laplace approximation itself,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 628, + 445, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 445, + 639 + ], + "score": 1.0, + "content": "which may be placing probability mass in low probability areas of the true posterior.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 655, + 211, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 213, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 213, + 669 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 108, + 680, + 503, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 693 + ], + "score": 1.0, + "content": "Most recent attempts to approximating the posterior of a neural network are based on formulating", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "score": 1.0, + "content": "an approximate distribution to the posterior and optimising the variational lower bound w.r.t. its", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 711, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "2We assume a uniform prior for now, such that the Hessians of the posterior and the log likelihood are equal.", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 494, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 494, + 732 + ], + "score": 1.0, + "content": "We discuss how we incorporate a non-zero Hessian of a prior into the Kronecker factors in the next section.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 352, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 353, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 353, + 95 + ], + "score": 1.0, + "content": "3.3 KRONECKER FACTORED LAPLACE APPROXIMATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 117 + ], + "score": 1.0, + "content": "So while it is desirable to model the covariance between the weights, some approximations are", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 504, + 127 + ], + "score": 1.0, + "content": "needed in order to remain computationally efficient. First, we assume the weights of the differ-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "ent layers to be independent. This corresponds to the block-diagonal approximation in KFAC and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "score": 1.0, + "content": "KFRA, which empirically preserves sufficient information about the curvature to obtain competitive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 504, + 160 + ], + "score": 1.0, + "content": "optimisation performance. For our purposes this means that our posterior factorises over the layers.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 102, + 506, + 160 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 177 + ], + "score": 1.0, + "content": "As discussed above, the Hessian of the log-likelihood for a single datapoint is Kronecker factored,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 279, + 190 + ], + "score": 1.0, + "content": "and we denote the two factor matrices as", + "type": "text" + }, + { + "bbox": [ + 280, + 175, + 352, + 189 + ], + "score": 0.92, + "content": "{ \\cal H } _ { \\lambda } = { \\mathcal Q } _ { \\lambda } \\otimes { \\mathcal H } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 176, + 505, + 190 + ], + "score": 1.0, + "content": ".2 By further assuming independence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 469, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 142, + 201 + ], + "score": 1.0, + "content": "between", + "type": "text" + }, + { + "bbox": [ + 142, + 188, + 152, + 199 + ], + "score": 0.82, + "content": "\\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 187, + 169, + 201 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 170, + 188, + 180, + 198 + ], + "score": 0.81, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 187, + 469, + 201 + ], + "score": 1.0, + "content": "in all layers, we can approximate the expected Hessian of each layer as:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 163, + 505, + 201 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 214, + 395, + 228 + ], + "lines": [ + { + "bbox": [ + 217, + 214, + 395, + 228 + ], + "spans": [ + { + "bbox": [ + 217, + 214, + 395, + 228 + ], + "score": 0.91, + "content": "\\mathbb { E } \\left[ H _ { \\lambda } \\right] = \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\otimes \\mathcal { H } _ { \\lambda } \\right] \\approx \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\right] \\otimes \\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]", + "type": "interline_equation", + "image_path": "d4e47d88af0576da23b9a0a7ce485ed4397c8ae1f2380f8a4bc9dfa977f43fd8.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 217, + 214, + 395, + 228 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "Hence, the Hessian of every layer is Kronecker factored over an entire dataset and the Laplace", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 248, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 259 + ], + "score": 1.0, + "content": "approximation can be approximated by a product of Gaussians. Each Gaussian has a Kronecker", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "score": 1.0, + "content": "factored covariance, corresponding to a matrix normal distribution (Gupta & Nagar, 1999), which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "considers the two Kronecker factors of the covariance to be the covariances of the rows and columns", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "of a matrix. The two factors are much smaller than the full covariance and allow for significantly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 292, + 495, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 495, + 303 + ], + "score": 1.0, + "content": "more efficient inversion and sampling (we review the matrix normal distribution in Appendix B).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 236, + 506, + 303 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 334, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 334, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 294, + 321 + ], + "score": 1.0, + "content": "Our resulting posterior for the weights in layer", + "type": "text" + }, + { + "bbox": [ + 295, + 309, + 302, + 318 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 307, + 334, + 321 + ], + "score": 1.0, + "content": "is then:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 307, + 334, + 321 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 333, + 367, + 349 + ], + "lines": [ + { + "bbox": [ + 244, + 333, + 367, + 349 + ], + "spans": [ + { + "bbox": [ + 244, + 333, + 367, + 349 + ], + "score": 0.91, + "content": "W _ { \\lambda } \\sim \\mathcal { M N } ( W _ { \\lambda } ^ { \\ast } , \\bar { \\mathcal { Q } } _ { \\lambda } ^ { - 1 } , \\bar { \\mathcal { H } } _ { \\lambda } ^ { - 1 } )", + "type": "interline_equation", + "image_path": "f8e7ca58ef6b39545d43d71a592e83d9d0fa209d6d79cfde44af829dd7be389d.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 244, + 333, + 367, + 349 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 506, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 382, + 370 + ], + "score": 1.0, + "content": "In contrast to optimisation methods, we do not need to approximate", + "type": "text" + }, + { + "bbox": [ + 382, + 357, + 411, + 369 + ], + "score": 0.92, + "content": "\\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "as it is only calculated", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "once. However, when it is possible to augment the data (e.g. randomised cropping of images), it", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 379, + 445, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 445, + 391 + ], + "score": 1.0, + "content": "may be advantageous. We provide a more detailed discussion of this in Appendix C.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 356, + 506, + 391 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 403, + 458, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 458, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 458, + 415 + ], + "score": 1.0, + "content": "3.4 INCORPORATING THE PRIOR AND REGULARISING THE CURVATURE FACTORS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "Just as the log posterior, the Hessian decomposes into a term depending on the data log likelihood", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 290, + 448 + ], + "score": 1.0, + "content": "and one on the prior. For the commonly used", + "type": "text" + }, + { + "bbox": [ + 290, + 435, + 303, + 447 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "-regularisation, corresponding to a Gaussian prior,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "the Hessian is equal to the precision of the prior times the identity matrix. We approximate this by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 457, + 465, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 465, + 470 + ], + "score": 1.0, + "content": "adding a multiple of the identity to each of the Kronecker factors from the log likelihood:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 424, + 505, + 470 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 481, + 478, + 515 + ], + "lines": [ + { + "bbox": [ + 133, + 481, + 478, + 515 + ], + "spans": [ + { + "bbox": [ + 133, + 481, + 478, + 515 + ], + "score": 0.93, + "content": "H _ { \\lambda } = N \\mathbb { E } \\left[ - \\frac { \\partial ^ { 2 } \\log p ( \\mathcal { D } | \\theta ) } { \\partial \\theta ^ { 2 } } \\right] + \\tau I \\approx ( \\sqrt { N } \\mathbb { E } \\left[ \\mathcal { Q } _ { \\lambda } \\right] + \\sqrt { \\tau } I ) \\otimes ( \\sqrt { N } \\mathbb { E } \\left[ \\mathcal { H } _ { \\lambda } \\right] + \\sqrt { \\tau } I )", + "type": "interline_equation", + "image_path": "6f831ef21c86d130330cc83051752a1f5eecb53501d281a26f3ec861b7202903.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 133, + 481, + 478, + 492.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 133, + 492.3333333333333, + 478, + 503.66666666666663 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 133, + 503.66666666666663, + 478, + 515.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 522, + 504, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 133, + 536 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 525, + 140, + 533 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 522, + 363, + 536 + ], + "score": 1.0, + "content": "is the precision of the Gaussian prior on the weights and", + "type": "text" + }, + { + "bbox": [ + 363, + 523, + 374, + 533 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "the size of the dataset. However,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "we can also treat them as hyperparameters and optimise them w.r.t. the predictive performance on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "a validation set. We emphasise that this can be done without retraining the network, so it does not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 556, + 373, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 373, + 568 + ], + "score": 1.0, + "content": "impose a large computational overhead and is trivial to parallelise.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 522, + 505, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 138, + 585 + ], + "score": 1.0, + "content": "Setting", + "type": "text" + }, + { + "bbox": [ + 138, + 573, + 149, + 582 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "to a larger value than the size of the dataset can be interpreted as including duplicates", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "of the data points as pseudo-observations. Adding a multiple of the uncertainty to the precision", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "matrix decreases the uncertainty about each parameter. This has a regularising effect both on our", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "approximation to the true Laplace, which may be overestimating the variance in certain directions", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "due to ignoring the covariances between the layers, as well as the Laplace approximation itself,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 628, + 445, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 445, + 639 + ], + "score": 1.0, + "content": "which may be placing probability mass in low probability areas of the true posterior.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 572, + 506, + 639 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 655, + 211, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 213, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 213, + 669 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 108, + 680, + 503, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 693 + ], + "score": 1.0, + "content": "Most recent attempts to approximating the posterior of a neural network are based on formulating", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "score": 1.0, + "content": "an approximate distribution to the posterior and optimising the variational lower bound w.r.t. its", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "parameters. (Graves, 2011; Blundell et al., 2015; Kingma et al., 2015) as well as the expectation", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "propagation based approaches of (Hernandez-Lobato & Adams, 2015) and (Ghosh et al., 2016) ´", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "assume independence between the individual weights which, particularly when optimising the KL", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "divergence, often lets the model underestimate the uncertainty about the weights. Gal & Ghahramani", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "(2016) interpret Dropout to approximate the posterior with a mixture of delta functions, assuming", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "independence between the columns. (Lakshminarayanan et al., 2016) suggest using an ensemble of", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 267, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 267, + 161 + ], + "score": 1.0, + "content": "networks for estimating the uncertainty.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 680, + 505, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "parameters. (Graves, 2011; Blundell et al., 2015; Kingma et al., 2015) as well as the expectation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "propagation based approaches of (Hernandez-Lobato & Adams, 2015) and (Ghosh et al., 2016) ´", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "assume independence between the individual weights which, particularly when optimising the KL", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "divergence, often lets the model underestimate the uncertainty about the weights. Gal & Ghahramani", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "(2016) interpret Dropout to approximate the posterior with a mixture of delta functions, assuming", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "independence between the columns. (Lakshminarayanan et al., 2016) suggest using an ensemble of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 267, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 267, + 161 + ], + "score": 1.0, + "content": "networks for estimating the uncertainty.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "Our work is a scalable approximation of (MacKay, 1992). 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Black dots are data points, the black line shows the noiseless", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "function. The red line shows the deterministic prediction of the network, the blue line the mean", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 616, + 488, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 488, + 628 + ], + "score": 1.0, + "content": "output. Each shade of blue visualises one additional standard deviation. Best viewed on screen.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "Fig. 1 shows the uncertainty obtained from the Kronecker factored and diagonal Laplace approxi-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 653, + 504, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 451, + 664 + ], + "score": 1.0, + "content": "mation applied to the same network, as well as from a full Laplace approximation and", + "type": "text" + }, + { + "bbox": [ + 451, + 653, + 504, + 664 + ], + "score": 0.28, + "content": "5 0 , 0 0 0 \\mathrm { H M C }", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 505, + 676 + ], + "score": 1.0, + "content": "(Neal, 1993) samples. The latter two methods are feasible only for such a small model and dataset.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 674, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 505, + 687 + ], + "score": 1.0, + "content": "For the diagonal and full Laplace approximation we use the Fisher identity and draw one sample per", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "data point. We set the hyperparameters of the Laplace approximations (see Section 3.4) using a grid", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 696, + 504, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 504, + 709 + ], + "score": 1.0, + "content": "search over the likelihood of 20 validation points that are sampled the same way as the training set.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 642, + 505, + 709 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "The regularised Laplace approximations all give an overall good fit to the HMC predictive poste-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 505, + 108 + ], + "score": 1.0, + "content": "rior. Their uncertainty is slightly higher close to the training data and increases more slowly away", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "from the data than that of the HMC posterior. The diagonal and full Laplace approximation re-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "quire stronger regularisation than our Kronecker factored one, as they have higher uncertainty when", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "not regularised. In particular the full Laplace approximation vastly overestimates the uncertainty", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "without additional regularisation, leading to a bad predictive mean (see Appendix E for the corre-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "sponding figures), as the Hessian of the log likelihood is underdetermined. This is commonly the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "case in deep learning, as the number of parameters is typically much larger than the number of data", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "points. Hence restricting the structure of the covariance is not only a computational necessity for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 484, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 484, + 194 + ], + "score": 1.0, + "content": "most architectures, but also allows for more precise estimation of the approximate covariance.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 108, + 212, + 295, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 298, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 298, + 225 + ], + "score": 1.0, + "content": "5.2 OUT-OF-DISTRIBUTION UNCERTAINTY", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "score": 1.0, + "content": "For a more realistic test, similar to (Louizos & Welling, 2017), we assess the uncertainty of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "predictions when classifying data from a different distribution than the training data. For this we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "train a network with two layers of 1024 hidden units and ReLU transfer functions to classify MNIST", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 266, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 234, + 284 + ], + "score": 1.0, + "content": "digits. We use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 234, + 268, + 255, + 280 + ], + "score": 0.9, + "content": "1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 266, + 506, + 284 + ], + "score": 1.0, + "content": "and momentum of 0.9 for 250 epochs. We apply Dropout with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 133, + 292 + ], + "score": 0.89, + "content": "p { = } 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "after each inner layer, as our chief interest is to compare against its uncertainty estimates. We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 290, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 290, + 151, + 307 + ], + "score": 1.0, + "content": "further use", + "type": "text" + }, + { + "bbox": [ + 151, + 293, + 164, + 304 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 290, + 285, + 307 + ], + "score": 1.0, + "content": "-regularisation with a factor of", + "type": "text" + }, + { + "bbox": [ + 285, + 291, + 307, + 303 + ], + "score": 0.9, + "content": "1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 290, + 506, + 307 + ], + "score": 1.0, + "content": "and randomly binarise the images during training", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 289, + 317 + ], + "score": 1.0, + "content": "according to their pixel intensities and draw", + "type": "text" + }, + { + "bbox": [ + 289, + 304, + 315, + 315 + ], + "score": 0.49, + "content": "1 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "such samples per datapoint for estimating the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "curvature factors. We use this network to classify the images in the notMNIST dataset4, which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 143, + 340 + ], + "score": 1.0, + "content": "contains", + "type": "text" + }, + { + "bbox": [ + 143, + 328, + 173, + 338 + ], + "score": 0.88, + "content": "2 8 \\times 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 327, + 307, + 340 + ], + "score": 1.0, + "content": "grey-scale images of the letters", + "type": "text" + }, + { + "bbox": [ + 307, + 328, + 320, + 338 + ], + "score": 0.37, + "content": "\\mathbf { \\dot { A } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 327, + 334, + 340 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 334, + 328, + 345, + 338 + ], + "score": 0.45, + "content": "\\mathbf { \\hat { J } } ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "from various computer fonts, i.e. not", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 405, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 405, + 351 + ], + "score": 1.0, + "content": "digits. An ideal classifier would make uniform predictions over its classes.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 296, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 296, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 296, + 368 + ], + "score": 1.0, + "content": "We compare the uncertainty obtained by pre-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 297, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 297, + 379 + ], + "score": 1.0, + "content": "dicting the digit class of the notMNIST images", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 297, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 297, + 390 + ], + "score": 1.0, + "content": "using 1. a deterministic forward pass through", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 298, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 298, + 402 + ], + "score": 1.0, + "content": "the Dropout trained network, 2. by sampling", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 398, + 297, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 297, + 412 + ], + "score": 1.0, + "content": "different Dropout masks and averaging the pre-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 410, + 297, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 297, + 423 + ], + "score": 1.0, + "content": "dictions, and by sampling different weight ma-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 421, + 297, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 297, + 433 + ], + "score": 1.0, + "content": "trices from 3. the matrix normal distribution", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 432, + 297, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 297, + 444 + ], + "score": 1.0, + "content": "obtained from our Kronecker factored Laplace", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 444, + 297, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 297, + 455 + ], + "score": 1.0, + "content": "approximation as well as 4. the diagonal one.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 454, + 297, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 297, + 465 + ], + "score": 1.0, + "content": "As an additional baseline similar to (Blundell", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 465, + 298, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 298, + 477 + ], + "score": 1.0, + "content": "et al., 2015; Graves, 2011), we compare to a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 475, + 297, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 297, + 488 + ], + "score": 1.0, + "content": "network with identical architecture with a fully", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 487, + 297, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 297, + 499 + ], + "score": 1.0, + "content": "factorised Gaussian (FFG) approximate pos-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 498, + 297, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 297, + 509 + ], + "score": 1.0, + "content": "terior on the weights and a standard normal", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 509, + 297, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 297, + 520 + ], + "score": 1.0, + "content": "prior. 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We use 100 sam-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 541, + 296, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 296, + 555 + ], + "score": 1.0, + "content": "ples for the stochastic forward passes and op-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 297, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 297, + 567 + ], + "score": 1.0, + "content": "timise the hyperparameters of the Laplace ap-", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 30 + }, + { + "type": "image", + "bbox": [ + 313, + 383, + 485, + 500 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 313, + 383, + 485, + 500 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 313, + 383, + 485, + 500 + ], + "spans": [ + { + "bbox": [ + 313, + 383, + 485, + 500 + ], + "score": 0.966, + "type": "image", + "image_path": "d894a7edf56530a450cd12d773fa589c25cc119d5b25ad9f5c7d7640b554c41f.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 313, + 383, + 485, + 396.0 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 313, + 396.0, + 485, + 409.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 313, + 409.0, + 485, + 422.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 313, + 422.0, + 485, + 435.0 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 313, + 435.0, + 485, + 448.0 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 313, + 448.0, + 485, + 461.0 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 313, + 461.0, + 485, + 474.0 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 313, + 474.0, + 485, + 487.0 + ], + "spans": [], + "index": 47 + }, + { + "bbox": [ + 313, + 487.0, + 485, + 500.0 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 511, + 504, + 543 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 304, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "Figure 2: Predictive entropy on notMNIST ob-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 303, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 303, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "tained from different methods for the forward pass", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 303, + 533, + 437, + 543 + ], + "spans": [ + { + "bbox": [ + 303, + 533, + 437, + 543 + ], + "score": 1.0, + "content": "on a network trained on MNIST.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50 + } + ], + "index": 47.0 + }, + { + "type": "text", + "bbox": [ + 108, + 564, + 383, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 384, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 384, + 577 + ], + "score": 1.0, + "content": "proximations w.r.t. the cross-entropy on the validation set of MNIST.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 107, + 581, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "We measure the uncertainty of the different methods as the entropy of the predictive distribution,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "which has a minimal value of 0 when a single class is predicted with certainty and a maximum of", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "about 2.3 for uniform predictions. Fig. 2 shows the inverse empirical cumulative distribution of the", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "entropy values obtained from the four methods. Consistent with the results in (Gal & Ghahramani,", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "score": 1.0, + "content": "2016), averaging the probabilities of multiple passes through the network yields predictions with", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "higher uncertainty than a deterministic pass that approximates the geometric average (Srivastava", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "et al., 2014). However, there still are some images that are predicted to be a digit with certainty. Our", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 656, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 671 + ], + "score": 1.0, + "content": "Kronecker factored Laplace approximation makes hardly any predictions with absolute certainty", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "and assigns high uncertainty to most of the letters as desired. The diagonal Laplace approxima-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "score": 1.0, + "content": "tion required stronger regularisation towards predicting deterministically, yet it performs similarly", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 691, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 702 + ], + "score": 1.0, + "content": "to Dropout. As shown in Table 1, however, the network makes predictions on the test set of MNIST", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 58 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 722, + 465, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 719, + 468, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 719, + 468, + 734 + ], + "score": 1.0, + "content": "4From: http://yaroslavvb.blogspot.nl/2011/09/notmnist-dataset.html", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "The regularised Laplace approximations all give an overall good fit to the HMC predictive poste-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 505, + 108 + ], + "score": 1.0, + "content": "rior. Their uncertainty is slightly higher close to the training data and increases more slowly away", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "from the data than that of the HMC posterior. The diagonal and full Laplace approximation re-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "quire stronger regularisation than our Kronecker factored one, as they have higher uncertainty when", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "not regularised. In particular the full Laplace approximation vastly overestimates the uncertainty", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "without additional regularisation, leading to a bad predictive mean (see Appendix E for the corre-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "sponding figures), as the Hessian of the log likelihood is underdetermined. This is commonly the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "case in deep learning, as the number of parameters is typically much larger than the number of data", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "points. Hence restricting the structure of the covariance is not only a computational necessity for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 484, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 484, + 194 + ], + "score": 1.0, + "content": "most architectures, but also allows for more precise estimation of the approximate covariance.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 82, + 506, + 194 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 212, + 295, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 298, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 298, + 225 + ], + "score": 1.0, + "content": "5.2 OUT-OF-DISTRIBUTION UNCERTAINTY", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "score": 1.0, + "content": "For a more realistic test, similar to (Louizos & Welling, 2017), we assess the uncertainty of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "predictions when classifying data from a different distribution than the training data. For this we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "train a network with two layers of 1024 hidden units and ReLU transfer functions to classify MNIST", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 266, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 234, + 284 + ], + "score": 1.0, + "content": "digits. We use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 234, + 268, + 255, + 280 + ], + "score": 0.9, + "content": "1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 266, + 506, + 284 + ], + "score": 1.0, + "content": "and momentum of 0.9 for 250 epochs. We apply Dropout with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 133, + 292 + ], + "score": 0.89, + "content": "p { = } 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "after each inner layer, as our chief interest is to compare against its uncertainty estimates. We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 290, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 290, + 151, + 307 + ], + "score": 1.0, + "content": "further use", + "type": "text" + }, + { + "bbox": [ + 151, + 293, + 164, + 304 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 290, + 285, + 307 + ], + "score": 1.0, + "content": "-regularisation with a factor of", + "type": "text" + }, + { + "bbox": [ + 285, + 291, + 307, + 303 + ], + "score": 0.9, + "content": "1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 290, + 506, + 307 + ], + "score": 1.0, + "content": "and randomly binarise the images during training", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 289, + 317 + ], + "score": 1.0, + "content": "according to their pixel intensities and draw", + "type": "text" + }, + { + "bbox": [ + 289, + 304, + 315, + 315 + ], + "score": 0.49, + "content": "1 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "such samples per datapoint for estimating the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "curvature factors. We use this network to classify the images in the notMNIST dataset4, which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 143, + 340 + ], + "score": 1.0, + "content": "contains", + "type": "text" + }, + { + "bbox": [ + 143, + 328, + 173, + 338 + ], + "score": 0.88, + "content": "2 8 \\times 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 327, + 307, + 340 + ], + "score": 1.0, + "content": "grey-scale images of the letters", + "type": "text" + }, + { + "bbox": [ + 307, + 328, + 320, + 338 + ], + "score": 0.37, + "content": "\\mathbf { \\dot { A } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 327, + 334, + 340 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 334, + 328, + 345, + 338 + ], + "score": 0.45, + "content": "\\mathbf { \\hat { J } } ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "from various computer fonts, i.e. not", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 405, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 405, + 351 + ], + "score": 1.0, + "content": "digits. An ideal classifier would make uniform predictions over its classes.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 234, + 506, + 351 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 296, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 296, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 296, + 368 + ], + "score": 1.0, + "content": "We compare the uncertainty obtained by pre-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 297, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 297, + 379 + ], + "score": 1.0, + "content": "dicting the digit class of the notMNIST images", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 297, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 297, + 390 + ], + "score": 1.0, + "content": "using 1. a deterministic forward pass through", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 298, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 298, + 402 + ], + "score": 1.0, + "content": "the Dropout trained network, 2. by sampling", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 398, + 297, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 297, + 412 + ], + "score": 1.0, + "content": "different Dropout masks and averaging the pre-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 410, + 297, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 297, + 423 + ], + "score": 1.0, + "content": "dictions, and by sampling different weight ma-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 421, + 297, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 297, + 433 + ], + "score": 1.0, + "content": "trices from 3. the matrix normal distribution", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 432, + 297, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 297, + 444 + ], + "score": 1.0, + "content": "obtained from our Kronecker factored Laplace", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 444, + 297, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 297, + 455 + ], + "score": 1.0, + "content": "approximation as well as 4. the diagonal one.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 454, + 297, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 297, + 465 + ], + "score": 1.0, + "content": "As an additional baseline similar to (Blundell", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 465, + 298, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 298, + 477 + ], + "score": 1.0, + "content": "et al., 2015; Graves, 2011), we compare to a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 475, + 297, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 297, + 488 + ], + "score": 1.0, + "content": "network with identical architecture with a fully", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 487, + 297, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 297, + 499 + ], + "score": 1.0, + "content": "factorised Gaussian (FFG) approximate pos-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 498, + 297, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 297, + 509 + ], + "score": 1.0, + "content": "terior on the weights and a standard normal", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 509, + 297, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 297, + 520 + ], + "score": 1.0, + "content": "prior. 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We use 100 sam-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 541, + 296, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 296, + 555 + ], + "score": 1.0, + "content": "ples for the stochastic forward passes and op-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 297, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 297, + 567 + ], + "score": 1.0, + "content": "timise the hyperparameters of the Laplace ap-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 563, + 384, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 384, + 577 + ], + "score": 1.0, + "content": "proximations w.r.t. the cross-entropy on the validation set of MNIST.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 354, + 298, + 567 + ] + }, + { + "type": "image", + "bbox": [ + 313, + 383, + 485, + 500 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 313, + 383, + 485, + 500 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 313, + 383, + 485, + 500 + ], + "spans": [ + { + "bbox": [ + 313, + 383, + 485, + 500 + ], + "score": 0.966, + "type": "image", + "image_path": "d894a7edf56530a450cd12d773fa589c25cc119d5b25ad9f5c7d7640b554c41f.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 313, + 383, + 485, + 396.0 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 313, + 396.0, + 485, + 409.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 313, + 409.0, + 485, + 422.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 313, + 422.0, + 485, + 435.0 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 313, + 435.0, + 485, + 448.0 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 313, + 448.0, + 485, + 461.0 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 313, + 461.0, + 485, + 474.0 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 313, + 474.0, + 485, + 487.0 + ], + "spans": [], + "index": 47 + }, + { + "bbox": [ + 313, + 487.0, + 485, + 500.0 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 511, + 504, + 543 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 304, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "Figure 2: Predictive entropy on notMNIST ob-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 303, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 303, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "tained from different methods for the forward pass", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 303, + 533, + 437, + 543 + ], + "spans": [ + { + "bbox": [ + 303, + 533, + 437, + 543 + ], + "score": 1.0, + "content": "on a network trained on MNIST.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50 + } + ], + "index": 47.0 + }, + { + "type": "text", + "bbox": [ + 108, + 564, + 383, + 575 + ], + "lines": [], + "index": 52, + "bbox_fs": [ + 105, + 563, + 384, + 577 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 581, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "We measure the uncertainty of the different methods as the entropy of the predictive distribution,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "which has a minimal value of 0 when a single class is predicted with certainty and a maximum of", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "about 2.3 for uniform predictions. Fig. 2 shows the inverse empirical cumulative distribution of the", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "entropy values obtained from the four methods. Consistent with the results in (Gal & Ghahramani,", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "score": 1.0, + "content": "2016), averaging the probabilities of multiple passes through the network yields predictions with", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "higher uncertainty than a deterministic pass that approximates the geometric average (Srivastava", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "et al., 2014). However, there still are some images that are predicted to be a digit with certainty. Our", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 656, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 671 + ], + "score": 1.0, + "content": "Kronecker factored Laplace approximation makes hardly any predictions with absolute certainty", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "and assigns high uncertainty to most of the letters as desired. The diagonal Laplace approxima-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "score": 1.0, + "content": "tion required stronger regularisation towards predicting deterministically, yet it performs similarly", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 691, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 702 + ], + "score": 1.0, + "content": "to Dropout. As shown in Table 1, however, the network makes predictions on the test set of MNIST", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "with similar accuracy to the deterministic forward pass and MC Dropout when using our approxi-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 452, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 452, + 106 + ], + "score": 1.0, + "content": "mation. The variational factorised Gaussian posterior has low uncertainty as expected.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 58, + "bbox_fs": [ + 105, + 580, + 506, + 702 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 503, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "with similar accuracy to the deterministic forward pass and MC Dropout when using our approxi-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 452, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 452, + 106 + ], + "score": 1.0, + "content": "mation. 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As first demonstrated in (Szegedy et al., 2013), neural", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "networks are prone to being fooled by gradient-based changes to their inputs. Li & Gal (2017)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 289, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 300 + ], + "score": 1.0, + "content": "suggest, and provide empirical support, that Bayesian models may be more robust to such attacks,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "since they implicitly form an infinitely large ensemble by integrating over the model parameters.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "For our experiments, we use the fully connected net trained on MNIST from the previous section", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 321, + 504, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 504, + 333 + ], + "score": 1.0, + "content": "and compare the sensitivity of the different prediction methods for two kinds of adversarial attacks.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 504, + 406 + ], + "lines": [ + { + "bbox": [ + 103, + 335, + 504, + 355 + ], + "spans": [ + { + "bbox": [ + 103, + 335, + 327, + 355 + ], + "score": 1.0, + "content": "First, we use the untargeted Fast Gradient Sign method", + "type": "text" + }, + { + "bbox": [ + 327, + 337, + 504, + 353 + ], + "score": 0.9, + "content": "\\begin{array} { r } { x _ { a d v } = x - \\eta \\mathrm { s g n } ( \\nabla _ { x } \\operatorname* { m a x } _ { y } \\log p ^ { ( M ) } ( y | x ) ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "suggested in (Goodfellow et al., 2014), which takes the gradient of the class predicted with maximal", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 360, + 501, + 377 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 199, + 377 + ], + "score": 1.0, + "content": "probability by method", + "type": "text" + }, + { + "bbox": [ + 200, + 362, + 212, + 372 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 360, + 277, + 377 + ], + "score": 1.0, + "content": "w.r.t. the input", + "type": "text" + }, + { + "bbox": [ + 278, + 364, + 285, + 372 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 360, + 495, + 377 + ], + "score": 1.0, + "content": "and reduces this probability with varying step size", + "type": "text" + }, + { + "bbox": [ + 495, + 364, + 501, + 374 + ], + "score": 0.71, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "score": 1.0, + "content": "This step size is rescaled by the difference between the maximal and minimal value per dimension", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "in the dataset. It is to be expected that this method generates examples away from the data manifold,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 395, + 398, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 398, + 407 + ], + "score": 1.0, + "content": "as there is no clear subset of the data that corresponds to e.g. ”not ones”.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "Fig. 3 shows the average predictive uncertainty and the accuracy on the original class on the MNIST", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 423, + 504, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 197, + 435 + ], + "score": 1.0, + "content": "test set as the step size", + "type": "text" + }, + { + "bbox": [ + 197, + 424, + 204, + 434 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 423, + 504, + 435 + ], + "score": 1.0, + "content": "increases. The Kronecker factored Laplace approximation achieves signifi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 434, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 504, + 446 + ], + "score": 1.0, + "content": "cantly higher uncertainty than any other prediction method as the images move away from the data.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "Both the diagonal and the Kronecker factored Laplace maintain higher accuracy than MC Dropout", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "on their original predictions. Interestingly, the deterministic forward pass appears to be most robust", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "in terms of accuracy, however it has much smaller uncertainty on the predictions it makes and will", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 478, + 491, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 491, + 489 + ], + "score": 1.0, + "content": "confidently predict a false class for most images, whereas the other methods are more uncertain.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 494, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "Furthermore, we perform a targeted attack that attempts to force the network to predict a specific", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 176, + 517 + ], + "score": 1.0, + "content": "class, in our case", + "type": "text" + }, + { + "bbox": [ + 176, + 506, + 187, + 516 + ], + "score": 0.69, + "content": "\\cdot _ { 0 } \\cdot \\mathrm { \\ }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "following (Li & Gal, 2017). Hence, for each method, we exclude all data points", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 284, + 528 + ], + "score": 1.0, + "content": "in the test set that are already predicted as", + "type": "text" + }, + { + "bbox": [ + 284, + 517, + 294, + 527 + ], + "score": 0.67, + "content": "\\cdot _ { 0 } \\cdot \\mathrm { \\ }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 516, + 505, + 528 + ], + "score": 1.0, + "content": ". The updates are of similar form to the untargeted", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 390, + 540 + ], + "score": 1.0, + "content": "attack, however they increase the probability of the pre-specified class", + "type": "text" + }, + { + "bbox": [ + 391, + 529, + 398, + 538 + ], + "score": 0.79, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "rather than decreasing the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 103, + 534, + 440, + 559 + ], + "spans": [ + { + "bbox": [ + 103, + 537, + 207, + 556 + ], + "score": 1.0, + "content": "current maximum as x(ty", + "type": "text" + }, + { + "bbox": [ + 191, + 538, + 367, + 554 + ], + "score": 0.92, + "content": "x _ { y } ^ { ( t + 1 ) } = x _ { y } ^ { ( \\bar { t } ) } + \\eta \\mathrm { s g n } \\big ( \\bar { \\nabla } _ { x } \\log \\bar { p ^ { ( M ) } } ( y | x _ { y } ^ { ( t ) } ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 534, + 399, + 559 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 399, + 538, + 435, + 554 + ], + "score": 0.93, + "content": "x _ { y } ^ { ( 0 ) } = x", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 534, + 440, + 559 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 193, + 573 + ], + "score": 1.0, + "content": "We use a step size of", + "type": "text" + }, + { + "bbox": [ + 193, + 558, + 228, + 572 + ], + "score": 0.92, + "content": "\\eta { = } 1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 558, + 505, + 573 + ], + "score": 1.0, + "content": "for the targeted attack. The uncertainty and accuracy on the original", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "and target class are shown in Fig. 4. Here, the Kronecker factored Laplace approximation has", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "slightly smaller uncertainty at its peak in comparison to the other methods, however it appears to be", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 284, + 605 + ], + "score": 1.0, + "content": "much more robust. It only misclassifies over", + "type": "text" + }, + { + "bbox": [ + 284, + 593, + 304, + 604 + ], + "score": 0.89, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "of the images after about 20 steps, whereas for the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 366, + 617 + ], + "score": 1.0, + "content": "other methods this is the case after roughly 10 steps and reaches", + "type": "text" + }, + { + "bbox": [ + 367, + 604, + 392, + 615 + ], + "score": 0.89, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "accuracy on the target class", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 614, + 495, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 495, + 628 + ], + "score": 1.0, + "content": "after almost 50 updates, whereas the other methods are fooled on all images after about 25 steps.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "In conjunction with the experiment on notMNIST, it appears that the Laplace approximation", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "achieves higher uncertainty than Dropout away from the data, as in the untargeted attack. In the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "targeted attack it exhibits smaller uncertainty than Dropout, yet it is more robust to having its pre-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 664, + 473, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 473, + 678 + ], + "score": 1.0, + "content": "diction changed. The diagonal Laplace approximation again performs similarly to Dropout.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49.5 + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 303, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 306, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 306, + 702 + ], + "score": 1.0, + "content": "5.4 UNCERTAINTY ON MISCLASSIFICATIONS", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "To highlight the scalability of our method, we apply it to a state-of-the-art convolutional network", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "architecture. 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As first demonstrated in (Szegedy et al., 2013), neural", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "networks are prone to being fooled by gradient-based changes to their inputs. Li & Gal (2017)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 289, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 300 + ], + "score": 1.0, + "content": "suggest, and provide empirical support, that Bayesian models may be more robust to such attacks,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "since they implicitly form an infinitely large ensemble by integrating over the model parameters.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "For our experiments, we use the fully connected net trained on MNIST from the previous section", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 321, + 504, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 504, + 333 + ], + "score": 1.0, + "content": "and compare the sensitivity of the different prediction methods for two kinds of adversarial attacks.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 254, + 506, + 333 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 504, + 406 + ], + "lines": [ + { + "bbox": [ + 103, + 335, + 504, + 355 + ], + "spans": [ + { + "bbox": [ + 103, + 335, + 327, + 355 + ], + "score": 1.0, + "content": "First, we use the untargeted Fast Gradient Sign method", + "type": "text" + }, + { + "bbox": [ + 327, + 337, + 504, + 353 + ], + "score": 0.9, + "content": "\\begin{array} { r } { x _ { a d v } = x - \\eta \\mathrm { s g n } ( \\nabla _ { x } \\operatorname* { m a x } _ { y } \\log p ^ { ( M ) } ( y | x ) ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "suggested in (Goodfellow et al., 2014), which takes the gradient of the class predicted with maximal", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 360, + 501, + 377 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 199, + 377 + ], + "score": 1.0, + "content": "probability by method", + "type": "text" + }, + { + "bbox": [ + 200, + 362, + 212, + 372 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 360, + 277, + 377 + ], + "score": 1.0, + "content": "w.r.t. the input", + "type": "text" + }, + { + "bbox": [ + 278, + 364, + 285, + 372 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 360, + 495, + 377 + ], + "score": 1.0, + "content": "and reduces this probability with varying step size", + "type": "text" + }, + { + "bbox": [ + 495, + 364, + 501, + 374 + ], + "score": 0.71, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "score": 1.0, + "content": "This step size is rescaled by the difference between the maximal and minimal value per dimension", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "in the dataset. It is to be expected that this method generates examples away from the data manifold,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 395, + 398, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 398, + 407 + ], + "score": 1.0, + "content": "as there is no clear subset of the data that corresponds to e.g. ”not ones”.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 103, + 335, + 506, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "Fig. 3 shows the average predictive uncertainty and the accuracy on the original class on the MNIST", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 423, + 504, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 197, + 435 + ], + "score": 1.0, + "content": "test set as the step size", + "type": "text" + }, + { + "bbox": [ + 197, + 424, + 204, + 434 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 423, + 504, + 435 + ], + "score": 1.0, + "content": "increases. The Kronecker factored Laplace approximation achieves signifi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 434, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 504, + 446 + ], + "score": 1.0, + "content": "cantly higher uncertainty than any other prediction method as the images move away from the data.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "Both the diagonal and the Kronecker factored Laplace maintain higher accuracy than MC Dropout", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "on their original predictions. Interestingly, the deterministic forward pass appears to be most robust", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "in terms of accuracy, however it has much smaller uncertainty on the predictions it makes and will", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 478, + 491, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 491, + 489 + ], + "score": 1.0, + "content": "confidently predict a false class for most images, whereas the other methods are more uncertain.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 411, + 506, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 494, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "Furthermore, we perform a targeted attack that attempts to force the network to predict a specific", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 176, + 517 + ], + "score": 1.0, + "content": "class, in our case", + "type": "text" + }, + { + "bbox": [ + 176, + 506, + 187, + 516 + ], + "score": 0.69, + "content": "\\cdot _ { 0 } \\cdot \\mathrm { \\ }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "following (Li & Gal, 2017). Hence, for each method, we exclude all data points", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 284, + 528 + ], + "score": 1.0, + "content": "in the test set that are already predicted as", + "type": "text" + }, + { + "bbox": [ + 284, + 517, + 294, + 527 + ], + "score": 0.67, + "content": "\\cdot _ { 0 } \\cdot \\mathrm { \\ }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 516, + 505, + 528 + ], + "score": 1.0, + "content": ". The updates are of similar form to the untargeted", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 390, + 540 + ], + "score": 1.0, + "content": "attack, however they increase the probability of the pre-specified class", + "type": "text" + }, + { + "bbox": [ + 391, + 529, + 398, + 538 + ], + "score": 0.79, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "rather than decreasing the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 103, + 534, + 440, + 559 + ], + "spans": [ + { + "bbox": [ + 103, + 537, + 207, + 556 + ], + "score": 1.0, + "content": "current maximum as x(ty", + "type": "text" + }, + { + "bbox": [ + 191, + 538, + 367, + 554 + ], + "score": 0.92, + "content": "x _ { y } ^ { ( t + 1 ) } = x _ { y } ^ { ( \\bar { t } ) } + \\eta \\mathrm { s g n } \\big ( \\bar { \\nabla } _ { x } \\log \\bar { p ^ { ( M ) } } ( y | x _ { y } ^ { ( t ) } ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 534, + 399, + 559 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 399, + 538, + 435, + 554 + ], + "score": 0.93, + "content": "x _ { y } ^ { ( 0 ) } = x", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 534, + 440, + 559 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 103, + 493, + 505, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 193, + 573 + ], + "score": 1.0, + "content": "We use a step size of", + "type": "text" + }, + { + "bbox": [ + 193, + 558, + 228, + 572 + ], + "score": 0.92, + "content": "\\eta { = } 1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 558, + 505, + 573 + ], + "score": 1.0, + "content": "for the targeted attack. The uncertainty and accuracy on the original", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "and target class are shown in Fig. 4. Here, the Kronecker factored Laplace approximation has", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "slightly smaller uncertainty at its peak in comparison to the other methods, however it appears to be", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 284, + 605 + ], + "score": 1.0, + "content": "much more robust. It only misclassifies over", + "type": "text" + }, + { + "bbox": [ + 284, + 593, + 304, + 604 + ], + "score": 0.89, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "of the images after about 20 steps, whereas for the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 366, + 617 + ], + "score": 1.0, + "content": "other methods this is the case after roughly 10 steps and reaches", + "type": "text" + }, + { + "bbox": [ + 367, + 604, + 392, + 615 + ], + "score": 0.89, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "accuracy on the target class", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 614, + 495, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 495, + 628 + ], + "score": 1.0, + "content": "after almost 50 updates, whereas the other methods are fooled on all images after about 25 steps.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 558, + 505, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "In conjunction with the experiment on notMNIST, it appears that the Laplace approximation", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "achieves higher uncertainty than Dropout away from the data, as in the untargeted attack. In the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "targeted attack it exhibits smaller uncertainty than Dropout, yet it is more robust to having its pre-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 664, + 473, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 473, + 678 + ], + "score": 1.0, + "content": "diction changed. The diagonal Laplace approximation again performs similarly to Dropout.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 632, + 505, + 678 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 303, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 306, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 306, + 702 + ], + "score": 1.0, + "content": "5.4 UNCERTAINTY ON MISCLASSIFICATIONS", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "To highlight the scalability of our method, we apply it to a state-of-the-art convolutional network", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "architecture. Recently, deep residual networks (He et al., 2016a;b) have been the most successful", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "score": 1.0, + "content": "ones among those. As demonstrated in (Grosse & Martens, 2016), Kronecker factored curvature", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "methods are applicable to convolutional layers by interpreting them as matrix-matrix multiplications.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 53.5, + "bbox_fs": [ + 106, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 125, + 79, + 486, + 191 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 79, + 486, + 191 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 125, + 79, + 486, + 191 + ], + "spans": [ + { + "bbox": [ + 125, + 79, + 486, + 191 + ], + "score": 0.874, + "type": "image", + "image_path": "5ed41153d9002457c121cab0e2d0510ad1bedc3708935fa9b5bfa70ad80fe8a9.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 125, + 79, + 486, + 116.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 125, + 116.33333333333334, + 486, + 153.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 125, + 153.66666666666669, + 486, + 191.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 199, + 504, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "Figure 5: Inverse ecdf of the predictive entropy from Wide Residual Networks trained with and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 211, + 504, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 504, + 224 + ], + "score": 1.0, + "content": "without Dropout on CIFAR100. For misclassifications, curves on top corresponding to higher un-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 222, + 408, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 408, + 234 + ], + "score": 1.0, + "content": "certainty are desirable, and curves on the bottom for correct classifications.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 504, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "score": 1.0, + "content": "ones among those. As demonstrated in (Grosse & Martens, 2016), Kronecker factored curvature", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "methods are applicable to convolutional layers by interpreting them as matrix-matrix multiplications.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "We compare our uncertainty estimates on wide residual networks (Zagoruyko & Komodakis, 2016),", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "a recent variation that achieved competitive performance on CIFAR100 (Krizhevsky & Hinton,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "2009) while, in contrast to most other residual architectures, including Dropout at specific points.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 332, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 347 + ], + "score": 1.0, + "content": "While this does not correspond to using Dropout in the Bayesian sense (Gal & Ghahramani, 2015),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 343, + 488, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 488, + 359 + ], + "score": 1.0, + "content": "it allows us to at least compare our method to the uncertainty estimates obtained from Dropout.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 504, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "We note that it is straightforward to incorporate batch normalisation (Ioffe & Szegedy, 2015) into the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "curvature backpropagation algorithms, so we apply a standard Laplace approximation to its param-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "eters as well. We are not aware of any interpretation of Dropout as performing Bayesian inference", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 394, + 479, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 479, + 406 + ], + "score": 1.0, + "content": "on the parameters of batch normalisation. Further implementation details are in Appendix G.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "Again, the accuracy of the prediction methods is comparable (see Table 2 in Appendix F). For cal-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "culating the curvature factors, we draw 5, 000 samples per image using the same data augmentation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 347, + 448 + ], + "score": 1.0, + "content": "as during training, effectively increasing the dataset size to", + "type": "text" + }, + { + "bbox": [ + 347, + 433, + 383, + 445 + ], + "score": 0.92, + "content": "2 . 5 \\times 1 0 ^ { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 433, + 506, + 448 + ], + "score": 1.0, + "content": ". 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Thus, high", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "uncertainty on misclassifications and low uncertainty on correct ones would be desirable, such that a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "system could return control to a human expert when it can not make a confident decision. In general,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 520, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 531 + ], + "score": 1.0, + "content": "the network tends to be more uncertain on its misclassifcations than its correct ones regardless of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "whether it was trained with or without Dropout and of the method used for prediction. Both Dropout", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "and the Laplace approximation similarly increase the uncertainty in the predictions, however this is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "irrespective of the correctness of the classification. Yet, our experiments show that the Kronecker", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "factored Laplace approximation can be scaled to modern convolutional networks and maintain good", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 574, + 465, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 465, + 586 + ], + "score": 1.0, + "content": "classification accuracy while having similar uncertainty about the predictions as Dropout.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "We had to use much stronger regularisation for the Laplace approximation on the wide residual", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "network, possibly because the block-diagonal approximation becomes more inaccurate on deep net-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "score": 1.0, + "content": "works, possibly because the number of parameters is much higher relative to the number of data.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "It would be interesting to see how the Laplace approximations behaves on a much larger dataset", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "like ImageNet for similarly sized networks, where we have a better ratio of data to parameters and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "curvature directions. However, even on a relatively small dataset like CIFAR we did not have to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 656, + 474, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 474, + 669 + ], + "score": 1.0, + "content": "regularise the Laplace approximation to the degree of the posterior becoming deterministic.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 105, + 711, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 505, + 725 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 505, + 725 + ], + "score": 1.0, + "content": "5We use the first 5, 000 images as a validation set to tune the hyperparameters of our Laplace approximation", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 399, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 399, + 732 + ], + "score": 1.0, + "content": "and the final 5, 000 ones for evaluating the predictive uncertainty on all methods.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 125, + 79, + 486, + 191 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 79, + 486, + 191 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 125, + 79, + 486, + 191 + ], + "spans": [ + { + "bbox": [ + 125, + 79, + 486, + 191 + ], + "score": 0.874, + "type": "image", + "image_path": "5ed41153d9002457c121cab0e2d0510ad1bedc3708935fa9b5bfa70ad80fe8a9.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 125, + 79, + 486, + 116.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 125, + 116.33333333333334, + 486, + 153.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 125, + 153.66666666666669, + 486, + 191.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 199, + 504, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "Figure 5: Inverse ecdf of the predictive entropy from Wide Residual Networks trained with and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 211, + 504, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 504, + 224 + ], + "score": 1.0, + "content": "without Dropout on CIFAR100. For misclassifications, curves on top corresponding to higher un-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 222, + 408, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 408, + 234 + ], + "score": 1.0, + "content": "certainty are desirable, and curves on the bottom for correct classifications.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 504, + 295 + ], + "lines": [], + "index": 6.5, + "bbox_fs": [ + 105, + 271, + 505, + 296 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "We compare our uncertainty estimates on wide residual networks (Zagoruyko & Komodakis, 2016),", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "a recent variation that achieved competitive performance on CIFAR100 (Krizhevsky & Hinton,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "2009) while, in contrast to most other residual architectures, including Dropout at specific points.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 332, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 347 + ], + "score": 1.0, + "content": "While this does not correspond to using Dropout in the Bayesian sense (Gal & Ghahramani, 2015),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 343, + 488, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 488, + 359 + ], + "score": 1.0, + "content": "it allows us to at least compare our method to the uncertainty estimates obtained from Dropout.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 300, + 506, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 504, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "We note that it is straightforward to incorporate batch normalisation (Ioffe & Szegedy, 2015) into the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "curvature backpropagation algorithms, so we apply a standard Laplace approximation to its param-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "eters as well. We are not aware of any interpretation of Dropout as performing Bayesian inference", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 394, + 479, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 479, + 406 + ], + "score": 1.0, + "content": "on the parameters of batch normalisation. Further implementation details are in Appendix G.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 361, + 506, + 406 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "Again, the accuracy of the prediction methods is comparable (see Table 2 in Appendix F). For cal-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "culating the curvature factors, we draw 5, 000 samples per image using the same data augmentation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 347, + 448 + ], + "score": 1.0, + "content": "as during training, effectively increasing the dataset size to", + "type": "text" + }, + { + "bbox": [ + 347, + 433, + 383, + 445 + ], + "score": 0.92, + "content": "2 . 5 \\times 1 0 ^ { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 433, + 506, + 448 + ], + "score": 1.0, + "content": ". The diagonal approximation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 446, + 475, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 475, + 457 + ], + "score": 1.0, + "content": "had to be regularised to the extent of becoming deterministic, so we omit it from the results.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 411, + 506, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 104, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "In Fig. 5 we compare the distribution of the predictive uncertainty on the test set.5 We distinguish", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "between the uncertainty on correct and incorrect classifications, as the mistakes of a system used", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "in practice may be less severe if the network can at least indicate that it is uncertain. Thus, high", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "uncertainty on misclassifications and low uncertainty on correct ones would be desirable, such that a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "system could return control to a human expert when it can not make a confident decision. In general,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 520, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 531 + ], + "score": 1.0, + "content": "the network tends to be more uncertain on its misclassifcations than its correct ones regardless of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "whether it was trained with or without Dropout and of the method used for prediction. Both Dropout", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "and the Laplace approximation similarly increase the uncertainty in the predictions, however this is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "irrespective of the correctness of the classification. Yet, our experiments show that the Kronecker", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "factored Laplace approximation can be scaled to modern convolutional networks and maintain good", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 574, + 465, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 465, + 586 + ], + "score": 1.0, + "content": "classification accuracy while having similar uncertainty about the predictions as Dropout.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 463, + 506, + 586 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "We had to use much stronger regularisation for the Laplace approximation on the wide residual", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "network, possibly because the block-diagonal approximation becomes more inaccurate on deep net-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "score": 1.0, + "content": "works, possibly because the number of parameters is much higher relative to the number of data.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "It would be interesting to see how the Laplace approximations behaves on a much larger dataset", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "like ImageNet for similarly sized networks, where we have a better ratio of data to parameters and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "curvature directions. However, even on a relatively small dataset like CIFAR we did not have to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 656, + 474, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 474, + 669 + ], + "score": 1.0, + "content": "regularise the Laplace approximation to the degree of the posterior becoming deterministic.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 590, + 506, + 669 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 195, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 197, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 197, + 97 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 504, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "We presented a scalable approximation to the Laplace approximation for the posterior of a neural", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 118, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 506, + 131 + ], + "score": 1.0, + "content": "network and provided experimental results suggesting that the uncertainty estimates are on par with", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "current alternatives like Dropout, if not better. It enables practitioners to obtain principled uncer-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 506, + 152 + ], + "score": 1.0, + "content": "tainty estimates from their models, even if they were trained in a maximum likelihood/MAP setting.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 156, + 505, + 233 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "There are many possible extensions to this work. One would be to automatically determine the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 165, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 181 + ], + "score": 1.0, + "content": "scale and regularisation hyperparameters of the Kronecker factored Laplace approximation using", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "the model evidence similar to how (MacKay, 1992) interpolates between the data log likelihood", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "and the width of the prior. The model evidence could further be used to perform Bayesian model", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "averaging on ensembles of neural networks, potentially improving their generalisation ability and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "uncertainty estimates. A challenging application would be active learning, where only little data is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 223, + 430, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 430, + 234 + ], + "score": 1.0, + "content": "available relative to the number of curvature directions that need to be estimated.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 108, + 251, + 225, + 262 + ], + "lines": [ + { + "bbox": [ + 108, + 251, + 226, + 264 + ], + "spans": [ + { + "bbox": [ + 108, + 251, + 226, + 264 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 108, + 275, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "This work was supported by the Alan Turing Institute under the EPSRC grant EP/N510129/1. We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "thank the anonymous reviewers for their feedback and Harshil Shah for his comments on an earlier", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 296, + 182, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 182, + 311 + ], + "score": 1.0, + "content": "draft of this paper.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 325, + 175, + 337 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 176, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 176, + 339 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 105, + 334, + 506, + 735 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 358 + ], + "score": 1.0, + "content": "Anoop Korattikara Balan, Vivek Rathod, Kevin P Murphy, and Max Welling. 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The model evidence could further be used to perform Bayesian model", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "averaging on ensembles of neural networks, potentially improving their generalisation ability and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "uncertainty estimates. 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We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "thank the anonymous reviewers for their feedback and Harshil Shah for his comments on an earlier", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 296, + 182, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 182, + 311 + ], + "score": 1.0, + "content": "draft of this paper.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 275, + 505, + 311 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 325, + 175, + 337 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 176, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 176, + 339 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "list", + "bbox": [ + 105, + 334, + 506, + 735 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 358 + ], + "score": 1.0, + "content": "Anoop Korattikara Balan, Vivek Rathod, Kevin P Murphy, and Max Welling. 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"content": "arXiv:1605.07146, 2016.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 706, + 506, + 730 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 79, + 205, + 101 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 208, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 208, + 103 + ], + "score": 1.0, + "content": "Appendices", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 115, + 414, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 415, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 415, + 131 + ], + "score": 1.0, + "content": "A DERIVATION OF THE ACTIVATION HESSIAN RECURSION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 105, + 142, + 505, + 165 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "Here, we provide the basic derivation of the factorisation of the diagonal blocks of the Hessian in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 154, + 449, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 295, + 165 + ], + "score": 1.0, + "content": "Eq. 1 and the recursive formula for calculating", + "type": "text" + }, + { + "bbox": [ + 295, + 154, + 305, + 164 + ], + "score": 0.82, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 154, + 449, + 165 + ], + "score": 1.0, + "content": "as presented in (Botev et al., 2017).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 108, + 170, + 476, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 170, + 477, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 305, + 183 + ], + "score": 1.0, + "content": "The Hessian of a neural network with parameters", + "type": "text" + 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269, + 386, + 302 + ], + "score": 0.94, + "content": "{ \\frac { \\partial E } { \\partial W _ { a , b } ^ { \\lambda } } } = \\sum _ { i } { \\frac { \\partial h _ { i } ^ { \\lambda } } { \\partial W _ { a , b } ^ { \\lambda } } } { \\frac { \\partial E } { \\partial h _ { i } ^ { \\lambda } } } = a _ { b } ^ { \\lambda - 1 } { \\frac { \\partial E } { \\partial h _ { a } ^ { \\lambda } } }", + "type": "interline_equation", + "image_path": "83dba684467e560f40061ae37edaec35ad0a2115e940c900c683886e03489863.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 269, + 386, + 285.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 225, + 285.5, + 386, + 302.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 312, + 482, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 483, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 142, + 326 + ], + "score": 1.0, + "content": "Keeping", + "type": "text" + }, + { 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} + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 495, + 413, + 525 + ], + "lines": [ + { + "bbox": [ + 198, + 495, + 413, + 525 + ], + "spans": [ + { + "bbox": [ + 198, + 495, + 413, + 525 + ], + "score": 0.92, + "content": "H _ { \\lambda } = \\frac { \\partial ^ { 2 } E } { \\partial \\operatorname { v e c } \\left( W _ { \\lambda } \\right) \\partial \\operatorname { v e c } \\left( W _ { \\lambda } \\right) } = \\left( a _ { \\lambda - 1 } a _ { \\lambda - 1 } ^ { \\mathsf { T } } \\right) \\otimes \\mathcal { H } _ { \\lambda }", + "type": "interline_equation", + "image_path": "36187593be22e70d34cda56a97545b1794bc291950b32db101df9d3a58cf5d5e.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 495, + 413, + 510.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 198, + 510.0, + 413, + 525.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 346, + 547 + ], + 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\\lambda } = \\mathrm { d i a g } \\left( \\mathbf f _ { \\lambda } ^ { \\prime } ( h _ { \\lambda } ) \\right) } } \\\\ { { D _ { \\lambda } = \\mathrm { d i a g } \\left( \\mathbf f _ { \\lambda } ^ { \\prime \\prime } ( h _ { \\lambda } ) { \\frac { \\partial E } { \\partial a _ { \\lambda } } } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "2d283728938d16ef2672bdb5f416bcd470208dcd734b67d9666da1668ae8810e.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 253, + 624, + 358, + 646.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 253, + 646.0, + 358, + 668.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 681, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 115, + 692 + ], + "score": 0.79, + "content": "\\mathbf { f ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 680, + 133, 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The recursion is initialised", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 692, + 354, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 354, + 705 + ], + "score": 1.0, + "content": "with the Hessian of the error w.r.t. the linear network outputs.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "score": 1.0, + "content": "For further details and on how to calculate the diagonal blocks of the Gauss-Newton and Fisher", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 720, + 431, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 431, + 732 + ], + "score": 1.0, + "content": "matrix, we refer the reader to (Botev et al., 2017) and (Martens & Grosse, 2015).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 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Strictly speaking, it is not essential to approximate", + "type": "text" + }, + { + "bbox": [ + 406, + 277, + 429, + 289 + ], + "score": 0.67, + "content": "\\mathbb { E } \\left[ \\mathcal { H } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "for the Kronecker", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "factored Laplace approximation, as in contrast to optimisation procedures the curvature only needs", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "score": 1.0, + "content": "to be calculated once and is thus not time critical. 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Thus, we make", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 355, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 504, + 366 + ], + "score": 1.0, + "content": "use of the minibatch approximation in our experiments — as we make use of data augmentation —", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 304, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 304, + 378 + ], + "score": 1.0, + "content": "in order to demonstrate its practical applicability.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "We note that E [H] can be calculated exactly by running KFRA (Botev et al., 2017) with a minibatch-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "size of one, and then averaging the results. 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This is more efficient in the sense that we only need to calculate two matrix-matrix", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 430, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 430, + 227 + ], + "score": 1.0, + "content": "products of small matrices, rather than a matrix-vector product with one big one.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 155, + 505, + 227 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 241, + 428, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 430, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 430, + 255 + ], + "score": 1.0, + "content": "C APPROXIMATION OF THE EXPECTED ACTIVATION HESSIAN", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 266, + 505, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 206, + 279 + ], + "score": 1.0, + "content": "While the square root of", + "type": "text" + }, + { + "bbox": [ + 207, + 267, + 221, + 278 + ], + "score": 0.9, + "content": "\\mathcal { Q } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 265, + 424, + 279 + ], + "score": 1.0, + "content": "is calculated during the forward pass on all layers,", + "type": "text" + }, + { + "bbox": [ + 424, + 267, + 434, + 277 + ], + "score": 0.77, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "requires an addi-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 405, + 290 + ], + "score": 1.0, + "content": "tional backward pass. Strictly speaking, it is not essential to approximate", + "type": "text" + }, + { + "bbox": [ + 406, + 277, + 429, + 289 + ], + "score": 0.67, + "content": "\\mathbb { E } \\left[ \\mathcal { H } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "for the Kronecker", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "factored Laplace approximation, as in contrast to optimisation procedures the curvature only needs", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "score": 1.0, + "content": "to be calculated once and is thus not time critical. For datasets of the scale of ImageNet and the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "score": 1.0, + "content": "networks used for such datasets, it would still be impractically slow to perform the calculation for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 320, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 335 + ], + "score": 1.0, + "content": "every data point individually. Furthermore, as most datasets are augmented during training, e.g. ran-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "dom cropping or reflections of images, the curvature of the network can be estimated using the same", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "augmentations, effectively increasing the size of the dataset by orders of magnitude. Thus, we make", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 355, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 504, + 366 + ], + "score": 1.0, + "content": "use of the minibatch approximation in our experiments — as we make use of data augmentation —", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 304, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 304, + 378 + ], + "score": 1.0, + "content": "in order to demonstrate its practical applicability.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 265, + 506, + 378 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "We note that E [H] can be calculated exactly by running KFRA (Botev et al., 2017) with a minibatch-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "size of one, and then averaging the results. KFAC (Martens & Grosse, 2015), in contrast, stochas-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "tically approximates the Fisher matrix, so even when run for every datapoint separately, it cannot", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 415, + 257, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 257, + 428 + ], + "score": 1.0, + "content": "calculate the curvature factor exactly.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 381, + 505, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 504, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "In the following, we also show figures for the adversarial experiments in which we calculate the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 443, + 330, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 330, + 455 + ], + "score": 1.0, + "content": "curvature per datapoint and without data augmentation:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 432, + 505, + 455 + ] + }, + { + "type": "image", + "bbox": [ + 109, + 465, + 503, + 587 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 465, + 503, + 587 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 465, + 503, + 587 + ], + "spans": [ + { + "bbox": [ + 109, + 465, + 503, + 587 + ], + "score": 0.927, + "type": "image", + "image_path": "bd401ad0b8e9d0a6e372927c134409d5b015ccf19eb9e8745ca37ac78fb7f7a9.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 109, + 465, + 503, + 505.6666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 109, + 505.6666666666667, + 503, + 546.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 109, + 546.3333333333334, + 503, + 587.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 597, + 504, + 621 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "Figure 6: Untargeted adversarial attack for Kronecker factored Laplace approximation with the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 608, + 488, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 488, + 621 + ], + "score": 1.0, + "content": "curvature calculated with and without data augmentation/approximating the activation Hessian.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + } + ], + "index": 30.25 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "score": 1.0, + "content": "Fig. 6 and Fig. 7 show how the Laplace approximation with the curvature estimated from 1000 ran-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "domly sampled binary MNIST images and the activation Hessian calculated with a minibatch size", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "of 100 performs in comparison to the curvature factor being calculated without any data augmen-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "tation with a batch size of 100 or exactly. 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The difference between approximating the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 266, + 452, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 452, + 279 + ], + "score": 1.0, + "content": "activation Hessian over a minibatch and calculating it exactly appears to be negligible.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 294, + 380, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 382, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 382, + 309 + ], + "score": 1.0, + "content": "D MEMORY AND COMPUTATIONAL REQUIREMENTS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 317, + 333 + ], + "score": 1.0, + "content": "If we denote the dimensionality of the input to layer", + "type": "text" + }, + { + "bbox": [ + 318, + 319, + 325, + 328 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 317, + 337, + 333 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 337, + 319, + 362, + 330 + ], + "score": 0.91, + "content": "D _ { \\lambda - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 317, + 432, + 333 + ], + "score": 1.0, + "content": "and its output as", + "type": "text" + }, + { + "bbox": [ + 432, + 319, + 446, + 330 + ], + "score": 0.89, + "content": "D _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 317, + 506, + 333 + ], + "score": 1.0, + "content": ", the curvature", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 329, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 329, + 382, + 347 + ], + "score": 1.0, + "content": "factors correspond to the two precision matrices with Dλ−1(Dλ−1+1)2", + "type": "text" + }, + { + "bbox": [ + 380, + 329, + 399, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 399, + 330, + 441, + 346 + ], + "score": 0.93, + "content": "\\frac { D _ { \\lambda } ( D _ { \\lambda } + 1 ) } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 329, + 506, + 347 + ], + "score": 1.0, + "content": "‘parameters’ to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "estimate, since they are symmetric. 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This is for computational efficiency and does not change the expectation.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 454, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 504, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 504, + 467 + ], + "score": 1.0, + "content": "One possibility to save computation time would be to sample a fixed set of weight matrices from the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "approximate posterior — in order to avoid solving the linear system on every forward pass — and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "treat the networks that they define as an ensemble. 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A further speed up can be achieved by dis-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "tilling the predictive distributions of the Laplace network into a smaller, deterministic feedforward", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 487, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 487, + 533 + ], + "score": 1.0, + "content": "network as successfully demonstrated in (Balan et al., 2015) for posterior samples using HMC.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 107, + 547, + 392, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 393, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 393, + 562 + ], + "score": 1.0, + "content": "E COMPLEMENTARY FIGURES FOR THE TOY DATASET", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 504, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "Fig. 8 shows the different Laplace approximations (Kronecker factored, diagonal, full) from the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "main text without any hyperparameter tuning. The figure of the uncertainty obtained from samples", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "using HMC is repeated. 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As previously dis-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "cussed in (Botev et al., 2017), the Hessian of a neural network is usually underdetermined as the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "number of data points is much smaller than the number of parameters — in our case we have 20 data", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 189, + 701 + ], + "score": 1.0, + "content": "points to estimate a", + "type": "text" + }, + { + "bbox": [ + 190, + 688, + 219, + 699 + ], + "score": 0.89, + "content": "7 8 \\times 7 8", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "precision matrix. 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Accuracy
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Accuracy
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The residual blocks compute a function of the form", + "type": "text" + }, + { + "bbox": [ + 464, + 512, + 505, + 524 + ], + "score": 0.91, + "content": "r e s ( x ) =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 149, + 536 + ], + "score": 0.96, + "content": "x + f _ { \\phi } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 523, + 180, + 536 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 181, + 523, + 192, + 536 + ], + "score": 0.89, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "typically is a sequence of convolutions, batch normalisation and elementwise", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "nonlinearities. This means that we would need to pass back two curvature matrices, one for each", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "summand. However, this would double the number of backpropagated matrices for each residual", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "connection, hence the computation time/memory requirements would grow exponentially in the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "number of residual blocks. 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Prediction MethodAccuracy
FFG Deterministic98.88% 98.86%
MC Dropout Diagonal Laplace98.85% 98.85%
KF Laplace98.80%
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Accuracy
Prediction MethodDropoutDeterministic
Deterministic79.12%79.18%
MC Dropout79.20%1
KF Laplace79.10%79.36%
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a/parse/train/SywMS6ZfM/SywMS6ZfM.md b/parse/train/SywMS6ZfM/SywMS6ZfM.md new file mode 100644 index 0000000000000000000000000000000000000000..8e86466b0dcf5f0659189f85e73f20f3df67eb92 --- /dev/null +++ b/parse/train/SywMS6ZfM/SywMS6ZfM.md @@ -0,0 +1,517 @@ +# DISTRIBUTIONAL INCLUSION VECTOR EMBEDDING FOR UNSUPERVISED HYPERNYMY DETECTION + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Modeling hypernymy, such as poodle is-a dog, is an important generalization aid to many NLP tasks, such as entailment, relation extraction, and question answering. Supervised learning from labeled hypernym sources, such as WordNet, limit the coverage of these models, which can be addressed by learning hypernyms from unlabeled text. Existing unsupervised methods either do not scale to large vocabularies or yield unacceptably poor accuracy. This paper introduces distributional inclusion vector embedding (DIVE), a simple-to-implement unsupervised method of hypernym discovery via per-word non-negative vector embeddings which preserve the inclusion property of word contexts. In experimental evaluations more comprehensive than any previous literature of which we are aware—evaluating on 11 datasets using multiple existing as well as newly proposed scoring functions— we find that our method provides up to double the precision of previous unsupervised methods, and the highest average performance, using a much more compact word representation, and yielding many new state-of-the-art results. In addition, the meaning of each dimension in DIVE is interpretable, which leads to a novel approach on word sense disambiguation as another promising application of DIVE. + +# 1 INTRODUCTION + +Numerous applications benefit from compactly representing context distributions, which assign meaning to objects under the rubric of distributional semantics. In natural language processing, distributional semantics has long been used to assign meanings to words (that is, to lexemes in the dictionary, not individual instances of word tokens). The meaning of a word in the distributional sense is often taken to be the set of textual contexts (nearby tokens) in which that word appears, represented as a large sparse bag of words (SBOW). Without any supervision, word2vec (Mikolov et al., 2013), among other approaches based on matrix factorization (Levy et al., 2015a), successfully compress the SBOW into a much lower dimensional embedding space, increasing the scalability and applicability of the embeddings while preserving (or even improving) the correlation of geometric embedding similarities with human word similarity judgments. + +While embedding models have achieved impressive results, context distributions capture more semantic features than just word similarity. The distributional inclusion hypothesis (DIH) (Weeds & Weir, 2003; Geffet & Dagan, 2005; Cimiano et al., 2005) posits that the context set of a word tends to be a subset of the contexts of its hypernyms. For a concrete example, most adjectives that can be applied to poodle can also be applied to dog, because dog is a hypernym of poodle. For instance, both can be obedient. However, the converse is not necessarily true — a dog can be straight-haired but a poodle cannot. Therefore, dog tends to have a broader context set than poodle. Many asymmetric scoring functions comparing SBOW based on DIH have been developed for automatic hypernymy detection (Weeds & Weir, 2003; Geffet & Dagan, 2005; Santus et al., 2017). + +Hypernymy detection plays a key role in many challenging NLP tasks, such as textual entailment (Sammons et al., 2011), coreference (Ponzetto & Strube, 2006), relation extraction (Demeester et al., 2016) and question answering (Huang et al., 2008). Leveraging the variety of contexts and inclusion properties in context distributions can greatly increase the ability to discover taxonomic structure among words (Santus et al., 2017). The inability to preserve these features limits the semantic representation power and downstream applicability of some popular existing unsupervised learning approaches such as word2vec. + +Several recently proposed methods aim to encode hypernym relations between words in dense embeddings, such as Gaussian embedding (Vilnis & McCallum, 2015; Athiwaratkun & Wilson, 2017), order embedding (Vendrov et al., 2016), H-feature detector (Roller & Erk, 2016), HyperScore (Nguyen et al., 2017), dual tensor (Glavas & Ponzetto, 2017), Poincar ˇ e embedding (Nickel ´ & Kiela, 2017), and LEAR (Vulic & Mrk ´ siˇ c, 2017). However, the methods focus on supervised ´ or semi-supervised setting (Vendrov et al., 2016; Roller & Erk, 2016; Nguyen et al., 2017; Glavasˇ & Ponzetto, 2017; Vulic & Mrk ´ siˇ c, 2017), do not learn from raw text (Nickel & Kiela, 2017) or ´ lack comprehensive experiments on the hypernym detection task (Vilnis & McCallum, 2015; Athiwaratkun & Wilson, 2017). + +Recent studies (Levy et al., 2015b; Santus et al., 2017) have underscored the difficulty of generalizing supervised hypernymy annotations to unseen pairs — classifiers often effectively memorize prototypical hypernyms (‘general’ words) and ignore relations between words. These findings motivate us to develop more accurate and scalable unsupervised embeddings to detect hypernymy and propose several scoring functions to analyze the embeddings from different perspectives. + +# 1.1 CONTRIBUTIONS + +• A novel unsupervised low-dimensional embedding method to model inclusion relations among word contexts via performing non-negative matrix factorization (NMF) on a weighted PMI matrix, which can be efficiently optimized using modified skip-grams. Several new asymmetric comparison functions to measure inclusion and generality properties and to evaluate different aspects of unsupervised embeddings. Extensive experiments on 11 datasets demonstrate the learned embeddings and comparison functions achieve state-of-the-art performances on unsupervised hypernym detection while requiring much less memory and compute than approaches based on the full SBOW. • A qualitative experiment illustrates DIVE can be used to solve word sense disambiguation, especially when efficiently modeling word senses at multiple granularities is desirable. + +# 2 METHOD + +The distributional inclusion hypothesis (DIH) suggests that the context set of a hypernym tends to contain the context set of its hyponyms. That is, when representing a word as the counts of contextual co-occurrences, the count in every dimension of hypernym $y$ tends to be larger than or equal to the corresponding count of its hyponym $x$ : + +$$ +x \preceq y \iff \forall c \in V , \# ( x , c ) \leq \# ( y , c ) , +$$ + +where $x \preceq y$ means $y$ is a hypernym of $x , V$ is the set of vocabulary, and $\# ( x , c )$ indicates the number of times that word $x$ and its context word $c$ co-occur in a small window with size $| W |$ in corpus $D$ . + +Our goal is to produce lower-dimensional embeddings that preserve the inclusion property that the embedding of hypernym $y$ is larger than or equal to the embedding of its hyponym $x$ in every dimension. Formally, the desirable property can be written as + +$$ +x \preceq y \iff \mathbf { x } [ i ] \leq \mathbf { y } [ i ] , \forall i \in \{ 1 , . . . , d _ { 0 } \} , +$$ + +where $d _ { 0 }$ is number of dimensions in the embedding space. We add additional non-negativity constraints, i.e. $x [ i ] \ge 0 , y [ i ] \ge 0 , \forall i$ , in order to increase the interpretability of the embeddings (the reason will be explained later in this section). + +This is a challenging task. In reality, there are a lot of noise and systematic biases which cause the violation of DIH in Equation (1) (i.e. $\# ( x , c ) > \# ( y , c )$ for some neighboring word $c$ ), but the general trend can be discovered by processing several thousands of neighboring words in SBOW together. After the compression, the same trend has to be estimated in a much smaller embedding space which discards most of the information in SBOW, so it is not surprising to see most of the unsupervised hypernymy detection studies use SBOW (Santus et al., 2017) and the existing unsupervised embeddings like Gaussian embedding have degraded accuracy (Vulic et al., 2016). ´ + +# 2.1 INCLUSION PRESERVING MATRIX FACTORIZATION + +Popular methods of unsupervised word embedding are usually based on matrix factorization (Levy et al., 2015a). The approaches first compute a co-occurrence statistic between the wth word and the cth context word as the $( w , c )$ th element of the matrix $M [ w , c ]$ . Next, the matrix $M$ is factorized such that $M [ w , c ] \approx \mathbf { w } ^ { T } \mathbf { \dot { c } }$ , where w is the low dimension embedding of $w$ th word and $\mathbf { c }$ is the cth context embedding. + +The statistic in $M [ w , c ]$ is usually related to pointwise mutual information: $P M I ( w , c ) ~ =$ $\textstyle \log ( { \frac { P ( w , c ) } { P ( w ) \cdot P ( c ) } } )$ , where $\begin{array} { r } { P ( w , c ) = \frac { \# ( w , c ) } { | D | } } \end{array}$ , $| D | = \sum _ { w \in V } \sum _ { c \in V } \# ( w , c )$ is number of co-occurrence word pairs in the corpus, $\begin{array} { r } { P ( w ) = \frac { \# ( w ) } { | D | } } \end{array}$ , $\# ( w ) = \sum _ { c \in V } \# ( w , c )$ is the frequency of the word $w$ times the window size $| W |$ , and similarly for $P ( c )$ . For example, $M [ w , c ]$ could be set as positive PMI (PPMI), $\operatorname* { m a x } ( P M I ( w , c ) , 0 )$ , or shifted PMI, $P M I ( w , c ) - \log ( k )$ , like skip-grams with negative sampling (SGNS) (Levy et al., 2015a). Intuitively, since $\dot { M } [ w , \bar { c ] } \approx \mathbf { w } ^ { T } \mathbf { c }$ , larger embedding values of w at every dimension seems to imply larger $\mathbf { \bar { w } } ^ { T } \mathbf { c }$ , larger $M [ w , c ]$ , larger $\bar { P } M I ( w , c )$ , and thus larger co-occurrence count $\# ( w , c )$ . However, the derivation has two flaws: (1) c could be negative and (2) lower $\# ( w , c )$ could still lead to larger $P M I ( w , c )$ as long as the $\# ( w )$ is small enough. + +To preserve DIH, we propose a novel word embedding method, distributional inclusion vector embedding $( D I V E )$ , which fixes the two flaws by performing non-negative factorization (NMF) on the matrix $M$ , where + +$$ +M [ w , c ] = \log ( \frac { P ( w , c ) } { P ( w ) \cdot P ( c ) } \cdot \frac { \# ( w ) } { k \cdot Z } ) = \log ( \frac { \# ( w , c ) | V | } { \# ( c ) k } ) , +$$ + +where $k$ is a constant which shifts PMI value like SGNS, $\begin{array} { r } { Z = \frac { | D | } { | V | } } \end{array}$ is the average word frequency, and $| V |$ is the vocabulary size. + +The design encourages the inclusion property in DIVE (i.e. Equation (2)) to be satisfied because the property implies that Equation (1) (DIH) holds if the matrix is reconstructed perfectly. The derivation is simple: Since context vector c is non-negative, if the embedding of hypernym $\mathbf { y }$ is greater than or equal to the embedding of its hyponym $\mathbf { X }$ in every dimension, $\mathbf { \bar { x } } ^ { T } \mathbf { c } \leq \mathbf { y } ^ { T } \mathbf { c }$ . Then, $\mathbf { \bar { \boldsymbol { M } } } [ \boldsymbol { x } , \boldsymbol { c } ] \le M [ \boldsymbol { y } , \boldsymbol { \bar { c } } ]$ tends to be true because $\mathbf { w } ^ { \hat { T } } \mathbf { c } \ \tilde { \approx } \ M [ w , c ]$ . This leads to $\# ( x , c ) \leq \# ( y , c )$ because M [w, c] = log( #(w,c)|V |#(c)k ) and only $\# ( w , c )$ change with $w$ . + +# 2.2 OPTIMIZATION + +Due to its appealing scalability properties during training time (Levy et al., 2015a), we optimize our embedding based on the skip-gram with negative sampling (SGNS) (Mikolov et al., 2013). The objective function of SGNS is + +$$ +l _ { S G N S } = \sum _ { w \in V } \sum _ { c \in V } \# ( w , c ) \log \sigma ( \mathbf { w } ^ { T } \mathbf { c } ) \ + \ \sum _ { w \in V } k ^ { \prime } \sum _ { c \in V } \# ( w , c ) \operatorname { \mathbb { E } } _ { c _ { N } \sim P _ { D } } [ \log \sigma ( - \mathbf { w } ^ { T } \mathbf { c } _ { \mathbf { N } } ) ] , +$$ + +where $\mathbf { w } \in \mathbb { R } , \mathbf { c } \in \mathbb { R } , \mathbf { c _ { N } } \in \mathbb { R } , k ^ { \prime }$ is a constant hyper-parameter indicating the ratio between positive and negative samples. + +Levy & Goldberg (2014) prove SGNS is equivalent to factorizing a shifted PMI matrix $M ^ { \prime }$ , where $\begin{array} { r } { M ^ { \prime } [ w , c ] = \log ( \frac { P ( w , c ) } { P ( w ) \cdot P ( c ) } \cdot \frac { 1 } { k ^ { \prime } } ) } \end{array}$ . By setting $\begin{array} { r } { k ^ { \prime } = \frac { k \cdot Z } { \# ( w ) } } \end{array}$ and applying non-negativity constraints to the embeddings, DIVE can be optimized using the similar objective function: + +$$ +l _ { D I V E } = \sum _ { w \in V } \sum _ { c \in V } \# ( w , c ) \log \sigma ( \mathbf { w } ^ { T } \mathbf { c } ) \ + k \sum _ { w \in V } \frac { Z } { \# ( w ) } \sum _ { c \in V } \# ( w , c ) \operatorname { \mathbb { E } } _ { c _ { N } \sim P _ { D } } [ \log \sigma ( - \mathbf { w } ^ { T } \mathbf { c } _ { \mathbf { N } } ) ] , +$$ + +where $\mathbf { w } \geq 0 , \mathbf { c } \geq 0 , \mathbf { c } _ { \mathbf { N } } \geq 0 ,$ $\sigma$ is the logistic sigmoid function, and $k$ is a constant hyper-parameter. $P _ { D }$ is the distribution of negative samples, which we set to be the corpus word frequency distribution in this paper. Equation (5) is optimized by ADAM (Kingma & Ba, 2015), a variant of stochastic gradient descent (SGD). The non-negativity constraint is implemented by projection (i.e., clipping any embedding which crosses the zero boundary after an update). + +![](images/bc81173fe9f03c1c3657656ec72c8d5249f2772d104addb772d049e2d14a7de5.jpg) + +Figure 1: The top 15 dimensions in the distributional inclusion vector embedding (DIVE) of the word core trained by the co-occurrence statistics of context words. The index of dimensions is sorted by the embedding values. The words in each row of the table are sorted by its embedding value in the dimension. + +
idTop 1-5 wordsTop 101-105 words
1element,gas,atom,rock,carbonmethane,llio
2system,rcieturvelope,gefunctional,rnt,ocsing,
3also, well,svral,rlyfall, eventually, main,ise,mosty
4partincorporate,ge,iead,oing,dd
5star,orbit,snbital,bright,posion,turieractio
6several,icdesignate,ist,iss,bch,i
7science,plosophy, thory,plosophr,rethical,dvocte,oic,bic,
8version,game,release,original,ilecassette, rtual,code,project, kb
9electron,ttric,citanode,wire,ac,perform,eistor
10tank, cylinder,heel,gialuminumtic,ott
11school,itildoctorateldert
12network,r,er,dtum,prltechnologyoutgnt,crooft,
13high,low,turergyatmosphric,tod,,io
14acid, carbon, product, ue,incph,monoided
15access,nd,rquire,llow,programsize,abilitytlly
+ +The optimization process provides an alternative angle to explain how DIVE preserves DIH. The gradients for the word embedding w is + +$$ +\frac { d l _ { D I V E } } { d \mathbf { w } } = \sum _ { c \in V } \# ( w , c ) ( 1 - \sigma ( \mathbf { w } ^ { T } \mathbf { c } ) ) \mathbf { c } - k \sum _ { c _ { N } \in V } \frac { \# ( c _ { N } ) } { | V | } \sigma ( \mathbf { w } ^ { T } \mathbf { c } _ { \mathbf { N } } ) \mathbf { c } _ { \mathbf { N } } . +$$ + +Assume hyponym $\mathbf { X }$ and hypernym y satisfy DIH in Equation (1) and the embeddings $\mathbf { x }$ and $\mathbf { y }$ are the same at some point during the gradient ascent. In the case, the gradients coming from negative sampling (the second term) decrease the same amount of embedding values for both $x$ and $y$ because $k$ is a constant hyper-parameter. However, the embedding of hypernym $\mathbf { y }$ would get higher or equal positive gradients from the first term than $\mathbf { x }$ in every dimension because $\# ( x , \bar { c } ) \leq \bar { \# } ( y , c )$ . This means Equation (1) tends to imply Equation (2). Combining the analysis from the matrix factorization viewpoint, DIH in Equation (1) is approximately equivalent to the inclusion property in DIVE (i.e. Equation (2)). + +# 2.3 PMI FILTERING + +For a frequent target word, there must be many neighboring words that incidentally appear near the target word without being semantically meaningful, especially when a large context window size is used. The unrelated context words cause noise in both the word vector and the context vector of DIVE. We address this issue by filtering out context words $c$ for each target word $w$ when the PMI of the co-occurring words is too small (i.e., $\begin{array} { r } { \log ( \frac { P ( w , c ) } { P ( w ) \cdot P ( c ) } ) < \log ( k _ { f } ) ) } \end{array}$ . That is, we set $\# ( w , c ) = 0$ in the objective function. This preprocessing step is similar with computing PPMI in SBOW (Bullinaria & Levy, 2007), where low PMI co-occurrences are removed from the count-based representation. + +# 2.4 INTERPRETABILITY + +After applying the non-negativity constraint, we observe that each dimension roughly corresponds to a topic, as previous findings suggest (Pauca et al., 2004; Murphy et al., 2012). This gives rise to a natural and intuitive interpretation of our word embeddings: the word embeddings can be seen as unnormalized probability distributions over topics. By removing the normalization of the target word frequency in the shifted PMI matrix, specific words have values in few dimensions (topics), while general words appear in more topics and correspondingly have high values in more dimensions, so the concreteness level of two words can be easily compared using the magnitude of their embeddings. In other words, general words have more diverse context distributions, so we need more dimensions to store the information in order to compress SBOW well (Nalisnick & Ravi, 2015). + +In Figure 1, we present three mentions of the word core and its surrounding contexts. These various context words increase the embedding values in different dimensions. Each dimension of the learned embeddings roughly corresponds to a topic, and the more general or representative words for each topic tend to have the higher value in the corresponding dimension (e.g. words in the second column of the table). The embedding is able to capture the common contexts where the word core appears. For example, the context of the first mention is related to the atom topic (dimension id 1) and the electron topic (id 9), while the second and third mention occur in the computer architecture topic (id 2) and education topic (id 11), respectively. + +# 3 EXPERIMENT SETUP + +We describe four experiments in Section 4-7. The first 3 experiments compare DIVE with other unsupervised embeddings and SBOW using different hypernymy scoring functions. In these experiments, unsupervised approaches refer to the methods that only train on plaintext corpus without using any hypernymy or lexicon annotation. The last experiment presents qualitative results on word sense disambiguation. + +# 3.1 DATASETS AND TESTING SETUP + +The SBOW and embeddings are tested on 11 datasets. The first 4 datasets come from the recent review of Santus et al. (2017): BLESS (Baroni & Lenci, 2011), EVALution (Santus et al., 2015), Lenci/Benotto (Benotto, 2015), and Weeds (Weeds et al., 2014). The next 4 datasets are downloaded from the code repository of the H-feature detector (Roller & Erk, 2016): Medical (i.e., Levy 2014) (Levy et al., 2014), LEDS (also referred to as ENTAILMENT or Baroni 2012) (Baroni et al., 2012), TM14 (i.e., Turney 2014) (Turney & Mohammad, 2015), and Kotlerman 2010 (Kotlerman et al., 2010). In addition, the performance on the test set of HyperNet (Shwartz et al., 2016) (using the random train/test split), the test set of WordNet (Vendrov et al., 2016), and all pairs in HyperLex (Vulic et al., 2016) are also evaluated. ´ + +The F1 and accuracy measurements are sometimes very similar even though the quality of prediction varies, so average precision $\operatorname { A P } @$ all is adopted as the main evaluation metric. The HyperLex dataset has a continuous score on each candidate word pair, so we adopt Spearman rank coefficient $\rho$ as suggested by the review study of Vulic et al. (2016). Any OOV (out-of-vocabulary) word en- ´ countered in the testing data is pushed to the bottom of the prediction list (effectively assuming the word pair does not have a hypernym relation). + +# 3.2 TRAINING SETUP + +We use WaCkypedia corpus (Baroni et al., 2009), a 2009 Wikipedia dump, to compute SBOW and train the embedding. For the datasets without Part of Speech (POS) information (i.e. Medical, LEDS, TM14, Kotlerman 2010, and HyperNet), the training data of SBOW and embeddings are raw text. For other datasets, we concatenate each token with the Part of Speech (POS) of the token before training the models except the case when we need to match the training setup of another paper. + +All words are lower cased. Stop words and rare words (occurs less than 10 times) are removed during our preprocessing step. The number of embedding dimensions in DIVE $d _ { 0 }$ is set to be 100. Other hyper-parameters used in the experiments are listed in the supplementary materials. The hyper-parameters of DIVE were decided based on the performance of HyperNet training set. To train embeddings more efficiently, we chunk the corpus into subsets/lines of 100 tokens instead of using sentence segmentation. Preliminary experiments show that this implementation simplification does not hurt the performance. + +Table 1: Comparison with previous unsupervised embeddings. All values are percentages. $\mathbf { A P } @$ all $( \% )$ for 10 datasets and Spearman $\rho$ $( \% )$ for HyperLex. Word2Vec $+ \mathrm { C }$ scores the word pairs using the cosine similarity on skip-grams (SGNS). $\mathrm { G E + C }$ and $\mathrm { G E + K L }$ computes cosine similarity and negative KL divergence on Gaussian embedding, respectively. + +
DatasetBLESSEVALutionLenciBenottoWeedsMedicalLEDS
Random Word2Vec+C5.3 9.226.6 25.441.2 40.851.4 51.68.5 11.250.5 71.8
GE+C GE+KL10.5 7.626.7 29.643.352.014.969.7
DIVE+C.△S45.151.315.764.6
16.333.050.465.525.383.5
DatasetTM14Kotlerman 2010HyperNetWordNetHyperLex
Random52.030.824.555.20
Word2Vec+C52.1
39.520.763.016.3
GE+C53.936.021.658.216.4
GE+KL52.039.423.754.49.6
DIVE+C·△S57.236.641.960.932.8
+ +In the following experiments, we train both SBOW and DIVE on only the first 512,000 lines (51.2 million tokens) because we find this way of training setting provides better performances (for both SBOW and DIVE) than training on the whole WaCkypedia or training on randomly sampled 512,000 lines. We suspect this is due to the corpus being sorted by the Wikipedia page titles, which makes some categorical words such as animal and mammal occur 3-4 times more frequently in the first 51.2 million tokens than the rest. The performances of training SBOW PPMI on the whole WaCkypedia is also provided for reference in Table 4 and Table 5. + +# 4 EXPERIMENT 1: COMPARISON WITH UNSUPERVISED EMBEDDINGS + +If a pair of words has the hypernym relation, the words tend to be similar and the hypernym should be more general than the hyponym. As in HyperScore (Nguyen et al., 2017), we score the hypernym candidates by multiplying two factors corresponding to these properties. The ${ \bf C } { \cdot } { \Delta } { \cal S }$ (i.e. the cosine similarity multiply the difference of summation) scoring function is defined as + +$$ +C \cdot \Delta S ( \mathbf { w } _ { q } \mathbf { w } _ { p } ) = \frac { \mathbf { w } _ { q } ^ { T } \mathbf { w } _ { p } } { | | \mathbf { w } _ { q } | | _ { 2 } \cdot | | \mathbf { w } _ { p } | | _ { 2 } } \cdot ( | | \mathbf { w } _ { p } | | _ { 1 } - | | \mathbf { w } _ { q } | | _ { 1 } ) , +$$ + +where ${ \bf w } _ { p }$ is the embedding of hypernym and ${ \bf w } _ { q }$ is the embedding of hyponym. + +As far as we know, Gaussian embedding (GE) is the only unsupervised embedding method which can capture the asymmetric relations between a hypernym and its hyponyms. Using the same training and testing setup, we use the code implemented by Athiwaratkun $\&$ Wilson $( 2 0 1 7 ) ^ { 1 }$ to train Gaussian embedding on the first 51.2 million tokens and test the embeddings on 11 datasets. Its hyper-parameters are determined using the same way as DIVE (i.e. maximizing the AP on HyperNet training set). We compare DIVE with $\mathrm { G E } ^ { 2 }$ in Table 1, and the performances of random scores and only measuring word similarity using skip-grams are also presented for reference. As we can see, DIVE is usually significantly better than other baselines. + +# 5 EXPERIMENT 2: HYPERNYMY SCORING FUNCTIONS ANALYSIS + +In Experiment 1, we show that there exists a scoring function $\left( \mathbf { C } { \cdot } \Delta \mathbf { S } \right)$ which detects hypernymy accurately using the embedding space of DIVE. Nevertheless, different scoring functions measure different signals in SBOW or embeddings. Since there are so many scoring functions and datasets available in the domain, we first introduce and test the performances of various scoring functions so as to select the representative ones for a more comprehensive evaluation of DIVE on the hypernymy detection tasks. We denote the embedding/context vector of the hypernym candidate and the hyponym candidate as ${ \bf w } _ { p }$ and ${ \bf w } _ { q }$ , respectively. The SBOW model which represents a word by the frequency of its neighboring words is denoted as SBOW Freq, while the SBOW which uses PPMI of its neighboring words as the features (Bullinaria & Levy, 2007) is denoted as SBOW PPMI. + +# 5.1 UNSUPERVISED SCORING FUNCTIONS + +# 5.1.1 SIMILARITY + +A hypernym tends to be similar to its hyponym, so we measure the cosine similarity between word vectors of the SBOW features (Levy et al., 2015b) or DIVE. We refer to the symmetric scoring function as Cosine or C for short in the following tables. We also train the original skip-grams with 100 dimensions and measure the cosine similarity between the resulting word2vec embeddings. This scoring function is referred to as Word2vec or W. + +# 5.1.2 GENERALITY + +The distributional informativeness hypothesis (Santus et al., 2014) observes that in many corpora, semantically ‘general’ words tend to appear more frequently and in more varied contexts. Thus, Santus et al. (2014) advocate using entropy of context distributions to capture the diversity of context. We adopt the two variations of the approach proposed by Santus et al. (2017): SLQS Row and SLQS Sub functions. We also refer to SLQS Row as $\Delta \mathrm { E }$ because it measures the entropy difference of context distributions. For SLQS Sub, the number of top context words is fixed as 100. + +Although effective at measuring diversity, the entropy totally ignores the frequency signal from the corpus. To leverage the information, we measure the generality of a word by its L1 norm $( | \mathbf { w } _ { p } | _ { 1 } )$ and L2 norm $( | | \mathbf { w } _ { p } | | _ { 2 } )$ . Recall that Equation (2) indicates that the embedding of the hypernym $\mathbf { y }$ should have a larger value at every dimension than the embedding of the hyponym x. When the inclusion property holds, $\begin{array} { r } { | { \bf y } | _ { 1 } = \sum _ { i } { \bf y } [ i ] \geq \sum _ { i } { \bf x } [ i ] = | { \bf x } | _ { 1 } } \end{array}$ and similarly $| | \mathbf { y } | | _ { 2 } \geq | | \mathbf { x } | | _ { 2 }$ . Thus, we propose two scoring functions, difference of vector summation $( | \mathbf { w } _ { p } | _ { 1 } - | \mathbf { w } _ { q } | _ { 1 } )$ and the difference of vector 2-norm $( | | \mathbf { \bar { w } } _ { p } | | _ { 2 } - | | \mathbf { w } _ { q } | | _ { 2 } )$ . Notice that when applying the difference of vector summations (denoted as $\Delta S$ ) to SBOW Freq, it is equivalent to computing the word frequency difference between the hypernym candidate pair. + +# 5.1.3 COMBINATION + +The combination of 2 similarity functions (Cosine and Word2vec) and the 3 generality functions (difference of entropy, summation, and 2-norm of vectors) leads to six different scoring functions as shown in Table 2, and ${ \bf C } { \cdot } { \Delta } { \cal S }$ is the same scoring function we used in Experiment 1. It should be noted that if we use skip-grams with negative sampling (word2vec) as the similarity measurement (i.e., $W \cdot \Delta \ \{ \mathrm { E } , \mathrm { S } , \mathrm { Q } \} ,$ ), the scores are determined by two embedding/feature spaces together (word2vec and DIVE/SBOW). + +# 5.1.4 INCLUSION + +Several scoring functions are proposed to measure inclusion properties of SBOW based on DIH. Weeds Precision (Weeds & Weir, 2003) and CDE (Clarke, 2009) both measure the magnitude of the intersection between feature vectors $( | \mathbf { w } _ { p } \cap \mathbf { w } _ { q } | )$ . For example, ${ \bf w } _ { p } \cap { \bf w } _ { q }$ is defined by the elementwise minimum in CDE. Then, both scoring functions divide the intersection by the magnitude of the potential hyponym vector $( | \mathbf { w } _ { q } | )$ . invCL (Lenci & Benotto, 2012) (A variant of CDE) is also tested. + +We choose these 3 functions because they have been shown to detect hypernymy well in a recent study (Santus et al., 2017). However, it is hard to confirm that their good performances come from the inclusion property between context distributions — it is also possible that the context vectors of more general words have higher chance to overlap with all other words due to their high frequency. For instance, considering a one dimension feature which stores only the frequency of words, the naive embedding could still have reasonable performance on the CDE function, but the embedding in fact only memorizes the general words without modeling relations between words (Levy et al., 2015b) and loses lots of inclusion signals in the word co-occurrence statistics. + +Table 2: Micro average AP@all $( \% )$ of 10 datasets using different scoring functions. The feature space is SBOW using word frequency. + +
Word2vec (W) 24.8Cosine (C) 26.7SLQS Sub 27.4SLQS Row (△E) 27.6Summation (△S) 31.5Two norm (△Q) 31.2
W.△EC.△EW.△SC.△SW.△QC.△Q
28.829.531.631.231.431.1
WeedsCDEinvCLAsymmetric L1(AL1)
19.031.130.728.2
+ +In order to measure the inclusion property without the interference of the word frequency signal from the SBOW or embeddings, we propose a new measurement called asymmetric $L _ { 1 }$ distance. We first get context distributions ${ \bf d } _ { p }$ and ${ \bf d } _ { q }$ by normalizing $\mathbf { w } _ { p }$ and ${ \bf w } _ { q }$ , respectively. Ideally, the context distribution of the hypernym ${ \bf d } _ { p }$ will include ${ \bf d } _ { q }$ . This suggests the hypernym distribution ${ \bf d } _ { p }$ is larger than context distribution of the hyponym with a proper scaling factor $a \mathbf { d } _ { q }$ (i.e., $\operatorname* { m a x } ( a \mathbf { d } _ { q } \bar { - }$ ${ \bf d } _ { p } , 0 )$ should be small). Furthermore, both distributions should be similar, so $a \mathbf { d } _ { q }$ should not be too different from ${ \bf d } _ { p }$ (i.e., $\operatorname* { m a x } ( \mathbf { d } _ { p } - a \mathbf { d } _ { q } , 0 )$ should also be small). Therefore, we define asymmetric L1 distance as + +$$ +A L _ { 1 } = \operatorname* { m i n } _ { a } \sum _ { c } w _ { 0 } \cdot \operatorname* { m a x } ( a \mathbf { d } _ { q } [ c ] - \mathbf { d } _ { p } [ c ] , 0 ) + \operatorname* { m a x } ( \mathbf { d } _ { p } [ c ] - a \mathbf { d } _ { q } [ c ] , 0 ) , +$$ + +where $w _ { 0 }$ is a constant which emphasizes the inclusion penalty. If $w _ { 0 } ~ = ~ 1$ and $a \ = \ 1$ , $A L _ { 1 }$ is equivalent to L1 distance. The lower $A L _ { 1 }$ distance implies a higher chance of observing the hypernym relation. We tried $w _ { 0 } = 5$ and $w _ { 0 } = 2 0$ . $w _ { 0 } = 2 0$ produces a worse micro-average AP@all on SBOW Freq, SBOW PPMI and DIVE, so we fix $w _ { 0 }$ to be 5 in all experiments. An efficient way to solve the optimization in $A L _ { 1 }$ is presented in the supplementary materials. + +# 5.2 RESULTS AND DISCUSSIONS + +We show the micro average $\mathbf { A P } @$ all on 10 datasets using different hypernymy scoring functions in Table 2. We can see the combination functions such as ${ \bf C } { \cdot } { \Delta } { \cal S }$ and ${ \bf W } { \cdot } { \Delta } { \cal S }$ perform the best overall. Among the unnormalized inclusion based scoring functions, CDE works the best. $A L _ { 1 }$ performs well compared with other functions which remove the frequency signal such as Word2vec, Cosine, and SLQS Row. The summation is the most robust generality measurement. In the table, the scoring functions are applied to SBOW Freq, but the performances of hypernymy scoring functions on the other feature spaces (e.g. DIVE) have a similar trend. + +# 6 EXPERIMENT 3: COMPARISON WITH SBOW + +# 6.1 COMPARISON WITH PREVIOUSLY REPORTED RESULTS + +In Table 3, DIVE with two of the best scoring functions $C { \cdot } \Delta s$ and ${ \bf W } { \cdot } { \Delta S }$ ) is compared with the previous unsupervised state-of-the-art approaches based on SBOW on different datasets. + +There are several reasons which might cause the large performance gaps in some datasets. In addition to the effectiveness of DIVE, some improvements come from our proposed scoring functions. The fact that every paper uses a different training corpus also affects the performances. Furthermore, Santus et al. (2017) select the scoring functions and feature space for the first 4 datasets based on $\mathbf { A P } @ 1 0 0$ , which we believe is too sensitive to the hyper-parameter settings of different methods. To isolate the impact of each factor, we perform a more comprehensive comparison next. + +# 6.2 PERFORMANCE ANALYSIS + +In this experiment, we examine whether DIVE successfully preserves the signals for hypernymy detection tasks, which are measured by the same scoring functions designed for SBOW. Summation difference $( \Delta \boldsymbol { S } )$ and CDE perform the best among generality and inclusion functions in Table 2, + +Table 3: Comparison with previous methods based on sparse bag of word (SBOW). All values are percentages. The results of invCL (Lenci & Benotto, 2012), APSyn (Santus et al., 2016), and CDE (Clarke, 2009) are selected because they have the best $\mathbf { A P } @ 1 0 0$ in the first 4 datasets (Santus et al., 2017). Cosine similarity (Levy et al., 2015b), balAPinc (Kotlerman et al., 2010) in 3 datasets (Turney & Mohammad, 2015), SLQS (Santus et al., 2014) in HyperNet dataset (Shwartz et al., 2016), and Freq ratio (FR) (Vulic et al., 2016) are compared. ´ + +
DatasetBLESS EVALutionLenciBenottoWeedsMedical
MetricAP@allF1
BaselinesinvCLAPSynCDECosine
5.135.338.244.123.1
DIVE+C·△S16.333.050.465.525.3
DIVE +W·△S18.632.351.568.625.7
DatasetLEDSTM14Kotlerman 2010HyperNetHyperLex
MetricAP@allF1Spearman p
BaselinesbalAPincSLQSFreq ratio
73563722.827.9
DIVE+C·△S83.557.236.641.932.8
DIVE+W·△S86.457.337.438.633.3
+ +respectively. $A L _ { 1 }$ could be used to examine the inclusion properties after removing the frequency signal. Therefore, we will present the results using these 3 scoring functions, along with ${ \bf W } { \cdot } { \Delta } S$ and ${ \bf C } { \cdot } { \Delta } { \cal S }$ . + +# 6.2.1 BASELINES + +In addition to classic representations such as SBOW Freq and SBOW PPMI, we compare distributional inclusion vector embedding (DIVE) with additional 4 baselines in Table 4. + +• SBOW PPMI with additional frequency weighting (PPMI w/ FW). Specifically, $\mathbf { w } [ c ] =$ $\begin{array} { r } { \operatorname* { m a x } ( \log ( \frac { P ( w , c ) } { P ( w ) * P ( c ) * \frac { Z } { \# ( w ) } } ) , 0 ) } \end{array}$ . This forms the matrix reconstructed by DIVE when $k = 1$ . +• DIVE without the PMI filter (DIVE w/o PMI) +• NMF on shifted PMI: Non-negative matrix factorization (NMF) on the shifted PMI without frequency weighting for DIVE (DIVE w/o FW). This is the same as applying the nonnegative constraint on the skip-gram model. +K-means (Freq NMF): The method first uses Mini-batch $\mathbf { k }$ -means (Sculley, 2010) to cluster words in skip-gram embedding space into 100 topics, and hashes each frequency count in SBOW into the corresponding topic. If running $\mathbf { k }$ -means on skip-grams is viewed as an approximation of clustering the SBOW context vectors, the method can be viewed as a kind of NMF (Ding et al., 2005). Let the $N \times N$ context matrix be denoted as $M _ { c }$ , where the $( i , j )$ th element stores the count of word $j$ appearing beside word $i$ . K-means hashing creates a $N \times 1 0 0$ matrix $G$ with orthonormal rows $\tilde { G } ^ { T } G = I )$ , where the $( i , k )$ th element is 0 if the word $i$ does not belong to cluster $k$ . The orthonormal $G$ is also an approximated solution of a type of NMF $( M _ { c } \overset { \mathbf { \backsimeq } } { \approx } F G ^ { T }$ ) (Ding et al., 2005). Hashing context vectors into topic vectors can be written as $M _ { c } G \approx F G ^ { T } G = F$ . + +In the experiment, we also tried to apply a constant $\log ( k )$ shifting to SBOW PPMI (i.e. $\operatorname* { m a x } ( P M I - \log ( k ) , 0 ) )$ ). We found that the performance degrades as $k$ increases. Similarly, applying PMI filter to SBOW PPMI (set context feature to be 0 if the value is lower than $\log ( k _ { f } ) )$ usually makes the performances worse, especially when $k _ { f }$ is large. Applying PMI filter to SBOW Freq only makes its performances closer to (but still much worse than) SBOW PPMI, so we omit this baseline as well. + +# 6.2.2 RESULTS AND DISCUSSIONS + +In Table 4, we first confirm the finding of the previous review study of Santus et al. (2017): there is no single hypernymy scoring function which always outperforms others. One of the main reasons is that different datasets collect negative samples differently. This is also why we evaluate our method on many datasets to make sure our conclusions hold in general. For example, if negative samples come from random word pairs (e.g. WordNet dataset), a symmetric similarity measure is already a pretty good scoring function. On the other hand, negative samples come from related or similar words in HyperNet, EVALution, Lenci/Benotto, and Weeds, so only computing generality difference leads to the best (or close to the best) performance. The negative samples in many datasets are composed of both random samples and similar words (such as BLESS), so the combination of similarity and generality difference yields the most stable results. + +Table 4: $\mathbf { A P } @$ all $( \% )$ of 10 datasets. The box at lower right corner compares the micro average AP across all 10 datasets. Numbers in different rows come from different feature or embedding spaces. Numbers in different columns come from different datasets and unsupervised scoring functions. We also present the micro average AP across the first 4 datasets (BLESS, EVALution, Lenci/Benotto and Weeds). All wiki means SBOW using PPMI features trained on the whole WaCkypedia. FW refers to frequency weighting on the shifted PMI matrix. + +
AP@all(%)BLESSEVALutionLenci/Benotto
CDEAL1△SW△SC△SCDE AL1△SW△SC△SCDEAL1△SW△SC△S
SBOWFreq6.37.3 5.6 5.611.0 17.25.9 15.335.3 30.432.6 27.736.233.036.351.847.651.051.851.1
PPMI13.65.127.534.131.934.347.239.750.851.152.0
PPMI w/FW6.25.05.5 12.45.836.036.332.936.452.043.150.951.950.7
All wiki12.15.26.9 12.513.4 28.527.130.329.931.047.139.948.548.751.1
DIVEFull9.37.6 6.0 5.618.616.330.027.534.933.046.743.251.351.550.4
w/o PMI7.86.916.77.132.832.235.732.3 32.535.447.644.950.951.649.7
w/o FW9.06.26.27.324.325.022.923.523.938.838.138.238.238.4
Kmean (Freq NMF)6.57.3 5.610.95.833.727.236.233.036.249.642.551.051.851.2
7.3 WeedsMicro Average (4 datasets)Medical
AP@all(%)CDE AL1△SCDEAL1CDEAL1△SW·△SC△S
SBOWFreq69.558.0 68.8W△S 68.2C△S 68.423.121.8△S 22.9W△S 25.0C△S 23.019.419.214.118.415.3
PPMI61.050.370.369.269.324.7 17.922.328.127.823.48.713.220.124.4
PPMI w/FW67.652.269.468.767.723.2 18.222.925.822.922.810.613.718.617.0
All wiki61.348.670.070.423.425.822.38.912.217.621.1
Full59.255.068.517.7 19.821.7 22.824.6 28.927.611.79.313.721.419.2
DIVE w/o PMI60.456.469.7 69.368.6 68.665.5 64.822.1 22.221.023.110.78.413.319.816.2
w/o FW49.247.3 45.145.144.918.917.322.7 17.228.0 16.817.510.99.87.47.67.7
Kmean (Freq NMF)69.451.1 68.868.268.922.519.322.924.923.012.610.914.018.114.6
LEDSTM14
AP@all(%)AL1 △SCDE AL1Kotlerman 2010
SBOWFreqCDE70.4 70.7W·△S 83.3C△S 73.355.653.2△S 54.9W·△S 55.7C.△S 55.0CDE 35.9AL1 40.5△S 34.5W.△S 37.0C.△S 35.4
SBOWPPMI82.7 84.450.2 72.286.584.556.252.354.457.639.130.933.037.036.3
All wiki83.149.7 67.982.981.454.750.552.657.0 55.154.938.531.232.235.435.3
DIVE83.372.786.483.555.352.655.257.357.235.331.633.637.436.6 MicroAverage(10datasets)
AP@all (%)74.7HyperNetWordNet
+ +DIVE performs similar or better on all the scoring functions compared with SBOW consistently across all datasets in Table 4, while using many fewer dimensions (see Table 6). Its results on combination scoring functions outperform SBOW Freq. Meanwhile, its results on $A L _ { 1 }$ outperform SBOW PPMI. The fact that combination scoring functions (i.e., W·∆S or $\mathbf { C } { \cdot } \Delta \mathbf { S }$ ) usually outperform generality functions suggests that only memorizing general words is not sufficient. The best average performance on 4 and 10 datasets are both produced by ${ \bf W } { \cdot } { \Delta } { \cal S }$ on DIVE. + +SBOW PPMI improves the combination functions from SBOW Freq but sacrifices AP on the inclusion functions. It generally hurts performance to change the frequency sampling of PPMI (PPMI w/ FW) or compute SBOW PPMI on the whole WaCkypedia (all wiki) instead of the first 51.2 million tokens. The similar trend can also be seen in Table 5. Note that $A L _ { 1 }$ completely fails in HyperLex dataset using SBOW PPMI, which suggests that PPMI might not necessarily preserve the distributional inclusion property, even though it can have good performance on combination functions. + +Removing the PMI filter from DIVE slightly drops the overall precision while removing frequency weights on shifted PMI (w/o FW) leads to poor performances. K-means (Freq NMF) produces similar AP compared with SBOW Freq, but has worse $A L _ { 1 }$ scores. Its best AP scores on different datasets are also significantly worse than the best AP of DIVE. This means that only making word2vec (skip-grams with negative sampling) non-negative or naively accumulating topic distribution in contexts cannot lead to satisfactory embeddings. + +Table 5: Spearman $\rho \left( \% \right)$ in HyperLex. + +
Spearman p (%)HyperLex
CDEAL1△SW·△SC.△S
SBOWFreq31.719.627.629.627.3
SBOW PPMI28.1-2.331.834.334.5
All wiki25.3-2.228.030.531.0
DIVE28.918.731.233.332.8
+ +Table 6: The average number of non-zero dimensions across all testing words in 10 datasets. + +
SBOW FreqSBOWPPMIDIVE
5799380820
+ +Table 7: Spectral clustering on the non-zero DIVE dimensions of the query word. When the number of clusters is set to be 2, we present the top 4 dimensions in each cluster, which have the highest values on the query word embedding. Otherwise, the top 2 dimensions are presented. CID refers to cluster ID. The top 5 words with the highest values of each dimension are presented. + +
QueryCIDTop 5 words in the top dimensions
rock1element,gas,atom,rock,carbon find, specie,species,animal,birdsea,lake,river,area,water point,side,line, front,circle
2band,song,album,music,rock early,work,century,late,beginwrite,john,guitar,band,author include,several, show, television,film
bank1county,area,city, town,west building,build,house, palace, siteseveral,main, province, include,consist sea,lake,river,area,water
2money, tax,price, pay, income united, states,country, world, europecompany,corporation,system,agency, service state,palestinian,israel,right, palestine
apple1food, fruit,vegetable,meat, potato war,german,ii,germany,worldgoddess,zeus,god,hero,sauron write, john,guitar,band,author
2version, game,release,original, file system,architecture,develop,base,languagecar,company,sell, manufacturer,model include,several, show, television,film
star1film, role, production, play, stage wear,blue,color, instrument,redcharacter,series,game,novel, fantasy write, john,guitar,band,author
2element,gas,atom,rock,carbon give, term,vector,mass,momentumstar,orbit,sun,orbital,planet light, image,lens,telescope,camera
tank1tank,cylinder,wheel, engine,steel acid,carbon,product, use, zincindustry, export, industrial, economy,company network,user,server, datum, protocol
2army,force,infantry,military,battle however,attempt,result, despite, failaircraft,navy,missile,ship,flight war, german, ii, germany, world
race1win,world,cup,play,championshiptwo,one,three,four,another
2 3railway, line, train,road, railcar,company, sell, manufacturer,model
population,language, ethnic, native, peoplefemale,age,woman,male,household
run1system,architecture, develop,base,languageaccess, need, require,allow, program
2railway,line,train,road,railalso,well, several, early, see
3game,team,season,win,leaguegame,player,run,deal,baseball
tablet1bc, source, greek, ancient, datebook,publish,write,work,edition
2use,system,design,term,methodversion, game,release,original, file
3system,blood,vessel,artery,intestinepatient, symptom, treatment,disorder, may
+ +# 7 EXPERIMENT 4: WORD SENSE DISAMBIGUATION + +In addition to hypernymy detection, Athiwaratkun & Wilson (2017) show that the mixture of Gaussian distributions can also be used to discover multiple senses of each word. In our qualitative experiment, we show that DIVE can achieve the similar goal without fixing the number of senses before training the embedding. + +Recall that each dimension roughly corresponds to one topic. Given a query word, the higher embedding value on a dimension implies higher likelihood to observe the word in the context of the topic. The embedding of a polysemy would have high values on different groups of topics/dimensions. This allows us to discover the senses by clustering the topics/dimensions of the polysemy. We use the embedding values as the feature each dimension, compute the pairwise similarity between dimensions, and apply spectral clustering (Stella & Shi, 2003) to group topics as shown in the Table 7. See more implementation details in the supplementary materials. + +In the word sense disambiguation tasks, it is usually challenging to determine how many senses/clusters each word should have. Many existing approaches fix the number of senses before training the embedding (Tian et al., 2014; Athiwaratkun & Wilson, 2017). Neelakantan et al. (2014) make the number of clusters approximately proportional to the diversity of the context, but the assumption does not always hold. Furthermore, the training process cannot capture different granularity of senses. For instance, race in the car context could share the same sense with the race in the game topic because they all mean contest, but the race in the car context actually refers to the specific contest of speed. Therefore, they can also be viewed as separate senses (like the results in Table 7). This means the correct number of clusters is not unique, and the methods, which fixes the cluster numbers, need to re-train the embedding many times to capture such granularity. + +In our approach, clustering dimensions is done after the training process of DIVE is completed, so it is fairly efficient to change the cluster numbers and hierarchical clustering is also an option. Similar to our method, Pelevina et al. (2016) also discover word senses by graph-based clustering. The main difference is that they cluster the top $n$ words which are most related to the query word instead of topics. However, choosing the hyper-parameter $n$ is difficult. Large $n$ would make graph clustering algorithm inefficient, while small $n$ would make less frequent senses difficult to discover. + +# 8 RELATED WORK + +Most previous unsupervised approaches focus on designing better hypernymy scoring functions for sparse bag of word (SBOW) features. They are well summarized in the recent study (Santus et al., 2017). Santus et al. (2017) also evaluate the influence of different contexts, such as changing the window size of contexts or incorporating dependency parsing information, but neglect scalability issues inherent to SBOW methods. + +A notable exception is the Gaussian embedding model (Vilnis & McCallum, 2015). The context distribution of each word is encoded as a multivariate Gaussian distribution, where the embeddings of hypernyms tend to have higher variance and overlap with the embedding of their hyponyms. However, since a Gaussian distribution is normalized, it is difficult to retain frequency information during the embedding process, and experiments on HyperLex (Vulic et al., 2016) demonstrate that a simple ´ baseline only relying on word frequency can achieve good results. Follow-up work models contexts by a mixture of Gaussians (Athiwaratkun & Wilson, 2017) relaxing the unimodality assumption but achieves little improvement on hypernym detection tasks. + +Kiela et al. (2015) show that images retrieved by a search engine can be a useful source of information to determine the generality of lexicons, but the resources might not be available for some corpora such as scientific literature. + +Order embedding (Vendrov et al., 2016) is a supervised approach to encode many annotated hypernym pairs (e.g. all of the whole WordNet (Miller, 1995)) into a compact embedding space, where the embedding of a hypernym should be smaller than the embedding of its hyponym in every dimension. Our method learns embedding from raw text, where a hypernym embedding should be larger than the embedding of its hyponym in every dimension. Thus, DIVE can be viewed as an unsupervised and reversed form of order embedding. + +Other semi-supervised hypernym detection methods aim to generalize from sets of annotated word pairs using raw text corpora. The goal of HyperScore (Nguyen et al., 2017) is similar to our model: the embedding of a hypernym should be similar to its hyponym but with higher magnitude. However, their training process relies heavily on annotated hypernym pairs, and the performance drops significantly when reducing the amount of supervision. In addition to context distributions, previous work also leverages training data to discover useful text pattern indicating is-a relation (Shwartz et al., 2016; Roller & Erk, 2016), but it remains challenging to increase recall of hypernym detection because commonsense facts like cat is-a animal might not appear in the corpus. + +Non-negative matrix factorization (NMF) has a long history in NLP, for example in the construction of topic models (Pauca et al., 2004). Non-negative sparse embedding (NNSE) (Murphy et al., 2012) and Faruqui et al. (2015) indicate that non-negativity can make embeddings more interpretable and improve word similarity evaluations. The sparse NMF is also shown to be effective in cross-lingual lexical entailment tasks but does not necessarily improve monolingual hypernymy detection (Vyas & + +Carpuat, 2016). In our study, a new type of NMF is proposed, and the comprehensive experimental analysis demonstrates its state-of-the-art performances on unsupervised hypernymy detection. + +# 9 CONCLUSIONS + +Compressing unsupervised SBOW models into a compact representation is challenging while preserving the inclusion, generality, and similarity signals which are important for hypernym detection. Our experiments suggest that simple baselines such as accumulating K-mean clusters and non-negative skip-grams do not lead to satisfactory performances in this task. + +To achieve this goal, we proposed an interpretable and scalable embedding method called distributional inclusion vector embedding (DIVE) by performing non-negative matrix factorization (NMF) on a weighted PMI matrix. We demonstrate that scoring functions which measure inclusion and generality properties in SBOW can also be applied to DIVE to detect hypernymy, and DIVE performs the best on average, slightly better than SBOW while using many fewer dimensions. + +Our experiments also indicate that unsupervised scoring functions, which combine similarity and generality measurements, work the best in general, but no one scoring function dominates across all datasets. A combination of unsupervised DIVE with the proposed scoring functions produces new state-of-the-art performances on many datasets under the unsupervised setup. + +Finally, a qualitative experiment shows that clusters of the topics discovered by DIVE often correspond to the word senses, which allow us to do word sense disambiguation without the need to know the number of senses before training the embeddings. + +# REFERENCES + +Ben Athiwaratkun and Andrew Gordon Wilson. Multimodal word distributions. In ACL, 2017. + +Marco Baroni and Alessandro Lenci. How we BLESSed distributional semantic evaluation. In Workshop on GEometrical Models of Natural Language Semantics (GEMS), 2011. + +Marco Baroni, Silvia Bernardini, Adriano Ferraresi, and Eros Zanchetta. The WaCky wide web: a collection of very large linguistically processed web-crawled corpora. Language resources and evaluation, 43(3):209–226, 2009. + +Marco Baroni, Raffaella Bernardi, Ngoc-Quynh Do, and Chung-chieh Shan. 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In EMNLP, 2003. + +Julie Weeds, Daoud Clarke, Jeremy Reffin, David Weir, and Bill Keller. Learning to distinguish hypernyms and co-hyponyms. In COLING, 2014. + +Table 8: Comparison with semi-supervised embeddings (with limited training data). All values are percentages. The number in parentheses beside each approach indicates the number of annotated hypernymy word pairs used to train the model. Semi-supervised embeddings include HyperScore (Nguyen et al., 2017) and H-feature (Roller & Erk, 2016). When we compare F1 with the results from other papers, we use 20 fold cross validation to determine prediction thresholds, as done by Roller & Erk (2016). Note that HyperScore ignores POS in the testing data, so we follow the setup when comparing with it. + +
DatasetHyperLexEVALution LenciBenotto WeedsMedical
MetricSpearman pAP@allF1
Baselines(#Training Hypernymy)HyperScore(1337)H-feature (897)
303944.858.526
DIVE+C·△S (0)34.533.852.970.025.3
+ +Table 9: We show the top 30 words with the highest embedding magnitude after dot product with the query embedding $\mathbf { q }$ (i.e. showing w such that $| \mathbf { w } ^ { T } \mathbf { q } | _ { 1 }$ is one of the top 30 highest values). The rows with the empty query word sort words based on $| \mathbf { w } | _ { 1 }$ . + +
QueryTop30 generalwords
use large american design studyname state life work leadsystem group may produce typeinclude power small control peoplebase death find great highcity form body write create
speciesspecie human gene fish evidencespecies bird tree disease breedanimal genus name live proteinfind family genetic food wildplant organism study cell similarmay suggest occur mammal fossil
systemsystem standard allow process networkuse type function code filedesign computer datum via developmentprovide application device base serviceoperate develop control program transportmodel method information software law
+ +# A SUPPLEMENTARY MATERIALS + +# A.1 COMPARISON WITH SEMI-SUPERVISED EMBEDDINGS + +In addition to the unsupervised approach, we also compare DIVE with semi-supervised approaches. When there are sufficient training data, there is no doubt that the semi-supervised embedding approaches such as HyperNet (Shwartz et al., 2016), H-feature detector (Roller & Erk, 2016), and HyperScore (Nguyen et al., 2017) can achieve better performance than all unsupervised methods. However, in many domains such as scientific literature, there are often not many annotated hypernymy pairs (e.g. Medical dataset (Levy et al., 2014)). + +Since we are comparing an unsupervised method with semi-supervised methods, it is hard to fairly control the experimental setups and tune the hyper-parameters. In Table 8, we only show several performances which are copied from the original paper when training data are limited3. As we can see, the performance from DIVE is roughly comparable to the previous semi-supervised approaches trained on small amount of hypernym pairs. This demonstrates the robustness of our approach and the difficulty of generalizing hypernymy annotations with semi-supervised approaches. + +# A.2 GENERALITY ESTIMATION AND HYPERNYM DIRECTIONALITY DETECTION + +In Table 9, we show the most general words in DIVE under different queries as constraints. We also present the accuracy of judging which word is a hypernym (more general) given word pairs with + +Table 10: Accuracy $( \% )$ of hypernym directionality prediction across 10 datasets. + +
Micro Average (1O datasets)
SBOW Freq+ SLQS Sub64.4SBOW Freq +△S66.8SBOW PPMI+△S66.8DIVE+△S67.0
+ +Table 11: Dataset sizes. N denotes the number of word pairs in the dataset, and OOV shows how many word pairs are not processed by all the methods in Table 4 and Table 5. + +
BLESSEVALutionLenci/BenottoWeedsAvg (4 datasets)
N 265540OV 1507N 13675OOV 2475N 5010OOV 1464N 29280OV 643N 48167OOv 6089
MedicalKotlerman 2010
NLEDSTM14HyperNet
12602OOV 3711NOOV 28NOOV 178NOOV 89NOOV 9424
27702188294017670
WordNetAvg(10 datasets)HyperLex
N OOVNN
8000 35969433726160OV 59
+ +hypernym relations in Table 10. The direction is classified correctly if the generality score is greater than 0 (hypernym is indeed predicted as the more general word). For instance, summation difference $( \Delta \mathsf { S } )$ classifies correctly if $\vert \mathbf { \bar { w } } _ { p } \vert _ { 1 } - \vert \mathbf { w } _ { q } \vert _ { 1 } > 0$ $( | \bar { \mathbf { w } _ { p } } | _ { 1 } > | \mathbf { w } _ { q } | _ { 1 } )$ . + +From the table, we can see that the simple summation difference performs better than SQLS Sub, and DIVE predicts directionality as well as SBOW. Notice that whenever we encounter OOV, the directionality is predicted randomly. If OOV is excluded, the accuracy of predicting directionality using unsupervised methods can reach around 0.7-0.75. + +# A.3 EXPERIMENTAL DETAILS FOR HYPERNYMY DETECTION + +In HyperNet and WordNet, some hypernym relations are determined between phrases instead of words. Phrase embeddings are composed by averaging word embeddings or SBOW features. For WordNet, we assume the Part of Speech (POS) tags of the words are the same as the phrase. All part-of-speech (POS) tags in the experiments come from NLTK. + +The window size $| W |$ of SBOW, DIVE, and GE are set as 20 (left 10 words and right 10 words). For DIVE, the number of epochs is 15, the learning rate is 0.001, the batch size is 128, the threshold in PMI filter $k _ { f }$ is set to be 30, and the ratio between negative and positive samples ${ \mathrm { ( k ) } }$ is 1.5. The hyper-parameters of DIVE were decided based on the performance of HyperNet training set. The window size of skip-grams (word2vec) is 10. The number of negative samples $( k ^ { \prime } )$ in skip-gram is set as 5. When composing skip gram into phrase embedding, average embedding is used. + +For Gaussian embedding (GE), the number of mixture is 1, the number of dimension is 100, the learning rate is 0.01, the lowest variance is 0.1, the highest variance is 100, the highest Gaussian mean is 10, and other hyper-parameters are the default value in https://github.com/ benathi/word2gm. The hyper-parameters of GE were also decided based on the performance of HyperNet training set. When determining the score between two phrases, we use the average score of every pair of tokens in two phrases. + +The number of testing pairs $_ \mathrm { N }$ and the number of OOV word pairs is presented in Table 11. + +# A.4 CLUSTERING DIMENSIONS FOR WORD SENSE DISAMBIGUATION + +We use all the default hyper-parameters of the spectral clustering library in Scikit-learn 0.18.2 (Pedregosa et al., 2011) except the number of clusters is set manually. A simple way to prepare the feature of each dimension $f ( c _ { i } )$ is to use the embedding values in that dimension $\mathbf { w } [ c _ { i } ]$ of all the words in our vocabulary $w \in V$ . That is, + +$$ +f ( c _ { i } ) = [ \mathbf { w } [ c _ { i } ] ] _ { w \in V } . +$$ + +However, clustering on the global features might group topics together based on the co-occurrence of words which are unrelated to the query words and we want to make the similarity dependent on the query word. For example, a country topic should be clustered together with a city topic if the query word is place, but it makes more sense to group the country topic with the money topic together if the query word is bank like we did in the word sense disambiguation experiment (Table 7). This means we want to focus on the geographical meaning of country when the query is related to geography, while focus on the economic meaning of country when the query is about economics. + +To create query dependent similarity measurement, we only consider the embedding of words which are related to the query word when preparing the features of dimensions. Specifically, given a query word $q$ , the feature vector of the $i$ th dimension $f ( c _ { i } , q )$ is defined as: + +$$ +f ( c _ { i } , q ) = \bigoplus _ { j = 1 } ^ { d _ { 0 } } \left[ \mathbf { w } [ c _ { i } ] \cdot \mathbf { w } _ { q } [ c _ { j } ] \right] _ { w \in C _ { j } ( n ) } , +$$ + +where ${ \bf w } _ { q } [ c _ { j } ]$ is the value of $j$ th dimension of query word embedding, $C _ { j } ( n )$ is the set of embeddings of top $n$ words in the $j$ th dimension, and the operator $\oplus$ means concatenation. This means instead of considering all the words in the vocabulary, we only take the top $n$ words of every dimension $j$ (n is fixed as 100 in the experiment), weight the feature based on how likely to observe query word in dimension $j$ $( \mathbf { w } _ { q } [ c _ { j } ] )$ , and concatenate all features together. That is, when measuring the similarity of dimensions, we only consider the aspects related to query word (e.g. mostly considering words related to facility and money when the query word is bank). + +After the features of all dimensions are collected, we normalize the feature of each dimension to have the norm 1, compute the pairwise similarity and run the spectral clustering to get the clustering results. + +# A.5 EFFICIENT WAY TO COMPUTE ASYMMETRIC L1 $( A L _ { 1 } )$ + +Recall that Equation (8) defines $A L _ { 1 }$ as follows: + +$$ +A L _ { 1 } = \mathcal { L } = \operatorname* { m i n } _ { a } \ \sum _ { c } w _ { 0 } \operatorname* { m a x } ( a \mathbf { d } _ { q } [ c ] - \mathbf { d } _ { p } [ c ] , 0 ) + \operatorname* { m a x } ( \mathbf { d } _ { p } [ c ] - a \mathbf { d } _ { q } [ c ] , 0 ) , +$$ + +where $\mathbf { d } _ { p } [ c ]$ is one of dimension in the feature vector of hypernym $\mathbf { d } _ { p }$ , $a \mathbf { d } _ { q }$ is the feature vector of hyponym after proper scaling. In Figure 2, an simple example is visualized to illustrate the intuition behind the distance function. + +By adding slack variables $\zeta$ and $\xi$ , the problem could be converted into a linear programming problem: + +$$ +\begin{array} { l } { \mathcal { L } = \displaystyle \operatorname* { m i n } _ { a , \zeta , \xi } ~ w _ { 0 } \sum _ { c } \zeta _ { c } + \sum _ { c } \xi _ { c } } \\ { ~ \zeta _ { c } \geq a \mathbf { d } _ { q } [ c ] - \mathbf { d } _ { p } [ c ] , ~ \zeta _ { c } \geq 0 } \\ { \displaystyle ~ \xi _ { c } \geq \mathbf { d } _ { p } [ c ] - a \mathbf { d } _ { q } [ c ] , ~ \xi _ { c } \geq 0 } \\ { ~ a \geq 0 , ~ } \end{array} +$$ + +so it can be simply solved by a general linear programming library. + +Nevertheless, the structure in the problem actually allows us to solve this optimization by a simple sorting. In this section, we are going to derive the efficient optimization algorithm. + +By introducing Lagrangian multiplier for the constraints, we can rewrite the problem as + +$$ +\begin{array} { l l } { { \mathcal { L } = \displaystyle \operatorname* { m i n } _ { a , \zeta , \xi } \operatorname* { m a x } _ { \alpha , \beta , \gamma , \delta } w _ { 0 } \sum _ { c } \zeta _ { c } + \sum _ { c } \xi _ { c } - \sum _ { c } \alpha _ { c } ( \zeta _ { c } - a { \bf d } _ { q } [ c ] + { \bf d } _ { p } [ c ] ) } } \\ { { \mathrm { ~ } } } \\ { { \mathrm { ~ } - \sum _ { c } \beta _ { c } ( \xi _ { c } - { \bf d } _ { p } [ c ] + a { \bf d } _ { q } [ c ] ) - \sum _ { c } \gamma _ { c } \zeta _ { c } - \sum _ { c } \delta _ { c } \xi _ { c } } } \\ { { \mathrm { ~ } } } \\ { { \zeta _ { c } \geq 0 , \xi _ { c } \geq 0 , \alpha _ { c } \geq 0 , \beta _ { c } \geq 0 , \gamma _ { c } \geq 0 , \delta _ { c } \geq 0 , a \geq 0 } } \end{array} +$$ + +![](images/8c0cf76ff70a7b6ae235a8767dbac4aa77b900b1aa3ec191dfbdfa9da0d93c69.jpg) +Figure 2: An example of $A L _ { 1 }$ distance. If the word pair indeed has the hypernym relation, the context distribution of hyponym $( \mathbf { d } _ { q } )$ tends to be included in the context distribution of hypernym $( \mathbf { d } _ { p } )$ after proper scaling according to DIH. Thus, the context words only appear beside the hyponym candidate $( a \mathbf { d } _ { q } [ c ] - \mathbf { d } _ { p } [ c ] )$ causes higher penalty. + +First, we eliminate the slack variables by taking derivatives with respect to them: + +$$ +\begin{array} { r l r } { \displaystyle \frac { \partial \mathcal { L } } { \partial \zeta _ { c } } = 0 = 1 - \beta _ { c } - \delta _ { c } } & { } & \\ { \displaystyle \delta _ { c } = 1 - \beta _ { c } , ~ \beta _ { c } \le 1 } & { } \\ { \displaystyle \frac { \partial \mathcal { L } } { \partial \xi _ { c } } = 0 = 1 - \gamma _ { c } - \alpha _ { c } } & { } \\ { \displaystyle \gamma _ { c } = w _ { 0 } - \alpha _ { c } , ~ \alpha _ { c } \le w _ { 0 } . } & { } \end{array} +$$ + +By substituting in these values for $\gamma _ { c }$ and $\delta _ { c }$ , we get rid of the slack variables and have a new Lagrangian: + +$$ +\begin{array} { c l l } { { } } & { { { \mathcal { L } } = \underset { a } { \mathrm { m i n } } \underset { \alpha , \beta } { \mathrm { m a x } } ~ - \sum _ { c } \alpha _ { c } ( - a \mathbf { d } _ { q } [ c ] + \mathbf { d } _ { p } [ c ] ) - \sum _ { c } \beta _ { c } ( - \mathbf { d } _ { p } [ c ] + a \mathbf { d } _ { q } [ c ] ) } } \\ { { } } & { { } } \\ { { } } & { { 0 \leq \alpha _ { c } \leq w _ { 0 } , 0 \leq \beta _ { c } \leq 1 , a \geq 0 } } \end{array} +$$ + +We can introduce a new dual variable $\lambda _ { c } = \alpha _ { c } - \beta _ { c } + 1$ and rewrite this as: + +$$ +\begin{array} { c } { \mathcal { L } = \displaystyle \underset { a } { \mathrm { m i n } } \displaystyle \operatorname* { m a x } _ { \lambda } \ \sum _ { c } ( \lambda _ { c } - 1 ) ( a \mathbf { d } _ { q } [ c ] - \mathbf { d } _ { p } [ c ] ) } \\ { 0 \leq \lambda _ { c } \leq w _ { 0 } + 1 , a \geq 0 } \end{array} +$$ + +Let’s remove the constraint on $a$ and replace with a dual variable $\eta$ : + +$$ +\begin{array} { c } { \mathcal { L } = \displaystyle \underset { a } { \mathrm { m i n } } \displaystyle \operatorname* { m a x } _ { \lambda } \ \sum _ { c } ( \lambda _ { c } - 1 ) ( a \mathbf { d } _ { q } [ c ] - \mathbf { d } _ { p } [ c ] ) - \eta a } \\ { 0 \leq \lambda _ { c } \leq w _ { 0 } + 1 , \eta \geq 0 } \end{array} +$$ + +Now let’s differentiate with respect to $a$ to get rid of the primal objective and add a new constraint: + +$$ +\begin{array} { c } { { \displaystyle \frac { \partial \mathcal { L } } { \partial a } = 0 = \sum _ { c } \lambda _ { c } \mathbf { d } _ { q } [ c ] - \sum _ { c } \mathbf { d } _ { q } [ c ] - \eta } } \\ { { \displaystyle \sum _ { c } \lambda _ { c } \mathbf { d } _ { q } [ c ] = \sum _ { c } \mathbf { d } _ { q } [ c ] + \eta } } \\ { { \displaystyle \mathcal { L } = \operatorname* { m a x } _ { \lambda } \ \sum _ { c } \mathbf { d } _ { p } [ c ] - \sum _ { c } \lambda _ { c } \mathbf { d } _ { p } [ c ] } } \\ { { \displaystyle \sum _ { c } \lambda _ { c } \mathbf { d } _ { q } [ c ] = \sum _ { c } \mathbf { d } _ { q } [ c ] + \eta } } \\ { { \displaystyle 0 \leq \lambda _ { c } \leq w _ { 0 } + 1 , \eta \geq 0 } } \end{array} +$$ + +Now we have some constant terms that are just the sums of $\mathbf { d } _ { p }$ and ${ \mathbf { d } } _ { q }$ , which will be 1 if they are distributions. + +$$ +\begin{array} { l } { \displaystyle \mathcal { L } = \displaystyle \operatorname* { m a x } _ { \lambda } ~ 1 - \sum _ { c } \lambda _ { c } \mathbf { d } _ { p } [ c ] } \\ { \displaystyle \sum _ { c } \lambda _ { c } \mathbf { d } _ { q } [ c ] = 1 + \eta } \\ { \displaystyle 0 \leq \lambda _ { c } \leq w _ { 0 } + 1 , \eta \geq 0 } \end{array} +$$ + +Now we introduce a new set of variables $\mu _ { c } = \lambda _ { c } \mathbf { d } _ { q } [ c ]$ and we can rewrite the objective as: + +$$ +\begin{array} { l } { \displaystyle \mathcal { L } = \underset { \mu } { \mathrm { m a x } } ~ 1 - \sum _ { c } \mu _ { c } \frac { \mathbf { d } _ { p } [ c ] } { \mathbf { d } _ { q } [ c ] } } \\ { \displaystyle \sum _ { c } \mu _ { c } = 1 + \eta } \\ { \displaystyle 0 \leq \mu _ { c } \leq ( w _ { 0 } + 1 ) \mathbf { d } _ { q } [ c ] , \eta \geq 0 } \end{array} +$$ + +Note that for terms where $\mathbf { d } _ { q } [ c ] = 0$ we can just set $\mathbf { d } _ { q } [ c ] = \epsilon$ for some very small epsilon, and in practice, our algorithm will not encounter these because it sorts. + +So $\mu$ we can think of as some fixed budget that we have to spend up until it adds up to 1, but it has a limit of how much we can spend for each coordinate, given by $( w _ { 0 } + 1 ) { \bf d } _ { q } [ c ]$ . Since we’re trying to minimize the term involving $\mu$ , we want to allocate as much budget as possible to the smallest terms in the summand, and then 0 to the rest once we’ve spent the budget. This also shows us that our optimal value for the dual variable $\eta$ is just 0 since we want to minimize the amount of budget we have to allocate. + +To make presentation easier, lets assume we sort the vectors in order of increasing $\begin{array} { r } { \frac { \mathbf { d } _ { p } [ c ] } { \mathbf { d } _ { q } [ c ] } } \end{array}$ , so that $\frac { \mathbf { d } _ { p } [ 1 ] } { \mathbf { d } _ { q } [ 1 ] }$ is the smallest element, etc. We can now give the following algorithm to find the optimal $\mu$ . + +$$ +\begin{array} { r l } & { \operatorname { i n i t } \ S = 0 , c = 1 , \mu = 0 } \\ & { \operatorname { w h i l e } \ S \le 1 : } \\ & { \ \mu _ { c } = \operatorname* { m i n } ( 1 - S , ( w _ { 0 } + 1 ) { \bf d } _ { q } [ c ] ) } \\ & { \ { S = S + \mu _ { c } } } \\ & { \ { c = c + 1 } } \end{array} +$$ + +At the end we can just plug in this optimal $\mu$ to the objective to get the value of our scoring function. \ No newline at end of file diff --git a/parse/train/SywMS6ZfM/SywMS6ZfM_content_list.json b/parse/train/SywMS6ZfM/SywMS6ZfM_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..568b92dd32d39164f8dd628c306bd5c4ed8d4c01 --- /dev/null +++ b/parse/train/SywMS6ZfM/SywMS6ZfM_content_list.json @@ -0,0 +1,2709 @@ +[ + { + "type": "text", + "text": "DISTRIBUTIONAL INCLUSION VECTOR EMBEDDING FOR UNSUPERVISED HYPERNYMY DETECTION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Modeling hypernymy, such as poodle is-a dog, is an important generalization aid to many NLP tasks, such as entailment, relation extraction, and question answering. Supervised learning from labeled hypernym sources, such as WordNet, limit the coverage of these models, which can be addressed by learning hypernyms from unlabeled text. Existing unsupervised methods either do not scale to large vocabularies or yield unacceptably poor accuracy. This paper introduces distributional inclusion vector embedding (DIVE), a simple-to-implement unsupervised method of hypernym discovery via per-word non-negative vector embeddings which preserve the inclusion property of word contexts. In experimental evaluations more comprehensive than any previous literature of which we are aware—evaluating on 11 datasets using multiple existing as well as newly proposed scoring functions— we find that our method provides up to double the precision of previous unsupervised methods, and the highest average performance, using a much more compact word representation, and yielding many new state-of-the-art results. In addition, the meaning of each dimension in DIVE is interpretable, which leads to a novel approach on word sense disambiguation as another promising application of DIVE. ", + "bbox": [ + 233, + 267, + 764, + 489 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 516, + 336, + 532 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Numerous applications benefit from compactly representing context distributions, which assign meaning to objects under the rubric of distributional semantics. In natural language processing, distributional semantics has long been used to assign meanings to words (that is, to lexemes in the dictionary, not individual instances of word tokens). The meaning of a word in the distributional sense is often taken to be the set of textual contexts (nearby tokens) in which that word appears, represented as a large sparse bag of words (SBOW). Without any supervision, word2vec (Mikolov et al., 2013), among other approaches based on matrix factorization (Levy et al., 2015a), successfully compress the SBOW into a much lower dimensional embedding space, increasing the scalability and applicability of the embeddings while preserving (or even improving) the correlation of geometric embedding similarities with human word similarity judgments. ", + "bbox": [ + 174, + 547, + 825, + 688 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While embedding models have achieved impressive results, context distributions capture more semantic features than just word similarity. The distributional inclusion hypothesis (DIH) (Weeds & Weir, 2003; Geffet & Dagan, 2005; Cimiano et al., 2005) posits that the context set of a word tends to be a subset of the contexts of its hypernyms. For a concrete example, most adjectives that can be applied to poodle can also be applied to dog, because dog is a hypernym of poodle. For instance, both can be obedient. However, the converse is not necessarily true — a dog can be straight-haired but a poodle cannot. Therefore, dog tends to have a broader context set than poodle. Many asymmetric scoring functions comparing SBOW based on DIH have been developed for automatic hypernymy detection (Weeds & Weir, 2003; Geffet & Dagan, 2005; Santus et al., 2017). ", + "bbox": [ + 174, + 694, + 825, + 819 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hypernymy detection plays a key role in many challenging NLP tasks, such as textual entailment (Sammons et al., 2011), coreference (Ponzetto & Strube, 2006), relation extraction (Demeester et al., 2016) and question answering (Huang et al., 2008). Leveraging the variety of contexts and inclusion properties in context distributions can greatly increase the ability to discover taxonomic structure among words (Santus et al., 2017). The inability to preserve these features limits the semantic representation power and downstream applicability of some popular existing unsupervised learning approaches such as word2vec. ", + "bbox": [ + 174, + 827, + 823, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Several recently proposed methods aim to encode hypernym relations between words in dense embeddings, such as Gaussian embedding (Vilnis & McCallum, 2015; Athiwaratkun & Wilson, 2017), order embedding (Vendrov et al., 2016), H-feature detector (Roller & Erk, 2016), HyperScore (Nguyen et al., 2017), dual tensor (Glavas & Ponzetto, 2017), Poincar ˇ e embedding (Nickel ´ & Kiela, 2017), and LEAR (Vulic & Mrk ´ siˇ c, 2017). However, the methods focus on supervised ´ or semi-supervised setting (Vendrov et al., 2016; Roller & Erk, 2016; Nguyen et al., 2017; Glavasˇ & Ponzetto, 2017; Vulic & Mrk ´ siˇ c, 2017), do not learn from raw text (Nickel & Kiela, 2017) or ´ lack comprehensive experiments on the hypernym detection task (Vilnis & McCallum, 2015; Athiwaratkun & Wilson, 2017). ", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Recent studies (Levy et al., 2015b; Santus et al., 2017) have underscored the difficulty of generalizing supervised hypernymy annotations to unseen pairs — classifiers often effectively memorize prototypical hypernyms (‘general’ words) and ignore relations between words. These findings motivate us to develop more accurate and scalable unsupervised embeddings to detect hypernymy and propose several scoring functions to analyze the embeddings from different perspectives. ", + "bbox": [ + 174, + 234, + 825, + 306 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 CONTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 176, + 324, + 331, + 338 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• A novel unsupervised low-dimensional embedding method to model inclusion relations among word contexts via performing non-negative matrix factorization (NMF) on a weighted PMI matrix, which can be efficiently optimized using modified skip-grams. Several new asymmetric comparison functions to measure inclusion and generality properties and to evaluate different aspects of unsupervised embeddings. Extensive experiments on 11 datasets demonstrate the learned embeddings and comparison functions achieve state-of-the-art performances on unsupervised hypernym detection while requiring much less memory and compute than approaches based on the full SBOW. • A qualitative experiment illustrates DIVE can be used to solve word sense disambiguation, especially when efficiently modeling word senses at multiple granularities is desirable. ", + "bbox": [ + 215, + 345, + 825, + 506 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 METHOD ", + "text_level": 1, + "bbox": [ + 174, + 527, + 282, + 545 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The distributional inclusion hypothesis (DIH) suggests that the context set of a hypernym tends to contain the context set of its hyponyms. That is, when representing a word as the counts of contextual co-occurrences, the count in every dimension of hypernym $y$ tends to be larger than or equal to the corresponding count of its hyponym $x$ : ", + "bbox": [ + 174, + 560, + 825, + 617 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/cabb88c7eb4931481c649a5a0f1622b626b6682768988690e96d662e6caa2e7c.jpg", + "text": "$$\nx \\preceq y \\iff \\forall c \\in V , \\# ( x , c ) \\leq \\# ( y , c ) ,\n$$", + "text_format": "latex", + "bbox": [ + 361, + 626, + 635, + 643 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $x \\preceq y$ means $y$ is a hypernym of $x , V$ is the set of vocabulary, and $\\# ( x , c )$ indicates the number of times that word $x$ and its context word $c$ co-occur in a small window with size $| W |$ in corpus $D$ . ", + "bbox": [ + 174, + 651, + 825, + 694 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our goal is to produce lower-dimensional embeddings that preserve the inclusion property that the embedding of hypernym $y$ is larger than or equal to the embedding of its hyponym $x$ in every dimension. Formally, the desirable property can be written as ", + "bbox": [ + 174, + 700, + 823, + 742 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/f95f554bff1355e3de20e8f7c782d362fd242b8512d6b4092a602786e11ff74c.jpg", + "text": "$$\nx \\preceq y \\iff \\mathbf { x } [ i ] \\leq \\mathbf { y } [ i ] , \\forall i \\in \\{ 1 , . . . , d _ { 0 } \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 357, + 751, + 638, + 768 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $d _ { 0 }$ is number of dimensions in the embedding space. We add additional non-negativity constraints, i.e. $x [ i ] \\ge 0 , y [ i ] \\ge 0 , \\forall i$ , in order to increase the interpretability of the embeddings (the reason will be explained later in this section). ", + "bbox": [ + 176, + 776, + 821, + 819 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This is a challenging task. In reality, there are a lot of noise and systematic biases which cause the violation of DIH in Equation (1) (i.e. $\\# ( x , c ) > \\# ( y , c )$ for some neighboring word $c$ ), but the general trend can be discovered by processing several thousands of neighboring words in SBOW together. After the compression, the same trend has to be estimated in a much smaller embedding space which discards most of the information in SBOW, so it is not surprising to see most of the unsupervised hypernymy detection studies use SBOW (Santus et al., 2017) and the existing unsupervised embeddings like Gaussian embedding have degraded accuracy (Vulic et al., 2016). ´ ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 INCLUSION PRESERVING MATRIX FACTORIZATION ", + "text_level": 1, + "bbox": [ + 173, + 103, + 566, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Popular methods of unsupervised word embedding are usually based on matrix factorization (Levy et al., 2015a). The approaches first compute a co-occurrence statistic between the wth word and the cth context word as the $( w , c )$ th element of the matrix $M [ w , c ]$ . Next, the matrix $M$ is factorized such that $M [ w , c ] \\approx \\mathbf { w } ^ { T } \\mathbf { \\dot { c } }$ , where w is the low dimension embedding of $w$ th word and $\\mathbf { c }$ is the cth context embedding. ", + "bbox": [ + 173, + 128, + 825, + 200 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The statistic in $M [ w , c ]$ is usually related to pointwise mutual information: $P M I ( w , c ) ~ =$ $\\textstyle \\log ( { \\frac { P ( w , c ) } { P ( w ) \\cdot P ( c ) } } )$ , where $\\begin{array} { r } { P ( w , c ) = \\frac { \\# ( w , c ) } { | D | } } \\end{array}$ , $| D | = \\sum _ { w \\in V } \\sum _ { c \\in V } \\# ( w , c )$ is number of co-occurrence word pairs in the corpus, $\\begin{array} { r } { P ( w ) = \\frac { \\# ( w ) } { | D | } } \\end{array}$ , $\\# ( w ) = \\sum _ { c \\in V } \\# ( w , c )$ is the frequency of the word $w$ times the window size $| W |$ , and similarly for $P ( c )$ . For example, $M [ w , c ]$ could be set as positive PMI (PPMI), $\\operatorname* { m a x } ( P M I ( w , c ) , 0 )$ , or shifted PMI, $P M I ( w , c ) - \\log ( k )$ , like skip-grams with negative sampling (SGNS) (Levy et al., 2015a). Intuitively, since $\\dot { M } [ w , \\bar { c ] } \\approx \\mathbf { w } ^ { T } \\mathbf { c }$ , larger embedding values of w at every dimension seems to imply larger $\\mathbf { \\bar { w } } ^ { T } \\mathbf { c }$ , larger $M [ w , c ]$ , larger $\\bar { P } M I ( w , c )$ , and thus larger co-occurrence count $\\# ( w , c )$ . However, the derivation has two flaws: (1) c could be negative and (2) lower $\\# ( w , c )$ could still lead to larger $P M I ( w , c )$ as long as the $\\# ( w )$ is small enough. ", + "bbox": [ + 173, + 205, + 825, + 359 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To preserve DIH, we propose a novel word embedding method, distributional inclusion vector embedding $( D I V E )$ , which fixes the two flaws by performing non-negative factorization (NMF) on the matrix $M$ , where ", + "bbox": [ + 176, + 364, + 823, + 407 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c6a13a703a825d9fb05300d20de10e363c749e520903d3ef71e43ee0faad41de.jpg", + "text": "$$\nM [ w , c ] = \\log ( \\frac { P ( w , c ) } { P ( w ) \\cdot P ( c ) } \\cdot \\frac { \\# ( w ) } { k \\cdot Z } ) = \\log ( \\frac { \\# ( w , c ) | V | } { \\# ( c ) k } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 411, + 692, + 446 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $k$ is a constant which shifts PMI value like SGNS, $\\begin{array} { r } { Z = \\frac { | D | } { | V | } } \\end{array}$ is the average word frequency, and $| V |$ is the vocabulary size. ", + "bbox": [ + 176, + 453, + 821, + 488 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The design encourages the inclusion property in DIVE (i.e. Equation (2)) to be satisfied because the property implies that Equation (1) (DIH) holds if the matrix is reconstructed perfectly. The derivation is simple: Since context vector c is non-negative, if the embedding of hypernym $\\mathbf { y }$ is greater than or equal to the embedding of its hyponym $\\mathbf { X }$ in every dimension, $\\mathbf { \\bar { x } } ^ { T } \\mathbf { c } \\leq \\mathbf { y } ^ { T } \\mathbf { c }$ . Then, $\\mathbf { \\bar { \\boldsymbol { M } } } [ \\boldsymbol { x } , \\boldsymbol { c } ] \\le M [ \\boldsymbol { y } , \\boldsymbol { \\bar { c } } ]$ tends to be true because $\\mathbf { w } ^ { \\hat { T } } \\mathbf { c } \\ \\tilde { \\approx } \\ M [ w , c ]$ . This leads to $\\# ( x , c ) \\leq \\# ( y , c )$ because M [w, c] = log( #(w,c)|V |#(c)k ) and only $\\# ( w , c )$ change with $w$ . ", + "bbox": [ + 173, + 493, + 825, + 585 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 174, + 601, + 320, + 614 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Due to its appealing scalability properties during training time (Levy et al., 2015a), we optimize our embedding based on the skip-gram with negative sampling (SGNS) (Mikolov et al., 2013). The objective function of SGNS is ", + "bbox": [ + 174, + 626, + 826, + 669 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/191c04333a96df1ef1afc25052fdcb51ca201e90fc4d067006831833bd67d25e.jpg", + "text": "$$\nl _ { S G N S } = \\sum _ { w \\in V } \\sum _ { c \\in V } \\# ( w , c ) \\log \\sigma ( \\mathbf { w } ^ { T } \\mathbf { c } ) \\ + \\ \\sum _ { w \\in V } k ^ { \\prime } \\sum _ { c \\in V } \\# ( w , c ) \\operatorname { \\mathbb { E } } _ { c _ { N } \\sim P _ { D } } [ \\log \\sigma ( - \\mathbf { w } ^ { T } \\mathbf { c } _ { \\mathbf { N } } ) ] ,\n$$", + "text_format": "latex", + "bbox": [ + 200, + 674, + 779, + 710 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { w } \\in \\mathbb { R } , \\mathbf { c } \\in \\mathbb { R } , \\mathbf { c _ { N } } \\in \\mathbb { R } , k ^ { \\prime }$ is a constant hyper-parameter indicating the ratio between positive and negative samples. ", + "bbox": [ + 171, + 715, + 825, + 744 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Levy & Goldberg (2014) prove SGNS is equivalent to factorizing a shifted PMI matrix $M ^ { \\prime }$ , where $\\begin{array} { r } { M ^ { \\prime } [ w , c ] = \\log ( \\frac { P ( w , c ) } { P ( w ) \\cdot P ( c ) } \\cdot \\frac { 1 } { k ^ { \\prime } } ) } \\end{array}$ . By setting $\\begin{array} { r } { k ^ { \\prime } = \\frac { k \\cdot Z } { \\# ( w ) } } \\end{array}$ and applying non-negativity constraints to the embeddings, DIVE can be optimized using the similar objective function: ", + "bbox": [ + 173, + 750, + 825, + 799 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/31d8eb839e946c7fe78479ef9e15c1f671c19ef400ec4c056d88de541a36f447.jpg", + "text": "$$\nl _ { D I V E } = \\sum _ { w \\in V } \\sum _ { c \\in V } \\# ( w , c ) \\log \\sigma ( \\mathbf { w } ^ { T } \\mathbf { c } ) \\ + k \\sum _ { w \\in V } \\frac { Z } { \\# ( w ) } \\sum _ { c \\in V } \\# ( w , c ) \\operatorname { \\mathbb { E } } _ { c _ { N } \\sim P _ { D } } [ \\log \\sigma ( - \\mathbf { w } ^ { T } \\mathbf { c } _ { \\mathbf { N } } ) ] ,\n$$", + "text_format": "latex", + "bbox": [ + 189, + 804, + 807, + 843 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { w } \\geq 0 , \\mathbf { c } \\geq 0 , \\mathbf { c } _ { \\mathbf { N } } \\geq 0 ,$ $\\sigma$ is the logistic sigmoid function, and $k$ is a constant hyper-parameter. $P _ { D }$ is the distribution of negative samples, which we set to be the corpus word frequency distribution in this paper. Equation (5) is optimized by ADAM (Kingma & Ba, 2015), a variant of stochastic gradient descent (SGD). The non-negativity constraint is implemented by projection (i.e., clipping any embedding which crosses the zero boundary after an update). ", + "bbox": [ + 173, + 853, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/bc81173fe9f03c1c3657656ec72c8d5249f2772d104addb772d049e2d14a7de5.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 215, + 99, + 789, + 237 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/407404929240f68823a0e9789e88a0ca607b4c3598e17ce527e3cb8e622ed8d8.jpg", + "table_caption": [ + "Figure 1: The top 15 dimensions in the distributional inclusion vector embedding (DIVE) of the word core trained by the co-occurrence statistics of context words. The index of dimensions is sorted by the embedding values. The words in each row of the table are sorted by its embedding value in the dimension. " + ], + "table_footnote": [], + "table_body": "
idTop 1-5 wordsTop 101-105 words
1element,gas,atom,rock,carbonmethane,llio
2system,rcieturvelope,gefunctional,rnt,ocsing,
3also, well,svral,rlyfall, eventually, main,ise,mosty
4partincorporate,ge,iead,oing,dd
5star,orbit,snbital,bright,posion,turieractio
6several,icdesignate,ist,iss,bch,i
7science,plosophy, thory,plosophr,rethical,dvocte,oic,bic,
8version,game,release,original,ilecassette, rtual,code,project, kb
9electron,ttric,citanode,wire,ac,perform,eistor
10tank, cylinder,heel,gialuminumtic,ott
11school,itildoctorateldert
12network,r,er,dtum,prltechnologyoutgnt,crooft,
13high,low,turergyatmosphric,tod,,io
14acid, carbon, product, ue,incph,monoided
15access,nd,rquire,llow,programsize,abilitytlly
", + "bbox": [ + 245, + 248, + 750, + 436 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The optimization process provides an alternative angle to explain how DIVE preserves DIH. The gradients for the word embedding w is ", + "bbox": [ + 176, + 545, + 823, + 574 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/74ae406659ce80afddc52ca68b3f6f54f32e122017fe6031b32c7d19e29deef6.jpg", + "text": "$$\n\\frac { d l _ { D I V E } } { d \\mathbf { w } } = \\sum _ { c \\in V } \\# ( w , c ) ( 1 - \\sigma ( \\mathbf { w } ^ { T } \\mathbf { c } ) ) \\mathbf { c } - k \\sum _ { c _ { N } \\in V } \\frac { \\# ( c _ { N } ) } { | V | } \\sigma ( \\mathbf { w } ^ { T } \\mathbf { c } _ { \\mathbf { N } } ) \\mathbf { c } _ { \\mathbf { N } } .\n$$", + "text_format": "latex", + "bbox": [ + 258, + 590, + 741, + 632 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Assume hyponym $\\mathbf { X }$ and hypernym y satisfy DIH in Equation (1) and the embeddings $\\mathbf { x }$ and $\\mathbf { y }$ are the same at some point during the gradient ascent. In the case, the gradients coming from negative sampling (the second term) decrease the same amount of embedding values for both $x$ and $y$ because $k$ is a constant hyper-parameter. However, the embedding of hypernym $\\mathbf { y }$ would get higher or equal positive gradients from the first term than $\\mathbf { x }$ in every dimension because $\\# ( x , \\bar { c } ) \\leq \\bar { \\# } ( y , c )$ . This means Equation (1) tends to imply Equation (2). Combining the analysis from the matrix factorization viewpoint, DIH in Equation (1) is approximately equivalent to the inclusion property in DIVE (i.e. Equation (2)). ", + "bbox": [ + 173, + 647, + 825, + 761 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 PMI FILTERING ", + "text_level": 1, + "bbox": [ + 174, + 790, + 326, + 804 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For a frequent target word, there must be many neighboring words that incidentally appear near the target word without being semantically meaningful, especially when a large context window size is used. The unrelated context words cause noise in both the word vector and the context vector of DIVE. We address this issue by filtering out context words $c$ for each target word $w$ when the PMI of the co-occurring words is too small (i.e., $\\begin{array} { r } { \\log ( \\frac { P ( w , c ) } { P ( w ) \\cdot P ( c ) } ) < \\log ( k _ { f } ) ) } \\end{array}$ . That is, we set $\\# ( w , c ) = 0$ in the objective function. This preprocessing step is similar with computing PPMI in SBOW (Bullinaria & Levy, 2007), where low PMI co-occurrences are removed from the count-based representation. ", + "bbox": [ + 173, + 820, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.4 INTERPRETABILITY ", + "text_level": 1, + "bbox": [ + 176, + 103, + 348, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "After applying the non-negativity constraint, we observe that each dimension roughly corresponds to a topic, as previous findings suggest (Pauca et al., 2004; Murphy et al., 2012). This gives rise to a natural and intuitive interpretation of our word embeddings: the word embeddings can be seen as unnormalized probability distributions over topics. By removing the normalization of the target word frequency in the shifted PMI matrix, specific words have values in few dimensions (topics), while general words appear in more topics and correspondingly have high values in more dimensions, so the concreteness level of two words can be easily compared using the magnitude of their embeddings. In other words, general words have more diverse context distributions, so we need more dimensions to store the information in order to compress SBOW well (Nalisnick & Ravi, 2015). ", + "bbox": [ + 174, + 131, + 825, + 256 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In Figure 1, we present three mentions of the word core and its surrounding contexts. These various context words increase the embedding values in different dimensions. Each dimension of the learned embeddings roughly corresponds to a topic, and the more general or representative words for each topic tend to have the higher value in the corresponding dimension (e.g. words in the second column of the table). The embedding is able to capture the common contexts where the word core appears. For example, the context of the first mention is related to the atom topic (dimension id 1) and the electron topic (id 9), while the second and third mention occur in the computer architecture topic (id 2) and education topic (id 11), respectively. ", + "bbox": [ + 174, + 262, + 825, + 375 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 EXPERIMENT SETUP ", + "text_level": 1, + "bbox": [ + 176, + 395, + 377, + 411 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We describe four experiments in Section 4-7. The first 3 experiments compare DIVE with other unsupervised embeddings and SBOW using different hypernymy scoring functions. In these experiments, unsupervised approaches refer to the methods that only train on plaintext corpus without using any hypernymy or lexicon annotation. The last experiment presents qualitative results on word sense disambiguation. ", + "bbox": [ + 174, + 428, + 825, + 498 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 DATASETS AND TESTING SETUP ", + "text_level": 1, + "bbox": [ + 176, + 517, + 436, + 531 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The SBOW and embeddings are tested on 11 datasets. The first 4 datasets come from the recent review of Santus et al. (2017): BLESS (Baroni & Lenci, 2011), EVALution (Santus et al., 2015), Lenci/Benotto (Benotto, 2015), and Weeds (Weeds et al., 2014). The next 4 datasets are downloaded from the code repository of the H-feature detector (Roller & Erk, 2016): Medical (i.e., Levy 2014) (Levy et al., 2014), LEDS (also referred to as ENTAILMENT or Baroni 2012) (Baroni et al., 2012), TM14 (i.e., Turney 2014) (Turney & Mohammad, 2015), and Kotlerman 2010 (Kotlerman et al., 2010). In addition, the performance on the test set of HyperNet (Shwartz et al., 2016) (using the random train/test split), the test set of WordNet (Vendrov et al., 2016), and all pairs in HyperLex (Vulic et al., 2016) are also evaluated. ´ ", + "bbox": [ + 174, + 542, + 825, + 669 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The F1 and accuracy measurements are sometimes very similar even though the quality of prediction varies, so average precision $\\operatorname { A P } @$ all is adopted as the main evaluation metric. The HyperLex dataset has a continuous score on each candidate word pair, so we adopt Spearman rank coefficient $\\rho$ as suggested by the review study of Vulic et al. (2016). Any OOV (out-of-vocabulary) word en- ´ countered in the testing data is pushed to the bottom of the prediction list (effectively assuming the word pair does not have a hypernym relation). ", + "bbox": [ + 174, + 676, + 825, + 760 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 TRAINING SETUP ", + "text_level": 1, + "bbox": [ + 176, + 777, + 336, + 792 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use WaCkypedia corpus (Baroni et al., 2009), a 2009 Wikipedia dump, to compute SBOW and train the embedding. For the datasets without Part of Speech (POS) information (i.e. Medical, LEDS, TM14, Kotlerman 2010, and HyperNet), the training data of SBOW and embeddings are raw text. For other datasets, we concatenate each token with the Part of Speech (POS) of the token before training the models except the case when we need to match the training setup of another paper. ", + "bbox": [ + 174, + 804, + 825, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "All words are lower cased. Stop words and rare words (occurs less than 10 times) are removed during our preprocessing step. The number of embedding dimensions in DIVE $d _ { 0 }$ is set to be 100. Other hyper-parameters used in the experiments are listed in the supplementary materials. The hyper-parameters of DIVE were decided based on the performance of HyperNet training set. To train embeddings more efficiently, we chunk the corpus into subsets/lines of 100 tokens instead of using sentence segmentation. Preliminary experiments show that this implementation simplification does not hurt the performance. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/c5ca2e4dea6995995f62106ba94a36f226f706b84ed9a9e5fbfd3d531bc54fbb.jpg", + "table_caption": [ + "Table 1: Comparison with previous unsupervised embeddings. All values are percentages. $\\mathbf { A P } @$ all $( \\% )$ for 10 datasets and Spearman $\\rho$ $( \\% )$ for HyperLex. Word2Vec $+ \\mathrm { C }$ scores the word pairs using the cosine similarity on skip-grams (SGNS). $\\mathrm { G E + C }$ and $\\mathrm { G E + K L }$ computes cosine similarity and negative KL divergence on Gaussian embedding, respectively. " + ], + "table_footnote": [], + "table_body": "
DatasetBLESSEVALutionLenciBenottoWeedsMedicalLEDS
Random Word2Vec+C5.3 9.226.6 25.441.2 40.851.4 51.68.5 11.250.5 71.8
GE+C GE+KL10.5 7.626.7 29.643.352.014.969.7
DIVE+C.△S45.151.315.764.6
16.333.050.465.525.383.5
DatasetTM14Kotlerman 2010HyperNetWordNetHyperLex
Random52.030.824.555.20
Word2Vec+C52.1
39.520.763.016.3
GE+C53.936.021.658.216.4
GE+KL52.039.423.754.49.6
DIVE+C·△S57.236.641.960.932.8
", + "bbox": [ + 238, + 167, + 761, + 308 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 329, + 825, + 386 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the following experiments, we train both SBOW and DIVE on only the first 512,000 lines (51.2 million tokens) because we find this way of training setting provides better performances (for both SBOW and DIVE) than training on the whole WaCkypedia or training on randomly sampled 512,000 lines. We suspect this is due to the corpus being sorted by the Wikipedia page titles, which makes some categorical words such as animal and mammal occur 3-4 times more frequently in the first 51.2 million tokens than the rest. The performances of training SBOW PPMI on the whole WaCkypedia is also provided for reference in Table 4 and Table 5. ", + "bbox": [ + 173, + 392, + 825, + 489 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENT 1: COMPARISON WITH UNSUPERVISED EMBEDDINGS ", + "text_level": 1, + "bbox": [ + 173, + 508, + 763, + 526 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "If a pair of words has the hypernym relation, the words tend to be similar and the hypernym should be more general than the hyponym. As in HyperScore (Nguyen et al., 2017), we score the hypernym candidates by multiplying two factors corresponding to these properties. The ${ \\bf C } { \\cdot } { \\Delta } { \\cal S }$ (i.e. the cosine similarity multiply the difference of summation) scoring function is defined as ", + "bbox": [ + 173, + 540, + 825, + 597 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/8aa8b98ee7aed28c95b3f7b4a287be312bd377fbc849a684efd09ab072a89ff3.jpg", + "text": "$$\nC \\cdot \\Delta S ( \\mathbf { w } _ { q } \\mathbf { w } _ { p } ) = \\frac { \\mathbf { w } _ { q } ^ { T } \\mathbf { w } _ { p } } { | | \\mathbf { w } _ { q } | | _ { 2 } \\cdot | | \\mathbf { w } _ { p } | | _ { 2 } } \\cdot ( | | \\mathbf { w } _ { p } | | _ { 1 } - | | \\mathbf { w } _ { q } | | _ { 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 599, + 692, + 636 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where ${ \\bf w } _ { p }$ is the embedding of hypernym and ${ \\bf w } _ { q }$ is the embedding of hyponym. ", + "bbox": [ + 174, + 638, + 694, + 654 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As far as we know, Gaussian embedding (GE) is the only unsupervised embedding method which can capture the asymmetric relations between a hypernym and its hyponyms. Using the same training and testing setup, we use the code implemented by Athiwaratkun $\\&$ Wilson $( 2 0 1 7 ) ^ { 1 }$ to train Gaussian embedding on the first 51.2 million tokens and test the embeddings on 11 datasets. Its hyper-parameters are determined using the same way as DIVE (i.e. maximizing the AP on HyperNet training set). We compare DIVE with $\\mathrm { G E } ^ { 2 }$ in Table 1, and the performances of random scores and only measuring word similarity using skip-grams are also presented for reference. As we can see, DIVE is usually significantly better than other baselines. ", + "bbox": [ + 173, + 659, + 825, + 771 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENT 2: HYPERNYMY SCORING FUNCTIONS ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 790, + 732, + 808 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Experiment 1, we show that there exists a scoring function $\\left( \\mathbf { C } { \\cdot } \\Delta \\mathbf { S } \\right)$ which detects hypernymy accurately using the embedding space of DIVE. Nevertheless, different scoring functions measure different signals in SBOW or embeddings. Since there are so many scoring functions and datasets available in the domain, we first introduce and test the performances of various scoring functions so as to select the representative ones for a more comprehensive evaluation of DIVE on the hypernymy detection tasks. We denote the embedding/context vector of the hypernym candidate and the hyponym candidate as ${ \\bf w } _ { p }$ and ${ \\bf w } _ { q }$ , respectively. The SBOW model which represents a word by the frequency of its neighboring words is denoted as SBOW Freq, while the SBOW which uses PPMI of its neighboring words as the features (Bullinaria & Levy, 2007) is denoted as SBOW PPMI. ", + "bbox": [ + 174, + 820, + 820, + 851 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 UNSUPERVISED SCORING FUNCTIONS ", + "text_level": 1, + "bbox": [ + 176, + 219, + 478, + 233 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1.1 SIMILARITY ", + "text_level": 1, + "bbox": [ + 174, + 246, + 313, + 260 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "A hypernym tends to be similar to its hyponym, so we measure the cosine similarity between word vectors of the SBOW features (Levy et al., 2015b) or DIVE. We refer to the symmetric scoring function as Cosine or C for short in the following tables. We also train the original skip-grams with 100 dimensions and measure the cosine similarity between the resulting word2vec embeddings. This scoring function is referred to as Word2vec or W. ", + "bbox": [ + 174, + 270, + 825, + 339 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1.2 GENERALITY ", + "text_level": 1, + "bbox": [ + 174, + 356, + 320, + 371 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The distributional informativeness hypothesis (Santus et al., 2014) observes that in many corpora, semantically ‘general’ words tend to appear more frequently and in more varied contexts. Thus, Santus et al. (2014) advocate using entropy of context distributions to capture the diversity of context. We adopt the two variations of the approach proposed by Santus et al. (2017): SLQS Row and SLQS Sub functions. We also refer to SLQS Row as $\\Delta \\mathrm { E }$ because it measures the entropy difference of context distributions. For SLQS Sub, the number of top context words is fixed as 100. ", + "bbox": [ + 174, + 381, + 825, + 465 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Although effective at measuring diversity, the entropy totally ignores the frequency signal from the corpus. To leverage the information, we measure the generality of a word by its L1 norm $( | \\mathbf { w } _ { p } | _ { 1 } )$ and L2 norm $( | | \\mathbf { w } _ { p } | | _ { 2 } )$ . Recall that Equation (2) indicates that the embedding of the hypernym $\\mathbf { y }$ should have a larger value at every dimension than the embedding of the hyponym x. When the inclusion property holds, $\\begin{array} { r } { | { \\bf y } | _ { 1 } = \\sum _ { i } { \\bf y } [ i ] \\geq \\sum _ { i } { \\bf x } [ i ] = | { \\bf x } | _ { 1 } } \\end{array}$ and similarly $| | \\mathbf { y } | | _ { 2 } \\geq | | \\mathbf { x } | | _ { 2 }$ . Thus, we propose two scoring functions, difference of vector summation $( | \\mathbf { w } _ { p } | _ { 1 } - | \\mathbf { w } _ { q } | _ { 1 } )$ and the difference of vector 2-norm $( | | \\mathbf { \\bar { w } } _ { p } | | _ { 2 } - | | \\mathbf { w } _ { q } | | _ { 2 } )$ . Notice that when applying the difference of vector summations (denoted as $\\Delta S$ ) to SBOW Freq, it is equivalent to computing the word frequency difference between the hypernym candidate pair. ", + "bbox": [ + 173, + 472, + 825, + 598 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1.3 COMBINATION ", + "text_level": 1, + "bbox": [ + 174, + 613, + 331, + 627 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The combination of 2 similarity functions (Cosine and Word2vec) and the 3 generality functions (difference of entropy, summation, and 2-norm of vectors) leads to six different scoring functions as shown in Table 2, and ${ \\bf C } { \\cdot } { \\Delta } { \\cal S }$ is the same scoring function we used in Experiment 1. It should be noted that if we use skip-grams with negative sampling (word2vec) as the similarity measurement (i.e., $W \\cdot \\Delta \\ \\{ \\mathrm { E } , \\mathrm { S } , \\mathrm { Q } \\} ,$ ), the scores are determined by two embedding/feature spaces together (word2vec and DIVE/SBOW). ", + "bbox": [ + 174, + 638, + 825, + 722 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1.4 INCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 738, + 307, + 752 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Several scoring functions are proposed to measure inclusion properties of SBOW based on DIH. Weeds Precision (Weeds & Weir, 2003) and CDE (Clarke, 2009) both measure the magnitude of the intersection between feature vectors $( | \\mathbf { w } _ { p } \\cap \\mathbf { w } _ { q } | )$ . For example, ${ \\bf w } _ { p } \\cap { \\bf w } _ { q }$ is defined by the elementwise minimum in CDE. Then, both scoring functions divide the intersection by the magnitude of the potential hyponym vector $( | \\mathbf { w } _ { q } | )$ . invCL (Lenci & Benotto, 2012) (A variant of CDE) is also tested. ", + "bbox": [ + 174, + 763, + 823, + 833 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We choose these 3 functions because they have been shown to detect hypernymy well in a recent study (Santus et al., 2017). However, it is hard to confirm that their good performances come from the inclusion property between context distributions — it is also possible that the context vectors of more general words have higher chance to overlap with all other words due to their high frequency. For instance, considering a one dimension feature which stores only the frequency of words, the naive embedding could still have reasonable performance on the CDE function, but the embedding in fact only memorizes the general words without modeling relations between words (Levy et al., 2015b) and loses lots of inclusion signals in the word co-occurrence statistics. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/15d7d8b07af0e2942e2146ff1af1c0e1c4cfd3865c3d67c5cfccd0c1032fd242.jpg", + "table_caption": [ + "Table 2: Micro average AP@all $( \\% )$ of 10 datasets using different scoring functions. The feature space is SBOW using word frequency. " + ], + "table_footnote": [], + "table_body": "
Word2vec (W) 24.8Cosine (C) 26.7SLQS Sub 27.4SLQS Row (△E) 27.6Summation (△S) 31.5Two norm (△Q) 31.2
W.△EC.△EW.△SC.△SW.△QC.△Q
28.829.531.631.231.431.1
WeedsCDEinvCLAsymmetric L1(AL1)
19.031.130.728.2
", + "bbox": [ + 209, + 140, + 789, + 212 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 238, + 821, + 265 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In order to measure the inclusion property without the interference of the word frequency signal from the SBOW or embeddings, we propose a new measurement called asymmetric $L _ { 1 }$ distance. We first get context distributions ${ \\bf d } _ { p }$ and ${ \\bf d } _ { q }$ by normalizing $\\mathbf { w } _ { p }$ and ${ \\bf w } _ { q }$ , respectively. Ideally, the context distribution of the hypernym ${ \\bf d } _ { p }$ will include ${ \\bf d } _ { q }$ . This suggests the hypernym distribution ${ \\bf d } _ { p }$ is larger than context distribution of the hyponym with a proper scaling factor $a \\mathbf { d } _ { q }$ (i.e., $\\operatorname* { m a x } ( a \\mathbf { d } _ { q } \\bar { - }$ ${ \\bf d } _ { p } , 0 )$ should be small). Furthermore, both distributions should be similar, so $a \\mathbf { d } _ { q }$ should not be too different from ${ \\bf d } _ { p }$ (i.e., $\\operatorname* { m a x } ( \\mathbf { d } _ { p } - a \\mathbf { d } _ { q } , 0 )$ should also be small). Therefore, we define asymmetric L1 distance as ", + "bbox": [ + 173, + 272, + 825, + 383 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/d9abf289974b7fa865cbebc67e0a06b58bb226409ab693c879e9697fdbff6caa.jpg", + "text": "$$\nA L _ { 1 } = \\operatorname* { m i n } _ { a } \\sum _ { c } w _ { 0 } \\cdot \\operatorname* { m a x } ( a \\mathbf { d } _ { q } [ c ] - \\mathbf { d } _ { p } [ c ] , 0 ) + \\operatorname* { m a x } ( \\mathbf { d } _ { p } [ c ] - a \\mathbf { d } _ { q } [ c ] , 0 ) ,\n$$", + "text_format": "latex", + "bbox": [ + 258, + 388, + 725, + 421 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $w _ { 0 }$ is a constant which emphasizes the inclusion penalty. If $w _ { 0 } ~ = ~ 1$ and $a \\ = \\ 1$ , $A L _ { 1 }$ is equivalent to L1 distance. The lower $A L _ { 1 }$ distance implies a higher chance of observing the hypernym relation. We tried $w _ { 0 } = 5$ and $w _ { 0 } = 2 0$ . $w _ { 0 } = 2 0$ produces a worse micro-average AP@all on SBOW Freq, SBOW PPMI and DIVE, so we fix $w _ { 0 }$ to be 5 in all experiments. An efficient way to solve the optimization in $A L _ { 1 }$ is presented in the supplementary materials. ", + "bbox": [ + 174, + 428, + 825, + 500 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 RESULTS AND DISCUSSIONS ", + "text_level": 1, + "bbox": [ + 176, + 516, + 411, + 530 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We show the micro average $\\mathbf { A P } @$ all on 10 datasets using different hypernymy scoring functions in Table 2. We can see the combination functions such as ${ \\bf C } { \\cdot } { \\Delta } { \\cal S }$ and ${ \\bf W } { \\cdot } { \\Delta } { \\cal S }$ perform the best overall. Among the unnormalized inclusion based scoring functions, CDE works the best. $A L _ { 1 }$ performs well compared with other functions which remove the frequency signal such as Word2vec, Cosine, and SLQS Row. The summation is the most robust generality measurement. In the table, the scoring functions are applied to SBOW Freq, but the performances of hypernymy scoring functions on the other feature spaces (e.g. DIVE) have a similar trend. ", + "bbox": [ + 174, + 541, + 825, + 640 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 EXPERIMENT 3: COMPARISON WITH SBOW ", + "text_level": 1, + "bbox": [ + 174, + 660, + 573, + 676 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.1 COMPARISON WITH PREVIOUSLY REPORTED RESULTS ", + "text_level": 1, + "bbox": [ + 173, + 693, + 584, + 707 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Table 3, DIVE with two of the best scoring functions $C { \\cdot } \\Delta s$ and ${ \\bf W } { \\cdot } { \\Delta S }$ ) is compared with the previous unsupervised state-of-the-art approaches based on SBOW on different datasets. ", + "bbox": [ + 173, + 718, + 823, + 747 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "There are several reasons which might cause the large performance gaps in some datasets. In addition to the effectiveness of DIVE, some improvements come from our proposed scoring functions. The fact that every paper uses a different training corpus also affects the performances. Furthermore, Santus et al. (2017) select the scoring functions and feature space for the first 4 datasets based on $\\mathbf { A P } @ 1 0 0$ , which we believe is too sensitive to the hyper-parameter settings of different methods. To isolate the impact of each factor, we perform a more comprehensive comparison next. ", + "bbox": [ + 174, + 753, + 825, + 838 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2 PERFORMANCE ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 856, + 395, + 869 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this experiment, we examine whether DIVE successfully preserves the signals for hypernymy detection tasks, which are measured by the same scoring functions designed for SBOW. Summation difference $( \\Delta \\boldsymbol { S } )$ and CDE perform the best among generality and inclusion functions in Table 2, ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3: Comparison with previous methods based on sparse bag of word (SBOW). All values are percentages. The results of invCL (Lenci & Benotto, 2012), APSyn (Santus et al., 2016), and CDE (Clarke, 2009) are selected because they have the best $\\mathbf { A P } @ 1 0 0$ in the first 4 datasets (Santus et al., 2017). Cosine similarity (Levy et al., 2015b), balAPinc (Kotlerman et al., 2010) in 3 datasets (Turney & Mohammad, 2015), SLQS (Santus et al., 2014) in HyperNet dataset (Shwartz et al., 2016), and Freq ratio (FR) (Vulic et al., 2016) are compared. ´ ", + "bbox": [ + 174, + 99, + 825, + 185 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/7915c44f621e73e33c0c328570c981738f85302a557c24457cb93e28d7217476.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
DatasetBLESS EVALutionLenciBenottoWeedsMedical
MetricAP@allF1
BaselinesinvCLAPSynCDECosine
5.135.338.244.123.1
DIVE+C·△S16.333.050.465.525.3
DIVE +W·△S18.632.351.568.625.7
DatasetLEDSTM14Kotlerman 2010HyperNetHyperLex
MetricAP@allF1Spearman p
BaselinesbalAPincSLQSFreq ratio
73563722.827.9
DIVE+C·△S83.557.236.641.932.8
DIVE+W·△S86.457.337.438.633.3
", + "bbox": [ + 259, + 195, + 736, + 337 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "respectively. $A L _ { 1 }$ could be used to examine the inclusion properties after removing the frequency signal. Therefore, we will present the results using these 3 scoring functions, along with ${ \\bf W } { \\cdot } { \\Delta } S$ and ${ \\bf C } { \\cdot } { \\Delta } { \\cal S }$ . ", + "bbox": [ + 176, + 362, + 823, + 404 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6.2.1 BASELINES", + "text_level": 1, + "bbox": [ + 174, + 419, + 308, + 434 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In addition to classic representations such as SBOW Freq and SBOW PPMI, we compare distributional inclusion vector embedding (DIVE) with additional 4 baselines in Table 4. ", + "bbox": [ + 174, + 443, + 823, + 472 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• SBOW PPMI with additional frequency weighting (PPMI w/ FW). Specifically, $\\mathbf { w } [ c ] =$ $\\begin{array} { r } { \\operatorname* { m a x } ( \\log ( \\frac { P ( w , c ) } { P ( w ) * P ( c ) * \\frac { Z } { \\# ( w ) } } ) , 0 ) } \\end{array}$ . This forms the matrix reconstructed by DIVE when $k = 1$ . \n• DIVE without the PMI filter (DIVE w/o PMI) \n• NMF on shifted PMI: Non-negative matrix factorization (NMF) on the shifted PMI without frequency weighting for DIVE (DIVE w/o FW). This is the same as applying the nonnegative constraint on the skip-gram model. \nK-means (Freq NMF): The method first uses Mini-batch $\\mathbf { k }$ -means (Sculley, 2010) to cluster words in skip-gram embedding space into 100 topics, and hashes each frequency count in SBOW into the corresponding topic. If running $\\mathbf { k }$ -means on skip-grams is viewed as an approximation of clustering the SBOW context vectors, the method can be viewed as a kind of NMF (Ding et al., 2005). Let the $N \\times N$ context matrix be denoted as $M _ { c }$ , where the $( i , j )$ th element stores the count of word $j$ appearing beside word $i$ . K-means hashing creates a $N \\times 1 0 0$ matrix $G$ with orthonormal rows $\\tilde { G } ^ { T } G = I )$ , where the $( i , k )$ th element is 0 if the word $i$ does not belong to cluster $k$ . The orthonormal $G$ is also an approximated solution of a type of NMF $( M _ { c } \\overset { \\mathbf { \\backsimeq } } { \\approx } F G ^ { T }$ ) (Ding et al., 2005). Hashing context vectors into topic vectors can be written as $M _ { c } G \\approx F G ^ { T } G = F$ . ", + "bbox": [ + 214, + 481, + 825, + 746 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In the experiment, we also tried to apply a constant $\\log ( k )$ shifting to SBOW PPMI (i.e. $\\operatorname* { m a x } ( P M I - \\log ( k ) , 0 ) )$ ). We found that the performance degrades as $k$ increases. Similarly, applying PMI filter to SBOW PPMI (set context feature to be 0 if the value is lower than $\\log ( k _ { f } ) )$ usually makes the performances worse, especially when $k _ { f }$ is large. Applying PMI filter to SBOW Freq only makes its performances closer to (but still much worse than) SBOW PPMI, so we omit this baseline as well. ", + "bbox": [ + 174, + 758, + 825, + 842 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6.2.2 RESULTS AND DISCUSSIONS ", + "text_level": 1, + "bbox": [ + 176, + 857, + 424, + 871 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In Table 4, we first confirm the finding of the previous review study of Santus et al. (2017): there is no single hypernymy scoring function which always outperforms others. One of the main reasons is that different datasets collect negative samples differently. This is also why we evaluate our method on many datasets to make sure our conclusions hold in general. For example, if negative samples come from random word pairs (e.g. WordNet dataset), a symmetric similarity measure is already a pretty good scoring function. On the other hand, negative samples come from related or similar words in HyperNet, EVALution, Lenci/Benotto, and Weeds, so only computing generality difference leads to the best (or close to the best) performance. The negative samples in many datasets are composed of both random samples and similar words (such as BLESS), so the combination of similarity and generality difference yields the most stable results. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/14b55133305f79ded6c7515ef1ded67d8693098e884c8b28338652bf52f0b2c6.jpg", + "table_caption": [ + "Table 4: $\\mathbf { A P } @$ all $( \\% )$ of 10 datasets. The box at lower right corner compares the micro average AP across all 10 datasets. Numbers in different rows come from different feature or embedding spaces. Numbers in different columns come from different datasets and unsupervised scoring functions. We also present the micro average AP across the first 4 datasets (BLESS, EVALution, Lenci/Benotto and Weeds). All wiki means SBOW using PPMI features trained on the whole WaCkypedia. FW refers to frequency weighting on the shifted PMI matrix. " + ], + "table_footnote": [], + "table_body": "
AP@all(%)BLESSEVALutionLenci/Benotto
CDEAL1△SW△SC△SCDE AL1△SW△SC△SCDEAL1△SW△SC△S
SBOWFreq6.37.3 5.6 5.611.0 17.25.9 15.335.3 30.432.6 27.736.233.036.351.847.651.051.851.1
PPMI13.65.127.534.131.934.347.239.750.851.152.0
PPMI w/FW6.25.05.5 12.45.836.036.332.936.452.043.150.951.950.7
All wiki12.15.26.9 12.513.4 28.527.130.329.931.047.139.948.548.751.1
DIVEFull9.37.6 6.0 5.618.616.330.027.534.933.046.743.251.351.550.4
w/o PMI7.86.916.77.132.832.235.732.3 32.535.447.644.950.951.649.7
w/o FW9.06.26.27.324.325.022.923.523.938.838.138.238.238.4
Kmean (Freq NMF)6.57.3 5.610.95.833.727.236.233.036.249.642.551.051.851.2
7.3 WeedsMicro Average (4 datasets)Medical
AP@all(%)CDE AL1△SCDEAL1CDEAL1△SW·△SC△S
SBOWFreq69.558.0 68.8W△S 68.2C△S 68.423.121.8△S 22.9W△S 25.0C△S 23.019.419.214.118.415.3
PPMI61.050.370.369.269.324.7 17.922.328.127.823.48.713.220.124.4
PPMI w/FW67.652.269.468.767.723.2 18.222.925.822.922.810.613.718.617.0
All wiki61.348.670.070.423.425.822.38.912.217.621.1
Full59.255.068.517.7 19.821.7 22.824.6 28.927.611.79.313.721.419.2
DIVE w/o PMI60.456.469.7 69.368.6 68.665.5 64.822.1 22.221.023.110.78.413.319.816.2
w/o FW49.247.3 45.145.144.918.917.322.7 17.228.0 16.817.510.99.87.47.67.7
Kmean (Freq NMF)69.451.1 68.868.268.922.519.322.924.923.012.610.914.018.114.6
LEDSTM14
AP@all(%)AL1 △SCDE AL1Kotlerman 2010
SBOWFreqCDE70.4 70.7W·△S 83.3C△S 73.355.653.2△S 54.9W·△S 55.7C.△S 55.0CDE 35.9AL1 40.5△S 34.5W.△S 37.0C.△S 35.4
SBOWPPMI82.7 84.450.2 72.286.584.556.252.354.457.639.130.933.037.036.3
All wiki83.149.7 67.982.981.454.750.552.657.0 55.154.938.531.232.235.435.3
DIVE83.372.786.483.555.352.655.257.357.235.331.633.637.436.6 MicroAverage(10datasets)
AP@all (%)74.7HyperNetWordNet
", + "bbox": [ + 176, + 195, + 823, + 494 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 554, + 825, + 652 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "DIVE performs similar or better on all the scoring functions compared with SBOW consistently across all datasets in Table 4, while using many fewer dimensions (see Table 6). Its results on combination scoring functions outperform SBOW Freq. Meanwhile, its results on $A L _ { 1 }$ outperform SBOW PPMI. The fact that combination scoring functions (i.e., W·∆S or $\\mathbf { C } { \\cdot } \\Delta \\mathbf { S }$ ) usually outperform generality functions suggests that only memorizing general words is not sufficient. The best average performance on 4 and 10 datasets are both produced by ${ \\bf W } { \\cdot } { \\Delta } { \\cal S }$ on DIVE. ", + "bbox": [ + 174, + 659, + 823, + 742 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "SBOW PPMI improves the combination functions from SBOW Freq but sacrifices AP on the inclusion functions. It generally hurts performance to change the frequency sampling of PPMI (PPMI w/ FW) or compute SBOW PPMI on the whole WaCkypedia (all wiki) instead of the first 51.2 million tokens. The similar trend can also be seen in Table 5. Note that $A L _ { 1 }$ completely fails in HyperLex dataset using SBOW PPMI, which suggests that PPMI might not necessarily preserve the distributional inclusion property, even though it can have good performance on combination functions. ", + "bbox": [ + 173, + 750, + 825, + 833 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Removing the PMI filter from DIVE slightly drops the overall precision while removing frequency weights on shifted PMI (w/o FW) leads to poor performances. K-means (Freq NMF) produces similar AP compared with SBOW Freq, but has worse $A L _ { 1 }$ scores. Its best AP scores on different datasets are also significantly worse than the best AP of DIVE. This means that only making word2vec (skip-grams with negative sampling) non-negative or naively accumulating topic distribution in contexts cannot lead to satisfactory embeddings. ", + "bbox": [ + 174, + 839, + 823, + 924 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/d2f6b59c7662383cdf7042bd246a8b24a9efa8ee9d769dc0770e202597b6bb21.jpg", + "table_caption": [ + "Table 5: Spearman $\\rho \\left( \\% \\right)$ in HyperLex. " + ], + "table_footnote": [], + "table_body": "
Spearman p (%)HyperLex
CDEAL1△SW·△SC.△S
SBOWFreq31.719.627.629.627.3
SBOW PPMI28.1-2.331.834.334.5
All wiki25.3-2.228.030.531.0
DIVE28.918.731.233.332.8
", + "bbox": [ + 186, + 127, + 486, + 193 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/6a5730fd56936ae1141a339855b3b312e943a22a9fa3dd62af50d2678c928f81.jpg", + "table_caption": [ + "Table 6: The average number of non-zero dimensions across all testing words in 10 datasets. " + ], + "table_footnote": [], + "table_body": "
SBOW FreqSBOWPPMIDIVE
5799380820
", + "bbox": [ + 524, + 151, + 797, + 183 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Table 7: Spectral clustering on the non-zero DIVE dimensions of the query word. When the number of clusters is set to be 2, we present the top 4 dimensions in each cluster, which have the highest values on the query word embedding. Otherwise, the top 2 dimensions are presented. CID refers to cluster ID. The top 5 words with the highest values of each dimension are presented. ", + "bbox": [ + 173, + 217, + 826, + 273 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/9fe846fb27dc83e25e6faa2374a99f9415236b010e8a162d9fcf7c1e7fc922e0.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
QueryCIDTop 5 words in the top dimensions
rock1element,gas,atom,rock,carbon find, specie,species,animal,birdsea,lake,river,area,water point,side,line, front,circle
2band,song,album,music,rock early,work,century,late,beginwrite,john,guitar,band,author include,several, show, television,film
bank1county,area,city, town,west building,build,house, palace, siteseveral,main, province, include,consist sea,lake,river,area,water
2money, tax,price, pay, income united, states,country, world, europecompany,corporation,system,agency, service state,palestinian,israel,right, palestine
apple1food, fruit,vegetable,meat, potato war,german,ii,germany,worldgoddess,zeus,god,hero,sauron write, john,guitar,band,author
2version, game,release,original, file system,architecture,develop,base,languagecar,company,sell, manufacturer,model include,several, show, television,film
star1film, role, production, play, stage wear,blue,color, instrument,redcharacter,series,game,novel, fantasy write, john,guitar,band,author
2element,gas,atom,rock,carbon give, term,vector,mass,momentumstar,orbit,sun,orbital,planet light, image,lens,telescope,camera
tank1tank,cylinder,wheel, engine,steel acid,carbon,product, use, zincindustry, export, industrial, economy,company network,user,server, datum, protocol
2army,force,infantry,military,battle however,attempt,result, despite, failaircraft,navy,missile,ship,flight war, german, ii, germany, world
race1win,world,cup,play,championshiptwo,one,three,four,another
2 3railway, line, train,road, railcar,company, sell, manufacturer,model
population,language, ethnic, native, peoplefemale,age,woman,male,household
run1system,architecture, develop,base,languageaccess, need, require,allow, program
2railway,line,train,road,railalso,well, several, early, see
3game,team,season,win,leaguegame,player,run,deal,baseball
tablet1bc, source, greek, ancient, datebook,publish,write,work,edition
2use,system,design,term,methodversion, game,release,original, file
3system,blood,vessel,artery,intestinepatient, symptom, treatment,disorder, may
", + "bbox": [ + 176, + 285, + 854, + 688 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "7 EXPERIMENT 4: WORD SENSE DISAMBIGUATION ", + "text_level": 1, + "bbox": [ + 174, + 723, + 616, + 741 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In addition to hypernymy detection, Athiwaratkun & Wilson (2017) show that the mixture of Gaussian distributions can also be used to discover multiple senses of each word. In our qualitative experiment, we show that DIVE can achieve the similar goal without fixing the number of senses before training the embedding. ", + "bbox": [ + 174, + 762, + 825, + 819 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Recall that each dimension roughly corresponds to one topic. Given a query word, the higher embedding value on a dimension implies higher likelihood to observe the word in the context of the topic. The embedding of a polysemy would have high values on different groups of topics/dimensions. This allows us to discover the senses by clustering the topics/dimensions of the polysemy. We use the embedding values as the feature each dimension, compute the pairwise similarity between dimensions, and apply spectral clustering (Stella & Shi, 2003) to group topics as shown in the Table 7. See more implementation details in the supplementary materials. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In the word sense disambiguation tasks, it is usually challenging to determine how many senses/clusters each word should have. Many existing approaches fix the number of senses before training the embedding (Tian et al., 2014; Athiwaratkun & Wilson, 2017). Neelakantan et al. (2014) make the number of clusters approximately proportional to the diversity of the context, but the assumption does not always hold. Furthermore, the training process cannot capture different granularity of senses. For instance, race in the car context could share the same sense with the race in the game topic because they all mean contest, but the race in the car context actually refers to the specific contest of speed. Therefore, they can also be viewed as separate senses (like the results in Table 7). This means the correct number of clusters is not unique, and the methods, which fixes the cluster numbers, need to re-train the embedding many times to capture such granularity. ", + "bbox": [ + 174, + 103, + 825, + 242 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In our approach, clustering dimensions is done after the training process of DIVE is completed, so it is fairly efficient to change the cluster numbers and hierarchical clustering is also an option. Similar to our method, Pelevina et al. (2016) also discover word senses by graph-based clustering. The main difference is that they cluster the top $n$ words which are most related to the query word instead of topics. However, choosing the hyper-parameter $n$ is difficult. Large $n$ would make graph clustering algorithm inefficient, while small $n$ would make less frequent senses difficult to discover. ", + "bbox": [ + 174, + 250, + 825, + 333 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "8 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 363, + 339, + 378 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Most previous unsupervised approaches focus on designing better hypernymy scoring functions for sparse bag of word (SBOW) features. They are well summarized in the recent study (Santus et al., 2017). Santus et al. (2017) also evaluate the influence of different contexts, such as changing the window size of contexts or incorporating dependency parsing information, but neglect scalability issues inherent to SBOW methods. ", + "bbox": [ + 174, + 401, + 825, + 469 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A notable exception is the Gaussian embedding model (Vilnis & McCallum, 2015). The context distribution of each word is encoded as a multivariate Gaussian distribution, where the embeddings of hypernyms tend to have higher variance and overlap with the embedding of their hyponyms. However, since a Gaussian distribution is normalized, it is difficult to retain frequency information during the embedding process, and experiments on HyperLex (Vulic et al., 2016) demonstrate that a simple ´ baseline only relying on word frequency can achieve good results. Follow-up work models contexts by a mixture of Gaussians (Athiwaratkun & Wilson, 2017) relaxing the unimodality assumption but achieves little improvement on hypernym detection tasks. ", + "bbox": [ + 174, + 477, + 825, + 589 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Kiela et al. (2015) show that images retrieved by a search engine can be a useful source of information to determine the generality of lexicons, but the resources might not be available for some corpora such as scientific literature. ", + "bbox": [ + 176, + 595, + 825, + 637 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Order embedding (Vendrov et al., 2016) is a supervised approach to encode many annotated hypernym pairs (e.g. all of the whole WordNet (Miller, 1995)) into a compact embedding space, where the embedding of a hypernym should be smaller than the embedding of its hyponym in every dimension. Our method learns embedding from raw text, where a hypernym embedding should be larger than the embedding of its hyponym in every dimension. Thus, DIVE can be viewed as an unsupervised and reversed form of order embedding. ", + "bbox": [ + 174, + 645, + 825, + 728 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Other semi-supervised hypernym detection methods aim to generalize from sets of annotated word pairs using raw text corpora. The goal of HyperScore (Nguyen et al., 2017) is similar to our model: the embedding of a hypernym should be similar to its hyponym but with higher magnitude. However, their training process relies heavily on annotated hypernym pairs, and the performance drops significantly when reducing the amount of supervision. In addition to context distributions, previous work also leverages training data to discover useful text pattern indicating is-a relation (Shwartz et al., 2016; Roller & Erk, 2016), but it remains challenging to increase recall of hypernym detection because commonsense facts like cat is-a animal might not appear in the corpus. ", + "bbox": [ + 174, + 736, + 823, + 847 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Non-negative matrix factorization (NMF) has a long history in NLP, for example in the construction of topic models (Pauca et al., 2004). Non-negative sparse embedding (NNSE) (Murphy et al., 2012) and Faruqui et al. (2015) indicate that non-negativity can make embeddings more interpretable and improve word similarity evaluations. The sparse NMF is also shown to be effective in cross-lingual lexical entailment tasks but does not necessarily improve monolingual hypernymy detection (Vyas & ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Carpuat, 2016). In our study, a new type of NMF is proposed, and the comprehensive experimental analysis demonstrates its state-of-the-art performances on unsupervised hypernymy detection. ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "9 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 152, + 328, + 169 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Compressing unsupervised SBOW models into a compact representation is challenging while preserving the inclusion, generality, and similarity signals which are important for hypernym detection. Our experiments suggest that simple baselines such as accumulating K-mean clusters and non-negative skip-grams do not lead to satisfactory performances in this task. ", + "bbox": [ + 174, + 184, + 823, + 239 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "To achieve this goal, we proposed an interpretable and scalable embedding method called distributional inclusion vector embedding (DIVE) by performing non-negative matrix factorization (NMF) on a weighted PMI matrix. We demonstrate that scoring functions which measure inclusion and generality properties in SBOW can also be applied to DIVE to detect hypernymy, and DIVE performs the best on average, slightly better than SBOW while using many fewer dimensions. ", + "bbox": [ + 174, + 247, + 823, + 318 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Our experiments also indicate that unsupervised scoring functions, which combine similarity and generality measurements, work the best in general, but no one scoring function dominates across all datasets. A combination of unsupervised DIVE with the proposed scoring functions produces new state-of-the-art performances on many datasets under the unsupervised setup. ", + "bbox": [ + 174, + 324, + 825, + 380 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Finally, a qualitative experiment shows that clusters of the topics discovered by DIVE often correspond to the word senses, which allow us to do word sense disambiguation without the need to know the number of senses before training the embeddings. ", + "bbox": [ + 174, + 387, + 823, + 429 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 450, + 285, + 465 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Ben Athiwaratkun and Andrew Gordon Wilson. Multimodal word distributions. In ACL, 2017. ", + "bbox": [ + 174, + 473, + 795, + 488 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Marco Baroni and Alessandro Lenci. How we BLESSed distributional semantic evaluation. In Workshop on GEometrical Models of Natural Language Semantics (GEMS), 2011. 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The number in parentheses beside each approach indicates the number of annotated hypernymy word pairs used to train the model. Semi-supervised embeddings include HyperScore (Nguyen et al., 2017) and H-feature (Roller & Erk, 2016). When we compare F1 with the results from other papers, we use 20 fold cross validation to determine prediction thresholds, as done by Roller & Erk (2016). Note that HyperScore ignores POS in the testing data, so we follow the setup when comparing with it. ", + "bbox": [ + 173, + 101, + 823, + 198 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/02c8df4710cfdd00fb6d9d3313f37aa6811ec9e74326e5bdebe89588e9a05e86.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
DatasetHyperLexEVALution LenciBenotto WeedsMedical
MetricSpearman pAP@allF1
Baselines(#Training Hypernymy)HyperScore(1337)H-feature (897)
303944.858.526
DIVE+C·△S (0)34.533.852.970.025.3
", + "bbox": [ + 232, + 210, + 766, + 268 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Table 9: We show the top 30 words with the highest embedding magnitude after dot product with the query embedding $\\mathbf { q }$ (i.e. showing w such that $| \\mathbf { w } ^ { T } \\mathbf { q } | _ { 1 }$ is one of the top 30 highest values). The rows with the empty query word sort words based on $| \\mathbf { w } | _ { 1 }$ . ", + "bbox": [ + 176, + 281, + 821, + 324 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/5f90f73a345229abcf82508d348fd99b46c47afa534c1e86245775e84f4028b3.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
QueryTop30 generalwords
use large american design studyname state life work leadsystem group may produce typeinclude power small control peoplebase death find great highcity form body write create
speciesspecie human gene fish evidencespecies bird tree disease breedanimal genus name live proteinfind family genetic food wildplant organism study cell similarmay suggest occur mammal fossil
systemsystem standard allow process networkuse type function code filedesign computer datum via developmentprovide application device base serviceoperate develop control program transportmodel method information software law
", + "bbox": [ + 230, + 335, + 766, + 540 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A SUPPLEMENTARY MATERIALS ", + "text_level": 1, + "bbox": [ + 176, + 565, + 457, + 580 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.1 COMPARISON WITH SEMI-SUPERVISED EMBEDDINGS ", + "text_level": 1, + "bbox": [ + 174, + 597, + 586, + 612 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In addition to the unsupervised approach, we also compare DIVE with semi-supervised approaches. When there are sufficient training data, there is no doubt that the semi-supervised embedding approaches such as HyperNet (Shwartz et al., 2016), H-feature detector (Roller & Erk, 2016), and HyperScore (Nguyen et al., 2017) can achieve better performance than all unsupervised methods. However, in many domains such as scientific literature, there are often not many annotated hypernymy pairs (e.g. Medical dataset (Levy et al., 2014)). ", + "bbox": [ + 173, + 623, + 825, + 708 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Since we are comparing an unsupervised method with semi-supervised methods, it is hard to fairly control the experimental setups and tune the hyper-parameters. In Table 8, we only show several performances which are copied from the original paper when training data are limited3. As we can see, the performance from DIVE is roughly comparable to the previous semi-supervised approaches trained on small amount of hypernym pairs. This demonstrates the robustness of our approach and the difficulty of generalizing hypernymy annotations with semi-supervised approaches. ", + "bbox": [ + 174, + 715, + 825, + 799 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.2 GENERALITY ESTIMATION AND HYPERNYM DIRECTIONALITY DETECTION", + "text_level": 1, + "bbox": [ + 176, + 818, + 728, + 832 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In Table 9, we show the most general words in DIVE under different queries as constraints. We also present the accuracy of judging which word is a hypernym (more general) given word pairs with ", + "bbox": [ + 176, + 843, + 823, + 872 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/0fada8b150dc51c71723913d4342e1a8600c79bf4dcc42b1fefca5a13b0646c7.jpg", + "table_caption": [ + "Table 10: Accuracy $( \\% )$ of hypernym directionality prediction across 10 datasets. " + ], + "table_footnote": [], + "table_body": "
Micro Average (1O datasets)
SBOW Freq+ SLQS Sub64.4SBOW Freq +△S66.8SBOW PPMI+△S66.8DIVE+△S67.0
", + "bbox": [ + 238, + 127, + 759, + 167 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Table 11: Dataset sizes. N denotes the number of word pairs in the dataset, and OOV shows how many word pairs are not processed by all the methods in Table 4 and Table 5. ", + "bbox": [ + 171, + 180, + 823, + 209 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/03a52ec0743b4a8493db03d0f0dda1cc99d4576f6afaf52573bf9a6d58916ccc.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
BLESSEVALutionLenci/BenottoWeedsAvg (4 datasets)
N 265540OV 1507N 13675OOV 2475N 5010OOV 1464N 29280OV 643N 48167OOv 6089
MedicalKotlerman 2010
NLEDSTM14HyperNet
12602OOV 3711NOOV 28NOOV 178NOOV 89NOOV 9424
27702188294017670
WordNetAvg(10 datasets)HyperLex
N OOVNN
8000 35969433726160OV 59
", + "bbox": [ + 222, + 220, + 776, + 338 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "hypernym relations in Table 10. The direction is classified correctly if the generality score is greater than 0 (hypernym is indeed predicted as the more general word). For instance, summation difference $( \\Delta \\mathsf { S } )$ classifies correctly if $\\vert \\mathbf { \\bar { w } } _ { p } \\vert _ { 1 } - \\vert \\mathbf { w } _ { q } \\vert _ { 1 } > 0$ $( | \\bar { \\mathbf { w } _ { p } } | _ { 1 } > | \\mathbf { w } _ { q } | _ { 1 } )$ . ", + "bbox": [ + 174, + 361, + 825, + 405 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "From the table, we can see that the simple summation difference performs better than SQLS Sub, and DIVE predicts directionality as well as SBOW. Notice that whenever we encounter OOV, the directionality is predicted randomly. If OOV is excluded, the accuracy of predicting directionality using unsupervised methods can reach around 0.7-0.75. ", + "bbox": [ + 174, + 410, + 825, + 467 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.3 EXPERIMENTAL DETAILS FOR HYPERNYMY DETECTION ", + "text_level": 1, + "bbox": [ + 174, + 484, + 602, + 497 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In HyperNet and WordNet, some hypernym relations are determined between phrases instead of words. Phrase embeddings are composed by averaging word embeddings or SBOW features. For WordNet, we assume the Part of Speech (POS) tags of the words are the same as the phrase. All part-of-speech (POS) tags in the experiments come from NLTK. ", + "bbox": [ + 174, + 508, + 825, + 565 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The window size $| W |$ of SBOW, DIVE, and GE are set as 20 (left 10 words and right 10 words). For DIVE, the number of epochs is 15, the learning rate is 0.001, the batch size is 128, the threshold in PMI filter $k _ { f }$ is set to be 30, and the ratio between negative and positive samples ${ \\mathrm { ( k ) } }$ is 1.5. The hyper-parameters of DIVE were decided based on the performance of HyperNet training set. The window size of skip-grams (word2vec) is 10. The number of negative samples $( k ^ { \\prime } )$ in skip-gram is set as 5. When composing skip gram into phrase embedding, average embedding is used. ", + "bbox": [ + 174, + 571, + 825, + 656 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For Gaussian embedding (GE), the number of mixture is 1, the number of dimension is 100, the learning rate is 0.01, the lowest variance is 0.1, the highest variance is 100, the highest Gaussian mean is 10, and other hyper-parameters are the default value in https://github.com/ benathi/word2gm. The hyper-parameters of GE were also decided based on the performance of HyperNet training set. When determining the score between two phrases, we use the average score of every pair of tokens in two phrases. ", + "bbox": [ + 174, + 662, + 825, + 746 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The number of testing pairs $_ \\mathrm { N }$ and the number of OOV word pairs is presented in Table 11. ", + "bbox": [ + 174, + 752, + 767, + 767 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.4 CLUSTERING DIMENSIONS FOR WORD SENSE DISAMBIGUATION ", + "text_level": 1, + "bbox": [ + 176, + 785, + 656, + 799 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We use all the default hyper-parameters of the spectral clustering library in Scikit-learn 0.18.2 (Pedregosa et al., 2011) except the number of clusters is set manually. A simple way to prepare the feature of each dimension $f ( c _ { i } )$ is to use the embedding values in that dimension $\\mathbf { w } [ c _ { i } ]$ of all the words in our vocabulary $w \\in V$ . That is, ", + "bbox": [ + 174, + 810, + 823, + 866 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/ecaff7fbfcf007a4303c943b3a12c48786dd863633c8754c2b34ad191d8a4925.jpg", + "text": "$$\nf ( c _ { i } ) = [ \\mathbf { w } [ c _ { i } ] ] _ { w \\in V } .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 872, + 568, + 891 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "However, clustering on the global features might group topics together based on the co-occurrence of words which are unrelated to the query words and we want to make the similarity dependent on the query word. For example, a country topic should be clustered together with a city topic if the query word is place, but it makes more sense to group the country topic with the money topic together if the query word is bank like we did in the word sense disambiguation experiment (Table 7). This means we want to focus on the geographical meaning of country when the query is related to geography, while focus on the economic meaning of country when the query is about economics. ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "To create query dependent similarity measurement, we only consider the embedding of words which are related to the query word when preparing the features of dimensions. Specifically, given a query word $q$ , the feature vector of the $i$ th dimension $f ( c _ { i } , q )$ is defined as: ", + "bbox": [ + 174, + 180, + 825, + 223 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/94f389557e3e78685fe0274c4322b5349bc3bb5f014ebe413c1ca8493a5b72bb.jpg", + "text": "$$\nf ( c _ { i } , q ) = \\bigoplus _ { j = 1 } ^ { d _ { 0 } } \\left[ \\mathbf { w } [ c _ { i } ] \\cdot \\mathbf { w } _ { q } [ c _ { j } ] \\right] _ { w \\in C _ { j } ( n ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 238, + 625, + 270 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where ${ \\bf w } _ { q } [ c _ { j } ]$ is the value of $j$ th dimension of query word embedding, $C _ { j } ( n )$ is the set of embeddings of top $n$ words in the $j$ th dimension, and the operator $\\oplus$ means concatenation. This means instead of considering all the words in the vocabulary, we only take the top $n$ words of every dimension $j$ (n is fixed as 100 in the experiment), weight the feature based on how likely to observe query word in dimension $j$ $( \\mathbf { w } _ { q } [ c _ { j } ] )$ , and concatenate all features together. That is, when measuring the similarity of dimensions, we only consider the aspects related to query word (e.g. mostly considering words related to facility and money when the query word is bank). ", + "bbox": [ + 173, + 284, + 825, + 383 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "After the features of all dimensions are collected, we normalize the feature of each dimension to have the norm 1, compute the pairwise similarity and run the spectral clustering to get the clustering results. ", + "bbox": [ + 176, + 388, + 825, + 431 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.5 EFFICIENT WAY TO COMPUTE ASYMMETRIC L1 $( A L _ { 1 } )$ ", + "text_level": 1, + "bbox": [ + 173, + 457, + 591, + 472 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Recall that Equation (8) defines $A L _ { 1 }$ as follows: ", + "bbox": [ + 173, + 486, + 491, + 501 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/c995e65ad636b1b72607d803255c3c7835cf59c95f6af220ccab4402fb8600dd.jpg", + "text": "$$\nA L _ { 1 } = \\mathcal { L } = \\operatorname* { m i n } _ { a } \\ \\sum _ { c } w _ { 0 } \\operatorname* { m a x } ( a \\mathbf { d } _ { q } [ c ] - \\mathbf { d } _ { p } [ c ] , 0 ) + \\operatorname* { m a x } ( \\mathbf { d } _ { p } [ c ] - a \\mathbf { d } _ { q } [ c ] , 0 ) ,\n$$", + "text_format": "latex", + "bbox": [ + 241, + 513, + 753, + 547 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\mathbf { d } _ { p } [ c ]$ is one of dimension in the feature vector of hypernym $\\mathbf { d } _ { p }$ , $a \\mathbf { d } _ { q }$ is the feature vector of hyponym after proper scaling. In Figure 2, an simple example is visualized to illustrate the intuition behind the distance function. ", + "bbox": [ + 173, + 561, + 823, + 603 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "By adding slack variables $\\zeta$ and $\\xi$ , the problem could be converted into a linear programming problem: ", + "bbox": [ + 173, + 611, + 823, + 640 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/47862d2ea91532e0b4c9155d87d84a66e6cbc039012de6262f9e44fec86a5e92.jpg", + "text": "$$\n\\begin{array} { l } { \\mathcal { L } = \\displaystyle \\operatorname* { m i n } _ { a , \\zeta , \\xi } ~ w _ { 0 } \\sum _ { c } \\zeta _ { c } + \\sum _ { c } \\xi _ { c } } \\\\ { ~ \\zeta _ { c } \\geq a \\mathbf { d } _ { q } [ c ] - \\mathbf { d } _ { p } [ c ] , ~ \\zeta _ { c } \\geq 0 } \\\\ { \\displaystyle ~ \\xi _ { c } \\geq \\mathbf { d } _ { p } [ c ] - a \\mathbf { d } _ { q } [ c ] , ~ \\xi _ { c } \\geq 0 } \\\\ { ~ a \\geq 0 , ~ } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 648, + 630, + 738 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "so it can be simply solved by a general linear programming library. ", + "bbox": [ + 173, + 751, + 612, + 766 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Nevertheless, the structure in the problem actually allows us to solve this optimization by a simple sorting. In this section, we are going to derive the efficient optimization algorithm. ", + "bbox": [ + 173, + 772, + 823, + 801 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "By introducing Lagrangian multiplier for the constraints, we can rewrite the problem as ", + "bbox": [ + 174, + 806, + 746, + 823 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/cee1d93bc1e8a082f4c0e3ee581dacdb9fc6b236c722cdfe9d5e776619b7df69.jpg", + "text": "$$\n\\begin{array} { l l } { { \\mathcal { L } = \\displaystyle \\operatorname* { m i n } _ { a , \\zeta , \\xi } \\operatorname* { m a x } _ { \\alpha , \\beta , \\gamma , \\delta } w _ { 0 } \\sum _ { c } \\zeta _ { c } + \\sum _ { c } \\xi _ { c } - \\sum _ { c } \\alpha _ { c } ( \\zeta _ { c } - a { \\bf d } _ { q } [ c ] + { \\bf d } _ { p } [ c ] ) } } \\\\ { { \\mathrm { ~ } } } \\\\ { { \\mathrm { ~ } - \\sum _ { c } \\beta _ { c } ( \\xi _ { c } - { \\bf d } _ { p } [ c ] + a { \\bf d } _ { q } [ c ] ) - \\sum _ { c } \\gamma _ { c } \\zeta _ { c } - \\sum _ { c } \\delta _ { c } \\xi _ { c } } } \\\\ { { \\mathrm { ~ } } } \\\\ { { \\zeta _ { c } \\geq 0 , \\xi _ { c } \\geq 0 , \\alpha _ { c } \\geq 0 , \\beta _ { c } \\geq 0 , \\gamma _ { c } \\geq 0 , \\delta _ { c } \\geq 0 , a \\geq 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 834, + 756, + 924 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/8c0cf76ff70a7b6ae235a8767dbac4aa77b900b1aa3ec191dfbdfa9da0d93c69.jpg", + "image_caption": [ + "Figure 2: An example of $A L _ { 1 }$ distance. If the word pair indeed has the hypernym relation, the context distribution of hyponym $( \\mathbf { d } _ { q } )$ tends to be included in the context distribution of hypernym $( \\mathbf { d } _ { p } )$ after proper scaling according to DIH. Thus, the context words only appear beside the hyponym candidate $( a \\mathbf { d } _ { q } [ c ] - \\mathbf { d } _ { p } [ c ] )$ causes higher penalty. " + ], + "image_footnote": [], + "bbox": [ + 316, + 108, + 687, + 321 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "First, we eliminate the slack variables by taking derivatives with respect to them: ", + "bbox": [ + 173, + 431, + 702, + 448 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/300855ebfb4e5c18b7c929eb083eb8d58a9ce454fa8eb6d882d4c6ce975c91e7.jpg", + "text": "$$\n\\begin{array} { r l r } { \\displaystyle \\frac { \\partial \\mathcal { L } } { \\partial \\zeta _ { c } } = 0 = 1 - \\beta _ { c } - \\delta _ { c } } & { } & \\\\ { \\displaystyle \\delta _ { c } = 1 - \\beta _ { c } , ~ \\beta _ { c } \\le 1 } & { } \\\\ { \\displaystyle \\frac { \\partial \\mathcal { L } } { \\partial \\xi _ { c } } = 0 = 1 - \\gamma _ { c } - \\alpha _ { c } } & { } \\\\ { \\displaystyle \\gamma _ { c } = w _ { 0 } - \\alpha _ { c } , ~ \\alpha _ { c } \\le w _ { 0 } . } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 388, + 463, + 611, + 569 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "By substituting in these values for $\\gamma _ { c }$ and $\\delta _ { c }$ , we get rid of the slack variables and have a new Lagrangian: ", + "bbox": [ + 173, + 608, + 823, + 638 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/515628505ea29477a1adcdcd85f267bc9d8bfa4c1de13ea02cef8e3cae9d9d64.jpg", + "text": "$$\n\\begin{array} { c l l } { { } } & { { { \\mathcal { L } } = \\underset { a } { \\mathrm { m i n } } \\underset { \\alpha , \\beta } { \\mathrm { m a x } } ~ - \\sum _ { c } \\alpha _ { c } ( - a \\mathbf { d } _ { q } [ c ] + \\mathbf { d } _ { p } [ c ] ) - \\sum _ { c } \\beta _ { c } ( - \\mathbf { d } _ { p } [ c ] + a \\mathbf { d } _ { q } [ c ] ) } } \\\\ { { } } & { { } } \\\\ { { } } & { { 0 \\leq \\alpha _ { c } \\leq w _ { 0 } , 0 \\leq \\beta _ { c } \\leq 1 , a \\geq 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 654, + 736, + 708 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We can introduce a new dual variable $\\lambda _ { c } = \\alpha _ { c } - \\beta _ { c } + 1$ and rewrite this as: ", + "bbox": [ + 173, + 729, + 671, + 747 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/cda601c4a87a38e22a502f1aa4f38ad75637e906f18e2a7efe9bab4deea1e612.jpg", + "text": "$$\n\\begin{array} { c } { \\mathcal { L } = \\displaystyle \\underset { a } { \\mathrm { m i n } } \\displaystyle \\operatorname* { m a x } _ { \\lambda } \\ \\sum _ { c } ( \\lambda _ { c } - 1 ) ( a \\mathbf { d } _ { q } [ c ] - \\mathbf { d } _ { p } [ c ] ) } \\\\ { 0 \\leq \\lambda _ { c } \\leq w _ { 0 } + 1 , a \\geq 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 761, + 650, + 815 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Let’s remove the constraint on $a$ and replace with a dual variable $\\eta$ : ", + "bbox": [ + 171, + 837, + 616, + 853 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/0e49eefeade4b5dbadf2f0408e6ebe516109c2fa2b554634a2cf57e4191a717d.jpg", + "text": "$$\n\\begin{array} { c } { \\mathcal { L } = \\displaystyle \\underset { a } { \\mathrm { m i n } } \\displaystyle \\operatorname* { m a x } _ { \\lambda } \\ \\sum _ { c } ( \\lambda _ { c } - 1 ) ( a \\mathbf { d } _ { q } [ c ] - \\mathbf { d } _ { p } [ c ] ) - \\eta a } \\\\ { 0 \\leq \\lambda _ { c } \\leq w _ { 0 } + 1 , \\eta \\geq 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 330, + 869, + 666, + 922 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Now let’s differentiate with respect to $a$ to get rid of the primal objective and add a new constraint: ", + "bbox": [ + 171, + 102, + 818, + 119 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/759cb34574383e8860d8bfd29395df07a25379fee15461f1db0196b82657f890.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\frac { \\partial \\mathcal { L } } { \\partial a } = 0 = \\sum _ { c } \\lambda _ { c } \\mathbf { d } _ { q } [ c ] - \\sum _ { c } \\mathbf { d } _ { q } [ c ] - \\eta } } \\\\ { { \\displaystyle \\sum _ { c } \\lambda _ { c } \\mathbf { d } _ { q } [ c ] = \\sum _ { c } \\mathbf { d } _ { q } [ c ] + \\eta } } \\\\ { { \\displaystyle \\mathcal { L } = \\operatorname* { m a x } _ { \\lambda } \\ \\sum _ { c } \\mathbf { d } _ { p } [ c ] - \\sum _ { c } \\lambda _ { c } \\mathbf { d } _ { p } [ c ] } } \\\\ { { \\displaystyle \\sum _ { c } \\lambda _ { c } \\mathbf { d } _ { q } [ c ] = \\sum _ { c } \\mathbf { d } _ { q } [ c ] + \\eta } } \\\\ { { \\displaystyle 0 \\leq \\lambda _ { c } \\leq w _ { 0 } + 1 , \\eta \\geq 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 125, + 642, + 284 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Now we have some constant terms that are just the sums of $\\mathbf { d } _ { p }$ and ${ \\mathbf { d } } _ { q }$ , which will be 1 if they are distributions. ", + "bbox": [ + 173, + 295, + 823, + 324 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/69ca4a33a34f8b2dbc575873ae5447066e3ac0489b70b1f1d8e17fa2050f21a7.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { L } = \\displaystyle \\operatorname* { m a x } _ { \\lambda } ~ 1 - \\sum _ { c } \\lambda _ { c } \\mathbf { d } _ { p } [ c ] } \\\\ { \\displaystyle \\sum _ { c } \\lambda _ { c } \\mathbf { d } _ { q } [ c ] = 1 + \\eta } \\\\ { \\displaystyle 0 \\leq \\lambda _ { c } \\leq w _ { 0 } + 1 , \\eta \\geq 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 383, + 327, + 614, + 415 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Now we introduce a new set of variables $\\mu _ { c } = \\lambda _ { c } \\mathbf { d } _ { q } [ c ]$ and we can rewrite the objective as: ", + "bbox": [ + 171, + 426, + 769, + 444 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/6db2e54f8a5abf69172e7896ef95554c3965e0ebf9a2ee9f037d8474f2bdf34e.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { L } = \\underset { \\mu } { \\mathrm { m a x } } ~ 1 - \\sum _ { c } \\mu _ { c } \\frac { \\mathbf { d } _ { p } [ c ] } { \\mathbf { d } _ { q } [ c ] } } \\\\ { \\displaystyle \\sum _ { c } \\mu _ { c } = 1 + \\eta } \\\\ { \\displaystyle 0 \\leq \\mu _ { c } \\leq ( w _ { 0 } + 1 ) \\mathbf { d } _ { q } [ c ] , \\eta \\geq 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 361, + 449, + 637, + 541 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Note that for terms where $\\mathbf { d } _ { q } [ c ] = 0$ we can just set $\\mathbf { d } _ { q } [ c ] = \\epsilon$ for some very small epsilon, and in practice, our algorithm will not encounter these because it sorts. ", + "bbox": [ + 171, + 553, + 823, + 582 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "So $\\mu$ we can think of as some fixed budget that we have to spend up until it adds up to 1, but it has a limit of how much we can spend for each coordinate, given by $( w _ { 0 } + 1 ) { \\bf d } _ { q } [ c ]$ . Since we’re trying to minimize the term involving $\\mu$ , we want to allocate as much budget as possible to the smallest terms in the summand, and then 0 to the rest once we’ve spent the budget. This also shows us that our optimal value for the dual variable $\\eta$ is just 0 since we want to minimize the amount of budget we have to allocate. ", + "bbox": [ + 173, + 588, + 825, + 672 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To make presentation easier, lets assume we sort the vectors in order of increasing $\\begin{array} { r } { \\frac { \\mathbf { d } _ { p } [ c ] } { \\mathbf { d } _ { q } [ c ] } } \\end{array}$ , so that $\\frac { \\mathbf { d } _ { p } [ 1 ] } { \\mathbf { d } _ { q } [ 1 ] }$ is the smallest element, etc. We can now give the following algorithm to find the optimal $\\mu$ . ", + "bbox": [ + 174, + 676, + 823, + 718 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/c9a07a1ac6a565d983629fc2d5888bcd57de4705664ae218e3755863c81d5e0d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\operatorname { i n i t } \\ S = 0 , c = 1 , \\mu = 0 } \\\\ & { \\operatorname { w h i l e } \\ S \\le 1 : } \\\\ & { \\ \\mu _ { c } = \\operatorname* { m i n } ( 1 - S , ( w _ { 0 } + 1 ) { \\bf d } _ { q } [ c ] ) } \\\\ & { \\ { S = S + \\mu _ { c } } } \\\\ & { \\ { c = c + 1 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 380, + 742, + 617, + 832 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "At the end we can just plug in this optimal $\\mu$ to the objective to get the value of our scoring function. ", + "bbox": [ + 169, + 844, + 820, + 861 + ], + "page_idx": 19 + } +] \ No newline at end of file diff --git a/parse/train/SywMS6ZfM/SywMS6ZfM_middle.json b/parse/train/SywMS6ZfM/SywMS6ZfM_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..4cba2871bea7253de886b359a72729d07dd7ce3e --- /dev/null +++ b/parse/train/SywMS6ZfM/SywMS6ZfM_middle.json @@ -0,0 +1,53715 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 97 + ], + "score": 1.0, + "content": "DISTRIBUTIONAL INCLUSION VECTOR EMBEDDING", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 451, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 451, + 118 + ], + "score": 1.0, + "content": "FOR UNSUPERVISED HYPERNYMY DETECTION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 212, + 468, + 388 + ], + "lines": [ + { + "bbox": [ + 141, + 211, + 469, + 224 + ], + "spans": [ + { + "bbox": [ + 141, + 211, + 469, + 224 + ], + "score": 1.0, + "content": "Modeling hypernymy, such as poodle is-a dog, is an important generalization aid", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 222, + 470, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 470, + 236 + ], + "score": 1.0, + "content": "to many NLP tasks, such as entailment, relation extraction, and question answer-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 470, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 470, + 247 + ], + "score": 1.0, + "content": "ing. 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9electron,ttric,citanode,wire,ac,perform,eistor
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11school,itildoctorateldert
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13high,low,turergyatmosphric,tod,,io
14acid, carbon, product, ue,incph,monoided
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This preprocessing step is similar with computing PPMI in SBOW (Bullinaria", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 720, + 497, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 497, + 734 + ], + "score": 1.0, + "content": "& Levy, 2007), where low PMI co-occurrences are removed from the count-based representation.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 103, + 650, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 213, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 215, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 215, + 94 + ], + "score": 1.0, + "content": "2.4 INTERPRETABILITY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 104, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 504, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 504, + 116 + ], + "score": 1.0, + "content": "After applying the non-negativity constraint, we observe that each dimension roughly corresponds", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 126 + ], + "score": 1.0, + "content": "to a topic, as previous findings suggest (Pauca et al., 2004; Murphy et al., 2012). This gives rise to a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "score": 1.0, + "content": "natural and intuitive interpretation of our word embeddings: the word embeddings can be seen as un-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "normalized probability distributions over topics. 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In other words, general words have more diverse context distributions, so we need more", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 191, + 491, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 491, + 203 + ], + "score": 1.0, + "content": "dimensions to store the information in order to compress SBOW well (Nalisnick & Ravi, 2015).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "In Figure 1, we present three mentions of the word core and its surrounding contexts. These various", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "context words increase the embedding values in different dimensions. Each dimension of the learned", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "embeddings roughly corresponds to a topic, and the more general or representative words for each", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "topic tend to have the higher value in the corresponding dimension (e.g. words in the second column", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "of the table). The embedding is able to capture the common contexts where the word core appears.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "For example, the context of the first mention is related to the atom topic (dimension id 1) and the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "electron topic (id 9), while the second and third mention occur in the computer architecture topic (id", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 281, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 281, + 298 + ], + "score": 1.0, + "content": "2) and education topic (id 11), respectively.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 313, + 231, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 232, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 232, + 328 + ], + "score": 1.0, + "content": "3 EXPERIMENT SETUP", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "We describe four experiments in Section 4-7. The first 3 experiments compare DIVE with other", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 350, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 504, + 363 + ], + "score": 1.0, + "content": "unsupervised embeddings and SBOW using different hypernymy scoring functions. In these ex-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "periments, unsupervised approaches refer to the methods that only train on plaintext corpus without", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 372, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 386 + ], + "score": 1.0, + "content": "using any hypernymy or lexicon annotation. The last experiment presents qualitative results on word", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 383, + 196, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 196, + 397 + ], + "score": 1.0, + "content": "sense disambiguation.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 108, + 410, + 267, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 269, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 269, + 423 + ], + "score": 1.0, + "content": "3.1 DATASETS AND TESTING SETUP", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "The SBOW and embeddings are tested on 11 datasets. The first 4 datasets come from the recent", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "review of Santus et al. (2017): BLESS (Baroni & Lenci, 2011), EVALution (Santus et al., 2015),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "Lenci/Benotto (Benotto, 2015), and Weeds (Weeds et al., 2014). The next 4 datasets are down-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 462, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 505, + 477 + ], + "score": 1.0, + "content": "loaded from the code repository of the H-feature detector (Roller & Erk, 2016): Medical (i.e., Levy", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "2014) (Levy et al., 2014), LEDS (also referred to as ENTAILMENT or Baroni 2012) (Baroni et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "2012), TM14 (i.e., Turney 2014) (Turney & Mohammad, 2015), and Kotlerman 2010 (Kotlerman", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 495, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 506, + 511 + ], + "score": 1.0, + "content": "et al., 2010). In addition, the performance on the test set of HyperNet (Shwartz et al., 2016) (using", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 506, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 521 + ], + "score": 1.0, + "content": "the random train/test split), the test set of WordNet (Vendrov et al., 2016), and all pairs in Hyper-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 519, + 277, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 277, + 531 + ], + "score": 1.0, + "content": "Lex (Vulic et al., 2016) are also evaluated. ´", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "The F1 and accuracy measurements are sometimes very similar even though the quality of predic-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 239, + 559 + ], + "score": 1.0, + "content": "tion varies, so average precision", + "type": "text" + }, + { + "bbox": [ + 239, + 547, + 261, + 557 + ], + "score": 0.32, + "content": "\\operatorname { A P } @", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "all is adopted as the main evaluation metric. The HyperLex", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "dataset has a continuous score on each candidate word pair, so we adopt Spearman rank coefficient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 113, + 580 + ], + "score": 0.74, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "as suggested by the review study of Vulic et al. (2016). Any OOV (out-of-vocabulary) word en- ´", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "countered in the testing data is pushed to the bottom of the prediction list (effectively assuming the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 591, + 292, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 292, + 603 + ], + "score": 1.0, + "content": "word pair does not have a hypernym relation).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 108, + 616, + 206, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 207, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 207, + 630 + ], + "score": 1.0, + "content": "3.2 TRAINING SETUP", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "We use WaCkypedia corpus (Baroni et al., 2009), a 2009 Wikipedia dump, to compute SBOW and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "train the embedding. 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The number of embedding dimensions in DIVE", + "type": "text" + }, + { + "bbox": [ + 426, + 710, + 438, + 721 + ], + "score": 0.86, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "is set to be 100.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "Other hyper-parameters used in the experiments are listed in the supplementary materials. The", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "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" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 213, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 215, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 215, + 94 + ], + "score": 1.0, + "content": "2.4 INTERPRETABILITY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 104, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 504, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 504, + 116 + ], + "score": 1.0, + "content": "After applying the non-negativity constraint, we observe that each dimension roughly corresponds", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 126 + ], + "score": 1.0, + "content": "to a topic, as previous findings suggest (Pauca et al., 2004; Murphy et al., 2012). This gives rise to a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "score": 1.0, + "content": "natural and intuitive interpretation of our word embeddings: the word embeddings can be seen as un-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "normalized probability distributions over topics. 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In other words, general words have more diverse context distributions, so we need more", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 191, + 491, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 491, + 203 + ], + "score": 1.0, + "content": "dimensions to store the information in order to compress SBOW well (Nalisnick & Ravi, 2015).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 104, + 506, + 203 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "In Figure 1, we present three mentions of the word core and its surrounding contexts. These various", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "context words increase the embedding values in different dimensions. Each dimension of the learned", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "embeddings roughly corresponds to a topic, and the more general or representative words for each", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "topic tend to have the higher value in the corresponding dimension (e.g. words in the second column", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "of the table). The embedding is able to capture the common contexts where the word core appears.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "For example, the context of the first mention is related to the atom topic (dimension id 1) and the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "electron topic (id 9), while the second and third mention occur in the computer architecture topic (id", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 281, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 281, + 298 + ], + "score": 1.0, + "content": "2) and education topic (id 11), respectively.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 208, + 505, + 298 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 313, + 231, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 232, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 232, + 328 + ], + "score": 1.0, + "content": "3 EXPERIMENT SETUP", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "We describe four experiments in Section 4-7. The first 3 experiments compare DIVE with other", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 350, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 504, + 363 + ], + "score": 1.0, + "content": "unsupervised embeddings and SBOW using different hypernymy scoring functions. In these ex-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "periments, unsupervised approaches refer to the methods that only train on plaintext corpus without", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 372, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 386 + ], + "score": 1.0, + "content": "using any hypernymy or lexicon annotation. The last experiment presents qualitative results on word", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 383, + 196, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 196, + 397 + ], + "score": 1.0, + "content": "sense disambiguation.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 339, + 506, + 397 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 410, + 267, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 269, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 269, + 423 + ], + "score": 1.0, + "content": "3.1 DATASETS AND TESTING SETUP", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "The SBOW and embeddings are tested on 11 datasets. The first 4 datasets come from the recent", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "review of Santus et al. (2017): BLESS (Baroni & Lenci, 2011), EVALution (Santus et al., 2015),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "Lenci/Benotto (Benotto, 2015), and Weeds (Weeds et al., 2014). The next 4 datasets are down-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 462, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 505, + 477 + ], + "score": 1.0, + "content": "loaded from the code repository of the H-feature detector (Roller & Erk, 2016): Medical (i.e., Levy", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "2014) (Levy et al., 2014), LEDS (also referred to as ENTAILMENT or Baroni 2012) (Baroni et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "2012), TM14 (i.e., Turney 2014) (Turney & Mohammad, 2015), and Kotlerman 2010 (Kotlerman", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 495, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 506, + 511 + ], + "score": 1.0, + "content": "et al., 2010). In addition, the performance on the test set of HyperNet (Shwartz et al., 2016) (using", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 506, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 521 + ], + "score": 1.0, + "content": "the random train/test split), the test set of WordNet (Vendrov et al., 2016), and all pairs in Hyper-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 519, + 277, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 277, + 531 + ], + "score": 1.0, + "content": "Lex (Vulic et al., 2016) are also evaluated. ´", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 430, + 506, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "The F1 and accuracy measurements are sometimes very similar even though the quality of predic-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 239, + 559 + ], + "score": 1.0, + "content": "tion varies, so average precision", + "type": "text" + }, + { + "bbox": [ + 239, + 547, + 261, + 557 + ], + "score": 0.32, + "content": "\\operatorname { A P } @", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "all is adopted as the main evaluation metric. The HyperLex", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "dataset has a continuous score on each candidate word pair, so we adopt Spearman rank coefficient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 113, + 580 + ], + "score": 0.74, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "as suggested by the review study of Vulic et al. (2016). Any OOV (out-of-vocabulary) word en- ´", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "countered in the testing data is pushed to the bottom of the prediction list (effectively assuming the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 591, + 292, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 292, + 603 + ], + "score": 1.0, + "content": "word pair does not have a hypernym relation).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 535, + 505, + 603 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 616, + 206, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 207, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 207, + 630 + ], + "score": 1.0, + "content": "3.2 TRAINING SETUP", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "We use WaCkypedia corpus (Baroni et al., 2009), a 2009 Wikipedia dump, to compute SBOW and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "train the embedding. For the datasets without Part of Speech (POS) information (i.e. Medical,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "LEDS, TM14, Kotlerman 2010, and HyperNet), the training data of SBOW and embeddings are raw", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "text. For other datasets, we concatenate each token with the Part of Speech (POS) of the token before", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 681, + 487, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 487, + 695 + ], + "score": 1.0, + "content": "training the models except the case when we need to match the training setup of another paper.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 637, + 505, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "All words are lower cased. Stop words and rare words (occurs less than 10 times) are removed", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 426, + 722 + ], + "score": 1.0, + "content": "during our preprocessing step. The number of embedding dimensions in DIVE", + "type": "text" + }, + { + "bbox": [ + 426, + 710, + 438, + 721 + ], + "score": 0.86, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "is set to be 100.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "Other hyper-parameters used in the experiments are listed in the supplementary materials. The", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "hyper-parameters of DIVE were decided based on the performance of HyperNet training set. To", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "train embeddings more efficiently, we chunk the corpus into subsets/lines of 100 tokens instead of", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "score": 1.0, + "content": "using sentence segmentation. 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All values are percentages.", + "type": "text" + }, + { + "bbox": [ + 470, + 81, + 493, + 91 + ], + "score": 0.51, + "content": "\\mathbf { A P } @", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 80, + 506, + 93 + ], + "score": 1.0, + "content": "all", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 90, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 107, + 92, + 123, + 102 + ], + "score": 0.82, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 90, + 246, + 104 + ], + "score": 1.0, + "content": "for 10 datasets and Spearman", + "type": "text" + }, + { + "bbox": [ + 247, + 93, + 254, + 103 + ], + "score": 0.6, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 92, + 271, + 102 + ], + "score": 0.47, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 90, + 377, + 104 + ], + "score": 1.0, + "content": "for HyperLex. Word2Vec", + "type": "text" + }, + { + "bbox": [ + 378, + 92, + 391, + 101 + ], + "score": 0.33, + "content": "+ \\mathrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 90, + 506, + 104 + ], + "score": 1.0, + "content": "scores the word pairs using", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 292, + 115 + ], + "score": 1.0, + "content": "the cosine similarity on skip-grams (SGNS).", + "type": "text" + }, + { + "bbox": [ + 293, + 102, + 320, + 113 + ], + "score": 0.82, + "content": "\\mathrm { G E + C }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 102, + 339, + 115 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 102, + 374, + 113 + ], + "score": 0.83, + "content": "\\mathrm { G E + K L }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "computes cosine similarity and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 357, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 357, + 126 + ], + "score": 1.0, + "content": "negative KL divergence on Gaussian embedding, respectively.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 146, + 133, + 466, + 244 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 133, + 466, + 244 + ], + "spans": [ + { + "bbox": [ + 146, + 133, + 466, + 244 + ], + "score": 0.981, + "html": "
DatasetBLESSEVALutionLenciBenottoWeedsMedicalLEDS
Random Word2Vec+C5.3 9.226.6 25.441.2 40.851.4 51.68.5 11.250.5 71.8
GE+C GE+KL10.5 7.626.7 29.643.352.014.969.7
DIVE+C.△S45.151.315.764.6
16.333.050.465.525.383.5
DatasetTM14Kotlerman 2010HyperNetWordNetHyperLex
Random52.030.824.555.20
Word2Vec+C52.1
39.520.763.016.3
GE+C53.936.021.658.216.4
GE+KL52.039.423.754.49.6
DIVE+C·△S57.236.641.960.932.8
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To", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "train embeddings more efficiently, we chunk the corpus into subsets/lines of 100 tokens instead of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "score": 1.0, + "content": "using sentence segmentation. 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We suspect this is due to the corpus being sorted by the Wikipedia page titles, which makes", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "some categorical words such as animal and mammal occur 3-4 times more frequently in the first 51.2", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "million tokens than the rest. The performances of training SBOW PPMI on the whole WaCkypedia", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 377, + 319, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 319, + 389 + ], + "score": 1.0, + "content": "is also provided for reference in Table 4 and Table 5.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 106, + 403, + 467, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 469, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 469, + 418 + ], + "score": 1.0, + "content": "4 EXPERIMENT 1: COMPARISON WITH UNSUPERVISED EMBEDDINGS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "If a pair of words has the hypernym relation, the words tend to be similar and the hypernym should", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "be more general than the hyponym. 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The", + "type": "text" + }, + { + "bbox": [ + 417, + 451, + 442, + 461 + ], + "score": 0.52, + "content": "{ \\bf C } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "(i.e. the cosine", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 421, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 421, + 474 + ], + "score": 1.0, + "content": "similarity multiply the difference of summation) scoring function is defined as", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 475, + 424, + 504 + ], + "lines": [ + { + "bbox": [ + 186, + 475, + 424, + 504 + ], + "spans": [ + { + "bbox": [ + 186, + 475, + 424, + 504 + ], + "score": 0.95, + "content": "C \\cdot \\Delta S ( \\mathbf { w } _ { q } \\mathbf { w } _ { p } ) = \\frac { \\mathbf { w } _ { q } ^ { T } \\mathbf { w } _ { p } } { | | \\mathbf { w } _ { q } | | _ { 2 } \\cdot | | \\mathbf { w } _ { p } | | _ { 2 } } \\cdot ( | | \\mathbf { w } _ { p } | | _ { 1 } - | | \\mathbf { w } _ { q } | | _ { 1 } ) ,", + "type": "interline_equation", + "image_path": "8aa8b98ee7aed28c95b3f7b4a287be312bd377fbc849a684efd09ab072a89ff3.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 475, + 424, + 489.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 186, + 489.5, + 424, + 504.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 425, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 426, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 133, + 521 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 507, + 146, + 518 + ], + "score": 0.88, + "content": "{ \\bf w } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 504, + 288, + 521 + ], + "score": 1.0, + "content": "is the embedding of hypernym and", + "type": "text" + }, + { + "bbox": [ + 288, + 507, + 302, + 518 + ], + "score": 0.89, + "content": "{ \\bf w } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 504, + 426, + 521 + ], + "score": 1.0, + "content": "is the embedding of hyponym.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "As far as we know, Gaussian embedding (GE) is the only unsupervised embedding method which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "can capture the asymmetric relations between a hypernym and its hyponyms. 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As we can", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 600, + 352, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 352, + 612 + ], + "score": 1.0, + "content": "see, DIVE is usually significantly better than other baselines.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 108, + 626, + 448, + 640 + ], + "lines": [ + { + "bbox": [ + 104, + 625, + 450, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 625, + 450, + 641 + ], + "score": 1.0, + "content": "5 EXPERIMENT 2: HYPERNYMY SCORING FUNCTIONS ANALYSIS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 502, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 365, + 666 + ], + "score": 1.0, + "content": "In Experiment 1, we show that there exists a scoring function", + "type": "text" + }, + { + "bbox": [ + 365, + 651, + 396, + 662 + ], + "score": 0.63, + "content": "\\left( \\mathbf { C } { \\cdot } \\Delta \\mathbf { S } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 649, + 504, + 666 + ], + "score": 1.0, + "content": "which detects hypernymy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 662, + 504, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 504, + 675 + ], + "score": 1.0, + "content": "accurately using the embedding space of DIVE. 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DatasetBLESSEVALutionLenciBenottoWeedsMedicalLEDS
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Random52.030.824.555.20
Word2Vec+C52.1
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GE+C53.936.021.658.216.4
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We suspect this is due to the corpus being sorted by the Wikipedia page titles, which makes", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "some categorical words such as animal and mammal occur 3-4 times more frequently in the first 51.2", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "million tokens than the rest. The performances of training SBOW PPMI on the whole WaCkypedia", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 377, + 319, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 319, + 389 + ], + "score": 1.0, + "content": "is also provided for reference in Table 4 and Table 5.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 311, + 506, + 389 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 403, + 467, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 469, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 469, + 418 + ], + "score": 1.0, + "content": "4 EXPERIMENT 1: COMPARISON WITH UNSUPERVISED EMBEDDINGS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "If a pair of words has the hypernym relation, the words tend to be similar and the hypernym should", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "be more general than the hyponym. As in HyperScore (Nguyen et al., 2017), we score the hypernym", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 417, + 464 + ], + "score": 1.0, + "content": "candidates by multiplying two factors corresponding to these properties. The", + "type": "text" + }, + { + "bbox": [ + 417, + 451, + 442, + 461 + ], + "score": 0.52, + "content": "{ \\bf C } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "(i.e. the cosine", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 421, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 421, + 474 + ], + "score": 1.0, + "content": "similarity multiply the difference of summation) scoring function is defined as", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 429, + 505, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 475, + 424, + 504 + ], + "lines": [ + { + "bbox": [ + 186, + 475, + 424, + 504 + ], + "spans": [ + { + "bbox": [ + 186, + 475, + 424, + 504 + ], + "score": 0.95, + "content": "C \\cdot \\Delta S ( \\mathbf { w } _ { q } \\mathbf { w } _ { p } ) = \\frac { \\mathbf { w } _ { q } ^ { T } \\mathbf { w } _ { p } } { | | \\mathbf { w } _ { q } | | _ { 2 } \\cdot | | \\mathbf { w } _ { p } | | _ { 2 } } \\cdot ( | | \\mathbf { w } _ { p } | | _ { 1 } - | | \\mathbf { w } _ { q } | | _ { 1 } ) ,", + "type": "interline_equation", + "image_path": "8aa8b98ee7aed28c95b3f7b4a287be312bd377fbc849a684efd09ab072a89ff3.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 475, + 424, + 489.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 186, + 489.5, + 424, + 504.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 425, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 426, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 133, + 521 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 507, + 146, + 518 + ], + "score": 0.88, + "content": "{ \\bf w } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 504, + 288, + 521 + ], + "score": 1.0, + "content": "is the embedding of hypernym and", + "type": "text" + }, + { + "bbox": [ + 288, + 507, + 302, + 518 + ], + "score": 0.89, + "content": "{ \\bf w } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 504, + 426, + 521 + ], + "score": 1.0, + "content": "is the embedding of hyponym.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 504, + 426, + 521 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "As far as we know, Gaussian embedding (GE) is the only unsupervised embedding method which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "can capture the asymmetric relations between a hypernym and its hyponyms. Using the same train-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 395, + 557 + ], + "score": 1.0, + "content": "ing and testing setup, we use the code implemented by Athiwaratkun", + "type": "text" + }, + { + "bbox": [ + 396, + 545, + 405, + 555 + ], + "score": 0.26, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 545, + 439, + 557 + ], + "score": 1.0, + "content": "Wilson", + "type": "text" + }, + { + "bbox": [ + 439, + 544, + 470, + 556 + ], + "score": 0.67, + "content": "( 2 0 1 7 ) ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "to train", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "Gaussian embedding on the first 51.2 million tokens and test the embeddings on 11 datasets. Its", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 504, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 504, + 579 + ], + "score": 1.0, + "content": "hyper-parameters are determined using the same way as DIVE (i.e. maximizing the AP on Hyper-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 278, + 590 + ], + "score": 1.0, + "content": "Net training set). We compare DIVE with", + "type": "text" + }, + { + "bbox": [ + 278, + 577, + 297, + 588 + ], + "score": 0.88, + "content": "\\mathrm { G E } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "in Table 1, and the performances of random scores", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "and only measuring word similarity using skip-grams are also presented for reference. As we can", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 600, + 352, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 352, + 612 + ], + "score": 1.0, + "content": "see, DIVE is usually significantly better than other baselines.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 523, + 506, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 626, + 448, + 640 + ], + "lines": [ + { + "bbox": [ + 104, + 625, + 450, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 625, + 450, + 641 + ], + "score": 1.0, + "content": "5 EXPERIMENT 2: HYPERNYMY SCORING FUNCTIONS ANALYSIS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 502, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 365, + 666 + ], + "score": 1.0, + "content": "In Experiment 1, we show that there exists a scoring function", + "type": "text" + }, + { + "bbox": [ + 365, + 651, + 396, + 662 + ], + "score": 0.63, + "content": "\\left( \\mathbf { C } { \\cdot } \\Delta \\mathbf { S } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 649, + 504, + 666 + ], + "score": 1.0, + "content": "which detects hypernymy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 662, + 504, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 504, + 675 + ], + "score": 1.0, + "content": "accurately using the embedding space of DIVE. Nevertheless, different scoring functions measure", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "different signals in SBOW or embeddings. Since there are so many scoring functions and datasets", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "available in the domain, we first introduce and test the performances of various scoring functions", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "so as to select the representative ones for a more comprehensive evaluation of DIVE on the hyper-", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "nymy detection tasks. We denote the embedding/context vector of the hypernym candidate and the", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 198, + 139 + ], + "score": 1.0, + "content": "hyponym candidate as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 198, + 128, + 212, + 139 + ], + "score": 0.88, + "content": "{ \\bf w } _ { p }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 212, + 127, + 230, + 139 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 230, + 128, + 243, + 138 + ], + "score": 0.87, + "content": "{ \\bf w } _ { q }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 243, + 127, + 505, + 139 + ], + "score": 1.0, + "content": ", respectively. The SBOW model which represents a word by the", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "frequency of its neighboring words is denoted as SBOW Freq, while the SBOW which uses PPMI", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 487, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 487, + 162 + ], + "score": 1.0, + "content": "of its neighboring words as the features (Bullinaria & Levy, 2007) is denoted as SBOW PPMI.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 649, + 504, + 675 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "different signals in SBOW or embeddings. Since there are so many scoring functions and datasets", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "available in the domain, we first introduce and test the performances of various scoring functions", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "so as to select the representative ones for a more comprehensive evaluation of DIVE on the hyper-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "nymy detection tasks. We denote the embedding/context vector of the hypernym candidate and the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 198, + 139 + ], + "score": 1.0, + "content": "hyponym candidate as", + "type": "text" + }, + { + "bbox": [ + 198, + 128, + 212, + 139 + ], + "score": 0.88, + "content": "{ \\bf w } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 127, + 230, + 139 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 128, + 243, + 138 + ], + "score": 0.87, + "content": "{ \\bf w } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 127, + 505, + 139 + ], + "score": 1.0, + "content": ", respectively. The SBOW model which represents a word by the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "frequency of its neighboring words is denoted as SBOW Freq, while the SBOW which uses PPMI", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 487, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 487, + 162 + ], + "score": 1.0, + "content": "of its neighboring words as the features (Bullinaria & Levy, 2007) is denoted as SBOW PPMI.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 108, + 174, + 293, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 173, + 295, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 295, + 187 + ], + "score": 1.0, + "content": "5.1 UNSUPERVISED SCORING FUNCTIONS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 195, + 192, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 194, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 194, + 208 + ], + "score": 1.0, + "content": "5.1.1 SIMILARITY", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "score": 1.0, + "content": "A hypernym tends to be similar to its hyponym, so we measure the cosine similarity between word", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "score": 1.0, + "content": "vectors of the SBOW features (Levy et al., 2015b) or DIVE. We refer to the symmetric scoring", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "function as Cosine or C for short in the following tables. We also train the original skip-grams with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "100 dimensions and measure the cosine similarity between the resulting word2vec embeddings. This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 305, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 305, + 270 + ], + "score": 1.0, + "content": "scoring function is referred to as Word2vec or W.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 282, + 196, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 199, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 199, + 296 + ], + "score": 1.0, + "content": "5.1.2 GENERALITY", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 316 + ], + "score": 1.0, + "content": "The distributional informativeness hypothesis (Santus et al., 2014) observes that in many corpora,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "semantically ‘general’ words tend to appear more frequently and in more varied contexts. Thus,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 323, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 338 + ], + "score": 1.0, + "content": "Santus et al. (2014) advocate using entropy of context distributions to capture the diversity of con-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "text. We adopt the two variations of the approach proposed by Santus et al. (2017): SLQS Row and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 317, + 359 + ], + "score": 1.0, + "content": "SLQS Sub functions. We also refer to SLQS Row as", + "type": "text" + }, + { + "bbox": [ + 317, + 347, + 333, + 357 + ], + "score": 0.68, + "content": "\\Delta \\mathrm { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 346, + 506, + 359 + ], + "score": 1.0, + "content": "because it measures the entropy difference", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 462, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 462, + 370 + ], + "score": 1.0, + "content": "of context distributions. For SLQS Sub, the number of top context words is fixed as 100.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 388 + ], + "score": 1.0, + "content": "Although effective at measuring diversity, the entropy totally ignores the frequency signal from the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 458, + 398 + ], + "score": 1.0, + "content": "corpus. To leverage the information, we measure the generality of a word by its L1 norm", + "type": "text" + }, + { + "bbox": [ + 459, + 385, + 487, + 397 + ], + "score": 0.9, + "content": "( | \\mathbf { w } _ { p } | _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 145, + 410 + ], + "score": 1.0, + "content": "L2 norm", + "type": "text" + }, + { + "bbox": [ + 145, + 396, + 179, + 408 + ], + "score": 0.88, + "content": "( | | \\mathbf { w } _ { p } | | _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 396, + 468, + 410 + ], + "score": 1.0, + "content": ". Recall that Equation (2) indicates that the embedding of the hypernym", + "type": "text" + }, + { + "bbox": [ + 468, + 398, + 475, + 408 + ], + "score": 0.46, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "should", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "have a larger value at every dimension than the embedding of the hyponym x. When the inclusion", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 171, + 432 + ], + "score": 1.0, + "content": "property holds,", + "type": "text" + }, + { + "bbox": [ + 171, + 417, + 308, + 431 + ], + "score": 0.92, + "content": "\\begin{array} { r } { | { \\bf y } | _ { 1 } = \\sum _ { i } { \\bf y } [ i ] \\geq \\sum _ { i } { \\bf x } [ i ] = | { \\bf x } | _ { 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 417, + 367, + 432 + ], + "score": 1.0, + "content": "and similarly", + "type": "text" + }, + { + "bbox": [ + 367, + 418, + 424, + 430 + ], + "score": 0.92, + "content": "| | \\mathbf { y } | | _ { 2 } \\geq | | \\mathbf { x } | | _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 417, + 506, + 432 + ], + "score": 1.0, + "content": ". Thus, we propose", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 327, + 443 + ], + "score": 1.0, + "content": "two scoring functions, difference of vector summation", + "type": "text" + }, + { + "bbox": [ + 327, + 430, + 390, + 441 + ], + "score": 0.91, + "content": "( | \\mathbf { w } _ { p } | _ { 1 } - | \\mathbf { w } _ { q } | _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "and the difference of vector", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 438, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 139, + 455 + ], + "score": 1.0, + "content": "2-norm", + "type": "text" + }, + { + "bbox": [ + 140, + 440, + 211, + 452 + ], + "score": 0.91, + "content": "( | | \\mathbf { \\bar { w } } _ { p } | | _ { 2 } - | | \\mathbf { w } _ { q } | | _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 438, + 506, + 455 + ], + "score": 1.0, + "content": ". Notice that when applying the difference of vector summations (denoted", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 118, + 465 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 452, + 134, + 462 + ], + "score": 0.73, + "content": "\\Delta S", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 450, + 505, + 465 + ], + "score": 1.0, + "content": ") to SBOW Freq, it is equivalent to computing the word frequency difference between the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 462, + 210, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 210, + 475 + ], + "score": 1.0, + "content": "hypernym candidate pair.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 486, + 203, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 204, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 204, + 499 + ], + "score": 1.0, + "content": "5.1.3 COMBINATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "score": 1.0, + "content": "The combination of 2 similarity functions (Cosine and Word2vec) and the 3 generality functions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "(difference of entropy, summation, and 2-norm of vectors) leads to six different scoring functions as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 193, + 541 + ], + "score": 1.0, + "content": "shown in Table 2, and", + "type": "text" + }, + { + "bbox": [ + 194, + 528, + 218, + 538 + ], + "score": 0.64, + "content": "{ \\bf C } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "is the same scoring function we used in Experiment 1. It should be noted", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "that if we use skip-grams with negative sampling (word2vec) as the similarity measurement (i.e.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 173, + 561 + ], + "score": 0.82, + "content": "W \\cdot \\Delta \\ \\{ \\mathrm { E } , \\mathrm { S } , \\mathrm { Q } \\} ,", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "), the scores are determined by two embedding/feature spaces together (word2vec", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 561, + 186, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 186, + 572 + ], + "score": 1.0, + "content": "and DIVE/SBOW).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 108, + 585, + 188, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 189, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 189, + 598 + ], + "score": 1.0, + "content": "5.1.4 INCLUSION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "Several scoring functions are proposed to measure inclusion properties of SBOW based on DIH.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "Weeds Precision (Weeds & Weir, 2003) and CDE (Clarke, 2009) both measure the magnitude of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 254, + 641 + ], + "score": 1.0, + "content": "intersection between feature vectors", + "type": "text" + }, + { + "bbox": [ + 254, + 627, + 300, + 639 + ], + "score": 0.91, + "content": "( | \\mathbf { w } _ { p } \\cap \\mathbf { w } _ { q } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 626, + 361, + 641 + ], + "score": 1.0, + "content": ". For example,", + "type": "text" + }, + { + "bbox": [ + 361, + 628, + 397, + 639 + ], + "score": 0.92, + "content": "{ \\bf w } _ { p } \\cap { \\bf w } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "is defined by the element-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "wise minimum in CDE. Then, both scoring functions divide the intersection by the magnitude of the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 211, + 662 + ], + "score": 1.0, + "content": "potential hyponym vector", + "type": "text" + }, + { + "bbox": [ + 212, + 649, + 236, + 661 + ], + "score": 0.88, + "content": "( | \\mathbf { w } _ { q } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 649, + 505, + 662 + ], + "score": 1.0, + "content": ". invCL (Lenci & Benotto, 2012) (A variant of CDE) is also tested.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We choose these 3 functions because they have been shown to detect hypernymy well in a recent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "study (Santus et al., 2017). However, it is hard to confirm that their good performances come from", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "the inclusion property between context distributions — it is also possible that the context vectors of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "score": 1.0, + "content": "more general words have higher chance to overlap with all other words due to their high frequency.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "For instance, considering a one dimension feature which stores only the frequency of words, the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "naive embedding could still have reasonable performance on the CDE function, but the embedding", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5 + } + ], + "page_idx": 6, + "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": [ + 303, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 7 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 81, + 506, + 162 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 174, + 293, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 173, + 295, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 295, + 187 + ], + "score": 1.0, + "content": "5.1 UNSUPERVISED SCORING FUNCTIONS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 195, + 192, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 194, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 194, + 208 + ], + "score": 1.0, + "content": "5.1.1 SIMILARITY", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "score": 1.0, + "content": "A hypernym tends to be similar to its hyponym, so we measure the cosine similarity between word", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "score": 1.0, + "content": "vectors of the SBOW features (Levy et al., 2015b) or DIVE. We refer to the symmetric scoring", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "function as Cosine or C for short in the following tables. We also train the original skip-grams with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "100 dimensions and measure the cosine similarity between the resulting word2vec embeddings. This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 305, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 305, + 270 + ], + "score": 1.0, + "content": "scoring function is referred to as Word2vec or W.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 214, + 505, + 270 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 282, + 196, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 199, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 199, + 296 + ], + "score": 1.0, + "content": "5.1.2 GENERALITY", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 316 + ], + "score": 1.0, + "content": "The distributional informativeness hypothesis (Santus et al., 2014) observes that in many corpora,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "semantically ‘general’ words tend to appear more frequently and in more varied contexts. Thus,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 323, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 338 + ], + "score": 1.0, + "content": "Santus et al. (2014) advocate using entropy of context distributions to capture the diversity of con-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "text. We adopt the two variations of the approach proposed by Santus et al. (2017): SLQS Row and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 317, + 359 + ], + "score": 1.0, + "content": "SLQS Sub functions. We also refer to SLQS Row as", + "type": "text" + }, + { + "bbox": [ + 317, + 347, + 333, + 357 + ], + "score": 0.68, + "content": "\\Delta \\mathrm { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 346, + 506, + 359 + ], + "score": 1.0, + "content": "because it measures the entropy difference", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 462, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 462, + 370 + ], + "score": 1.0, + "content": "of context distributions. For SLQS Sub, the number of top context words is fixed as 100.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 301, + 506, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 388 + ], + "score": 1.0, + "content": "Although effective at measuring diversity, the entropy totally ignores the frequency signal from the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 458, + 398 + ], + "score": 1.0, + "content": "corpus. To leverage the information, we measure the generality of a word by its L1 norm", + "type": "text" + }, + { + "bbox": [ + 459, + 385, + 487, + 397 + ], + "score": 0.9, + "content": "( | \\mathbf { w } _ { p } | _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 145, + 410 + ], + "score": 1.0, + "content": "L2 norm", + "type": "text" + }, + { + "bbox": [ + 145, + 396, + 179, + 408 + ], + "score": 0.88, + "content": "( | | \\mathbf { w } _ { p } | | _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 396, + 468, + 410 + ], + "score": 1.0, + "content": ". Recall that Equation (2) indicates that the embedding of the hypernym", + "type": "text" + }, + { + "bbox": [ + 468, + 398, + 475, + 408 + ], + "score": 0.46, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "should", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "have a larger value at every dimension than the embedding of the hyponym x. When the inclusion", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 171, + 432 + ], + "score": 1.0, + "content": "property holds,", + "type": "text" + }, + { + "bbox": [ + 171, + 417, + 308, + 431 + ], + "score": 0.92, + "content": "\\begin{array} { r } { | { \\bf y } | _ { 1 } = \\sum _ { i } { \\bf y } [ i ] \\geq \\sum _ { i } { \\bf x } [ i ] = | { \\bf x } | _ { 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 417, + 367, + 432 + ], + "score": 1.0, + "content": "and similarly", + "type": "text" + }, + { + "bbox": [ + 367, + 418, + 424, + 430 + ], + "score": 0.92, + "content": "| | \\mathbf { y } | | _ { 2 } \\geq | | \\mathbf { x } | | _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 417, + 506, + 432 + ], + "score": 1.0, + "content": ". Thus, we propose", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 327, + 443 + ], + "score": 1.0, + "content": "two scoring functions, difference of vector summation", + "type": "text" + }, + { + "bbox": [ + 327, + 430, + 390, + 441 + ], + "score": 0.91, + "content": "( | \\mathbf { w } _ { p } | _ { 1 } - | \\mathbf { w } _ { q } | _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "and the difference of vector", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 438, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 139, + 455 + ], + "score": 1.0, + "content": "2-norm", + "type": "text" + }, + { + "bbox": [ + 140, + 440, + 211, + 452 + ], + "score": 0.91, + "content": "( | | \\mathbf { \\bar { w } } _ { p } | | _ { 2 } - | | \\mathbf { w } _ { q } | | _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 438, + 506, + 455 + ], + "score": 1.0, + "content": ". Notice that when applying the difference of vector summations (denoted", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 118, + 465 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 452, + 134, + 462 + ], + "score": 0.73, + "content": "\\Delta S", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 450, + 505, + 465 + ], + "score": 1.0, + "content": ") to SBOW Freq, it is equivalent to computing the word frequency difference between the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 462, + 210, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 210, + 475 + ], + "score": 1.0, + "content": "hypernym candidate pair.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 373, + 506, + 475 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 486, + 203, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 204, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 204, + 499 + ], + "score": 1.0, + "content": "5.1.3 COMBINATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "score": 1.0, + "content": "The combination of 2 similarity functions (Cosine and Word2vec) and the 3 generality functions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "(difference of entropy, summation, and 2-norm of vectors) leads to six different scoring functions as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 193, + 541 + ], + "score": 1.0, + "content": "shown in Table 2, and", + "type": "text" + }, + { + "bbox": [ + 194, + 528, + 218, + 538 + ], + "score": 0.64, + "content": "{ \\bf C } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "is the same scoring function we used in Experiment 1. It should be noted", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "that if we use skip-grams with negative sampling (word2vec) as the similarity measurement (i.e.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 173, + 561 + ], + "score": 0.82, + "content": "W \\cdot \\Delta \\ \\{ \\mathrm { E } , \\mathrm { S } , \\mathrm { Q } \\} ,", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "), the scores are determined by two embedding/feature spaces together (word2vec", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 561, + 186, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 186, + 572 + ], + "score": 1.0, + "content": "and DIVE/SBOW).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 506, + 506, + 572 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 585, + 188, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 189, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 189, + 598 + ], + "score": 1.0, + "content": "5.1.4 INCLUSION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "Several scoring functions are proposed to measure inclusion properties of SBOW based on DIH.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "Weeds Precision (Weeds & Weir, 2003) and CDE (Clarke, 2009) both measure the magnitude of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 254, + 641 + ], + "score": 1.0, + "content": "intersection between feature vectors", + "type": "text" + }, + { + "bbox": [ + 254, + 627, + 300, + 639 + ], + "score": 0.91, + "content": "( | \\mathbf { w } _ { p } \\cap \\mathbf { w } _ { q } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 626, + 361, + 641 + ], + "score": 1.0, + "content": ". For example,", + "type": "text" + }, + { + "bbox": [ + 361, + 628, + 397, + 639 + ], + "score": 0.92, + "content": "{ \\bf w } _ { p } \\cap { \\bf w } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "is defined by the element-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "wise minimum in CDE. Then, both scoring functions divide the intersection by the magnitude of the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 211, + 662 + ], + "score": 1.0, + "content": "potential hyponym vector", + "type": "text" + }, + { + "bbox": [ + 212, + 649, + 236, + 661 + ], + "score": 0.88, + "content": "( | \\mathbf { w } _ { q } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 649, + 505, + 662 + ], + "score": 1.0, + "content": ". invCL (Lenci & Benotto, 2012) (A variant of CDE) is also tested.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 604, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We choose these 3 functions because they have been shown to detect hypernymy well in a recent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "study (Santus et al., 2017). However, it is hard to confirm that their good performances come from", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "the inclusion property between context distributions — it is also possible that the context vectors of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "score": 1.0, + "content": "more general words have higher chance to overlap with all other words due to their high frequency.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "For instance, considering a one dimension feature which stores only the frequency of words, the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "naive embedding could still have reasonable performance on the CDE function, but the embedding", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 187, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 202 + ], + "score": 1.0, + "content": "in fact only memorizes the general words without modeling relations between words (Levy et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 199, + 420, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 420, + 212 + ], + "score": 1.0, + "content": "2015b) and loses lots of inclusion signals in the word co-occurrence statistics.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 665, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 111, + 483, + 168 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 502, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 240, + 93 + ], + "score": 1.0, + "content": "Table 2: Micro average AP@all", + "type": "text" + }, + { + "bbox": [ + 240, + 81, + 256, + 91 + ], + "score": 0.75, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 78, + 505, + 93 + ], + "score": 1.0, + "content": "of 10 datasets using different scoring functions. 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Word2vec (W) 24.8Cosine (C) 26.7SLQS Sub 27.4SLQS Row (△E) 27.6Summation (△S) 31.5Two norm (△Q) 31.2
W.△EC.△EW.△SC.△SW.△QC.△Q
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WeedsCDEinvCLAsymmetric L1(AL1)
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Ideally, the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 249, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 253, + 263 + ], + "score": 1.0, + "content": "context distribution of the hypernym", + "type": "text" + }, + { + "bbox": [ + 254, + 250, + 265, + 262 + ], + "score": 0.87, + "content": "{ \\bf d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 249, + 315, + 263 + ], + "score": 1.0, + "content": "will include", + "type": "text" + }, + { + "bbox": [ + 315, + 250, + 327, + 262 + ], + "score": 0.87, + "content": "{ \\bf d } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 249, + 492, + 263 + ], + "score": 1.0, + "content": ". This suggests the hypernym distribution", + "type": "text" + }, + { + "bbox": [ + 492, + 250, + 504, + 262 + ], + "score": 0.83, + "content": "{ \\bf d } _ { p }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 258, + 504, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 417, + 275 + ], + "score": 1.0, + "content": "is larger than context distribution of the hyponym with a proper scaling factor", + "type": "text" + }, + { + "bbox": [ + 417, + 261, + 434, + 272 + ], + "score": 0.88, + "content": "a \\mathbf { d } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 258, + 455, + 275 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 456, + 261, + 504, + 273 + ], + "score": 0.88, + "content": "\\operatorname* { m a x } ( a \\mathbf { d } _ { q } \\bar { - }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 131, + 284 + ], + "score": 0.88, + "content": "{ \\bf d } _ { p } , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 271, + 416, + 284 + ], + "score": 1.0, + "content": "should be small). Furthermore, both distributions should be similar, so", + "type": "text" + }, + { + "bbox": [ + 416, + 273, + 432, + 284 + ], + "score": 0.88, + "content": "a \\mathbf { d } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "should not be too", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 281, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 281, + 165, + 296 + ], + "score": 1.0, + "content": "different from", + "type": "text" + }, + { + "bbox": [ + 165, + 283, + 177, + 295 + ], + "score": 0.88, + "content": "{ \\bf d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 281, + 200, + 296 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 200, + 282, + 275, + 295 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } ( \\mathbf { d } _ { p } - a \\mathbf { d } _ { q } , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 281, + 505, + 296 + ], + "score": 1.0, + "content": "should also be small). 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We can see the combination functions such as", + "type": "text" + }, + { + "bbox": [ + 331, + 441, + 356, + 451 + ], + "score": 0.63, + "content": "{ \\bf C } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 440, + 374, + 453 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 375, + 441, + 402, + 452 + ], + "score": 0.49, + "content": "{ \\bf W } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "perform the best overall.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 444, + 464 + ], + "score": 1.0, + "content": "Among the unnormalized inclusion based scoring functions, CDE works the best.", + "type": "text" + }, + { + "bbox": [ + 444, + 452, + 464, + 463 + ], + "score": 0.9, + "content": "A L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "performs", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "well compared with other functions which remove the frequency signal such as Word2vec, Cosine,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "and SLQS Row. The summation is the most robust generality measurement. 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Furthermore, both distributions should be similar, so", + "type": "text" + }, + { + "bbox": [ + 416, + 273, + 432, + 284 + ], + "score": 0.88, + "content": "a \\mathbf { d } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "should not be too", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 281, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 281, + 165, + 296 + ], + "score": 1.0, + "content": "different from", + "type": "text" + }, + { + "bbox": [ + 165, + 283, + 177, + 295 + ], + "score": 0.88, + "content": "{ \\bf d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 281, + 200, + 296 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 200, + 282, + 275, + 295 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } ( \\mathbf { d } _ { p } - a \\mathbf { d } _ { q } , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 281, + 505, + 296 + ], + "score": 1.0, + "content": "should also be small). Therefore, we define asymmetric", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 293, + 167, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 167, + 305 + ], + "score": 1.0, + "content": "L1 distance as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 217, + 505, + 305 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 308, + 444, + 334 + ], + "lines": [ + { + "bbox": [ + 158, + 308, + 444, + 334 + ], + "spans": [ + { + "bbox": [ + 158, + 308, + 444, + 334 + ], + "score": 0.9, + "content": "A L _ { 1 } = \\operatorname* { m i n } _ { a } \\sum _ { c } w _ { 0 } \\cdot \\operatorname* { m a x } ( a \\mathbf { d } _ { q } [ c ] - \\mathbf { d } _ { p } [ c ] , 0 ) + \\operatorname* { m a x } ( \\mathbf { d } _ { p } [ c ] - a \\mathbf { d } _ { q } [ c ] , 0 ) ,", + "type": "interline_equation", + "image_path": "d9abf289974b7fa865cbebc67e0a06b58bb226409ab693c879e9697fdbff6caa.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 158, + 308, + 444, + 334 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 504, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 135, + 352 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 342, + 147, + 351 + ], + "score": 0.85, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 339, + 390, + 352 + ], + "score": 1.0, + "content": "is a constant which emphasizes the inclusion penalty. If", + "type": "text" + }, + { + "bbox": [ + 390, + 340, + 427, + 351 + ], + "score": 0.92, + "content": "w _ { 0 } ~ = ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 339, + 448, + 352 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 448, + 340, + 478, + 350 + ], + "score": 0.85, + "content": "a \\ = \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 339, + 483, + 352 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 484, + 340, + 504, + 351 + ], + "score": 0.78, + "content": "A L _ { 1 }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 275, + 363 + ], + "score": 1.0, + "content": "is equivalent to L1 distance. The lower", + "type": "text" + }, + { + "bbox": [ + 275, + 351, + 295, + 362 + ], + "score": 0.9, + "content": "A L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "distance implies a higher chance of observing the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 227, + 375 + ], + "score": 1.0, + "content": "hypernym relation. We tried", + "type": "text" + }, + { + "bbox": [ + 227, + 362, + 262, + 373 + ], + "score": 0.91, + "content": "w _ { 0 } = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 361, + 282, + 375 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 282, + 362, + 322, + 373 + ], + "score": 0.87, + "content": "w _ { 0 } = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 361, + 329, + 375 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 329, + 362, + 369, + 373 + ], + "score": 0.88, + "content": "w _ { 0 } = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "produces a worse micro-average", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 360, + 385 + ], + "score": 1.0, + "content": "AP@all on SBOW Freq, SBOW PPMI and DIVE, so we fix", + "type": "text" + }, + { + "bbox": [ + 360, + 375, + 373, + 384 + ], + "score": 0.86, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "to be 5 in all experiments. An", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 384, + 470, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 272, + 396 + ], + "score": 1.0, + "content": "efficient way to solve the optimization in", + "type": "text" + }, + { + "bbox": [ + 272, + 384, + 291, + 395 + ], + "score": 0.91, + "content": "A L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 384, + 470, + 396 + ], + "score": 1.0, + "content": "is presented in the supplementary materials.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 339, + 505, + 396 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 409, + 252, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 254, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 254, + 423 + ], + "score": 1.0, + "content": "5.2 RESULTS AND DISCUSSIONS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 219, + 443 + ], + "score": 1.0, + "content": "We show the micro average", + "type": "text" + }, + { + "bbox": [ + 220, + 430, + 242, + 441 + ], + "score": 0.57, + "content": "\\mathbf { A P } @", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "all on 10 datasets using different hypernymy scoring functions in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 331, + 453 + ], + "score": 1.0, + "content": "Table 2. We can see the combination functions such as", + "type": "text" + }, + { + "bbox": [ + 331, + 441, + 356, + 451 + ], + "score": 0.63, + "content": "{ \\bf C } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 440, + 374, + 453 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 375, + 441, + 402, + 452 + ], + "score": 0.49, + "content": "{ \\bf W } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "perform the best overall.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 444, + 464 + ], + "score": 1.0, + "content": "Among the unnormalized inclusion based scoring functions, CDE works the best.", + "type": "text" + }, + { + "bbox": [ + 444, + 452, + 464, + 463 + ], + "score": 0.9, + "content": "A L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "performs", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "well compared with other functions which remove the frequency signal such as Word2vec, Cosine,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "and SLQS Row. The summation is the most robust generality measurement. In the table, the scoring", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "functions are applied to SBOW Freq, but the performances of hypernymy scoring functions on the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 321, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 321, + 508 + ], + "score": 1.0, + "content": "other feature spaces (e.g. DIVE) have a similar trend.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 429, + 505, + 508 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 523, + 351, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 351, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 351, + 538 + ], + "score": 1.0, + "content": "6 EXPERIMENT 3: COMPARISON WITH SBOW", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 106, + 549, + 358, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 359, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 359, + 561 + ], + "score": 1.0, + "content": "6.1 COMPARISON WITH PREVIOUSLY REPORTED RESULTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 340, + 582 + ], + "score": 1.0, + "content": "In Table 3, DIVE with two of the best scoring functions", + "type": "text" + }, + { + "bbox": [ + 341, + 570, + 366, + 580 + ], + "score": 0.42, + "content": "C { \\cdot } \\Delta s", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 568, + 384, + 582 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 569, + 412, + 580 + ], + "score": 0.37, + "content": "{ \\bf W } { \\cdot } { \\Delta S }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 568, + 505, + 582 + ], + "score": 1.0, + "content": ") is compared with the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 581, + 462, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 462, + 594 + ], + "score": 1.0, + "content": "previous unsupervised state-of-the-art approaches based on SBOW on different datasets.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 568, + 505, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "There are several reasons which might cause the large performance gaps in some datasets. In addi-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "tion to the effectiveness of DIVE, some improvements come from our proposed scoring functions.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 618, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 633 + ], + "score": 1.0, + "content": "The fact that every paper uses a different training corpus also affects the performances. Furthermore,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 630, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 643 + ], + "score": 1.0, + "content": "Santus et al. (2017) select the scoring functions and feature space for the first 4 datasets based on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 145, + 652 + ], + "score": 0.8, + "content": "\\mathbf { A P } @ 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 641, + 505, + 654 + ], + "score": 1.0, + "content": ", which we believe is too sensitive to the hyper-parameter settings of different methods. To", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 653, + 450, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 450, + 664 + ], + "score": 1.0, + "content": "isolate the impact of each factor, we perform a more comprehensive comparison next.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 596, + 506, + 664 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 678, + 242, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 243, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 243, + 690 + ], + "score": 1.0, + "content": "6.2 PERFORMANCE ANALYSIS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 697, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 505, + 713 + ], + "score": 1.0, + "content": "In this experiment, we examine whether DIVE successfully preserves the signals for hypernymy", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "detection tasks, which are measured by the same scoring functions designed for SBOW. Summation", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 150, + 733 + ], + "score": 1.0, + "content": "difference", + "type": "text" + }, + { + "bbox": [ + 151, + 721, + 172, + 732 + ], + "score": 0.4, + "content": "( \\Delta \\boldsymbol { S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "and CDE perform the best among generality and inclusion functions in Table 2,", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 697, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 79, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 3: Comparison with previous methods based on sparse bag of word (SBOW). All values", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "are percentages. The results of invCL (Lenci & Benotto, 2012), APSyn (Santus et al., 2016), and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 353, + 115 + ], + "score": 1.0, + "content": "CDE (Clarke, 2009) are selected because they have the best", + "type": "text" + }, + { + "bbox": [ + 353, + 102, + 392, + 113 + ], + "score": 0.63, + "content": "\\mathbf { A P } @ 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "in the first 4 datasets (San-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "tus et al., 2017). Cosine similarity (Levy et al., 2015b), balAPinc (Kotlerman et al., 2010) in 3", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 123, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 505, + 136 + ], + "score": 1.0, + "content": "datasets (Turney & Mohammad, 2015), SLQS (Santus et al., 2014) in HyperNet dataset (Shwartz", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 375, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 375, + 148 + ], + "score": 1.0, + "content": "et al., 2016), and Freq ratio (FR) (Vulic et al., 2016) are compared. ´", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 159, + 155, + 451, + 267 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 159, + 155, + 451, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 159, + 155, + 451, + 267 + ], + "spans": [ + { + "bbox": [ + 159, + 155, + 451, + 267 + ], + "score": 0.984, + "html": "
DatasetBLESS EVALutionLenciBenottoWeedsMedical
MetricAP@allF1
BaselinesinvCLAPSynCDECosine
5.135.338.244.123.1
DIVE+C·△S16.333.050.465.525.3
DIVE +W·△S18.632.351.568.625.7
DatasetLEDSTM14Kotlerman 2010HyperNetHyperLex
MetricAP@allF1Spearman p
BaselinesbalAPincSLQSFreq ratio
73563722.827.9
DIVE+C·△S83.557.236.641.932.8
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Therefore, we will present the results using these 3 scoring functions, along with", + "type": "text" + }, + { + "bbox": [ + 460, + 298, + 487, + 309 + ], + "score": 0.42, + "content": "{ \\bf W } { \\cdot } { \\Delta } S", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 307, + 136, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 131, + 320 + ], + "score": 0.56, + "content": "{ \\bf C } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 307, + 136, + 321 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 332, + 189, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 332, + 190, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 190, + 345 + ], + "score": 1.0, + "content": "6.2.1 BASELINES", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 504, + 364 + ], + "score": 1.0, + "content": "In addition to classic representations such as SBOW Freq and SBOW PPMI, we compare distribu-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 363, + 432, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 432, + 374 + ], + "score": 1.0, + "content": "tional inclusion vector embedding (DIVE) with additional 4 baselines in Table 4.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 131, + 381, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 131, + 382, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 131, + 382, + 473, + 397 + ], + "score": 1.0, + "content": "• SBOW PPMI with additional frequency weighting (PPMI w/ FW). 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Hashing context vectors into topic vectors can be written", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 140, + 578, + 246, + 592 + ], + "spans": [ + { + "bbox": [ + 140, + 578, + 153, + 592 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 154, + 579, + 240, + 591 + ], + "score": 0.92, + "content": "M _ { c } G \\approx F G ^ { T } G = F", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 578, + 246, + 592 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 504, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 343, + 613 + ], + "score": 1.0, + "content": "In the experiment, we also tried to apply a constant", + "type": "text" + }, + { + "bbox": [ + 343, + 601, + 371, + 613 + ], + "score": 0.91, + "content": "\\log ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 600, + 504, + 613 + ], + "score": 1.0, + "content": "shifting to SBOW PPMI (i.e.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 611, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 107, + 612, + 208, + 624 + ], + "score": 0.87, + "content": "\\operatorname* { m a x } ( P M I - \\log ( k ) , 0 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 611, + 394, + 626 + ], + "score": 1.0, + "content": "). We found that the performance degrades as", + "type": "text" + }, + { + "bbox": [ + 395, + 613, + 402, + 622 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 611, + 505, + 626 + ], + "score": 1.0, + "content": "increases. Similarly, ap-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 622, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 622, + 468, + 637 + ], + "score": 1.0, + "content": "plying PMI filter to SBOW PPMI (set context feature to be 0 if the value is lower than", + "type": "text" + }, + { + "bbox": [ + 468, + 623, + 502, + 635 + ], + "score": 0.89, + "content": "\\log ( k _ { f } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 622, + 506, + 637 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 333, + 646 + ], + "score": 1.0, + "content": "usually makes the performances worse, especially when", + "type": "text" + }, + { + "bbox": [ + 333, + 634, + 345, + 646 + ], + "score": 0.89, + "content": "k _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "is large. Applying PMI filter to SBOW", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "Freq only makes its performances closer to (but still much worse than) SBOW PPMI, so we omit", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 655, + 191, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 191, + 668 + ], + "score": 1.0, + "content": "this baseline as well.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 108, + 679, + 260, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 679, + 262, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 262, + 692 + ], + "score": 1.0, + "content": "6.2.2 RESULTS AND DISCUSSIONS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "In Table 4, we first confirm the finding of the previous review study of Santus et al. (2017): there is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "no single hypernymy scoring function which always outperforms others. One of the main reasons", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "is that different datasets collect negative samples differently. 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All values", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "are percentages. The results of invCL (Lenci & Benotto, 2012), APSyn (Santus et al., 2016), and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 353, + 115 + ], + "score": 1.0, + "content": "CDE (Clarke, 2009) are selected because they have the best", + "type": "text" + }, + { + "bbox": [ + 353, + 102, + 392, + 113 + ], + "score": 0.63, + "content": "\\mathbf { A P } @ 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "in the first 4 datasets (San-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "tus et al., 2017). Cosine similarity (Levy et al., 2015b), balAPinc (Kotlerman et al., 2010) in 3", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 123, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 505, + 136 + ], + "score": 1.0, + "content": "datasets (Turney & Mohammad, 2015), SLQS (Santus et al., 2014) in HyperNet dataset (Shwartz", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 375, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 375, + 148 + ], + "score": 1.0, + "content": "et al., 2016), and Freq ratio (FR) (Vulic et al., 2016) are compared. ´", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 79, + 505, + 148 + ] + }, + { + "type": "table", + "bbox": [ + 159, + 155, + 451, + 267 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 159, + 155, + 451, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 159, + 155, + 451, + 267 + ], + "spans": [ + { + "bbox": [ + 159, + 155, + 451, + 267 + ], + "score": 0.984, + "html": "
DatasetBLESS EVALutionLenciBenottoWeedsMedical
MetricAP@allF1
BaselinesinvCLAPSynCDECosine
5.135.338.244.123.1
DIVE+C·△S16.333.050.465.525.3
DIVE +W·△S18.632.351.568.625.7
DatasetLEDSTM14Kotlerman 2010HyperNetHyperLex
MetricAP@allF1Spearman p
BaselinesbalAPincSLQSFreq ratio
73563722.827.9
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Therefore, we will present the results using these 3 scoring functions, along with", + "type": "text" + }, + { + "bbox": [ + 460, + 298, + 487, + 309 + ], + "score": 0.42, + "content": "{ \\bf W } { \\cdot } { \\Delta } S", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 307, + 136, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 131, + 320 + ], + "score": 0.56, + "content": "{ \\bf C } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 307, + 136, + 321 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 285, + 506, + 321 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 332, + 189, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 332, + 190, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 190, + 345 + ], + "score": 1.0, + "content": "6.2.1 BASELINES", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 504, + 364 + ], + "score": 1.0, + "content": "In addition to classic representations such as SBOW Freq and SBOW PPMI, we compare distribu-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 363, + 432, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 432, + 374 + ], + "score": 1.0, + "content": "tional inclusion vector embedding (DIVE) with additional 4 baselines in Table 4.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 351, + 504, + 374 + ] + }, + { + "type": "list", + "bbox": [ + 131, + 381, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 131, + 382, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 131, + 382, + 473, + 397 + ], + "score": 1.0, + "content": "• SBOW PPMI with additional frequency weighting (PPMI w/ FW). 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K-means hashing creates a", + "type": "text" + }, + { + "bbox": [ + 426, + 537, + 464, + 547 + ], + "score": 0.89, + "content": "N \\times 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 537, + 495, + 548 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 495, + 537, + 504, + 547 + ], + "score": 0.76, + "content": "G", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 141, + 546, + 239, + 560 + ], + "score": 1.0, + "content": "with orthonormal rows", + "type": "text" + }, + { + "bbox": [ + 240, + 547, + 285, + 558 + ], + "score": 0.89, + "content": "\\tilde { G } ^ { T } G = I )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 546, + 333, + 560 + ], + "score": 1.0, + "content": ", where the", + "type": "text" + }, + { + "bbox": [ + 333, + 547, + 354, + 559 + ], + "score": 0.89, + "content": "( i , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 546, + 462, + 560 + ], + "score": 1.0, + "content": "th element is 0 if the word", + "type": "text" + }, + { + "bbox": [ + 462, + 548, + 468, + 557 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "does not", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 141, + 557, + 211, + 571 + ], + "score": 1.0, + "content": "belong to cluster", + "type": "text" + }, + { + "bbox": [ + 211, + 559, + 218, + 568 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 557, + 291, + 571 + ], + "score": 1.0, + "content": ". The orthonormal", + "type": "text" + }, + { + "bbox": [ + 292, + 559, + 301, + 568 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "is also an approximated solution of a type of NMF", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 144, + 568, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 144, + 569, + 196, + 580 + ], + "score": 0.92, + "content": "( M _ { c } \\overset { \\mathbf { \\backsimeq } } { \\approx } F G ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 568, + 506, + 582 + ], + "score": 1.0, + "content": ") (Ding et al., 2005). Hashing context vectors into topic vectors can be written", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 140, + 578, + 246, + 592 + ], + "spans": [ + { + "bbox": [ + 140, + 578, + 153, + 592 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 154, + 579, + 240, + 591 + ], + "score": 0.92, + "content": "M _ { c } G \\approx F G ^ { T } G = F", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 578, + 246, + 592 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + } + ], + "index": 23, + "bbox_fs": [ + 131, + 382, + 506, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 504, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 343, + 613 + ], + "score": 1.0, + "content": "In the experiment, we also tried to apply a constant", + "type": "text" + }, + { + "bbox": [ + 343, + 601, + 371, + 613 + ], + "score": 0.91, + "content": "\\log ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 600, + 504, + 613 + ], + "score": 1.0, + "content": "shifting to SBOW PPMI (i.e.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 611, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 107, + 612, + 208, + 624 + ], + "score": 0.87, + "content": "\\operatorname* { m a x } ( P M I - \\log ( k ) , 0 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 611, + 394, + 626 + ], + "score": 1.0, + "content": "). We found that the performance degrades as", + "type": "text" + }, + { + "bbox": [ + 395, + 613, + 402, + 622 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 611, + 505, + 626 + ], + "score": 1.0, + "content": "increases. Similarly, ap-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 622, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 622, + 468, + 637 + ], + "score": 1.0, + "content": "plying PMI filter to SBOW PPMI (set context feature to be 0 if the value is lower than", + "type": "text" + }, + { + "bbox": [ + 468, + 623, + 502, + 635 + ], + "score": 0.89, + "content": "\\log ( k _ { f } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 622, + 506, + 637 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 333, + 646 + ], + "score": 1.0, + "content": "usually makes the performances worse, especially when", + "type": "text" + }, + { + "bbox": [ + 333, + 634, + 345, + 646 + ], + "score": 0.89, + "content": "k _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "is large. Applying PMI filter to SBOW", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "Freq only makes its performances closer to (but still much worse than) SBOW PPMI, so we omit", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 655, + 191, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 191, + 668 + ], + "score": 1.0, + "content": "this baseline as well.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 600, + 506, + 668 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 679, + 260, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 679, + 262, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 262, + 692 + ], + "score": 1.0, + "content": "6.2.2 RESULTS AND DISCUSSIONS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "In Table 4, we first confirm the finding of the previous review study of Santus et al. (2017): there is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "no single hypernymy scoring function which always outperforms others. One of the main reasons", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "is that different datasets collect negative samples differently. This is also why we evaluate our", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "method on many datasets to make sure our conclusions hold in general. For example, if negative", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "samples come from random word pairs (e.g. WordNet dataset), a symmetric similarity measure is", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "already a pretty good scoring function. On the other hand, negative samples come from related or", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 472, + 504, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 504, + 486 + ], + "score": 1.0, + "content": "similar words in HyperNet, EVALution, Lenci/Benotto, and Weeds, so only computing generality", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "difference leads to the best (or close to the best) performance. The negative samples in many datasets", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 495, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 506 + ], + "score": 1.0, + "content": "are composed of both random samples and similar words (such as BLESS), so the combination of", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 506, + 369, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 369, + 518 + ], + "score": 1.0, + "content": "similarity and generality difference yields the most stable results.", + "type": "text", + "cross_page": true + } + ], + "index": 15 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 155, + 504, + 392 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 142, + 93 + ], + "score": 1.0, + "content": "Table 4:", + "type": "text" + }, + { + "bbox": [ + 142, + 81, + 164, + 91 + ], + "score": 0.43, + "content": "\\mathbf { A P } @", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 79, + 177, + 93 + ], + "score": 1.0, + "content": "all", + "type": "text" + }, + { + "bbox": [ + 177, + 81, + 192, + 91 + ], + "score": 0.77, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "of 10 datasets. The box at lower right corner compares the micro average AP", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 505, + 104 + ], + "score": 1.0, + "content": "across all 10 datasets. Numbers in different rows come from different feature or embedding spaces.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 114 + ], + "score": 1.0, + "content": "Numbers in different columns come from different datasets and unsupervised scoring functions. We", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "also present the micro average AP across the first 4 datasets (BLESS, EVALution, Lenci/Benotto", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "score": 1.0, + "content": "and Weeds). All wiki means SBOW using PPMI features trained on the whole WaCkypedia. 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AP@all(%)BLESSEVALutionLenci/Benotto
CDEAL1△SW△SC△SCDE AL1△SW△SC△SCDEAL1△SW△SC△S
SBOWFreq6.37.3 5.6 5.611.0 17.25.9 15.335.3 30.432.6 27.736.233.036.351.847.651.051.851.1
PPMI13.65.127.534.131.934.347.239.750.851.152.0
PPMI w/FW6.25.05.5 12.45.836.036.332.936.452.043.150.951.950.7
All wiki12.15.26.9 12.513.4 28.527.130.329.931.047.139.948.548.751.1
DIVEFull9.37.6 6.0 5.618.616.330.027.534.933.046.743.251.351.550.4
w/o PMI7.86.916.77.132.832.235.732.3 32.535.447.644.950.951.649.7
w/o FW9.06.26.27.324.325.022.923.523.938.838.138.238.238.4
Kmean (Freq NMF)6.57.3 5.610.95.833.727.236.233.036.249.642.551.051.851.2
7.3 WeedsMicro Average (4 datasets)Medical
AP@all(%)CDE AL1△SCDEAL1CDEAL1△SW·△SC△S
SBOWFreq69.558.0 68.8W△S 68.2C△S 68.423.121.8△S 22.9W△S 25.0C△S 23.019.419.214.118.415.3
PPMI61.050.370.369.269.324.7 17.922.328.127.823.48.713.220.124.4
PPMI w/FW67.652.269.468.767.723.2 18.222.925.822.922.810.613.718.617.0
All wiki61.348.670.070.423.425.822.38.912.217.621.1
Full59.255.068.517.7 19.821.7 22.824.6 28.927.611.79.313.721.419.2
DIVE w/o PMI60.456.469.7 69.368.6 68.665.5 64.822.1 22.221.023.110.78.413.319.816.2
w/o FW49.247.3 45.145.144.918.917.322.7 17.228.0 16.817.510.99.87.47.67.7
Kmean (Freq NMF)69.451.1 68.868.268.922.519.322.924.923.012.610.914.018.114.6
LEDSTM14
AP@all(%)AL1 △SCDE AL1Kotlerman 2010
SBOWFreqCDE70.4 70.7W·△S 83.3C△S 73.355.653.2△S 54.9W·△S 55.7C.△S 55.0CDE 35.9AL1 40.5△S 34.5W.△S 37.0C.△S 35.4
SBOWPPMI82.7 84.450.2 72.286.584.556.252.354.457.639.130.933.037.036.3
All wiki83.149.7 67.982.981.454.750.552.657.0 55.154.938.531.232.235.435.3
DIVE83.372.786.483.555.352.655.257.357.235.331.633.637.436.6 MicroAverage(10datasets)
AP@all (%)74.7HyperNetWordNet
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The negative samples in many datasets", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 495, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 506 + ], + "score": 1.0, + "content": "are composed of both random samples and similar words (such as BLESS), so the combination of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 506, + 369, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 369, + 518 + ], + "score": 1.0, + "content": "similarity and generality difference yields the most stable results.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "DIVE performs similar or better on all the scoring functions compared with SBOW consistently", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "across all datasets in Table 4, while using many fewer dimensions (see Table 6). Its results on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 436, + 557 + ], + "score": 1.0, + "content": "combination scoring functions outperform SBOW Freq. Meanwhile, its results on", + "type": "text" + }, + { + "bbox": [ + 437, + 545, + 456, + 555 + ], + "score": 0.9, + "content": "A L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "outperform", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 398, + 568 + ], + "score": 1.0, + "content": "SBOW PPMI. 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AP@all(%)BLESSEVALutionLenci/Benotto
CDEAL1△SW△SC△SCDE AL1△SW△SC△SCDEAL1△SW△SC△S
SBOWFreq6.37.3 5.6 5.611.0 17.25.9 15.335.3 30.432.6 27.736.233.036.351.847.651.051.851.1
PPMI13.65.127.534.131.934.347.239.750.851.152.0
PPMI w/FW6.25.05.5 12.45.836.036.332.936.452.043.150.951.950.7
All wiki12.15.26.9 12.513.4 28.527.130.329.931.047.139.948.548.751.1
DIVEFull9.37.6 6.0 5.618.616.330.027.534.933.046.743.251.351.550.4
w/o PMI7.86.916.77.132.832.235.732.3 32.535.447.644.950.951.649.7
w/o FW9.06.26.27.324.325.022.923.523.938.838.138.238.238.4
Kmean (Freq NMF)6.57.3 5.610.95.833.727.236.233.036.249.642.551.051.851.2
7.3 WeedsMicro Average (4 datasets)Medical
AP@all(%)CDE AL1△SCDEAL1CDEAL1△SW·△SC△S
SBOWFreq69.558.0 68.8W△S 68.2C△S 68.423.121.8△S 22.9W△S 25.0C△S 23.019.419.214.118.415.3
PPMI61.050.370.369.269.324.7 17.922.328.127.823.48.713.220.124.4
PPMI w/FW67.652.269.468.767.723.2 18.222.925.822.922.810.613.718.617.0
All wiki61.348.670.070.423.425.822.38.912.217.621.1
Full59.255.068.517.7 19.821.7 22.824.6 28.927.611.79.313.721.419.2
DIVE w/o PMI60.456.469.7 69.368.6 68.665.5 64.822.1 22.221.023.110.78.413.319.816.2
w/o FW49.247.3 45.145.144.918.917.322.7 17.228.0 16.817.510.99.87.47.67.7
Kmean (Freq NMF)69.451.1 68.868.268.922.519.322.924.923.012.610.914.018.114.6
LEDSTM14
AP@all(%)AL1 △SCDE AL1Kotlerman 2010
SBOWFreqCDE70.4 70.7W·△S 83.3C△S 73.355.653.2△S 54.9W·△S 55.7C.△S 55.0CDE 35.9AL1 40.5△S 34.5W.△S 37.0C.△S 35.4
SBOWPPMI82.7 84.450.2 72.286.584.556.252.354.457.639.130.933.037.036.3
All wiki83.149.7 67.982.981.454.750.552.657.0 55.154.938.531.232.235.435.3
DIVE83.372.786.483.555.352.655.257.357.235.331.633.637.436.6 MicroAverage(10datasets)
AP@all (%)74.7HyperNetWordNet
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Its results on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 436, + 557 + ], + "score": 1.0, + "content": "combination scoring functions outperform SBOW Freq. Meanwhile, its results on", + "type": "text" + }, + { + "bbox": [ + 437, + 545, + 456, + 555 + ], + "score": 0.9, + "content": "A L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "outperform", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 398, + 568 + ], + "score": 1.0, + "content": "SBOW PPMI. 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The best average", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 577, + 399, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 329, + 589 + ], + "score": 1.0, + "content": "performance on 4 and 10 datasets are both produced by", + "type": "text" + }, + { + "bbox": [ + 330, + 577, + 357, + 588 + ], + "score": 0.27, + "content": "{ \\bf W } { \\cdot } { \\Delta } { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 577, + 399, + 589 + ], + "score": 1.0, + "content": "on DIVE.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 522, + 506, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "SBOW PPMI improves the combination functions from SBOW Freq but sacrifices AP on the inclu-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "sion functions. It generally hurts performance to change the frequency sampling of PPMI (PPMI w/", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "FW) or compute SBOW PPMI on the whole WaCkypedia (all wiki) instead of the first 51.2 million", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 363, + 639 + ], + "score": 1.0, + "content": "tokens. The similar trend can also be seen in Table 5. Note that", + "type": "text" + }, + { + "bbox": [ + 364, + 627, + 384, + 638 + ], + "score": 0.91, + "content": "A L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "completely fails in HyperLex", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "dataset using SBOW PPMI, which suggests that PPMI might not necessarily preserve the distribu-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 650, + 487, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 487, + 661 + ], + "score": 1.0, + "content": "tional inclusion property, even though it can have good performance on combination functions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 594, + 505, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "Removing the PMI filter from DIVE slightly drops the overall precision while removing frequency", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "weights on shifted PMI (w/o FW) leads to poor performances. K-means (Freq NMF) produces", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 333, + 700 + ], + "score": 1.0, + "content": "similar AP compared with SBOW Freq, but has worse", + "type": "text" + }, + { + "bbox": [ + 333, + 688, + 352, + 699 + ], + "score": 0.9, + "content": "A L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "scores. Its best AP scores on differ-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 506, + 713 + ], + "score": 1.0, + "content": "ent datasets are also significantly worse than the best AP of DIVE. This means that only making", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "word2vec (skip-grams with negative sampling) non-negative or naively accumulating topic distribu-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 719, + 330, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 330, + 734 + ], + "score": 1.0, + "content": "tion in contexts cannot lead to satisfactory embeddings.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 664, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 114, + 101, + 298, + 153 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 128, + 80, + 284, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 79, + 285, + 94 + ], + "spans": [ + { + "bbox": [ + 127, + 79, + 204, + 94 + ], + "score": 1.0, + "content": "Table 5: Spearman", + "type": "text" + }, + { + "bbox": [ + 205, + 81, + 229, + 92 + ], + "score": 0.69, + "content": "\\rho \\left( \\% \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 79, + 285, + 94 + ], + "score": 1.0, + "content": "in HyperLex.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 114, + 101, + 298, + 153 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 101, + 298, + 153 + ], + "spans": [ + { + "bbox": [ + 114, + 101, + 298, + 153 + ], + "score": 0.975, + "html": "
Spearman p (%)HyperLex
CDEAL1△SW·△SC.△S
SBOWFreq31.719.627.629.627.3
SBOW PPMI28.1-2.331.834.334.5
All wiki25.3-2.228.030.531.0
DIVE28.918.731.233.332.8
", + "type": "table", + "image_path": "d2f6b59c7662383cdf7042bd246a8b24a9efa8ee9d769dc0770e202597b6bb21.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 114, + 101, + 298, + 114.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 114, + 114.0, + 298, + 127.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 114, + 127.0, + 298, + 140.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 114, + 140.0, + 298, + 153.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.25 + }, + { + "type": "table", + "bbox": [ + 321, + 120, + 488, + 145 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 88, + 504, + 111 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 304, + 88, + 504, + 100 + ], + "spans": [ + { + "bbox": [ + 304, + 88, + 504, + 100 + ], + "score": 1.0, + "content": "Table 6: The average number of non-zero dimen-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 304, + 100, + 482, + 111 + ], + "spans": [ + { + "bbox": [ + 304, + 100, + 482, + 111 + ], + "score": 1.0, + "content": "sions across all testing words in 10 datasets.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "table_body", + "bbox": [ + 321, + 120, + 488, + 145 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 321, + 120, + 488, + 145 + ], + "spans": [ + { + "bbox": [ + 321, + 120, + 488, + 145 + ], + "score": 0.964, + "html": "
SBOW FreqSBOWPPMIDIVE
5799380820
", + "type": "table", + "image_path": "6a5730fd56936ae1141a339855b3b312e943a22a9fa3dd62af50d2678c928f81.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 321, + 120, + 488, + 145 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 6.25 + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 506, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "Table 7: Spectral clustering on the non-zero DIVE dimensions of the query word. When the number", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "of clusters is set to be 2, we present the top 4 dimensions in each cluster, which have the highest", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "values on the query word embedding. Otherwise, the top 2 dimensions are presented. CID refers to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 446, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 446, + 218 + ], + "score": 1.0, + "content": "cluster ID. The top 5 words with the highest values of each dimension are presented.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "table", + "bbox": [ + 108, + 226, + 523, + 545 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 226, + 523, + 545 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 108, + 226, + 523, + 545 + ], + "spans": [ + { + "bbox": [ + 108, + 226, + 523, + 545 + ], + "score": 0.984, + "html": "
QueryCIDTop 5 words in the top dimensions
rock1element,gas,atom,rock,carbon find, specie,species,animal,birdsea,lake,river,area,water point,side,line, front,circle
2band,song,album,music,rock early,work,century,late,beginwrite,john,guitar,band,author include,several, show, television,film
bank1county,area,city, town,west building,build,house, palace, siteseveral,main, province, include,consist sea,lake,river,area,water
2money, tax,price, pay, income united, states,country, world, europecompany,corporation,system,agency, service state,palestinian,israel,right, palestine
apple1food, fruit,vegetable,meat, potato war,german,ii,germany,worldgoddess,zeus,god,hero,sauron write, john,guitar,band,author
2version, game,release,original, file system,architecture,develop,base,languagecar,company,sell, manufacturer,model include,several, show, television,film
star1film, role, production, play, stage wear,blue,color, instrument,redcharacter,series,game,novel, fantasy write, john,guitar,band,author
2element,gas,atom,rock,carbon give, term,vector,mass,momentumstar,orbit,sun,orbital,planet light, image,lens,telescope,camera
tank1tank,cylinder,wheel, engine,steel acid,carbon,product, use, zincindustry, export, industrial, economy,company network,user,server, datum, protocol
2army,force,infantry,military,battle however,attempt,result, despite, failaircraft,navy,missile,ship,flight war, german, ii, germany, world
race1win,world,cup,play,championshiptwo,one,three,four,another
2 3railway, line, train,road, railcar,company, sell, manufacturer,model
population,language, ethnic, native, peoplefemale,age,woman,male,household
run1system,architecture, develop,base,languageaccess, need, require,allow, program
2railway,line,train,road,railalso,well, several, early, see
3game,team,season,win,leaguegame,player,run,deal,baseball
tablet1bc, source, greek, ancient, datebook,publish,write,work,edition
2use,system,design,term,methodversion, game,release,original, file
3system,blood,vessel,artery,intestinepatient, symptom, treatment,disorder, may
", + "type": "table", + "image_path": "9fe846fb27dc83e25e6faa2374a99f9415236b010e8a162d9fcf7c1e7fc922e0.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 108, + 226, + 523, + 332.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 108, + 332.3333333333333, + 523, + 438.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 108, + 438.66666666666663, + 523, + 545.0 + ], + "spans": [], + "index": 14 + } + ] + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 573, + 377, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 379, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 379, + 588 + ], + "score": 1.0, + "content": "7 EXPERIMENT 4: WORD SENSE DISAMBIGUATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "In addition to hypernymy detection, Athiwaratkun & Wilson (2017) show that the mixture of Gaus-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "sian distributions can also be used to discover multiple senses of each word. In our qualitative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "experiment, we show that DIVE can achieve the similar goal without fixing the number of senses", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 637, + 231, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 231, + 651 + ], + "score": 1.0, + "content": "before training the embedding.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "Recall that each dimension roughly corresponds to one topic. Given a query word, the higher embed-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "ding value on a dimension implies higher likelihood to observe the word in the context of the topic.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "The embedding of a polysemy would have high values on different groups of topics/dimensions.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "This allows us to discover the senses by clustering the topics/dimensions of the polysemy. We use", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "the embedding values as the feature each dimension, compute the pairwise similarity between di-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "mensions, and apply spectral clustering (Stella & Shi, 2003) to group topics as shown in the Table 7.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 721, + 367, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 367, + 733 + ], + "score": 1.0, + "content": "See more implementation details in the supplementary materials.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + } + ], + "page_idx": 10, + "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": [ + 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": "table", + "bbox": [ + 114, + 101, + 298, + 153 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 128, + 80, + 284, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 79, + 285, + 94 + ], + "spans": [ + { + "bbox": [ + 127, + 79, + 204, + 94 + ], + "score": 1.0, + "content": "Table 5: Spearman", + "type": "text" + }, + { + "bbox": [ + 205, + 81, + 229, + 92 + ], + "score": 0.69, + "content": "\\rho \\left( \\% \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 79, + 285, + 94 + ], + "score": 1.0, + "content": "in HyperLex.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 114, + 101, + 298, + 153 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 101, + 298, + 153 + ], + "spans": [ + { + "bbox": [ + 114, + 101, + 298, + 153 + ], + "score": 0.975, + "html": "
Spearman p (%)HyperLex
CDEAL1△SW·△SC.△S
SBOWFreq31.719.627.629.627.3
SBOW PPMI28.1-2.331.834.334.5
All wiki25.3-2.228.030.531.0
DIVE28.918.731.233.332.8
", + "type": "table", + "image_path": "d2f6b59c7662383cdf7042bd246a8b24a9efa8ee9d769dc0770e202597b6bb21.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 114, + 101, + 298, + 114.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 114, + 114.0, + 298, + 127.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 114, + 127.0, + 298, + 140.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 114, + 140.0, + 298, + 153.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.25 + }, + { + "type": "table", + "bbox": [ + 321, + 120, + 488, + 145 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 88, + 504, + 111 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 304, + 88, + 504, + 100 + ], + "spans": [ + { + "bbox": [ + 304, + 88, + 504, + 100 + ], + "score": 1.0, + "content": "Table 6: The average number of non-zero dimen-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 304, + 100, + 482, + 111 + ], + "spans": [ + { + "bbox": [ + 304, + 100, + 482, + 111 + ], + "score": 1.0, + "content": "sions across all testing words in 10 datasets.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "table_body", + "bbox": [ + 321, + 120, + 488, + 145 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 321, + 120, + 488, + 145 + ], + "spans": [ + { + "bbox": [ + 321, + 120, + 488, + 145 + ], + "score": 0.964, + "html": "
SBOW FreqSBOWPPMIDIVE
5799380820
", + "type": "table", + "image_path": "6a5730fd56936ae1141a339855b3b312e943a22a9fa3dd62af50d2678c928f81.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 321, + 120, + 488, + 145 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 6.25 + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 506, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "Table 7: Spectral clustering on the non-zero DIVE dimensions of the query word. When the number", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "of clusters is set to be 2, we present the top 4 dimensions in each cluster, which have the highest", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "values on the query word embedding. Otherwise, the top 2 dimensions are presented. CID refers to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 446, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 446, + 218 + ], + "score": 1.0, + "content": "cluster ID. The top 5 words with the highest values of each dimension are presented.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 172, + 506, + 218 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 226, + 523, + 545 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 226, + 523, + 545 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 108, + 226, + 523, + 545 + ], + "spans": [ + { + "bbox": [ + 108, + 226, + 523, + 545 + ], + "score": 0.984, + "html": "
QueryCIDTop 5 words in the top dimensions
rock1element,gas,atom,rock,carbon find, specie,species,animal,birdsea,lake,river,area,water point,side,line, front,circle
2band,song,album,music,rock early,work,century,late,beginwrite,john,guitar,band,author include,several, show, television,film
bank1county,area,city, town,west building,build,house, palace, siteseveral,main, province, include,consist sea,lake,river,area,water
2money, tax,price, pay, income united, states,country, world, europecompany,corporation,system,agency, service state,palestinian,israel,right, palestine
apple1food, fruit,vegetable,meat, potato war,german,ii,germany,worldgoddess,zeus,god,hero,sauron write, john,guitar,band,author
2version, game,release,original, file system,architecture,develop,base,languagecar,company,sell, manufacturer,model include,several, show, television,film
star1film, role, production, play, stage wear,blue,color, instrument,redcharacter,series,game,novel, fantasy write, john,guitar,band,author
2element,gas,atom,rock,carbon give, term,vector,mass,momentumstar,orbit,sun,orbital,planet light, image,lens,telescope,camera
tank1tank,cylinder,wheel, engine,steel acid,carbon,product, use, zincindustry, export, industrial, economy,company network,user,server, datum, protocol
2army,force,infantry,military,battle however,attempt,result, despite, failaircraft,navy,missile,ship,flight war, german, ii, germany, world
race1win,world,cup,play,championshiptwo,one,three,four,another
2 3railway, line, train,road, railcar,company, sell, manufacturer,model
population,language, ethnic, native, peoplefemale,age,woman,male,household
run1system,architecture, develop,base,languageaccess, need, require,allow, program
2railway,line,train,road,railalso,well, several, early, see
3game,team,season,win,leaguegame,player,run,deal,baseball
tablet1bc, source, greek, ancient, datebook,publish,write,work,edition
2use,system,design,term,methodversion, game,release,original, file
3system,blood,vessel,artery,intestinepatient, symptom, treatment,disorder, may
", + "type": "table", + "image_path": "9fe846fb27dc83e25e6faa2374a99f9415236b010e8a162d9fcf7c1e7fc922e0.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 108, + 226, + 523, + 332.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 108, + 332.3333333333333, + 523, + 438.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 108, + 438.66666666666663, + 523, + 545.0 + ], + "spans": [], + "index": 14 + } + ] + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 573, + 377, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 379, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 379, + 588 + ], + "score": 1.0, + "content": "7 EXPERIMENT 4: WORD SENSE DISAMBIGUATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "In addition to hypernymy detection, Athiwaratkun & Wilson (2017) show that the mixture of Gaus-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "sian distributions can also be used to discover multiple senses of each word. In our qualitative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "experiment, we show that DIVE can achieve the similar goal without fixing the number of senses", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 637, + 231, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 231, + 651 + ], + "score": 1.0, + "content": "before training the embedding.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 605, + 505, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "Recall that each dimension roughly corresponds to one topic. Given a query word, the higher embed-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "ding value on a dimension implies higher likelihood to observe the word in the context of the topic.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "The embedding of a polysemy would have high values on different groups of topics/dimensions.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "This allows us to discover the senses by clustering the topics/dimensions of the polysemy. We use", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "the embedding values as the feature each dimension, compute the pairwise similarity between di-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "mensions, and apply spectral clustering (Stella & Shi, 2003) to group topics as shown in the Table 7.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 721, + 367, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 367, + 733 + ], + "score": 1.0, + "content": "See more implementation details in the supplementary materials.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 654, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "In the word sense disambiguation tasks, it is usually challenging to determine how many", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "senses/clusters each word should have. Many existing approaches fix the number of senses be-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "fore training the embedding (Tian et al., 2014; Athiwaratkun & Wilson, 2017). Neelakantan et al.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "(2014) make the number of clusters approximately proportional to the diversity of the context, but", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "the assumption does not always hold. Furthermore, the training process cannot capture different", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "granularity of senses. For instance, race in the car context could share the same sense with the race", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "in the game topic because they all mean contest, but the race in the car context actually refers to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "specific contest of speed. Therefore, they can also be viewed as separate senses (like the results in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "Table 7). This means the correct number of clusters is not unique, and the methods, which fixes the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 458, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 458, + 194 + ], + "score": 1.0, + "content": "cluster numbers, need to re-train the embedding many times to capture such granularity.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "In our approach, clustering dimensions is done after the training process of DIVE is completed, so it", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "is fairly efficient to change the cluster numbers and hierarchical clustering is also an option. Similar", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "to our method, Pelevina et al. (2016) also discover word senses by graph-based clustering. The main", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 257, + 243 + ], + "score": 1.0, + "content": "difference is that they cluster the top", + "type": "text" + }, + { + "bbox": [ + 257, + 233, + 264, + 241 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "words which are most related to the query word instead of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 298, + 256 + ], + "score": 1.0, + "content": "topics. However, choosing the hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 298, + 244, + 305, + 252 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 240, + 378, + 256 + ], + "score": 1.0, + "content": "is difficult. Large", + "type": "text" + }, + { + "bbox": [ + 378, + 244, + 385, + 252 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 240, + 506, + 256 + ], + "score": 1.0, + "content": "would make graph clustering", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 464, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 241, + 265 + ], + "score": 1.0, + "content": "algorithm inefficient, while small", + "type": "text" + }, + { + "bbox": [ + 242, + 255, + 249, + 263 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 253, + 464, + 265 + ], + "score": 1.0, + "content": "would make less frequent senses difficult to discover.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 288, + 208, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 210, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 210, + 303 + ], + "score": 1.0, + "content": "8 RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "Most previous unsupervised approaches focus on designing better hypernymy scoring functions for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "sparse bag of word (SBOW) features. They are well summarized in the recent study (Santus et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "2017). Santus et al. (2017) also evaluate the influence of different contexts, such as changing the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 349, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 365 + ], + "score": 1.0, + "content": "window size of contexts or incorporating dependency parsing information, but neglect scalability", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 247, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 247, + 373 + ], + "score": 1.0, + "content": "issues inherent to SBOW methods.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "A notable exception is the Gaussian embedding model (Vilnis & McCallum, 2015). The context dis-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 390, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 401 + ], + "score": 1.0, + "content": "tribution of each word is encoded as a multivariate Gaussian distribution, where the embeddings of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "hypernyms tend to have higher variance and overlap with the embedding of their hyponyms. How-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "ever, since a Gaussian distribution is normalized, it is difficult to retain frequency information during", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "the embedding process, and experiments on HyperLex (Vulic et al., 2016) demonstrate that a simple ´", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "baseline only relying on word frequency can achieve good results. Follow-up work models contexts", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "by a mixture of Gaussians (Athiwaratkun & Wilson, 2017) relaxing the unimodality assumption but", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 456, + 338, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 338, + 468 + ], + "score": 1.0, + "content": "achieves little improvement on hypernym detection tasks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 108, + 472, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 504, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 504, + 484 + ], + "score": 1.0, + "content": "Kiela et al. (2015) show that images retrieved by a search engine can be a useful source of infor-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "mation to determine the generality of lexicons, but the resources might not be available for some", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 495, + 250, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 250, + 507 + ], + "score": 1.0, + "content": "corpora such as scientific literature.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "Order embedding (Vendrov et al., 2016) is a supervised approach to encode many annotated hyper-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "nym pairs (e.g. all of the whole WordNet (Miller, 1995)) into a compact embedding space, where the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "embedding of a hypernym should be smaller than the embedding of its hyponym in every dimension.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "score": 1.0, + "content": "Our method learns embedding from raw text, where a hypernym embedding should be larger than", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "the embedding of its hyponym in every dimension. Thus, DIVE can be viewed as an unsupervised", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 565, + 265, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 265, + 579 + ], + "score": 1.0, + "content": "and reversed form of order embedding.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "Other semi-supervised hypernym detection methods aim to generalize from sets of annotated word", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "pairs using raw text corpora. The goal of HyperScore (Nguyen et al., 2017) is similar to our model:", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "the embedding of a hypernym should be similar to its hyponym but with higher magnitude. How-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "ever, their training process relies heavily on annotated hypernym pairs, and the performance drops", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "significantly when reducing the amount of supervision. In addition to context distributions, previ-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "ous work also leverages training data to discover useful text pattern indicating is-a relation (Shwartz", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "et al., 2016; Roller & Erk, 2016), but it remains challenging to increase recall of hypernym detection", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 424, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 424, + 673 + ], + "score": 1.0, + "content": "because commonsense facts like cat is-a animal might not appear in the corpus.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "Non-negative matrix factorization (NMF) has a long history in NLP, for example in the construction", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "of topic models (Pauca et al., 2004). Non-negative sparse embedding (NNSE) (Murphy et al., 2012)", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "and Faruqui et al. (2015) indicate that non-negativity can make embeddings more interpretable and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "improve word similarity evaluations. The sparse NMF is also shown to be effective in cross-lingual", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "lexical entailment tasks but does not necessarily improve monolingual hypernymy detection (Vyas &", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49 + } + ], + "page_idx": 11, + "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": [ + 301, + 751, + 311, + 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": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "In the word sense disambiguation tasks, it is usually challenging to determine how many", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "senses/clusters each word should have. 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Therefore, they can also be viewed as separate senses (like the results in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "Table 7). This means the correct number of clusters is not unique, and the methods, which fixes the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 458, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 458, + 194 + ], + "score": 1.0, + "content": "cluster numbers, need to re-train the embedding many times to capture such granularity.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 82, + 506, + 194 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "In our approach, clustering dimensions is done after the training process of DIVE is completed, so it", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "is fairly efficient to change the cluster numbers and hierarchical clustering is also an option. Similar", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "to our method, Pelevina et al. (2016) also discover word senses by graph-based clustering. The main", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 257, + 243 + ], + "score": 1.0, + "content": "difference is that they cluster the top", + "type": "text" + }, + { + "bbox": [ + 257, + 233, + 264, + 241 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "words which are most related to the query word instead of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 298, + 256 + ], + "score": 1.0, + "content": "topics. However, choosing the hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 298, + 244, + 305, + 252 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 240, + 378, + 256 + ], + "score": 1.0, + "content": "is difficult. Large", + "type": "text" + }, + { + "bbox": [ + 378, + 244, + 385, + 252 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 240, + 506, + 256 + ], + "score": 1.0, + "content": "would make graph clustering", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 464, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 241, + 265 + ], + "score": 1.0, + "content": "algorithm inefficient, while small", + "type": "text" + }, + { + "bbox": [ + 242, + 255, + 249, + 263 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 253, + 464, + 265 + ], + "score": 1.0, + "content": "would make less frequent senses difficult to discover.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 198, + 506, + 265 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 288, + 208, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 210, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 210, + 303 + ], + "score": 1.0, + "content": "8 RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "Most previous unsupervised approaches focus on designing better hypernymy scoring functions for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "sparse bag of word (SBOW) features. They are well summarized in the recent study (Santus et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "2017). Santus et al. (2017) also evaluate the influence of different contexts, such as changing the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 349, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 365 + ], + "score": 1.0, + "content": "window size of contexts or incorporating dependency parsing information, but neglect scalability", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 247, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 247, + 373 + ], + "score": 1.0, + "content": "issues inherent to SBOW methods.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 318, + 505, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "A notable exception is the Gaussian embedding model (Vilnis & McCallum, 2015). The context dis-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 390, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 401 + ], + "score": 1.0, + "content": "tribution of each word is encoded as a multivariate Gaussian distribution, where the embeddings of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "hypernyms tend to have higher variance and overlap with the embedding of their hyponyms. How-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "ever, since a Gaussian distribution is normalized, it is difficult to retain frequency information during", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "the embedding process, and experiments on HyperLex (Vulic et al., 2016) demonstrate that a simple ´", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "baseline only relying on word frequency can achieve good results. Follow-up work models contexts", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "by a mixture of Gaussians (Athiwaratkun & Wilson, 2017) relaxing the unimodality assumption but", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 456, + 338, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 338, + 468 + ], + "score": 1.0, + "content": "achieves little improvement on hypernym detection tasks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 379, + 506, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 472, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 504, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 504, + 484 + ], + "score": 1.0, + "content": "Kiela et al. (2015) show that images retrieved by a search engine can be a useful source of infor-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "mation to determine the generality of lexicons, but the resources might not be available for some", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 495, + 250, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 250, + 507 + ], + "score": 1.0, + "content": "corpora such as scientific literature.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 472, + 505, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "Order embedding (Vendrov et al., 2016) is a supervised approach to encode many annotated hyper-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "nym pairs (e.g. all of the whole WordNet (Miller, 1995)) into a compact embedding space, where the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "embedding of a hypernym should be smaller than the embedding of its hyponym in every dimension.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "score": 1.0, + "content": "Our method learns embedding from raw text, where a hypernym embedding should be larger than", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "the embedding of its hyponym in every dimension. 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DatasetHyperLexEVALution LenciBenotto WeedsMedical
MetricSpearman pAP@allF1
Baselines(#Training Hypernymy)HyperScore(1337)H-feature (897)
303944.858.526
DIVE+C·△S (0)34.533.852.970.025.3
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QueryTop30 generalwords
use large american design studyname state life work leadsystem group may produce typeinclude power small control peoplebase death find great highcity form body write create
speciesspecie human gene fish evidencespecies bird tree disease breedanimal genus name live proteinfind family genetic food wildplant organism study cell similarmay suggest occur mammal fossil
systemsystem standard allow process networkuse type function code filedesign computer datum via developmentprovide application device base serviceoperate develop control program transportmodel method information software law
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DatasetHyperLexEVALution LenciBenotto WeedsMedical
MetricSpearman pAP@allF1
Baselines(#Training Hypernymy)HyperScore(1337)H-feature (897)
303944.858.526
DIVE+C·△S (0)34.533.852.970.025.3
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QueryTop30 generalwords
use large american design studyname state life work leadsystem group may produce typeinclude power small control peoplebase death find great highcity form body write create
speciesspecie human gene fish evidencespecies bird tree disease breedanimal genus name live proteinfind family genetic food wildplant organism study cell similarmay suggest occur mammal fossil
systemsystem standard allow process networkuse type function code filedesign computer datum via developmentprovide application device base serviceoperate develop control program transportmodel method information software law
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Micro Average (1O datasets)
SBOW Freq+ SLQS Sub64.4SBOW Freq +△S66.8SBOW PPMI+△S66.8DIVE+△S67.0
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BLESSEVALutionLenci/BenottoWeedsAvg (4 datasets)
N 265540OV 1507N 13675OOV 2475N 5010OOV 1464N 29280OV 643N 48167OOv 6089
MedicalKotlerman 2010
NLEDSTM14HyperNet
12602OOV 3711NOOV 28NOOV 178NOOV 89NOOV 9424
27702188294017670
WordNetAvg(10 datasets)HyperLex
N OOVNN
8000 35969433726160OV 59
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Micro Average (1O datasets)
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BLESSEVALutionLenci/BenottoWeedsAvg (4 datasets)
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NLEDSTM14HyperNet
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All", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 437, + 365, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 365, + 448 + ], + "score": 1.0, + "content": "part-of-speech (POS) tags in the experiments come from NLTK.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 404, + 506, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 504, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 178, + 465 + ], + "score": 1.0, + "content": "The window size", + "type": "text" + }, + { + "bbox": [ + 179, + 453, + 196, + 465 + ], + "score": 0.89, + "content": "| W |", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 454, + 504, + 465 + ], + "score": 1.0, + "content": "of SBOW, DIVE, and GE are set as 20 (left 10 words and right 10 words).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 465, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 476 + ], + "score": 1.0, + "content": "For DIVE, the number of epochs is 15, the learning rate is 0.001, the batch size is 128, the threshold", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 475, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 159, + 488 + ], + "score": 1.0, + "content": "in PMI filter", + "type": "text" + }, + { + "bbox": [ + 159, + 476, + 171, + 488 + ], + "score": 0.89, + "content": "k _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 475, + 444, + 488 + ], + "score": 1.0, + "content": "is set to be 30, and the ratio between negative and positive samples", + "type": "text" + }, + { + "bbox": [ + 445, + 476, + 457, + 487 + ], + "score": 0.32, + "content": "{ \\mathrm { ( k ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 475, + 504, + 488 + ], + "score": 1.0, + "content": "is 1.5. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "hyper-parameters of DIVE were decided based on the performance of HyperNet training set. The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 425, + 510 + ], + "score": 1.0, + "content": "window size of skip-grams (word2vec) is 10. The number of negative samples", + "type": "text" + }, + { + "bbox": [ + 425, + 498, + 441, + 509 + ], + "score": 0.79, + "content": "( k ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "in skip-gram is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 508, + 464, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 464, + 521 + ], + "score": 1.0, + "content": "set as 5. When composing skip gram into phrase embedding, average embedding is used.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 453, + 506, + 521 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "For Gaussian embedding (GE), the number of mixture is 1, the number of dimension is 100, the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "learning rate is 0.01, the lowest variance is 0.1, the highest variance is 100, the highest Gaus-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "sian mean is 10, and other hyper-parameters are the default value in https://github.com/", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "benathi/word2gm. 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A simple way to prepare the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 664, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 215, + 676 + ], + "score": 1.0, + "content": "feature of each dimension", + "type": "text" + }, + { + "bbox": [ + 215, + 664, + 237, + 676 + ], + "score": 0.91, + "content": "f ( c _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 665, + 442, + 676 + ], + "score": 1.0, + "content": "is to use the embedding values in that dimension", + "type": "text" + }, + { + "bbox": [ + 442, + 664, + 464, + 676 + ], + "score": 0.91, + "content": "\\mathbf { w } [ c _ { i } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 665, + 506, + 676 + ], + "score": 1.0, + "content": "of all the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 675, + 272, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 205, + 688 + ], + "score": 1.0, + "content": "words in our vocabulary", + "type": "text" + }, + { + "bbox": [ + 206, + 676, + 234, + 686 + ], + "score": 0.9, + "content": "w \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 675, + 272, + 688 + ], + "score": 1.0, + "content": ". 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2system,rcieturvelope,gefunctional,rnt,ocsing,
3also, well,svral,rlyfall, eventually, main,ise,mosty
4partincorporate,ge,iead,oing,dd
5star,orbit,snbital,bright,posion,turieractio
6several,icdesignate,ist,iss,bch,i
7science,plosophy, thory,plosophr,rethical,dvocte,oic,bic,
8version,game,release,original,ilecassette, rtual,code,project, kb
9electron,ttric,citanode,wire,ac,perform,eistor
10tank, cylinder,heel,gialuminumtic,ott
11school,itildoctorateldert
12network,r,er,dtum,prltechnologyoutgnt,crooft,
13high,low,turergyatmosphric,tod,,io
14acid, carbon, product, ue,incph,monoided
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DatasetBLESSEVALutionLenciBenottoWeedsMedicalLEDS
Random Word2Vec+C5.3 9.226.6 25.441.2 40.851.4 51.68.5 11.250.5 71.8
GE+C GE+KL10.5 7.626.7 29.643.352.014.969.7
DIVE+C.△S45.151.315.764.6
16.333.050.465.525.383.5
DatasetTM14Kotlerman 2010HyperNetWordNetHyperLex
Random52.030.824.555.20
Word2Vec+C52.1
39.520.763.016.3
GE+C53.936.021.658.216.4
GE+KL52.039.423.754.49.6
DIVE+C·△S57.236.641.960.932.8
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DatasetBLESS EVALutionLenciBenottoWeedsMedical
MetricAP@allF1
BaselinesinvCLAPSynCDECosine
5.135.338.244.123.1
DIVE+C·△S16.333.050.465.525.3
DIVE +W·△S18.632.351.568.625.7
DatasetLEDSTM14Kotlerman 2010HyperNetHyperLex
MetricAP@allF1Spearman p
BaselinesbalAPincSLQSFreq ratio
73563722.827.9
DIVE+C·△S83.557.236.641.932.8
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AP@all(%)BLESSEVALutionLenci/Benotto
CDEAL1△SW△SC△SCDE AL1△SW△SC△SCDEAL1△SW△SC△S
SBOWFreq6.37.3 5.6 5.611.0 17.25.9 15.335.3 30.432.6 27.736.233.036.351.847.651.051.851.1
PPMI13.65.127.534.131.934.347.239.750.851.152.0
PPMI w/FW6.25.05.5 12.45.836.036.332.936.452.043.150.951.950.7
All wiki12.15.26.9 12.513.4 28.527.130.329.931.047.139.948.548.751.1
DIVEFull9.37.6 6.0 5.618.616.330.027.534.933.046.743.251.351.550.4
w/o PMI7.86.916.77.132.832.235.732.3 32.535.447.644.950.951.649.7
w/o FW9.06.26.27.324.325.022.923.523.938.838.138.238.238.4
Kmean (Freq NMF)6.57.3 5.610.95.833.727.236.233.036.249.642.551.051.851.2
7.3 WeedsMicro Average (4 datasets)Medical
AP@all(%)CDE AL1△SCDEAL1CDEAL1△SW·△SC△S
SBOWFreq69.558.0 68.8W△S 68.2C△S 68.423.121.8△S 22.9W△S 25.0C△S 23.019.419.214.118.415.3
PPMI61.050.370.369.269.324.7 17.922.328.127.823.48.713.220.124.4
PPMI w/FW67.652.269.468.767.723.2 18.222.925.822.922.810.613.718.617.0
All wiki61.348.670.070.423.425.822.38.912.217.621.1
Full59.255.068.517.7 19.821.7 22.824.6 28.927.611.79.313.721.419.2
DIVE w/o PMI60.456.469.7 69.368.6 68.665.5 64.822.1 22.221.023.110.78.413.319.816.2
w/o FW49.247.3 45.145.144.918.917.322.7 17.228.0 16.817.510.99.87.47.67.7
Kmean (Freq NMF)69.451.1 68.868.268.922.519.322.924.923.012.610.914.018.114.6
LEDSTM14
AP@all(%)AL1 △SCDE AL1Kotlerman 2010
SBOWFreqCDE70.4 70.7W·△S 83.3C△S 73.355.653.2△S 54.9W·△S 55.7C.△S 55.0CDE 35.9AL1 40.5△S 34.5W.△S 37.0C.△S 35.4
SBOWPPMI82.7 84.450.2 72.286.584.556.252.354.457.639.130.933.037.036.3
All wiki83.149.7 67.982.981.454.750.552.657.0 55.154.938.531.232.235.435.3
DIVE83.372.786.483.555.352.655.257.357.235.331.633.637.436.6 MicroAverage(10datasets)
AP@all (%)74.7HyperNetWordNet
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QueryCIDTop 5 words in the top dimensions
rock1element,gas,atom,rock,carbon find, specie,species,animal,birdsea,lake,river,area,water point,side,line, front,circle
2band,song,album,music,rock early,work,century,late,beginwrite,john,guitar,band,author include,several, show, television,film
bank1county,area,city, town,west building,build,house, palace, siteseveral,main, province, include,consist sea,lake,river,area,water
2money, tax,price, pay, income united, states,country, world, europecompany,corporation,system,agency, service state,palestinian,israel,right, palestine
apple1food, fruit,vegetable,meat, potato war,german,ii,germany,worldgoddess,zeus,god,hero,sauron write, john,guitar,band,author
2version, game,release,original, file system,architecture,develop,base,languagecar,company,sell, manufacturer,model include,several, show, television,film
star1film, role, production, play, stage wear,blue,color, instrument,redcharacter,series,game,novel, fantasy write, john,guitar,band,author
2element,gas,atom,rock,carbon give, term,vector,mass,momentumstar,orbit,sun,orbital,planet light, image,lens,telescope,camera
tank1tank,cylinder,wheel, engine,steel acid,carbon,product, use, zincindustry, export, industrial, economy,company network,user,server, datum, protocol
2army,force,infantry,military,battle however,attempt,result, despite, failaircraft,navy,missile,ship,flight war, german, ii, germany, world
race1win,world,cup,play,championshiptwo,one,three,four,another
2 3railway, line, train,road, railcar,company, sell, manufacturer,model
population,language, ethnic, native, peoplefemale,age,woman,male,household
run1system,architecture, develop,base,languageaccess, need, require,allow, program
2railway,line,train,road,railalso,well, several, early, see
3game,team,season,win,leaguegame,player,run,deal,baseball
tablet1bc, source, greek, ancient, datebook,publish,write,work,edition
2use,system,design,term,methodversion, game,release,original, file
3system,blood,vessel,artery,intestinepatient, symptom, treatment,disorder, may
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QueryTop30 generalwords
use large american design studyname state life work leadsystem group may produce typeinclude power small control peoplebase death find great highcity form body write create
speciesspecie human gene fish evidencespecies bird tree disease breedanimal genus name live proteinfind family genetic food wildplant organism study cell similarmay suggest occur mammal fossil
systemsystem standard allow process networkuse type function code filedesign computer datum via developmentprovide application device base serviceoperate develop control program transportmodel method information software law
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DatasetHyperLexEVALution LenciBenotto WeedsMedical
MetricSpearman pAP@allF1
Baselines(#Training Hypernymy)HyperScore(1337)H-feature (897)
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BLESSEVALutionLenci/BenottoWeedsAvg (4 datasets)
N 265540OV 1507N 13675OOV 2475N 5010OOV 1464N 29280OV 643N 48167OOv 6089
MedicalKotlerman 2010
NLEDSTM14HyperNet
12602OOV 3711NOOV 28NOOV 178NOOV 89NOOV 9424
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WordNetAvg(10 datasets)HyperLex
N OOVNN
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Micro Average (1O datasets)
SBOW Freq+ SLQS Sub64.4SBOW Freq +△S66.8SBOW PPMI+△S66.8DIVE+△S67.0
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a/parse/train/Wga_hrCa3P3/Wga_hrCa3P3_content_list.json b/parse/train/Wga_hrCa3P3/Wga_hrCa3P3_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d80c806b606ef751c10d20ca81d34d8c51117241 --- /dev/null +++ b/parse/train/Wga_hrCa3P3/Wga_hrCa3P3_content_list.json @@ -0,0 +1,2554 @@ +[ + { + "type": "text", + "text": "CONTRASTIVE LEARNING WITH ADVERSARIAL PERTURBATIONS FOR CONDITIONAL TEXT GENERATION ", + "text_level": 1, + "bbox": [ + 174, + 99, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Seanie Lee1∗, Dong Bok Lee1∗, Sung Ju Hwang1,2 KAIST1, AITRICS2, South Korea {lsnfamily02, markhi, sjhwang82}@kaist.ac.kr ", + "bbox": [ + 181, + 169, + 611, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 250, + 544, + 265 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, sequence-to-sequence (seq2seq) models with the Transformer architecture have achieved remarkable performance on various conditional text generation tasks, such as machine translation. However, most of them are trained with teacher forcing with the ground truth label given at each time step, without being exposed to incorrectly generated tokens during training, which hurts its generalization to unseen inputs, that is known as the “exposure bias” problem. In this work, we propose to mitigate the conditional text generation problem by contrasting positive pairs with negative pairs, such that the model is exposed to various valid or incorrect perturbations of the inputs, for improved generalization. However, training the model with na¨ıve contrastive learning framework using random non-target sequences as negative examples is suboptimal, since they are easily distinguishable from the correct output, especially so with models pretrained with large text corpora. Also, generating positive examples requires domain-specific augmentation heuristics which may not generalize over diverse domains. To tackle this problem, we propose a principled method to generate positive and negative samples for contrastive learning of seq2seq models. Specifically, we generate negative examples by adding small perturbations to the input sequence to minimize its conditional likelihood, and positive examples by adding large perturbations while enforcing it to have a high conditional likelihood. Such “hard” positive and negative pairs generated using our method guides the model to better distinguish correct outputs from incorrect ones. We empirically show that our proposed method significantly improves the generalization of the seq2seq on three text generation tasks — machine translation, text summarization, and question generation. ", + "bbox": [ + 233, + 280, + 764, + 599 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 627, + 336, + 642 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The sequence-to-sequence (seq2seq) models (Sutskever et al., 2014), which learn to map an arbitrary-length input sequence to another arbitrary-length output sequence, have successfully tackled a wide range of language generation tasks. Early seq2seq models have used recurrent neural networks to encode and decode sequences, leveraging attention mechanism (Bahdanau et al., 2015) that allows the decoder to attend to a specific token in the input sequence to capture long-term dependencies between the source and target sequences. Recently, the Transformer (Vaswani et al., 2017), which is an all-attention model that effectively captures long-term relationships between tokens in the input sequence as well as across input and output sequences, has become the de facto standard for most of the text generation tasks due to its impressive performance. Moreover, Transformerbased language models trained on large text corpora (Dong et al., 2019; Raffel et al., 2020; Lewis et al., 2020) have shown to significantly improve the model performance on text generation tasks. ", + "bbox": [ + 174, + 659, + 825, + 810 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, a crucial limitation of seq2seq models is that they are mostly trained only with teacher forcing, where ground truth is provided at each time step and thus never exposed to incorrectly generated tokens during training (Fig. 1-(a)), which hurts its generalization. This problem is known as the “exposure bias” problem (Ranzato et al., 2016) and often results in the generation of lowquality texts on unseen inputs. Several prior works tackle the problem, such as using reinforcement learning (RL) to maximize non-differentiable reward (Bahdanau et al., 2017; Paulus et al., 2018). ", + "bbox": [ + 173, + 818, + 825, + 901 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/3152b03d6604aaa99473aaf4ecace4d924367395828fc4b243fd8556ad897871.jpg", + "image_caption": [ + "Figure 1: Concept. (a) Training seq2seq with teacher forcing. (b) Na¨ıve contrastive learning with randomly sampled negative examples. (c) Our method, CLAPS, which generates hard negative and positive examples. " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 820, + 238 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Another approach is to use RL or gumbel softmax (Jang et al., 2017) to match the distribution of generated sentences to that of the ground truth, in which case the reward is the discriminator output from a Generative Adversarial Network (GAN) (Zhang et al., 2018; 2017; Yu et al., 2017). Although the aforementioned approaches improve the performance of the seq2seq models on text generation tasks, they either require a vast amount of effort in tuning hyperparameters or stabilize training. ", + "bbox": [ + 173, + 277, + 823, + 348 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we propose to mitigate the exposure bias problem with a simple yet effective approach, in which we contrast a positive pair of input and output sequence to negative pairs, to expose the model to various valid or incorrect sentences. Na¨ıvely, we can construct negative pairs by simply using random nontarget sequences from the batch (Chen et al., 2020). However, such a na¨ıve construction yields meaningless negative examples that are already well-discriminated in the embedding space (Fig. 1-(b)), which we highlight as the reason why existing methods (Chen et al., 2020) require large batch size. This is clearly shown in Fig. 2, where a large portion of positive-negative pairs can be easily discriminated without any training, which gets worse as the batch size decreases as it will reduce the chance to have meaningfully difficult examples in the batch. Moreover, discriminating positive and na¨ıve negative pairs becomes even more easier for models pretrained on large text corpora. ", + "bbox": [ + 174, + 354, + 593, + 534 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/876f3fbd65a3f50ad0917ec686970cfba6abaa84b103c8f8f2f38c365038e298.jpg", + "image_caption": [ + "Figure 2: Accuracy of classifying a positive pair from negative pairs varying batch size without any training. " + ], + "image_footnote": [], + "bbox": [ + 607, + 410, + 821, + 530 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 535, + 820, + 563 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To resolve this issue, we propose principled approaches to automatically generate negative and positive pairs for constrastive learning, which we refer to as Contrastive Learning with Adversarial Perturbation for Seq2seq learning (CLAPS). Specifically, we generate a negative example by adding a small perturbation to the hidden representation of the target sequence, such that its conditional likelihood is minimized (Denoted as the red circle in Fig. 1-(c)). Conversely, we construct an additional positive example (Denoted as green circle in Fig. 1-(c)) by adding a large amount of perturbation to the hidden representation of target sequence such that the perturbed sample is far away from the source sequence in the embedding space, while enforcing it to have high conditional likelihood by minimizing Kullback-Leibler (KL) divergence between the original conditional distribution and perturbed conditional distribution. This will yield a negative example that is very close to the original representation of target sequence in the embedding space but is largely dissimilar in the semantics, while the generated positive example is far away from the original input sequence but has the same semantic as the target sequence. This will generate difficult examples that the model fails to correctly discriminate (Fig. 1-(c), Fig.2), helping it learn with more meaningful pairs. ", + "bbox": [ + 173, + 570, + 825, + 765 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To verify the efficacy of our method, we empirically show that it significantly improves the performance of seq2seq model on three conditional text generation tasks, namely machine translation, text summarization and question generation. Our contribution in this work is threefold: ", + "bbox": [ + 174, + 771, + 825, + 813 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• To mitigate the exposure bias problem, we propose a contrastive learning framework for conditional sequence generation, which contrasts a positive pair of source and target sentence to negative pairs in the latent embedding space, to expose the model to various valid or incorrect outputs. • To tackle the ineffectiveness of conventional approach for constructing negative and positive examples for contrastive learning, we propose a principled method to automatically generate negative and positive pairs, that are more difficult and allows to learn more meaningful representations. ", + "bbox": [ + 173, + 823, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We show that our proposed method, CLAPS, significantly improves the performance of seq2seq model on three different tasks: machine translation, text summarization, and question generation. ", + "bbox": [ + 174, + 103, + 823, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 152, + 344, + 169 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Exposure Bias There are several prior works to tackle the exposure bias (Ranzato et al., 2016). Bengio et al. (2015) introduce scheduled sampling where the model is initially guided with the true previous tokens but uses the tokens generated by the seq2seq model as the conditional input for the next token, as training goes on. Paulus et al. (2018); Bahdanau et al. (2017) leverage RL to maximize non-differentiable rewards, so it enables to penalize the model for incorrectly generated sentences. Another works (Zhang et al., 2017; 2018; Yu et al., 2017) train GANs to match the distribution of generated sequences to that of ground truth. Since sampling tokens from the generator is not differentiable, they resort RL or gumbel-softmax to train the networks in end-to-end fashion. However, they require either a large amount of effort to tune hyperparameters or stabilize training. However, Choshen et al. (2020) show that RL for machine translation does not optimize the expected reward and the performance gain is attributed to the unrelated effects such as increasing the peakiness of the output distribution. Moreover, (Caccia et al., 2019) show that by tuning the temperature parameter, the language models trained with MLE can be tuned to outperform GAN-based text generation models. ", + "bbox": [ + 174, + 185, + 825, + 380 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Adversarial Perturbation Many existing works, such as (Madry et al., 2018), address the robustness of neural networks to adversarial examples, which are generated by applying a small perturbations to the input samples. While adversarial robustness has been mostly explored in image domains, Miyato et al. (2017) adopted adversarial training to text domains. However instead of targeting robustness to perturbed samples, they utilize the adversarial examples as augmented data, and enforce consistency across the predictions across original unlabeled example and its perturbation, for semisupervised learning. Recently Zhu et al. (2019) and Jiang et al. (2020) leverage adversarial training to induce the smoothness of text classifiers, to prevent overfitting to training samples. While they are relevant to ours, these methods do not have the notion of positive and negative examples as they do not consider contrastive learning, and only target text classification. Moreover, they are computationally prohibitive since they use PGD for adversarial training, which requires iterative optimization for each individual sample. Recently, Aghajanyan et al. (2020) propose a simpler yet effective method based on Gaussian noise perturbation to regularize neural networks without expensive PGD steps, which is shown to outperform methods from Zhu et al. (2019) and Jiang et al. (2020). Although our work is similar to these prior works in that we add perturbations to the text embeddings, note that we used the adversarially-generated samples as negative examples of our contrastive learning framework rather than trying to learn the model to be robust to them. ", + "bbox": [ + 174, + 387, + 825, + 622 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Contrastive Learning Contrastive learning has been widely used. It is to learn a representation by contrasting positive pairs and negative pairs. Chopra et al. (2005); Weinberger & Saul (2009); Schroff et al. (2015) leverage a triplet loss to separate positive examples from negative examples in metric learning. Chen et al. (2020) show that contrastive learning can boost the performance of selfsupervised and semi-supervised learning in computer vison tasks. In natural language processing (NLP), contrastive learning has been widely used. In Word2Vec (Mikolov et al., 2013), neighbouring words are predicted from context with noise-contrastive estimation (Gutmann & Hyvarinen, 2012). ¨ Beyond word representation, Logeswaran & Lee (2018) sample two contiguous sentences for positive pairs and the sentences from other document as negative pairs. They constrast positive and negative pairs to learn sentence representation. Moreover, contrastive learning has been investigated in various NLP tasks — language modeling (Huang et al., 2018), unsupervised word alignment (Liu & Sun, 2015), caption generation (Mao et al., 2016; Vedantam et al., 2017), and machine translation (Yang et al., 2019). ", + "bbox": [ + 174, + 630, + 825, + 810 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 METHOD ", + "text_level": 1, + "bbox": [ + 174, + 832, + 282, + 848 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 BACKGROUND: CONDITIONAL TEXT GENERATION", + "text_level": 1, + "bbox": [ + 174, + 864, + 563, + 880 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The goal of conditional text generation with a seq2seq model is to generate an output text sequence $\\mathbf { y } ^ { ( i ) } = ( y _ { 1 } ^ { ( i ) } , \\dots , y _ { T } ^ { ( i ) } )$ with length $T$ conditioned on the input text sequence $\\mathbf { x } ^ { ( i ) } \\stackrel { - } { = } ( x _ { 1 } ^ { ( i ) } , \\cdot \\cdot \\cdot , x _ { L } ^ { ( i ) } )$ ", + "bbox": [ + 173, + 892, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "with length $L$ . A typical approach to the conditional text generation is to leverage the encoderdecoder architecture to parameterize the conditional distribution. We maximize the conditional log likelihood $\\log p _ { \\theta } ( \\mathbf { y } | \\mathbf { x } )$ for a given $N$ observations $\\{ ( \\mathbf { x } ^ { ( i ) } , \\mathbf { y } ^ { ( i ) } ) \\} _ { i = 1 } ^ { N }$ as follows: ", + "bbox": [ + 176, + 103, + 823, + 148 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/028420f1e6633eaa7a03bb9d07ed106d1aa423de0544a77cea99791d4ecd31cf.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\mathcal { L } _ { M L E } ( \\theta ) = \\sum _ { i = 1 } ^ { N } \\log p _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } ) } } \\\\ { { { } } } \\\\ { { \\displaystyle p _ { \\theta } ( y _ { 1 } ^ { ( i ) } , \\dots , y _ { T } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } ) } } \\\\ { { { } } } \\\\ { { \\displaystyle p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } ) = \\mathrm { s o f t m a x } ( \\mathbf { W h } _ { t } ^ { ( i ) } + \\mathbf { b } ) } } \\\\ { { { } } } \\\\ { { { \\mathbf { h } } _ { t } ^ { ( i ) } = g ( y _ { t - 1 } ^ { ( i ) } , \\mathbf { M } ^ { ( i ) } ; \\theta ) , ~ \\mathbf { M } ^ { ( i ) } = f ( \\mathbf { x } ^ { ( i ) } ; \\theta ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 157, + 702, + 292 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $f , g$ denote the encoder and the decoder respectively and $\\mathbf { M } ^ { ( i ) } = [ \\mathbf { m } _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot \\mathbf { m } _ { L } ^ { ( i ) } ] \\in \\mathbb { R } ^ { d \\times L }$ is the concatenation of the hidden representations of the source tokens $\\mathbf { x } ^ { ( i ) }$ . ", + "bbox": [ + 174, + 303, + 823, + 335 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 CONTRASTIVE LEARNING WITH ADVERSARIAL PERTURBATIONS FOR SEQ2SEQ ", + "text_level": 1, + "bbox": [ + 171, + 357, + 766, + 372 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Since most of the seq2seq models are trained with teacher forcing where the ground truth tokens are provided to maximize Eq. 1, they are never exposed to incorrectly generated tokens during training, which is known as the “expousre bias” problem. In order to tackle the problem, we propose a contrastive learning framework to expose the model to various valid or incorrect output sequences for a given input sentence. Following the contrastive learning framework (Chen et al., 2020), we can train the model to learn the representations of the ground truth sentence by contrasting the positive pairs with the negative pairs, where we select the negative pairs as a random non-target output sequence from the same batch. As shown in Fig. 3-(a), we project the source and target text sequences onto the latent embedding space. Then we maximize the similarity between the pair of source and target sequence, while minimizing the similarity between the negative pairs as follows: ", + "bbox": [ + 173, + 383, + 825, + 523 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6874ca9691671928d41da0f69e1f32c29b7e14648f1969a782970d086f1fe5f2.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { c o n t } ( \\theta ) = \\displaystyle \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp \\left( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( i ) } ) / \\tau \\right) } { \\sum _ { \\mathbf { z } _ { \\mathbf { y } } ^ { ( j ) } \\in S } \\exp \\left( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( j ) } ) / \\tau \\right) } } \\\\ & { \\qquad \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } = \\xi ( \\mathbf { M } ^ { ( i ) } ; \\theta ) , \\mathbf { z } _ { \\mathbf { y } } ^ { ( i ) } = \\xi ( \\mathbf { H } ^ { ( i ) } ; \\theta ) } \\\\ & { \\mathbf { \\cdot } _ { 1 } \\cdot \\cdot \\mathbf { \\cdot v } _ { T } ] ; \\theta ) : = \\mathbf { A } \\mathrm { v g P o o l } ( [ \\mathbf { u } _ { 1 } \\cdot \\cdot \\cdot \\mathbf { u } _ { T } ] ) , \\mathrm { w h e r e } \\mathbf { u } _ { t } = \\mathrm { R e L U } ( \\mathbf { W } ^ { ( 1 ) } \\mathbf { v } _ { t } + \\mathbf { b } ^ { ( 1 ) } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 534, + 758, + 627 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\xi$ denotes the composition of affine transformation with the ReLU (Nair & Hinton, 2010) and average pooling to compute the fixed sized representation of a sentence $\\textbf { z } \\in \\ \\mathbb { R } ^ { d }$ , $\\begin{array} { r l } { \\mathbf { H } ^ { ( i ) } } & { { } = } \\end{array}$ $[ { \\bf h } _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot { \\bf h } _ { T } ^ { \\bar { ( i ) } } ] \\in \\mathbb { R } ^ { \\bar { d } \\times T }$ is a concatenation of the decoder hidden states of the target sentence $\\mathbf { y } ^ { ( i ) }$ across all the time steps. Furthermore, $S = \\{ \\mathbf { z } _ { \\mathbf { y } } ^ { ( j ) } : j \\neq i \\}$ is a set of hidden representations of target sentences (the objects other than circles in Fig. 3-(a)) that are randomly sampled and not paired with the source sentence $\\mathbf { x } ^ { ( i ) }$ , and $\\sin ( \\cdot , \\cdot )$ is a cosine similarity function. ", + "bbox": [ + 173, + 641, + 825, + 737 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "However, training the model with na¨ıve contrastive learning framework using random non-target sequences as negative examples is highly suboptimal, as described in the introduction and shown in Fig. 1. Many of such na¨ıve negative examples are often located far away from the positive examples in the embedding space from the beginning, when using the pretrained language model. Therefore, simply using the examples from the same batch as done in Chen et al. (2020) will result in trivial negative examples and require very large batch size to enable sampling meaningful negative pairs within the same batch. Moreover, generating positive examples for text sequences is not a trivial problem either since for text domains, we do not have a well-defined set of augmentation methods that preserves the input semantics, unlike with the image domains. To tackle such difficulties, we propose a principled method to automatically construct the adversarial negative and positive examples, such that the samples are difficult for the model to classify correctly. These adversarial positive/negative pairs can guide the model to learn a more accurate representation of the target text sequence, by identifying which features make the output positive or negative (See Fig. 1-(c)). ", + "bbox": [ + 173, + 742, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/bf63475b3841ed84076bdf161013a042e960c1737dd40ffe340a99b5ed4b73a6.jpg", + "image_caption": [ + "Figure 3: Generation of imposters and distant-targets with perturbation. (a) We add small perturbation $\\delta _ { t }$ to $\\mathbf { h } _ { t }$ for $\\tilde { \\mathbf { z } } _ { \\mathbf { y } }$ so that its conditional likelihood is minimized to generate an invalid sentence. (b) We add large perturbation $\\zeta _ { t }$ to $\\mathbf { h } _ { t }$ for $\\hat { \\mathbf { z } } _ { \\mathbf { y } }$ by maximizing the distance from $\\mathbf { z _ { x } }$ , the representation of source sentence but enforcing its likelihood high to preserve the original semantics. " + ], + "image_footnote": [], + "bbox": [ + 178, + 104, + 818, + 280 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 GENERATION OF IMPOSTERS ", + "text_level": 1, + "bbox": [ + 176, + 343, + 415, + 358 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As shown in Fig. 3-(b), to generate a negative example, we add a small perturbation $\\begin{array} { r l } { \\delta ^ { ( i ) } = } \\end{array}$ $[ \\delta _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot \\delta _ { T } ^ { ( i ) } ] \\in \\mathbb { R } ^ { d \\times T }$ to the $\\mathbf { H } ^ { ( i ) }$ , which is the hidden representation of target sequence $\\mathbf { y } ^ { ( i ) }$ , such that its conditional likelihood is minimized as follows: ", + "bbox": [ + 176, + 369, + 825, + 415 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/01640fade6423fd64c07385f4b9eab2406db67d65b2800c6cfbb502730e57cbc.jpg", + "text": "$$\np _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } ; \\mathbf { H } ^ { ( i ) } + \\delta ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } ; \\mathbf { h } _ { t } ^ { ( i ) } + \\delta _ { t } )\n$$", + "text_format": "latex", + "bbox": [ + 289, + 446, + 673, + 488 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } ; \\mathbf { h } _ { t } ^ { ( i ) } + \\delta _ { t } ) = \\mathrm { s o f t m a x } \\{ \\mathbf { W } ( \\mathbf { h } _ { t } ^ { ( i ) } + \\delta _ { t } ) + \\mathbf { b } \\}$ where $\\delta _ { t } \\in \\mathbb { R } ^ { d }$ ", + "bbox": [ + 256, + 491, + 758, + 511 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The exact minimization of the conditional log likelihood with respect to $\\delta$ is intractable for deep neural networks. Following Goodfellow et al. (2015), we approximate it by linearizing $\\log p _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } )$ around $\\mathbf { H } ^ { ( i ) }$ as follows: ", + "bbox": [ + 176, + 512, + 821, + 556 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8b627f41f339e09ff089f11e61eee5714eecefa7e59bee9d36e1ba24191eb17c.jpg", + "text": "$$\n\\tilde { \\mathbf { H } } ^ { ( i ) } = \\mathbf { H } ^ { ( i ) } - \\epsilon \\frac { g } { | | g | | _ { 2 } } , \\mathrm { w h e r e } g = \\nabla _ { \\mathbf { H } ^ { ( i ) } } \\log p _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } )\n$$", + "text_format": "latex", + "bbox": [ + 305, + 555, + 692, + 587 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We add small perturbation to the hidden representation of each token of target sentence $\\mathbf { y } ^ { ( i ) }$ such that its conditional likelihood is minimized. Thus, the perturbed $\\tilde { \\mathbf { H } } ^ { ( i ) }$ , which we call an imposter (inspired by Weinberger & Saul (2009)), is semantically very dissimilar to $\\mathbf { y } ^ { ( i ) }$ , but very close to the hidden representation $\\mathbf { H } ^ { ( i ) }$ in the embedding space (Fig. 3-(a)). This will make it non-trivial for the sequence-to-sequence model to distinguish it from the representation of true target sequence $\\mathbf { y } ^ { ( i ) }$ . Please note while adversarial perturbations are generated similarly as in Miyato et al. (2017), we use them in a completely different way. While they train the model to be invariant to adversarial samples within the $\\epsilon$ -ball, we push them far away from the source sentence while pulling the ground truth target sentence to the input sentence. In other words, we use the perturbed representation as an additional negative sample for contrastive learning as follows: ", + "bbox": [ + 173, + 594, + 825, + 741 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4901d914e010b64017527b65b8fde9ac90b02b05ca90e2d78b06242af3333ab0.jpg", + "text": "$$\n\\mathcal { L } _ { c o n t - n e g } ( \\theta ) = \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp ( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( i ) } ) / \\tau ) } { \\sum _ { \\mathbf { z } _ { \\mathbf { y } } ^ { ( k ) } \\in S \\cup \\{ \\bar { \\mathbf { z } } _ { \\mathbf { y } } ^ { ( i ) } \\} } \\exp ( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( k ) } ) / \\tau ) } , \\mathrm { w h e r e ~ } \\tilde { \\mathbf { z } } _ { \\mathbf { y } } ^ { ( i ) } = \\xi ( \\tilde { \\mathbf { H } } ^ { ( i ) } ; \\theta _ { \\mathbf { z } _ { \\mathbf { x } } } ^ { ( i ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 196, + 742, + 771, + 790 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Alternatively, we can generate an imposter by perturbing the hidden representation of target sentence $\\mathbf { y }$ so that its conditional likelihood is minimized but very close to the source sentence $\\mathbf { x }$ in the embedding space. However, we empirically find that such a variation yields less performance gain. ", + "bbox": [ + 174, + 790, + 826, + 833 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 GENERATION OF DISTANT-TARGETS ", + "text_level": 1, + "bbox": [ + 176, + 847, + 465, + 863 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Moreover, as shown in Fig. 3-(c), we construct an additional positive pair of source sequence $\\mathbf { x } ^ { ( i ) }$ by adding large perturbation $\\boldsymbol { \\zeta } ^ { ( i ) } = [ \\zeta _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot \\zeta _ { T } ^ { ( i ) } ] \\in \\mathbb { R } ^ { d \\times T }$ to $\\mathbf { H } ^ { ( i ) }$ the hidden state of target sequence $\\mathbf { y } ^ { ( i ) }$ , such that cosine similarity from $\\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) }$ is minimized, but the conditional likelihood is enforced to remain high. However, the exact computation of $\\zeta ^ { ( i ) }$ with such constraints is intractable. We approximate it with the following two separate stages. First, we add perturbation to $\\mathbf { H } ^ { ( i ) }$ such that it minimizes the contrastive learning objective $\\mathcal { L } _ { c o n t } ( \\theta )$ as shown in Eq. 6. Then we add another perturbation to minimize the KL divergence between perturbed conditional distribution $p _ { \\theta } ( \\hat { y } _ { t } ^ { ( i ) } | \\hat { \\mathbf { y } } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } )$ and the original conditional distribution $p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } )$ as shown in Eq. 7, where $\\overline { { \\mathbf { H } } } = [ \\overline { { \\mathbf { h } } } _ { 1 } \\cdot \\cdot \\cdot \\overline { { \\mathbf { h } } } _ { T } ] \\in \\mathbb { R } ^ { d \\times T }$ , $\\hat { \\mathbf { H } } = [ \\hat { \\mathbf { h } } _ { 1 } \\cdot \\cdot \\cdot \\hat { \\mathbf { h } } _ { T } ] \\in \\mathbb { R } ^ { d \\times T }$ , and $\\eta \\in \\mathbb { R }$ . Note that $\\theta ^ { * }$ denotes the copied of the model parameter $\\theta$ and is considered to be constant to prevent it from being updated through back-propagation. ", + "bbox": [ + 174, + 873, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 223 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c32b228a53fdade8ca5f9b88e86f501f3ba86e9d7508f87caa72834feec5acfc.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\overline { { { \\bf { H } } } } } ^ { ( i ) } = { \\bf { H } } ^ { ( i ) } - \\eta \\frac { { \\bf { g } } } { | | { \\bf { g } } | | _ { 2 } } } \\mathrm { { w h e r e } ~ } { \\bf { g } } = \\nabla _ { { \\bf { H } } ^ { ( i ) } } \\mathcal { L } _ { c o n t } ( \\theta ) } \\\\ { { p _ { \\theta } } ( \\hat { y } _ { t } ^ { ( i ) } | \\hat { \\bf { y } } _ { < t } ^ { ( i ) } , { \\bf { x } } ^ { ( i ) } ) = \\mathrm { s o f t m a x } ( \\mathbf { W } { \\bf { \\overline { { { h } } } } } _ { t } ^ { ( i ) } + { \\bf { b } } ) } \\\\ { \\mathcal { L } _ { K L } ( \\theta ) = \\displaystyle { \\sum _ { i = 1 } ^ { N } \\sum _ { t = 1 } ^ { T } D _ { K L } \\big ( p _ { \\theta ^ { * } } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , { \\bf { x } } ^ { ( i ) } ) | | p _ { \\theta } ( \\hat { y } _ { t } ^ { ( i ) } | \\hat { \\bf { y } } _ { < t } ^ { ( i ) } , { \\bf { x } } ^ { ( i ) } ) } \\\\ { \\hat { \\bf { H } } ^ { ( i ) } = { \\bf { \\overline { { { H } } } } } ^ { ( i ) } - \\eta \\frac { { \\bf { f } } } { | | { \\bf { f } } | | _ { 2 } } } , \\mathrm { { w h e r e } ~ } { \\bf { f } } = \\nabla _ { { \\bf { \\overline { { { H } } } } } _ { 1 } ^ { ( i ) } } \\mathcal { L } _ { K L } ( \\theta ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 224, + 740, + 351 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We consider the perturbed hidden state $\\hat { \\mathbf { H } } ^ { ( i ) }$ as an additional positive example for source sequence $\\mathbf { x } ^ { ( i ) }$ , which we refer to as a distant-target. We can use a distant-target to augment contrastive learning and minimize ${ \\mathcal { L } } _ { K L } ( \\theta )$ as follows: ", + "bbox": [ + 176, + 353, + 823, + 397 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ef8bcc3c01b19cdea67005df2afa15741a648b523a3f6740e634e05423550ddd.jpg", + "text": "$$\n\\mathcal { L } _ { c o n t - p o s } ( \\theta ) = \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp ( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\hat { \\mathbf { z } } _ { \\mathbf { y } } ^ { ( i ) } ) / \\tau ) } { \\sum _ { \\mathbf { z } _ { \\mathbf { y } } ^ { ( k ) } \\in S \\cup \\{ \\hat { \\mathbf { z } } _ { \\mathbf { y } } ^ { ( i ) } \\} } \\exp ( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( k ) } ) / \\tau ) } , \\mathrm { w h e r e ~ } \\hat { \\mathbf { z } } _ { \\mathbf { y } } ^ { ( i ) } = \\xi ( \\hat { \\mathbf { H } } ^ { ( i ) } ; \\theta )\n$$", + "text_format": "latex", + "bbox": [ + 197, + 400, + 781, + 446 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "CLAPS objective Incorporating the loss on the imposter and the distant target introduced above, we estimate the parameters of the seq2seq model $\\theta$ by maximizing the following objective, where $\\alpha , \\beta$ are hyperparameters which control the importance of contrastive learning and KL divergence: ", + "bbox": [ + 174, + 455, + 825, + 497 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/8334bb5fbc43c156e8dd70c4e27c8121bcebf670467085851ce1d4c5aad2f090.jpg", + "text": "$$\n\\displaystyle \\operatorname* { m a x } _ { \\theta } \\mathcal { L } _ { M L E } ( \\theta ) - \\alpha \\mathcal { L } _ { K L } ( \\theta ) + \\beta \\{ \\mathcal { L } _ { c o n t - n e g } ( \\theta ) + \\mathcal { L } _ { c o n t - p o s } ( \\theta ) \\}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 498, + 715, + 522 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For all the experiments, we set $\\alpha$ and $\\beta$ as 1, which we search through cross-validation. Note that after training is done, we remove the pooling layer $\\xi$ and generate text with the decoder $g$ , given an input encoded with the encoder $f$ . ", + "bbox": [ + 173, + 523, + 826, + 566 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENT ", + "text_level": 1, + "bbox": [ + 174, + 585, + 316, + 602 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We validate our method on benchmark datasets on three conditional text generation tasks. ", + "bbox": [ + 171, + 616, + 759, + 631 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 TASKS ", + "text_level": 1, + "bbox": [ + 174, + 646, + 261, + 661 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Machine Translation (MT) For machine translation, we use WMT16 Romanian-English parallel corpus (WMT’16 RO-EN) to train the model. We tokenize the pairs of source and target sequences with the same tokenizer as Raffel et al. (2020). We finetune the pretrained T5-small model for 20 epochs with the batch size of 128 and Adafactor (Shazeer & Stern, 2018). For contrastive learning, we set the norm of perturbation, $\\eta$ and $\\epsilon$ as 3.0. ", + "bbox": [ + 174, + 671, + 825, + 742 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Text Summarization (Sum.) For text summarization, we use XSum dataset (Narayan et al., 2018) of which summaries are highly abstractive, thus extractive summarization models under-perform abstractive models. We follow the most of the experimental settings for machine translation as described above, except that we set the norm of perturbation, $\\eta$ and $\\epsilon$ as 1.0 and 1.0, respectively. ", + "bbox": [ + 173, + 750, + 825, + 806 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Question Generation (QG) For question generation, we aim to generate a question from a given answer and paragraph, i.e., we model conditional distribution $p _ { \\theta } ( \\mathbf { y } | \\mathbf { x } , \\mathbf { a } )$ where $\\mathbf x , \\mathbf y$ , a denote a paragraph, question and answer, respectively. We concatenate the answer and paragraph with special tokens to generate the question conditioned on both of the answer and paragraph. As the previous experimental settings, we finetune T5-small model on SQuAD dataset (Rajpurkar et al., 2016) for 20 epochs with batch size 128 and set the norm of perturbation, $\\eta$ as 3.0 and $\\epsilon$ as 1.0. Since the test set of $\\mathrm { S Q u A D }$ is only accessible via leader board, we randomly split the validation set into a validation set and a test set. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/eca812764f69606ee590496ff08461b1a9116a4e9cda69a4e8589c35584cd217.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: BLEU scores on WMT’16 RO-EN and SQuAD for machine translation and question generation. EM/F1 scores with BERT-base QA model for question generation. " + ], + "table_body": "
MethodAug.BLEU-1BLEU-2BLEU-3BLEU-4BLEUF1/EM
Question Generation - SQuAD
Harvesting-QG=20.9015.16=66.05/54.62
T5-MLE41.2630.3023.3818.5421.0067.64/55.91
α-T5-MLE(α=0.7)40.8229.7922.8417.9920.5068.04/56.30
α-T5-MLE(α= 2.0)37.3527.2020.7916.3618.4165.74/54.76
T5-SSMBAPos.41.6730.5923.5318.5721.0768.47/56.37
T5-WordDropout ContrastiveNeg.41.3730.5023.5818.7121.1968.16/56.41
R3F-41.0030.1523.2618.4420.9765.84/54.10
T5-MLE-contrastive-41.2330.2823.3318.4520.9167.32/55.25
T5-CLAPS w/o negativePos.41.8730.9323.9018.9221.38
T5-CLAPS w/o positiveNeg.41.6530.6923.7118.8121.2568.26/56.41
T5-CLAPSPos.+Neg.42.3331.2924.2219.1921.5569.01/57.06
ERNIE-GEN (Xiao et al.,2020)=26.95=
Info-HCVAE (Lee et al.,2020)-==-=81.51/71.18
MachineTranslation-WMT'16RO-EN
Transformer50.3637.1828.4222.2126.17
Scratch-T5-MLE51.6237.2227.2621.1325.34
Scratch-CLAPSPos.+Neg.53.4239.5730.2423.5927.61
T5-MLE57.7644.4535.1228.2132.43
α-T5-MLE(α = 0.7)57.6344.2333.8427.9032.14
α-T5-MLE(α = 2.0)-56.0342.5933.2926.4530.72
T5-SSMBAPos.58.2344.8735.5028.4832.81
T5-WordDropout ContrastiveNeg.57.7744.4535.1228.2132.44
R3F-58.0744.8635.5728.6632.99
T5-MLE-contrastive-57.6444.1234.7427.7932.03
T5-CLAPS w/o negativePos.
T5-CLAPS w/o positiveNeg.58.81 57.9045.52 44.6036.20 35.2729.23 28.3433.50
T5-CLAPSPos.+Neg.58.9845.7236.3929.4132.55 33.96
Conneau & Lample (2019)1----38.5
", + "bbox": [ + 187, + 104, + 805, + 469 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 EXPERIMENTAL SETUPS ", + "text_level": 1, + "bbox": [ + 174, + 513, + 383, + 529 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Implementation Details For the encoder $f$ , and decoder $g$ , we use T5-small model, which is based on transformer with the hidden dimension, $d = 5 1 2$ . We set the temperature, $\\tau$ as 0.1 for all the experiments. At test time, we use beam search of width 4 to generate the target sequences. Common Baselines We compare our method against relevant baselines. ", + "bbox": [ + 173, + 546, + 825, + 602 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1. T5-MLE: A pretrained T5 model fine-tuned to maximize $\\mathcal { L } _ { M L E } ( \\theta )$ . \n2. Scratch-T5-MLE: A random initialized Transformer model that has the identical architecture to T5, trained by maximizing $\\mathcal { L } _ { M L E } ( \\theta )$ . \n3. $\\alpha$ -T5-MLE: T5 model trained with MLE, with varying temperature $\\alpha$ in the softmax function when decoding the target sentences, as done in Caccia et al. (2019) \n4. T5-SSMBA: This is the T5 model trained to maximize $\\mathcal { L } _ { M L E } ( \\theta )$ , with additional examples generated by the technique proposed in $\\mathrm { N g }$ et al. (2020). which are generated by corrupting the target sequences and reconstructs them using a masked language model, BERT. \n5. T5-WordDropout Contrastive: This is a T5 model trained with the contrastive learning framework proposed in Yang et al. (2019), which heuristically generates negative examples by removing the most frequent word from the target sequence. We pretrain T5-small to maximize $\\mathcal { L } _ { M L E } ( \\theta )$ and further train the model to assign higher probability to the ground truth target sentence than a negative example with max-margin loss. \n6. R3F: This is a T5 model that minimizes the negative log likelihood and symmetric KL-divergence between original conditional log likelihood $p _ { \\boldsymbol { \\theta } } ( \\mathbf { y } | \\mathbf { x } )$ and $p _ { \\boldsymbol { \\theta } } ( \\mathbf { y } | \\tilde { \\mathbf { x } } )$ to enforce the function to be smooth, where $\\tilde { \\mathbf { x } } =$ WordEmbedding $\\mathbf { \\Phi } _ { \\cdot } ( \\mathbf { x } ) + \\mathbf { z } , \\mathbf { z } = ( z _ { 1 } , \\dots , z _ { L } ) , z _ { i } \\overset { i . i . d } { \\sim } \\mathcal { N } ( \\mathbf { 0 } , \\mathrm { d i a g } ( \\sigma _ { 1 } , \\dots , \\sigma _ { d } ) )$ . \n7. T5-MLE-contrastive: This is a naive constrastive learning framework with positive/negative pairs, which maximizes the contrastive learning objective from Eq. 2. \n8. T5-CLAPS w/o positive (negative): Our proposed model which jointly maximizes the log likelihood and the contrastive learning objective with imposters but does not use any distant-targets or imposters. ", + "bbox": [ + 171, + 606, + 826, + 925 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/3f75a9b48bbd7789a73222d10343aa6cfa8ce55575632d446aae69e15e6c7e44.jpg", + "table_caption": [ + "Table 2: Rouge and Meteor on Xsum test set for text summarization. " + ], + "table_footnote": [], + "table_body": "
MethodAug.Rouge-1Rouge-2Rouge-LMETEOR
Text Summarization - XSum
PTGEN-COVG一28.108.0221.7212.46
CONVS2S31.8911.5425.7513.20
Scratch-T5-MLE31.4411.0725.1813.01
Stcratch-CLAPSPos.+Neg.33.5212.5926.9114.18
T5-MLE36.1014.7229.1615.78
α-T5-MLE(α= 0.7)=36.6815.1029.7215.78
α-T5-MLE (α= 2.0)=34.1813.5327.3514.51
T5-SSMBAPos.36.5814.8129.6815.38
T5-WordDropout ContrastiveNeg.36.8815.1129.7915.77
R3F-36.9615.1229.7615.68
T5-MLE-contrastive-36.3414.8129.4115.85
T5-CLAPS w/o negativePos.37.4915.3130.4216.36
T5-CLAPS w/o positiveNeg.37.7215.4930.7416.06
T5-CLAPSPos.+Neg.37.8915.7830.5916.38
PEGASUS (Zhang et al.,2020)=47.2124.5639.25-
", + "bbox": [ + 200, + 128, + 794, + 378 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "9. T5-CLAPS: Our full model which jointly maximizes the log likelihood, contrastive learning objective, and KL-divergence as described in the Eq. 9. ", + "bbox": [ + 174, + 388, + 818, + 416 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "10. Scratch-CLAPS: Our full model as T5-CLAPS but with randomly initialized T5 architecture. ", + "bbox": [ + 169, + 419, + 813, + 433 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Task specific baselines For machine translation, we use the Transformer (Vaswani et al., 2017) which consists of 6 layers of self-attention layer with 8 multi-head attention and 512 dimension, as an additional baseline. For QG, we additionally compare our models against Harvesting-QG (Du & Cardie, 2018), which is a LSTM model with copy mechanism. For text summarization, we use PTGEN-COVG (See et al., 2017) as a baseline, which uses copy mechanism and coverage to handle out of vocabulary word and prevent word repetition, and CONVS2S (Narayan et al., 2018) which uses convolutional networks as the encoder and decoder. ", + "bbox": [ + 173, + 446, + 825, + 544 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Evaluation Metric Following the conventional evaluation metrics, we adopt n-gram BLEU and BLEU (Papineni et al., 2002) for MT and QG. For text summarization, we use Rouge (Lin & Hovy, 2002) and Meteor (Banerjee & Lavie, 2005). As an additional performance measure for question generation, we evaluate a BERT QA model on the SQuAD test set, where the QA model is trained with the questions generated by each QG methods from the contexts and answers of HarvestingQA dataset (Du & Cardie, 2018), and report the F1 and Exact Match (EM). ", + "bbox": [ + 174, + 551, + 825, + 635 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 655, + 392, + 670 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Quantitative Results We compare our model with the baseline models on WMT’16 RO-En, XSum, SQuAD dataset for machine translation, text summarization and question generation, respectively. Table 1 shows that our proposed method CLAPS significantly outperforms the other baseline, with the performance gain of more than $1 \\%$ on all tasks according to the BLEU scores. Moreover our proposed method improves the performance of the randomly initialized T5 model (Scratch-CLAPS). For question generation, our proposed method also improves F1/EM as well as BLEU scores. It shows that our proposed model is able to generate semantically valid questions that are beneficial for training the QA model. Note that naively constructing the negative examples for contrastive learning on the both tasks, by randomly shuffling the association of $\\displaystyle ( \\mathbf { x } , \\mathbf { y } )$ from a given mini-batch, degrades the performance. Increasing the batch size to a large value, using larger memory, may increase its performance as observed in SimCLR (Chen et al., 2020). However, such an approach will be highly sample-inefficient. On the contrary, our model outperforms all the other baseline models on Xsum dataset for text summarization, as shown in Table 2. For summarization, we observe that contrastive learning with imposters alone can improve the performance by a large margin. ", + "bbox": [ + 174, + 678, + 825, + 873 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Visualization To examine our model with proposed contrastive learning framework learns meaningful representation of sentences, we encode a pair of sequences $\\displaystyle ( \\mathbf { x } , \\mathbf { y } )$ into M, H with encoder $f$ and decoder $g$ . Then, we add perturbations to $\\mathbf { H }$ to construct an imposter $\\tilde { \\bf H }$ and an additional positive ", + "bbox": [ + 176, + 880, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/733b12af4b7726f4664721ec16f095a20ba6155c9de025921b94f7a542040235.jpg", + "image_caption": [ + "Figure 4: Visualization. (a) Embedding space without contrastive learning. (b) Embedding space with our proposed contrastive learning, CLAPS. " + ], + "image_footnote": [], + "bbox": [ + 176, + 97, + 495, + 209 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/5e3c4a270d4e7b2ff7183698fa63a4acccc75293c5c75e53e1466d0f3ddb0a3a.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
(MT) Lupta lui Hilary a fost mai atractiva. =>(GT): Hillary's struggle was more attractive =>(Dist.): Hilary's fight was more attractive =>(Imp.): Thearies'batle fight has attractive appealing
(QG)...Von Miller... recording five solo tackles,.. =>(GT): How many solo tackles did Von Miler make at Super Bowl 50? =>(Dist.): How many solo tackles did Von Miller record at Super Bowl 50? =>(Imp.):What much tackle did was Miller record at Super Bowl 50?
(Sum.) Pieces from the board game...have been found in... China.... =>(GT): An ancient board game has been found in a Chinese Tomb. =>(Dist.):An ancient board game has been discovered in a Chinese Tomb. =>(Imp.):America's gained vast Africa most well geographical countries, 22
", + "bbox": [ + 506, + 99, + 818, + 219 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 3: Greedy decoding from hidden representation of imposters and distant-targets. The answer span is highlighted for QG. ", + "bbox": [ + 503, + 222, + 823, + 260 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "example $\\hat { \\bf H }$ as shown in Eq. 3 and 6, 7. We apply average pooling to $\\mathbf { M } , \\mathbf { H } , \\tilde { \\mathbf { H } }$ , and $\\hat { \\bf H }$ and project them onto two dimensional space with t-SNE (Maaten & Hinton, 2008). As shown in Fig. 4-(b), the model pushes away the imposter from the embedding of target sequence and pulls the embedding of the distant-targets to the embedding of the source sequence. For the model without contrastive learning, however, the embeddings of both target sequences and distant targets are far away from those of source sequences and the imposters are very close to them as shown in Fig. 4-(a). ", + "bbox": [ + 173, + 265, + 825, + 348 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Qualitative Examples For qualitative analysis, we examine the texts that are represented by the distant-target and imposter from our method, CLAPS. To decode them into output sequences, we apply affine transformation and softmax to $\\tilde { \\bf H }$ and $\\hat { \\bf H }$ and select the most likely token at each time step. As shown in Table 3, the distant-target example (Dist.), preserves the semantic of the original target sequence (GT) with a single word replaced by a synonym (colored in green). However, the imposters (Imp.) have completely different semantics, and often are gramatically incorrect (colored in red). This shows that the model are exposed to those various valid or incorrect sentences with our proposed contrastive learning framework with adversarial perturbations. ", + "bbox": [ + 174, + 356, + 825, + 469 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Human Evaluation We further conduct a human evaluation of the 20 summaries and 20 questions generated by our CLAPS and T5-MLE trained for text summarization and QG task. Specifically, 20 human judges perform blind quality assessment of two sentences generated by the two models, that are presented in a random order. For text summarization, $70 \\%$ of the human annotators chose the sentences generated by our model as better than the baseline, and for QG, $85 \\%$ favored the sentences generated by our model over that of the baseline. ", + "bbox": [ + 174, + 476, + 825, + 559 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 580, + 315, + 595 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To mitigate the exposure bias problem in sequence-to-sequence learning, we proposed a contrastive learning framework which maximizes the similarity between ground truth input and output sequence, and minimize the similarity between the input and an incorrect output sequence. Moreover, since conventional approach to sample random non-target examples from the batch as negative examples for contrastive learning results in trivial pairs that are well-discriminated from the beginning, we propose a new principled approach to automatically construct “hard” negative and positive examples, where the former is semantically dissimilar but close to the input embedding, and the latter is far from the input embedding but semantically similar. This adversarial learning enables the model to learn both the correct and incorrect variations of the input, and generalize better to unseen inputs. We empirically showed that our method improved the performance of seq2seq model on machine translation, question generation, and text summarization tasks. While we specifically targeted the exposure bias problem with seq2seq models for conditional text generation, our method may be applicable to seq2seq learning for tasks from other domains, such as automatic speech recognition, text-to-speech generation, or video captioning. ", + "bbox": [ + 173, + 611, + 825, + 805 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Acknowledgements This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2020- 0-00153), Samsung Advanced Institute of Technology (SAIT), Samsung Electronics (IO201214- 08145-01), Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST)), and the Engineering Research Center Program through the National Research Foundation of Korea (NRF) funded by the Korean Government MSIT (NRF2018R1A5A1059921). 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", + "bbox": [ + 174, + 103, + 823, + 145 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/2b71e65c3fe58092de022d67ed3ff174f0151afc934e03d9a6621e87327881e5.jpg", + "table_caption": [ + "Table 4: The statistics and the data source of WMT’16 RO-EN, Xsum, and SQuAD. " + ], + "table_footnote": [], + "table_body": "
DatasetsTrain (#)Valid (#)Test (#)Source
WMT'16 RO-EN610,3201,9991,999Romanian-English Parallel corpus.
Xsum204,04511,33211,334One-sentence summary of BBC news articles.
SQuAD86.5885,1925,378Crowd-sourced questions from Wikipedia paragraph
", + "bbox": [ + 173, + 127, + 839, + 207 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 231, + 421, + 247 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Dataset For machine translation, text summarization, and question generation, we use WMT’16 RO-EN, Xsum, SQuAD dataset, for each task. The number of train/validation/test set and its source is shown in Table 4. Note that the number of validation and test set for SQuAD is different from the original dataset. Since the original test set is only accessible via the leader board of $\\mathrm { S Q u A D ^ { 1 } }$ , we split the original validation set into our new validation and test set, following the conventions of question generation communities. ", + "bbox": [ + 173, + 262, + 826, + 345 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Preprocessing For machine translation, we download the raw text2, not the tokenized text, and use the same T5-tokenizer as Raffel et al. (2020) to tokenize both Romanian and English sentences. We limit the input and output length to 128 tokens. For text summarization, we also use the T5-tokenizer as before, and limit the input length to 512 tokens and output length to 128 tokens. For question generation, we set the maximum length of question as 64 tokens and input which is concatenation of answer and context as 384 tokens. ", + "bbox": [ + 173, + 361, + 825, + 445 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Implementation We finetune the pretrained T5-small model provided from the transformers library (Wolf et al., 2019)3 with Adafactor optimizer. We set the batch size 128 and follow the default setting of Adafactor optimizer to finetune the T5-small models. However, the number of negative examples from the batch is 16 or 32 (total batch size divided by the number of GPUs), because we split the batch into smaller batches and distribute them to each GPU machines. We use 8 GPUs for text summarization, and 4 GPUs for machine translation and question generation. The dimension of hidden state of T5 model, $d$ is 512, so we set the hidden size of $\\mathbf { z }$ as the same. ", + "bbox": [ + 173, + 460, + 825, + 558 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Evaluation We use beam search with beam width 4 to generate the target sentences from the source sentences of the test set. Some of the examples are shown in Table 5,6,A. After the generation, we convert the tokens into the raw texts and compare them to the raw text of ground truth target sentences with the automatic evaluation metrics. For n-gram BLEU and Meteor, we use the implementation by Sharma et al. $( 2 0 1 7 ) ^ { 4 }$ . For BLEU score, we adopt the implementation by Post $( 2 0 1 8 ) ^ { 5 }$ . ", + "bbox": [ + 173, + 574, + 825, + 657 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Article: The US military says a strike targeting Taliban in the northern city of Kunduz may have caused ”collateral damage”. Offering his ”deepest condolences”, Mr Obama said he expected a ”full accounting of the facts” and would then make a definitive judgement. . . . ", + "bbox": [ + 191, + 185, + 799, + 224 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "GT: President Barack Obama says the US has launched a ”full investigation” into air strikes that killed 19 people at an MSF-run Afghan hospital on Saturday. ", + "bbox": [ + 191, + 237, + 779, + 263 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "CLAPS: US President Barack Obama has called for an inquiry into air strikes in Afghanistan that killed dozens of medical workers. ", + "bbox": [ + 189, + 276, + 785, + 301 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Article: Forecasts were for quarterly growth of between $0 . 5 \\%$ and $0 . 7 \\%$ . Official statistics also showed that household consumption expenditure boosted the quarterly growth numbers. But economist Shane Oliver told the BBC the numbers were “well below potential”. On an annual basis the economy expanded $2 . 3 \\%$ , beating expectations for $2 . 1 \\%$ . Economic growth in the March quarter of 2014 was $2 . 9 \\%$ . “The March quarter GDP [gross domestic product] growth was far better than feared just a few days ago,” said Mr Oliver, who is chief economist with AMP Capital in Sydney. “However, Australia is still not out of the woods, as annual growth at $2 . 3 \\%$ is well below potential, and a full $0 . 8 \\%$ percentage points of the $0 . 9 \\%$ growth came from higher inventories and trade.” He said domestic demand remained “very weak with consumer spending and home construction only just offsetting the ongoing slump in mining investment”. ... ", + "bbox": [ + 191, + 308, + 789, + 450 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "GT: Australia’s economy grew at a better-than-expected $0 . 9 \\%$ in the first quarter of 2015, compared to the previous quarter, boosted by mining together with financial and insurance services. ", + "bbox": [ + 191, + 462, + 794, + 488 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "CLAPS: Australia’s economy grew faster than expected in the first three months of the year, according to official figures. ", + "bbox": [ + 191, + 501, + 754, + 527 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Article: After the problems last week, many doubt the system will cope. \nTransport for London (TfL) remains confident, although it admits there will be breakdowns. \nThe trick will be in getting the system back up and running quickly. So here’s some friendly advice for tourists and Olympic visitors to try and make the transport experience as easy as possible. \nIf anyone thinks of any more please post below. ", + "bbox": [ + 191, + 531, + 797, + 598 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "GT: The busiest summer ever looms for London’s transport system. ", + "bbox": [ + 191, + 609, + 602, + 623 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "CLAPS: London’s transport system has been a pretty busy week. ", + "bbox": [ + 191, + 636, + 589, + 650 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Article: The outgoing vice-president spoke during a state dinner and took the opportunity to praise America’s northern neighbour. ”The world is going to spend a lot of time looking to you, Mr Prime Minister”, he told the Canadian leader. Mr Biden has been highly critical of US President-elect Donald Trump. ”Vive le Canada because we need you very, very badly,” he told the dinner guests. He went on to describe the self-doubt that liberal leaders across the world are currently experiencing after several political defeats. But he praised ”genuine leaders” including German Chancellor Angela Merkel, saying such statesmen and women are in short supply. Mr Trudeau reportedly became emotional during Mr Biden’s remarks when the American spoke of his late father, former Prime Minister Pierre Trudeau. ”You’re a successful father when your children turn out better than you,” Mr Biden said. ... ", + "bbox": [ + 189, + 654, + 802, + 786 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "GT: US Vice-President Joe Biden told an audience in Ottawa that the world needs ”genuine leaders” such as Canadian Prime Minister Justin Trudeau. ", + "bbox": [ + 186, + 796, + 802, + 821 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "CLAPS: Vice-President Joe Biden has praised Canadian Prime Minister Vive le Canada for his visit to the country. ", + "bbox": [ + 191, + 835, + 727, + 861 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Article: The Swedish giant asked customers who bought \nany model of the Mysingso chair to return it for a full refund. \nThe global recall comes after Ikea received reports from \nFinland, Germany, the US, Denmark and Australia \nthat users had received injuries to their fingers that needed medical treatment. \nIkea’s statement said the chair had a ”risk of falling or finger entrapment”. \nIt said: ”After washing the fabric seat it is possible to re-assemble the chair incorrectly leading to risks of falls or finger entrapments. \n”Ikea has received five incident reports in which a Mysingso beach chair collapsed during use due to incorrect re-assembly. \nAll five reports included injuries to fingers and required medical attention. \nIt added that a full investigation had led to an improved design \n”to further mitigate the risks of incorrect re-assembly and injuries” \nand the updated chair would be available from next month. \nIkea has more than 300 stores in 27 countries. ", + "bbox": [ + 191, + 137, + 723, + 333 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "GT: Ikea is recalling a beach chair sold in the UK after reports that it can collapse and cause injury. ", + "bbox": [ + 191, + 345, + 576, + 372 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/800e27d488f6ec17d024e0e94b876d9a6fdd1c82267ad0b6d8c6b2db649bf07b.jpg", + "table_caption": [ + "CLAPS: Ikea is recalling a popular beach chair that collapsed during use because of incorrect re-assemblies. " + ], + "table_footnote": [], + "table_body": "
Article: Spending on the NHS should also be paid for by a dedicated tax marked on every payslip,the former health minister suggested. Under Mr Lamb's plan,taxes would not be increased as the new levy would be offset by deductions to income tax or national insurance. He has warned the NHS faces collapse without an urgent cash injection. The plans are not yet party policy and will not be put to this year's conference in Bournemouth.But Mr Lamb,the party's health spokesman, told party members he was "very interested in the idea of a dedicated NH S and care contribution - separating it out from the rest of taxation, clearly identified on your payslip."And Iam really interested in the idea of the right for local areas to raise additional funds for the NHS and care if theychoose.’The Lib Dems say he would like to implement the ideas across theUK,although,as health and social care are devolved, it is unclear how this would be enforced. Mr Lamb - who lost out to Tim Farron in a leadership election in July - proposes a cross-party commission to explore the ideas. Heintends to consult health bodies and professionals, patients,trade unions and academics.Ministers have pledged f2bn in this financial year for the NHS,and an extra f8bn by 2020. But MrLamb told the BBC that this was insuffcient and, having "seen the books”as a minister in the last government, he feared the NHS could face a funding shortfall of f3Obn by 2020. "The bottom line is with rising demand because of an ageing population we need more investment,” he said.MrLamb also warned that the social care system was "on its knees" and could collapse without a cash injection of f5bn. "I've been in the department. Ihave seen the books and Iam deeply concerned. If we carry on regardless, the system will crash." Taxpayers are already shown how much they have contributed to the health service in annual personal tax statements.An attempt to establish a cross-party commission on social care before the 2O1O election -led in part by MrLamb -collapsed in acrimony.
", + "bbox": [ + 191, + 415, + 803, + 808 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "GT: English councils should be allowed to put up taxes to fund the NHS, Norman Lamb has told the Lib Dem conference. ", + "bbox": [ + 191, + 818, + 640, + 844 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "CLAPS:A new levy on the NHS and social care should be introduced by the Liberal Democrats, Norman Lamb has said. ", + "bbox": [ + 191, + 857, + 570, + 883 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Article: Yorkshire, Lancashire and Derbyshire have been worst affected, \nafter $2 { \\cdot } 5 \\mathrm { c m }$ fell overnight, with $1 0 \\mathrm { c m }$ reported on higher ground. \nPassengers waiting to depart Manchester Airport have reported \nbeing stuck on the runway for hours due to a lack of de-icers. \nLeeds Bradford Airport suspended all morning flights but has since reopened. \nManchester Airport reported ”minor delays to departing aircraft” \n- but passengers told the BBC they had been stuck on board outbound flights. \nShirley Hale said her Jet2 flight to Tenerife \nhad been waiting to depart for over four hours. \n”We have been told that there are not enough de-icers at the airport,” \nshe said. The airport apologised and said de-icing was the responsibility of \nairlines and their ground teams. More than 100 schools were closed across East Lancashire and Oldham, with 80 shut in West Yorkshire. \nBBC Weather said Buxton in Derbyshire saw up to $1 7 \\mathrm { c m }$ of snow, \nthe deepest measured on Friday. The avalanche risk in the Peak District was \ncurrently extremely high, Buxton Mountain Rescue Team said. \nParts of Staffordshire have been affected, with several centimetres of \nsnow reported in Flash, England’s highest village. \nCommuters have been urged to allow extra journey time, \nand the Met Office has issued snow and ice warnings. \nMore on the snow and other stories in West Yorkshire Weather \nupdates for Lancashire and Greater Manchester BBC Weather \npresenter Kay Crewdson said conditions were due to slowly \nimprove into Saturday. Molly Greenwood reported $1 0 \\mathrm { c m }$ of snow \nin the Huddersfield area. ”Don’t think I’m going anywhere,” she said. \nZulfi Hussain said the snow was causing ”traffic chaos” in Woodhall Road, \nCalverley, near Leeds. Elliott Hudson, another West Yorkshire resident, \nsaid: ”Looks like I have woken up in Narnia.” \nWest Yorkshire’s Liversedge FC, who have had to cancel every home \ngame for the last four months due to bad weather, \ntweeted a picture of snow with the caption: \n”It’s not looking good for Liversedge FC’s home game with Worksop Town tomorrow.” The A628 Woodhead, A57 Snake Pass and A537 Cat and \nFiddle roads are all affected, with delays reported on \nthe M65 motorway. Highways England said the A57 eastbound \nin Great Manchester is closed between M67/A560 and B6174 \ndue to severe weather conditions. It said teams were working \nto clear the road. Tony Hallwood, from Leeds Bradford Airport, \nsaid it reopened at about 09:00 GMT after crews used ploughs \nto clear snow from the runway. He said: ”We are asking passengers \nto make their way to the airport as early as they can given the difficult \nconditions.” Bus operators are also reporting delays to all services \nacross West Yorkshire. Oldham Council has said 48 schools had closed \nthis morning as a result of the snow and severe weather. \nDrivers are also being asked to take extra care after snow \nfell overnight in some parts of Northern Ireland. \nA Met Office yellow warning for ice and snow in \nnorthern England and Wales ended at 15:00. ", + "bbox": [ + 192, + 130, + 802, + 851 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "GT: Heavy snowfall has caused travel disruption in parts of northern England. ", + "bbox": [ + 192, + 848, + 736, + 864 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "CLAPS: Flights have been disrupted after a large avalanche hit parts of England. ", + "bbox": [ + 191, + 878, + 756, + 893 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Article: But once the votes are counted, what can residents expect to pay in co Below are the figures for a Band D property for every council area \nin Wales for the current financial year of 2017/18, \nhow much that has gone up by for the current year, \nand what the average property in the area actually pays. \nThey are grouped here by police force region - \ncouncil tax includes the police precept which is added to \nthe overall bill paid by homes. Local government is not fully \nfunded by council tax. Much of the funding for councils comes \nin the form of grants from the Welsh Government, \nwhich in turn gets its funding from the UK government in London. \nIn 2017/18 a total of $\\pm 4 . 1$ bn is being divided among Wales’ 22 councils. The lions share of council cash goes on schools \n- with social services following behind, as shown in the graph above. \nResidents pay council tax based on which band their property is in, \nbased on its worth. Band D has historically been used as \nthe standard for comparing council tax levels between and across local \nauthorities. It is used to charge tax to a property that, in Wales, \nwas worth between $\\varepsilon 9 1 \\mathrm { , 0 0 1 }$ to $\\mathcal { L } 1 2 3 \\mathrm { , 0 0 0 }$ on April 2003 values. \nCouncil tax gets lower the cheaper a property is, \nand higher the more expensive a property is. \nCouncil tax figures source: Welsh Government ", + "bbox": [ + 191, + 155, + 669, + 440 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "GT: Voters will go to the polls on Thursday to determine who will represent them on local councils. ", + "bbox": [ + 189, + 452, + 802, + 467 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "CLAPS: The people of Wales are voting in a referendum on whether or not to pay council tax. ", + "bbox": [ + 191, + 478, + 769, + 492 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Article: The side’s appearance in France will be its first at a major tournament since the 1958 World Cup. Players and coaches left their base at the Vale Resort, Vale of Glamorgan, on Saturday and headed to Cardiff Airport. After a send-off from pupils from Ysgol Treganna, Cardiff, the team took off for a friendly in Sweden on Sunday. They will then head to France ahead of the team’s first game of the tournament against Slovakia on 11 June. ", + "bbox": [ + 191, + 497, + 599, + 602 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "GT: Wales’ football team has departed the country as their Euro 2016 preparations reach a climax. ", + "bbox": [ + 189, + 614, + 792, + 627 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "CLAPS: Wales’ Euro 2016 squad have arrived in France for the first time since 1958. ", + "bbox": [ + 192, + 640, + 715, + 654 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Article: The 40-year-old, from the South Bank area of Teesside, was discovered on the A66 in the early hours ”in a distressed state” with wounds to his groin after the attack. The road, from Greystones Roundabout to Church Lane in Middlesbrough, was shut earlier while searches of the area were carried out. It has now reopened. A 22-year-old man was arrested on suspicion of assault and later bailed. Cleveland Police said the injured man had been placed in an induced coma in hospital. The force said in a statement: ”Police can confirm that the man found this morning on the A66 had wounds to his groin area. ”Officers are continuing to investigate and are appealing for anyone with information to contact them.” ", + "bbox": [ + 189, + 659, + 661, + 803 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "GT: A man has been found by the side of a road with his penis cut off. ", + "bbox": [ + 191, + 814, + 627, + 828 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "CLAPS: A man is in an induced coma after being found with serious injuries on a Teesside road. ", + "bbox": [ + 187, + 840, + 789, + 854 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/7b06a4e4b6294dac117f7c75305a69fc62563eaa746f2d988b91f3db74056b40.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Article:In July,a majorbugwas discovered in the software that could let hijackers access data on up to a billion phones. Manufacturers have been slow to roll out a fix because many variations of Android are widely used. One Android expert said it was "about time" phone makers issued security fixes more quickly. Android has been working to patch a vulnerability,known as Stagefright, which could let hackers access a phone's data simply by sending somebody a video message."My guess is that this is the single largest software update the world has ever seen,”said Adrian Ludwig, Android's lead engineer for security,at hacking conference Black Hat. LG,Samsung and Google have all said a number of their handsets will get the fix,with further updates every month. Android is an open source operating system,with the software freelyavailable for phone manufacturers to modify and use on their handsets. The Google-led project does provide security fixes for the software, but phone manufacturers are responsible for sending the updates to their devices. Some phones running old versions of Android are no longer updated by the manufacturer. Many companies also deploy customised versions of Android which take time to rebuild with the security changes. Apple and BlackBerry can patch security problems more quickly because they develop both the software and the hardware for their devices. BlackBerry's software is reviewed by mobile networks before being sent to handsets,while Apple can push updates to its phones whenever it wants. "The very nature of Android is that manufacturers add their own software on top,so there have been delays in software roll-outs,”said Jack Parsons,editor of Android Magazine. "In the US it's even worse because mobile carriers often add their own software too,adding another layer of bureaucracy holding up security fixes. "There's no real villain here,that's just how the system works. But there will always be security concerns with software,so it's right that some of the manufacturers are stepping up to deal with this now."
", + "bbox": [ + 191, + 156, + 669, + 553 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "GT: Samsung, LG and Google have pledged to provide monthly security updates for smartphones running the Android operating system. ", + "bbox": [ + 191, + 563, + 617, + 589 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "CLAPS: The world’s largest software update is to be issued by Google-led Android. ", + "bbox": [ + 191, + 601, + 684, + 613 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Article: The move follows a claim by Crossmaglen Rangers player Aaron Cunningham that he was the victim of verbal abuse during the 2 December Ulster football final. The Ulster Council carried out an investigation and BBC Sport understands one Kilcoo player is to be banned for six months and another for four months. Kilcoo said they had not been notified, and the players could appeal. The two suspensions have yet to be officially confirmed by the Ulster Council. It is believed the case was the first time an allegation of racial abuse had been lodged with the provincial governing body. When an investigation was announced, Ulster GAA president Aogan O Fearghail, said anyone found ´ guilty of racism would be dealt with severely. Kilcoo released a statement saying the club condemned abuse and would co-operate with the Ulster Council’s investigation. The Gaelic Athletic Association, which governs the sport in Ireland, is to discuss how to deal with racism at its annual congress in March. ", + "bbox": [ + 189, + 617, + 632, + 818 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "GT: Two Kilcoo players are to be suspended by Ulster GAA chiefs following allegations of racial abuse. ", + "bbox": [ + 186, + 829, + 803, + 842 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "CLAPS: Two Kilcoo players have been suspended by the Ulster GAA for alleged racial abuse. ", + "bbox": [ + 191, + 853, + 743, + 867 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Table 6: Generated Questions by CLAPS from SQuAD. Answer spans are highlighted. ", + "bbox": [ + 209, + 112, + 782, + 127 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Context: ... The Broncos finished the regular season with a 12-4 record, and denied the New England Patriots a chance to defend their title from Super Bowl XLIX by defeating them 20-18 in the AFC Championship Game. They joined the Patriots, Dallas Cowboys, and Pittsburgh Steelers as one of four teams that have made eight appearances in the Super Bowl. ", + "bbox": [ + 191, + 130, + 776, + 183 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "GT: How many appearances have the Denver Broncos made in the Super Bowl? ", + "bbox": [ + 191, + 194, + 671, + 207 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "CLAPS: How many Super Bowl appearances have the Broncos made? ", + "bbox": [ + 189, + 219, + 619, + 232 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Context: In late November 2015, reports surfaced stating that “multiple acts” would perform during the halftime show. On December 3, the league confirmed that the show would be headlined by the British rock group Coldplay. On January 7, 2016, Pepsi confirmed to the Associated Press that Beyonce, who headlined the Super Bowl XLVII halftime show and ´ collaborated with Coldplay on the single “Hymn for the Weekend”, would be making an appearance. Bruno Mars, who headlined the Super Bowl XLVIII halftime show, and Mark Ronson also performed. ", + "bbox": [ + 189, + 238, + 803, + 315 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "GT: What nationality is the band Coldplay? ", + "bbox": [ + 191, + 327, + 455, + 340 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "CLAPS: What nationality was Coldplay? ", + "bbox": [ + 192, + 353, + 442, + 366 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Context: There are 13 natural reserves in Warsaw - among others, Bielany Forest, Kabaty Woods, Czerniakow Lake. About 15 kilometres (9 miles) from Warsaw, the Vistula river’s environment ´ changes strikingly and features a perfectly preserved ecosystem, with a habitat of animals that includes the otter, beaver and hundreds of bird species. There are also several lakes in Warsaw - mainly the oxbow lakes, like Czerniakow Lake, the lakes in the Łazienki or Wilan ´ ow Parks, ´ Kamionek Lake. There are lot of small lakes in the parks, but only a few are permanent - the majority are emptied before winter to clean them of plants and sediments. ", + "bbox": [ + 189, + 371, + 800, + 462 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "GT: What animals does the Vistula river’s ecosystem include? ", + "bbox": [ + 191, + 473, + 565, + 486 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "CLAPS: What animals are included in the Vistula river’s habitat? ", + "bbox": [ + 191, + 498, + 586, + 512 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Context: ”The FSO Car Factory was established in 1951. A number of vehicles have been assembled there over the decades, including the Warszawa, Syrena, Fiat 125p (under license from Fiat, later renamed FSO $1 2 5 \\mathrm { p }$ when the license expired) and the Polonez. The last two models listed were also sent abroad and assembled in a number of other countries, including Egypt and Colombia. In 1995 the factory was purchased by the South Korean car manufacturer Daewoo, which assembled the Tico, Espero,Nubia, Tacuma, Leganza, Lanos and Matiz there for the European market. In 2005 the factory was sold to AvtoZAZ, a Ukrainian car manufacturer which assembled there the Chevrolet Aveo. The license for the production of the Aveo expired in February 2011 and has since not been renewed. Currently the company is defunct.” ", + "bbox": [ + 189, + 517, + 763, + 645 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "GT: Who bought the factory in 2005? ", + "bbox": [ + 192, + 657, + 419, + 670 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "CLAPS: To whom was the factory sold in 2005? ", + "bbox": [ + 191, + 683, + 486, + 695 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/25495fb249c5e2407e119bfa513b3ed0e5b15790e940265f930ec7b91992eb0c.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Context: The Scotland Act 1998,which was passed by the Parliament of the United Kingdom and given royal assent by Queen Elizabeth II on 19 November 1998, governs the functions and role of the Scottish Parliament and delimits its legislative competence. The Scotland Act 2O12 extends the devolved competencies. For the purposes of parliamentary sovereignty, the Parliament of the United Kingdom at Westminster continues to constitute the supreme legislature of Scotland. However, under the terms of the Scotland Act, Westminster agreed to devolve some of its responsibilities over Scottish domestic policy to the Scottish Parliament. Such devolved mattersinclude education, health,agriculture and justice.
", + "bbox": [ + 189, + 700, + 805, + 881 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "GT: What does the Scotland Act of 2012 extend? ", + "bbox": [ + 191, + 892, + 490, + 906 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "CLAPS: What does the Scotland Act 2012 extend? ", + "bbox": [ + 189, + 917, + 500, + 931 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Context: Stage 1 is the first, or introductory stage of the bill, \nwhere the minister or member in charge of the bill will formally introduce it to Parliament together with its accompanying documents-Explanatory Notes, a Policy Memorandum \nsetting out the policy underlying the bill, \nand a Financial Memorandum setting out the costs and savings associated with it. \nStatements from the Presiding Officer and \nthe member in charge of the bill are also lodged indicating whether the bill is \nwithin the legislative competence of the Parliament. \nStage 1 usually takes place, initially, in the relevant committee or committees \nand is then submitted to the whole Parliament \nfor a full debate in the chamber on the general principles of the bill. \nIf the whole Parliament agrees in a vote to the general principles of the bill, it then proceeds to Stage 2. ", + "bbox": [ + 192, + 122, + 803, + 276 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "GT: Where are bills typically gestated in Stage 1? ", + "bbox": [ + 192, + 286, + 495, + 300 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "CLAPS: Where does Stage 1 usually take place? ", + "bbox": [ + 192, + 311, + 486, + 325 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Context: Moderate and reformist Islamists who accept and \nwork within the democratic process include parties like the Tunisian Ennahda Movement. Jamaat-e-Islami of Pakistan is basically a socio-political \nand democratic Vanguard party but has also gained political influence \nthrough military coup d’etat in past. ´ \nThe Islamist groups like Hezbollah in Lebanon and Hamas in Palestine participate in democratic and political process as well as armed attacks, \nseeking to abolish the state of Israel. \nRadical Islamist organizations like al-Qaeda and the Egyptian Islamic Jihad, \nand groups such as the Taliban, entirely reject democracy, \noften declaring as kuffar those Muslims who support it (see takfirism), \nas well as calling for violent/offensive jihad \nor urging and conducting attacks on a religious basis. ", + "bbox": [ + 192, + 330, + 728, + 496 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "GT: Where does Hamas originate? ", + "bbox": [ + 192, + 507, + 406, + 520 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "CLAPS: Where are Hamas located? ", + "bbox": [ + 192, + 531, + 411, + 545 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Context: Sayyid Abul Ala Maududi was an important early twentieth-century figure \nin the Islamic revival in India, and then after independence from Britain, in Pakistan. \nTrained as a lawyer he chose the profession of journalism, \nand wrote about contemporary issues and most importantly about Islam and Islamic law. \nMaududi founded the Jamaat-e-Islami party in 1941 and remained its leader until 1972. \nHowever, Maududi had much more impact through his writing \nthan through his political organising. His extremely influential books (translated into many languages) placed Islam in a modern context, and influenced not only conservative ulema \nbut liberal modernizer Islamists such as al-Faruqi, \nwhose ¨Islamization of Knowledgecarried forward some of Maududi’s key principles. ¨ ", + "bbox": [ + 192, + 550, + 803, + 679 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "GT: Where did Maududi exert the most impact? ", + "bbox": [ + 194, + 690, + 482, + 703 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "CLAPS: How did Maududi have more impact on Islam than his political organising? ", + "bbox": [ + 191, + 715, + 699, + 728 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/80b24ca374042e62491798a3f12350d6388e0d9042c2b08e9ba845f25d5fbdc3.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Context: ByLike many other mainline Protestant denominations in the United States, the United Methodist Church has experienced significant membership losses in recent decades. At the time of its formation,the UMC had about 11 million members in nearly 42.000 congregations. In 1975,membership dropped below 1O million for the first time. In 2005,there were about 8 million members in over 34,OoO congregations. Membership is concentrated primarily in the Midwest and in the South. Texas has the largest number of members,with about 1 million.
", + "bbox": [ + 189, + 732, + 792, + 901 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Context: Celoron’s expedition force consisted of about 200 Troupes de la marine and 30 Indians. The expedition covered about 3,000 miles $( 4 , 8 0 0 \\mathrm { k m } )$ between June and November 1749. \nIt went up the St. Lawrence, continued along the northern shore of Lake Ontario, \ncrossed the portage at Niagara, and followed the southern shore of Lake Erie. \nAt the Chautauqua Portage (near present-day Barcelona, New York), \nthe expedition moved inland to the Allegheny River, which it followed to the site of present-day Pittsburgh. There Celoron buried lead plates engraved with the French claim to the Ohio Country. ´ \nWhenever he encountered British merchants or fur-traders, \nCeloron informed them of the French claims on the territory and told them to leave. ", + "bbox": [ + 191, + 143, + 807, + 256 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "GT: How did Celeron handle business on trip? ", + "bbox": [ + 194, + 265, + 460, + 279 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "CLAPS: What did Celoron do when he encountered the British? ", + "bbox": [ + 192, + 290, + 563, + 301 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Context: Like many cities in Central and Eastern Europe, infrastructure in Warsaw suffered considerably during its time as an Eastern Bloc economy - though it is worth mentioning that the initial Three-Year Plan to rebuild Poland (especially Warsaw) was a major success, but what followed was very much the opposite. However, over the past decade Warsaw has seen many improvements due to solid economic growth, an increase in foreign investment as well as funding from the European Union. In particular, the city’s metro, roads, sidewalks, health care facilities and sanitation facilities have improved markedly. answer:improved markedly ", + "bbox": [ + 191, + 306, + 666, + 431 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "GT: Warsaw’s sidewalks and sanitation facilities are some examples of things which have what? ", + "bbox": [ + 191, + 440, + 746, + 454 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "CLAPS: What has happened to Warsaw’s infrastructure in the past decade? ", + "bbox": [ + 192, + 465, + 630, + 478 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Context: Several commemorative events take place every year. \nGatherings of thousands of people on the banks of the Vistula \non Midsummer’s Night for a festival called Wianki (Polish for Wreaths) have become a tradition and a yearly event in the programme of cultural events in Warsaw. The festival traces its roots to a peaceful pagan ritual \nwhere maidens would float their wreaths of herbs on the water to predict when they would be married, and to whom. \nBy the 19th century this tradition had become a festive event, \nand it continues today. The city council organize concerts and other events. Each Midsummer’s Eve, apart from the official floating of wreaths, \njumping over fires, looking for the fern flower, \nthere are musical performances, dignitaries’ speeches, fairs \nand fireworks by the river bank. ", + "bbox": [ + 191, + 483, + 643, + 642 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "GT: How man people gather along the banks of the Vistula for the Wianki festival? ", + "bbox": [ + 191, + 652, + 673, + 665 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "CLAPS: How many people gather on the banks of the Vistula on Midsummer’s Night for a festival called Wianki? ", + "bbox": [ + 192, + 678, + 684, + 702 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Context: The origin of the legendary figure is not fully known. The best-known legend, by Artur Oppman, is that long ago two of Triton’s daughters set out on a journey through the depths of the oceans and seas. One of them decided to stay on the coast of Denmark and can be seen sitting at the entrance to the port of Copenhagen. The second mermaid reached the mouth of the Vistula River and plunged into its waters. She stopped to rest on a sandy beach by the village of Warszowa, where fishermen came to admire her beauty and listen to her beautiful voice. A greedy merchant also heard her songs; he followed the fishermen and captured the mermaid. ", + "bbox": [ + 191, + 707, + 632, + 830 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "GT: What did a greedy merchant do to the mermaid? ", + "bbox": [ + 191, + 840, + 501, + 853 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "CLAPS: What did Oppman do to the mermaid? ", + "bbox": [ + 192, + 864, + 472, + 877 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Context: Warsaw remained the capital of the Polish-Lithuanian \nCommonwealth until 1796, when it was annexed by the Kingdom of Prussia to become the capital of the province of South Prussia. \nLiberated by Napoleon’s army in 1806, Warsaw was made \nthe capital of the newly created Duchy of Warsaw. \nFollowing the Congress of Vienna of 1815, Warsaw became the centre of the Congress Poland, a constitutional monarchy under a personal union \nwith Imperial Russia. The Royal University of Warsaw was established in 1816. ", + "bbox": [ + 192, + 218, + 718, + 330 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "GT: How long was Warsaw the capital of the Polish-Lithuanian Commonwealth? ", + "bbox": [ + 194, + 343, + 725, + 357 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "CLAPS: How long did Warsaw remain the capital of the Polish-Lithuanian Commonwealth? ", + "bbox": [ + 191, + 371, + 803, + 385 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Context: John Paul II’s visits to his native country in 1979 and 1983 brought support to the budding solidarity movement and encouraged the growing anti-communist fervor there. In 1979, less than a year after becoming pope, John Paul celebrated Mass in Victory Square in Warsaw and ended his sermon with a call to ”renew the face” of Poland: Let Thy Spirit descend! Let Thy Spirit descend and renew the face of the land! This land! These words were very meaningful for the Polish citizens who understood them as the incentive for the democratic changes. GT: What is St. John’s Cathedral an example of, stylistically? ", + "bbox": [ + 191, + 390, + 653, + 530 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "CLAPS: St. John’s Cathedral is a typical example of what style? ", + "bbox": [ + 192, + 542, + 622, + 558 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Context: Gothic architecture is represented in the majestic churches \nbut also at the burgher houses and fortifications. \nThe most significant buildings are St. John’s Cathedral (14th century), \nthe temple is a typical example of the so-called Masovian gothic style, \nSt. Mary’s Church (1411), a town house of Burbach family (14th century), \nGunpowder Tower (after 1379) and the Royal Castle Curia Maior (140720131410). ˘ The most notable examples of Renaissance architecture in the city \nare the house of Baryczko merchant family (1562), building called ”The Negro” (early 17th century) and Salwator tenement (1632). The most interesting examples of mannerist architecture are the Royal Castle (159620131619) and ˘ \nthe Jesuit Church (160920131626) at Old Town. ˘ \nAmong the first structures of the early baroque the most important \nare St. Hyacinth’s Church (1603-1639) and Sigismund’s Column (1644). ", + "bbox": [ + 192, + 563, + 736, + 750 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "GT: What is St. John’s Cathedral an example of, stylistically? ", + "bbox": [ + 194, + 761, + 604, + 776 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "CLAPS: St. John’s Cathedral is a typical example of what style? ", + "bbox": [ + 192, + 789, + 622, + 804 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/d0af8068649fcf125d5cdad79581add0f0880f64c684d2642727d391a51da88c.jpg", + "table_caption": [ + "Table 7: Translation of Romanian by CLAPS from WMT’16 RO-EN. " + ], + "table_footnote": [], + "table_body": "
RO: De partea cealalta, 47% dintre alegatoriirepublicani afirma ca ar fi,nemultumiti” sau ,suparati” daca favoritul Jeb Bush castigä cursa pentru nominalizare.
GT: On the flip side,47 percent of Republican Voters say they would be "dissatisfied" or "upset"if establishment favorite Jeb Bush becomes the nominee.
CLAPS: On the other hand,47% of Republican voters say they would be "unsatisfied"or "uneasy" if the winer
Jeb Bush wins the nomination race. RO:Datoria va deveni o problema importanta.
GT: Debt will become a big issue.
CLAPS: Debt will become an important issue.
RO: Se pare ca Moreno a avut noroc si a scapat de cartonasul rosu si de concesia unei lovituri de pedeapsa.
GT: Moreno appeared fortunate to escape a red card and the concession of a spot-kick.
CLAPS: It seems that Moreno has been lucky and escaped the red card and the concession of a penalty kick.
RO: Astfel de decizii nu se iau niciodata usor.
GT: Such decisions are never taken lightly.
CLAPS: Such decisions are never made easily.
RO: Toate astea au cu certitudine un cost.
GT: All that stuff sure does take a toll.
CLAPS: All of this certainly has a cost.
RO:Astfel de decizii nu se iau niciodata usor.
GT: Such decisions are never taken lightly.
CLAPS: Such decisions are never made easily. RO:Deci, nu am mai gasit niciun sens de a continua pe acest drum.
GT: So,I had no reason to continue on this path.
CLAPS: So there is no point in continuing on this road.
RO: Profitul a crescut cu 5%, pana la 12,3 miliarde dolari.
GT: Revenue rose 5 percent, to $12.3 billion.
CLAPS: Profit has increased by between 5% and $12.3 billion.
RO:,Este imposibil ca toata lumea sa vina in Europa”- Dalai Lama - RT news
GT: "It is impossible for everyone to come to Europe" - Dalai Lama - RT News
CLAPS: "It is impossible for everyone to come to Europe" - the Dalai Lama - RT news
RO: Spune povestea detinutului asa cum isi doreste acesta,obtine acces.
GT: Tellthe inmate's story the way he likes it, get access.
CLAPS: It tells the detainee's story as he wants, it gets acess.
RO: Peter Moody a afirmat cä stewarzii Racing Victoria sau incercat sa infiltreze un spion la grajdurile sale anul trecut si a amenintat cä se retrage imediat din curse.
GT: Peter Moody has alleged Racing Victoria stewards attmpted to plant a spy in his stables last year and threatened to quit racing immediately.
CLAPS: Peter Moody said stewarzia Racing Victoria or tried to infiltrate a spy at her track last year and
", + "bbox": [ + 176, + 151, + 821, + 862 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "RO: Corbyn, Tsipras și Syriza în Grecia, Podemos în Spania, chiar Bernie Sanders în Statele Unite își alimentează retorica populistă din frustrările acumulate în societățile oocidentale. ", + "bbox": [ + 176, + 93, + 813, + 119 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "GT: Corbyn, Tsipras and Syriza in Greece, Podemos in Spain, even Bernie Sanders in the United States feed their populist rhetoric with the frustrations accumulated in the Western world. ", + "bbox": [ + 179, + 122, + 810, + 148 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "CLAPS: Corbyn, Tsipras and Syriza in Greece, Podemos in Spain, even Bernie Sanders in the United States are fuelling their populist rhetoric from the frustrations gained in the occident societies. ", + "bbox": [ + 181, + 151, + 802, + 178 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "RO: Pentru România, chiar și cota voluntară propusă de București, în cuantum de circa 1500 de suflete, ne depășește cu mult bunele intenții de solidaritate cu Uniunea Europeană exprimate în ultimele luni. ", + "bbox": [ + 181, + 183, + 774, + 210 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "GT: For Romania, even the voluntary quota proposed by Bucharest, amounting to about 1,500 souls, surpasses by ", + "bbox": [ + 181, + 214, + 810, + 241 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/d66afa1b2388e54c0336b32250ab4945d60a715d935430fb19fd532a8a85d39d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
far our good intentions of solidarity with the European Union expressed in recent months.
CLAPS: For Romania, even Bucharest's proposed voluntary quota,amounting to around 1 50O souls,goes far beyond our good intentions of solidarity with the European Union expressed in recent months.
RO: 7.0oo de euro pe an pentru a inchiria Clubul Pogor.
GT: 7,00o Euro per year to rent Pogor Club.
CLAPS: 7,0o0 euros per year to rent the Pogor Club.
RO: Crina a fost internata impreuna cubunicaei la spitalul municipal din Pascani,iar tot atunci tatal fetei, Costica Balcan, a plecat la munca in orasul Alexandria, judetul Teleorman.
GT: Crina and her grandmother were admited to the hospital of the municipalityof Pascani,and on the same day her father, Costica Balcan went to work in the city of Alexandria, Teleorman County.
CLAPS:The crisis was admited with her grandmother to the municipal hospital in Pascani,and the girl's father, Costica Balcan, also went to work in the town of Alexandria, Teleorman County.
RO: In tara, parintii Crinei, Alina si Costica,au continuat separat cautarile, fiecare pe unde a putut.
GT: I was just a little girl when I found out.
CLAPS:Iwas small when I found out.
RO: Cu o saptamana inainte de intalnire nici nu prea mai putea sä doarma.
GT: A week before the meeting he hadn't really been able to sleep.
CLAPS: A week before the meeting, we could not even sleep much longer.
RO: "Casa iti vezi copilul dupa atatia ani,esteceva",ne spune onepoata,venita si eacu Costica de la Ruginoasa. GT: "It's something extraordinary to see your baby after so many years",says a niece who came with Costica from
Ruginoasa. CLAPS: "To see your child after so many years,it is something," says a litle girl, who also comes with Costica
from Ruginoasa.
RO: In Piata Uniri, subcerul inceputului de toamna,familia din Ruginoasa isiuneste din nou destinul cu fata lor din Palermo. GT: In Piata Uniri,beneath the autumn sky,the family from Ruginoasa joins theirdestiny with their daughter in
Palermo. CLAPS: In the Square of the Union,under the skies ofthe early autumn,the family of Ruginoasa is once again
uniting their destiny with their mother in Palermo.
RO: "Sper sa am o amintire frumoasa cu ei, pentru ca acum sunt atat de multe lucruri de spus", ne marturiseste Crina. GT: "I hope to gain a beautiful memory after meting them, because now there are so many things to say,"
confesses Crina. CLAPS: "Ihope to have a nice memory with them,because there is so much to say now," Crina tels us.
RO: In weekendul care a trecut,Alina si Costica si-au vazut pentru prima data fata pierduta pe holurile spitalului
din Iasi,in urma cu 20 ani. GT: Last weekend,Alina and Costica saw the girl who was lost on thehals of the hospitalin Iasi for the first time
in 20 years. CLAPS: Forthe firsttime in the weekend,Alina and Costica saw their girllostin hospital rooms in Jasmine 20
yearsago. RO: "A durat pana la urma cam oluna de zile pana sa il gasim pe tatal biologic al fetei", ne explica comisarul sef
Romica Ichim. GT: Then they gave me some pointers,only to find that the person I was directed to wasn't he person I was looking for.
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(a) Training seq2seq with teacher forcing. (b) Na¨ıve contrastive learning with randomly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 201, + 496, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 496, + 214 + ], + "score": 1.0, + "content": "sampled negative examples. (c) Our method, CLAPS, which generates hard negative and positive examples.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 220, + 504, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "Another approach is to use RL or gumbel softmax (Jang et al., 2017) to match the distribution of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "generated sentences to that of the ground truth, in which case the reward is the discriminator output", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "from a Generative Adversarial Network (GAN) (Zhang et al., 2018; 2017; Yu et al., 2017). Although", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 252, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 267 + ], + "score": 1.0, + "content": "the aforementioned approaches improve the performance of the seq2seq models on text generation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 264, + 488, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 488, + 278 + ], + "score": 1.0, + "content": "tasks, they either require a vast amount of effort in tuning hyperparameters or stabilize training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 363, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 364, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 364, + 294 + ], + "score": 1.0, + "content": "In this work, we propose to mitigate the exposure bias problem", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 291, + 364, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 364, + 305 + ], + "score": 1.0, + "content": "with a simple yet effective approach, in which we contrast a pos-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 303, + 364, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 364, + 316 + ], + "score": 1.0, + "content": "itive pair of input and output sequence to negative pairs, to ex-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 314, + 364, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 364, + 327 + ], + "score": 1.0, + "content": "pose the model to various valid or incorrect sentences. Na¨ıvely,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 326, + 364, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 364, + 338 + ], + "score": 1.0, + "content": "we can construct negative pairs by simply using random non-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 336, + 365, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 365, + 349 + ], + "score": 1.0, + "content": "target sequences from the batch (Chen et al., 2020). However,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 348, + 364, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 364, + 360 + ], + "score": 1.0, + "content": "such a na¨ıve construction yields meaningless negative exam-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 358, + 364, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 364, + 370 + ], + "score": 1.0, + "content": "ples that are already well-discriminated in the embedding space", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 369, + 364, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 364, + 381 + ], + "score": 1.0, + "content": "(Fig. 1-(b)), which we highlight as the reason why existing meth-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 380, + 364, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 364, + 392 + ], + "score": 1.0, + "content": "ods (Chen et al., 2020) require large batch size. This is clearly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 390, + 364, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 364, + 403 + ], + "score": 1.0, + "content": "shown in Fig. 2, where a large portion of positive-negative pairs", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 402, + 364, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 364, + 415 + ], + "score": 1.0, + "content": "can be easily discriminated without any training, which gets", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 413, + 364, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 364, + 425 + ], + "score": 1.0, + "content": "worse as the batch size decreases as it will reduce the chance", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16 + }, + { + "type": "image", + "bbox": [ + 372, + 325, + 503, + 420 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 371, + 284, + 504, + 324 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 371, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 371, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "Figure 2: Accuracy of classify-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 372, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 372, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "ing a positive pair from negative", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 370, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 370, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "pairs varying batch size without any", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 371, + 313, + 408, + 326 + ], + "spans": [ + { + "bbox": [ + 371, + 313, + 408, + 326 + ], + "score": 1.0, + "content": "training.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "image_body", + "bbox": [ + 372, + 325, + 503, + 420 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 372, + 325, + 503, + 420 + ], + "spans": [ + { + "bbox": [ + 372, + 325, + 503, + 420 + ], + "score": 0.964, + "type": "image", + "image_path": "876f3fbd65a3f50ad0917ec686970cfba6abaa84b103c8f8f2f38c365038e298.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 372, + 325, + 503, + 372.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 372, + 372.5, + 503, + 420.0 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "index": 26.0 + }, + { + "type": "text", + "bbox": [ + 108, + 424, + 502, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 504, + 436 + ], + "score": 1.0, + "content": "to have meaningfully difficult examples in the batch. Moreover, discriminating positive and na¨ıve", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 434, + 447, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 447, + 448 + ], + "score": 1.0, + "content": "negative pairs becomes even more easier for models pretrained on large text corpora.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "To resolve this issue, we propose principled approaches to automatically generate negative and posi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "tive pairs for constrastive learning, which we refer to as Contrastive Learning with Adversarial Per-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "turbation for Seq2seq learning (CLAPS). Specifically, we generate a negative example by adding a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "small perturbation to the hidden representation of the target sequence, such that its conditional like-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "lihood is minimized (Denoted as the red circle in Fig. 1-(c)). Conversely, we construct an additional", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "positive example (Denoted as green circle in Fig. 1-(c)) by adding a large amount of perturbation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "to the hidden representation of target sequence such that the perturbed sample is far away from the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 528, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 504, + 541 + ], + "score": 1.0, + "content": "source sequence in the embedding space, while enforcing it to have high conditional likelihood by", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "minimizing Kullback-Leibler (KL) divergence between the original conditional distribution and per-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "turbed conditional distribution. This will yield a negative example that is very close to the original", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "representation of target sequence in the embedding space but is largely dissimilar in the semantics,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "while the generated positive example is far away from the original input sequence but has the same", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "semantic as the target sequence. This will generate difficult examples that the model fails to correctly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 594, + 412, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 412, + 607 + ], + "score": 1.0, + "content": "discriminate (Fig. 1-(c), Fig.2), helping it learn with more meaningful pairs.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "To verify the efficacy of our method, we empirically show that it significantly improves the perfor-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "mance of seq2seq model on three conditional text generation tasks, namely machine translation, text", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 633, + 438, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 438, + 645 + ], + "score": 1.0, + "content": "summarization and question generation. Our contribution in this work is threefold:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 106, + 652, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 651, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 665 + ], + "score": 1.0, + "content": "• To mitigate the exposure bias problem, we propose a contrastive learning framework for con-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 115, + 662, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 115, + 662, + 505, + 677 + ], + "score": 1.0, + "content": "ditional sequence generation, which contrasts a positive pair of source and target sentence to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 115, + 674, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 115, + 674, + 505, + 686 + ], + "score": 1.0, + "content": "negative pairs in the latent embedding space, to expose the model to various valid or incorrect", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 116, + 686, + 151, + 697 + ], + "spans": [ + { + "bbox": [ + 116, + 686, + 151, + 697 + ], + "score": 1.0, + "content": "outputs.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "• To tackle the ineffectiveness of conventional approach for constructing negative and positive ex-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 115, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 115, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "amples for contrastive learning, we propose a principled method to automatically generate nega-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 116, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 116, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "tive and positive pairs, that are more difficult and allows to learn more meaningful representations.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 51 + } + ], + "page_idx": 1, + "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, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 81, + 502, + 189 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 81, + 502, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 81, + 502, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 502, + 189 + ], + "score": 0.963, + "type": "image", + "image_path": "3152b03d6604aaa99473aaf4ecace4d924367395828fc4b243fd8556ad897871.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 81, + 502, + 117.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 117.0, + 502, + 153.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 153.0, + 502, + 189.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 192, + 502, + 212 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 190, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 504, + 204 + ], + "score": 1.0, + "content": "Figure 1: Concept. (a) Training seq2seq with teacher forcing. (b) Na¨ıve contrastive learning with randomly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 201, + 496, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 496, + 214 + ], + "score": 1.0, + "content": "sampled negative examples. (c) Our method, CLAPS, which generates hard negative and positive examples.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 220, + 504, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "Another approach is to use RL or gumbel softmax (Jang et al., 2017) to match the distribution of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "generated sentences to that of the ground truth, in which case the reward is the discriminator output", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "from a Generative Adversarial Network (GAN) (Zhang et al., 2018; 2017; Yu et al., 2017). Although", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 252, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 267 + ], + "score": 1.0, + "content": "the aforementioned approaches improve the performance of the seq2seq models on text generation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 264, + 488, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 488, + 278 + ], + "score": 1.0, + "content": "tasks, they either require a vast amount of effort in tuning hyperparameters or stabilize training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 220, + 506, + 278 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 363, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 364, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 364, + 294 + ], + "score": 1.0, + "content": "In this work, we propose to mitigate the exposure bias problem", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 291, + 364, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 364, + 305 + ], + "score": 1.0, + "content": "with a simple yet effective approach, in which we contrast a pos-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 303, + 364, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 364, + 316 + ], + "score": 1.0, + "content": "itive pair of input and output sequence to negative pairs, to ex-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 314, + 364, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 364, + 327 + ], + "score": 1.0, + "content": "pose the model to various valid or incorrect sentences. Na¨ıvely,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 326, + 364, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 364, + 338 + ], + "score": 1.0, + "content": "we can construct negative pairs by simply using random non-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 336, + 365, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 365, + 349 + ], + "score": 1.0, + "content": "target sequences from the batch (Chen et al., 2020). However,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 348, + 364, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 364, + 360 + ], + "score": 1.0, + "content": "such a na¨ıve construction yields meaningless negative exam-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 358, + 364, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 364, + 370 + ], + "score": 1.0, + "content": "ples that are already well-discriminated in the embedding space", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 369, + 364, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 364, + 381 + ], + "score": 1.0, + "content": "(Fig. 1-(b)), which we highlight as the reason why existing meth-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 380, + 364, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 364, + 392 + ], + "score": 1.0, + "content": "ods (Chen et al., 2020) require large batch size. This is clearly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 390, + 364, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 364, + 403 + ], + "score": 1.0, + "content": "shown in Fig. 2, where a large portion of positive-negative pairs", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 402, + 364, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 364, + 415 + ], + "score": 1.0, + "content": "can be easily discriminated without any training, which gets", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 413, + 364, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 364, + 425 + ], + "score": 1.0, + "content": "worse as the batch size decreases as it will reduce the chance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 424, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 504, + 436 + ], + "score": 1.0, + "content": "to have meaningfully difficult examples in the batch. Moreover, discriminating positive and na¨ıve", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 434, + 447, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 447, + 448 + ], + "score": 1.0, + "content": "negative pairs becomes even more easier for models pretrained on large text corpora.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 281, + 365, + 425 + ] + }, + { + "type": "image", + "bbox": [ + 372, + 325, + 503, + 420 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 371, + 284, + 504, + 324 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 371, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 371, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "Figure 2: Accuracy of classify-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 372, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 372, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "ing a positive pair from negative", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 370, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 370, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "pairs varying batch size without any", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 371, + 313, + 408, + 326 + ], + "spans": [ + { + "bbox": [ + 371, + 313, + 408, + 326 + ], + "score": 1.0, + "content": "training.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "image_body", + "bbox": [ + 372, + 325, + 503, + 420 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 372, + 325, + 503, + 420 + ], + "spans": [ + { + "bbox": [ + 372, + 325, + 503, + 420 + ], + "score": 0.964, + "type": "image", + "image_path": "876f3fbd65a3f50ad0917ec686970cfba6abaa84b103c8f8f2f38c365038e298.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 372, + 325, + 503, + 372.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 372, + 372.5, + 503, + 420.0 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "index": 26.0 + }, + { + "type": "text", + "bbox": [ + 108, + 424, + 502, + 446 + ], + "lines": [], + "index": 29.5, + "bbox_fs": [ + 106, + 424, + 504, + 448 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "To resolve this issue, we propose principled approaches to automatically generate negative and posi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "tive pairs for constrastive learning, which we refer to as Contrastive Learning with Adversarial Per-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "turbation for Seq2seq learning (CLAPS). Specifically, we generate a negative example by adding a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "small perturbation to the hidden representation of the target sequence, such that its conditional like-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "lihood is minimized (Denoted as the red circle in Fig. 1-(c)). Conversely, we construct an additional", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "positive example (Denoted as green circle in Fig. 1-(c)) by adding a large amount of perturbation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "to the hidden representation of target sequence such that the perturbed sample is far away from the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 528, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 504, + 541 + ], + "score": 1.0, + "content": "source sequence in the embedding space, while enforcing it to have high conditional likelihood by", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "minimizing Kullback-Leibler (KL) divergence between the original conditional distribution and per-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "turbed conditional distribution. This will yield a negative example that is very close to the original", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "representation of target sequence in the embedding space but is largely dissimilar in the semantics,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "while the generated positive example is far away from the original input sequence but has the same", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "semantic as the target sequence. This will generate difficult examples that the model fails to correctly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 594, + 412, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 412, + 607 + ], + "score": 1.0, + "content": "discriminate (Fig. 1-(c), Fig.2), helping it learn with more meaningful pairs.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 450, + 506, + 607 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "To verify the efficacy of our method, we empirically show that it significantly improves the perfor-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "mance of seq2seq model on three conditional text generation tasks, namely machine translation, text", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 633, + 438, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 438, + 645 + ], + "score": 1.0, + "content": "summarization and question generation. Our contribution in this work is threefold:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 610, + 505, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 652, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 651, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 665 + ], + "score": 1.0, + "content": "• To mitigate the exposure bias problem, we propose a contrastive learning framework for con-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 115, + 662, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 115, + 662, + 505, + 677 + ], + "score": 1.0, + "content": "ditional sequence generation, which contrasts a positive pair of source and target sentence to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 115, + 674, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 115, + 674, + 505, + 686 + ], + "score": 1.0, + "content": "negative pairs in the latent embedding space, to expose the model to various valid or incorrect", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 116, + 686, + 151, + 697 + ], + "spans": [ + { + "bbox": [ + 116, + 686, + 151, + 697 + ], + "score": 1.0, + "content": "outputs.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "• To tackle the ineffectiveness of conventional approach for constructing negative and positive ex-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 115, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 115, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "amples for contrastive learning, we propose a principled method to automatically generate nega-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 116, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 116, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "tive and positive pairs, that are more difficult and allows to learn more meaningful representations.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 651, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 506, + 96 + ], + "score": 1.0, + "content": "• We show that our proposed method, CLAPS, significantly improves the performance of seq2seq", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 115, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 115, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "model on three different tasks: machine translation, text summarization, and question generation.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 108, + 121, + 211, + 134 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 213, + 136 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 213, + 136 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 147, + 505, + 301 + ], + "lines": [ + { + "bbox": [ + 106, + 148, + 504, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 504, + 159 + ], + "score": 1.0, + "content": "Exposure Bias There are several prior works to tackle the exposure bias (Ranzato et al., 2016).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "Bengio et al. (2015) introduce scheduled sampling where the model is initially guided with the true", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "previous tokens but uses the tokens generated by the seq2seq model as the conditional input for the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "next token, as training goes on. Paulus et al. (2018); Bahdanau et al. (2017) leverage RL to maximize", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "non-differentiable rewards, so it enables to penalize the model for incorrectly generated sentences.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 202, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 506, + 215 + ], + "score": 1.0, + "content": "Another works (Zhang et al., 2017; 2018; Yu et al., 2017) train GANs to match the distribution of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "generated sequences to that of ground truth. Since sampling tokens from the generator is not differ-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 223, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 104, + 223, + 506, + 238 + ], + "score": 1.0, + "content": "entiable, they resort RL or gumbel-softmax to train the networks in end-to-end fashion. However,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "they require either a large amount of effort to tune hyperparameters or stabilize training. However,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 259 + ], + "score": 1.0, + "content": "Choshen et al. (2020) show that RL for machine translation does not optimize the expected reward", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "and the performance gain is attributed to the unrelated effects such as increasing the peakiness of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "the output distribution. Moreover, (Caccia et al., 2019) show that by tuning the temperature param-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "eter, the language models trained with MLE can be tuned to outperform GAN-based text generation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 140, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 140, + 301 + ], + "score": 1.0, + "content": "models.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "Adversarial Perturbation Many existing works, such as (Madry et al., 2018), address the robust-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "ness of neural networks to adversarial examples, which are generated by applying a small perturba-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "tions to the input samples. While adversarial robustness has been mostly explored in image domains,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "Miyato et al. (2017) adopted adversarial training to text domains. However instead of targeting ro-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "bustness to perturbed samples, they utilize the adversarial examples as augmented data, and enforce", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 362, + 504, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 504, + 374 + ], + "score": 1.0, + "content": "consistency across the predictions across original unlabeled example and its perturbation, for semi-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 387 + ], + "score": 1.0, + "content": "supervised learning. Recently Zhu et al. (2019) and Jiang et al. (2020) leverage adversarial training", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 398 + ], + "score": 1.0, + "content": "to induce the smoothness of text classifiers, to prevent overfitting to training samples. While they", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 407 + ], + "score": 1.0, + "content": "are relevant to ours, these methods do not have the notion of positive and negative examples as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 404, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 420 + ], + "score": 1.0, + "content": "they do not consider contrastive learning, and only target text classification. Moreover, they are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "computationally prohibitive since they use PGD for adversarial training, which requires iterative", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "optimization for each individual sample. Recently, Aghajanyan et al. 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A typical approach to the conditional text generation is to leverage the encoder-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "decoder architecture to parameterize the conditional distribution. 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\\theta ) , ~ \\mathbf { M } ^ { ( i ) } = f ( \\mathbf { x } ^ { ( i ) } ; \\theta ) } } \\end{array}", + "type": "interline_equation", + "image_path": "028420f1e6633eaa7a03bb9d07ed106d1aa423de0544a77cea99791d4ecd31cf.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 180, + 125, + 430, + 160.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 180, + 160.66666666666666, + 430, + 196.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 180, + 196.33333333333331, + 430, + 231.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 504, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 133, + 256 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 243, + 150, + 254 + ], + "score": 0.91, + "content": "f , g", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 237, + 363, + 256 + ], + "score": 1.0, + "content": "denote the encoder and the decoder respectively and", + "type": "text" + }, + { + "bbox": [ + 363, + 239, + 494, + 254 + ], + "score": 0.93, + "content": "\\mathbf { M } ^ { ( i ) } = [ \\mathbf { m } _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot \\mathbf { m } _ { L } ^ { ( i ) } ] \\in \\mathbb { R } ^ { d \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 237, + 506, + 256 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 253, + 400, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 379, + 267 + ], + "score": 1.0, + "content": "the concatenation of the hidden representations of the source tokens", + "type": "text" + }, + { + "bbox": [ + 379, + 254, + 396, + 264 + ], + "score": 0.88, + "content": "\\mathbf { x } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 253, + 400, + 267 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 105, + 283, + 469, + 295 + ], + "lines": [ + { + "bbox": [ + 104, + 282, + 471, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 282, + 471, + 297 + ], + "score": 1.0, + "content": "3.2 CONTRASTIVE LEARNING WITH ADVERSARIAL PERTURBATIONS FOR SEQ2SEQ", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "Since most of the seq2seq models are trained with teacher forcing where the ground truth tokens are", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "provided to maximize Eq. 1, they are never exposed to incorrectly generated tokens during training,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "which is known as the “expousre bias” problem. In order to tackle the problem, we propose a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "contrastive learning framework to expose the model to various valid or incorrect output sequences", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "for a given input sentence. Following the contrastive learning framework (Chen et al., 2020), we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "can train the model to learn the representations of the ground truth sentence by contrasting the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "positive pairs with the negative pairs, where we select the negative pairs as a random non-target", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "output sequence from the same batch. As shown in Fig. 3-(a), we project the source and target text", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "sequences onto the latent embedding space. Then we maximize the similarity between the pair of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 404, + 501, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 501, + 416 + ], + "score": 1.0, + "content": "source and target sequence, while minimizing the similarity between the negative pairs as follows:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 423, + 464, + 497 + ], + "lines": [ + { + "bbox": [ + 165, + 423, + 464, + 497 + ], + "spans": [ + { + "bbox": [ + 165, + 423, + 464, + 497 + ], + "score": 0.69, + "content": "\\begin{array} { r l } & { \\mathcal { L } _ { c o n t } ( \\theta ) = \\displaystyle \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp \\left( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( i ) } ) / \\tau \\right) } { \\sum _ { \\mathbf { z } _ { \\mathbf { y } } ^ { ( j ) } \\in S } \\exp \\left( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( j ) } ) / \\tau \\right) } } \\\\ & { \\qquad \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } = \\xi ( \\mathbf { M } ^ { ( i ) } ; \\theta ) , \\mathbf { z } _ { \\mathbf { y } } ^ { ( i ) } = \\xi ( \\mathbf { H } ^ { ( i ) } ; \\theta ) } \\\\ & { \\mathbf { \\cdot } _ { 1 } \\cdot \\cdot \\mathbf { \\cdot v } _ { T } ] ; \\theta ) : = \\mathbf { A } \\mathrm { v g P o o l } ( [ \\mathbf { u } _ { 1 } \\cdot \\cdot \\cdot \\mathbf { u } _ { T } ] ) , \\mathrm { w h e r e } \\mathbf { u } _ { t } = \\mathrm { R e L U } ( \\mathbf { W } ^ { ( 1 ) } \\mathbf { v } _ { t } + \\mathbf { b } ^ { ( 1 ) } ) } \\end{array}", + "type": "interline_equation", + "image_path": "6874ca9691671928d41da0f69e1f32c29b7e14648f1969a782970d086f1fe5f2.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 165, + 423, + 464, + 447.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 165, + 447.6666666666667, + 464, + 472.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 165, + 472.33333333333337, + 464, + 497.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 508, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 134, + 521 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 510, + 141, + 521 + ], + "score": 0.83, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "denotes the composition of affine transformation with the ReLU (Nair & Hinton, 2010)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 520, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 428, + 534 + ], + "score": 1.0, + "content": "and average pooling to compute the fixed sized representation of a sentence", + "type": "text" + }, + { + "bbox": [ + 428, + 520, + 465, + 532 + ], + "score": 0.81, + "content": "\\textbf { z } \\in \\ \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 520, + 470, + 534 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 470, + 520, + 504, + 532 + ], + "score": 0.74, + "content": "\\begin{array} { r l } { \\mathbf { H } ^ { ( i ) } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 531, + 504, + 549 + ], + "spans": [ + { + "bbox": [ + 107, + 532, + 199, + 548 + ], + "score": 0.93, + "content": "[ { \\bf h } _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot { \\bf h } _ { T } ^ { \\bar { ( i ) } } ] \\in \\mathbb { R } ^ { \\bar { d } \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 531, + 486, + 549 + ], + "score": 1.0, + "content": "is a concatenation of the decoder hidden states of the target sentence", + "type": "text" + }, + { + "bbox": [ + 486, + 533, + 504, + 546 + ], + "score": 0.87, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 546, + 507, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 261, + 564 + ], + "score": 1.0, + "content": "across all the time steps. Furthermore,", + "type": "text" + }, + { + "bbox": [ + 261, + 546, + 337, + 561 + ], + "score": 0.93, + "content": "S = \\{ \\mathbf { z } _ { \\mathbf { y } } ^ { ( j ) } : j \\neq i \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 546, + 507, + 564 + ], + "score": 1.0, + "content": "is a set of hidden representations of target", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "sentences (the objects other than circles in Fig. 3-(a)) that are randomly sampled and not paired with", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 571, + 382, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 186, + 585 + ], + "score": 1.0, + "content": "the source sentence", + "type": "text" + }, + { + "bbox": [ + 186, + 571, + 203, + 582 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 571, + 224, + 585 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 224, + 572, + 257, + 584 + ], + "score": 0.91, + "content": "\\sin ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 571, + 382, + 585 + ], + "score": 1.0, + "content": "is a cosine similarity function.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 588, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "However, training the model with na¨ıve contrastive learning framework using random non-target", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "sequences as negative examples is highly suboptimal, as described in the introduction and shown in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "Fig. 1. Many of such na¨ıve negative examples are often located far away from the positive examples", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 622, + 504, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 504, + 634 + ], + "score": 1.0, + "content": "in the embedding space from the beginning, when using the pretrained language model. Therefore,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "simply using the examples from the same batch as done in Chen et al. (2020) will result in trivial", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "negative examples and require very large batch size to enable sampling meaningful negative pairs", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "within the same batch. Moreover, generating positive examples for text sequences is not a trivial", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "problem either since for text domains, we do not have a well-defined set of augmentation methods", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "that preserves the input semantics, unlike with the image domains. To tackle such difficulties, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "propose a principled method to automatically construct the adversarial negative and positive ex-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "amples, such that the samples are difficult for the model to classify correctly. These adversarial", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "positive/negative pairs can guide the model to learn a more accurate representation of the target text", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 481, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 481, + 733 + ], + "score": 1.0, + "content": "sequence, by identifying which features make the output positive or negative (See Fig. 1-(c)).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34 + } + ], + "page_idx": 3, + "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": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 156, + 96 + ], + "score": 1.0, + "content": "with length", + "type": "text" + }, + { + "bbox": [ + 156, + 83, + 164, + 92 + ], + "score": 0.68, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 82, + 506, + 96 + ], + "score": 1.0, + "content": ". A typical approach to the conditional text generation is to leverage the encoder-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "decoder architecture to parameterize the conditional distribution. 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\\theta ) , ~ \\mathbf { M } ^ { ( i ) } = f ( \\mathbf { x } ^ { ( i ) } ; \\theta ) } } \\end{array}", + "type": "interline_equation", + "image_path": "028420f1e6633eaa7a03bb9d07ed106d1aa423de0544a77cea99791d4ecd31cf.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 180, + 125, + 430, + 160.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 180, + 160.66666666666666, + 430, + 196.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 180, + 196.33333333333331, + 430, + 231.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 504, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 133, + 256 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 243, + 150, + 254 + ], + "score": 0.91, + "content": "f , g", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 237, + 363, + 256 + ], + "score": 1.0, + "content": "denote the encoder and the decoder respectively and", + "type": "text" + }, + { + "bbox": [ + 363, + 239, + 494, + 254 + ], + "score": 0.93, + "content": "\\mathbf { M } ^ { ( i ) } = [ \\mathbf { m } _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot \\mathbf { m } _ { L } ^ { ( i ) } ] \\in \\mathbb { R } ^ { d \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 237, + 506, + 256 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 253, + 400, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 379, + 267 + ], + "score": 1.0, + "content": "the concatenation of the hidden representations of the source tokens", + "type": "text" + }, + { + "bbox": [ + 379, + 254, + 396, + 264 + ], + "score": 0.88, + "content": "\\mathbf { x } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 253, + 400, + 267 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 237, + 506, + 267 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 283, + 469, + 295 + ], + "lines": [ + { + "bbox": [ + 104, + 282, + 471, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 282, + 471, + 297 + ], + "score": 1.0, + "content": "3.2 CONTRASTIVE LEARNING WITH ADVERSARIAL PERTURBATIONS FOR SEQ2SEQ", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "Since most of the seq2seq models are trained with teacher forcing where the ground truth tokens are", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "provided to maximize Eq. 1, they are never exposed to incorrectly generated tokens during training,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "which is known as the “expousre bias” problem. In order to tackle the problem, we propose a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "contrastive learning framework to expose the model to various valid or incorrect output sequences", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "for a given input sentence. Following the contrastive learning framework (Chen et al., 2020), we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "can train the model to learn the representations of the ground truth sentence by contrasting the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "positive pairs with the negative pairs, where we select the negative pairs as a random non-target", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "output sequence from the same batch. As shown in Fig. 3-(a), we project the source and target text", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "sequences onto the latent embedding space. Then we maximize the similarity between the pair of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 404, + 501, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 501, + 416 + ], + "score": 1.0, + "content": "source and target sequence, while minimizing the similarity between the negative pairs as follows:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 306, + 506, + 416 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 423, + 464, + 497 + ], + "lines": [ + { + "bbox": [ + 165, + 423, + 464, + 497 + ], + "spans": [ + { + "bbox": [ + 165, + 423, + 464, + 497 + ], + "score": 0.69, + "content": "\\begin{array} { r l } & { \\mathcal { L } _ { c o n t } ( \\theta ) = \\displaystyle \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp \\left( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( i ) } ) / \\tau \\right) } { \\sum _ { \\mathbf { z } _ { \\mathbf { y } } ^ { ( j ) } \\in S } \\exp \\left( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( j ) } ) / \\tau \\right) } } \\\\ & { \\qquad \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } = \\xi ( \\mathbf { M } ^ { ( i ) } ; \\theta ) , \\mathbf { z } _ { \\mathbf { y } } ^ { ( i ) } = \\xi ( \\mathbf { H } ^ { ( i ) } ; \\theta ) } \\\\ & { \\mathbf { \\cdot } _ { 1 } \\cdot \\cdot \\mathbf { \\cdot v } _ { T } ] ; \\theta ) : = \\mathbf { A } \\mathrm { v g P o o l } ( [ \\mathbf { u } _ { 1 } \\cdot \\cdot \\cdot \\mathbf { u } _ { T } ] ) , \\mathrm { w h e r e } \\mathbf { u } _ { t } = \\mathrm { R e L U } ( \\mathbf { W } ^ { ( 1 ) } \\mathbf { v } _ { t } + \\mathbf { b } ^ { ( 1 ) } ) } \\end{array}", + "type": "interline_equation", + "image_path": "6874ca9691671928d41da0f69e1f32c29b7e14648f1969a782970d086f1fe5f2.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 165, + 423, + 464, + 447.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 165, + 447.6666666666667, + 464, + 472.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 165, + 472.33333333333337, + 464, + 497.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 508, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 134, + 521 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 510, + 141, + 521 + ], + "score": 0.83, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "denotes the composition of affine transformation with the ReLU (Nair & Hinton, 2010)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 520, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 428, + 534 + ], + "score": 1.0, + "content": "and average pooling to compute the fixed sized representation of a sentence", + "type": "text" + }, + { + "bbox": [ + 428, + 520, + 465, + 532 + ], + "score": 0.81, + "content": "\\textbf { z } \\in \\ \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 520, + 470, + 534 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 470, + 520, + 504, + 532 + ], + "score": 0.74, + "content": "\\begin{array} { r l } { \\mathbf { H } ^ { ( i ) } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 531, + 504, + 549 + ], + "spans": [ + { + "bbox": [ + 107, + 532, + 199, + 548 + ], + "score": 0.93, + "content": "[ { \\bf h } _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot { \\bf h } _ { T } ^ { \\bar { ( i ) } } ] \\in \\mathbb { R } ^ { \\bar { d } \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 531, + 486, + 549 + ], + "score": 1.0, + "content": "is a concatenation of the decoder hidden states of the target sentence", + "type": "text" + }, + { + "bbox": [ + 486, + 533, + 504, + 546 + ], + "score": 0.87, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 546, + 507, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 261, + 564 + ], + "score": 1.0, + "content": "across all the time steps. Furthermore,", + "type": "text" + }, + { + "bbox": [ + 261, + 546, + 337, + 561 + ], + "score": 0.93, + "content": "S = \\{ \\mathbf { z } _ { \\mathbf { y } } ^ { ( j ) } : j \\neq i \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 546, + 507, + 564 + ], + "score": 1.0, + "content": "is a set of hidden representations of target", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "sentences (the objects other than circles in Fig. 3-(a)) that are randomly sampled and not paired with", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 571, + 382, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 186, + 585 + ], + "score": 1.0, + "content": "the source sentence", + "type": "text" + }, + { + "bbox": [ + 186, + 571, + 203, + 582 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 571, + 224, + 585 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 224, + 572, + 257, + 584 + ], + "score": 0.91, + "content": "\\sin ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 571, + 382, + 585 + ], + "score": 1.0, + "content": "is a cosine similarity function.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 509, + 507, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 588, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "However, training the model with na¨ıve contrastive learning framework using random non-target", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "sequences as negative examples is highly suboptimal, as described in the introduction and shown in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "Fig. 1. Many of such na¨ıve negative examples are often located far away from the positive examples", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 622, + 504, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 504, + 634 + ], + "score": 1.0, + "content": "in the embedding space from the beginning, when using the pretrained language model. Therefore,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "simply using the examples from the same batch as done in Chen et al. (2020) will result in trivial", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "negative examples and require very large batch size to enable sampling meaningful negative pairs", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "within the same batch. Moreover, generating positive examples for text sequences is not a trivial", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "problem either since for text domains, we do not have a well-defined set of augmentation methods", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "that preserves the input semantics, unlike with the image domains. To tackle such difficulties, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "propose a principled method to automatically construct the adversarial negative and positive ex-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "amples, such that the samples are difficult for the model to classify correctly. These adversarial", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "positive/negative pairs can guide the model to learn a more accurate representation of the target text", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 481, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 481, + 733 + ], + "score": 1.0, + "content": "sequence, by identifying which features make the output positive or negative (See Fig. 1-(c)).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 589, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 83, + 501, + 222 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 83, + 501, + 222 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 83, + 501, + 222 + ], + "spans": [ + { + "bbox": [ + 109, + 83, + 501, + 222 + ], + "score": 0.97, + "type": "image", + "image_path": "bf63475b3841ed84076bdf161013a042e960c1737dd40ffe340a99b5ed4b73a6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 83, + 501, + 129.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 129.33333333333334, + 501, + 175.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 175.66666666666669, + 501, + 222.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 226, + 505, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "Figure 3: Generation of imposters and distant-targets with perturbation. (a) We add small perturbation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 107, + 237, + 115, + 246 + ], + "score": 0.87, + "content": "\\delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 236, + 125, + 248 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 125, + 237, + 136, + 246 + ], + "score": 0.88, + "content": "\\mathbf { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 236, + 149, + 248 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 149, + 237, + 160, + 247 + ], + "score": 0.86, + "content": "\\tilde { \\mathbf { z } } _ { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "so that its conditional likelihood is minimized to generate an invalid sentence. (b) We add large", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 246, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 153, + 258 + ], + "score": 1.0, + "content": "perturbation", + "type": "text" + }, + { + "bbox": [ + 154, + 247, + 162, + 257 + ], + "score": 0.86, + "content": "\\zeta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 246, + 173, + 258 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 174, + 246, + 184, + 256 + ], + "score": 0.87, + "content": "\\mathbf { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 246, + 199, + 258 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 199, + 247, + 210, + 257 + ], + "score": 0.87, + "content": "\\hat { \\mathbf { z } } _ { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 246, + 336, + 258 + ], + "score": 1.0, + "content": "by maximizing the distance from", + "type": "text" + }, + { + "bbox": [ + 337, + 248, + 348, + 256 + ], + "score": 0.86, + "content": "\\mathbf { z _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 246, + 506, + 258 + ], + "score": 1.0, + "content": ", the representation of source sentence but", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 256, + 336, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 336, + 268 + ], + "score": 1.0, + "content": "enforcing its likelihood high to preserve the original semantics.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 272, + 254, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 255, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 255, + 286 + ], + "score": 1.0, + "content": "3.3 GENERATION OF IMPOSTERS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 293, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 104, + 291, + 504, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 473, + 307 + ], + "score": 1.0, + "content": "As shown in Fig. 3-(b), to generate a negative example, we add a small perturbation", + "type": "text" + }, + { + "bbox": [ + 473, + 292, + 504, + 304 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\delta ^ { ( i ) } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 302, + 507, + 321 + ], + "spans": [ + { + "bbox": [ + 107, + 304, + 194, + 320 + ], + "score": 0.93, + "content": "[ \\delta _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot \\delta _ { T } ^ { ( i ) } ] \\in \\mathbb { R } ^ { d \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 302, + 221, + 321 + ], + "score": 1.0, + "content": "to the", + "type": "text" + }, + { + "bbox": [ + 222, + 306, + 241, + 317 + ], + "score": 0.89, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 302, + 462, + 321 + ], + "score": 1.0, + "content": ", which is the hidden representation of target sequence", + "type": "text" + }, + { + "bbox": [ + 463, + 306, + 480, + 319 + ], + "score": 0.89, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 302, + 507, + 321 + ], + "score": 1.0, + "content": ", such", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 317, + 326, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 326, + 330 + ], + "score": 1.0, + "content": "that its conditional likelihood is minimized as follows:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 354, + 412, + 387 + ], + "lines": [ + { + "bbox": [ + 177, + 354, + 412, + 387 + ], + "spans": [ + { + "bbox": [ + 177, + 354, + 412, + 387 + ], + "score": 0.87, + "content": "p _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } ; \\mathbf { H } ^ { ( i ) } + \\delta ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } ; \\mathbf { h } _ { t } ^ { ( i ) } + \\delta _ { t } )", + "type": "interline_equation", + "image_path": "01640fade6423fd64c07385f4b9eab2406db67d65b2800c6cfbb502730e57cbc.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 177, + 354, + 412, + 370.5 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 177, + 370.5, + 412, + 387.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 157, + 389, + 464, + 405 + ], + "lines": [ + { + "bbox": [ + 159, + 385, + 464, + 409 + ], + "spans": [ + { + "bbox": [ + 159, + 389, + 399, + 405 + ], + "score": 0.81, + "content": "p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } ; \\mathbf { h } _ { t } ^ { ( i ) } + \\delta _ { t } ) = \\mathrm { s o f t m a x } \\{ \\mathbf { W } ( \\mathbf { h } _ { t } ^ { ( i ) } + \\delta _ { t } ) + \\mathbf { b } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 385, + 430, + 409 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 430, + 390, + 464, + 403 + ], + "score": 0.91, + "content": "\\delta _ { t } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 406, + 503, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 390, + 418 + ], + "score": 1.0, + "content": "The exact minimization of the conditional log likelihood with respect to", + "type": "text" + }, + { + "bbox": [ + 390, + 406, + 397, + 416 + ], + "score": 0.81, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "is intractable for deep neu-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 417, + 504, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 436, + 431 + ], + "score": 1.0, + "content": "ral networks. Following Goodfellow et al. (2015), we approximate it by linearizing", + "type": "text" + }, + { + "bbox": [ + 437, + 417, + 504, + 430 + ], + "score": 0.92, + "content": "\\log p _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } )", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 429, + 204, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 136, + 443 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 137, + 429, + 156, + 441 + ], + "score": 0.9, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 429, + 204, + 443 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 440, + 424, + 465 + ], + "lines": [ + { + "bbox": [ + 187, + 440, + 424, + 465 + ], + "spans": [ + { + "bbox": [ + 187, + 440, + 424, + 465 + ], + "score": 0.91, + "content": "\\tilde { \\mathbf { H } } ^ { ( i ) } = \\mathbf { H } ^ { ( i ) } - \\epsilon \\frac { g } { | | g | | _ { 2 } } , \\mathrm { w h e r e } g = \\nabla _ { \\mathbf { H } ^ { ( i ) } } \\log p _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } )", + "type": "interline_equation", + "image_path": "8b627f41f339e09ff089f11e61eee5714eecefa7e59bee9d36e1ba24191eb17c.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 187, + 440, + 424, + 465 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 465, + 485 + ], + "score": 1.0, + "content": "We add small perturbation to the hidden representation of each token of target sentence", + "type": "text" + }, + { + "bbox": [ + 465, + 471, + 482, + 484 + ], + "score": 0.9, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 471, + 506, + 485 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 371, + 498 + ], + "score": 1.0, + "content": "that its conditional likelihood is minimized. Thus, the perturbed", + "type": "text" + }, + { + "bbox": [ + 371, + 483, + 390, + 496 + ], + "score": 0.88, + "content": "\\tilde { \\mathbf { H } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 483, + 506, + 498 + ], + "score": 1.0, + "content": ", which we call an imposter", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 412, + 510 + ], + "score": 1.0, + "content": "(inspired by Weinberger & Saul (2009)), is semantically very dissimilar to", + "type": "text" + }, + { + "bbox": [ + 412, + 496, + 429, + 509 + ], + "score": 0.91, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 496, + 506, + 510 + ], + "score": 1.0, + "content": ", but very close to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 507, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 212, + 523 + ], + "score": 1.0, + "content": "the hidden representation", + "type": "text" + }, + { + "bbox": [ + 212, + 508, + 232, + 520 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 507, + 506, + 523 + ], + "score": 1.0, + "content": "in the embedding space (Fig. 3-(a)). This will make it non-trivial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 519, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 534 + ], + "score": 1.0, + "content": "for the sequence-to-sequence model to distinguish it from the representation of true target sequence", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 530, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 123, + 544 + ], + "score": 0.9, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 530, + 505, + 547 + ], + "score": 1.0, + "content": ". Please note while adversarial perturbations are generated similarly as in Miyato et al. (2017),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "we use them in a completely different way. While they train the model to be invariant to adversarial", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 183, + 568 + ], + "score": 1.0, + "content": "samples within the", + "type": "text" + }, + { + "bbox": [ + 183, + 556, + 189, + 564 + ], + "score": 0.74, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "-ball, we push them far away from the source sentence while pulling the ground", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "truth target sentence to the input sentence. In other words, we use the perturbed representation as an", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 576, + 356, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 356, + 589 + ], + "score": 1.0, + "content": "additional negative sample for contrastive learning as follows:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 588, + 472, + 626 + ], + "lines": [ + { + "bbox": [ + 120, + 588, + 472, + 626 + ], + "spans": [ + { + "bbox": [ + 120, + 588, + 472, + 626 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { c o n t - n e g } ( \\theta ) = \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp ( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( i ) } ) / \\tau ) } { \\sum _ { \\mathbf { z } _ { \\mathbf { y } } ^ { ( k ) } \\in S \\cup \\{ \\bar { \\mathbf { z } } _ { \\mathbf { y } } ^ { ( i ) } \\} } \\exp ( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( k ) } ) / \\tau ) } , \\mathrm { w h e r e ~ } \\tilde { \\mathbf { z } } _ { \\mathbf { y } } ^ { ( i ) } = \\xi ( \\tilde { \\mathbf { H } } ^ { ( i ) } ; \\theta _ { \\mathbf { z } _ { \\mathbf { x } } } ^ { ( i ) } ) ,", + "type": "interline_equation", + "image_path": "4901d914e010b64017527b65b8fde9ac90b02b05ca90e2d78b06242af3333ab0.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 120, + 588, + 472, + 600.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 120, + 600.6666666666666, + 472, + 613.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 120, + 613.3333333333333, + 472, + 625.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 506, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "Alternatively, we can generate an imposter by perturbing the hidden representation of target sentence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 114, + 649 + ], + "score": 0.35, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 637, + 468, + 649 + ], + "score": 1.0, + "content": "so that its conditional likelihood is minimized but very close to the source sentence", + "type": "text" + }, + { + "bbox": [ + 468, + 639, + 477, + 647 + ], + "score": 0.48, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 648, + 503, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 503, + 662 + ], + "score": 1.0, + "content": "embedding space. However, we empirically find that such a variation yields less performance gain.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 108, + 671, + 285, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 671, + 286, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 286, + 685 + ], + "score": 1.0, + "content": "3.4 GENERATION OF DISTANT-TARGETS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 692, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 503, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 487, + 705 + ], + "score": 1.0, + "content": "Moreover, as shown in Fig. 3-(c), we construct an additional positive pair of source sequence", + "type": "text" + }, + { + "bbox": [ + 487, + 691, + 503, + 703 + ], + "score": 0.87, + "content": "\\mathbf { x } ^ { ( i ) }", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 102, + 700, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 102, + 700, + 226, + 723 + ], + "score": 1.0, + "content": "by adding large perturbation", + "type": "text" + }, + { + "bbox": [ + 227, + 703, + 349, + 719 + ], + "score": 0.92, + "content": "\\boldsymbol { \\zeta } ^ { ( i ) } = [ \\zeta _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot \\zeta _ { T } ^ { ( i ) } ] \\in \\mathbb { R } ^ { d \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 700, + 362, + 723 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 363, + 705, + 382, + 717 + ], + "score": 0.9, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 700, + 506, + 723 + ], + "score": 1.0, + "content": "the hidden state of target se-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 102, + 712, + 505, + 738 + ], + "spans": [ + { + "bbox": [ + 102, + 712, + 137, + 738 + ], + "score": 1.0, + "content": "quence", + "type": "text" + }, + { + "bbox": [ + 138, + 719, + 155, + 732 + ], + "score": 0.9, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 712, + 294, + 738 + ], + "score": 1.0, + "content": ", such that cosine similarity from", + "type": "text" + }, + { + "bbox": [ + 294, + 718, + 310, + 731 + ], + "score": 0.9, + "content": "\\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 712, + 505, + 738 + ], + "score": 1.0, + "content": "is minimized, but the conditional likelihood is", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + } + ], + "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, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 83, + 501, + 222 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 83, + 501, + 222 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 83, + 501, + 222 + ], + "spans": [ + { + "bbox": [ + 109, + 83, + 501, + 222 + ], + "score": 0.97, + "type": "image", + "image_path": "bf63475b3841ed84076bdf161013a042e960c1737dd40ffe340a99b5ed4b73a6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 83, + 501, + 129.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 129.33333333333334, + 501, + 175.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 175.66666666666669, + 501, + 222.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 226, + 505, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "Figure 3: Generation of imposters and distant-targets with perturbation. (a) We add small perturbation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 107, + 237, + 115, + 246 + ], + "score": 0.87, + "content": "\\delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 236, + 125, + 248 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 125, + 237, + 136, + 246 + ], + "score": 0.88, + "content": "\\mathbf { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 236, + 149, + 248 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 149, + 237, + 160, + 247 + ], + "score": 0.86, + "content": "\\tilde { \\mathbf { z } } _ { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "so that its conditional likelihood is minimized to generate an invalid sentence. (b) We add large", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 246, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 153, + 258 + ], + "score": 1.0, + "content": "perturbation", + "type": "text" + }, + { + "bbox": [ + 154, + 247, + 162, + 257 + ], + "score": 0.86, + "content": "\\zeta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 246, + 173, + 258 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 174, + 246, + 184, + 256 + ], + "score": 0.87, + "content": "\\mathbf { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 246, + 199, + 258 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 199, + 247, + 210, + 257 + ], + "score": 0.87, + "content": "\\hat { \\mathbf { z } } _ { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 246, + 336, + 258 + ], + "score": 1.0, + "content": "by maximizing the distance from", + "type": "text" + }, + { + "bbox": [ + 337, + 248, + 348, + 256 + ], + "score": 0.86, + "content": "\\mathbf { z _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 246, + 506, + 258 + ], + "score": 1.0, + "content": ", the representation of source sentence but", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 256, + 336, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 336, + 268 + ], + "score": 1.0, + "content": "enforcing its likelihood high to preserve the original semantics.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 272, + 254, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 255, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 255, + 286 + ], + "score": 1.0, + "content": "3.3 GENERATION OF IMPOSTERS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 293, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 104, + 291, + 504, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 473, + 307 + ], + "score": 1.0, + "content": "As shown in Fig. 3-(b), to generate a negative example, we add a small perturbation", + "type": "text" + }, + { + "bbox": [ + 473, + 292, + 504, + 304 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\delta ^ { ( i ) } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 302, + 507, + 321 + ], + "spans": [ + { + "bbox": [ + 107, + 304, + 194, + 320 + ], + "score": 0.93, + "content": "[ \\delta _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot \\delta _ { T } ^ { ( i ) } ] \\in \\mathbb { R } ^ { d \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 302, + 221, + 321 + ], + "score": 1.0, + "content": "to the", + "type": "text" + }, + { + "bbox": [ + 222, + 306, + 241, + 317 + ], + "score": 0.89, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 302, + 462, + 321 + ], + "score": 1.0, + "content": ", which is the hidden representation of target sequence", + "type": "text" + }, + { + "bbox": [ + 463, + 306, + 480, + 319 + ], + "score": 0.89, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 302, + 507, + 321 + ], + "score": 1.0, + "content": ", such", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 317, + 326, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 326, + 330 + ], + "score": 1.0, + "content": "that its conditional likelihood is minimized as follows:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 291, + 507, + 330 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 354, + 412, + 387 + ], + "lines": [ + { + "bbox": [ + 177, + 354, + 412, + 387 + ], + "spans": [ + { + "bbox": [ + 177, + 354, + 412, + 387 + ], + "score": 0.87, + "content": "p _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } ; \\mathbf { H } ^ { ( i ) } + \\delta ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } ; \\mathbf { h } _ { t } ^ { ( i ) } + \\delta _ { t } )", + "type": "interline_equation", + "image_path": "01640fade6423fd64c07385f4b9eab2406db67d65b2800c6cfbb502730e57cbc.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 177, + 354, + 412, + 370.5 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 177, + 370.5, + 412, + 387.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 157, + 389, + 464, + 405 + ], + "lines": [ + { + "bbox": [ + 159, + 385, + 464, + 409 + ], + "spans": [ + { + "bbox": [ + 159, + 389, + 399, + 405 + ], + "score": 0.81, + "content": "p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } ; \\mathbf { h } _ { t } ^ { ( i ) } + \\delta _ { t } ) = \\mathrm { s o f t m a x } \\{ \\mathbf { W } ( \\mathbf { h } _ { t } ^ { ( i ) } + \\delta _ { t } ) + \\mathbf { b } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 385, + 430, + 409 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 430, + 390, + 464, + 403 + ], + "score": 0.91, + "content": "\\delta _ { t } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 159, + 385, + 464, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 406, + 503, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 390, + 418 + ], + "score": 1.0, + "content": "The exact minimization of the conditional log likelihood with respect to", + "type": "text" + }, + { + "bbox": [ + 390, + 406, + 397, + 416 + ], + "score": 0.81, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "is intractable for deep neu-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 417, + 504, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 436, + 431 + ], + "score": 1.0, + "content": "ral networks. Following Goodfellow et al. (2015), we approximate it by linearizing", + "type": "text" + }, + { + "bbox": [ + 437, + 417, + 504, + 430 + ], + "score": 0.92, + "content": "\\log p _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } )", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 429, + 204, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 136, + 443 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 137, + 429, + 156, + 441 + ], + "score": 0.9, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 429, + 204, + 443 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 406, + 505, + 443 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 440, + 424, + 465 + ], + "lines": [ + { + "bbox": [ + 187, + 440, + 424, + 465 + ], + "spans": [ + { + "bbox": [ + 187, + 440, + 424, + 465 + ], + "score": 0.91, + "content": "\\tilde { \\mathbf { H } } ^ { ( i ) } = \\mathbf { H } ^ { ( i ) } - \\epsilon \\frac { g } { | | g | | _ { 2 } } , \\mathrm { w h e r e } g = \\nabla _ { \\mathbf { H } ^ { ( i ) } } \\log p _ { \\theta } ( \\mathbf { y } ^ { ( i ) } | \\mathbf { x } ^ { ( i ) } )", + "type": "interline_equation", + "image_path": "8b627f41f339e09ff089f11e61eee5714eecefa7e59bee9d36e1ba24191eb17c.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 187, + 440, + 424, + 465 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 465, + 485 + ], + "score": 1.0, + "content": "We add small perturbation to the hidden representation of each token of target sentence", + "type": "text" + }, + { + "bbox": [ + 465, + 471, + 482, + 484 + ], + "score": 0.9, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 471, + 506, + 485 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 371, + 498 + ], + "score": 1.0, + "content": "that its conditional likelihood is minimized. Thus, the perturbed", + "type": "text" + }, + { + "bbox": [ + 371, + 483, + 390, + 496 + ], + "score": 0.88, + "content": "\\tilde { \\mathbf { H } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 483, + 506, + 498 + ], + "score": 1.0, + "content": ", which we call an imposter", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 412, + 510 + ], + "score": 1.0, + "content": "(inspired by Weinberger & Saul (2009)), is semantically very dissimilar to", + "type": "text" + }, + { + "bbox": [ + 412, + 496, + 429, + 509 + ], + "score": 0.91, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 496, + 506, + 510 + ], + "score": 1.0, + "content": ", but very close to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 507, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 212, + 523 + ], + "score": 1.0, + "content": "the hidden representation", + "type": "text" + }, + { + "bbox": [ + 212, + 508, + 232, + 520 + ], + "score": 0.88, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 507, + 506, + 523 + ], + "score": 1.0, + "content": "in the embedding space (Fig. 3-(a)). This will make it non-trivial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 519, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 534 + ], + "score": 1.0, + "content": "for the sequence-to-sequence model to distinguish it from the representation of true target sequence", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 530, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 123, + 544 + ], + "score": 0.9, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 530, + 505, + 547 + ], + "score": 1.0, + "content": ". Please note while adversarial perturbations are generated similarly as in Miyato et al. (2017),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "we use them in a completely different way. While they train the model to be invariant to adversarial", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 183, + 568 + ], + "score": 1.0, + "content": "samples within the", + "type": "text" + }, + { + "bbox": [ + 183, + 556, + 189, + 564 + ], + "score": 0.74, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "-ball, we push them far away from the source sentence while pulling the ground", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "truth target sentence to the input sentence. In other words, we use the perturbed representation as an", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 576, + 356, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 356, + 589 + ], + "score": 1.0, + "content": "additional negative sample for contrastive learning as follows:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 471, + 506, + 589 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 588, + 472, + 626 + ], + "lines": [ + { + "bbox": [ + 120, + 588, + 472, + 626 + ], + "spans": [ + { + "bbox": [ + 120, + 588, + 472, + 626 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { c o n t - n e g } ( \\theta ) = \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp ( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( i ) } ) / \\tau ) } { \\sum _ { \\mathbf { z } _ { \\mathbf { y } } ^ { ( k ) } \\in S \\cup \\{ \\bar { \\mathbf { z } } _ { \\mathbf { y } } ^ { ( i ) } \\} } \\exp ( \\sin ( \\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) } , \\mathbf { z } _ { \\mathbf { y } } ^ { ( k ) } ) / \\tau ) } , \\mathrm { w h e r e ~ } \\tilde { \\mathbf { z } } _ { \\mathbf { y } } ^ { ( i ) } = \\xi ( \\tilde { \\mathbf { H } } ^ { ( i ) } ; \\theta _ { \\mathbf { z } _ { \\mathbf { x } } } ^ { ( i ) } ) ,", + "type": "interline_equation", + "image_path": "4901d914e010b64017527b65b8fde9ac90b02b05ca90e2d78b06242af3333ab0.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 120, + 588, + 472, + 600.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 120, + 600.6666666666666, + 472, + 613.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 120, + 613.3333333333333, + 472, + 625.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 506, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "Alternatively, we can generate an imposter by perturbing the hidden representation of target sentence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 114, + 649 + ], + "score": 0.35, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 637, + 468, + 649 + ], + "score": 1.0, + "content": "so that its conditional likelihood is minimized but very close to the source sentence", + "type": "text" + }, + { + "bbox": [ + 468, + 639, + 477, + 647 + ], + "score": 0.48, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 648, + 503, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 503, + 662 + ], + "score": 1.0, + "content": "embedding space. However, we empirically find that such a variation yields less performance gain.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 626, + 506, + 662 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 671, + 285, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 671, + 286, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 286, + 685 + ], + "score": 1.0, + "content": "3.4 GENERATION OF DISTANT-TARGETS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 692, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 503, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 487, + 705 + ], + "score": 1.0, + "content": "Moreover, as shown in Fig. 3-(c), we construct an additional positive pair of source sequence", + "type": "text" + }, + { + "bbox": [ + 487, + 691, + 503, + 703 + ], + "score": 0.87, + "content": "\\mathbf { x } ^ { ( i ) }", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 102, + 700, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 102, + 700, + 226, + 723 + ], + "score": 1.0, + "content": "by adding large perturbation", + "type": "text" + }, + { + "bbox": [ + 227, + 703, + 349, + 719 + ], + "score": 0.92, + "content": "\\boldsymbol { \\zeta } ^ { ( i ) } = [ \\zeta _ { 1 } ^ { ( i ) } \\cdot \\cdot \\cdot \\zeta _ { T } ^ { ( i ) } ] \\in \\mathbb { R } ^ { d \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 700, + 362, + 723 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 363, + 705, + 382, + 717 + ], + "score": 0.9, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 700, + 506, + 723 + ], + "score": 1.0, + "content": "the hidden state of target se-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 102, + 712, + 505, + 738 + ], + "spans": [ + { + "bbox": [ + 102, + 712, + 137, + 738 + ], + "score": 1.0, + "content": "quence", + "type": "text" + }, + { + "bbox": [ + 138, + 719, + 155, + 732 + ], + "score": 0.9, + "content": "\\mathbf { y } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 712, + 294, + 738 + ], + "score": 1.0, + "content": ", such that cosine similarity from", + "type": "text" + }, + { + "bbox": [ + 294, + 718, + 310, + 731 + ], + "score": 0.9, + "content": "\\mathbf { z } _ { \\mathbf { x } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 712, + 505, + 738 + ], + "score": 1.0, + "content": "is minimized, but the conditional likelihood is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 369, + 95 + ], + "score": 1.0, + "content": "enforced to remain high. However, the exact computation of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 369, + 81, + 385, + 94 + ], + "score": 0.91, + "content": "\\zeta ^ { ( i ) }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 386, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "with such constraints is in-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "tractable. We approximate it with the following two separate stages. First, we add perturbation", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 117, + 119 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 117, + 104, + 137, + 116 + ], + "score": 0.9, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 137, + 103, + 365, + 119 + ], + "score": 1.0, + "content": "such that it minimizes the contrastive learning objective", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 366, + 105, + 402, + 118 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { c o n t } ( \\theta )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 402, + 103, + 505, + 119 + ], + "score": 1.0, + "content": "as shown in Eq. 6. Then", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "we add another perturbation to minimize the KL divergence between perturbed conditional distribu-", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 125, + 507, + 145 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 124, + 145 + ], + "score": 1.0, + "content": "tion", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 125, + 127, + 196, + 142 + ], + "score": 0.92, + "content": "p _ { \\theta } ( \\hat { y } _ { t } ^ { ( i ) } | \\hat { \\mathbf { y } } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 196, + 125, + 357, + 145 + ], + "score": 1.0, + "content": "and the original conditional distribution", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 358, + 127, + 429, + 142 + ], + "score": 0.96, + "content": "p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 429, + 125, + 507, + 145 + ], + "score": 1.0, + "content": "as shown in Eq. 7,", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 140, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 104, + 140, + 134, + 157 + ], + "score": 1.0, + "content": "where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 134, + 142, + 240, + 156 + ], + "score": 0.76, + "content": "\\overline { { \\mathbf { H } } } = [ \\overline { { \\mathbf { h } } } _ { 1 } \\cdot \\cdot \\cdot \\overline { { \\mathbf { h } } } _ { T } ] \\in \\mathbb { R } ^ { d \\times T }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 240, + 140, + 244, + 157 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 244, + 141, + 350, + 156 + ], + "score": 0.85, + "content": "\\hat { \\mathbf { H } } = [ \\hat { \\mathbf { h } } _ { 1 } \\cdot \\cdot \\cdot \\hat { \\mathbf { h } } _ { T } ] \\in \\mathbb { R } ^ { d \\times T }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 350, + 140, + 371, + 157 + ], + "score": 1.0, + "content": ", and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 372, + 144, + 399, + 155 + ], + "score": 0.9, + "content": "\\eta \\in \\mathbb { R }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 399, + 140, + 444, + 157 + ], + "score": 1.0, + "content": ". Note that", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 444, + 144, + 455, + 154 + ], + "score": 0.86, + "content": "\\theta ^ { * }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 456, + 140, + 506, + 157 + ], + "score": 1.0, + "content": "denotes the", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 232, + 168 + ], + "score": 1.0, + "content": "copied of the model parameter", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 233, + 155, + 239, + 164 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 239, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "and is considered to be constant to prevent it from being updated", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 215, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 215, + 179 + ], + "score": 1.0, + "content": "through back-propagation.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 36, + "bbox_fs": [ + 102, + 691, + 506, + 738 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 369, + 95 + ], + "score": 1.0, + "content": "enforced to remain high. However, the exact computation of", + "type": "text" + }, + { + "bbox": [ + 369, + 81, + 385, + 94 + ], + "score": 0.91, + "content": "\\zeta ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "with such constraints is in-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "tractable. We approximate it with the following two separate stages. First, we add perturbation", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 117, + 119 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 104, + 137, + 116 + ], + "score": 0.9, + "content": "\\mathbf { H } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 103, + 365, + 119 + ], + "score": 1.0, + "content": "such that it minimizes the contrastive learning objective", + "type": "text" + }, + { + "bbox": [ + 366, + 105, + 402, + 118 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { c o n t } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 103, + 505, + 119 + ], + "score": 1.0, + "content": "as shown in Eq. 6. Then", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "we add another perturbation to minimize the KL divergence between perturbed conditional distribu-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 125, + 507, + 145 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 124, + 145 + ], + "score": 1.0, + "content": "tion", + "type": "text" + }, + { + "bbox": [ + 125, + 127, + 196, + 142 + ], + "score": 0.92, + "content": "p _ { \\theta } ( \\hat { y } _ { t } ^ { ( i ) } | \\hat { \\mathbf { y } } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 125, + 357, + 145 + ], + "score": 1.0, + "content": "and the original conditional distribution", + "type": "text" + }, + { + "bbox": [ + 358, + 127, + 429, + 142 + ], + "score": 0.96, + "content": "p _ { \\theta } ( y _ { t } ^ { ( i ) } | \\mathbf { y } _ { < t } ^ { ( i ) } , \\mathbf { x } ^ { ( i ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 125, + 507, + 145 + ], + "score": 1.0, + "content": "as shown in Eq. 7,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 140, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 104, + 140, + 134, + 157 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 142, + 240, + 156 + ], + "score": 0.76, + "content": "\\overline { { \\mathbf { H } } } = [ \\overline { { \\mathbf { h } } } _ { 1 } \\cdot \\cdot \\cdot \\overline { { \\mathbf { h } } } _ { T } ] \\in \\mathbb { R } ^ { d \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 140, + 244, + 157 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 244, + 141, + 350, + 156 + ], + "score": 0.85, + "content": "\\hat { \\mathbf { H } } = [ \\hat { \\mathbf { h } } _ { 1 } \\cdot \\cdot \\cdot \\hat { \\mathbf { h } } _ { T } ] \\in \\mathbb { R } ^ { d \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 140, + 371, + 157 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 372, + 144, + 399, + 155 + ], + "score": 0.9, + "content": "\\eta \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 140, + 444, + 157 + ], + "score": 1.0, + "content": ". 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Note that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 311, + 439 + ], + "score": 1.0, + "content": "after training is done, we remove the pooling layer", + "type": "text" + }, + { + "bbox": [ + 311, + 427, + 318, + 438 + ], + "score": 0.84, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 426, + 458, + 439 + ], + "score": 1.0, + "content": "and generate text with the decoder", + "type": "text" + }, + { + "bbox": [ + 459, + 428, + 465, + 438 + ], + "score": 0.76, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 426, + 505, + 439 + ], + "score": 1.0, + "content": ", given an", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 438, + 245, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 234, + 450 + ], + "score": 1.0, + "content": "input encoded with the encoder", + "type": "text" + }, + { + "bbox": [ + 234, + 438, + 241, + 449 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 438, + 245, + 450 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 107, + 464, + 194, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 195, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 195, + 478 + ], + "score": 1.0, + "content": "4 EXPERIMENT", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 105, + 488, + 465, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 465, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 465, + 501 + ], + "score": 1.0, + "content": "We validate our method on benchmark datasets on three conditional text generation tasks.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 512, + 160, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 160, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 160, + 525 + ], + "score": 1.0, + "content": "4.1 TASKS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "Machine Translation (MT) For machine translation, we use WMT16 Romanian-English parallel", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "score": 1.0, + "content": "corpus (WMT’16 RO-EN) to train the model. We tokenize the pairs of source and target sequences", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "with the same tokenizer as Raffel et al. (2020). We finetune the pretrained T5-small model for 20", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 564, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 564, + 506, + 581 + ], + "score": 1.0, + "content": "epochs with the batch size of 128 and Adafactor (Shazeer & Stern, 2018). 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We follow the most of the experimental settings for machine translation as", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 627, + 495, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 351, + 639 + ], + "score": 1.0, + "content": "described above, except that we set the norm of perturbation,", + "type": "text" + }, + { + "bbox": [ + 352, + 629, + 358, + 639 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 627, + 376, + 639 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 376, + 629, + 382, + 637 + ], + "score": 0.62, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 627, + 495, + 639 + ], + "score": 1.0, + "content": "as 1.0 and 1.0, respectively.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "Question Generation (QG) For question generation, we aim to generate a question from a given", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 363, + 667 + ], + "score": 1.0, + "content": "answer and paragraph, i.e., we model conditional distribution", + "type": "text" + }, + { + "bbox": [ + 363, + 655, + 407, + 667 + ], + "score": 0.92, + "content": "p _ { \\theta } ( \\mathbf { y } | \\mathbf { x } , \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 655, + 437, + 667 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 437, + 657, + 456, + 666 + ], + "score": 0.52, + "content": "\\mathbf x , \\mathbf y", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 655, + 506, + 667 + ], + "score": 1.0, + "content": ", a denote a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "paragraph, question and answer, respectively. 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We tokenize the pairs of source and target sequences", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "with the same tokenizer as Raffel et al. (2020). We finetune the pretrained T5-small model for 20", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 564, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 564, + 506, + 581 + ], + "score": 1.0, + "content": "epochs with the batch size of 128 and Adafactor (Shazeer & Stern, 2018). 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As the previous", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "experimental settings, we finetune T5-small model on SQuAD dataset (Rajpurkar et al., 2016) for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 371, + 711 + ], + "score": 1.0, + "content": "20 epochs with batch size 128 and set the norm of perturbation,", + "type": "text" + }, + { + "bbox": [ + 371, + 701, + 379, + 710 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 698, + 425, + 711 + ], + "score": 1.0, + "content": "as 3.0 and", + "type": "text" + }, + { + "bbox": [ + 425, + 700, + 432, + 709 + ], + "score": 0.68, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "as 1.0. Since the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 149, + 722 + ], + "score": 1.0, + "content": "test set of", + "type": "text" + }, + { + "bbox": [ + 149, + 710, + 183, + 721 + ], + "score": 0.46, + "content": "\\mathrm { S Q u A D }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "is only accessible via leader board, we randomly split the validation set into a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 721, + 218, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 218, + 732 + ], + "score": 1.0, + "content": "validation set and a test set.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 644, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 115, + 83, + 493, + 372 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 115, + 83, + 493, + 372 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 83, + 493, + 372 + ], + "spans": [ + { + "bbox": [ + 115, + 83, + 493, + 372 + ], + "score": 0.986, + "html": "
MethodAug.BLEU-1BLEU-2BLEU-3BLEU-4BLEUF1/EM
Question Generation - SQuAD
Harvesting-QG=20.9015.16=66.05/54.62
T5-MLE41.2630.3023.3818.5421.0067.64/55.91
α-T5-MLE(α=0.7)40.8229.7922.8417.9920.5068.04/56.30
α-T5-MLE(α= 2.0)37.3527.2020.7916.3618.4165.74/54.76
T5-SSMBAPos.41.6730.5923.5318.5721.0768.47/56.37
T5-WordDropout ContrastiveNeg.41.3730.5023.5818.7121.1968.16/56.41
R3F-41.0030.1523.2618.4420.9765.84/54.10
T5-MLE-contrastive-41.2330.2823.3318.4520.9167.32/55.25
T5-CLAPS w/o negativePos.41.8730.9323.9018.9221.38
T5-CLAPS w/o positiveNeg.41.6530.6923.7118.8121.2568.26/56.41
T5-CLAPSPos.+Neg.42.3331.2924.2219.1921.5569.01/57.06
ERNIE-GEN (Xiao et al.,2020)=26.95=
Info-HCVAE (Lee et al.,2020)-==-=81.51/71.18
MachineTranslation-WMT'16RO-EN
Transformer50.3637.1828.4222.2126.17
Scratch-T5-MLE51.6237.2227.2621.1325.34
Scratch-CLAPSPos.+Neg.53.4239.5730.2423.5927.61
T5-MLE57.7644.4535.1228.2132.43
α-T5-MLE(α = 0.7)57.6344.2333.8427.9032.14
α-T5-MLE(α = 2.0)-56.0342.5933.2926.4530.72
T5-SSMBAPos.58.2344.8735.5028.4832.81
T5-WordDropout ContrastiveNeg.57.7744.4535.1228.2132.44
R3F-58.0744.8635.5728.6632.99
T5-MLE-contrastive-57.6444.1234.7427.7932.03
T5-CLAPS w/o negativePos.
T5-CLAPS w/o positiveNeg.58.81 57.9045.52 44.6036.20 35.2729.23 28.3433.50
T5-CLAPSPos.+Neg.58.9845.7236.3929.4132.55 33.96
Conneau & Lample (2019)1----38.5
", + "type": "table", + "image_path": "eca812764f69606ee590496ff08461b1a9116a4e9cda69a4e8589c35584cd217.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 83, + 493, + 179.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 179.33333333333331, + 493, + 275.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 275.66666666666663, + 493, + 371.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 107, + 374, + 502, + 394 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 372, + 504, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 504, + 386 + ], + "score": 1.0, + "content": "Table 1: BLEU scores on WMT’16 RO-EN and SQuAD for machine translation and question generation.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 383, + 347, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 347, + 396 + ], + "score": 1.0, + "content": "EM/F1 scores with BERT-base QA model for question generation.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 107, + 407, + 235, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 235, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 235, + 419 + ], + "score": 1.0, + "content": "4.2 EXPERIMENTAL SETUPS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 273, + 445 + ], + "score": 1.0, + "content": "Implementation Details For the encoder", + "type": "text" + }, + { + "bbox": [ + 274, + 434, + 281, + 445 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 433, + 335, + 445 + ], + "score": 1.0, + "content": ", and decoder", + "type": "text" + }, + { + "bbox": [ + 335, + 435, + 342, + 444 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 433, + 505, + 445 + ], + "score": 1.0, + "content": ", we use T5-small model, which is based", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 283, + 456 + ], + "score": 1.0, + "content": "on transformer with the hidden dimension,", + "type": "text" + }, + { + "bbox": [ + 283, + 444, + 321, + 454 + ], + "score": 0.89, + "content": "d = 5 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 444, + 426, + 456 + ], + "score": 1.0, + "content": ". 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Common", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 466, + 355, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 355, + 478 + ], + "score": 1.0, + "content": "Baselines We compare our method against relevant baselines.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 105, + 480, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 395, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 350, + 495 + ], + "score": 1.0, + "content": "1. 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Scratch-T5-MLE: A random initialized Transformer model that has the identical architecture to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 118, + 506, + 272, + 519 + ], + "spans": [ + { + "bbox": [ + 118, + 506, + 227, + 519 + ], + "score": 1.0, + "content": "T5, trained by maximizing", + "type": "text" + }, + { + "bbox": [ + 228, + 506, + 269, + 518 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { M L E } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 506, + 272, + 519 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 118, + 531 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 119, + 520, + 127, + 529 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 519, + 398, + 531 + ], + "score": 1.0, + "content": "-T5-MLE: T5 model trained with MLE, with varying temperature", + "type": "text" + }, + { + "bbox": [ + 398, + 521, + 406, + 529 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "in the softmax function", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 119, + 530, + 389, + 542 + ], + "spans": [ + { + "bbox": [ + 119, + 530, + 389, + 542 + ], + "score": 1.0, + "content": "when decoding the target sentences, as done in Caccia et al. 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(2020). which are generated by corrupting the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 118, + 565, + 438, + 577 + ], + "spans": [ + { + "bbox": [ + 118, + 565, + 438, + 577 + ], + "score": 1.0, + "content": "target sequences and reconstructs them using a masked language model, BERT.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "5. T5-WordDropout Contrastive: This is a T5 model trained with the contrastive learning frame-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 118, + 588, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 118, + 588, + 505, + 603 + ], + "score": 1.0, + "content": "work proposed in Yang et al. (2019), which heuristically generates negative examples by re-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 118, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 118, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "moving the most frequent word from the target sequence. 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R3F: This is a T5 model that minimizes the negative log likelihood and symmetric KL-divergence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 118, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 118, + 646, + 297, + 658 + ], + "score": 1.0, + "content": "between original conditional log likelihood", + "type": "text" + }, + { + "bbox": [ + 297, + 646, + 331, + 658 + ], + "score": 0.92, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { y } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 646, + 350, + 658 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 351, + 646, + 384, + 658 + ], + "score": 0.92, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { y } | \\tilde { \\mathbf { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "to enforce the function to be", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 114, + 655, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 114, + 655, + 180, + 677 + ], + "score": 1.0, + "content": "smooth, where", + "type": "text" + }, + { + "bbox": [ + 180, + 661, + 200, + 672 + ], + "score": 0.42, + "content": "\\tilde { \\mathbf { x } } =", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 655, + 267, + 677 + ], + "score": 1.0, + "content": "WordEmbedding", + "type": "text" + }, + { + "bbox": [ + 267, + 658, + 500, + 674 + ], + "score": 0.8, + "content": "\\mathbf { \\Phi } _ { \\cdot } ( \\mathbf { x } ) + \\mathbf { z } , \\mathbf { z } = ( z _ { 1 } , \\dots , z _ { L } ) , z _ { i } \\overset { i . i . d } { \\sim } \\mathcal { N } ( \\mathbf { 0 } , \\mathrm { d i a g } ( \\sigma _ { 1 } , \\dots , \\sigma _ { d } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 655, + 506, + 677 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 674, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 107, + 674, + 505, + 687 + ], + "score": 1.0, + "content": "7. T5-MLE-contrastive: This is a naive constrastive learning framework with positive/negative", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 118, + 686, + 398, + 699 + ], + "spans": [ + { + "bbox": [ + 118, + 686, + 398, + 699 + ], + "score": 1.0, + "content": "pairs, which maximizes the contrastive learning objective from Eq. 2.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "8. T5-CLAPS w/o positive (negative): Our proposed model which jointly maximizes the log like-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 117, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 117, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "lihood and the contrastive learning objective with imposters but does not use any distant-targets", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 118, + 721, + 173, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 721, + 173, + 734 + ], + "score": 1.0, + "content": "or imposters.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 20 + } + ], + "page_idx": 6, + "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, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 115, + 83, + 493, + 372 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 115, + 83, + 493, + 372 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 83, + 493, + 372 + ], + "spans": [ + { + "bbox": [ + 115, + 83, + 493, + 372 + ], + "score": 0.986, + "html": "
MethodAug.BLEU-1BLEU-2BLEU-3BLEU-4BLEUF1/EM
Question Generation - SQuAD
Harvesting-QG=20.9015.16=66.05/54.62
T5-MLE41.2630.3023.3818.5421.0067.64/55.91
α-T5-MLE(α=0.7)40.8229.7922.8417.9920.5068.04/56.30
α-T5-MLE(α= 2.0)37.3527.2020.7916.3618.4165.74/54.76
T5-SSMBAPos.41.6730.5923.5318.5721.0768.47/56.37
T5-WordDropout ContrastiveNeg.41.3730.5023.5818.7121.1968.16/56.41
R3F-41.0030.1523.2618.4420.9765.84/54.10
T5-MLE-contrastive-41.2330.2823.3318.4520.9167.32/55.25
T5-CLAPS w/o negativePos.41.8730.9323.9018.9221.38
T5-CLAPS w/o positiveNeg.41.6530.6923.7118.8121.2568.26/56.41
T5-CLAPSPos.+Neg.42.3331.2924.2219.1921.5569.01/57.06
ERNIE-GEN (Xiao et al.,2020)=26.95=
Info-HCVAE (Lee et al.,2020)-==-=81.51/71.18
MachineTranslation-WMT'16RO-EN
Transformer50.3637.1828.4222.2126.17
Scratch-T5-MLE51.6237.2227.2621.1325.34
Scratch-CLAPSPos.+Neg.53.4239.5730.2423.5927.61
T5-MLE57.7644.4535.1228.2132.43
α-T5-MLE(α = 0.7)57.6344.2333.8427.9032.14
α-T5-MLE(α = 2.0)-56.0342.5933.2926.4530.72
T5-SSMBAPos.58.2344.8735.5028.4832.81
T5-WordDropout ContrastiveNeg.57.7744.4535.1228.2132.44
R3F-58.0744.8635.5728.6632.99
T5-MLE-contrastive-57.6444.1234.7427.7932.03
T5-CLAPS w/o negativePos.
T5-CLAPS w/o positiveNeg.58.81 57.9045.52 44.6036.20 35.2729.23 28.3433.50
T5-CLAPSPos.+Neg.58.9845.7236.3929.4132.55 33.96
Conneau & Lample (2019)1----38.5
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We set the temperature,", + "type": "text" + }, + { + "bbox": [ + 426, + 446, + 434, + 454 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "as 0.1 for all the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "experiments. At test time, we use beam search of width 4 to generate the target sequences. 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MethodAug.Rouge-1Rouge-2Rouge-LMETEOR
Text Summarization - XSum
PTGEN-COVG一28.108.0221.7212.46
CONVS2S31.8911.5425.7513.20
Scratch-T5-MLE31.4411.0725.1813.01
Stcratch-CLAPSPos.+Neg.33.5212.5926.9114.18
T5-MLE36.1014.7229.1615.78
α-T5-MLE(α= 0.7)=36.6815.1029.7215.78
α-T5-MLE (α= 2.0)=34.1813.5327.3514.51
T5-SSMBAPos.36.5814.8129.6815.38
T5-WordDropout ContrastiveNeg.36.8815.1129.7915.77
R3F-36.9615.1229.7615.68
T5-MLE-contrastive-36.3414.8129.4115.85
T5-CLAPS w/o negativePos.37.4915.3130.4216.36
T5-CLAPS w/o positiveNeg.37.7215.4930.7416.06
T5-CLAPSPos.+Neg.37.8915.7830.5916.38
PEGASUS (Zhang et al.,2020)=47.2124.5639.25-
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T5-CLAPS: Our full model which jointly maximizes the log likelihood, contrastive learning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 118, + 318, + 342, + 331 + ], + "spans": [ + { + "bbox": [ + 118, + 318, + 342, + 331 + ], + "score": 1.0, + "content": "objective, and KL-divergence as described in the Eq. 9.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 104, + 332, + 498, + 343 + ], + "lines": [ + { + "bbox": [ + 101, + 330, + 499, + 345 + ], + "spans": [ + { + "bbox": [ + 101, + 330, + 499, + 345 + ], + "score": 1.0, + "content": "10. Scratch-CLAPS: Our full model as T5-CLAPS but with randomly initialized T5 architecture.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "Task specific baselines For machine translation, we use the Transformer (Vaswani et al., 2017)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "which consists of 6 layers of self-attention layer with 8 multi-head attention and 512 dimension, as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "an additional baseline. For QG, we additionally compare our models against Harvesting-QG (Du", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "& Cardie, 2018), which is a LSTM model with copy mechanism. For text summarization, we use", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "score": 1.0, + "content": "PTGEN-COVG (See et al., 2017) as a baseline, which uses copy mechanism and coverage to handle", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "out of vocabulary word and prevent word repetition, and CONVS2S (Narayan et al., 2018) which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 421, + 334, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 334, + 432 + ], + "score": 1.0, + "content": "uses convolutional networks as the encoder and decoder.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "Evaluation Metric Following the conventional evaluation metrics, we adopt n-gram BLEU and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "BLEU (Papineni et al., 2002) for MT and QG. For text summarization, we use Rouge (Lin & Hovy,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "2002) and Meteor (Banerjee & Lavie, 2005). 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MethodAug.Rouge-1Rouge-2Rouge-LMETEOR
Text Summarization - XSum
PTGEN-COVG一28.108.0221.7212.46
CONVS2S31.8911.5425.7513.20
Scratch-T5-MLE31.4411.0725.1813.01
Stcratch-CLAPSPos.+Neg.33.5212.5926.9114.18
T5-MLE36.1014.7229.1615.78
α-T5-MLE(α= 0.7)=36.6815.1029.7215.78
α-T5-MLE (α= 2.0)=34.1813.5327.3514.51
T5-SSMBAPos.36.5814.8129.6815.38
T5-WordDropout ContrastiveNeg.36.8815.1129.7915.77
R3F-36.9615.1229.7615.68
T5-MLE-contrastive-36.3414.8129.4115.85
T5-CLAPS w/o negativePos.37.4915.3130.4216.36
T5-CLAPS w/o positiveNeg.37.7215.4930.7416.06
T5-CLAPSPos.+Neg.37.8915.7830.5916.38
PEGASUS (Zhang et al.,2020)=47.2124.5639.25-
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T5-CLAPS: Our full model which jointly maximizes the log likelihood, contrastive learning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 118, + 318, + 342, + 331 + ], + "spans": [ + { + "bbox": [ + 118, + 318, + 342, + 331 + ], + "score": 1.0, + "content": "objective, and KL-divergence as described in the Eq. 9.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 306, + 503, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 332, + 498, + 343 + ], + "lines": [ + { + "bbox": [ + 101, + 330, + 499, + 345 + ], + "spans": [ + { + "bbox": [ + 101, + 330, + 499, + 345 + ], + "score": 1.0, + "content": "10. Scratch-CLAPS: Our full model as T5-CLAPS but with randomly initialized T5 architecture.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 101, + 330, + 499, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "Task specific baselines For machine translation, we use the Transformer (Vaswani et al., 2017)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "which consists of 6 layers of self-attention layer with 8 multi-head attention and 512 dimension, as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "an additional baseline. For QG, we additionally compare our models against Harvesting-QG (Du", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "& Cardie, 2018), which is a LSTM model with copy mechanism. For text summarization, we use", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "score": 1.0, + "content": "PTGEN-COVG (See et al., 2017) as a baseline, which uses copy mechanism and coverage to handle", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "out of vocabulary word and prevent word repetition, and CONVS2S (Narayan et al., 2018) which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 421, + 334, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 334, + 432 + ], + "score": 1.0, + "content": "uses convolutional networks as the encoder and decoder.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 354, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "Evaluation Metric Following the conventional evaluation metrics, we adopt n-gram BLEU and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "BLEU (Papineni et al., 2002) for MT and QG. For text summarization, we use Rouge (Lin & Hovy,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "2002) and Meteor (Banerjee & Lavie, 2005). As an additional performance measure for question", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "generation, we evaluate a BERT QA model on the SQuAD test set, where the QA model is trained", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "with the questions generated by each QG methods from the contexts and answers of HarvestingQA", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 492, + 392, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 392, + 504 + ], + "score": 1.0, + "content": "dataset (Du & Cardie, 2018), and report the F1 and Exact Match (EM).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 437, + 505, + 504 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 519, + 240, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 241, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 241, + 532 + ], + "score": 1.0, + "content": "4.3 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 537, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Quantitative Results We compare our model with the baseline models on WMT’16 RO-En, XSum,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "score": 1.0, + "content": "SQuAD dataset for machine translation, text summarization and question generation, respectively.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "Table 1 shows that our proposed method CLAPS significantly outperforms the other baseline, with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 250, + 583 + ], + "score": 1.0, + "content": "the performance gain of more than", + "type": "text" + }, + { + "bbox": [ + 251, + 570, + 266, + 581 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "on all tasks according to the BLEU scores. Moreover our", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "proposed method improves the performance of the randomly initialized T5 model (Scratch-CLAPS).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "For question generation, our proposed method also improves F1/EM as well as BLEU scores. 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(a) Embedding space with-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 187, + 307, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 307, + 199 + ], + "score": 1.0, + "content": "out contrastive learning. (b) Embedding space with our", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 249, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 249, + 209 + ], + "score": 1.0, + "content": "proposed contrastive learning, CLAPS.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + } + ], + "index": 4.75 + }, + { + "type": "table", + "bbox": [ + 310, + 79, + 501, + 174 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 310, + 79, + 501, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 310, + 79, + 501, + 174 + ], + "spans": [ + { + "bbox": [ + 310, + 79, + 501, + 174 + ], + "score": 0.842, + "html": "
(MT) Lupta lui Hilary a fost mai atractiva. =>(GT): Hillary's struggle was more attractive =>(Dist.): Hilary's fight was more attractive =>(Imp.): Thearies'batle fight has attractive appealing
(QG)...Von Miller... recording five solo tackles,.. =>(GT): How many solo tackles did Von Miler make at Super Bowl 50? =>(Dist.): How many solo tackles did Von Miller record at Super Bowl 50? =>(Imp.):What much tackle did was Miller record at Super Bowl 50?
(Sum.) Pieces from the board game...have been found in... China.... =>(GT): An ancient board game has been found in a Chinese Tomb. =>(Dist.):An ancient board game has been discovered in a Chinese Tomb. =>(Imp.):America's gained vast Africa most well geographical countries, 22
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As shown in Fig. 4-(b), the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 231, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 246 + ], + "score": 1.0, + "content": "model pushes away the imposter from the embedding of target sequence and pulls the embedding", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 242, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 256 + ], + "score": 1.0, + "content": "of the distant-targets to the embedding of the source sequence. 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As shown in Table 3, the distant-target example (Dist.), preserves the semantic of the original", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "score": 1.0, + "content": "target sequence (GT) with a single word replaced by a synonym (colored in green). However, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "imposters (Imp.) have completely different semantics, and often are gramatically incorrect (colored", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "in red). 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(a) Embedding space with-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 187, + 307, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 307, + 199 + ], + "score": 1.0, + "content": "out contrastive learning. (b) Embedding space with our", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 249, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 249, + 209 + ], + "score": 1.0, + "content": "proposed contrastive learning, CLAPS.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + } + ], + "index": 4.75 + }, + { + "type": "table", + "bbox": [ + 310, + 79, + 501, + 174 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 310, + 79, + 501, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 310, + 79, + 501, + 174 + ], + "spans": [ + { + "bbox": [ + 310, + 79, + 501, + 174 + ], + "score": 0.842, + "html": "
(MT) Lupta lui Hilary a fost mai atractiva. =>(GT): Hillary's struggle was more attractive =>(Dist.): Hilary's fight was more attractive =>(Imp.): Thearies'batle fight has attractive appealing
(QG)...Von Miller... recording five solo tackles,.. =>(GT): How many solo tackles did Von Miler make at Super Bowl 50? =>(Dist.): How many solo tackles did Von Miller record at Super Bowl 50? =>(Imp.):What much tackle did was Miller record at Super Bowl 50?
(Sum.) Pieces from the board game...have been found in... China.... =>(GT): An ancient board game has been found in a Chinese Tomb. =>(Dist.):An ancient board game has been discovered in a Chinese Tomb. =>(Imp.):America's gained vast Africa most well geographical countries, 22
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The answer span is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 307, + 195, + 381, + 207 + ], + "spans": [ + { + "bbox": [ + 307, + 195, + 381, + 207 + ], + "score": 1.0, + "content": "highlighted for QG.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 307, + 175, + 505, + 207 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 142, + 223 + ], + "score": 1.0, + "content": "example", + "type": "text" + }, + { + "bbox": [ + 143, + 208, + 154, + 221 + ], + "score": 0.51, + "content": "\\hat { \\bf H }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 209, + 384, + 223 + ], + "score": 1.0, + "content": "as shown in Eq. 3 and 6, 7. We apply average pooling to", + "type": "text" + }, + { + "bbox": [ + 384, + 208, + 424, + 222 + ], + "score": 0.9, + "content": "\\mathbf { M } , \\mathbf { H } , \\tilde { \\mathbf { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 209, + 446, + 223 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 446, + 208, + 457, + 221 + ], + "score": 0.61, + "content": "\\hat { \\bf H }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "and project", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "them onto two dimensional space with t-SNE (Maaten & Hinton, 2008). As shown in Fig. 4-(b), the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 231, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 246 + ], + "score": 1.0, + "content": "model pushes away the imposter from the embedding of target sequence and pulls the embedding", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 242, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 256 + ], + "score": 1.0, + "content": "of the distant-targets to the embedding of the source sequence. 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To decode them into output sequences, we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 281, + 318 + ], + "score": 1.0, + "content": "apply affine transformation and softmax to", + "type": "text" + }, + { + "bbox": [ + 281, + 304, + 292, + 316 + ], + "score": 0.66, + "content": "\\tilde { \\bf H }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 305, + 310, + 318 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 303, + 321, + 316 + ], + "score": 0.6, + "content": "\\hat { \\bf H }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "and select the most likely token at each time", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "step. 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This shows that the model are exposed to those various valid or incorrect sentences with our", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 361, + 396, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 396, + 373 + ], + "score": 1.0, + "content": "proposed contrastive learning framework with adversarial perturbations.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 282, + 506, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 377, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "Human Evaluation We further conduct a human evaluation of the 20 summaries and 20 questions", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 506, + 402 + ], + "score": 1.0, + "content": "generated by our CLAPS and T5-MLE trained for text summarization and QG task. 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For text summarization,", + "type": "text" + }, + { + "bbox": [ + 356, + 410, + 378, + 421 + ], + "score": 0.87, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 409, + 505, + 423 + ], + "score": 1.0, + "content": "of the human annotators chose", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 433, + 434 + ], + "score": 1.0, + "content": "the sentences generated by our model as better than the baseline, and for QG,", + "type": "text" + }, + { + "bbox": [ + 433, + 421, + 454, + 432 + ], + "score": 0.84, + "content": "85 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "favored the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 432, + 344, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 344, + 444 + ], + "score": 1.0, + "content": "sentences generated by our model over that of the baseline.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 376, + 506, + 444 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 193, + 472 + ], + "lines": [ + { + "bbox": [ + 104, + 459, + 196, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 196, + 475 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "To mitigate the exposure bias problem in sequence-to-sequence learning, we proposed a contrastive", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "learning framework which maximizes the similarity between ground truth input and output se-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "quence, and minimize the similarity between the input and an incorrect output sequence. 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DatasetsTrain (#)Valid (#)Test (#)Source
WMT'16 RO-EN610,3201,9991,999Romanian-English Parallel corpus.
Xsum204,04511,33211,334One-sentence summary of BBC news articles.
SQuAD86.5885,1925,378Crowd-sourced questions from Wikipedia paragraph
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For question", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "generation, we set the maximum length of question as 64 tokens and input which is concatenation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 256, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 256, + 354 + ], + "score": 1.0, + "content": "of answer and context as 384 tokens.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "Implementation We finetune the pretrained T5-small model provided from the transformers li-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "brary (Wolf et al., 2019)3 with Adafactor optimizer. 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Some of the examples are shown in Table 5,6,A. After the gener-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 477, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 504, + 488 + ], + "score": 1.0, + "content": "ation, we convert the tokens into the raw texts and compare them to the raw text of ground truth", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 488, + 504, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 504, + 500 + ], + "score": 1.0, + "content": "target sentences with the automatic evaluation metrics. 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DatasetsTrain (#)Valid (#)Test (#)Source
WMT'16 RO-EN610,3201,9991,999Romanian-English Parallel corpus.
Xsum204,04511,33211,334One-sentence summary of BBC news articles.
SQuAD86.5885,1925,378Crowd-sourced questions from Wikipedia paragraph
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Since the original test set is only accessible via the leader board of", + "type": "text" + }, + { + "bbox": [ + 464, + 241, + 501, + 253 + ], + "score": 0.58, + "content": "\\mathrm { S Q u A D ^ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 241, + 505, + 253 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 253, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 264 + ], + "score": 1.0, + "content": "we split the original validation set into our new validation and test set, following the conventions of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 264, + 244, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 244, + 275 + ], + "score": 1.0, + "content": "question generation communities.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 208, + 506, + 275 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "Preprocessing For machine translation, we download the raw text2, not the tokenized text, and use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "the same T5-tokenizer as Raffel et al. (2020) to tokenize both Romanian and English sentences. We", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "limit the input and output length to 128 tokens. For text summarization, we also use the T5-tokenizer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "as before, and limit the input length to 512 tokens and output length to 128 tokens. For question", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "generation, we set the maximum length of question as 64 tokens and input which is concatenation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 256, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 256, + 354 + ], + "score": 1.0, + "content": "of answer and context as 384 tokens.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 287, + 505, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "Implementation We finetune the pretrained T5-small model provided from the transformers li-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "brary (Wolf et al., 2019)3 with Adafactor optimizer. We set the batch size 128 and follow the default", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 388, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 399 + ], + "score": 1.0, + "content": "setting of Adafactor optimizer to finetune the T5-small models. However, the number of negative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "examples from the batch is 16 or 32 (total batch size divided by the number of GPUs), because we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "split the batch into smaller batches and distribute them to each GPU machines. We use 8 GPUs for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 420, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 432 + ], + "score": 1.0, + "content": "text summarization, and 4 GPUs for machine translation and question generation. The dimension of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 431, + 419, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 211, + 443 + ], + "score": 1.0, + "content": "hidden state of T5 model,", + "type": "text" + }, + { + "bbox": [ + 211, + 432, + 218, + 441 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 431, + 360, + 443 + ], + "score": 1.0, + "content": "is 512, so we set the hidden size of", + "type": "text" + }, + { + "bbox": [ + 360, + 433, + 367, + 441 + ], + "score": 0.55, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 431, + 419, + 443 + ], + "score": 1.0, + "content": "as the same.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 365, + 506, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "Evaluation We use beam search with beam width 4 to generate the target sentences from the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "source sentences of the test set. Some of the examples are shown in Table 5,6,A. After the gener-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 477, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 504, + 488 + ], + "score": 1.0, + "content": "ation, we convert the tokens into the raw texts and compare them to the raw text of ground truth", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 488, + 504, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 504, + 500 + ], + "score": 1.0, + "content": "target sentences with the automatic evaluation metrics. For n-gram BLEU and Meteor, we use the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 241, + 511 + ], + "score": 1.0, + "content": "implementation by Sharma et al.", + "type": "text" + }, + { + "bbox": [ + 242, + 498, + 273, + 510 + ], + "score": 0.77, + "content": "( 2 0 1 7 ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 498, + 506, + 511 + ], + "score": 1.0, + "content": ". 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"bbox": [ + 117, + 433, + 459, + 444 + ], + "score": 1.0, + "content": "Transport for London (TfL) remains confident, although it admits there will be breakdowns.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 116, + 442, + 487, + 455 + ], + "spans": [ + { + "bbox": [ + 116, + 442, + 487, + 455 + ], + "score": 1.0, + "content": "The trick will be in getting the system back up and running quickly. 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Article: Spending on the NHS should also be paid for by a dedicated tax marked on every payslip,the former health minister suggested. Under Mr Lamb's plan,taxes would not be increased as the new levy would be offset by deductions to income tax or national insurance. He has warned the NHS faces collapse without an urgent cash injection. The plans are not yet party policy and will not be put to this year's conference in Bournemouth.But Mr Lamb,the party's health spokesman, told party members he was "very interested in the idea of a dedicated NH S and care contribution - separating it out from the rest of taxation, clearly identified on your payslip."And Iam really interested in the idea of the right for local areas to raise additional funds for the NHS and care if theychoose.’The Lib Dems say he would like to implement the ideas across theUK,although,as health and social care are devolved, it is unclear how this would be enforced. Mr Lamb - who lost out to Tim Farron in a leadership election in July - proposes a cross-party commission to explore the ideas. Heintends to consult health bodies and professionals, patients,trade unions and academics.Ministers have pledged f2bn in this financial year for the NHS,and an extra f8bn by 2020. But MrLamb told the BBC that this was insuffcient and, having "seen the books”as a minister in the last government, he feared the NHS could face a funding shortfall of f3Obn by 2020. "The bottom line is with rising demand because of an ageing population we need more investment,” he said.MrLamb also warned that the social care system was "on its knees" and could collapse without a cash injection of f5bn. "I've been in the department. Ihave seen the books and Iam deeply concerned. If we carry on regardless, the system will crash." Taxpayers are already shown how much they have contributed to the health service in annual personal tax statements.An attempt to establish a cross-party commission on social care before the 2O1O election -led in part by MrLamb -collapsed in acrimony.
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Article: Spending on the NHS should also be paid for by a dedicated tax marked on every payslip,the former health minister suggested. Under Mr Lamb's plan,taxes would not be increased as the new levy would be offset by deductions to income tax or national insurance. He has warned the NHS faces collapse without an urgent cash injection. The plans are not yet party policy and will not be put to this year's conference in Bournemouth.But Mr Lamb,the party's health spokesman, told party members he was "very interested in the idea of a dedicated NH S and care contribution - separating it out from the rest of taxation, clearly identified on your payslip."And Iam really interested in the idea of the right for local areas to raise additional funds for the NHS and care if theychoose.’The Lib Dems say he would like to implement the ideas across theUK,although,as health and social care are devolved, it is unclear how this would be enforced. Mr Lamb - who lost out to Tim Farron in a leadership election in July - proposes a cross-party commission to explore the ideas. Heintends to consult health bodies and professionals, patients,trade unions and academics.Ministers have pledged f2bn in this financial year for the NHS,and an extra f8bn by 2020. But MrLamb told the BBC that this was insuffcient and, having "seen the books”as a minister in the last government, he feared the NHS could face a funding shortfall of f3Obn by 2020. "The bottom line is with rising demand because of an ageing population we need more investment,” he said.MrLamb also warned that the social care system was "on its knees" and could collapse without a cash injection of f5bn. "I've been in the department. Ihave seen the books and Iam deeply concerned. If we carry on regardless, the system will crash." Taxpayers are already shown how much they have contributed to the health service in annual personal tax statements.An attempt to establish a cross-party commission on social care before the 2O1O election -led in part by MrLamb -collapsed in acrimony.
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More than 100 schools were closed across East", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 117, + 241, + 361, + 255 + ], + "spans": [ + { + "bbox": [ + 117, + 241, + 361, + 255 + ], + "score": 1.0, + "content": "Lancashire and Oldham, with 80 shut in West Yorkshire.", + "type": "text" + } + ], + "index": 12, + "is_list_end_line": true + }, + { + "bbox": [ + 116, + 252, + 402, + 267 + ], + "spans": [ + { + "bbox": [ + 116, + 252, + 337, + 267 + ], + "score": 1.0, + "content": "BBC Weather said Buxton in Derbyshire saw up to", + "type": "text" + }, + { + "bbox": [ + 337, + 254, + 362, + 264 + ], + "score": 0.46, + "content": "1 7 \\mathrm { c m }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 252, + 402, + 267 + ], + "score": 1.0, + "content": "of snow,", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 266, + 443, + 278 + ], + "spans": [ + { + "bbox": [ + 117, + 266, + 443, + 278 + ], + "score": 1.0, + "content": "the deepest measured on Friday. 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Elliott Hudson, another West Yorkshire resident,", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 416, + 315, + 430 + ], + "spans": [ + { + "bbox": [ + 117, + 416, + 315, + 430 + ], + "score": 1.0, + "content": "said: ”Looks like I have woken up in Narnia.”", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 428, + 414, + 442 + ], + "spans": [ + { + "bbox": [ + 117, + 428, + 414, + 442 + ], + "score": 1.0, + "content": "West Yorkshire’s Liversedge FC, who have had to cancel every home", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 115, + 438, + 333, + 456 + ], + "spans": [ + { + "bbox": [ + 115, + 438, + 333, + 456 + ], + "score": 1.0, + "content": "game for the last four months due to bad weather,", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 451, + 304, + 465 + ], + "spans": [ + { + "bbox": [ + 117, + 451, + 304, + 465 + ], + "score": 1.0, + "content": "tweeted a picture of snow with the caption:", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 461, + 491, + 477 + ], + "spans": [ + { + "bbox": [ + 117, + 461, + 491, + 477 + ], + "score": 1.0, + "content": "”It’s not looking good for Liversedge FC’s home game with Worksop Town tomorrow.”", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 117, + 474, + 365, + 488 + ], + "spans": [ + { + "bbox": [ + 117, + 474, + 365, + 488 + ], + "score": 1.0, + "content": "The A628 Woodhead, A57 Snake Pass and A537 Cat and", + "type": "text" + } + ], + "index": 32, + "is_list_end_line": true + }, + { + "bbox": [ + 116, + 485, + 344, + 500 + ], + "spans": [ + { + "bbox": [ + 116, + 485, + 344, + 500 + ], + "score": 1.0, + "content": "Fiddle roads are all affected, with delays reported on", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 497, + 390, + 511 + ], + "spans": [ + { + "bbox": [ + 117, + 497, + 390, + 511 + ], + "score": 1.0, + "content": "the M65 motorway. Highways England said the A57 eastbound", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 509, + 383, + 523 + ], + "spans": [ + { + "bbox": [ + 117, + 509, + 383, + 523 + ], + "score": 1.0, + "content": "in Great Manchester is closed between M67/A560 and B6174", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 115, + 519, + 381, + 536 + ], + "spans": [ + { + "bbox": [ + 115, + 519, + 381, + 536 + ], + "score": 1.0, + "content": "due to severe weather conditions. It said teams were working", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 533, + 392, + 546 + ], + "spans": [ + { + "bbox": [ + 117, + 533, + 392, + 546 + ], + "score": 1.0, + "content": "to clear the road. 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Local government is not fully", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 116, + 205, + 354, + 217 + ], + "spans": [ + { + "bbox": [ + 116, + 205, + 354, + 217 + ], + "score": 1.0, + "content": "funded by council tax. 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Band D has historically been used as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 116, + 286, + 382, + 299 + ], + "spans": [ + { + "bbox": [ + 116, + 286, + 382, + 299 + ], + "score": 1.0, + "content": "the standard for comparing council tax levels between and across local", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 116, + 297, + 352, + 309 + ], + "spans": [ + { + "bbox": [ + 116, + 297, + 352, + 309 + ], + "score": 1.0, + "content": "authorities. 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Article:In July,a majorbugwas discovered in the software that could let hijackers access data on up to a billion phones. Manufacturers have been slow to roll out a fix because many variations of Android are widely used. One Android expert said it was "about time" phone makers issued security fixes more quickly. Android has been working to patch a vulnerability,known as Stagefright, which could let hackers access a phone's data simply by sending somebody a video message."My guess is that this is the single largest software update the world has ever seen,”said Adrian Ludwig, Android's lead engineer for security,at hacking conference Black Hat. LG,Samsung and Google have all said a number of their handsets will get the fix,with further updates every month. Android is an open source operating system,with the software freelyavailable for phone manufacturers to modify and use on their handsets. The Google-led project does provide security fixes for the software, but phone manufacturers are responsible for sending the updates to their devices. Some phones running old versions of Android are no longer updated by the manufacturer. Many companies also deploy customised versions of Android which take time to rebuild with the security changes. Apple and BlackBerry can patch security problems more quickly because they develop both the software and the hardware for their devices. BlackBerry's software is reviewed by mobile networks before being sent to handsets,while Apple can push updates to its phones whenever it wants. "The very nature of Android is that manufacturers add their own software on top,so there have been delays in software roll-outs,”said Jack Parsons,editor of Android Magazine. "In the US it's even worse because mobile carriers often add their own software too,adding another layer of bureaucracy holding up security fixes. "There's no real villain here,that's just how the system works. But there will always be security concerns with software,so it's right that some of the manufacturers are stepping up to deal with this now."
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Article:In July,a majorbugwas discovered in the software that could let hijackers access data on up to a billion phones. Manufacturers have been slow to roll out a fix because many variations of Android are widely used. One Android expert said it was "about time" phone makers issued security fixes more quickly. Android has been working to patch a vulnerability,known as Stagefright, which could let hackers access a phone's data simply by sending somebody a video message."My guess is that this is the single largest software update the world has ever seen,”said Adrian Ludwig, Android's lead engineer for security,at hacking conference Black Hat. LG,Samsung and Google have all said a number of their handsets will get the fix,with further updates every month. Android is an open source operating system,with the software freelyavailable for phone manufacturers to modify and use on their handsets. The Google-led project does provide security fixes for the software, but phone manufacturers are responsible for sending the updates to their devices. Some phones running old versions of Android are no longer updated by the manufacturer. Many companies also deploy customised versions of Android which take time to rebuild with the security changes. Apple and BlackBerry can patch security problems more quickly because they develop both the software and the hardware for their devices. BlackBerry's software is reviewed by mobile networks before being sent to handsets,while Apple can push updates to its phones whenever it wants. "The very nature of Android is that manufacturers add their own software on top,so there have been delays in software roll-outs,”said Jack Parsons,editor of Android Magazine. "In the US it's even worse because mobile carriers often add their own software too,adding another layer of bureaucracy holding up security fixes. "There's no real villain here,that's just how the system works. But there will always be security concerns with software,so it's right that some of the manufacturers are stepping up to deal with this now."
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Context: The Scotland Act 1998,which was passed by the Parliament of the United Kingdom and given royal assent by Queen Elizabeth II on 19 November 1998, governs the functions and role of the Scottish Parliament and delimits its legislative competence. The Scotland Act 2O12 extends the devolved competencies. For the purposes of parliamentary sovereignty, the Parliament of the United Kingdom at Westminster continues to constitute the supreme legislature of Scotland. However, under the terms of the Scotland Act, Westminster agreed to devolve some of its responsibilities over Scottish domestic policy to the Scottish Parliament. Such devolved mattersinclude education, health,agriculture and justice.
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Context: The Scotland Act 1998,which was passed by the Parliament of the United Kingdom and given royal assent by Queen Elizabeth II on 19 November 1998, governs the functions and role of the Scottish Parliament and delimits its legislative competence. The Scotland Act 2O12 extends the devolved competencies. For the purposes of parliamentary sovereignty, the Parliament of the United Kingdom at Westminster continues to constitute the supreme legislature of Scotland. However, under the terms of the Scotland Act, Westminster agreed to devolve some of its responsibilities over Scottish domestic policy to the Scottish Parliament. Such devolved mattersinclude education, health,agriculture and justice.
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Context: ByLike many other mainline Protestant denominations in the United States, the United Methodist Church has experienced significant membership losses in recent decades. At the time of its formation,the UMC had about 11 million members in nearly 42.000 congregations. In 1975,membership dropped below 1O million for the first time. In 2005,there were about 8 million members in over 34,OoO congregations. Membership is concentrated primarily in the Midwest and in the South. Texas has the largest number of members,with about 1 million.
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RO: De partea cealalta, 47% dintre alegatoriirepublicani afirma ca ar fi,nemultumiti” sau ,suparati” daca favoritul Jeb Bush castigä cursa pentru nominalizare.
GT: On the flip side,47 percent of Republican Voters say they would be "dissatisfied" or "upset"if establishment favorite Jeb Bush becomes the nominee.
CLAPS: On the other hand,47% of Republican voters say they would be "unsatisfied"or "uneasy" if the winer
Jeb Bush wins the nomination race. RO:Datoria va deveni o problema importanta.
GT: Debt will become a big issue.
CLAPS: Debt will become an important issue.
RO: Se pare ca Moreno a avut noroc si a scapat de cartonasul rosu si de concesia unei lovituri de pedeapsa.
GT: Moreno appeared fortunate to escape a red card and the concession of a spot-kick.
CLAPS: It seems that Moreno has been lucky and escaped the red card and the concession of a penalty kick.
RO: Astfel de decizii nu se iau niciodata usor.
GT: Such decisions are never taken lightly.
CLAPS: Such decisions are never made easily.
RO: Toate astea au cu certitudine un cost.
GT: All that stuff sure does take a toll.
CLAPS: All of this certainly has a cost.
RO:Astfel de decizii nu se iau niciodata usor.
GT: Such decisions are never taken lightly.
CLAPS: Such decisions are never made easily. RO:Deci, nu am mai gasit niciun sens de a continua pe acest drum.
GT: So,I had no reason to continue on this path.
CLAPS: So there is no point in continuing on this road.
RO: Profitul a crescut cu 5%, pana la 12,3 miliarde dolari.
GT: Revenue rose 5 percent, to $12.3 billion.
CLAPS: Profit has increased by between 5% and $12.3 billion.
RO:,Este imposibil ca toata lumea sa vina in Europa”- Dalai Lama - RT news
GT: "It is impossible for everyone to come to Europe" - Dalai Lama - RT News
CLAPS: "It is impossible for everyone to come to Europe" - the Dalai Lama - RT news
RO: Spune povestea detinutului asa cum isi doreste acesta,obtine acces.
GT: Tellthe inmate's story the way he likes it, get access.
CLAPS: It tells the detainee's story as he wants, it gets acess.
RO: Peter Moody a afirmat cä stewarzii Racing Victoria sau incercat sa infiltreze un spion la grajdurile sale anul trecut si a amenintat cä se retrage imediat din curse.
GT: Peter Moody has alleged Racing Victoria stewards attmpted to plant a spy in his stables last year and threatened to quit racing immediately.
CLAPS: Peter Moody said stewarzia Racing Victoria or tried to infiltrate a spy at her track last year and
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RO: De partea cealalta, 47% dintre alegatoriirepublicani afirma ca ar fi,nemultumiti” sau ,suparati” daca favoritul Jeb Bush castigä cursa pentru nominalizare.
GT: On the flip side,47 percent of Republican Voters say they would be "dissatisfied" or "upset"if establishment favorite Jeb Bush becomes the nominee.
CLAPS: On the other hand,47% of Republican voters say they would be "unsatisfied"or "uneasy" if the winer
Jeb Bush wins the nomination race. RO:Datoria va deveni o problema importanta.
GT: Debt will become a big issue.
CLAPS: Debt will become an important issue.
RO: Se pare ca Moreno a avut noroc si a scapat de cartonasul rosu si de concesia unei lovituri de pedeapsa.
GT: Moreno appeared fortunate to escape a red card and the concession of a spot-kick.
CLAPS: It seems that Moreno has been lucky and escaped the red card and the concession of a penalty kick.
RO: Astfel de decizii nu se iau niciodata usor.
GT: Such decisions are never taken lightly.
CLAPS: Such decisions are never made easily.
RO: Toate astea au cu certitudine un cost.
GT: All that stuff sure does take a toll.
CLAPS: All of this certainly has a cost.
RO:Astfel de decizii nu se iau niciodata usor.
GT: Such decisions are never taken lightly.
CLAPS: Such decisions are never made easily. RO:Deci, nu am mai gasit niciun sens de a continua pe acest drum.
GT: So,I had no reason to continue on this path.
CLAPS: So there is no point in continuing on this road.
RO: Profitul a crescut cu 5%, pana la 12,3 miliarde dolari.
GT: Revenue rose 5 percent, to $12.3 billion.
CLAPS: Profit has increased by between 5% and $12.3 billion.
RO:,Este imposibil ca toata lumea sa vina in Europa”- Dalai Lama - RT news
GT: "It is impossible for everyone to come to Europe" - Dalai Lama - RT News
CLAPS: "It is impossible for everyone to come to Europe" - the Dalai Lama - RT news
RO: Spune povestea detinutului asa cum isi doreste acesta,obtine acces.
GT: Tellthe inmate's story the way he likes it, get access.
CLAPS: It tells the detainee's story as he wants, it gets acess.
RO: Peter Moody a afirmat cä stewarzii Racing Victoria sau incercat sa infiltreze un spion la grajdurile sale anul trecut si a amenintat cä se retrage imediat din curse.
GT: Peter Moody has alleged Racing Victoria stewards attmpted to plant a spy in his stables last year and threatened to quit racing immediately.
CLAPS: Peter Moody said stewarzia Racing Victoria or tried to infiltrate a spy at her track last year and
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far our good intentions of solidarity with the European Union expressed in recent months.
CLAPS: For Romania, even Bucharest's proposed voluntary quota,amounting to around 1 50O souls,goes far beyond our good intentions of solidarity with the European Union expressed in recent months.
RO: 7.0oo de euro pe an pentru a inchiria Clubul Pogor.
GT: 7,00o Euro per year to rent Pogor Club.
CLAPS: 7,0o0 euros per year to rent the Pogor Club.
RO: Crina a fost internata impreuna cubunicaei la spitalul municipal din Pascani,iar tot atunci tatal fetei, Costica Balcan, a plecat la munca in orasul Alexandria, judetul Teleorman.
GT: Crina and her grandmother were admited to the hospital of the municipalityof Pascani,and on the same day her father, Costica Balcan went to work in the city of Alexandria, Teleorman County.
CLAPS:The crisis was admited with her grandmother to the municipal hospital in Pascani,and the girl's father, Costica Balcan, also went to work in the town of Alexandria, Teleorman County.
RO: In tara, parintii Crinei, Alina si Costica,au continuat separat cautarile, fiecare pe unde a putut.
GT: I was just a little girl when I found out.
CLAPS:Iwas small when I found out.
RO: Cu o saptamana inainte de intalnire nici nu prea mai putea sä doarma.
GT: A week before the meeting he hadn't really been able to sleep.
CLAPS: A week before the meeting, we could not even sleep much longer.
RO: "Casa iti vezi copilul dupa atatia ani,esteceva",ne spune onepoata,venita si eacu Costica de la Ruginoasa. GT: "It's something extraordinary to see your baby after so many years",says a niece who came with Costica from
Ruginoasa. CLAPS: "To see your child after so many years,it is something," says a litle girl, who also comes with Costica
from Ruginoasa.
RO: In Piata Uniri, subcerul inceputului de toamna,familia din Ruginoasa isiuneste din nou destinul cu fata lor din Palermo. GT: In Piata Uniri,beneath the autumn sky,the family from Ruginoasa joins theirdestiny with their daughter in
Palermo. CLAPS: In the Square of the Union,under the skies ofthe early autumn,the family of Ruginoasa is once again
uniting their destiny with their mother in Palermo.
RO: "Sper sa am o amintire frumoasa cu ei, pentru ca acum sunt atat de multe lucruri de spus", ne marturiseste Crina. GT: "I hope to gain a beautiful memory after meting them, because now there are so many things to say,"
confesses Crina. CLAPS: "Ihope to have a nice memory with them,because there is so much to say now," Crina tels us.
RO: In weekendul care a trecut,Alina si Costica si-au vazut pentru prima data fata pierduta pe holurile spitalului
din Iasi,in urma cu 20 ani. GT: Last weekend,Alina and Costica saw the girl who was lost on thehals of the hospitalin Iasi for the first time
in 20 years. CLAPS: Forthe firsttime in the weekend,Alina and Costica saw their girllostin hospital rooms in Jasmine 20
yearsago. RO: "A durat pana la urma cam oluna de zile pana sa il gasim pe tatal biologic al fetei", ne explica comisarul sef
Romica Ichim. GT: Then they gave me some pointers,only to find that the person I was directed to wasn't he person I was looking for.
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far our good intentions of solidarity with the European Union expressed in recent months.
CLAPS: For Romania, even Bucharest's proposed voluntary quota,amounting to around 1 50O souls,goes far beyond our good intentions of solidarity with the European Union expressed in recent months.
RO: 7.0oo de euro pe an pentru a inchiria Clubul Pogor.
GT: 7,00o Euro per year to rent Pogor Club.
CLAPS: 7,0o0 euros per year to rent the Pogor Club.
RO: Crina a fost internata impreuna cubunicaei la spitalul municipal din Pascani,iar tot atunci tatal fetei, Costica Balcan, a plecat la munca in orasul Alexandria, judetul Teleorman.
GT: Crina and her grandmother were admited to the hospital of the municipalityof Pascani,and on the same day her father, Costica Balcan went to work in the city of Alexandria, Teleorman County.
CLAPS:The crisis was admited with her grandmother to the municipal hospital in Pascani,and the girl's father, Costica Balcan, also went to work in the town of Alexandria, Teleorman County.
RO: In tara, parintii Crinei, Alina si Costica,au continuat separat cautarile, fiecare pe unde a putut.
GT: I was just a little girl when I found out.
CLAPS:Iwas small when I found out.
RO: Cu o saptamana inainte de intalnire nici nu prea mai putea sä doarma.
GT: A week before the meeting he hadn't really been able to sleep.
CLAPS: A week before the meeting, we could not even sleep much longer.
RO: "Casa iti vezi copilul dupa atatia ani,esteceva",ne spune onepoata,venita si eacu Costica de la Ruginoasa. GT: "It's something extraordinary to see your baby after so many years",says a niece who came with Costica from
Ruginoasa. CLAPS: "To see your child after so many years,it is something," says a litle girl, who also comes with Costica
from Ruginoasa.
RO: In Piata Uniri, subcerul inceputului de toamna,familia din Ruginoasa isiuneste din nou destinul cu fata lor din Palermo. GT: In Piata Uniri,beneath the autumn sky,the family from Ruginoasa joins theirdestiny with their daughter in
Palermo. CLAPS: In the Square of the Union,under the skies ofthe early autumn,the family of Ruginoasa is once again
uniting their destiny with their mother in Palermo.
RO: "Sper sa am o amintire frumoasa cu ei, pentru ca acum sunt atat de multe lucruri de spus", ne marturiseste Crina. GT: "I hope to gain a beautiful memory after meting them, because now there are so many things to say,"
confesses Crina. CLAPS: "Ihope to have a nice memory with them,because there is so much to say now," Crina tels us.
RO: In weekendul care a trecut,Alina si Costica si-au vazut pentru prima data fata pierduta pe holurile spitalului
din Iasi,in urma cu 20 ani. GT: Last weekend,Alina and Costica saw the girl who was lost on thehals of the hospitalin Iasi for the first time
in 20 years. CLAPS: Forthe firsttime in the weekend,Alina and Costica saw their girllostin hospital rooms in Jasmine 20
yearsago. RO: "A durat pana la urma cam oluna de zile pana sa il gasim pe tatal biologic al fetei", ne explica comisarul sef
Romica Ichim. GT: Then they gave me some pointers,only to find that the person I was directed to wasn't he person I was looking for.
", + "type": "table", + "image_path": "d66afa1b2388e54c0336b32250ab4945d60a715d935430fb19fd532a8a85d39d.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 107, + 182, + 502, + 365.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 107, + 365.66666666666663, + 502, + 549.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 107, + 549.3333333333333, + 502, + 732.9999999999999 + ], + "spans": [], + "index": 11 + } + ] + } + ], + "index": 10 + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/Wi5KUNlqWty/Wi5KUNlqWty.md b/parse/train/Wi5KUNlqWty/Wi5KUNlqWty.md new file mode 100644 index 0000000000000000000000000000000000000000..388a61d1faf9ef8a41ebd727f8bb0266518a9f93 --- /dev/null +++ b/parse/train/Wi5KUNlqWty/Wi5KUNlqWty.md @@ -0,0 +1,564 @@ +# HOW TO FIND YOUR FRIENDLY NEIGHBORHOOD: GRAPH ATTENTION DESIGN WITH SELF-SUPERVISION + +Dongkwan Kim & Alice Oh +KAIST, Republic of Korea +dongkwan.kim@kaist.ac.kr, alice.oh@kaist.edu + +# ABSTRACT + +Attention mechanism in graph neural networks is designed to assign larger weights to important neighbor nodes for better representation. However, what graph attention learns is not understood well, particularly when graphs are noisy. In this paper, we propose a self-supervised graph attention network (SuperGAT), an improved graph attention model for noisy graphs. Specifically, we exploit two attention forms compatible with a self-supervised task to predict edges, whose presence and absence contain the inherent information about the importance of the relationships between nodes. By encoding edges, SuperGAT learns more expressive attention in distinguishing mislinked neighbors. We find two graph characteristics influence the effectiveness of attention forms and self-supervision: homophily and average degree. Thus, our recipe provides guidance on which attention design to use when those two graph characteristics are known. Our experiment on 17 real-world datasets demonstrates that our recipe generalizes across 15 datasets of them, and our models designed by recipe show improved performance over baselines. + +# 1 INTRODUCTION + +Graphs are widely used in various domains, such as social networks, biology, and chemistry. Since their patterns are complex and irregular, learning to represent graphs is challenging (Bruna et al., 2014; Henaff et al., 2015; Defferrard et al., 2016; Duvenaud et al., 2015; Atwood & Towsley, 2016). Recently, graph neural networks (GNNs) have shown a significant performance improvement by generating features of the center node by aggregating those of its neighbors (Zhou et al., 2018; Wu et al., 2020). However, real-world graphs are often noisy with connections between unrelated nodes, and this causes GNNs to learn suboptimal representations. Graph attention networks (GATs) (Velickovi ˇ c et al., 2018) adopt self-attention to alleviate this issue. Similar to attention in ´ sequential data (Luong et al., 2015; Bahdanau et al., 2015; Vaswani et al., 2017), graph attention captures the relational importance of a graph, in other words, the degree of importance of each of the neighbors to represent the center node. GATs have shown performance improvements in node classification, but they are inconsistent in the degree of improvement across datasets, and there is little understanding of what graph attention actually learns. + +Hence, there is still room for graph attention to improve, and we start by assessing and learning the relational importance for each graph via self-supervised attention. We leverage edges that explicitly encode information about the importance of relations provided by a graph. If node $i$ and $j$ are linked, they are more relevant to each other than others, and if node $i$ and $j$ are not linked, they are not important to each other. Although conventional attention is trained without direct supervision, if we have prior knowledge about what to attend, we can supervise attention using them (Knyazev et al., 2019; Yu et al., 2017). Specifically, we exploit a self-supervised task, using the attention value as input to predict the likelihood that an edge exists between nodes. + +To encode edges in graph attention, we first analyze what graph attention learns and how it relates to the presence of edges. In this analysis, we focus on two commonly used attention mechanisms, GAT’s original single-layer neural network (GO) and dot-product (DP), as building blocks of our proposed model, self-supervised graph attention network (SuperGAT). We observe that DP attention shows better performance than GO attention in the task to predict link with attention value. On the other hand, GO attention outperforms DP attention in capturing label-agreement between a target node and its neighbors. Based on our analysis, we propose two variants of SuperGAT, scaled dotproduct (SD) and mixed GO and DP (MX), to emphasize the strength of GO and DP. + +Then, which graph attention models the relational importance best and produces the best node representations? We find that it depends on the average degree and homophily of the graph. We generate synthetic graph datasets with various degrees and homophily, and analyze how the choice of attention affects node classification performance. Based on this result, we propose a recipe to design graph attention with edge self-supervision that works most effectively for given graph characteristics. We conduct experiments on a total of 17 real-world datasets and demonstrate that our recipe can be generalized across them. In addition, we show that models developed by our method improve performance over baselines. + +We present the following contributions. First, we present models with self-supervised attention using edge information. Second, we analyze the classic attention forms GO and DP using label-agreement and link prediction tasks, and this analysis reveals that GO is better at label agreement and DP at link prediction. Third, we propose recipes to design graph attention concerning homophily and average degree and confirm its validity through experiments on real-world datasets. We make our code available for future research (https://github.com/dongkwan-kim/SuperGAT). + +# 2 RELATED WORK + +Deep neural networks are actively studied in modeling graphs, for example the graph convolutional networks (Kipf & Welling, 2017) which approximate spectral graph convolution (Bruna et al., 2014; Defferrard et al., 2016). A representative work in a non-spectral way is the graph attention networks (GATs) (Velickovi ˇ c et al., 2018) which model relations in graphs using self-attention mech- ´ anism (Vaswani et al., 2017). Similar to attention in sequence data (Bahdanau et al., 2015; Luong et al., 2015; Vaswani et al., 2017), variants of attention in graph neural networks (Thekumparampil et al., 2018; Zhang et al., 2018; Wang et al., 2019a; Gao & Ji, 2019; Zhang et al., 2020; Hou et al., 2020) are trained without direct supervision. Our work is motivated by studies that improve attention’s expressive power by giving direct supervision (Knyazev et al., 2019; Yu et al., 2017). Specifically, we employ a self-supervised task to predict edge presence from attention value. This is in line with two branches of recent GNN research: self-supervision and graph structure learning. + +Recent studies about self-supervised learning for GNNs propose tasks leveraging the inherent information in the graph structure: clustering, partitioning, context prediction after node masking, and completion after attribute masking (Hu et al., 2020b; Hui et al., 2020; Sun et al., 2020; You et al., 2020). To the best of our knowledge, ours is the first study to analyze self-supervised learning of graph attention with edge information. Our self-supervised task is similar to link prediction (Liben-Nowell & Kleinberg, 2007), which is a well-studied problem and recently tackled by neural networks (Zhang & Chen, 2017; 2018). Our DP attention to predict links is motivated by graph autoencoder (GAE) (Kipf & Welling, 2016) and its extensions (Pan et al., 2018; Park et al., 2019) reconstructing edges by applying a dot-product decoder to node representations. + +Graph structure learning is an approach to learn the underlying graph structure while jointly learning downstream tasks (Jiang et al., 2019; Franceschi et al., 2019; Klicpera et al., 2019; Stretcu et al., 2019; Zheng et al., 2020). Since real-world graphs often have noisy edges, encoding structure information contributes to learn better representation. However, recent models with graph structure learning suffer from high memory and computational complexity. Some studies target all spaces where edges can exist, so they require $O ( | V | ^ { 2 } )$ space and computational complexity (Jiang et al., 2019; Franceschi et al., 2019). Others using iterative training (or co-training) between the GNNs and the structure learning model are time-intensive in training (Franceschi et al., 2019; Stretcu et al., 2019). We moderate this problem using graph attention, which consists of parallelizable operations, and our model is built on it without additional parameters. Our model learns attention values that are predictive of edges, and this can be seen as a new paradigm of learning the graph structure. + +# 3 MODEL + +In this section, we review the original GAT (Velickovi ˇ c et al., 2018) and then describe our self-´ supervised GAT (SuperGAT) models. + +![](images/477ee301413365f6f6b7c07290d71e4ea5c193f303f00ce1ad7030218957e85a.jpg) +Figure 1: Overview of attention mechanism of SuperGATs: GO, DP, MX, and SD. Blue circles $( e _ { i j } )$ represent the unnormalized attention before softmax and red diamonds $( \phi _ { i j } )$ indicate the probability of edge between node $i$ and $j$ . The attention mechanism of the original GAT (Velickovi ˇ c et al., 2018) ´ is in the dashed rectangle. + +Notation For a graph $\mathcal { G } = ( V , E )$ , $N$ is the number of nodes and $F ^ { l }$ is the number of features at layer $l$ . Graph attention layer takes a set of features $\mathbb { H } ^ { l } = \{ h _ { 1 } ^ { l } , \dots , h _ { N } ^ { l } \} , h _ { i } ^ { l } \in \mathbb { R } ^ { F ^ { l } }$ as input and produces output features $\mathbb { H } ^ { l + 1 } = \{ h _ { 1 } ^ { l + 1 } , \dots , h _ { N } ^ { l + 1 } \}$ . To compute $\boldsymbol { h } _ { i } ^ { l + 1 }$ , the model multiplies the weight matrix $W ^ { l + 1 } \in \mathbb { R } ^ { F ^ { l + 1 } \times F ^ { l } }$ to $\mathbb { H } ^ { l }$ , linearly combines the features of its first-order neighbors (including itself) $j \in \mathbb { N } _ { i } \cup \{ i \}$ by attention coefficients $\alpha _ { i j } ^ { l + 1 }$ , and finally applies a non-linear activation $\rho$ That is $\begin{array} { r } { \pmb { h } _ { i } ^ { l + 1 } = \rho \left( \sum _ { j \in \mathbb { N } _ { i } \cup \{ i \} } \alpha _ { i j } ^ { l + 1 } \pmb { W } ^ { l + 1 } \pmb { h } _ { j } ^ { l } \right) } \end{array}$ . We can compute $\alpha _ { i j } ^ { l + 1 } = \mathrm { s o f t m a x } _ { j } ( \mathrm { L R e L U } ( e _ { i j } ^ { l + 1 } ) )$ by normalizing $e _ { i j } ^ { l + 1 } = a _ { e } ( W ^ { l + 1 } \pmb { h } _ { i } ^ { l } , W ^ { l + 1 } \pmb { h } _ { j } ^ { l } )$ with softmax on $\mathbb { N } _ { i } \cup \{ i \}$ under leaky ReLU activation (Maas et al., 2013), where ae is a function of the form RF l+1 $\mathbb { R } ^ { F ^ { l + 1 } } \times \mathbb { R } ^ { F ^ { l + 1 } } \mathbb { R }$ . + +Graph Attention Forms Among two widely used attention mechanisms, the original GAT (GO) computes the coefficients by single-layer feed-forward network parameterized by al+1 ∈ R2Fl+1. The other is the dot-product (DP) attention, (Luong et al., 2015; Vaswani et al., 2017) motivated by prior work on node representation learning, and it adopts the same mathematical expression for link prediction score (Tang et al., 2015; Kipf & Welling, 2016), + +$$ +e _ { i j , \mathrm { G } 0 } ^ { l + 1 } = ( \boldsymbol { a } ^ { l + 1 } ) ^ { \top } \left[ \boldsymbol { W } ^ { l + 1 } h _ { i } ^ { l } \Vert \boldsymbol { W } ^ { l + 1 } h _ { j } ^ { l } \right] \quad \mathrm { a n d } \quad e _ { i j , \mathrm { D P } } ^ { l + 1 } = ( \boldsymbol { W } ^ { l + 1 } h _ { i } ^ { l } ) ^ { \top } \cdot \boldsymbol { W } ^ { l + 1 } h _ { j } ^ { l } . +$$ + +From now on, we call GAT that uses GO and DP as $\mathrm { G A T _ { G O } }$ and $\mathrm { G A T _ { D P } }$ , respectively. + +Self-supervised Graph Attention Network We propose SuperGAT with the idea of guiding attention with the presence or absence of an edge between a node pair. We exploit the link prediction task to self-supervise attention with labels from edges: for a pair $i$ and $j$ , 1 if an edge exists and 0 otherwise. We introduce $a _ { \phi }$ with sigmoid $\sigma$ to infer the probability $\phi _ { i j }$ of an edge between $i$ and $j$ . + +$$ +a _ { \phi } : \mathbb { R } ^ { F } \times \mathbb { R } ^ { F } \mathbb { R } \quad \mathrm { a n d } \quad \phi _ { i j } = P ( ( j , i ) \in E ) = \sigma ( a _ { \phi } ( W h _ { i } , W h _ { j } ) ) +$$ + +We employ four types (GO, DP, SD, and MX) of SuperGAT based on GO and DP attention. For $a _ { \phi }$ , the form of which is the same as $a _ { e }$ in $\mathrm { G A T _ { G O } }$ and $\mathrm { G A T _ { D P } }$ , we name them SuperGATGO and SuperGATDP respectively. For more advanced versions, we describe SuperGATSD (Scaled Dotproduct) and SuperGATMX (Mixed GO and DP) by unnormalized attention $e _ { i j }$ and probability $\phi _ { i j }$ that an edge exist between $i$ and $j$ . + +$$ +\begin{array} { r l } & { \mathrm { S u p e r G A T } _ { \mathrm { S D } } \colon e _ { i j , \mathrm { S D } } = e _ { i j , \mathrm { D P } } / \sqrt { F } , \quad \phi _ { i j , \mathrm { S D } } = \sigma ( e _ { i j , \mathrm { S D } } ) . } \\ & { \mathrm { S u p e r G A T } _ { \mathrm { M X } } \colon e _ { i j , \mathrm { M X } } = e _ { i j , \mathrm { G O } } \cdot \sigma ( e _ { i j , \mathrm { D P } } ) , \quad \phi _ { i j , \mathrm { M X } } = \sigma ( e _ { i j , \mathrm { D P } } ) . } \end{array} +$$ + +SuperGATSD divides the dot-product of nodes by a square root of dimension as Transformer (Vaswani et al., 2017). This prevents some large values to dominate the entire attention after softmax. SuperGATMX multiplies GO and DP attention with sigmoid. The motivation of this form comes from the gating mechanism of Gated Recurrent Units (Cho et al., 2014). Since DP attention with the sigmoid represents the probability of an edge, it can softly drop neighbors that are not likely linked while implicitly assigning importance to the remaining nodes. + +Training samples are a set of edges $E$ and the complementary set $E ^ { c } = ( V \times V ) \setminus E$ . However, if the number of nodes is large, it is not efficient to use all possible negative cases in $E ^ { c }$ . So, we use negative sampling as in training word or graph embeddings (Mikolov et al., 2013; Tang et al., 2015; Grover & Leskovec, 2016), arbitrarily choosing a total of $p _ { n } \cdot | E |$ negative samples $E ^ { - }$ from $E ^ { c }$ where the negative sampling ratio $p _ { n } \in \mathbb { R } ^ { + }$ is a hyperparameter. SuperGAT is capable of modeling graphs that are sparse with a sufficiently large number of negative samples (i.e., $\mathsf { | } V \times V | \gg | E | )$ , but this is generally not a problem because most real-world graphs are sparse (Chung, 2010). + +We define the optimization objective of layer $l$ as a binary cross-entropy loss $\mathcal { L } _ { E } ^ { l }$ , + +$$ +\begin{array} { r } { \mathcal { L } _ { E } ^ { l } = - \frac { 1 } { \vert E \cup E ^ { - } \vert } \sum _ { ( j , i ) \in E \cup E ^ { - } } \mathbf { 1 } _ { ( j , i ) = 1 } \cdot \log \phi _ { i j } ^ { l } + \mathbf { 1 } _ { ( j , i ) = 0 } \cdot \log \left( 1 - \phi _ { i j } ^ { l } \right) , } \end{array} +$$ + +where 1· is an indicator function. We use a subset of $E \cup E ^ { - }$ sampled by probability $p _ { e } \in ( 0 , 1 ]$ (also a hyperparameter) at each training iteration for a regularization effect from randomness. + +Finally, we combine cross-entropy loss on node labels $( { \mathcal { L } } _ { V } )$ , self-supervised graph attention losses for all $L$ layers $( \mathcal { L } _ { E } ^ { l } )$ , and L2 regularization loss, with mixing coefficients $\lambda _ { E }$ and $\lambda _ { 2 }$ . + +$$ +\begin{array} { r } { \mathcal { L } = \mathcal { L } _ { V } + \lambda _ { E } \cdot \sum _ { l = 1 } ^ { L } \mathcal { L } _ { E } ^ { l } + \lambda _ { 2 } \cdot \| \pmb { W } \| _ { 2 } . } \end{array} +$$ + +We use the same form of multi-head attention in GAT and take the mean of each head’s attention value before the sigmoid to compute $\phi _ { i j }$ . Note that SuperGAT has equivalent time and space complexity as GAT. To compute $\mathcal { L } _ { E } ^ { l }$ for one head, we need additional operations in terms of $O ( F ^ { l } \cdot | E \cup E ^ { - } | )$ , and we do not need extra parameters. + +# 4 EXPERIMENTS + +Our primary research objective is to design graph attentions that are effective with edge selfsupervision. To do this, we pose four specific research questions. We first analyze what basic graph attentions (GO and DP) learn (RQ1 and 2) and how that can be improved with edge self-supervision (RQ3 and 4). We describe each research question and the corresponding experiment design below. + +RQ1. Does graph attention learn label-agreement? First, we evaluate what the graph attentions of $\mathrm { G A T _ { G O } }$ and $\mathrm { G A T _ { D P } }$ learn without edge supervision. For this, we present ground-truth of relational importance and a metric to assess graph attention with ground-truth. Wang et al. (2019a) showed that node representations in the connected component converge to the same value in deep GATs. If there is an edge between nodes with different labels, then it will be hard to distinguish the two corresponding labels with GAT of sufficiently many layers; that is, ideal attention should give all weights to label-agreed neighbors. In that sense, we choose label-agreement between nodes as ground-truth of importance. + +We compare label-agreement and graph attention based on Kullback–Leibler divergence of the normalized attention $\pmb { \alpha } _ { k } = [ \alpha _ { k k } , \alpha _ { k 1 } , . . . , \alpha _ { k J } ]$ with label agreement distribution for the center node $k$ and its neighbors 1 to $J$ . The label agreement distribution, $\ell _ { k } = [ \ell _ { k k } , \ell _ { k 1 } , \dots , \ell _ { k J } ]$ is defined by, + +$$ +\begin{array} { r } { \ell _ { k j } = \hat { \ell } _ { k j } / \sum _ { s } \hat { \ell } _ { k s } , \quad \hat { \ell } _ { k j } = 1 \ ( \mathrm { i f } \ k \ \mathrm { a n d } \ j \ \mathrm { h a v e \ t h e \ s a m e \ l a b e l ) \ { o r \ } 0 \ ( o t h e r w i s e ) . } } \end{array} +$$ + +We employ KL divergence in Eq. 8, whose value becomes small when attention captures well the label-agreement between a node and its neighbors. + +$$ +\begin{array} { r } { \mathrm { K L D } ( \alpha _ { k } , \ell _ { k } ) = \sum _ { j \in \mathbb { N } _ { k } \cup \{ k \} } \alpha _ { k j } \log ( \alpha _ { k j } / \ell _ { k j } ) } \end{array} +$$ + +RQ2. Is graph attention predictive of edge presence? To evaluate how well edge information is encoded in SuperGAT, we conduct link prediction experiments with SuperGATGO and SuperGATDP using $\phi _ { i j }$ of the last layer as a predictor. We measure the performance by AUC over multiple runs. Since link prediction performance depends on the mixing coefficient $\lambda _ { E }$ in Eq. 6, we adopt multiple $\lambda _ { E } \in \{ 1 0 ^ { \overset { . } { - } 3 } , 1 0 ^ { - 2 } , \overset { . } { \cdot } \overset { . } { \cdot } , 1 0 ^ { 3 } \}$ . We train with an incomplete set of edges, and test with the missing edges and the same number of negative samples. At the same time, node classification performance is measured with the same settings to see how learning edge presence affects node classification. + +RQ3. Which graph attention should we use for given graphs? The above two research questions explore what different graph attention learns with or without supervision of edge presence. Then, which graph attention is effective among them for given graphs? We hypothesize that different graph attention will have different abilities to model graphs under various homophily and average degree. We choose these two properties among various graph statistics because they determine the quality and quantity of labels in our self-supervised task. From the perspective of supervised learning of graph attention with edge labels, the learning result depends on how noisy labels are (i.e., how low the homophily is) and how many labels exist (i.e., how high the average degree is). So, we generate 144 synthetic graphs (Section 4.1) controlling 9 homophily $( 0 . 1 - 0 . 9 )$ and 16 average degree $( 1 - 1 0 0 )$ and perform the node classification task in the transductive setting with GCN, GATGO, SuperGATSD, and $\mathrm { S u p e r G A T _ { M X } }$ . + +In RQ3, there are also practical reasons to use the average degree and homophily, out of many graph properties (e.g., diameter, degree sequence, degree distribution, average clustering coefficient). First, the graph property can be computed efficiently even for large graphs. Second, there should be an algorithm that can generate graphs by controlling the property of interest only. Third, the property should be a scalar value because if the synthetic graph space is too wide, it would be impossible to conduct an experiment with sufficient coverage. Average degree and homophily satisfy the above conditions and are suitable for our experiment, unlike some of the other graph properties. + +RQ4. Does design choice based on RQ3 generalize to real-world datasets? Experiments on synthetic datasets provide an understanding of graph attention models’ performance, but they are oversimplified versions of real-world graphs. Can design choice from synthetic datasets be generalized to real-world datasets, considering more complex structures and rich features in real-world graphs? To answer this question, we conduct experiments on 17 real-world datasets with the various average degree (1.8 – 35.8) and homophily (0.16 – 0.91), and compare them with synthetic graph experiments in RQ3. + +# 4.1 DATASETS + +Real-world datasets We use a total of 17 real-world datasets (Cora, CiteSeer, PubMed, Cora-ML, Cora-Full, DBLP, ogbn-arxiv, CS, Physics, Photo, Computers, Wiki-CS, Four-Univ, Chameleon, Crocodile, Flickr, and PPI) in diverse domains (citation, co-authorship, co-purchase, web page, and biology) and scales $( 2 \mathrm { k } - 1 6 9 \mathrm { k }$ nodes). We try to use their original settings as much as possible. To verify research questions 1 and 2, we choose four classic benchmarks: Cora, CiteSeer, PubMed in the transductive setting, and PPI in the inductive setting. See appendix A.1 for detailed description, splits, statistics (including degree and homophily), and references. + +Synthetic datasets We generate random partition graphs of $n$ nodes per class and $c$ classes (Fortunato, 2010), using NetworkX library (Hagberg et al., 2008). A random partition graph is a graph of communities controlled by two probabilities $p _ { i n }$ and $p _ { o u t }$ . If the nodes have the same class labels, they are connected with $p _ { i n }$ , and otherwise, they are connected with $p _ { o u t }$ . To generate a graph with an average degree of $d _ { a v g } = n \cdot \delta$ , we choose $p _ { i n }$ and $p _ { o u t }$ by $p _ { i n } + ( c - 1 ) \cdot p _ { o u t } = \delta$ . The input features of nodes are sampled from overlapping multi-Gaussian distributions (Abu-El-Haija et al., 2019). We set $n$ to 500, $c$ to 10, and choose $d _ { a v g }$ between 1 and 100, $p _ { i n }$ from $\{ 0 . 1 \delta , 0 . 2 \delta , \ldots , 0 . 9 \delta \}$ . We use 20 samples per class for training, 500 for validation and 1000 for test. + +# 4.2 EXPERIMENTAL SET-UP + +We follow the experimental set-up of GAT with minor adjustments. All parameters are initialized by Glorot initialization (Glorot & Bengio, 2010) and optimized by Adam (Kingma & Ba, 2014). We apply L2 regularization, dropout (Srivastava et al., 2014) to features and attention coefficients, and early stopping on validation loss and accuracy. We use ELU (Clevert et al., 2016) as a nonlinear activation $\rho$ . Unless specified, we employ a two-layer SuperGAT with $F = 8$ features and $K = 8$ attention heads (total 64 features). All models are implemented in PyTorch (Paszke et al., 2019) and PyTorch Geometric (Fey & Lenssen, 2019). See appendix A.5 for detailed model and hyperparameter configurations. + +![](images/5655dcd1d4856266ece3055fb81f4694564136af3508e8f265ea50b58ed87e85.jpg) +Figure 2: Distribution of KL divergence between normalized attention and label-agreement on all nodes and layers for Cora dataset (Left: two-layer GAT, Right: four-layer GAT). + +Baselines For all datasets, we compare our model against representative graph neural models: graph convolutional network (GCN) (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017), and graph attention network (GAT) (Velickovi ˇ c et al., 2018). Furthermore, for Cora, CiteSeer, and ´ PubMed, we choose recent graph neural architectures that learn aggregation coefficients (or discrete structures) over edges: constrained graph attention network (CGAT1) (Wang et al., 2019a), graph learning-convolutional network (GLCN) (Jiang et al., 2019), learning discrete structure (LDS) (Franceschi et al., 2019), graph agreement model (GAM) (Stretcu et al., 2019), and NeuralSparse (NS in short) (Zheng et al., 2020). For the PPI, we use CGAT as an additional baseline. + +# 5 RESULTS + +This section describes the experimental results which answer the research questions in Section 4. We include a qualitative analysis of attention, quantitative comparisons on node classification and link prediction, and recipes of graph attention design. The results for sensitivity analysis of important hyper-parameters are in the appendix B.5. + +# Does graph attention learn label-agreement? GO learns label-agreement better than DP. + +We draw box plots of KL divergence between attention and label agreement distributions of twolayer and four-layer GAT with GO and DP attention for Cora dataset in Figure 2. We see similar patterns in other datasets and place their plots in the appendix B.3. At the rightmost of each subfigure, we draw the KLD distribution when uniform attention is given to all neighborhoods. Note that the maximum value of KLD of each node is different since the degree of nodes is different. Also, the KLD distribution shows a long-tail shape like a degree distribution of real-world graphs. + +There are three observations regarding distributions of KLD. First, we observe that the KLD distribution of GO attention shows a pattern similar to the uniform attention for all citation datasets. This implies that trained GO attention is similar to the uniform distribution, which is in line with previously reported results in the case of entropy2 (Wang et al., 2019b). Second, KLD values of DP attention tend to be larger than those of GO attention for the last layer, resulting in bigger long-tails. This mismatch between the learned distribution of DP attention and the label agreement distribution suggests that DP attention does not learn label-agreement in the neighborhood. Third, the deeper the model (more than two), the larger the KLD value of DP attention in the last layer. This is because the variance of DP attention increases as the layer gets deeper, as explained below in Proposition 1. + +Proposition 1. For $l + 1 t h \ G A T$ layer, $i f$ W and $\textbf { \em a }$ are independent and identically drawn from zero-mean uniform distribution with variance $\sigma _ { w } ^ { 2 }$ and $\sigma _ { a } ^ { 2 }$ respectively, assuming that parameters are independent to input features $h ^ { l }$ and elements of $h ^ { l }$ are independent to each other, + +$$ +\begin{array} { r } { V a r [ e _ { i j , G o } ^ { l + 1 } ] = 2 F ^ { l + 1 } \sigma _ { w } ^ { 2 } \sigma _ { a } ^ { 2 } \mathbb { E } ( \Vert h ^ { l } \Vert _ { 2 } ^ { 2 } ) ~ a n d ~ V a r [ e _ { i j , D P } ^ { l + 1 } ] \geq F ^ { l + 1 } \sigma _ { w } ^ { 4 } \left( \frac { 4 } { 5 } \mathbb { E } \left( ( h _ { i } ^ { l } ) ^ { \top } h _ { j } ^ { l } ) ^ { 2 } \right) + V a r ( ( h _ { i } ^ { l } ) ^ { \top } h _ { j } ^ { l } ) \right) } \end{array} +$$ + +The proof is given in the appendix B.1. While the variance of GO depends on the norm of features only, the variance of DP depends on the expectation of the square of input’s dot-product and variance of input’s dot-product. Stacking GAT layers, the more features of $i$ and $j$ correlate with each other, the larger the input’s dot-product will be. After DP attention is normalized by softmax, which intensifies the larger values among them, normalized DP attention attends to only a small portion of the neighbors and learns a biased representation. + +![](images/8879eb0d78cf1cb5bd50b0a8f96799c8a3680def49289c4b6ddce0786606c356.jpg) +Figure 3: Test performance on node classification and link prediction for GO and DP attentions against the mixing coefficient $\lambda _ { E }$ . We report accuracy (Cora, CiteSeer, PubMed) and micro f1-score (PPI) for node classification, and AUC for link prediction. + +![](images/b6414dea89a6faa66f22beaef6a90fd6cf1b9ed48b5f745505be88d742a3e778.jpg) +Figure 4: Mean test accuracy gains (of 5 runs) against $\mathrm { G A T _ { G O } }$ on synthetic datasets, varying homophily and average degree of the input graph. + +# Is graph attention predictive for edge presence? DP predicts edge presence better than $G O$ + +In Figure 3, we report the mean AUC over multiple runs (5 for PPI and 10 for others) for link prediction (red lines) and node classification (gray lines). As the mixing coefficient $\lambda _ { E }$ increases, the link prediction score increases in all datasets and attentions. This is a natural result considering that $\lambda _ { E }$ is the weight factor of self-supervised graph attention loss ( $\mathcal { L } _ { E }$ in Equation 6). For three out of four datasets, DP attention outperforms GO for link prediction for all range of $\lambda _ { E }$ in our experiment. Surprisingly, even for small $\lambda _ { E }$ , DP attention shows around 80 AUC, much higher than the performance of GO attention. PPI is an exception where GO attention shows higher performance for small $\lambda _ { E }$ than DP, but the difference is slight. The results of this experiment demonstrate that DP attention is more suitable than GO attention in encoding edges. + +This figure also includes node classification performance. For all datasets except PubMed, we observe a trade-off between node classification and link prediction; that is, node classification performance decreases in SuperGATGO and SuperGATDP as $\lambda _ { E }$ increases and thus link prediction performance increases. PubMed also shows a decrease in performance at the largest $\lambda _ { E }$ we have tested. This implies that it is hard to learn the relational importance from edges by simply optimizing graph attention for link prediction. + +# Which graph attention should we use for given graphs? It depends on homophily and average degree of the graph. + +In Figure 4, we draw the mean test accuracy gains (over 5 runs) against $\mathrm { G A T _ { G O } }$ as the average degree increases from 1 to 100, for different values of homophily, on 64 synthetic graphs with GCN, SuperGATSD, $\mathrm { S u p e r G A T _ { M X } }$ . See full results from appendix B.2. We define homophily $h$ as the average ratio of neighbors with the same label as the center node (Pei et al., 2020). That is $\begin{array} { r } { h = \frac { 1 } { | V | } \bar { \sum } _ { i \in V } \left( \sum _ { j \in \mathbb { N } _ { i } } \mathbf { \bar { 1 } } _ { l ( i ) = l ( j ) } / | \mathbb { N } _ { i } | \right) } \end{array}$ , where $l ( i )$ is the label of node $i$ . The expectation of homophily for random partition graphs is analytically $p _ { i n } / \delta$ , and we just adopt this value to label the homophily of graphs in Figure 4. + +We make the following observations from this figure. First, if the homophily is low $( \le ~ 0 . 2 )$ , SuperGATSD performs best among models because DP attention tends to focus on a small number of neighbors. This result empirically confirms what we analytically found in Proposition 1. Second, even when homophily is low, the performance gain of SuperGAT against GAT increases as the average degree increases to a certain level (around 10), meaning relation modeling can benefit from self-supervision if there are sufficiently many edges providing supervision. This is in agreement with prior study of label noise for deep neural networks where they find that the absolute amount of data with correct labels affects the learning quality more than the ratio between data with noisy and correct labels (Rolnick et al., 2017). Third, if the average degree and homophily are high enough, there is no difference between all models, including GCNs. If there are more correct edges beyond a certain amount, we can learn fine representation without self-supervision. Most importantly, if the average degree is not too low or high and homophily is above 0.2, Supe $\mathbf { G A T _ { M X } }$ performs better than or similar to SuperGATSD. This implies that we can take advantage of both GO attention to learn label-agreement and DP attention to learn edge presence by mixing GO and DP. Note that many of the real-world graphs belong to this range of graph characteristics. + +Table 1: Summary of classification accuracies of GCN, GraphSAGE, GAT, SuperGATSD, and Super $\mathbf { G A T _ { M X } }$ for real-world datasets (30 runs for ogbn-arxiv and Flickr, and 100 runs for others). + +
Modelogbn-arxiv CSPhysicsCora-ML Cora-Full DBLPCham.Four-Univ Wiki-CSPhotoComp.FlickrCroco.
GCN33.3±1.291.5±0.292.5±0.285.0±0.459.5±0.277.8±0.533.1±0.974.8±0.674.0±1.091.6±0.684.5±1.451.2±0.432.6±0.4
GraphSAGE54.6±0.390.0±0.192.2±0.183.7±0.459.2±0.278.7±0.641.0±0.974.4±0.677.5±0.590.4±1.183.0±1.450.7±0.253.0±1.0
GAT54.1±0.589.5±0.291.2±0.683.2±0.658.7±0.378.2±1.540.8±0.774.2±0.777.6±0.691.8±0.685.7±0.950.9±0.253.3±1.0
SuperGATsD54.50.388.80.491.6±0.584.50.455.80.679.40.841.60.776.20.877.9±0.786.8±2.582.2±0.945.1±1.253.3±0.9
SuperGATMX55.1±0.290.20.291.90.584.70.459.60280.70.742.00.875.30.677.9±0.591.8±0.9 85.7±1.1 50.8±0.253.3±0.9
+ +Table 2: Summary of classification accuracies with 100 random seeds for Cora, CiteSeer, and PubMed. We mark with daggers $( \dag )$ the reprinted results from the respective papers. + +
ModelCoraCiteSeerPubMed
GCN+81.570.379.0
GraphSAGE82.1 ± 0.671.9 ± 0.978.0 ± 0.7
CGAT81.4 ± 1.170.1 ± 0.978.1 ±1.0
GLCN+85.572.078.3
LDSt84.175-
GCN + GAM†86.273.586.0
GCN + NSt83.7 ±1.474.1 ± 1.41
GATt83.0±0.772.5 ± 0.779.0± 0.4
SuperGATsD82.70.672.5±0.881.30.5
SuperGATMX84.3*0.672.6±0.881.70.5
+ +Table 3: Summary of micro f1- scores with 30 random seeds for PPI. + +
ModelPPI
GCN GraphSAGE61.5 ± 0.4 59.0 ±1.2
CGAT GAT68.3 ± 1.7 72.2 ± 0.6
SuperGATsD74.40.4
SuperGATMX67.21.2
+ +\*\* p-value < .0001 \* p-value < .0005 ↓ Worse than GATGO Color Best graph attention (See Fig. 5) + +The results on synthetic graphs imply that understanding of graph domains should be preceded to design graph attention. That is, by knowing the average degree and homophily of the graphs, we can choose the optimal graph attention in our design space. + +# Does design choice based on RQ3 generalize to real-world datasets? It does for 15 of 17 realworld datasets. + +In Figure 5, we plot the best-performed graph attention for synthetic graphs with square points in the plane of average degree and homophily. The size is the performance gain of SuperGAT against GAT, and the color indicates the best model. If the difference is not statistically significant ( $\dot { p }$ -value $\geq . 0 5 )$ between GAT and SuperGAT, and between SuperGATMX and $\mathrm { S u p e r G A T _ { S D } }$ , we mark as GAT-Any and SuperGAT-Any, respectively. We call this plot a recipe since it introduces the optimal attention to a specific region’s graph. + +Now we map the results of 17 real-world datasets in Figure 5 according to their average degree and homophily. Average degree and homophily can be found in the appendix A.1, and experimental results of graph attention models are summarized in Tables 1, 2 and 3. We report the mean and standard deviation of performance over multiple seeds (30 for graphs with more than $5 0 \mathrm { k }$ nodes and 100 for others). We put unpaired t-test results of SuperGAT with $\mathrm { G A T _ { G O } }$ with asterisks. + +We find that the graph attention recipe based on synthetic experiments can generalize across realworld graphs. PPI and Four-Univ (N) are surrounded by squares of SuperGATSD $( \_ )$ at the bottom of the plane. Wiki-CS (N), located in the SuperGAT-Any region $( \_ )$ , also show no difference in performance between SuperGATs. Nine datasets (N), which SuperGATMX shows the highest performance, are located in the MX regions () or within the margin of two squares. Note that there are two MX regions: lower-middle average degree (2.5 - 7.5) and high homophily (0.8 - 0.9), and uppermiddle average degree $( 7 . 5 - 5 0 )$ and lower-middle homophily $( 0 . 3 \textrm { - } 0 . 5 )$ . There are five datasets with no significant performance change across graph attention $\mathbf { \Psi } ( \mathbf { \blacktriangle } )$ . CiteSeer, Photo, and Computers are within a margin of one square from the GAT-Any region $( \sqcup )$ ; however, Flickr and Crocodile are in the SuperGAT-Any region. To find out the cause of this irregularity, we examine the distribution of degree and per-node homophily (appendix A.4). We observe a more complex mixture distribution of homophily and average degree in Flickr and Crocodile, and this seems to be equivalent to mixing graphs of different characteristics, resulting in inconsistent results with our attention design recipe. + +![](images/c2ba813c72d3a1b9241f24df38a8a48755c3ca74c3c2ce39298546d3f985bb01.jpg) +Figure 5: The best-performed graph attention design for synthetic and real-world graphs with various average degree and homophily. + +Comparison with baselines For a total of 17 datasets, SuperGAT outperforms GCN for 13 datasets, GAT for 12 datasets, GraphSAGE for 16 datasets. Interestingly, for CS, Physics, CoraML, and Flickr, in which our model performs worse than GCN, GAT also cannot surpass GCN. It is not yet known when the degree-normalized aggregation of GCN outperforms the attentionbased aggregation, and more research is needed to figure out how to embed the degree information into graph attention. Tables 2 and 3 show performance comparisons between SuperGAT and recent GNNs for Cora, CiteSeer, PubMed, and PPI. Our model performs better for CiteSeer $( 0 . 6 \% { \mathfrak { p } } )$ and PubMed $( 3 . 4 \% )$ than GLCN, which gives regularization to all relations in a graph. $\mathrm { G C N + }$ NS (NeuralSparse) performs better than our model for CiteSeer $( 1 . 5 \% { \mathfrak { p } } )$ but not for Cora $( 0 . 6 \% \mathfrak { p } )$ . CGAT modified for semi-supervised learning shows lower performance than GAT. Although LDS and GAM which use iterative training show better performance except for LDS on Cora, these models require significantly more computation for the iterative training. For example, $\mathrm { G C N } + \mathrm { G A M }$ compared to our model needs more $\times 3 4$ more training time for Cora, $\times 7 2$ for CiteSeer, and $\times 8 2$ for PubMed. See appendix A.7 and B.4 for the experimental set-up and the result of wall-clock time analysis. + +# 6 CONCLUSION + +We proposed novel graph neural architecture designs to self-supervise graph attention following the input graph’s characteristics. We first assessed what graph attention is learning and analyzed the effect of edge self-supervision to link prediction and node classification performance. This analysis showed two widely used attention mechanisms (original GAT and dot-product) have difficulty encoding label-agreement and edge presence simultaneously. To address this problem, we suggested several graph attention forms that balance these two factors and argued that graph attention should be designed depending on the input graph’s average degree and homophily. Our experiments demonstrated that our graph attention recipe generalizes across various real-world datasets such that the models designed according to the recipe outperform other baseline models. + +# ACKNOWLEDGMENTS + +This research was supported by the Engineering Research Center Program through the National Research Foundation of Korea (NRF) funded by the Korean Government MSIT (NRF2018R1A5A1059921) + +# REFERENCES + +Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. 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For multi-label graphs (PPI), we extend the homophily to the average of the ratio of shared labels on neighbors over nodes, i.e., $\begin{array} { r } { h = \frac { 1 } { | V | } \sum _ { i \in V } \left( \sum _ { j \in \mathbb { N } _ { i } } \left| \mathbb { C } _ { i } \cap \mathbb { C } _ { j } \right| / ( | \tilde { \mathbb { N } _ { i } } | \cdot | \mathbb { C } | ) \right) } \end{array}$ , where $\mathbb { C } _ { i }$ is a set of labels for node $i$ and $\mathbb { C }$ is a set of all labels. + +# A.1.1 CITATION NETWORK + +We use a total of 7 citation network datasets. Nodes are documents, and edges are citations. The task for all citation network datasets is to classify each paper’s topic. + +Cora, CiteSeer, PubMed We use three benchmark datasets for semi-supervised node classification tasks in the transductive setting (Sen et al., 2008; Yang et al., 2016). The features of the nodes are bag-of-words representations of documents. We follow the train/validation/test split of previous work (Kipf & Welling, 2017). We use 20 samples per class for training, 500 samples for the validation, and 1000 samples for the test. + +Cora-ML, Cora-Full, DBLP These are other citation network datasets from Bojchevski & Gunnemann (2018). Node features are bag-of-words representations of documents. For CoraFull ¨ with features more than 5000, we reduce the dimension to 500 by performing PCA. We use the split setting in Shchur et al. (2018): 20 samples per class for training, 30 samples per class for validation, the rest for the test. + +ogbn-arxiv The ogbn-arxiv is a recently proposed large-scale dataset of citation networks (Hu et al., 2020a; Wang et al., 2020). Nodes represent arXiv papers, and edges indicate citations between papers, and node features are mean vectors of skip-gram word embeddings of their titles and abstracts. We use the public split by publication dates provided by the original paper. + +# A.1.2 CO-AUTHOR NETWORK + +CS, Physics The CS and Physics are co-author networks in each domain (Shchur et al., 2018). Nodes are authors, and edges mean whether two authors co-authored a paper. Node features are paper keywords from the author’s papers, and we reduce the original dimension (6805 and 8415) to 500 using PCA. The split is the 20-per-class/30-per-class/rest from Shchur et al. (2018). The goal of this task is to classify each author’s respective field of study. + +# A.1.3 AMAZON CO-PURCHASE + +Photo, Computers The Photo and Computers are parts of the Amazon co-purchase graph (McAuley et al., 2015; Shchur et al., 2018). Nodes are goods, and edges indicate whether two goods are frequently purchased together, and node features are a bag-of-words representation of product reviews. The split is the 20-per-class/30-per-class/rest from Shchur et al. (2018). The task is to classify the categories of goods. + +# A.1.4 WEB PAGE NETWORK + +Wiki-CS The Wiki-CS dataset is computer science related page networks in Wikipedia (Mernyei & Cangea, 2020). Nodes represent articles about computer science, and edges represent hyperlinks between articles. The features of nodes are mean vectors of GloVe word embeddings of articles. There are 20 standard splits, and we experiment with five random seeds for each split (total 100 runs). The task is to classify the main category of articles. + +Chameleon, Crocodile These datasets are Wikipedia page networks about specific topics, Chameleon and Crocodile (Rozemberczki et al., 2019). Nodes are articles, and edges are mutual links between them. Node features are a bag-of-words representation with informative nouns in the article. The number of features is 13183, but we use a reduced dimension of 500 by PCA. The split is 20-per-class/30-per-class/rest from Shchur et al. (2018). The original dataset is for the regression task to predict monthly traffic, but we group values into six bins and make it a classification problem. + +Table 4: Average degree and homophily of real-world graphs. + +
DatasetDegreeHomophily
Four-Univ1.83 ± 1.710.16
PPI28.0 ± 39.260.17
Chameleon15.85 ± 18.200.21
Crocodile15.48 ± 15.970.26
Flickr10.08 ± 31.750.32
Cora-Full6.41 ± 8.790.59
ogbn-arxiv7.68 ± 9.050.63
Wiki-CS26.40 ± 36.040.68
CiteSeer2.78 ± 3.390.72
PubMed4.50 ± 7.430.79
Cora-ML5.45 ± 8.240.81
DBLP5.97 ± 9.350.81
Computers35.76 ± 70.310.81
Cora3.90 ± 5.230.83
CS8.93 ± 9.110.83
Photo31.13 ± 47.270.85
Physics14.38 ± 15.570.91
+ +Four-Univ The Four-Univ dataset is a web page networks from computer science departments of diverse universities (Craven et al., 1998). Nodes are web pages, edges are hyperlinks between them, and node features are TF-IDF vectors of web page’s contents. There are five graphs consists of four universities (Cornell, Texas, Washington, and Wisconsin) and a miscellaneous graph from other universities. As the original authors suggested3, we use three graphs of universities and a miscellaneous graph for training, another one graph for validation (Cornell), and the other one graph for the test (Texas). Classification labels are types of web pages (student, faculty, staff, department, course, and project). + +# A.1.5 FLICKR + +The Flickr dataset is a graph of images from Flickr (McAuley & Leskovec, 2012; Zeng et al., 2020). Nodes are images, and edges indicate whether two images share common properties such as geographic location, gallery, and users commented. Node features are a bag-of-words representation of images. We use labels and split in in Zeng et al. (2020). For labels, they construct seven classes by manually merging 81 image tags. + +# A.1.6 PROTEIN-PROTEIN INTERACTION + +The protein-protein interaction (PPI) dataset (Zitnik & Leskovec, 2017; Hamilton et al., 2017; Subramanian et al., 2005) is a well-known benchmark in the inductive setting. A graph is given for human tissue, the nodes are proteins, the node’s features are biological signatures like genes, and the edges illustrate proteins’ interactions. The dataset consists of 20 training graphs, two validation graphs, and two test graphs. This dataset has multi-labels of gene ontology sets. + +# A.2 LINK PREDICTION + +For link prediction, we split $5 \%$ and $10 \%$ of edges for validation and test set, respectively. We fix the negative edges for the test set and sample negative edges for the training set at each iteration. + +Table 5: Statistics of the real-world datasets. + +
Dataset#Nodes#Edges#Features#ClassesSplit#Training Nodes# Val.Nodes#TestNodes
Four-Univ4518342620006fixed4014 (3 Gs)248 (1 G)256 (1G)
PPI5694481871650121fixed44906 (20 Gs)6514 (2 Gs)5524 (2 Gs)
Chameleon2277361015006random1201801977
Crocodile116311800205006random12018011331
Flickr892504498785007fixed446252231222313
Cora-Full197936342150070random1395204916349
ogbn-arxiv169343116624312840fixed909412979948603
Wiki-CS1170129711030010fixed58017695847
CiteSeer3327473237036fixed1205001000
PubMed19717443385003fixed605001000
Cora-ML2995815828797random1402102645
DBLP177165286716394random8012017516
Computers1375224586176710random20030013252
Cora2708542914337fixed1405001000
CS183338189450015random30045017583
Photo76501190817458random1602407250
Physics344932479625005random10015034243
+ +# A.3 SYNTHETIC DATASET + +To the best of our knowledge, our synthetic datasets are not used in recent literature. Therefore, we give some small examples of synthetic datasets to see qualitatively how the average degree and homophily vary according to $\delta$ and $p _ { i n }$ . Specifically, we draw 2D t-SNE plot of node features and edges in Figure 6. In this figure, we can observe that the average degree $( d _ { a v g } = n \cdot \delta )$ ) increases as $\delta$ increases, and homophily $( h = p _ { i n } / \delta )$ increases as $p _ { i n }$ increases. Note that these are raw input features sampled from the 2D Gaussian distribution, not learned node representation. We use the code from the prior work4 (Abu-El-Haija et al., 2019) and apply normalization by standard score. We choose $\delta$ from $\{ 0 . 0 2 5 , 0 . 2 \}$ , $p _ { i n }$ from $\{ 0 . 1 \delta , 0 . 5 \delta , 0 . 9 \delta \}$ , and fix $n = 1 0 0$ and $c = 5$ . + +![](images/e3b9079245f2af111484609e3b99820050dc228cc53143bc13cefd45bf64735d.jpg) +Figure 6: t-SNE plots of node features and edges for synthetic graph examples. Hyperparameters are $\delta \in \{ 0 . 0 2 5 ~ ( \bar { \mathrm { T o p } } ) , 0 . 2 ~ ( \mathrm { B o t t o m } ) \}$ and $p _ { i n } \in \{ 0 . 1 \delta$ (Left), $0 . 5 \delta$ (Center), $\bar { 0 } . 9 \delta$ (Right)}. + +# A.4 DISTRIBUTION OF DEGREE AND HOMOPHILY OF DATASETS + +In Figure 7, we draw kernel density estimation plots of per-node homophily and degree of nodes in real-world datasets. We define per-node homophily as the ratio of neighbors with the same label as the center node, that is, $\begin{array} { r } { h _ { i } \ = \ \stackrel { \bullet } { \sum } _ { j \in \mathbb { N } _ { i } } \mathbf { 1 } _ { l ( i ) = l ( j ) } \big / \big | \mathbb { N } _ { i } \big | } \end{array}$ . Note that we define homophily as $h =$ ${ \frac { 1 } { | V | } } \sum _ { i \in V } h _ { i }$ in Section 5. + +![](images/e1ca30b2f78b42ac0edcb91746459d800064846123a965e68588b2e9ed0211f0.jpg) +Figure 7: Kernel density estimate plot of distribution of degree and per-node homophily in realworld graphs. + +We focus more on the part outside of where the degree is 1 (0 with log scale), and the per-node homophily is 0. These are leaf nodes incorrectly connected and does not significantly affect the learning overall graph representation. In most datasets, only the largest mode exists, or there are some small modes around it. However, in Flickr and Crocodile, we can observe that the interval between modes is wide. More specifically, Crocodile’s modes are in the area of (high degree, low per-node homophily) and (low degree, high per-node homophily), and Flickr’s modes cover most homophily at a specific degree. Note that we can regard a mixture of distribution as a mixture of different sub-graphs. We argue that this is the reason our recipe does not fit for these two datasets. + +# A.5 MODEL & HYPERPARAMETER CONFIGURATIONS + +Model Since we experiment with numbers of datasets, we maintain almost the same configurations across datasets. We do not use other methods such as residual connections, deeper layers, batch normalization, edge augmentation, and more hidden features, although we have confirmed from previous studies that these techniques contribute to performance improvement. For example, prior work has shown f1-score close to 100 for PPI. To clearly see the difference between the various graph attention designs, we intentionally keep a simple model configuration. + +Table 6: Hyperparameters for experiments on real-world datasets. + +
DatasetModel入2EpePn
CoraSuperGATsD0.00782993594510.882669370.80.5
SuperGATMX0.00822886497311.346574530.80.5
CiteSeerSuperGATsD0.048238086570.090739928280.80.3
SuperGATMX0.041613218320.013081692730.80.5
PubMedSuperGATsD0.000203092756318.825603330.60.7
SuperGATMX2.19E-0410.45205180.60.5
PPISuperGATsD3.39E-070.00103435184210.5
SuperGATmX1.00E-071.79E-0610.5
+ +Table 7: Summary of classification accuracies with 10 random seeds for Cora, CiteSeer, and PubMed in the full-supervised setting. We mark with asterisks the reprinted results from the respective papers. + +
ModelCoraCiteSeerPubMed
GAT*87.2 ± 0.377.3 ± 0.387.0 ± 0.3
CGAT*88.2 ± 0.378.9 ± 0.287.4 ± 0.3
CGAT (Our Impl.: 2-layer w/ 64-features)88.9 ± 0.378.9 ± 0.286.9 ± 0.2
SuperGATMx (2-layer w/ 64-features)88.7 ± 0.279.1 ± 0.287.0 ± 0.1
+ +Hyperparameter For real-world datasets, we tune two hyperparameters (mixing coefficients $\lambda _ { 2 }$ and $\lambda _ { E }$ ) by Bayesian optimization for the mean performance of 3 random seeds. We choose negative sampling ratio $p _ { n }$ from $\{ 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 \}$ , and edge sampling ratio $p _ { e }$ from $\{ 0 . 6 , 0 . 8 , 1 . 0 \}$ . We fix dropout probability to 0.0 for PPI, 0.2 for ogbn-arxiv, 0.6 for others. We set learning rate to 0.05 (ogbn-arxiv), 0.01 (PubMed, PPI, Wiki-CS, Photo, Computers, CS, Physics, Crocodile, Cora-Full, DBLP), 0.005 (Cora, CiteSeer, Cora-ML, Chameleon), 0.001 (Four-Univ). For ogbn-arxiv, we set the number of features per head to 16 and the number of heads in the last layer to one; otherwise, we use eight features per head and eight heads in the last layer. + +For synthetic datasets, we choose $\lambda _ { E }$ from $\{ 1 0 ^ { - 5 } , 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 1 0 ^ { - 2 } , 1 0 ^ { - 1 } , 1 , 1 0 , 1 0 ^ { 2 } \}$ and $\lambda _ { 2 }$ from $\{ 1 0 ^ { - 7 } , 1 0 ^ { - 5 } , 1 0 ^ { - 3 } \}$ . We fix learning rate to 0.01, dropout probability to 0.2, $p _ { n }$ to 0.5, and, $p _ { e }$ to 0.8 for all synthetic graphs. + +Table 6 describes hyperparameters for SuperGAT on four real-world datasets. For other datasets and experiments, please see the code (./SuperGAT/args.yaml). + +# A.6 CGAT IMPLEMENTATION + +CGAT (Wang et al., 2019a) has two auxiliary losses: graph structure based constraint $\mathcal { L } _ { g }$ and class boundary constraint $\mathcal { L } _ { b }$ . We borrow their notation for this section: $\nu$ as a set of nodes, $\mathbb { N } _ { i }$ as a set of one-hop neighbors, $\mathbb { N } _ { i } ^ { + }$ as a set of neighbors that share labels, $\mathbb { N } _ { i } ^ { - }$ as a set of neighbors that do not share labels, $\zeta$ · as a margin between attention values, and $\phi \left( v _ { i } , v _ { k } \right)$ as unnormalized attention value. + +$$ +\begin{array} { r l } & { \mathcal { L } _ { g } = \displaystyle \sum _ { i \in \mathcal { V } } \displaystyle \sum _ { j \in \mathbb { N } _ { i } \setminus \mathbb { N } _ { i } ^ { - } } \sum _ { k \in ( \mathcal { V } \setminus \mathbb { N } _ { i } ) } \operatorname* { m a x } \left( 0 , \phi \left( v _ { i } , v _ { k } \right) + \zeta _ { g } - \phi \left( v _ { i } , v _ { j } \right) \right) } \\ & { \mathcal { L } _ { b } = \displaystyle \sum _ { i \in \mathcal { V } } \sum _ { j \in \mathbb { N } _ { i } ^ { + } } \sum _ { k \in \mathbb { N } _ { i } ^ { - } } \operatorname* { m a x } \left( 0 , \phi \left( v _ { i } , v _ { k } \right) + \zeta _ { b } - \phi \left( v _ { i } , v _ { j } \right) \right) } \end{array} +$$ + +Since label information is included in these two losses, they are difficult to use in semi-supervised settings that provide few labeled samples. In fact, in the CGAT paper, they conduct experiments in full-supervised settings; that is, they use all nodes in training except validation and test nodes. + +So, we only use $\mathcal { L } _ { g }$ modified for semi-supervised learning. + +$$ +\mathcal { L } _ { g } ^ { \mathrm { S S L } } = \sum _ { i \in \mathcal { V } } \sum _ { j \in \mathbb { N } _ { i } } \sum _ { k \in ( \mathcal { V } \backslash \mathbb { N } _ { i } ) } \operatorname* { m a x } \left( 0 , \phi \left( v _ { i } , v _ { k } \right) + \zeta _ { g } - \phi \left( v _ { i } , v _ { j } \right) \right) +$$ + +With $\mathcal { L } _ { c }$ , the multi-class cross-entropy on node labels, CGAT’s optimization objective is + +$$ +\mathcal { L } = \mathcal { L } _ { c } + \lambda _ { g } \mathcal { L } _ { g } + \lambda _ { b } \mathcal { L } _ { b } , +$$ + +and our modified CGAT’s loss is + +$$ +\begin{array} { r } { \mathcal { L } = \mathcal { L } _ { c } + \lambda _ { g } \mathcal { L } _ { g } ^ { \mathrm { S S L } } . } \end{array} +$$ + +In addition to losses, CGAT proposes top- $\mathbf { \nabla } \cdot \mathbf { k }$ softmax and node importance based negative sampling (NINS). Top- $\mathbf { \nabla } \cdot \mathbf { k }$ softmax picks up nodes with top- $\mathbf { \nabla } \cdot \mathbf { k }$ attention values among neighbors. NINS adopts importance sampling when choosing negative sample nodes. + +Since the code for CGAT has not been released, we implement our own version. In all experiments in our paper, we use only the modified losses (Equation 14) and top- $\mathbf { \nabla } \cdot \mathbf { k }$ softmax due to the training and implementation complexity of NINS. For PPI, even if we do not assume a semi-supervised setting, we use the same loss because we could not accurately implement multi-label cases for Equation 10 and 11 with only CGAT’s description. + +To verify the functionality of our implementation, we report the results of a full-supervised setting with the original loss (Equation 13) like CGAT paper, in Table 7. Our implementation of CGAT shows almost the same performance reported in the original paper. In addition, SuperGAT and CGAT showed almost similar performance in a full-supervised setting. Note that the original paper employs two hidden layers with hidden dimensions as 32 for Cora, 64 for CiteSeer, and three hidden layers with hidden dimensions 32 for PubMed, where models in our experiments are all two-layer with 64 features. + +# A.7 WALL-CLOCK TIME EXPERIMENTAL SET-UP + +To demonstrate our model’s efficiency, we measure the mean wall-clock time of the entire training process of three runs using a single GPU (GeForce GTX 1080Ti). We compare our model with GAT (Velickovi ˇ c et al., 2018) and GAM (Stretcu et al., 2019). GAT is the basic model using a ´ simpler attention mechanism than ours, and GAM is the state-of-the-art model using co-training with the auxiliary model. + +For GAT and SuperGAT, we use our implementation (including hyperparameter settings) in PyTorch (Paszke et al., 2019). For GAM, we adopt the code in TensorFlow (Abadi et al., 2015) from the authors 5 and choose $\mathrm { G C N + G A M }$ model, which showed the best performance. We retain the default settings in the code but use the hyperparameters reported in the paper, if possible. With this setting, $\mathrm { G C N + G A M }$ on PubMed is not finished after 24 hours; therefore, we manually early-stop the training at the best accuracy. + +# B RESULTS + +# B.1 PROOF OF PROPOSITION + +Proposition 2. For $l + 1 t h \ G A T$ layer, if $W$ and $\textbf { \em a }$ are independent and identically drawn from zero-mean uniform distribution with variance $\sigma _ { w } ^ { 2 }$ and $\sigma _ { a } ^ { 2 }$ respectively, assuming that parameters are independent to input features $h ^ { l }$ and elements of $h ^ { l }$ are independent to each other, + +$$ +\begin{array} { r } { V a r [ e _ { i j , G o } ^ { l + 1 } ] = 2 F ^ { l + 1 } \sigma _ { w } ^ { 2 } \sigma _ { a } ^ { 2 } \mathbb { E } ( \Vert h ^ { l } \Vert _ { 2 } ^ { 2 } ) ~ a n d ~ V a r [ e _ { i j , D P } ^ { l + 1 } ] \geq F ^ { l + 1 } \sigma _ { w } ^ { 4 } \left( \frac { 4 } { 5 } \mathbb { E } \left( ( h _ { i } ^ { l } ) ^ { \top } h _ { j } ^ { l } ) ^ { 2 } \right) + V a r ( ( h _ { i } ^ { l } ) ^ { \top } h _ { j } ^ { l } ) \right) } \end{array} +$$ + +$h ^ { \prime } = W h ^ { l }$ $\begin{array} { r } { \pmb { h } _ { i , k } ^ { \prime } = \sum _ { r = 1 } ^ { F ^ { l + 1 } } \pmb { W } _ { k r } \pmb { h } _ { i , r } ^ { l } } \end{array}$ + +Note that, + +$$ +e _ { i j , \mathrm { G O } } ^ { l + 1 } = { \pmb { a } } ^ { \top } [ { \pmb { h } } _ { i } ^ { \prime } \| { \pmb { h } } _ { j } ^ { \prime } ] \quad \mathrm { a n d } \quad e _ { i j , \mathrm { D P } } ^ { l + 1 } = { \pmb { h } } _ { i } ^ { \prime \top } { \pmb { h } } _ { j } ^ { \prime } +$$ + +First, we compute $\mathbb { E } ( a ^ { 2 } )$ and $\mathbb { E } ( h ^ { \prime 2 } )$ + +$$ +\mathbb { E } ( a ^ { 2 } ) = \operatorname { V a r } ( a ) + \mathbb { E } ( a ) ^ { 2 } = \sigma _ { a } ^ { 2 } +$$ + +$$ +\begin{array} { r l } & { \mathbb { E } ( h _ { k } ^ { \prime 2 } ) = \mathbb { E } \left( \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { \cdot , r } ^ { l } \right) ^ { 2 } \right) } \\ & { \qquad = \mathbb { E } \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } ^ { 2 } ( h _ { \cdot , r } ^ { l } ) ^ { 2 } \right) } \\ & { \qquad = \mathbb { E } \left( W _ { k , \cdot } ^ { 2 } \right) \mathbb { E } \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { \cdot , r } ^ { l } ) ^ { 2 } \right) } \\ & { \qquad = \sigma _ { w } ^ { 2 } \mathbb { E } \left( \| h ^ { l } \| _ { 2 } ^ { 2 } \right) } \end{array} +$$ + +For the variance of $e _ { i j , \mathrm { G O } } ^ { l + 1 }$ + +$$ +\begin{array} { r l } & { \mathrm { V a r } ( e _ { i j , \mathrm { G J } } ^ { l + 1 } ) = \mathrm { V a r } ( a ^ { \top } [ h _ { i } ^ { \prime } | | h _ { j } ^ { \prime } ] ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad = \mathrm { V a r } \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } \left( a _ { r } h _ { i , r } ^ { \prime } + a _ { r + F ^ { l + 1 } } h _ { j , r } ^ { \prime } \right) \right) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad = 2 F ^ { l + 1 } \left( \mathbb { E } \left( a ^ { 2 } \right) \mathbb { E } \left( h ^ { \prime 2 } \right) - \mathbb { E } \left( a \right) ^ { 2 } \mathbb { E } \left( h ^ { \prime } \right) ^ { 2 } \right) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad = 2 F ^ { l + 1 } \mathbb { E } \left( a ^ { 2 } \right) \mathbb { E } \left( h ^ { \prime 2 } \right) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad = 2 F ^ { l + 1 } \sigma _ { a } ^ { 2 } \sigma _ { w } ^ { 2 } \mathbb { E } \left( \| h ^ { l } \| _ { 2 } ^ { 2 } \right) } \end{array} +$$ + +Now we compute $\mathbb { E } ( h _ { i } ^ { \prime } h _ { j } ^ { \prime } )$ and $\mathbb { E } ( h _ { i } ^ { \prime 2 } h _ { j } ^ { \prime 2 } )$ , + +$$ +\begin{array} { r l } & { \mathbb { E } ( h _ { i , k } ^ { \prime } h _ { j , k } ^ { \prime } ) = \mathbb { E } \left( \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { i , r } ^ { l } \right) \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { j , r } ^ { l } \right) \right) } \\ & { \quad \quad \quad \quad = \mathbb { E } \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } ^ { 2 } h _ { i , r } ^ { l } h _ { j , r } ^ { l } \right) } \\ & { \quad \quad \quad = \mathbb { E } ( W _ { k , \cdot } ^ { 2 } ) \mathbb { E } \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } h _ { i , r } ^ { l } h _ { j , r } ^ { l } \right) } \\ & { \quad \quad \quad = \sigma _ { w } ^ { 2 } \mathbb { E } ( ( h _ { i } ^ { l } ) ^ { \top } h _ { j } ^ { l } ) } \end{array} +$$ + +$$ +\begin{array} { r l } & { \mathbb { E } ( h _ { i , k } ^ { \ell } h _ { j , k } ^ { \ell } ) } \\ & { = \mathbb { E } \left( \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { i , r } ^ { \ell } \right) ^ { 2 } \left( \sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { j , r } ^ { l } \right) ^ { 2 } \right) } \\ & { = \mathbb { E } ( W _ { k , \cdot } ^ { 4 } ) \mathbb { E } \left( \displaystyle \sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \right) + \mathbb { E } ( W _ { k , \cdot } ^ { 2 } ) ^ { 2 } \mathbb { E } \left( \displaystyle \sum _ { s \neq t } ^ { F ^ { l + 1 } } ( h _ { i , s } ^ { l } h _ { j , t } ^ { l } ) ^ { 2 } + 2 ( h _ { i , s } ^ { l } h _ { j , s } ^ { l } ) ( h _ { i , t } ^ { l } h _ { j , t } ^ { l } ) \right) } \\ & { = \mathbb { E } ( W _ { k , \cdot } ^ { 4 } ) \mathbb { E } \left( \displaystyle \sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \right) + \mathbb { E } ( W _ { k , \cdot } ^ { 2 } ) ^ { 2 } \mathbb { E } \left( \displaystyle \sum _ { r } ^ { r } ( h _ { i , r } ^ { l } ) ^ { 2 } \displaystyle \sum _ { r } ^ { r } ( h _ { j , r } ^ { l } ) ^ { 2 } - \displaystyle \sum _ { r } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \right) } \end{array} +$$ + +$$ +\begin{array} { r l r } { { + 2 \mathbb { R } ( W _ { k , * } ^ { 2 } ) ^ { 2 } \mathbb { R } ( ( \sum _ { r } h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } - \sum _ { r } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } ) } } \\ & { } & { = \frac { 9 } { 5 } \sigma _ { \mathrm { w } } ^ { 4 } \mathbb { E } ( \sum _ { r = 1 } ^ { r + 1 } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } ) + \sigma _ { \mathrm { w } } ^ { 4 } \mathbb { E } ( \| h _ { i } ^ { i } \| _ { 2 } ^ { 2 } \| h _ { j } ^ { i } \| _ { 2 } ^ { 2 } - 3 \sum _ { r = 1 } ^ { r + 1 } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } ) + 2 \sigma _ { \mathrm { w } } ^ { 4 } \mathbb { E } ( ( ( h _ { i } ^ { i } ) ^ { \top } h _ { j , r } ^ { i } ) ^ { 2 } } \\ & { } & { ( 3 7 ) } \\ & { } & { = \sigma _ { \mathrm { w } } ^ { 4 } \mathbb { R } ( \| h _ { i } ^ { i } \| _ { 2 } ^ { 2 } \| h _ { j } ^ { i } \| _ { 2 } ^ { 2 } - \frac { 6 } { 5 } \sum _ { r = 1 } ^ { r + 1 } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } + 2 ( ( h _ { i } ^ { i } ) ^ { \top } h _ { j } ^ { i } ) ^ { 2 } ) } \\ & { } & { ( 3 8 ) } \\ & { } & = \sigma _ { \mathrm { w } } ^ { 4 } \mathbb { E } ( \frac { 4 } { 5 } ( h _ { i } ^ { i } ) ^ { \top } h _ { j } ^ { i } ) ^ { 2 } + \frac { 1 } { 1 0 } \sum _ { s \not = 1 } ^ { r } ( ( h _ { i , s } ^ { i } h _ { j , s } ^ { i } + h _ { i , r } ^ { i } h _ { j , s } ^ { i } ) ^ { 2 } + 8 ( h _ { i , r } ^ { i } h _ { j , s } ^ { i } ) ^ { 2 } ) + \sigma _ { \mathrm { w } } ^ { 4 } \mathbb { E } ( ( h \end{array} +$$ + +Note that for the zero-mean uniform distribution $\boldsymbol { \mathcal { U } } ( - \boldsymbol { u } , \boldsymbol { u } )$ with variance $\sigma _ { w } ^ { 2 }$ + +$$ +\mathbb { E } ( W _ { \cdot , \cdot } ^ { 2 } ) = \operatorname { V a r } ( W _ { \cdot , \cdot } ) + \mathbb { E } ( W _ { \cdot , \cdot } ) ^ { 2 } = \sigma _ { w } ^ { 2 } +$$ + +$$ +\begin{array} { c } { { \displaystyle \sigma _ { w } ^ { 2 } = \frac 1 { 1 2 } ( u - ( - u ) ) ^ { 2 } = \frac 1 3 u ^ { 2 } } } \\ { { \displaystyle \mathbb { E } ( W _ { \cdot , \cdot } ^ { 4 } ) = \frac 1 5 \sum _ { r = 0 } ^ { 4 } ( - u ) ^ { r } u ^ { 4 - r } = \frac 1 5 u ^ { 4 } = \frac 9 5 \sigma _ { w } ^ { 4 } } } \end{array} +$$ + +For the variance of $e _ { i j , \mathrm { D P } } ^ { l + 1 }$ + +$$ +\begin{array} { r l } & { \mathrm { ~ \forall i a r ~ } ( c _ { \delta , ( 1 , 1 ) } ^ { i ( 1 , 1 ) } ) } \\ & { = \mathrm { ~ \operatorname* { W i a r } ~ } ( h _ { i } ^ { i } , h _ { j } ^ { i } ) } \\ & { = \mathrm { ~ \operatorname* { W i a r } ~ } ( \sum _ { r ^ { \prime \prime } = 1 } ^ { r ^ { \prime \prime } + 1 } h _ { i , r } ^ { i } h _ { j , x } ^ { i } ) } \\ & { = E ^ { i + 1 } \mathrm { ~ \operatorname* { W a r } ~ } ( h _ { i } ^ { j } , h _ { j } ^ { i } ) } \\ & { = F ^ { i + 1 } \mathrm { ~ { \mathbb { E } } ~ } [ ( h _ { i } ^ { i } h _ { j } ^ { i } ) ^ { 2 } ) - \mathrm { ~ { \mathbb { E } } } ( h _ { i } ^ { i } h _ { j } ^ { j } ) ^ { 2 } ] } \\ & { = F ^ { i + 1 } \mathrm { ~ { \mathbb { C } } ~ } \mathrm { e } ^ { i } \mathrm { ~ { W } ~ } ( \frac { 4 } { 5 } ( h _ { i } ^ { i } h _ { j } ^ { \top } h _ { j } ^ { i } ) ^ { 2 } + \frac { 1 } { 1 0 } \sum _ { \phi \in \mathbb { R } _ { j } } ( h _ { i , \phi } ^ { i } h _ { j , + } ^ { i } + h _ { i , } ^ { i } h _ { j , \phi } ^ { i } ) ^ { 2 } + \mathrm { { \mathbb { S } } } ( h _ { i , \phi } ^ { i } h _ { j , + } ^ { i } ) ^ { 2 } ) } \\ & { \quad + F ^ { i + 1 } \sigma _ { \phi } ^ { i } \mathbb { E } ( \mathbb { E } ( ( h _ { i } ^ { i } h _ { j } ^ { \top } h _ { j } ^ { i } ) ^ { 2 } - \mathbb { E } [ ( h _ { i } ^ { i } h _ { j } ^ { \top } h _ { j } ^ { i } ) ^ { 2 } ] ) } \\ & { \geq F ^ { i + 1 } \sigma _ { \phi } ^ { i } ( \frac { 4 } { 5 } \mathbb { E } ( ( h _ { i } ^ { i } h _ { j } ^ { \top } ) ^ { 2 } h _ { j } ^ { i } ) + \mathrm { { \mathbb { S } } } \{ ( h _ { i } ^ { i } ) ^ { 2 } h _ { j } ^ { i } \} ) } \end{array} +$$ + +Table 8: Summary of classification accuracies (of 5 runs) for synthetic datasets. + +
Avg.degree Homophily
GCN 0.10.2 0.30.4 0.50.6 0.7 0.8 0.9
1 32.7 ± 1.135.1±0.434.7 ±0.440.8±0.443.4±0.346.2± 0.5 49.7±0.554.8±0.9 56.2±0.3
1.5 28.7± 0.530.0±0.634.6±0.440.8± 0.441.7 ± 0.446.9 ± 0.5 52.1±0.654.5± 0.563.6± 0.5
2.5 24.6±0.227.9 ±0.931.0±0.339.7±0.644.0± 0.651.4± 0.8 56.6±0.565.2 ±0.674.5± 0.4
3.5 25.8±0.830.5±0.831.9± 0.738.4± 0.445.0 ±1.049.8 ± 0.560.8±0.770.3±0.379.5± 0.3
5 25.0±0.326.3± 0.533.0 ± 1.139.8±0.750.2 ±0.558.6 ± 1.168.8±0.378.0 ±1.988.5±0.2
7.5 26.1± 0.529.8±0.634.0±0.643.8±0.952.0±0.470.7±3.274.6± 2.488.6±0.895.0±0.2
10 25.4± 0.629.2 ± 0.537.1± 0.547.6 ± 1.058.1±0.973.1 ± 2.885.3± 2.692.7 ± 1.197.9± 0.5
12.5 24.0±0.427.6±0.539.6±0.848.9±0.465.8 ± 3.177.8 ± 2.488.5±1.595.0 ±0.999.2± 0.0
15 22.1 ± 0.528.2±0.644.0± 0.753.0±0.569.9 ± 4.779.1 ± 2.092.2 ±1.898.0±0.499.6±0.3
20 24.3±0.630.9±0.741.9 ± 0.958.1 ± 1.275.0 ± 2.986.7 ± 1.196.1±0.398.9±0.199.8 ±0.1
25 26.6± 1.031.1± 0.344.9±0.262.4± 0.780.1± 2.691.0 ± 1.597.5± 0.599.5± 0.2100.0 ± 0.0
32.5 26.3± 0.733.8±0.751.8±0.867.5 ± 2.983.5 ± 1.995.7 ± 0.398.4±0.2100.0±0.0100.0±0.0
40 23.7±0.534.2± 0.553.4± 0.472.8 ± 1.587.6± 0.996.1 ± 0.699.7 ± 0.199.9 ± 0.0100.0 ± 0.0
50 25.0 ± 1.136.4± 1.055.1 ± 0.582.7 ± 3.091.2 ± 0.798.6±0.499.8±0.099.9 ±0.0100.0±0.0
75 25.9±0.640.6± 0.668.0 ± 1.087.9± 2.697.1± 0.699.6±0.1100.0 ±0.0100.0 ± 0.0100.0 ±0.0
100 25.1 ± 0.542.0±0.972.4 ± 1.492.6± 0.498.6± 0.2100.0 ±0.0100.0±0.0100.0±0.0100.0±0.0
GATGO 0.10.20.30.40.50.60.70.80.9
1 33.7±0.936.7 ±0.835.2± 1.240.1± 0.743.0± 0.846.3± 0.649.4± 0.553.8 ±1.456.1 ± 0.5
1.5 28.8 ±1.033.0±0.534.8±0.940.7± 0.442.2 ± 0.946.7± 1.052.2 ±0.853.6 ± 1.462.8 ± 0.5
2.5 28.8±1.230.6±0.632.3± 0.540.1 ± 0.844.0 ± 1.352.2± 0.655.5±1.464.6 ± 1.373.1± 0.6
3.5 28.0±0.433.4± 1.633.5± 0.640.1 ± 1.046.2 ± 0.650.8± 0.662.5±0.671.7± 0.678.0 ± 0.5
5 28.0±1.330.0±0.836.9± 0.642.1± 0.453.6± 0.761.0 ± 1.170.4±0.578.7 ±1.287.5± 0.6
7.5 30.0±0.931.6 ± 2.040.4± 0.545.0 ±1.059.5 ± 0.672.3 ± 1.079.4±0.690.4± 0.595.0 ± 0.5
10 31.0 ± 1.434.0 ±1.243.1 ± 0.654.6±0.566.1 ± 0.677.4 ± 1.590.6± 1.494.7 ±0.798.4± 0.5
12.5 29.8 ± 1.435.5± 1.747.3 ± 0.958.6±1.772.1 ± 1.486.3± 0.590.7±0.796.2 ±0.398.9±0.2
15 31.9 ±1.634.6 ± 1.649.5± 0.962.0±0.777.2 ± 1.585.9 ± 0.394.4± 0.998.5± 0.599.4± 0.1
2034.4 ±1.838.3± 1.654.1 ± 1.970.0 ± 1.183.9 ±1.093.7± 0.197.5±0.399.0 ±0.399.7 ± 0.0
2535.8±2.042.0± 2.457.7 ± 0.677.7 ± 0.987.9 ±1.295.0±0.798.7±0.699.6±0.399.9 ± 0.1
32.537.4 ±1.244.7 ± 1.166.5 ± 1.979.9 ± 1.291.4 ± 1.498.0± 0.599.0±0.399.8 ± 0.1100.0 ± 0.0
4037.5± 2.045.1± 0.966.5 ± 1.185.7 ± 1.593.5 ± 1.097.6± 0.699.5± 0.199.9 ± 0.199.9± 0.1
5038.7± 1.849.5± 2.368.9± 2.589.5± 0.896.0± 0.899.0± 0.499.7 ± 0.299.8 ± 0.1100.0 ±0.0
7539.6±3.053.5 ±1.777.6 ± 2.692.8± 2.398.0 ±1.299.5 ± 0.399.8±0.2100.0±0.0100.0 ± 0.0
10041.3 ± 1.256.6 ± 1.481.1 ± 3.595.9 ± 1.299.0 ± 0.499.8 ± 0.199.9 ± 0.1100.0 ± 0.1100.0 ± 0.0
SuperGATsD0.10.20.30.40.50.60.70.80.9
138.5±1.040.5±1.539.5 ±1.642.3±0.944.0± 0.547.1 ± 0.550.1 ± 1.053.0±0.355.1± 1.0
1.536.7 ± 1.237.8 ± 1.842.0± 0.442.9±0.845.0± 0.747.8 ± 0.552.9 ±0.754.7 ± 1.263.0±0.4
2.535.7±1.837.3 ± 1.639.2 ± 1.447.8 ±1.349.0± 0.356.1 ± 0.559.4±0.465.9 ±0.870.6 ± 1.2
3.535.7± 1.039.2 ± 1.941.4 ± 0.544.0±0.751.0± 0.355.0 ± 1.265.1 ± 1.173.5 ± 1.078.1±0.6
537.0± 1.539.0 ± 1.545.0 ±1.748.1 ± 1.057.5± 0.666.1 ± 0.974.4 ± 0.281.0 ± 1.088.3 ± 0.8
7.536.8 ± 1.538.7±0.747.3± 0.551.1 ±0.861.2 ± 0.675.5± 0.582.3±0.991.0±0.295.2± 0.3
1039.4 ± 1.142.3 ±0.750.5± 1.258.6±0.370.0± 0.580.1 ±0.590.8±0.395.5± 0.698.7±0.3
12.538.5± 1.042.0±0.750.5± 0.662.4± 0.573.6± 0.486.8± 0.492.9± 0.498.0± 0.699.3± 0.3
1540.4 ± 1.142.8± 0.753.6± 0.367.4± 0.479.7± 0.389.0 ± 1.096.3±0.399.1 ± 0.299.6± 0.3
2037.8±0.842.1 ± 0.956.9 ±0.970.0 ± 0.586.1 ±0.895.6± 0.399.1± 0.199.6± 0.299.7± 0.1
2540.0± 1.048.8±1.259.5±0.578.7±0.290.7 ±0.297.5± 0.199.4 ± 0.1100.0±0.099.9 ± 0.1
32.539.7 ± 1.148.5±0.769.4± 0.482.7 ± 0.594.3 ±0.299.3± 0.199.7 ± 0.199.9 ± 0.199.9 ±0.0
4044.2 ± 1.148.7± 1.369.2 ± 0.788.3± 0.497.1 ± 0.199.7± 0.199.9±0.099.8± 0.2100.0 ±0.0
5044.3 ± 0.753.2 ± 0.673.4 ±1.291.2 ± 0.497.7 ±0.2100.0 ±0.0100.0 ±0.0100.0±0.0100.0 ±0.0
7544.8± 0.756.1± 0.982.7±0.695.7 ±0.299.8±0.299.9 ± 0.1100.0±0.0100.0±0.0100.0±0.0
10043.1 ± 1.360.7±0.787.4± 0.398.3± 0.199.9 ± 0.0100.0 ±0.0100.0 ±0.0100.0 ±0.0100.0 ± 0.0
SuperGATMX0.10.20.30.40.50.60.70.80.9
133.4±0.736.3±0.835.4±0.740.5±0.343.4±0.647.0±0.850.0±0.554.2 ± 1.156.4±0.4
1.528.8±0.731.5 ± 1.036.0±0.841.1 ± 1.141.3 ±0.647.5± 0.852.3±0.754.8±0.563.4± 0.3
2.526.6±0.930.5±1.333.6±0.842.7 ± 1.544.7 ± 0.552.2 ± 0.556.5 ± 0.866.7 ±0.874.5 ± 0.7
3.527.7± 0.933.9 ± 1.737.9± 0.742.4± 0.649.7 ±0.854.2 ± 1.165.3 ± 1.173.2± 0.779.4 ± 1.5
528.4 ± 2.133.4±0.840.9±0.846.2± 0.456.4±0.264.1 ± 1.874.6± 1.483.1 ± 1.090.6±0.9
7.531.1 ± 0.735.0±0.946.5 ± 1.151.9 ± 0.764.3 ± 1.575.2 ± 1.682.1 ± 1.392.6 ± 1.895.8±0.2
1032.6± 1.741.5 ± 1.448.9 ± 1.261.7 ± 1.170.4 ± 1.381.3 ± 1.191.3±0.795.1 ± 0.899.2± 0.3
12.534.2 ± 3.141.1 ± 0.853.3 ± 1.065.9 ± 1.177.2 ± 1.488.1 ±0.792.6± 2.697.5 ± 0.999.3±0.2
1536.4 ± 1.339.9 ± 2.153.8±1.568.5±0.881.0±0.490.2 ± 1.096.1 ±0.999.2 ± 0.299.6± 0.1
2035.2 ± 3.141.0 ± 2.162.3 ±1.273.9 ± 0.986.3±0.695.2 ± 1.398.6±0.799.6± 0.199.9± 0.2
2536.8 ± 1.946.6± 1.762.7 ± 1.181.9 ± 1.790.7±0.896.8 ± 1.599.3±0.399.8±0.2100.0 ± 0.0
32.536.7± 0.846.8± 0.568.9± 0.683.8±0.793.7 ± 1.098.4±0.799.8± 0.1100.0±0.0100.0 ± 0.0
4039.8± 2.448.8 ±1.772.1 ±0.887.5 ±1.796.6±0.898.9±0.399.9 ± 0.1100.0±0.0100.0±0.0
5042.1 ± 1.353.6± 2.375.1 ± 1.992.7 ± 1.097.1 ± 0.999.6± 0.2100.0±0.0100.0±0.0100.0±0.0
7541.9 ± 1.155.9± 2.480.8±1.295.0 ± 1.199.5± 0.299.9 ±0.1100.0 ±0.0100.0 ±0.0100.0 ±0.0
10039.6± 2.059.2 ± 1.884.2 ± 3.196.9 ± 0.799.7 ± 0.1100.0± 0.0100.0 ±0.0100.0± 0.0100.0±0.0
+ +# B.2 FULL RESULT OF SYNTHETIC GRAPH EXPERIMENTS + +In Table 8, we report all results of synthetic graph experiments. We experiment on a total of 144 synthetic graphs controlling 9 homophily (0.1, 0.2, ..., 0.9) and 16 average degree (1, 1.5, 2.5, 3.5, 5, 7.5, 10, 12.5, 15, 20, 25, 32.5, 40, 50, 75, 100). + +![](images/762ab1e4ccc2a8a192f78d09ad595597707a91b0e7af9118f15205550b082f32.jpg) +Figure 8: Distribution of KL divergence between normalized attention and label-agreement on all nodes and layers for Cora, CiteSeer, PubMed, and PPI (Left: two-layer GAT, Right: four-layer GAT). + +Table 9: Mean wall-clock time (seconds) of three runs of the training process on real-world datasets. + +
ModelCoraCiteSeerPubMed
GAT11.3 ± 2.720.4 ± 6.721.1 ± 2.0
GCN+GAM709.3 ± 235.91099.3 ± 812.56923.3 ± 7042.0
SuperGATMX30.8 ± 0.519.3 ± 1.1151.4 ± 11.4
SuperGATMx+ MPNS20.8 ± 0.215.2 ± 0.184.6 ± 0.9
+ +B.3 LABEL-AGREEMENT STUDY FOR OTHER DATASETS AND DEEPER MODELS + +In Figure 8, we draw box plots of KL divergence between attention distribution and label agreement distribution for all nodes and layers of two-layer GATs and four-layer GATs. As shown in the paper, we can see that DP attention does not capture label-agreement rather than GO attention. Also, the degree of this phenomenon becomes stronger as the layer goes down. + +# B.4 WALL-CLOCK TIME RESULT + +In Table 9, we report the mean wall-clock time (over three runs) of the training of GAT, GAM, and SuperGATMX. In SuperGAT, we find that negative sampling of edges is the bottleneck of training. So, we additionally implement Super $\mathrm { { G A T _ { M X } + } }$ MPNS, which employs multi-processing when sampling negative edges. There are three observations in this experiment. $\mathrm { G C N + G A M }$ is highly time-intensive in the training stage $( \times 5 3 . 9 - \times 3 2 8 . 1 $ versus GAT) for all datasets. Compared to GAT, our model needs $\times 2 . 7$ more training time for Cora and $\times 7 . 2$ for PubMed, and we reduce the time by applying multi-processing to negative sampling $( \times 1 . 8$ for Cora and $\times 4 . 0$ for PubMed). For CiteSeer, we can see that SuperGATMX ends faster than GAT because of faster convergence and fewer epochs. + +![](images/6ec406148c025a72f669e3a67c42fc1338ea4afbb04e4ed791e0359c3a63475d.jpg) + +![](images/56755856dbd1ff30185098ee8776c616cb831305772ef54070eb5d474e61a6b8.jpg) +Figure 9: Test performance on node classification against the mixing coefficient $\lambda _ { E }$ for SuperGATMX (Cora, CiteSeer, PubMed) and SuperGATSD (PPI). + +![](images/5d9a2c6a1a46bac033c6f3d14b2694ecc02b0f618b9abfcf3da86eb6929ce4de.jpg) +Figure 10: Test performance on node classification against the negative sampling ratio $p _ { n }$ for SuperGATMX (Cora, CiteSeer, PubMed) and SuperGATSD (PPI). +Figure 11: Test performance on node classification against the edge sampling ratio $p _ { e }$ for SuperGATMX (Cora, CiteSeer, PubMed) and SuperGATSD (PPI). + +# B.5 SENSITIVITY ANALYSIS OF HYPER-PARAMETERS + +We analyze sensitivity of mixing coefficient of losses $\lambda _ { E }$ , negative sampling ratio $p _ { n }$ , and edge sampling ratio $p _ { e }$ . We plot mean node classification performance (over 5 runs) against each hyperparameter in Figure 9, 10, and 11 respectively. We use the best model for each dataset: SuperGATMX for citation networks and SuperGATSD for PPI. + +For $\lambda _ { E }$ , there is a specific range that maximizes test performance in all datasets. Performance on PPI is the largest when $\lambda _ { E }$ is $\bar { 1 } 0 ^ { - 3 }$ , but the difference is relatively small comparing to others. We observe that there is an optimal level of the edge supervision for each dataset, and using too large $\lambda _ { E }$ degrades node classification performance. + +For $p _ { n }$ , using too many negative samples has been shown to decrease performance. The optimal number of negative samples is different for each dataset, and all are less than the number of positive samples $( p _ { n } < 1 . 0 )$ . Note that as $p _ { n }$ increases, the required GPU memory also increases. When $p _ { n } = 5 . 0$ , the model and data for PPI could not be accommodated by one single GPU (GeForce GTX 1080Ti). + +When $p _ { e }$ changes, the performance also changes, but the pattern is different by datasets. For Cora and PubMed, the performance against $p _ { e }$ shows the convex curve. Performance for CiteSeer generally decreases as $p _ { e }$ increases, but there are intervals the performance change of which is nearly zero. In the case of PPI, there are no noticeable changes against $p _ { e }$ . \ No newline at end of file diff --git a/parse/train/Wi5KUNlqWty/Wi5KUNlqWty_content_list.json b/parse/train/Wi5KUNlqWty/Wi5KUNlqWty_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..9930f68f73e9dceb3589211f487de634eab755d1 --- /dev/null +++ b/parse/train/Wi5KUNlqWty/Wi5KUNlqWty_content_list.json @@ -0,0 +1,2901 @@ +[ + { + "type": "text", + "text": "HOW TO FIND YOUR FRIENDLY NEIGHBORHOOD: GRAPH ATTENTION DESIGN WITH SELF-SUPERVISION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Dongkwan Kim & Alice Oh \nKAIST, Republic of Korea \ndongkwan.kim@kaist.ac.kr, alice.oh@kaist.edu ", + "bbox": [ + 183, + 169, + 616, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 241, + 544, + 257 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Attention mechanism in graph neural networks is designed to assign larger weights to important neighbor nodes for better representation. However, what graph attention learns is not understood well, particularly when graphs are noisy. In this paper, we propose a self-supervised graph attention network (SuperGAT), an improved graph attention model for noisy graphs. Specifically, we exploit two attention forms compatible with a self-supervised task to predict edges, whose presence and absence contain the inherent information about the importance of the relationships between nodes. By encoding edges, SuperGAT learns more expressive attention in distinguishing mislinked neighbors. We find two graph characteristics influence the effectiveness of attention forms and self-supervision: homophily and average degree. Thus, our recipe provides guidance on which attention design to use when those two graph characteristics are known. Our experiment on 17 real-world datasets demonstrates that our recipe generalizes across 15 datasets of them, and our models designed by recipe show improved performance over baselines. ", + "bbox": [ + 233, + 271, + 764, + 478 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 503, + 336, + 518 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graphs are widely used in various domains, such as social networks, biology, and chemistry. Since their patterns are complex and irregular, learning to represent graphs is challenging (Bruna et al., 2014; Henaff et al., 2015; Defferrard et al., 2016; Duvenaud et al., 2015; Atwood & Towsley, 2016). Recently, graph neural networks (GNNs) have shown a significant performance improvement by generating features of the center node by aggregating those of its neighbors (Zhou et al., 2018; Wu et al., 2020). However, real-world graphs are often noisy with connections between unrelated nodes, and this causes GNNs to learn suboptimal representations. Graph attention networks (GATs) (Velickovi ˇ c et al., 2018) adopt self-attention to alleviate this issue. Similar to attention in ´ sequential data (Luong et al., 2015; Bahdanau et al., 2015; Vaswani et al., 2017), graph attention captures the relational importance of a graph, in other words, the degree of importance of each of the neighbors to represent the center node. GATs have shown performance improvements in node classification, but they are inconsistent in the degree of improvement across datasets, and there is little understanding of what graph attention actually learns. ", + "bbox": [ + 174, + 535, + 825, + 714 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hence, there is still room for graph attention to improve, and we start by assessing and learning the relational importance for each graph via self-supervised attention. We leverage edges that explicitly encode information about the importance of relations provided by a graph. If node $i$ and $j$ are linked, they are more relevant to each other than others, and if node $i$ and $j$ are not linked, they are not important to each other. Although conventional attention is trained without direct supervision, if we have prior knowledge about what to attend, we can supervise attention using them (Knyazev et al., 2019; Yu et al., 2017). Specifically, we exploit a self-supervised task, using the attention value as input to predict the likelihood that an edge exists between nodes. ", + "bbox": [ + 174, + 722, + 825, + 833 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To encode edges in graph attention, we first analyze what graph attention learns and how it relates to the presence of edges. In this analysis, we focus on two commonly used attention mechanisms, GAT’s original single-layer neural network (GO) and dot-product (DP), as building blocks of our proposed model, self-supervised graph attention network (SuperGAT). We observe that DP attention shows better performance than GO attention in the task to predict link with attention value. On the other hand, GO attention outperforms DP attention in capturing label-agreement between a target node and its neighbors. Based on our analysis, we propose two variants of SuperGAT, scaled dotproduct (SD) and mixed GO and DP (MX), to emphasize the strength of GO and DP. ", + "bbox": [ + 174, + 840, + 823, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 820, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Then, which graph attention models the relational importance best and produces the best node representations? We find that it depends on the average degree and homophily of the graph. We generate synthetic graph datasets with various degrees and homophily, and analyze how the choice of attention affects node classification performance. Based on this result, we propose a recipe to design graph attention with edge self-supervision that works most effectively for given graph characteristics. We conduct experiments on a total of 17 real-world datasets and demonstrate that our recipe can be generalized across them. In addition, we show that models developed by our method improve performance over baselines. ", + "bbox": [ + 174, + 138, + 823, + 250 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We present the following contributions. First, we present models with self-supervised attention using edge information. Second, we analyze the classic attention forms GO and DP using label-agreement and link prediction tasks, and this analysis reveals that GO is better at label agreement and DP at link prediction. Third, we propose recipes to design graph attention concerning homophily and average degree and confirm its validity through experiments on real-world datasets. We make our code available for future research (https://github.com/dongkwan-kim/SuperGAT). ", + "bbox": [ + 174, + 257, + 825, + 340 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 362, + 343, + 378 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Deep neural networks are actively studied in modeling graphs, for example the graph convolutional networks (Kipf & Welling, 2017) which approximate spectral graph convolution (Bruna et al., 2014; Defferrard et al., 2016). A representative work in a non-spectral way is the graph attention networks (GATs) (Velickovi ˇ c et al., 2018) which model relations in graphs using self-attention mech- ´ anism (Vaswani et al., 2017). Similar to attention in sequence data (Bahdanau et al., 2015; Luong et al., 2015; Vaswani et al., 2017), variants of attention in graph neural networks (Thekumparampil et al., 2018; Zhang et al., 2018; Wang et al., 2019a; Gao & Ji, 2019; Zhang et al., 2020; Hou et al., 2020) are trained without direct supervision. Our work is motivated by studies that improve attention’s expressive power by giving direct supervision (Knyazev et al., 2019; Yu et al., 2017). Specifically, we employ a self-supervised task to predict edge presence from attention value. This is in line with two branches of recent GNN research: self-supervision and graph structure learning. ", + "bbox": [ + 174, + 395, + 825, + 549 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Recent studies about self-supervised learning for GNNs propose tasks leveraging the inherent information in the graph structure: clustering, partitioning, context prediction after node masking, and completion after attribute masking (Hu et al., 2020b; Hui et al., 2020; Sun et al., 2020; You et al., 2020). To the best of our knowledge, ours is the first study to analyze self-supervised learning of graph attention with edge information. Our self-supervised task is similar to link prediction (Liben-Nowell & Kleinberg, 2007), which is a well-studied problem and recently tackled by neural networks (Zhang & Chen, 2017; 2018). Our DP attention to predict links is motivated by graph autoencoder (GAE) (Kipf & Welling, 2016) and its extensions (Pan et al., 2018; Park et al., 2019) reconstructing edges by applying a dot-product decoder to node representations. ", + "bbox": [ + 174, + 555, + 825, + 681 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Graph structure learning is an approach to learn the underlying graph structure while jointly learning downstream tasks (Jiang et al., 2019; Franceschi et al., 2019; Klicpera et al., 2019; Stretcu et al., 2019; Zheng et al., 2020). Since real-world graphs often have noisy edges, encoding structure information contributes to learn better representation. However, recent models with graph structure learning suffer from high memory and computational complexity. Some studies target all spaces where edges can exist, so they require $O ( | V | ^ { 2 } )$ space and computational complexity (Jiang et al., 2019; Franceschi et al., 2019). Others using iterative training (or co-training) between the GNNs and the structure learning model are time-intensive in training (Franceschi et al., 2019; Stretcu et al., 2019). We moderate this problem using graph attention, which consists of parallelizable operations, and our model is built on it without additional parameters. Our model learns attention values that are predictive of edges, and this can be seen as a new paradigm of learning the graph structure. ", + "bbox": [ + 174, + 688, + 825, + 840 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 MODEL ", + "text_level": 1, + "bbox": [ + 176, + 863, + 267, + 878 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we review the original GAT (Velickovi ˇ c et al., 2018) and then describe our self-´ supervised GAT (SuperGAT) models. ", + "bbox": [ + 176, + 895, + 821, + 922 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/477ee301413365f6f6b7c07290d71e4ea5c193f303f00ce1ad7030218957e85a.jpg", + "image_caption": [ + "Figure 1: Overview of attention mechanism of SuperGATs: GO, DP, MX, and SD. Blue circles $( e _ { i j } )$ represent the unnormalized attention before softmax and red diamonds $( \\phi _ { i j } )$ indicate the probability of edge between node $i$ and $j$ . The attention mechanism of the original GAT (Velickovi ˇ c et al., 2018) ´ is in the dashed rectangle. " + ], + "image_footnote": [], + "bbox": [ + 179, + 102, + 789, + 258 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Notation For a graph $\\mathcal { G } = ( V , E )$ , $N$ is the number of nodes and $F ^ { l }$ is the number of features at layer $l$ . Graph attention layer takes a set of features $\\mathbb { H } ^ { l } = \\{ h _ { 1 } ^ { l } , \\dots , h _ { N } ^ { l } \\} , h _ { i } ^ { l } \\in \\mathbb { R } ^ { F ^ { l } }$ as input and produces output features $\\mathbb { H } ^ { l + 1 } = \\{ h _ { 1 } ^ { l + 1 } , \\dots , h _ { N } ^ { l + 1 } \\}$ . To compute $\\boldsymbol { h } _ { i } ^ { l + 1 }$ , the model multiplies the weight matrix $W ^ { l + 1 } \\in \\mathbb { R } ^ { F ^ { l + 1 } \\times F ^ { l } }$ to $\\mathbb { H } ^ { l }$ , linearly combines the features of its first-order neighbors (including itself) $j \\in \\mathbb { N } _ { i } \\cup \\{ i \\}$ by attention coefficients $\\alpha _ { i j } ^ { l + 1 }$ , and finally applies a non-linear activation $\\rho$ That is $\\begin{array} { r } { \\pmb { h } _ { i } ^ { l + 1 } = \\rho \\left( \\sum _ { j \\in \\mathbb { N } _ { i } \\cup \\{ i \\} } \\alpha _ { i j } ^ { l + 1 } \\pmb { W } ^ { l + 1 } \\pmb { h } _ { j } ^ { l } \\right) } \\end{array}$ . We can compute $\\alpha _ { i j } ^ { l + 1 } = \\mathrm { s o f t m a x } _ { j } ( \\mathrm { L R e L U } ( e _ { i j } ^ { l + 1 } ) )$ by normalizing $e _ { i j } ^ { l + 1 } = a _ { e } ( W ^ { l + 1 } \\pmb { h } _ { i } ^ { l } , W ^ { l + 1 } \\pmb { h } _ { j } ^ { l } )$ with softmax on $\\mathbb { N } _ { i } \\cup \\{ i \\}$ under leaky ReLU activation (Maas et al., 2013), where ae is a function of the form RF l+1 $\\mathbb { R } ^ { F ^ { l + 1 } } \\times \\mathbb { R } ^ { F ^ { l + 1 } } \\mathbb { R }$ . ", + "bbox": [ + 173, + 349, + 825, + 496 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Graph Attention Forms Among two widely used attention mechanisms, the original GAT (GO) computes the coefficients by single-layer feed-forward network parameterized by al+1 ∈ R2Fl+1. The other is the dot-product (DP) attention, (Luong et al., 2015; Vaswani et al., 2017) motivated by prior work on node representation learning, and it adopts the same mathematical expression for link prediction score (Tang et al., 2015; Kipf & Welling, 2016), ", + "bbox": [ + 173, + 508, + 825, + 582 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/25615df015f00494a888b0df41fb5d1b9aa54f60e9c4e5cb6b3f60780017f609.jpg", + "text": "$$\ne _ { i j , \\mathrm { G } 0 } ^ { l + 1 } = ( \\boldsymbol { a } ^ { l + 1 } ) ^ { \\top } \\left[ \\boldsymbol { W } ^ { l + 1 } h _ { i } ^ { l } \\Vert \\boldsymbol { W } ^ { l + 1 } h _ { j } ^ { l } \\right] \\quad \\mathrm { a n d } \\quad e _ { i j , \\mathrm { D P } } ^ { l + 1 } = ( \\boldsymbol { W } ^ { l + 1 } h _ { i } ^ { l } ) ^ { \\top } \\cdot \\boldsymbol { W } ^ { l + 1 } h _ { j } ^ { l } .\n$$", + "text_format": "latex", + "bbox": [ + 225, + 585, + 771, + 607 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "From now on, we call GAT that uses GO and DP as $\\mathrm { G A T _ { G O } }$ and $\\mathrm { G A T _ { D P } }$ , respectively. ", + "bbox": [ + 173, + 609, + 735, + 626 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Self-supervised Graph Attention Network We propose SuperGAT with the idea of guiding attention with the presence or absence of an edge between a node pair. We exploit the link prediction task to self-supervise attention with labels from edges: for a pair $i$ and $j$ , 1 if an edge exists and 0 otherwise. We introduce $a _ { \\phi }$ with sigmoid $\\sigma$ to infer the probability $\\phi _ { i j }$ of an edge between $i$ and $j$ . ", + "bbox": [ + 173, + 640, + 825, + 696 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/954dbc358c1a3257183d41e82de095b09128af8368424742e110ea51f6e41192.jpg", + "text": "$$\na _ { \\phi } : \\mathbb { R } ^ { F } \\times \\mathbb { R } ^ { F } \\mathbb { R } \\quad \\mathrm { a n d } \\quad \\phi _ { i j } = P ( ( j , i ) \\in E ) = \\sigma ( a _ { \\phi } ( W h _ { i } , W h _ { j } ) )\n$$", + "text_format": "latex", + "bbox": [ + 254, + 702, + 741, + 720 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We employ four types (GO, DP, SD, and MX) of SuperGAT based on GO and DP attention. For $a _ { \\phi }$ , the form of which is the same as $a _ { e }$ in $\\mathrm { G A T _ { G O } }$ and $\\mathrm { G A T _ { D P } }$ , we name them SuperGATGO and SuperGATDP respectively. For more advanced versions, we describe SuperGATSD (Scaled Dotproduct) and SuperGATMX (Mixed GO and DP) by unnormalized attention $e _ { i j }$ and probability $\\phi _ { i j }$ that an edge exist between $i$ and $j$ . ", + "bbox": [ + 174, + 723, + 825, + 795 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/87cd3a3ec912e0853c13c0784eb415cfeb10d6cf11883e6f860518c307fbed3c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { S u p e r G A T } _ { \\mathrm { S D } } \\colon e _ { i j , \\mathrm { S D } } = e _ { i j , \\mathrm { D P } } / \\sqrt { F } , \\quad \\phi _ { i j , \\mathrm { S D } } = \\sigma ( e _ { i j , \\mathrm { S D } } ) . } \\\\ & { \\mathrm { S u p e r G A T } _ { \\mathrm { M X } } \\colon e _ { i j , \\mathrm { M X } } = e _ { i j , \\mathrm { G O } } \\cdot \\sigma ( e _ { i j , \\mathrm { D P } } ) , \\quad \\phi _ { i j , \\mathrm { M X } } = \\sigma ( e _ { i j , \\mathrm { D P } } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 797, + 714, + 838 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "SuperGATSD divides the dot-product of nodes by a square root of dimension as Transformer (Vaswani et al., 2017). This prevents some large values to dominate the entire attention after softmax. SuperGATMX multiplies GO and DP attention with sigmoid. The motivation of this form comes from the gating mechanism of Gated Recurrent Units (Cho et al., 2014). Since DP attention with the sigmoid represents the probability of an edge, it can softly drop neighbors that are not likely linked while implicitly assigning importance to the remaining nodes. ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Training samples are a set of edges $E$ and the complementary set $E ^ { c } = ( V \\times V ) \\setminus E$ . However, if the number of nodes is large, it is not efficient to use all possible negative cases in $E ^ { c }$ . So, we use negative sampling as in training word or graph embeddings (Mikolov et al., 2013; Tang et al., 2015; Grover & Leskovec, 2016), arbitrarily choosing a total of $p _ { n } \\cdot | E |$ negative samples $E ^ { - }$ from $E ^ { c }$ where the negative sampling ratio $p _ { n } \\in \\mathbb { R } ^ { + }$ is a hyperparameter. SuperGAT is capable of modeling graphs that are sparse with a sufficiently large number of negative samples (i.e., $\\mathsf { | } V \\times V | \\gg | E | )$ , but this is generally not a problem because most real-world graphs are sparse (Chung, 2010). ", + "bbox": [ + 173, + 102, + 826, + 202 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We define the optimization objective of layer $l$ as a binary cross-entropy loss $\\mathcal { L } _ { E } ^ { l }$ , ", + "bbox": [ + 173, + 208, + 707, + 223 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f95d77b197178a44c18529809f1b5757ef98cf00db6518580a3452e935fd53b9.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { E } ^ { l } = - \\frac { 1 } { \\vert E \\cup E ^ { - } \\vert } \\sum _ { ( j , i ) \\in E \\cup E ^ { - } } \\mathbf { 1 } _ { ( j , i ) = 1 } \\cdot \\log \\phi _ { i j } ^ { l } + \\mathbf { 1 } _ { ( j , i ) = 0 } \\cdot \\log \\left( 1 - \\phi _ { i j } ^ { l } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 232, + 743, + 255 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where 1· is an indicator function. We use a subset of $E \\cup E ^ { - }$ sampled by probability $p _ { e } \\in ( 0 , 1 ]$ (also a hyperparameter) at each training iteration for a regularization effect from randomness. ", + "bbox": [ + 173, + 263, + 825, + 292 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Finally, we combine cross-entropy loss on node labels $( { \\mathcal { L } } _ { V } )$ , self-supervised graph attention losses for all $L$ layers $( \\mathcal { L } _ { E } ^ { l } )$ , and L2 regularization loss, with mixing coefficients $\\lambda _ { E }$ and $\\lambda _ { 2 }$ . ", + "bbox": [ + 173, + 299, + 825, + 328 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8a5c6c253f910068dd2470ba5100afe1db5d26388e6e9b8624ca3b997796fc7e.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } = \\mathcal { L } _ { V } + \\lambda _ { E } \\cdot \\sum _ { l = 1 } ^ { L } \\mathcal { L } _ { E } ^ { l } + \\lambda _ { 2 } \\cdot \\| \\pmb { W } \\| _ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 362, + 337, + 635, + 358 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We use the same form of multi-head attention in GAT and take the mean of each head’s attention value before the sigmoid to compute $\\phi _ { i j }$ . Note that SuperGAT has equivalent time and space complexity as GAT. To compute $\\mathcal { L } _ { E } ^ { l }$ for one head, we need additional operations in terms of $O ( F ^ { l } \\cdot | E \\cup E ^ { - } | )$ , and we do not need extra parameters. ", + "bbox": [ + 174, + 366, + 825, + 426 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 448, + 326, + 463 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our primary research objective is to design graph attentions that are effective with edge selfsupervision. To do this, we pose four specific research questions. We first analyze what basic graph attentions (GO and DP) learn (RQ1 and 2) and how that can be improved with edge self-supervision (RQ3 and 4). We describe each research question and the corresponding experiment design below. ", + "bbox": [ + 174, + 481, + 825, + 537 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "RQ1. Does graph attention learn label-agreement? First, we evaluate what the graph attentions of $\\mathrm { G A T _ { G O } }$ and $\\mathrm { G A T _ { D P } }$ learn without edge supervision. For this, we present ground-truth of relational importance and a metric to assess graph attention with ground-truth. Wang et al. (2019a) showed that node representations in the connected component converge to the same value in deep GATs. If there is an edge between nodes with different labels, then it will be hard to distinguish the two corresponding labels with GAT of sufficiently many layers; that is, ideal attention should give all weights to label-agreed neighbors. In that sense, we choose label-agreement between nodes as ground-truth of importance. ", + "bbox": [ + 173, + 554, + 825, + 666 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We compare label-agreement and graph attention based on Kullback–Leibler divergence of the normalized attention $\\pmb { \\alpha } _ { k } = [ \\alpha _ { k k } , \\alpha _ { k 1 } , . . . , \\alpha _ { k J } ]$ with label agreement distribution for the center node $k$ and its neighbors 1 to $J$ . The label agreement distribution, $\\ell _ { k } = [ \\ell _ { k k } , \\ell _ { k 1 } , \\dots , \\ell _ { k J } ]$ is defined by, ", + "bbox": [ + 174, + 672, + 825, + 717 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0cfdb7c2638e9bd22eac6e1d84d990c6e5c48688aa247584c3a744d062f1859e.jpg", + "text": "$$\n\\begin{array} { r } { \\ell _ { k j } = \\hat { \\ell } _ { k j } / \\sum _ { s } \\hat { \\ell } _ { k s } , \\quad \\hat { \\ell } _ { k j } = 1 \\ ( \\mathrm { i f } \\ k \\ \\mathrm { a n d } \\ j \\ \\mathrm { h a v e \\ t h e \\ s a m e \\ l a b e l ) \\ { o r \\ } 0 \\ ( o t h e r w i s e ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 724, + 764, + 744 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We employ KL divergence in Eq. 8, whose value becomes small when attention captures well the label-agreement between a node and its neighbors. ", + "bbox": [ + 171, + 752, + 821, + 781 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e5ec21531b26537c5cde7a98dfe304c1d14df930e8da819b7d309ea9e46a949f.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { K L D } ( \\alpha _ { k } , \\ell _ { k } ) = \\sum _ { j \\in \\mathbb { N } _ { k } \\cup \\{ k \\} } \\alpha _ { k j } \\log ( \\alpha _ { k j } / \\ell _ { k j } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 790, + 655, + 810 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "RQ2. Is graph attention predictive of edge presence? To evaluate how well edge information is encoded in SuperGAT, we conduct link prediction experiments with SuperGATGO and SuperGATDP using $\\phi _ { i j }$ of the last layer as a predictor. We measure the performance by AUC over multiple runs. Since link prediction performance depends on the mixing coefficient $\\lambda _ { E }$ in Eq. 6, we adopt multiple $\\lambda _ { E } \\in \\{ 1 0 ^ { \\overset { . } { - } 3 } , 1 0 ^ { - 2 } , \\overset { . } { \\cdot } \\overset { . } { \\cdot } , 1 0 ^ { 3 } \\}$ . We train with an incomplete set of edges, and test with the missing edges and the same number of negative samples. At the same time, node classification performance is measured with the same settings to see how learning edge presence affects node classification. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "RQ3. Which graph attention should we use for given graphs? The above two research questions explore what different graph attention learns with or without supervision of edge presence. Then, which graph attention is effective among them for given graphs? We hypothesize that different graph attention will have different abilities to model graphs under various homophily and average degree. We choose these two properties among various graph statistics because they determine the quality and quantity of labels in our self-supervised task. From the perspective of supervised learning of graph attention with edge labels, the learning result depends on how noisy labels are (i.e., how low the homophily is) and how many labels exist (i.e., how high the average degree is). So, we generate 144 synthetic graphs (Section 4.1) controlling 9 homophily $( 0 . 1 - 0 . 9 )$ and 16 average degree $( 1 - 1 0 0 )$ and perform the node classification task in the transductive setting with GCN, GATGO, SuperGATSD, and $\\mathrm { S u p e r G A T _ { M X } }$ . ", + "bbox": [ + 174, + 103, + 825, + 257 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In RQ3, there are also practical reasons to use the average degree and homophily, out of many graph properties (e.g., diameter, degree sequence, degree distribution, average clustering coefficient). First, the graph property can be computed efficiently even for large graphs. Second, there should be an algorithm that can generate graphs by controlling the property of interest only. Third, the property should be a scalar value because if the synthetic graph space is too wide, it would be impossible to conduct an experiment with sufficient coverage. Average degree and homophily satisfy the above conditions and are suitable for our experiment, unlike some of the other graph properties. ", + "bbox": [ + 174, + 263, + 825, + 361 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "RQ4. Does design choice based on RQ3 generalize to real-world datasets? Experiments on synthetic datasets provide an understanding of graph attention models’ performance, but they are oversimplified versions of real-world graphs. Can design choice from synthetic datasets be generalized to real-world datasets, considering more complex structures and rich features in real-world graphs? To answer this question, we conduct experiments on 17 real-world datasets with the various average degree (1.8 – 35.8) and homophily (0.16 – 0.91), and compare them with synthetic graph experiments in RQ3. ", + "bbox": [ + 174, + 382, + 825, + 479 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 502, + 287, + 517 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Real-world datasets We use a total of 17 real-world datasets (Cora, CiteSeer, PubMed, Cora-ML, Cora-Full, DBLP, ogbn-arxiv, CS, Physics, Photo, Computers, Wiki-CS, Four-Univ, Chameleon, Crocodile, Flickr, and PPI) in diverse domains (citation, co-authorship, co-purchase, web page, and biology) and scales $( 2 \\mathrm { k } - 1 6 9 \\mathrm { k }$ nodes). We try to use their original settings as much as possible. To verify research questions 1 and 2, we choose four classic benchmarks: Cora, CiteSeer, PubMed in the transductive setting, and PPI in the inductive setting. See appendix A.1 for detailed description, splits, statistics (including degree and homophily), and references. ", + "bbox": [ + 174, + 530, + 825, + 628 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Synthetic datasets We generate random partition graphs of $n$ nodes per class and $c$ classes (Fortunato, 2010), using NetworkX library (Hagberg et al., 2008). A random partition graph is a graph of communities controlled by two probabilities $p _ { i n }$ and $p _ { o u t }$ . If the nodes have the same class labels, they are connected with $p _ { i n }$ , and otherwise, they are connected with $p _ { o u t }$ . To generate a graph with an average degree of $d _ { a v g } = n \\cdot \\delta$ , we choose $p _ { i n }$ and $p _ { o u t }$ by $p _ { i n } + ( c - 1 ) \\cdot p _ { o u t } = \\delta$ . The input features of nodes are sampled from overlapping multi-Gaussian distributions (Abu-El-Haija et al., 2019). We set $n$ to 500, $c$ to 10, and choose $d _ { a v g }$ between 1 and 100, $p _ { i n }$ from $\\{ 0 . 1 \\delta , 0 . 2 \\delta , \\ldots , 0 . 9 \\delta \\}$ . We use 20 samples per class for training, 500 for validation and 1000 for test. ", + "bbox": [ + 174, + 650, + 825, + 761 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 EXPERIMENTAL SET-UP ", + "text_level": 1, + "bbox": [ + 176, + 784, + 380, + 797 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We follow the experimental set-up of GAT with minor adjustments. All parameters are initialized by Glorot initialization (Glorot & Bengio, 2010) and optimized by Adam (Kingma & Ba, 2014). We apply L2 regularization, dropout (Srivastava et al., 2014) to features and attention coefficients, and early stopping on validation loss and accuracy. We use ELU (Clevert et al., 2016) as a nonlinear activation $\\rho$ . Unless specified, we employ a two-layer SuperGAT with $F = 8$ features and $K = 8$ attention heads (total 64 features). All models are implemented in PyTorch (Paszke et al., 2019) and PyTorch Geometric (Fey & Lenssen, 2019). See appendix A.5 for detailed model and hyperparameter configurations. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/5655dcd1d4856266ece3055fb81f4694564136af3508e8f265ea50b58ed87e85.jpg", + "image_caption": [ + "Figure 2: Distribution of KL divergence between normalized attention and label-agreement on all nodes and layers for Cora dataset (Left: two-layer GAT, Right: four-layer GAT). " + ], + "image_footnote": [], + "bbox": [ + 183, + 101, + 810, + 194 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines For all datasets, we compare our model against representative graph neural models: graph convolutional network (GCN) (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017), and graph attention network (GAT) (Velickovi ˇ c et al., 2018). Furthermore, for Cora, CiteSeer, and ´ PubMed, we choose recent graph neural architectures that learn aggregation coefficients (or discrete structures) over edges: constrained graph attention network (CGAT1) (Wang et al., 2019a), graph learning-convolutional network (GLCN) (Jiang et al., 2019), learning discrete structure (LDS) (Franceschi et al., 2019), graph agreement model (GAM) (Stretcu et al., 2019), and NeuralSparse (NS in short) (Zheng et al., 2020). For the PPI, we use CGAT as an additional baseline. ", + "bbox": [ + 173, + 246, + 825, + 357 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 377, + 281, + 392 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This section describes the experimental results which answer the research questions in Section 4. We include a qualitative analysis of attention, quantitative comparisons on node classification and link prediction, and recipes of graph attention design. The results for sensitivity analysis of important hyper-parameters are in the appendix B.5. ", + "bbox": [ + 174, + 407, + 825, + 463 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Does graph attention learn label-agreement? GO learns label-agreement better than DP. ", + "text_level": 1, + "bbox": [ + 173, + 478, + 782, + 493 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We draw box plots of KL divergence between attention and label agreement distributions of twolayer and four-layer GAT with GO and DP attention for Cora dataset in Figure 2. We see similar patterns in other datasets and place their plots in the appendix B.3. At the rightmost of each subfigure, we draw the KLD distribution when uniform attention is given to all neighborhoods. Note that the maximum value of KLD of each node is different since the degree of nodes is different. Also, the KLD distribution shows a long-tail shape like a degree distribution of real-world graphs. ", + "bbox": [ + 174, + 500, + 825, + 583 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "There are three observations regarding distributions of KLD. First, we observe that the KLD distribution of GO attention shows a pattern similar to the uniform attention for all citation datasets. This implies that trained GO attention is similar to the uniform distribution, which is in line with previously reported results in the case of entropy2 (Wang et al., 2019b). Second, KLD values of DP attention tend to be larger than those of GO attention for the last layer, resulting in bigger long-tails. This mismatch between the learned distribution of DP attention and the label agreement distribution suggests that DP attention does not learn label-agreement in the neighborhood. Third, the deeper the model (more than two), the larger the KLD value of DP attention in the last layer. This is because the variance of DP attention increases as the layer gets deeper, as explained below in Proposition 1. ", + "bbox": [ + 173, + 589, + 825, + 715 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 1. For $l + 1 t h \\ G A T$ layer, $i f$ W and $\\textbf { \\em a }$ are independent and identically drawn from zero-mean uniform distribution with variance $\\sigma _ { w } ^ { 2 }$ and $\\sigma _ { a } ^ { 2 }$ respectively, assuming that parameters are independent to input features $h ^ { l }$ and elements of $h ^ { l }$ are independent to each other, ", + "bbox": [ + 174, + 718, + 823, + 760 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/84b7db261b5ef5a176d392c4a621ee3a65e4a98f5366a3c40c84e36bc980a895.jpg", + "text": "$$\n\\begin{array} { r } { V a r [ e _ { i j , G o } ^ { l + 1 } ] = 2 F ^ { l + 1 } \\sigma _ { w } ^ { 2 } \\sigma _ { a } ^ { 2 } \\mathbb { E } ( \\Vert h ^ { l } \\Vert _ { 2 } ^ { 2 } ) ~ a n d ~ V a r [ e _ { i j , D P } ^ { l + 1 } ] \\geq F ^ { l + 1 } \\sigma _ { w } ^ { 4 } \\left( \\frac { 4 } { 5 } \\mathbb { E } \\left( ( h _ { i } ^ { l } ) ^ { \\top } h _ { j } ^ { l } ) ^ { 2 } \\right) + V a r ( ( h _ { i } ^ { l } ) ^ { \\top } h _ { j } ^ { l } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 186, + 761, + 790, + 781 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The proof is given in the appendix B.1. While the variance of GO depends on the norm of features only, the variance of DP depends on the expectation of the square of input’s dot-product and variance of input’s dot-product. Stacking GAT layers, the more features of $i$ and $j$ correlate with each other, the larger the input’s dot-product will be. After DP attention is normalized by softmax, which intensifies the larger values among them, normalized DP attention attends to only a small portion of the neighbors and learns a biased representation. ", + "bbox": [ + 173, + 791, + 825, + 876 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/8879eb0d78cf1cb5bd50b0a8f96799c8a3680def49289c4b6ddce0786606c356.jpg", + "image_caption": [ + "Figure 3: Test performance on node classification and link prediction for GO and DP attentions against the mixing coefficient $\\lambda _ { E }$ . We report accuracy (Cora, CiteSeer, PubMed) and micro f1-score (PPI) for node classification, and AUC for link prediction. " + ], + "image_footnote": [], + "bbox": [ + 194, + 99, + 795, + 189 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/b6414dea89a6faa66f22beaef6a90fd6cf1b9ed48b5f745505be88d742a3e778.jpg", + "image_caption": [ + "Figure 4: Mean test accuracy gains (of 5 runs) against $\\mathrm { G A T _ { G O } }$ on synthetic datasets, varying homophily and average degree of the input graph. " + ], + "image_footnote": [], + "bbox": [ + 192, + 241, + 797, + 353 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Is graph attention predictive for edge presence? DP predicts edge presence better than $G O$ ", + "text_level": 1, + "bbox": [ + 171, + 397, + 802, + 411 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Figure 3, we report the mean AUC over multiple runs (5 for PPI and 10 for others) for link prediction (red lines) and node classification (gray lines). As the mixing coefficient $\\lambda _ { E }$ increases, the link prediction score increases in all datasets and attentions. This is a natural result considering that $\\lambda _ { E }$ is the weight factor of self-supervised graph attention loss ( $\\mathcal { L } _ { E }$ in Equation 6). For three out of four datasets, DP attention outperforms GO for link prediction for all range of $\\lambda _ { E }$ in our experiment. Surprisingly, even for small $\\lambda _ { E }$ , DP attention shows around 80 AUC, much higher than the performance of GO attention. PPI is an exception where GO attention shows higher performance for small $\\lambda _ { E }$ than DP, but the difference is slight. The results of this experiment demonstrate that DP attention is more suitable than GO attention in encoding edges. ", + "bbox": [ + 173, + 417, + 825, + 544 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "This figure also includes node classification performance. For all datasets except PubMed, we observe a trade-off between node classification and link prediction; that is, node classification performance decreases in SuperGATGO and SuperGATDP as $\\lambda _ { E }$ increases and thus link prediction performance increases. PubMed also shows a decrease in performance at the largest $\\lambda _ { E }$ we have tested. This implies that it is hard to learn the relational importance from edges by simply optimizing graph attention for link prediction. ", + "bbox": [ + 174, + 550, + 825, + 633 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Which graph attention should we use for given graphs? It depends on homophily and average degree of the graph. ", + "text_level": 1, + "bbox": [ + 176, + 648, + 821, + 678 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Figure 4, we draw the mean test accuracy gains (over 5 runs) against $\\mathrm { G A T _ { G O } }$ as the average degree increases from 1 to 100, for different values of homophily, on 64 synthetic graphs with GCN, SuperGATSD, $\\mathrm { S u p e r G A T _ { M X } }$ . See full results from appendix B.2. We define homophily $h$ as the average ratio of neighbors with the same label as the center node (Pei et al., 2020). That is $\\begin{array} { r } { h = \\frac { 1 } { | V | } \\bar { \\sum } _ { i \\in V } \\left( \\sum _ { j \\in \\mathbb { N } _ { i } } \\mathbf { \\bar { 1 } } _ { l ( i ) = l ( j ) } / | \\mathbb { N } _ { i } | \\right) } \\end{array}$ , where $l ( i )$ is the label of node $i$ . The expectation of homophily for random partition graphs is analytically $p _ { i n } / \\delta$ , and we just adopt this value to label the homophily of graphs in Figure 4. ", + "bbox": [ + 173, + 684, + 825, + 792 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We make the following observations from this figure. First, if the homophily is low $( \\le ~ 0 . 2 )$ , SuperGATSD performs best among models because DP attention tends to focus on a small number of neighbors. This result empirically confirms what we analytically found in Proposition 1. Second, even when homophily is low, the performance gain of SuperGAT against GAT increases as the average degree increases to a certain level (around 10), meaning relation modeling can benefit from self-supervision if there are sufficiently many edges providing supervision. This is in agreement with prior study of label noise for deep neural networks where they find that the absolute amount of data with correct labels affects the learning quality more than the ratio between data with noisy and correct labels (Rolnick et al., 2017). Third, if the average degree and homophily are high enough, there is no difference between all models, including GCNs. If there are more correct edges beyond a certain amount, we can learn fine representation without self-supervision. Most importantly, if the average degree is not too low or high and homophily is above 0.2, Supe $\\mathbf { G A T _ { M X } }$ performs better than or similar to SuperGATSD. This implies that we can take advantage of both GO attention to learn label-agreement and DP attention to learn edge presence by mixing GO and DP. Note that many of the real-world graphs belong to this range of graph characteristics. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/c18bfa8f5f13dbb2cd6e26a20308396201641ee0857658baa81b89fdb5923d62.jpg", + "table_caption": [ + "Table 1: Summary of classification accuracies of GCN, GraphSAGE, GAT, SuperGATSD, and Super $\\mathbf { G A T _ { M X } }$ for real-world datasets (30 runs for ogbn-arxiv and Flickr, and 100 runs for others). " + ], + "table_footnote": [], + "table_body": "
Modelogbn-arxiv CSPhysicsCora-ML Cora-Full DBLPCham.Four-Univ Wiki-CSPhotoComp.FlickrCroco.
GCN33.3±1.291.5±0.292.5±0.285.0±0.459.5±0.277.8±0.533.1±0.974.8±0.674.0±1.091.6±0.684.5±1.451.2±0.432.6±0.4
GraphSAGE54.6±0.390.0±0.192.2±0.183.7±0.459.2±0.278.7±0.641.0±0.974.4±0.677.5±0.590.4±1.183.0±1.450.7±0.253.0±1.0
GAT54.1±0.589.5±0.291.2±0.683.2±0.658.7±0.378.2±1.540.8±0.774.2±0.777.6±0.691.8±0.685.7±0.950.9±0.253.3±1.0
SuperGATsD54.50.388.80.491.6±0.584.50.455.80.679.40.841.60.776.20.877.9±0.786.8±2.582.2±0.945.1±1.253.3±0.9
SuperGATMX55.1±0.290.20.291.90.584.70.459.60280.70.742.00.875.30.677.9±0.591.8±0.9 85.7±1.1 50.8±0.253.3±0.9
", + "bbox": [ + 174, + 133, + 825, + 213 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/d0ede1ee46ded55281489bf28ad58304c1fd9990756fb71d1a79729565cfd1b0.jpg", + "table_caption": [ + "Table 2: Summary of classification accuracies with 100 random seeds for Cora, CiteSeer, and PubMed. We mark with daggers $( \\dag )$ the reprinted results from the respective papers. " + ], + "table_footnote": [], + "table_body": "
ModelCoraCiteSeerPubMed
GCN+81.570.379.0
GraphSAGE82.1 ± 0.671.9 ± 0.978.0 ± 0.7
CGAT81.4 ± 1.170.1 ± 0.978.1 ±1.0
GLCN+85.572.078.3
LDSt84.175-
GCN + GAM†86.273.586.0
GCN + NSt83.7 ±1.474.1 ± 1.41
GATt83.0±0.772.5 ± 0.779.0± 0.4
SuperGATsD82.70.672.5±0.881.30.5
SuperGATMX84.3*0.672.6±0.881.70.5
", + "bbox": [ + 187, + 268, + 547, + 424 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/6b8e829d315a73eb7f9c8e66783c50228909b4cdde31b8e3840a5466ed7cd7b7.jpg", + "table_caption": [ + "Table 3: Summary of micro f1- scores with 30 random seeds for PPI. " + ], + "table_footnote": [ + "\\*\\* p-value < .0001 \\* p-value < .0005 ↓ Worse than GATGO Color Best graph attention (See Fig. 5) " + ], + "table_body": "
ModelPPI
GCN GraphSAGE61.5 ± 0.4 59.0 ±1.2
CGAT GAT68.3 ± 1.7 72.2 ± 0.6
SuperGATsD74.40.4
SuperGATMX67.21.2
", + "bbox": [ + 609, + 267, + 797, + 367 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 444, + 825, + 530 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The results on synthetic graphs imply that understanding of graph domains should be preceded to design graph attention. That is, by knowing the average degree and homophily of the graphs, we can choose the optimal graph attention in our design space. ", + "bbox": [ + 174, + 536, + 825, + 579 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Does design choice based on RQ3 generalize to real-world datasets? It does for 15 of 17 realworld datasets. ", + "text_level": 1, + "bbox": [ + 176, + 595, + 821, + 625 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Figure 5, we plot the best-performed graph attention for synthetic graphs with square points in the plane of average degree and homophily. The size is the performance gain of SuperGAT against GAT, and the color indicates the best model. If the difference is not statistically significant ( $\\dot { p }$ -value $\\geq . 0 5 )$ between GAT and SuperGAT, and between SuperGATMX and $\\mathrm { S u p e r G A T _ { S D } }$ , we mark as GAT-Any and SuperGAT-Any, respectively. We call this plot a recipe since it introduces the optimal attention to a specific region’s graph. ", + "bbox": [ + 174, + 631, + 825, + 715 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Now we map the results of 17 real-world datasets in Figure 5 according to their average degree and homophily. Average degree and homophily can be found in the appendix A.1, and experimental results of graph attention models are summarized in Tables 1, 2 and 3. We report the mean and standard deviation of performance over multiple seeds (30 for graphs with more than $5 0 \\mathrm { k }$ nodes and 100 for others). We put unpaired t-test results of SuperGAT with $\\mathrm { G A T _ { G O } }$ with asterisks. ", + "bbox": [ + 174, + 722, + 825, + 791 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We find that the graph attention recipe based on synthetic experiments can generalize across realworld graphs. PPI and Four-Univ (N) are surrounded by squares of SuperGATSD $( \\_ )$ at the bottom of the plane. Wiki-CS (N), located in the SuperGAT-Any region $( \\_ )$ , also show no difference in performance between SuperGATs. Nine datasets (N), which SuperGATMX shows the highest performance, are located in the MX regions (\u0004) or within the margin of two squares. Note that there are two MX regions: lower-middle average degree (2.5 - 7.5) and high homophily (0.8 - 0.9), and uppermiddle average degree $( 7 . 5 - 5 0 )$ and lower-middle homophily $( 0 . 3 \\textrm { - } 0 . 5 )$ . There are five datasets with no significant performance change across graph attention $\\mathbf { \\Psi } ( \\mathbf { \\blacktriangle } )$ . CiteSeer, Photo, and Computers are within a margin of one square from the GAT-Any region $( \\sqcup )$ ; however, Flickr and Crocodile are in the SuperGAT-Any region. To find out the cause of this irregularity, we examine the distribution of degree and per-node homophily (appendix A.4). We observe a more complex mixture distribution of homophily and average degree in Flickr and Crocodile, and this seems to be equivalent to mixing graphs of different characteristics, resulting in inconsistent results with our attention design recipe. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/c2ba813c72d3a1b9241f24df38a8a48755c3ca74c3c2ce39298546d3f985bb01.jpg", + "image_caption": [ + "Figure 5: The best-performed graph attention design for synthetic and real-world graphs with various average degree and homophily. " + ], + "image_footnote": [], + "bbox": [ + 279, + 109, + 710, + 330 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 387, + 825, + 443 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Comparison with baselines For a total of 17 datasets, SuperGAT outperforms GCN for 13 datasets, GAT for 12 datasets, GraphSAGE for 16 datasets. Interestingly, for CS, Physics, CoraML, and Flickr, in which our model performs worse than GCN, GAT also cannot surpass GCN. It is not yet known when the degree-normalized aggregation of GCN outperforms the attentionbased aggregation, and more research is needed to figure out how to embed the degree information into graph attention. Tables 2 and 3 show performance comparisons between SuperGAT and recent GNNs for Cora, CiteSeer, PubMed, and PPI. Our model performs better for CiteSeer $( 0 . 6 \\% { \\mathfrak { p } } )$ and PubMed $( 3 . 4 \\% )$ than GLCN, which gives regularization to all relations in a graph. $\\mathrm { G C N + }$ NS (NeuralSparse) performs better than our model for CiteSeer $( 1 . 5 \\% { \\mathfrak { p } } )$ but not for Cora $( 0 . 6 \\% \\mathfrak { p } )$ . CGAT modified for semi-supervised learning shows lower performance than GAT. Although LDS and GAM which use iterative training show better performance except for LDS on Cora, these models require significantly more computation for the iterative training. For example, $\\mathrm { G C N } + \\mathrm { G A M }$ compared to our model needs more $\\times 3 4$ more training time for Cora, $\\times 7 2$ for CiteSeer, and $\\times 8 2$ for PubMed. See appendix A.7 and B.4 for the experimental set-up and the result of wall-clock time analysis. ", + "bbox": [ + 174, + 458, + 825, + 666 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 685, + 318, + 702 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We proposed novel graph neural architecture designs to self-supervise graph attention following the input graph’s characteristics. We first assessed what graph attention is learning and analyzed the effect of edge self-supervision to link prediction and node classification performance. This analysis showed two widely used attention mechanisms (original GAT and dot-product) have difficulty encoding label-agreement and edge presence simultaneously. To address this problem, we suggested several graph attention forms that balance these two factors and argued that graph attention should be designed depending on the input graph’s average degree and homophily. Our experiments demonstrated that our graph attention recipe generalizes across various real-world datasets such that the models designed according to the recipe outperform other baseline models. ", + "bbox": [ + 174, + 717, + 825, + 842 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 858, + 326, + 871 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This research was supported by the Engineering Research Center Program through the National Research Foundation of Korea (NRF) funded by the Korean Government MSIT (NRF2018R1A5A1059921) ", + "bbox": [ + 176, + 882, + 823, + 922 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 103, + 285, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. 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", + "bbox": [ + 171, + 207, + 823, + 234 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A EXPERIMENTAL SET-UP ", + "text_level": 1, + "bbox": [ + 176, + 102, + 408, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1 REAL-WORLD DATASET ", + "text_level": 1, + "bbox": [ + 176, + 133, + 383, + 148 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this section, we describe details (including nodes, edges, features, labels, and splits) of real-world datasets. We report statistics of real-world datasets in Tables 4 and 5. For multi-label graphs (PPI), we extend the homophily to the average of the ratio of shared labels on neighbors over nodes, i.e., $\\begin{array} { r } { h = \\frac { 1 } { | V | } \\sum _ { i \\in V } \\left( \\sum _ { j \\in \\mathbb { N } _ { i } } \\left| \\mathbb { C } _ { i } \\cap \\mathbb { C } _ { j } \\right| / ( | \\tilde { \\mathbb { N } _ { i } } | \\cdot | \\mathbb { C } | ) \\right) } \\end{array}$ , where $\\mathbb { C } _ { i }$ is a set of labels for node $i$ and $\\mathbb { C }$ is a set of all labels. ", + "bbox": [ + 173, + 160, + 825, + 238 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1.1 CITATION NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 253, + 377, + 268 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We use a total of 7 citation network datasets. Nodes are documents, and edges are citations. The task for all citation network datasets is to classify each paper’s topic. ", + "bbox": [ + 173, + 279, + 823, + 306 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Cora, CiteSeer, PubMed We use three benchmark datasets for semi-supervised node classification tasks in the transductive setting (Sen et al., 2008; Yang et al., 2016). The features of the nodes are bag-of-words representations of documents. We follow the train/validation/test split of previous work (Kipf & Welling, 2017). We use 20 samples per class for training, 500 samples for the validation, and 1000 samples for the test. ", + "bbox": [ + 174, + 321, + 825, + 392 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Cora-ML, Cora-Full, DBLP These are other citation network datasets from Bojchevski & Gunnemann (2018). Node features are bag-of-words representations of documents. For CoraFull ¨ with features more than 5000, we reduce the dimension to 500 by performing PCA. We use the split setting in Shchur et al. (2018): 20 samples per class for training, 30 samples per class for validation, the rest for the test. ", + "bbox": [ + 174, + 407, + 825, + 478 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "ogbn-arxiv The ogbn-arxiv is a recently proposed large-scale dataset of citation networks (Hu et al., 2020a; Wang et al., 2020). Nodes represent arXiv papers, and edges indicate citations between papers, and node features are mean vectors of skip-gram word embeddings of their titles and abstracts. We use the public split by publication dates provided by the original paper. ", + "bbox": [ + 174, + 493, + 825, + 550 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1.2 CO-AUTHOR NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 565, + 397, + 580 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "CS, Physics The CS and Physics are co-author networks in each domain (Shchur et al., 2018). Nodes are authors, and edges mean whether two authors co-authored a paper. Node features are paper keywords from the author’s papers, and we reduce the original dimension (6805 and 8415) to 500 using PCA. The split is the 20-per-class/30-per-class/rest from Shchur et al. (2018). The goal of this task is to classify each author’s respective field of study. ", + "bbox": [ + 174, + 589, + 825, + 660 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1.3 AMAZON CO-PURCHASE ", + "text_level": 1, + "bbox": [ + 176, + 675, + 403, + 689 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Photo, Computers The Photo and Computers are parts of the Amazon co-purchase graph (McAuley et al., 2015; Shchur et al., 2018). Nodes are goods, and edges indicate whether two goods are frequently purchased together, and node features are a bag-of-words representation of product reviews. The split is the 20-per-class/30-per-class/rest from Shchur et al. (2018). The task is to classify the categories of goods. ", + "bbox": [ + 174, + 700, + 825, + 770 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1.4 WEB PAGE NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 785, + 385, + 799 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Wiki-CS The Wiki-CS dataset is computer science related page networks in Wikipedia (Mernyei & Cangea, 2020). Nodes represent articles about computer science, and edges represent hyperlinks between articles. The features of nodes are mean vectors of GloVe word embeddings of articles. There are 20 standard splits, and we experiment with five random seeds for each split (total 100 runs). The task is to classify the main category of articles. ", + "bbox": [ + 174, + 809, + 823, + 880 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Chameleon, Crocodile These datasets are Wikipedia page networks about specific topics, Chameleon and Crocodile (Rozemberczki et al., 2019). Nodes are articles, and edges are mutual links between them. Node features are a bag-of-words representation with informative nouns in the article. The number of features is 13183, but we use a reduced dimension of 500 by PCA. The split is 20-per-class/30-per-class/rest from Shchur et al. (2018). The original dataset is for the regression task to predict monthly traffic, but we group values into six bins and make it a classification problem. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/b9159c1c04784a741ef8d2cffa3058a7116e2fd1ad941983f278094e78f11a0c.jpg", + "table_caption": [ + "Table 4: Average degree and homophily of real-world graphs. " + ], + "table_footnote": [], + "table_body": "
DatasetDegreeHomophily
Four-Univ1.83 ± 1.710.16
PPI28.0 ± 39.260.17
Chameleon15.85 ± 18.200.21
Crocodile15.48 ± 15.970.26
Flickr10.08 ± 31.750.32
Cora-Full6.41 ± 8.790.59
ogbn-arxiv7.68 ± 9.050.63
Wiki-CS26.40 ± 36.040.68
CiteSeer2.78 ± 3.390.72
PubMed4.50 ± 7.430.79
Cora-ML5.45 ± 8.240.81
DBLP5.97 ± 9.350.81
Computers35.76 ± 70.310.81
Cora3.90 ± 5.230.83
CS8.93 ± 9.110.83
Photo31.13 ± 47.270.85
Physics14.38 ± 15.570.91
", + "bbox": [ + 343, + 127, + 650, + 381 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 409, + 825, + 464 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Four-Univ The Four-Univ dataset is a web page networks from computer science departments of diverse universities (Craven et al., 1998). Nodes are web pages, edges are hyperlinks between them, and node features are TF-IDF vectors of web page’s contents. There are five graphs consists of four universities (Cornell, Texas, Washington, and Wisconsin) and a miscellaneous graph from other universities. As the original authors suggested3, we use three graphs of universities and a miscellaneous graph for training, another one graph for validation (Cornell), and the other one graph for the test (Texas). Classification labels are types of web pages (student, faculty, staff, department, course, and project). ", + "bbox": [ + 173, + 482, + 825, + 594 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.1.5 FLICKR ", + "text_level": 1, + "bbox": [ + 176, + 612, + 285, + 626 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The Flickr dataset is a graph of images from Flickr (McAuley & Leskovec, 2012; Zeng et al., 2020). Nodes are images, and edges indicate whether two images share common properties such as geographic location, gallery, and users commented. Node features are a bag-of-words representation of images. We use labels and split in in Zeng et al. (2020). For labels, they construct seven classes by manually merging 81 image tags. ", + "bbox": [ + 174, + 636, + 825, + 707 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.1.6 PROTEIN-PROTEIN INTERACTION ", + "text_level": 1, + "bbox": [ + 176, + 724, + 462, + 739 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The protein-protein interaction (PPI) dataset (Zitnik & Leskovec, 2017; Hamilton et al., 2017; Subramanian et al., 2005) is a well-known benchmark in the inductive setting. A graph is given for human tissue, the nodes are proteins, the node’s features are biological signatures like genes, and the edges illustrate proteins’ interactions. The dataset consists of 20 training graphs, two validation graphs, and two test graphs. This dataset has multi-labels of gene ontology sets. ", + "bbox": [ + 174, + 750, + 825, + 820 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.2 LINK PREDICTION", + "text_level": 1, + "bbox": [ + 176, + 839, + 346, + 853 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For link prediction, we split $5 \\%$ and $10 \\%$ of edges for validation and test set, respectively. We fix the negative edges for the test set and sample negative edges for the training set at each iteration. ", + "bbox": [ + 176, + 867, + 823, + 895 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/aee96c742ee30bb26f1dd23a049f3d87cbbd778892185bc9e41a65f3aebeb4ee.jpg", + "table_caption": [ + "Table 5: Statistics of the real-world datasets. " + ], + "table_footnote": [], + "table_body": "
Dataset#Nodes#Edges#Features#ClassesSplit#Training Nodes# Val.Nodes#TestNodes
Four-Univ4518342620006fixed4014 (3 Gs)248 (1 G)256 (1G)
PPI5694481871650121fixed44906 (20 Gs)6514 (2 Gs)5524 (2 Gs)
Chameleon2277361015006random1201801977
Crocodile116311800205006random12018011331
Flickr892504498785007fixed446252231222313
Cora-Full197936342150070random1395204916349
ogbn-arxiv169343116624312840fixed909412979948603
Wiki-CS1170129711030010fixed58017695847
CiteSeer3327473237036fixed1205001000
PubMed19717443385003fixed605001000
Cora-ML2995815828797random1402102645
DBLP177165286716394random8012017516
Computers1375224586176710random20030013252
Cora2708542914337fixed1405001000
CS183338189450015random30045017583
Photo76501190817458random1602407250
Physics344932479625005random10015034243
", + "bbox": [ + 173, + 126, + 825, + 327 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.3 SYNTHETIC DATASET ", + "text_level": 1, + "bbox": [ + 176, + 349, + 367, + 364 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "To the best of our knowledge, our synthetic datasets are not used in recent literature. Therefore, we give some small examples of synthetic datasets to see qualitatively how the average degree and homophily vary according to $\\delta$ and $p _ { i n }$ . Specifically, we draw 2D t-SNE plot of node features and edges in Figure 6. In this figure, we can observe that the average degree $( d _ { a v g } = n \\cdot \\delta )$ ) increases as $\\delta$ increases, and homophily $( h = p _ { i n } / \\delta )$ increases as $p _ { i n }$ increases. Note that these are raw input features sampled from the 2D Gaussian distribution, not learned node representation. We use the code from the prior work4 (Abu-El-Haija et al., 2019) and apply normalization by standard score. We choose $\\delta$ from $\\{ 0 . 0 2 5 , 0 . 2 \\}$ , $p _ { i n }$ from $\\{ 0 . 1 \\delta , 0 . 5 \\delta , 0 . 9 \\delta \\}$ , and fix $n = 1 0 0$ and $c = 5$ . ", + "bbox": [ + 173, + 376, + 825, + 488 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/e3b9079245f2af111484609e3b99820050dc228cc53143bc13cefd45bf64735d.jpg", + "image_caption": [ + "Figure 6: t-SNE plots of node features and edges for synthetic graph examples. Hyperparameters are $\\delta \\in \\{ 0 . 0 2 5 ~ ( \\bar { \\mathrm { T o p } } ) , 0 . 2 ~ ( \\mathrm { B o t t o m } ) \\}$ and $p _ { i n } \\in \\{ 0 . 1 \\delta$ (Left), $0 . 5 \\delta$ (Center), $\\bar { 0 } . 9 \\delta$ (Right)}. " + ], + "image_footnote": [], + "bbox": [ + 202, + 505, + 795, + 729 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.4 DISTRIBUTION OF DEGREE AND HOMOPHILY OF DATASETS ", + "text_level": 1, + "bbox": [ + 176, + 799, + 630, + 814 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In Figure 7, we draw kernel density estimation plots of per-node homophily and degree of nodes in real-world datasets. We define per-node homophily as the ratio of neighbors with the same label as the center node, that is, $\\begin{array} { r } { h _ { i } \\ = \\ \\stackrel { \\bullet } { \\sum } _ { j \\in \\mathbb { N } _ { i } } \\mathbf { 1 } _ { l ( i ) = l ( j ) } \\big / \\big | \\mathbb { N } _ { i } \\big | } \\end{array}$ . Note that we define homophily as $h =$ ${ \\frac { 1 } { | V | } } \\sum _ { i \\in V } h _ { i }$ in Section 5. ", + "bbox": [ + 173, + 824, + 825, + 886 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/e1ca30b2f78b42ac0edcb91746459d800064846123a965e68588b2e9ed0211f0.jpg", + "image_caption": [ + "Figure 7: Kernel density estimate plot of distribution of degree and per-node homophily in realworld graphs. " + ], + "image_footnote": [], + "bbox": [ + 256, + 92, + 741, + 645 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We focus more on the part outside of where the degree is 1 (0 with log scale), and the per-node homophily is 0. These are leaf nodes incorrectly connected and does not significantly affect the learning overall graph representation. In most datasets, only the largest mode exists, or there are some small modes around it. However, in Flickr and Crocodile, we can observe that the interval between modes is wide. More specifically, Crocodile’s modes are in the area of (high degree, low per-node homophily) and (low degree, high per-node homophily), and Flickr’s modes cover most homophily at a specific degree. Note that we can regard a mixture of distribution as a mixture of different sub-graphs. We argue that this is the reason our recipe does not fit for these two datasets. ", + "bbox": [ + 174, + 712, + 825, + 823 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.5 MODEL & HYPERPARAMETER CONFIGURATIONS ", + "text_level": 1, + "bbox": [ + 173, + 840, + 558, + 856 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Model Since we experiment with numbers of datasets, we maintain almost the same configurations across datasets. We do not use other methods such as residual connections, deeper layers, batch normalization, edge augmentation, and more hidden features, although we have confirmed from previous studies that these techniques contribute to performance improvement. For example, prior work has shown f1-score close to 100 for PPI. To clearly see the difference between the various graph attention designs, we intentionally keep a simple model configuration. ", + "bbox": [ + 174, + 867, + 823, + 922 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/0d364200ccfa2fd4606d979c99bf0438c7a89755e8f3ecd91abe33df7bee7536.jpg", + "table_caption": [ + "Table 6: Hyperparameters for experiments on real-world datasets. " + ], + "table_footnote": [], + "table_body": "
DatasetModel入2EpePn
CoraSuperGATsD0.00782993594510.882669370.80.5
SuperGATMX0.00822886497311.346574530.80.5
CiteSeerSuperGATsD0.048238086570.090739928280.80.3
SuperGATMX0.041613218320.013081692730.80.5
PubMedSuperGATsD0.000203092756318.825603330.60.7
SuperGATMX2.19E-0410.45205180.60.5
PPISuperGATsD3.39E-070.00103435184210.5
SuperGATmX1.00E-071.79E-0610.5
", + "bbox": [ + 232, + 127, + 766, + 258 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/a77d48fd59f9ce3b5cf8313d2e0f38c5fd22084a45cf3b6831ec7df4a40415da.jpg", + "table_caption": [ + "Table 7: Summary of classification accuracies with 10 random seeds for Cora, CiteSeer, and PubMed in the full-supervised setting. We mark with asterisks the reprinted results from the respective papers. " + ], + "table_footnote": [], + "table_body": "
ModelCoraCiteSeerPubMed
GAT*87.2 ± 0.377.3 ± 0.387.0 ± 0.3
CGAT*88.2 ± 0.378.9 ± 0.287.4 ± 0.3
CGAT (Our Impl.: 2-layer w/ 64-features)88.9 ± 0.378.9 ± 0.286.9 ± 0.2
SuperGATMx (2-layer w/ 64-features)88.7 ± 0.279.1 ± 0.287.0 ± 0.1
", + "bbox": [ + 215, + 315, + 782, + 390 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 421, + 821, + 450 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Hyperparameter For real-world datasets, we tune two hyperparameters (mixing coefficients $\\lambda _ { 2 }$ and $\\lambda _ { E }$ ) by Bayesian optimization for the mean performance of 3 random seeds. We choose negative sampling ratio $p _ { n }$ from $\\{ 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 \\}$ , and edge sampling ratio $p _ { e }$ from $\\{ 0 . 6 , 0 . 8 , 1 . 0 \\}$ . We fix dropout probability to 0.0 for PPI, 0.2 for ogbn-arxiv, 0.6 for others. We set learning rate to 0.05 (ogbn-arxiv), 0.01 (PubMed, PPI, Wiki-CS, Photo, Computers, CS, Physics, Crocodile, Cora-Full, DBLP), 0.005 (Cora, CiteSeer, Cora-ML, Chameleon), 0.001 (Four-Univ). For ogbn-arxiv, we set the number of features per head to 16 and the number of heads in the last layer to one; otherwise, we use eight features per head and eight heads in the last layer. ", + "bbox": [ + 173, + 472, + 825, + 584 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For synthetic datasets, we choose $\\lambda _ { E }$ from $\\{ 1 0 ^ { - 5 } , 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 1 0 ^ { - 2 } , 1 0 ^ { - 1 } , 1 , 1 0 , 1 0 ^ { 2 } \\}$ and $\\lambda _ { 2 }$ from $\\{ 1 0 ^ { - 7 } , 1 0 ^ { - 5 } , 1 0 ^ { - 3 } \\}$ . We fix learning rate to 0.01, dropout probability to 0.2, $p _ { n }$ to 0.5, and, $p _ { e }$ to 0.8 for all synthetic graphs. ", + "bbox": [ + 176, + 589, + 823, + 633 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Table 6 describes hyperparameters for SuperGAT on four real-world datasets. For other datasets and experiments, please see the code (./SuperGAT/args.yaml). ", + "bbox": [ + 173, + 640, + 823, + 669 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.6 CGAT IMPLEMENTATION ", + "text_level": 1, + "bbox": [ + 174, + 693, + 395, + 707 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "CGAT (Wang et al., 2019a) has two auxiliary losses: graph structure based constraint $\\mathcal { L } _ { g }$ and class boundary constraint $\\mathcal { L } _ { b }$ . We borrow their notation for this section: $\\nu$ as a set of nodes, $\\mathbb { N } _ { i }$ as a set of one-hop neighbors, $\\mathbb { N } _ { i } ^ { + }$ as a set of neighbors that share labels, $\\mathbb { N } _ { i } ^ { - }$ as a set of neighbors that do not share labels, $\\zeta$ · as a margin between attention values, and $\\phi \\left( v _ { i } , v _ { k } \\right)$ as unnormalized attention value. ", + "bbox": [ + 174, + 720, + 825, + 777 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/bdc18dee3eebd9aa6f7b0c94514e28757d07a625598ea87c807cbce8887b6f3c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { g } = \\displaystyle \\sum _ { i \\in \\mathcal { V } } \\displaystyle \\sum _ { j \\in \\mathbb { N } _ { i } \\setminus \\mathbb { N } _ { i } ^ { - } } \\sum _ { k \\in ( \\mathcal { V } \\setminus \\mathbb { N } _ { i } ) } \\operatorname* { m a x } \\left( 0 , \\phi \\left( v _ { i } , v _ { k } \\right) + \\zeta _ { g } - \\phi \\left( v _ { i } , v _ { j } \\right) \\right) } \\\\ & { \\mathcal { L } _ { b } = \\displaystyle \\sum _ { i \\in \\mathcal { V } } \\sum _ { j \\in \\mathbb { N } _ { i } ^ { + } } \\sum _ { k \\in \\mathbb { N } _ { i } ^ { - } } \\operatorname* { m a x } \\left( 0 , \\phi \\left( v _ { i } , v _ { k } \\right) + \\zeta _ { b } - \\phi \\left( v _ { i } , v _ { j } \\right) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 790, + 710, + 871 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Since label information is included in these two losses, they are difficult to use in semi-supervised settings that provide few labeled samples. In fact, in the CGAT paper, they conduct experiments in full-supervised settings; that is, they use all nodes in training except validation and test nodes. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "So, we only use $\\mathcal { L } _ { g }$ modified for semi-supervised learning. ", + "bbox": [ + 173, + 103, + 560, + 119 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/e81f2265a8a62b545c154da359d998c8e5dc2fe7fbc90b21a0b78edd55af149f.jpg", + "text": "$$\n\\mathcal { L } _ { g } ^ { \\mathrm { S S L } } = \\sum _ { i \\in \\mathcal { V } } \\sum _ { j \\in \\mathbb { N } _ { i } } \\sum _ { k \\in ( \\mathcal { V } \\backslash \\mathbb { N } _ { i } ) } \\operatorname* { m a x } \\left( 0 , \\phi \\left( v _ { i } , v _ { k } \\right) + \\zeta _ { g } - \\phi \\left( v _ { i } , v _ { j } \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 292, + 125, + 702, + 162 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "With $\\mathcal { L } _ { c }$ , the multi-class cross-entropy on node labels, CGAT’s optimization objective is ", + "bbox": [ + 169, + 175, + 751, + 190 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/263deb5b254c25fb593a2f7fc3e2ce853d11c6b1c430c5f06fe94863042c6ce3.jpg", + "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { c } + \\lambda _ { g } \\mathcal { L } _ { g } + \\lambda _ { b } \\mathcal { L } _ { b } ,\n$$", + "text_format": "latex", + "bbox": [ + 413, + 198, + 583, + 215 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "and our modified CGAT’s loss is ", + "bbox": [ + 174, + 220, + 390, + 236 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/fc919f51eb7cf286a4398f0fdc0ce69ab31acd658c7020440037c5a575b645e1.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } = \\mathcal { L } _ { c } + \\lambda _ { g } \\mathcal { L } _ { g } ^ { \\mathrm { S S L } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 434, + 241, + 563, + 261 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In addition to losses, CGAT proposes top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ softmax and node importance based negative sampling (NINS). Top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ softmax picks up nodes with top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ attention values among neighbors. NINS adopts importance sampling when choosing negative sample nodes. ", + "bbox": [ + 174, + 273, + 823, + 316 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Since the code for CGAT has not been released, we implement our own version. In all experiments in our paper, we use only the modified losses (Equation 14) and top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ softmax due to the training and implementation complexity of NINS. For PPI, even if we do not assume a semi-supervised setting, we use the same loss because we could not accurately implement multi-label cases for Equation 10 and 11 with only CGAT’s description. ", + "bbox": [ + 174, + 323, + 825, + 392 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To verify the functionality of our implementation, we report the results of a full-supervised setting with the original loss (Equation 13) like CGAT paper, in Table 7. Our implementation of CGAT shows almost the same performance reported in the original paper. In addition, SuperGAT and CGAT showed almost similar performance in a full-supervised setting. Note that the original paper employs two hidden layers with hidden dimensions as 32 for Cora, 64 for CiteSeer, and three hidden layers with hidden dimensions 32 for PubMed, where models in our experiments are all two-layer with 64 features. ", + "bbox": [ + 174, + 398, + 825, + 497 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "A.7 WALL-CLOCK TIME EXPERIMENTAL SET-UP ", + "text_level": 1, + "bbox": [ + 176, + 513, + 526, + 529 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To demonstrate our model’s efficiency, we measure the mean wall-clock time of the entire training process of three runs using a single GPU (GeForce GTX 1080Ti). We compare our model with GAT (Velickovi ˇ c et al., 2018) and GAM (Stretcu et al., 2019). GAT is the basic model using a ´ simpler attention mechanism than ours, and GAM is the state-of-the-art model using co-training with the auxiliary model. ", + "bbox": [ + 174, + 540, + 825, + 609 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For GAT and SuperGAT, we use our implementation (including hyperparameter settings) in PyTorch (Paszke et al., 2019). For GAM, we adopt the code in TensorFlow (Abadi et al., 2015) from the authors 5 and choose $\\mathrm { G C N + G A M }$ model, which showed the best performance. We retain the default settings in the code but use the hyperparameters reported in the paper, if possible. With this setting, $\\mathrm { G C N + G A M }$ on PubMed is not finished after 24 hours; therefore, we manually early-stop the training at the best accuracy. ", + "bbox": [ + 174, + 617, + 825, + 700 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "B RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 285, + 118 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "B.1 PROOF OF PROPOSITION ", + "text_level": 1, + "bbox": [ + 174, + 137, + 387, + 152 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proposition 2. For $l + 1 t h \\ G A T$ layer, if $W$ and $\\textbf { \\em a }$ are independent and identically drawn from zero-mean uniform distribution with variance $\\sigma _ { w } ^ { 2 }$ and $\\sigma _ { a } ^ { 2 }$ respectively, assuming that parameters are independent to input features $h ^ { l }$ and elements of $h ^ { l }$ are independent to each other, ", + "bbox": [ + 174, + 165, + 825, + 209 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/7e6447134fa9e9cbf09c4177539a4816dab582fe00c8e95a442836527d9a8325.jpg", + "text": "$$\n\\begin{array} { r } { V a r [ e _ { i j , G o } ^ { l + 1 } ] = 2 F ^ { l + 1 } \\sigma _ { w } ^ { 2 } \\sigma _ { a } ^ { 2 } \\mathbb { E } ( \\Vert h ^ { l } \\Vert _ { 2 } ^ { 2 } ) ~ a n d ~ V a r [ e _ { i j , D P } ^ { l + 1 } ] \\geq F ^ { l + 1 } \\sigma _ { w } ^ { 4 } \\left( \\frac { 4 } { 5 } \\mathbb { E } \\left( ( h _ { i } ^ { l } ) ^ { \\top } h _ { j } ^ { l } ) ^ { 2 } \\right) + V a r ( ( h _ { i } ^ { l } ) ^ { \\top } h _ { j } ^ { l } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 194, + 222, + 803, + 241 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "$h ^ { \\prime } = W h ^ { l }$ $\\begin{array} { r } { \\pmb { h } _ { i , k } ^ { \\prime } = \\sum _ { r = 1 } ^ { F ^ { l + 1 } } \\pmb { W } _ { k r } \\pmb { h } _ { i , r } ^ { l } } \\end{array}$ ", + "bbox": [ + 173, + 292, + 526, + 314 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Note that, ", + "bbox": [ + 173, + 320, + 240, + 334 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/6ba96dd5ada71191fc66893632737b5b47550c3f67e74e274213f489b03c41ac.jpg", + "text": "$$\ne _ { i j , \\mathrm { G O } } ^ { l + 1 } = { \\pmb { a } } ^ { \\top } [ { \\pmb { h } } _ { i } ^ { \\prime } \\| { \\pmb { h } } _ { j } ^ { \\prime } ] \\quad \\mathrm { a n d } \\quad e _ { i j , \\mathrm { D P } } ^ { l + 1 } = { \\pmb { h } } _ { i } ^ { \\prime \\top } { \\pmb { h } } _ { j } ^ { \\prime }\n$$", + "text_format": "latex", + "bbox": [ + 351, + 344, + 647, + 367 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "First, we compute $\\mathbb { E } ( a ^ { 2 } )$ and $\\mathbb { E } ( h ^ { \\prime 2 } )$ ", + "bbox": [ + 173, + 387, + 415, + 404 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/90991e231cfed5525b41c2eed7f6554e36b7446f911812f683731adc08b91e7d.jpg", + "text": "$$\n\\mathbb { E } ( a ^ { 2 } ) = \\operatorname { V a r } ( a ) + \\mathbb { E } ( a ) ^ { 2 } = \\sigma _ { a } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 393, + 416, + 602, + 435 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/fc3406d8fe57a84211576ca728cd8ea431ee32bf050dbbc4e0f0da1ead74b81b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } ( h _ { k } ^ { \\prime 2 } ) = \\mathbb { E } \\left( \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { \\cdot , r } ^ { l } \\right) ^ { 2 } \\right) } \\\\ & { \\qquad = \\mathbb { E } \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } ^ { 2 } ( h _ { \\cdot , r } ^ { l } ) ^ { 2 } \\right) } \\\\ & { \\qquad = \\mathbb { E } \\left( W _ { k , \\cdot } ^ { 2 } \\right) \\mathbb { E } \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { \\cdot , r } ^ { l } ) ^ { 2 } \\right) } \\\\ & { \\qquad = \\sigma _ { w } ^ { 2 } \\mathbb { E } \\left( \\| h ^ { l } \\| _ { 2 } ^ { 2 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 364, + 464, + 630, + 577 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For the variance of $e _ { i j , \\mathrm { G O } } ^ { l + 1 }$ ", + "bbox": [ + 173, + 595, + 346, + 614 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/9d0b392ae75d18bfeb11ef9f7f979e5dabfade4b5f915dd263510119a4bc09d7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { V a r } ( e _ { i j , \\mathrm { G J } } ^ { l + 1 } ) = \\mathrm { V a r } ( a ^ { \\top } [ h _ { i } ^ { \\prime } | | h _ { j } ^ { \\prime } ] ) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad = \\mathrm { V a r } \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } \\left( a _ { r } h _ { i , r } ^ { \\prime } + a _ { r + F ^ { l + 1 } } h _ { j , r } ^ { \\prime } \\right) \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad = 2 F ^ { l + 1 } \\left( \\mathbb { E } \\left( a ^ { 2 } \\right) \\mathbb { E } \\left( h ^ { \\prime 2 } \\right) - \\mathbb { E } \\left( a \\right) ^ { 2 } \\mathbb { E } \\left( h ^ { \\prime } \\right) ^ { 2 } \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad = 2 F ^ { l + 1 } \\mathbb { E } \\left( a ^ { 2 } \\right) \\mathbb { E } \\left( h ^ { \\prime 2 } \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad = 2 F ^ { l + 1 } \\sigma _ { a } ^ { 2 } \\sigma _ { w } ^ { 2 } \\mathbb { E } \\left( \\| h ^ { l } \\| _ { 2 } ^ { 2 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 627, + 684, + 770 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Now we compute $\\mathbb { E } ( h _ { i } ^ { \\prime } h _ { j } ^ { \\prime } )$ and $\\mathbb { E } ( h _ { i } ^ { \\prime 2 } h _ { j } ^ { \\prime 2 } )$ , ", + "bbox": [ + 174, + 786, + 450, + 805 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/c16b2b7db2596ee2a18b74513541bee562da3f239e7c10d8671ed1098a927289.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } ( h _ { i , k } ^ { \\prime } h _ { j , k } ^ { \\prime } ) = \\mathbb { E } \\left( \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { i , r } ^ { l } \\right) \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { j , r } ^ { l } \\right) \\right) } \\\\ & { \\quad \\quad \\quad \\quad = \\mathbb { E } \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } ^ { 2 } h _ { i , r } ^ { l } h _ { j , r } ^ { l } \\right) } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } ( W _ { k , \\cdot } ^ { 2 } ) \\mathbb { E } \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } h _ { i , r } ^ { l } h _ { j , r } ^ { l } \\right) } \\\\ & { \\quad \\quad \\quad = \\sigma _ { w } ^ { 2 } \\mathbb { E } ( ( h _ { i } ^ { l } ) ^ { \\top } h _ { j } ^ { l } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 818, + 696, + 925 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/78c23b06e35ecc114379a54f70b71effdabf1355d7b03027cfa3fd2c0b2e3d0a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } ( h _ { i , k } ^ { \\ell } h _ { j , k } ^ { \\ell } ) } \\\\ & { = \\mathbb { E } \\left( \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { i , r } ^ { \\ell } \\right) ^ { 2 } \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { j , r } ^ { l } \\right) ^ { 2 } \\right) } \\\\ & { = \\mathbb { E } ( W _ { k , \\cdot } ^ { 4 } ) \\mathbb { E } \\left( \\displaystyle \\sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right) + \\mathbb { E } ( W _ { k , \\cdot } ^ { 2 } ) ^ { 2 } \\mathbb { E } \\left( \\displaystyle \\sum _ { s \\neq t } ^ { F ^ { l + 1 } } ( h _ { i , s } ^ { l } h _ { j , t } ^ { l } ) ^ { 2 } + 2 ( h _ { i , s } ^ { l } h _ { j , s } ^ { l } ) ( h _ { i , t } ^ { l } h _ { j , t } ^ { l } ) \\right) } \\\\ & { = \\mathbb { E } ( W _ { k , \\cdot } ^ { 4 } ) \\mathbb { E } \\left( \\displaystyle \\sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right) + \\mathbb { E } ( W _ { k , \\cdot } ^ { 2 } ) ^ { 2 } \\mathbb { E } \\left( \\displaystyle \\sum _ { r } ^ { r } ( h _ { i , r } ^ { l } ) ^ { 2 } \\displaystyle \\sum _ { r } ^ { r } ( h _ { j , r } ^ { l } ) ^ { 2 } - \\displaystyle \\sum _ { r } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 173, + 98, + 785, + 272 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/52c023c6080b13c1dac8f05b82b90c20250a84c89da56f59d71b5c847e8de762.jpg", + "text": "$$\n\\begin{array} { r l r } { { + 2 \\mathbb { R } ( W _ { k , * } ^ { 2 } ) ^ { 2 } \\mathbb { R } ( ( \\sum _ { r } h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } - \\sum _ { r } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } ) } } \\\\ & { } & { = \\frac { 9 } { 5 } \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( \\sum _ { r = 1 } ^ { r + 1 } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } ) + \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( \\| h _ { i } ^ { i } \\| _ { 2 } ^ { 2 } \\| h _ { j } ^ { i } \\| _ { 2 } ^ { 2 } - 3 \\sum _ { r = 1 } ^ { r + 1 } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } ) + 2 \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( ( ( h _ { i } ^ { i } ) ^ { \\top } h _ { j , r } ^ { i } ) ^ { 2 } } \\\\ & { } & { ( 3 7 ) } \\\\ & { } & { = \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { R } ( \\| h _ { i } ^ { i } \\| _ { 2 } ^ { 2 } \\| h _ { j } ^ { i } \\| _ { 2 } ^ { 2 } - \\frac { 6 } { 5 } \\sum _ { r = 1 } ^ { r + 1 } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } + 2 ( ( h _ { i } ^ { i } ) ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } ) } \\\\ & { } & { ( 3 8 ) } \\\\ & { } & = \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( \\frac { 4 } { 5 } ( h _ { i } ^ { i } ) ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } + \\frac { 1 } { 1 0 } \\sum _ { s \\not = 1 } ^ { r } ( ( h _ { i , s } ^ { i } h _ { j , s } ^ { i } + h _ { i , r } ^ { i } h _ { j , s } ^ { i } ) ^ { 2 } + 8 ( h _ { i , r } ^ { i } h _ { j , s } ^ { i } ) ^ { 2 } ) + \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( ( h \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 174, + 289, + 826, + 505 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Note that for the zero-mean uniform distribution $\\boldsymbol { \\mathcal { U } } ( - \\boldsymbol { u } , \\boldsymbol { u } )$ with variance $\\sigma _ { w } ^ { 2 }$ ", + "bbox": [ + 173, + 529, + 678, + 545 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/3071a40330b35a882c2773e4c35cbd06d7596b21c846ca4510c99263ee8e9c36.jpg", + "text": "$$\n\\mathbb { E } ( W _ { \\cdot , \\cdot } ^ { 2 } ) = \\operatorname { V a r } ( W _ { \\cdot , \\cdot } ) + \\mathbb { E } ( W _ { \\cdot , \\cdot } ) ^ { 2 } = \\sigma _ { w } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 364, + 546, + 632, + 565 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/84a20c86feba795c9ec3703ff698f323dcb430e3a12be8a20f6bf1af98208186.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\sigma _ { w } ^ { 2 } = \\frac 1 { 1 2 } ( u - ( - u ) ) ^ { 2 } = \\frac 1 3 u ^ { 2 } } } \\\\ { { \\displaystyle \\mathbb { E } ( W _ { \\cdot , \\cdot } ^ { 4 } ) = \\frac 1 5 \\sum _ { r = 0 } ^ { 4 } ( - u ) ^ { r } u ^ { 4 - r } = \\frac 1 5 u ^ { 4 } = \\frac 9 5 \\sigma _ { w } ^ { 4 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 570, + 647, + 646 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For the variance of $e _ { i j , \\mathrm { D P } } ^ { l + 1 }$ ", + "bbox": [ + 173, + 655, + 344, + 672 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/57632c9b644a8152b5789930512298cb6268805641b8d47466acb5879bbca639.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { ~ \\forall i a r ~ } ( c _ { \\delta , ( 1 , 1 ) } ^ { i ( 1 , 1 ) } ) } \\\\ & { = \\mathrm { ~ \\operatorname* { W i a r } ~ } ( h _ { i } ^ { i } , h _ { j } ^ { i } ) } \\\\ & { = \\mathrm { ~ \\operatorname* { W i a r } ~ } ( \\sum _ { r ^ { \\prime \\prime } = 1 } ^ { r ^ { \\prime \\prime } + 1 } h _ { i , r } ^ { i } h _ { j , x } ^ { i } ) } \\\\ & { = E ^ { i + 1 } \\mathrm { ~ \\operatorname* { W a r } ~ } ( h _ { i } ^ { j } , h _ { j } ^ { i } ) } \\\\ & { = F ^ { i + 1 } \\mathrm { ~ { \\mathbb { E } } ~ } [ ( h _ { i } ^ { i } h _ { j } ^ { i } ) ^ { 2 } ) - \\mathrm { ~ { \\mathbb { E } } } ( h _ { i } ^ { i } h _ { j } ^ { j } ) ^ { 2 } ] } \\\\ & { = F ^ { i + 1 } \\mathrm { ~ { \\mathbb { C } } ~ } \\mathrm { e } ^ { i } \\mathrm { ~ { W } ~ } ( \\frac { 4 } { 5 } ( h _ { i } ^ { i } h _ { j } ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } + \\frac { 1 } { 1 0 } \\sum _ { \\phi \\in \\mathbb { R } _ { j } } ( h _ { i , \\phi } ^ { i } h _ { j , + } ^ { i } + h _ { i , } ^ { i } h _ { j , \\phi } ^ { i } ) ^ { 2 } + \\mathrm { { \\mathbb { S } } } ( h _ { i , \\phi } ^ { i } h _ { j , + } ^ { i } ) ^ { 2 } ) } \\\\ & { \\quad + F ^ { i + 1 } \\sigma _ { \\phi } ^ { i } \\mathbb { E } ( \\mathbb { E } ( ( h _ { i } ^ { i } h _ { j } ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } - \\mathbb { E } [ ( h _ { i } ^ { i } h _ { j } ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } ] ) } \\\\ & { \\geq F ^ { i + 1 } \\sigma _ { \\phi } ^ { i } ( \\frac { 4 } { 5 } \\mathbb { E } ( ( h _ { i } ^ { i } h _ { j } ^ { \\top } ) ^ { 2 } h _ { j } ^ { i } ) + \\mathrm { { \\mathbb { S } } } \\{ ( h _ { i } ^ { i } ) ^ { 2 } h _ { j } ^ { i } \\} ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 674, + 769, + 904 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/a34e155770e1a710ad6b921241c6bfbaab4ff3ab46ff112c69f3724f0fdce018.jpg", + "table_caption": [ + "Table 8: Summary of classification accuracies (of 5 runs) for synthetic datasets. " + ], + "table_footnote": [], + "table_body": "
Avg.degree Homophily
GCN 0.10.2 0.30.4 0.50.6 0.7 0.8 0.9
1 32.7 ± 1.135.1±0.434.7 ±0.440.8±0.443.4±0.346.2± 0.5 49.7±0.554.8±0.9 56.2±0.3
1.5 28.7± 0.530.0±0.634.6±0.440.8± 0.441.7 ± 0.446.9 ± 0.5 52.1±0.654.5± 0.563.6± 0.5
2.5 24.6±0.227.9 ±0.931.0±0.339.7±0.644.0± 0.651.4± 0.8 56.6±0.565.2 ±0.674.5± 0.4
3.5 25.8±0.830.5±0.831.9± 0.738.4± 0.445.0 ±1.049.8 ± 0.560.8±0.770.3±0.379.5± 0.3
5 25.0±0.326.3± 0.533.0 ± 1.139.8±0.750.2 ±0.558.6 ± 1.168.8±0.378.0 ±1.988.5±0.2
7.5 26.1± 0.529.8±0.634.0±0.643.8±0.952.0±0.470.7±3.274.6± 2.488.6±0.895.0±0.2
10 25.4± 0.629.2 ± 0.537.1± 0.547.6 ± 1.058.1±0.973.1 ± 2.885.3± 2.692.7 ± 1.197.9± 0.5
12.5 24.0±0.427.6±0.539.6±0.848.9±0.465.8 ± 3.177.8 ± 2.488.5±1.595.0 ±0.999.2± 0.0
15 22.1 ± 0.528.2±0.644.0± 0.753.0±0.569.9 ± 4.779.1 ± 2.092.2 ±1.898.0±0.499.6±0.3
20 24.3±0.630.9±0.741.9 ± 0.958.1 ± 1.275.0 ± 2.986.7 ± 1.196.1±0.398.9±0.199.8 ±0.1
25 26.6± 1.031.1± 0.344.9±0.262.4± 0.780.1± 2.691.0 ± 1.597.5± 0.599.5± 0.2100.0 ± 0.0
32.5 26.3± 0.733.8±0.751.8±0.867.5 ± 2.983.5 ± 1.995.7 ± 0.398.4±0.2100.0±0.0100.0±0.0
40 23.7±0.534.2± 0.553.4± 0.472.8 ± 1.587.6± 0.996.1 ± 0.699.7 ± 0.199.9 ± 0.0100.0 ± 0.0
50 25.0 ± 1.136.4± 1.055.1 ± 0.582.7 ± 3.091.2 ± 0.798.6±0.499.8±0.099.9 ±0.0100.0±0.0
75 25.9±0.640.6± 0.668.0 ± 1.087.9± 2.697.1± 0.699.6±0.1100.0 ±0.0100.0 ± 0.0100.0 ±0.0
100 25.1 ± 0.542.0±0.972.4 ± 1.492.6± 0.498.6± 0.2100.0 ±0.0100.0±0.0100.0±0.0100.0±0.0
GATGO 0.10.20.30.40.50.60.70.80.9
1 33.7±0.936.7 ±0.835.2± 1.240.1± 0.743.0± 0.846.3± 0.649.4± 0.553.8 ±1.456.1 ± 0.5
1.5 28.8 ±1.033.0±0.534.8±0.940.7± 0.442.2 ± 0.946.7± 1.052.2 ±0.853.6 ± 1.462.8 ± 0.5
2.5 28.8±1.230.6±0.632.3± 0.540.1 ± 0.844.0 ± 1.352.2± 0.655.5±1.464.6 ± 1.373.1± 0.6
3.5 28.0±0.433.4± 1.633.5± 0.640.1 ± 1.046.2 ± 0.650.8± 0.662.5±0.671.7± 0.678.0 ± 0.5
5 28.0±1.330.0±0.836.9± 0.642.1± 0.453.6± 0.761.0 ± 1.170.4±0.578.7 ±1.287.5± 0.6
7.5 30.0±0.931.6 ± 2.040.4± 0.545.0 ±1.059.5 ± 0.672.3 ± 1.079.4±0.690.4± 0.595.0 ± 0.5
10 31.0 ± 1.434.0 ±1.243.1 ± 0.654.6±0.566.1 ± 0.677.4 ± 1.590.6± 1.494.7 ±0.798.4± 0.5
12.5 29.8 ± 1.435.5± 1.747.3 ± 0.958.6±1.772.1 ± 1.486.3± 0.590.7±0.796.2 ±0.398.9±0.2
15 31.9 ±1.634.6 ± 1.649.5± 0.962.0±0.777.2 ± 1.585.9 ± 0.394.4± 0.998.5± 0.599.4± 0.1
2034.4 ±1.838.3± 1.654.1 ± 1.970.0 ± 1.183.9 ±1.093.7± 0.197.5±0.399.0 ±0.399.7 ± 0.0
2535.8±2.042.0± 2.457.7 ± 0.677.7 ± 0.987.9 ±1.295.0±0.798.7±0.699.6±0.399.9 ± 0.1
32.537.4 ±1.244.7 ± 1.166.5 ± 1.979.9 ± 1.291.4 ± 1.498.0± 0.599.0±0.399.8 ± 0.1100.0 ± 0.0
4037.5± 2.045.1± 0.966.5 ± 1.185.7 ± 1.593.5 ± 1.097.6± 0.699.5± 0.199.9 ± 0.199.9± 0.1
5038.7± 1.849.5± 2.368.9± 2.589.5± 0.896.0± 0.899.0± 0.499.7 ± 0.299.8 ± 0.1100.0 ±0.0
7539.6±3.053.5 ±1.777.6 ± 2.692.8± 2.398.0 ±1.299.5 ± 0.399.8±0.2100.0±0.0100.0 ± 0.0
10041.3 ± 1.256.6 ± 1.481.1 ± 3.595.9 ± 1.299.0 ± 0.499.8 ± 0.199.9 ± 0.1100.0 ± 0.1100.0 ± 0.0
SuperGATsD0.10.20.30.40.50.60.70.80.9
138.5±1.040.5±1.539.5 ±1.642.3±0.944.0± 0.547.1 ± 0.550.1 ± 1.053.0±0.355.1± 1.0
1.536.7 ± 1.237.8 ± 1.842.0± 0.442.9±0.845.0± 0.747.8 ± 0.552.9 ±0.754.7 ± 1.263.0±0.4
2.535.7±1.837.3 ± 1.639.2 ± 1.447.8 ±1.349.0± 0.356.1 ± 0.559.4±0.465.9 ±0.870.6 ± 1.2
3.535.7± 1.039.2 ± 1.941.4 ± 0.544.0±0.751.0± 0.355.0 ± 1.265.1 ± 1.173.5 ± 1.078.1±0.6
537.0± 1.539.0 ± 1.545.0 ±1.748.1 ± 1.057.5± 0.666.1 ± 0.974.4 ± 0.281.0 ± 1.088.3 ± 0.8
7.536.8 ± 1.538.7±0.747.3± 0.551.1 ±0.861.2 ± 0.675.5± 0.582.3±0.991.0±0.295.2± 0.3
1039.4 ± 1.142.3 ±0.750.5± 1.258.6±0.370.0± 0.580.1 ±0.590.8±0.395.5± 0.698.7±0.3
12.538.5± 1.042.0±0.750.5± 0.662.4± 0.573.6± 0.486.8± 0.492.9± 0.498.0± 0.699.3± 0.3
1540.4 ± 1.142.8± 0.753.6± 0.367.4± 0.479.7± 0.389.0 ± 1.096.3±0.399.1 ± 0.299.6± 0.3
2037.8±0.842.1 ± 0.956.9 ±0.970.0 ± 0.586.1 ±0.895.6± 0.399.1± 0.199.6± 0.299.7± 0.1
2540.0± 1.048.8±1.259.5±0.578.7±0.290.7 ±0.297.5± 0.199.4 ± 0.1100.0±0.099.9 ± 0.1
32.539.7 ± 1.148.5±0.769.4± 0.482.7 ± 0.594.3 ±0.299.3± 0.199.7 ± 0.199.9 ± 0.199.9 ±0.0
4044.2 ± 1.148.7± 1.369.2 ± 0.788.3± 0.497.1 ± 0.199.7± 0.199.9±0.099.8± 0.2100.0 ±0.0
5044.3 ± 0.753.2 ± 0.673.4 ±1.291.2 ± 0.497.7 ±0.2100.0 ±0.0100.0 ±0.0100.0±0.0100.0 ±0.0
7544.8± 0.756.1± 0.982.7±0.695.7 ±0.299.8±0.299.9 ± 0.1100.0±0.0100.0±0.0100.0±0.0
10043.1 ± 1.360.7±0.787.4± 0.398.3± 0.199.9 ± 0.0100.0 ±0.0100.0 ±0.0100.0 ±0.0100.0 ± 0.0
SuperGATMX0.10.20.30.40.50.60.70.80.9
133.4±0.736.3±0.835.4±0.740.5±0.343.4±0.647.0±0.850.0±0.554.2 ± 1.156.4±0.4
1.528.8±0.731.5 ± 1.036.0±0.841.1 ± 1.141.3 ±0.647.5± 0.852.3±0.754.8±0.563.4± 0.3
2.526.6±0.930.5±1.333.6±0.842.7 ± 1.544.7 ± 0.552.2 ± 0.556.5 ± 0.866.7 ±0.874.5 ± 0.7
3.527.7± 0.933.9 ± 1.737.9± 0.742.4± 0.649.7 ±0.854.2 ± 1.165.3 ± 1.173.2± 0.779.4 ± 1.5
528.4 ± 2.133.4±0.840.9±0.846.2± 0.456.4±0.264.1 ± 1.874.6± 1.483.1 ± 1.090.6±0.9
7.531.1 ± 0.735.0±0.946.5 ± 1.151.9 ± 0.764.3 ± 1.575.2 ± 1.682.1 ± 1.392.6 ± 1.895.8±0.2
1032.6± 1.741.5 ± 1.448.9 ± 1.261.7 ± 1.170.4 ± 1.381.3 ± 1.191.3±0.795.1 ± 0.899.2± 0.3
12.534.2 ± 3.141.1 ± 0.853.3 ± 1.065.9 ± 1.177.2 ± 1.488.1 ±0.792.6± 2.697.5 ± 0.999.3±0.2
1536.4 ± 1.339.9 ± 2.153.8±1.568.5±0.881.0±0.490.2 ± 1.096.1 ±0.999.2 ± 0.299.6± 0.1
2035.2 ± 3.141.0 ± 2.162.3 ±1.273.9 ± 0.986.3±0.695.2 ± 1.398.6±0.799.6± 0.199.9± 0.2
2536.8 ± 1.946.6± 1.762.7 ± 1.181.9 ± 1.790.7±0.896.8 ± 1.599.3±0.399.8±0.2100.0 ± 0.0
32.536.7± 0.846.8± 0.568.9± 0.683.8±0.793.7 ± 1.098.4±0.799.8± 0.1100.0±0.0100.0 ± 0.0
4039.8± 2.448.8 ±1.772.1 ±0.887.5 ±1.796.6±0.898.9±0.399.9 ± 0.1100.0±0.0100.0±0.0
5042.1 ± 1.353.6± 2.375.1 ± 1.992.7 ± 1.097.1 ± 0.999.6± 0.2100.0±0.0100.0±0.0100.0±0.0
7541.9 ± 1.155.9± 2.480.8±1.295.0 ± 1.199.5± 0.299.9 ±0.1100.0 ±0.0100.0 ±0.0100.0 ±0.0
10039.6± 2.059.2 ± 1.884.2 ± 3.196.9 ± 0.799.7 ± 0.1100.0± 0.0100.0 ±0.0100.0± 0.0100.0±0.0
", + "bbox": [ + 173, + 122, + 826, + 828 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "B.2 FULL RESULT OF SYNTHETIC GRAPH EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 854, + 581, + 869 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "In Table 8, we report all results of synthetic graph experiments. We experiment on a total of 144 synthetic graphs controlling 9 homophily (0.1, 0.2, ..., 0.9) and 16 average degree (1, 1.5, 2.5, 3.5, 5, 7.5, 10, 12.5, 15, 20, 25, 32.5, 40, 50, 75, 100). ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/762ab1e4ccc2a8a192f78d09ad595597707a91b0e7af9118f15205550b082f32.jpg", + "image_caption": [ + "Figure 8: Distribution of KL divergence between normalized attention and label-agreement on all nodes and layers for Cora, CiteSeer, PubMed, and PPI (Left: two-layer GAT, Right: four-layer GAT). " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 820, + 474 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/cadacca41726db9ff9f2055da3e0c4697e4c6f1bf589d44cc440f5ca54822b74.jpg", + "table_caption": [ + "Table 9: Mean wall-clock time (seconds) of three runs of the training process on real-world datasets. " + ], + "table_footnote": [], + "table_body": "
ModelCoraCiteSeerPubMed
GAT11.3 ± 2.720.4 ± 6.721.1 ± 2.0
GCN+GAM709.3 ± 235.91099.3 ± 812.56923.3 ± 7042.0
SuperGATMX30.8 ± 0.519.3 ± 1.1151.4 ± 11.4
SuperGATMx+ MPNS20.8 ± 0.215.2 ± 0.184.6 ± 0.9
", + "bbox": [ + 230, + 569, + 766, + 643 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "B.3 LABEL-AGREEMENT STUDY FOR OTHER DATASETS AND DEEPER MODELS ", + "bbox": [ + 174, + 670, + 732, + 685 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In Figure 8, we draw box plots of KL divergence between attention distribution and label agreement distribution for all nodes and layers of two-layer GATs and four-layer GATs. As shown in the paper, we can see that DP attention does not capture label-agreement rather than GO attention. Also, the degree of this phenomenon becomes stronger as the layer goes down. ", + "bbox": [ + 174, + 696, + 825, + 753 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "B.4 WALL-CLOCK TIME RESULT ", + "text_level": 1, + "bbox": [ + 176, + 771, + 415, + 786 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In Table 9, we report the mean wall-clock time (over three runs) of the training of GAT, GAM, and SuperGATMX. In SuperGAT, we find that negative sampling of edges is the bottleneck of training. So, we additionally implement Super $\\mathrm { { G A T _ { M X } + } }$ MPNS, which employs multi-processing when sampling negative edges. There are three observations in this experiment. $\\mathrm { G C N + G A M }$ is highly time-intensive in the training stage $( \\times 5 3 . 9 - \\times 3 2 8 . 1 $ versus GAT) for all datasets. Compared to GAT, our model needs $\\times 2 . 7$ more training time for Cora and $\\times 7 . 2$ for PubMed, and we reduce the time by applying multi-processing to negative sampling $( \\times 1 . 8$ for Cora and $\\times 4 . 0$ for PubMed). For CiteSeer, we can see that SuperGATMX ends faster than GAT because of faster convergence and fewer epochs. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/6ec406148c025a72f669e3a67c42fc1338ea4afbb04e4ed791e0359c3a63475d.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 173, + 102, + 821, + 181 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/56755856dbd1ff30185098ee8776c616cb831305772ef54070eb5d474e61a6b8.jpg", + "image_caption": [ + "Figure 9: Test performance on node classification against the mixing coefficient $\\lambda _ { E }$ for SuperGATMX (Cora, CiteSeer, PubMed) and SuperGATSD (PPI). " + ], + "image_footnote": [], + "bbox": [ + 173, + 223, + 823, + 301 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/5d9a2c6a1a46bac033c6f3d14b2694ecc02b0f618b9abfcf3da86eb6929ce4de.jpg", + "image_caption": [ + "Figure 10: Test performance on node classification against the negative sampling ratio $p _ { n }$ for SuperGATMX (Cora, CiteSeer, PubMed) and SuperGATSD (PPI). ", + "Figure 11: Test performance on node classification against the edge sampling ratio $p _ { e }$ for SuperGATMX (Cora, CiteSeer, PubMed) and SuperGATSD (PPI). " + ], + "image_footnote": [], + "bbox": [ + 173, + 342, + 823, + 421 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "B.5 SENSITIVITY ANALYSIS OF HYPER-PARAMETERS ", + "text_level": 1, + "bbox": [ + 174, + 488, + 558, + 502 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We analyze sensitivity of mixing coefficient of losses $\\lambda _ { E }$ , negative sampling ratio $p _ { n }$ , and edge sampling ratio $p _ { e }$ . We plot mean node classification performance (over 5 runs) against each hyperparameter in Figure 9, 10, and 11 respectively. We use the best model for each dataset: SuperGATMX for citation networks and SuperGATSD for PPI. ", + "bbox": [ + 174, + 513, + 825, + 570 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "For $\\lambda _ { E }$ , there is a specific range that maximizes test performance in all datasets. Performance on PPI is the largest when $\\lambda _ { E }$ is $\\bar { 1 } 0 ^ { - 3 }$ , but the difference is relatively small comparing to others. We observe that there is an optimal level of the edge supervision for each dataset, and using too large $\\lambda _ { E }$ degrades node classification performance. ", + "bbox": [ + 174, + 577, + 825, + 632 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "For $p _ { n }$ , using too many negative samples has been shown to decrease performance. The optimal number of negative samples is different for each dataset, and all are less than the number of positive samples $( p _ { n } < 1 . 0 )$ . Note that as $p _ { n }$ increases, the required GPU memory also increases. When $p _ { n } = 5 . 0$ , the model and data for PPI could not be accommodated by one single GPU (GeForce GTX 1080Ti). ", + "bbox": [ + 173, + 640, + 825, + 709 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "When $p _ { e }$ changes, the performance also changes, but the pattern is different by datasets. For Cora and PubMed, the performance against $p _ { e }$ shows the convex curve. Performance for CiteSeer generally decreases as $p _ { e }$ increases, but there are intervals the performance change of which is nearly zero. In the case of PPI, there are no noticeable changes against $p _ { e }$ . 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Although conventional attention is trained without direct supervision,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "if we have prior knowledge about what to attend, we can supervise attention using them (Knyazev", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "et al., 2019; Yu et al., 2017). Specifically, we exploit a self-supervised task, using the attention value", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 377, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 377, + 661 + ], + "score": 1.0, + "content": "as input to predict the likelihood that an edge exists between nodes.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "To encode edges in graph attention, we first analyze what graph attention learns and how it relates", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "to the presence of edges. 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However, what", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 237, + 470, + 250 + ], + "spans": [ + { + "bbox": [ + 141, + 237, + 470, + 250 + ], + "score": 1.0, + "content": "graph attention learns is not understood well, particularly when graphs are noisy.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 248, + 469, + 261 + ], + "spans": [ + { + "bbox": [ + 141, + 248, + 469, + 261 + ], + "score": 1.0, + "content": "In this paper, we propose a self-supervised graph attention network (SuperGAT),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 259, + 470, + 271 + ], + "spans": [ + { + "bbox": [ + 141, + 259, + 470, + 271 + ], + "score": 1.0, + "content": "an improved graph attention model for noisy graphs. Specifically, we exploit two", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 271, + 469, + 281 + ], + "spans": [ + { + "bbox": [ + 142, + 271, + 469, + 281 + ], + "score": 1.0, + "content": "attention forms compatible with a self-supervised task to predict edges, whose", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 281, + 470, + 293 + ], + "spans": [ + { + "bbox": [ + 141, + 281, + 470, + 293 + ], + "score": 1.0, + "content": "presence and absence contain the inherent information about the importance of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 293, + 470, + 304 + ], + "spans": [ + { + "bbox": [ + 142, + 293, + 470, + 304 + ], + "score": 1.0, + "content": "the relationships between nodes. By encoding edges, SuperGAT learns more", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 303, + 470, + 315 + ], + "spans": [ + { + "bbox": [ + 141, + 303, + 470, + 315 + ], + "score": 1.0, + "content": "expressive attention in distinguishing mislinked neighbors. We find two graph", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 314, + 469, + 326 + ], + "spans": [ + { + "bbox": [ + 141, + 314, + 469, + 326 + ], + "score": 1.0, + "content": "characteristics influence the effectiveness of attention forms and self-supervision:", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 325, + 470, + 337 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 470, + 337 + ], + "score": 1.0, + "content": "homophily and average degree. Thus, our recipe provides guidance on which at-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 336, + 469, + 348 + ], + "spans": [ + { + "bbox": [ + 142, + 336, + 469, + 348 + ], + "score": 1.0, + "content": "tention design to use when those two graph characteristics are known. Our experi-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 347, + 469, + 358 + ], + "spans": [ + { + "bbox": [ + 141, + 347, + 469, + 358 + ], + "score": 1.0, + "content": "ment on 17 real-world datasets demonstrates that our recipe generalizes across 15", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 357, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 469, + 370 + ], + "score": 1.0, + "content": "datasets of them, and our models designed by recipe show improved performance", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 369, + 204, + 380 + ], + "spans": [ + { + "bbox": [ + 142, + 369, + 204, + 380 + ], + "score": 1.0, + "content": "over baselines.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 214, + 470, + 380 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 399, + 206, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 208, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 208, + 415 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "Graphs are widely used in various domains, such as social networks, biology, and chemistry. Since", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "their patterns are complex and irregular, learning to represent graphs is challenging (Bruna et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 444, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 460 + ], + "score": 1.0, + "content": "2014; Henaff et al., 2015; Defferrard et al., 2016; Duvenaud et al., 2015; Atwood & Towsley,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "2016). Recently, graph neural networks (GNNs) have shown a significant performance improve-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "ment by generating features of the center node by aggregating those of its neighbors (Zhou et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "2018; Wu et al., 2020). However, real-world graphs are often noisy with connections between un-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 490, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 501 + ], + "score": 1.0, + "content": "related nodes, and this causes GNNs to learn suboptimal representations. Graph attention networks", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "(GATs) (Velickovi ˇ c et al., 2018) adopt self-attention to alleviate this issue. Similar to attention in ´", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "sequential data (Luong et al., 2015; Bahdanau et al., 2015; Vaswani et al., 2017), graph attention", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "captures the relational importance of a graph, in other words, the degree of importance of each of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "the neighbors to represent the center node. GATs have shown performance improvements in node", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "classification, but they are inconsistent in the degree of improvement across datasets, and there is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 556, + 343, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 343, + 568 + ], + "score": 1.0, + "content": "little understanding of what graph attention actually learns.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 423, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "Hence, there is still room for graph attention to improve, and we start by assessing and learning the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 504, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 504, + 595 + ], + "score": 1.0, + "content": "relational importance for each graph via self-supervised attention. We leverage edges that explicitly", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 456, + 606 + ], + "score": 1.0, + "content": "encode information about the importance of relations provided by a graph. If node", + "type": "text" + }, + { + "bbox": [ + 456, + 595, + 461, + 604 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 594, + 481, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 482, + 595, + 488, + 606 + ], + "score": 0.79, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 380, + 618 + ], + "score": 1.0, + "content": "linked, they are more relevant to each other than others, and if node", + "type": "text" + }, + { + "bbox": [ + 380, + 606, + 385, + 615 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 604, + 403, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 403, + 606, + 409, + 616 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "are not linked, they are", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "not important to each other. Although conventional attention is trained without direct supervision,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "if we have prior knowledge about what to attend, we can supervise attention using them (Knyazev", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "et al., 2019; Yu et al., 2017). Specifically, we exploit a self-supervised task, using the attention value", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 377, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 377, + 661 + ], + "score": 1.0, + "content": "as input to predict the likelihood that an edge exists between nodes.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 571, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "To encode edges in graph attention, we first analyze what graph attention learns and how it relates", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "to the presence of edges. In this analysis, we focus on two commonly used attention mechanisms,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "GAT’s original single-layer neural network (GO) and dot-product (DP), as building blocks of our", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "proposed model, self-supervised graph attention network (SuperGAT). We observe that DP attention", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "shows better performance than GO attention in the task to predict link with attention value. On the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "other hand, GO attention outperforms DP attention in capturing label-agreement between a target", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "node and its neighbors. Based on our analysis, we propose two variants of SuperGAT, scaled dot-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 447, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 447, + 106 + ], + "score": 1.0, + "content": "product (SD) and mixed GO and DP (MX), to emphasize the strength of GO and DP.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 666, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 502, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "node and its neighbors. Based on our analysis, we propose two variants of SuperGAT, scaled dot-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 447, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 447, + 106 + ], + "score": 1.0, + "content": "product (SD) and mixed GO and DP (MX), to emphasize the strength of GO and DP.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 504, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "Then, which graph attention models the relational importance best and produces the best node repre-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "score": 1.0, + "content": "sentations? We find that it depends on the average degree and homophily of the graph. We generate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "synthetic graph datasets with various degrees and homophily, and analyze how the choice of atten-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "tion affects node classification performance. Based on this result, we propose a recipe to design", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "graph attention with edge self-supervision that works most effectively for given graph characteris-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "tics. We conduct experiments on a total of 17 real-world datasets and demonstrate that our recipe", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "can be generalized across them. In addition, we show that models developed by our method improve", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 221, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 221, + 199 + ], + "score": 1.0, + "content": "performance over baselines.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 218 + ], + "score": 1.0, + "content": "We present the following contributions. First, we present models with self-supervised attention using", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "edge information. Second, we analyze the classic attention forms GO and DP using label-agreement", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "score": 1.0, + "content": "and link prediction tasks, and this analysis reveals that GO is better at label agreement and DP at", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "link prediction. Third, we propose recipes to design graph attention concerning homophily and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "average degree and confirm its validity through experiments on real-world datasets. We make our", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 493, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 493, + 271 + ], + "score": 1.0, + "content": "code available for future research (https://github.com/dongkwan-kim/SuperGAT).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 287, + 210, + 300 + ], + "lines": [ + { + "bbox": [ + 104, + 286, + 213, + 303 + ], + "spans": [ + { + "bbox": [ + 104, + 286, + 213, + 303 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "Deep neural networks are actively studied in modeling graphs, for example the graph convolutional", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "networks (Kipf & Welling, 2017) which approximate spectral graph convolution (Bruna et al., 2014;", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "score": 1.0, + "content": "Defferrard et al., 2016). A representative work in a non-spectral way is the graph attention net-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "works (GATs) (Velickovi ˇ c et al., 2018) which model relations in graphs using self-attention mech- ´", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 371 + ], + "score": 1.0, + "content": "anism (Vaswani et al., 2017). Similar to attention in sequence data (Bahdanau et al., 2015; Luong", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "et al., 2015; Vaswani et al., 2017), variants of attention in graph neural networks (Thekumpara-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "mpil et al., 2018; Zhang et al., 2018; Wang et al., 2019a; Gao & Ji, 2019; Zhang et al., 2020; Hou", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "et al., 2020) are trained without direct supervision. Our work is motivated by studies that improve", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "attention’s expressive power by giving direct supervision (Knyazev et al., 2019; Yu et al., 2017).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "Specifically, we employ a self-supervised task to predict edge presence from attention value. This is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 423, + 493, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 493, + 437 + ], + "score": 1.0, + "content": "in line with two branches of recent GNN research: self-supervision and graph structure learning.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "Recent studies about self-supervised learning for GNNs propose tasks leveraging the inherent in-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "formation in the graph structure: clustering, partitioning, context prediction after node masking,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "and completion after attribute masking (Hu et al., 2020b; Hui et al., 2020; Sun et al., 2020; You", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "et al., 2020). To the best of our knowledge, ours is the first study to analyze self-supervised learn-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "ing of graph attention with edge information. Our self-supervised task is similar to link predic-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "tion (Liben-Nowell & Kleinberg, 2007), which is a well-studied problem and recently tackled by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "neural networks (Zhang & Chen, 2017; 2018). Our DP attention to predict links is motivated by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "score": 1.0, + "content": "graph autoencoder (GAE) (Kipf & Welling, 2016) and its extensions (Pan et al., 2018; Park et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 454, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 454, + 542 + ], + "score": 1.0, + "content": "2019) reconstructing edges by applying a dot-product decoder to node representations.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "score": 1.0, + "content": "Graph structure learning is an approach to learn the underlying graph structure while jointly learning", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "downstream tasks (Jiang et al., 2019; Franceschi et al., 2019; Klicpera et al., 2019; Stretcu et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 565, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 581 + ], + "score": 1.0, + "content": "2019; Zheng et al., 2020). Since real-world graphs often have noisy edges, encoding structure", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 578, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 504, + 591 + ], + "score": 1.0, + "content": "information contributes to learn better representation. However, recent models with graph structure", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "learning suffer from high memory and computational complexity. Some studies target all spaces", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 264, + 613 + ], + "score": 1.0, + "content": "where edges can exist, so they require", + "type": "text" + }, + { + "bbox": [ + 265, + 600, + 299, + 612 + ], + "score": 0.93, + "content": "O ( | V | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "space and computational complexity (Jiang et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "2019; Franceschi et al., 2019). Others using iterative training (or co-training) between the GNNs", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "and the structure learning model are time-intensive in training (Franceschi et al., 2019; Stretcu et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "2019). We moderate this problem using graph attention, which consists of parallelizable operations,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "and our model is built on it without additional parameters. Our model learns attention values that", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 654, + 486, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 486, + 668 + ], + "score": 1.0, + "content": "are predictive of edges, and this can be seen as a new paradigm of learning the graph structure.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42 + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 164, + 696 + ], + "lines": [ + { + "bbox": [ + 104, + 682, + 167, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 167, + 699 + ], + "score": 1.0, + "content": "3 MODEL", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "In this section, we review the original GAT (Velickovi ˇ c et al., 2018) and then describe our self-´", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 259, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 259, + 732 + ], + "score": 1.0, + "content": "supervised GAT (SuperGAT) models.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + } + ], + "page_idx": 1, + "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, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 502, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 504, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 504, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "Then, which graph attention models the relational importance best and produces the best node repre-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "score": 1.0, + "content": "sentations? We find that it depends on the average degree and homophily of the graph. We generate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "synthetic graph datasets with various degrees and homophily, and analyze how the choice of atten-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "tion affects node classification performance. Based on this result, we propose a recipe to design", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "graph attention with edge self-supervision that works most effectively for given graph characteris-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "tics. We conduct experiments on a total of 17 real-world datasets and demonstrate that our recipe", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "can be generalized across them. In addition, we show that models developed by our method improve", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 221, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 221, + 199 + ], + "score": 1.0, + "content": "performance over baselines.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 110, + 506, + 199 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 218 + ], + "score": 1.0, + "content": "We present the following contributions. First, we present models with self-supervised attention using", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "edge information. Second, we analyze the classic attention forms GO and DP using label-agreement", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "score": 1.0, + "content": "and link prediction tasks, and this analysis reveals that GO is better at label agreement and DP at", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "link prediction. Third, we propose recipes to design graph attention concerning homophily and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "average degree and confirm its validity through experiments on real-world datasets. We make our", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 493, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 493, + 271 + ], + "score": 1.0, + "content": "code available for future research (https://github.com/dongkwan-kim/SuperGAT).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 203, + 506, + 271 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 287, + 210, + 300 + ], + "lines": [ + { + "bbox": [ + 104, + 286, + 213, + 303 + ], + "spans": [ + { + "bbox": [ + 104, + 286, + 213, + 303 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "Deep neural networks are actively studied in modeling graphs, for example the graph convolutional", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "networks (Kipf & Welling, 2017) which approximate spectral graph convolution (Bruna et al., 2014;", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "score": 1.0, + "content": "Defferrard et al., 2016). A representative work in a non-spectral way is the graph attention net-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "works (GATs) (Velickovi ˇ c et al., 2018) which model relations in graphs using self-attention mech- ´", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 371 + ], + "score": 1.0, + "content": "anism (Vaswani et al., 2017). Similar to attention in sequence data (Bahdanau et al., 2015; Luong", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "et al., 2015; Vaswani et al., 2017), variants of attention in graph neural networks (Thekumpara-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "mpil et al., 2018; Zhang et al., 2018; Wang et al., 2019a; Gao & Ji, 2019; Zhang et al., 2020; Hou", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "et al., 2020) are trained without direct supervision. Our work is motivated by studies that improve", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "attention’s expressive power by giving direct supervision (Knyazev et al., 2019; Yu et al., 2017).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "Specifically, we employ a self-supervised task to predict edge presence from attention value. This is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 423, + 493, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 493, + 437 + ], + "score": 1.0, + "content": "in line with two branches of recent GNN research: self-supervision and graph structure learning.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 313, + 506, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "Recent studies about self-supervised learning for GNNs propose tasks leveraging the inherent in-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "formation in the graph structure: clustering, partitioning, context prediction after node masking,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "and completion after attribute masking (Hu et al., 2020b; Hui et al., 2020; Sun et al., 2020; You", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "et al., 2020). To the best of our knowledge, ours is the first study to analyze self-supervised learn-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "ing of graph attention with edge information. Our self-supervised task is similar to link predic-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "tion (Liben-Nowell & Kleinberg, 2007), which is a well-studied problem and recently tackled by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "neural networks (Zhang & Chen, 2017; 2018). Our DP attention to predict links is motivated by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "score": 1.0, + "content": "graph autoencoder (GAE) (Kipf & Welling, 2016) and its extensions (Pan et al., 2018; Park et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 454, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 454, + 542 + ], + "score": 1.0, + "content": "2019) reconstructing edges by applying a dot-product decoder to node representations.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 441, + 506, + 542 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "score": 1.0, + "content": "Graph structure learning is an approach to learn the underlying graph structure while jointly learning", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "downstream tasks (Jiang et al., 2019; Franceschi et al., 2019; Klicpera et al., 2019; Stretcu et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 565, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 581 + ], + "score": 1.0, + "content": "2019; Zheng et al., 2020). Since real-world graphs often have noisy edges, encoding structure", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 578, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 504, + 591 + ], + "score": 1.0, + "content": "information contributes to learn better representation. However, recent models with graph structure", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "learning suffer from high memory and computational complexity. Some studies target all spaces", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 264, + 613 + ], + "score": 1.0, + "content": "where edges can exist, so they require", + "type": "text" + }, + { + "bbox": [ + 265, + 600, + 299, + 612 + ], + "score": 0.93, + "content": "O ( | V | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "space and computational complexity (Jiang et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "2019; Franceschi et al., 2019). Others using iterative training (or co-training) between the GNNs", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "and the structure learning model are time-intensive in training (Franceschi et al., 2019; Stretcu et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "2019). We moderate this problem using graph attention, which consists of parallelizable operations,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "and our model is built on it without additional parameters. Our model learns attention values that", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 654, + 486, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 486, + 668 + ], + "score": 1.0, + "content": "are predictive of edges, and this can be seen as a new paradigm of learning the graph structure.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 544, + 506, + 668 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 164, + 696 + ], + "lines": [ + { + "bbox": [ + 104, + 682, + 167, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 167, + 699 + ], + "score": 1.0, + "content": "3 MODEL", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "In this section, we review the original GAT (Velickovi ˇ c et al., 2018) and then describe our self-´", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 259, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 259, + 732 + ], + "score": 1.0, + "content": "supervised GAT (SuperGAT) models.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5, + "bbox_fs": [ + 106, + 709, + 504, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 81, + 483, + 205 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 81, + 483, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 81, + 483, + 205 + ], + "spans": [ + { + "bbox": [ + 110, + 81, + 483, + 205 + ], + "score": 0.963, + "type": "image", + "image_path": "477ee301413365f6f6b7c07290d71e4ea5c193f303f00ce1ad7030218957e85a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 81, + 483, + 122.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 122.33333333333334, + 483, + 163.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 163.66666666666669, + 483, + 205.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 214, + 506, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 213, + 504, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 484, + 228 + ], + "score": 1.0, + "content": "Figure 1: Overview of attention mechanism of SuperGATs: GO, DP, MX, and SD. Blue circles", + "type": "text" + }, + { + "bbox": [ + 484, + 216, + 504, + 227 + ], + "score": 0.86, + "content": "( e _ { i j } )", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 389, + 239 + ], + "score": 1.0, + "content": "represent the unnormalized attention before softmax and red diamonds", + "type": "text" + }, + { + "bbox": [ + 389, + 226, + 409, + 238 + ], + "score": 0.88, + "content": "( \\phi _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "indicate the probability", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 194, + 250 + ], + "score": 1.0, + "content": "of edge between node", + "type": "text" + }, + { + "bbox": [ + 195, + 238, + 199, + 247 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 237, + 216, + 250 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 217, + 237, + 222, + 248 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 237, + 506, + 250 + ], + "score": 1.0, + "content": ". 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Graph attention layer takes a set of features", + "type": "text" + }, + { + "bbox": [ + 309, + 290, + 435, + 305 + ], + "score": 0.91, + "content": "\\mathbb { H } ^ { l } = \\{ h _ { 1 } ^ { l } , \\dots , h _ { N } ^ { l } \\} , h _ { i } ^ { l } \\in \\mathbb { R } ^ { F ^ { l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 291, + 506, + 307 + ], + "score": 1.0, + "content": "as input and pro-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 102, + 302, + 507, + 323 + ], + "spans": [ + { + "bbox": [ + 102, + 302, + 192, + 323 + ], + "score": 1.0, + "content": "duces output features", + "type": "text" + }, + { + "bbox": [ + 193, + 304, + 300, + 318 + ], + "score": 0.93, + "content": "\\mathbb { H } ^ { l + 1 } = \\{ h _ { 1 } ^ { l + 1 } , \\dots , h _ { N } ^ { l + 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 302, + 353, + 323 + ], + "score": 1.0, + "content": ". To compute", + "type": "text" + }, + { + "bbox": [ + 353, + 305, + 374, + 318 + ], + "score": 0.91, + "content": "\\boldsymbol { h } _ { i } ^ { l + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 302, + 507, + 323 + ], + "score": 1.0, + "content": ", the model multiplies the weight", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 103, + 313, + 507, + 335 + ], + "spans": [ + { + "bbox": [ + 103, + 313, + 135, + 335 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 136, + 317, + 214, + 330 + ], + "score": 0.93, + "content": "W ^ { l + 1 } \\in \\mathbb { R } ^ { F ^ { l + 1 } \\times F ^ { l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 313, + 226, + 335 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 226, + 319, + 238, + 330 + ], + "score": 0.87, + "content": "\\mathbb { H } ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 313, + 507, + 335 + ], + "score": 1.0, + "content": ", linearly combines the features of its first-order neighbors (includ-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 103, + 327, + 501, + 351 + ], + "spans": [ + { + "bbox": [ + 103, + 327, + 147, + 351 + ], + "score": 1.0, + "content": "ing itself)", + "type": "text" + }, + { + "bbox": [ + 148, + 332, + 201, + 344 + ], + "score": 0.92, + "content": "j \\in \\mathbb { N } _ { i } \\cup \\{ i \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 327, + 302, + 351 + ], + "score": 1.0, + "content": "by attention coefficients", + "type": "text" + }, + { + "bbox": [ + 302, + 330, + 322, + 346 + ], + "score": 0.92, + "content": "\\alpha _ { i j } ^ { l + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 327, + 495, + 351 + ], + "score": 1.0, + "content": ", and finally applies a non-linear activation", + "type": "text" + }, + { + "bbox": [ + 495, + 335, + 501, + 344 + ], + "score": 0.75, + "content": "\\rho", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 102, + 341, + 503, + 370 + ], + "spans": [ + { + "bbox": [ + 102, + 341, + 136, + 370 + ], + "score": 1.0, + "content": "That is", + "type": "text" + }, + { + "bbox": [ + 137, + 345, + 293, + 364 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\pmb { h } _ { i } ^ { l + 1 } = \\rho \\left( \\sum _ { j \\in \\mathbb { N } _ { i } \\cup \\{ i \\} } \\alpha _ { i j } ^ { l + 1 } \\pmb { W } ^ { l + 1 } \\pmb { h } _ { j } ^ { l } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 341, + 367, + 370 + ], + "score": 1.0, + "content": ". We can compute", + "type": "text" + }, + { + "bbox": [ + 367, + 347, + 503, + 362 + ], + "score": 0.89, + "content": "\\alpha _ { i j } ^ { l + 1 } = \\mathrm { s o f t m a x } _ { j } ( \\mathrm { L R e L U } ( e _ { i j } ^ { l + 1 } ) )", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 102, + 359, + 509, + 384 + ], + "spans": [ + { + "bbox": [ + 102, + 359, + 169, + 384 + ], + "score": 1.0, + "content": "by normalizing", + "type": "text" + }, + { + "bbox": [ + 170, + 364, + 295, + 379 + ], + "score": 0.91, + "content": "e _ { i j } ^ { l + 1 } = a _ { e } ( W ^ { l + 1 } \\pmb { h } _ { i } ^ { l } , W ^ { l + 1 } \\pmb { h } _ { j } ^ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 359, + 363, + 384 + ], + "score": 1.0, + "content": "with softmax on", + "type": "text" + }, + { + "bbox": [ + 363, + 365, + 399, + 377 + ], + "score": 0.92, + "content": "\\mathbb { N } _ { i } \\cup \\{ i \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 359, + 509, + 384 + ], + "score": 1.0, + "content": "under leaky ReLU activa-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 103, + 375, + 434, + 396 + ], + "spans": [ + { + "bbox": [ + 103, + 375, + 373, + 396 + ], + "score": 1.0, + "content": "tion (Maas et al., 2013), where ae is a function of the form RF l+1", + "type": "text" + }, + { + "bbox": [ + 343, + 378, + 429, + 391 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { F ^ { l + 1 } } \\times \\mathbb { R } ^ { F ^ { l + 1 } } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 379, + 434, + 393 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "Graph Attention Forms Among two widely used attention mechanisms, the original GAT (GO)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 102, + 413, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 102, + 413, + 506, + 432 + ], + "score": 1.0, + "content": "computes the coefficients by single-layer feed-forward network parameterized by al+1 ∈ R2Fl+1.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "The other is the dot-product (DP) attention, (Luong et al., 2015; 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For", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 118, + 597 + ], + "score": 0.84, + "content": "a _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 584, + 260, + 599 + ], + "score": 1.0, + "content": ", the form of which is the same as", + "type": "text" + }, + { + "bbox": [ + 260, + 587, + 271, + 596 + ], + "score": 0.85, + "content": "a _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 584, + 283, + 599 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 284, + 586, + 315, + 596 + ], + "score": 0.39, + "content": "\\mathrm { G A T _ { G O } }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 584, + 334, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 334, + 586, + 364, + 596 + ], + "score": 0.69, + "content": "\\mathrm { G A T _ { D P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 584, + 506, + 599 + ], + "score": 1.0, + "content": ", we name them SuperGATGO and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "SuperGATDP respectively. 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For", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 118, + 597 + ], + "score": 0.84, + "content": "a _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 584, + 260, + 599 + ], + "score": 1.0, + "content": ", the form of which is the same as", + "type": "text" + }, + { + "bbox": [ + 260, + 587, + 271, + 596 + ], + "score": 0.85, + "content": "a _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 584, + 283, + 599 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 284, + 586, + 315, + 596 + ], + "score": 0.39, + "content": "\\mathrm { G A T _ { G O } }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 584, + 334, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 334, + 586, + 364, + 596 + ], + "score": 0.69, + "content": "\\mathrm { G A T _ { D P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 584, + 506, + 599 + ], + "score": 1.0, + "content": ", we name them SuperGATGO and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "SuperGATDP respectively. 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This prevents some large values to dominate the entire attention", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "after softmax. SuperGATMX multiplies GO and DP attention with sigmoid. The motivation of this", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "form comes from the gating mechanism of Gated Recurrent Units (Cho et al., 2014). Since DP", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "attention with the sigmoid represents the probability of an edge, it can softly drop neighbors that are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 424, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 424, + 734 + ], + "score": 1.0, + "content": "not likely linked while implicitly assigning importance to the remaining nodes.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 666, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 249, + 96 + ], + "score": 1.0, + "content": "Training samples are a set of edges", + "type": "text" + }, + { + "bbox": [ + 250, + 83, + 259, + 92 + ], + "score": 0.81, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 82, + 370, + 96 + ], + "score": 1.0, + "content": "and the complementary set", + "type": "text" + }, + { + "bbox": [ + 370, + 82, + 451, + 95 + ], + "score": 0.93, + "content": "E ^ { c } = ( V \\times V ) \\setminus E", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 82, + 506, + 96 + ], + "score": 1.0, + "content": ". However, if", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 440, + 107 + ], + "score": 1.0, + "content": "the number of nodes is large, it is not efficient to use all possible negative cases in", + "type": "text" + }, + { + "bbox": [ + 441, + 94, + 453, + 104 + ], + "score": 0.86, + "content": "E ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 93, + 506, + 107 + ], + "score": 1.0, + "content": ". So, we use", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "negative sampling as in training word or graph embeddings (Mikolov et al., 2013; Tang et al., 2015;", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 343, + 128 + ], + "score": 1.0, + "content": "Grover & Leskovec, 2016), arbitrarily choosing a total of", + "type": "text" + }, + { + "bbox": [ + 343, + 115, + 376, + 127 + ], + "score": 0.92, + "content": "p _ { n } \\cdot | E |", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 114, + 450, + 128 + ], + "score": 1.0, + "content": "negative samples", + "type": "text" + }, + { + "bbox": [ + 450, + 115, + 466, + 126 + ], + "score": 0.89, + "content": "E ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 114, + 490, + 128 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 491, + 116, + 504, + 125 + ], + "score": 0.86, + "content": "E ^ { c }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 243, + 140 + ], + "score": 1.0, + "content": "where the negative sampling ratio", + "type": "text" + }, + { + "bbox": [ + 244, + 126, + 281, + 138 + ], + "score": 0.92, + "content": "p _ { n } \\in \\mathbb { R } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "is a hyperparameter. SuperGAT is capable of modeling", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 432, + 151 + ], + "score": 1.0, + "content": "graphs that are sparse with a sufficiently large number of negative samples (i.e.,", + "type": "text" + }, + { + "bbox": [ + 432, + 137, + 500, + 149 + ], + "score": 0.91, + "content": "\\mathsf { | } V \\times V | \\gg | E | )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 137, + 506, + 151 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 479, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 479, + 161 + ], + "score": 1.0, + "content": "but this is generally not a problem because most real-world graphs are sparse (Chung, 2010).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 433, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 433, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 288, + 180 + ], + "score": 1.0, + "content": "We define the optimization objective of layer", + "type": "text" + }, + { + "bbox": [ + 289, + 166, + 293, + 175 + ], + "score": 0.63, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 162, + 414, + 180 + ], + "score": 1.0, + "content": "as a binary cross-entropy loss", + "type": "text" + }, + { + "bbox": [ + 414, + 164, + 429, + 177 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { E } ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 162, + 433, + 180 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 184, + 455, + 202 + ], + "lines": [ + { + "bbox": [ + 155, + 184, + 455, + 202 + ], + "spans": [ + { + "bbox": [ + 155, + 184, + 455, + 202 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathcal { L } _ { E } ^ { l } = - \\frac { 1 } { \\vert E \\cup E ^ { - } \\vert } \\sum _ { ( j , i ) \\in E \\cup E ^ { - } } \\mathbf { 1 } _ { ( j , i ) = 1 } \\cdot \\log \\phi _ { i j } ^ { l } + \\mathbf { 1 } _ { ( j , i ) = 0 } \\cdot \\log \\left( 1 - \\phi _ { i j } ^ { l } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "f95d77b197178a44c18529809f1b5757ef98cf00db6518580a3452e935fd53b9.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 155, + 184, + 455, + 202 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 209, + 505, + 232 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 504, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 325, + 223 + ], + "score": 1.0, + "content": "where 1· is an indicator function. 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Note that SuperGAT has equivalent time and space", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 247, + 327 + ], + "score": 1.0, + "content": "complexity as GAT. To compute", + "type": "text" + }, + { + "bbox": [ + 247, + 313, + 262, + 326 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { E } ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "for one head, we need additional operations in terms of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 324, + 336, + 339 + ], + "spans": [ + { + "bbox": [ + 107, + 325, + 180, + 338 + ], + "score": 0.92, + "content": "O ( F ^ { l } \\cdot | E \\cup E ^ { - } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 324, + 336, + 339 + ], + "score": 1.0, + "content": ", and we do not need extra parameters.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 355, + 200, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 201, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 201, + 369 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "Our primary research objective is to design graph attentions that are effective with edge self-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "supervision. To do this, we pose four specific research questions. We first analyze what basic graph", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 404, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 504, + 415 + ], + "score": 1.0, + "content": "attentions (GO and DP) learn (RQ1 and 2) and how that can be improved with edge self-supervision", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 501, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 501, + 428 + ], + "score": 1.0, + "content": "(RQ3 and 4). We describe each research question and the corresponding experiment design below.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "RQ1. Does graph attention learn label-agreement? First, we evaluate what the graph attentions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 117, + 464 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 451, + 148, + 462 + ], + "score": 0.31, + "content": "\\mathrm { G A T _ { G O } }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 450, + 165, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 165, + 451, + 195, + 462 + ], + "score": 0.4, + "content": "\\mathrm { G A T _ { D P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "learn without edge supervision. For this, we present ground-truth of relational", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "importance and a metric to assess graph attention with ground-truth. Wang et al. (2019a) showed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "that node representations in the connected component converge to the same value in deep GATs.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "If there is an edge between nodes with different labels, then it will be hard to distinguish the two", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "corresponding labels with GAT of sufficiently many layers; that is, ideal attention should give all", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "weights to label-agreed neighbors. In that sense, we choose label-agreement between nodes as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 517, + 220, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 220, + 529 + ], + "score": 1.0, + "content": "ground-truth of importance.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "We compare label-agreement and graph attention based on Kullback–Leibler divergence of the nor-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 543, + 504, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 178, + 558 + ], + "score": 1.0, + "content": "malized attention", + "type": "text" + }, + { + "bbox": [ + 178, + 545, + 284, + 556 + ], + "score": 0.9, + "content": "\\pmb { \\alpha } _ { k } = [ \\alpha _ { k k } , \\alpha _ { k 1 } , . . . , \\alpha _ { k J } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 543, + 497, + 558 + ], + "score": 1.0, + "content": "with label agreement distribution for the center node", + "type": "text" + }, + { + "bbox": [ + 497, + 545, + 504, + 554 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 495, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 195, + 568 + ], + "score": 1.0, + "content": "and its neighbors 1 to", + "type": "text" + }, + { + "bbox": [ + 195, + 556, + 203, + 565 + ], + "score": 0.79, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 556, + 341, + 568 + ], + "score": 1.0, + "content": ". The label agreement distribution,", + "type": "text" + }, + { + "bbox": [ + 341, + 555, + 438, + 568 + ], + "score": 0.93, + "content": "\\ell _ { k } = [ \\ell _ { k k } , \\ell _ { k 1 } , \\dots , \\ell _ { k J } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 556, + 495, + 568 + ], + "score": 1.0, + "content": "is defined by,", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 574, + 468, + 590 + ], + "lines": [ + { + "bbox": [ + 144, + 574, + 468, + 590 + ], + "spans": [ + { + "bbox": [ + 144, + 574, + 468, + 590 + ], + "score": 0.51, + "content": "\\begin{array} { r } { \\ell _ { k j } = \\hat { \\ell } _ { k j } / \\sum _ { s } \\hat { \\ell } _ { k s } , \\quad \\hat { \\ell } _ { k j } = 1 \\ ( \\mathrm { i f } \\ k \\ \\mathrm { a n d } \\ j \\ \\mathrm { h a v e \\ t h e \\ s a m e \\ l a b e l ) \\ { o r \\ } 0 \\ ( o t h e r w i s e ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "0cfdb7c2638e9bd22eac6e1d84d990c6e5c48688aa247584c3a744d062f1859e.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 144, + 574, + 468, + 590 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 596, + 503, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "We employ KL divergence in Eq. 8, whose value becomes small when attention captures well the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 311, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 311, + 621 + ], + "score": 1.0, + "content": "label-agreement between a node and its neighbors.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 626, + 401, + 642 + ], + "lines": [ + { + "bbox": [ + 210, + 626, + 401, + 642 + ], + "spans": [ + { + "bbox": [ + 210, + 626, + 401, + 642 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathrm { K L D } ( \\alpha _ { k } , \\ell _ { k } ) = \\sum _ { j \\in \\mathbb { N } _ { k } \\cup \\{ k \\} } \\alpha _ { k j } \\log ( \\alpha _ { k j } / \\ell _ { k j } ) } \\end{array}", + "type": "interline_equation", + "image_path": "e5ec21531b26537c5cde7a98dfe304c1d14df930e8da819b7d309ea9e46a949f.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 210, + 626, + 401, + 642 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "RQ2. Is graph attention predictive of edge presence? To evaluate how well edge information is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "encoded in SuperGAT, we conduct link prediction experiments with SuperGATGO and SuperGATDP", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 131, + 689 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 131, + 677, + 145, + 689 + ], + "score": 0.89, + "content": "\\phi _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "of the last layer as a predictor. 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We train with an incomplete set of edges, and test with the missing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "edges and the same number of negative samples. 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However, if", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 440, + 107 + ], + "score": 1.0, + "content": "the number of nodes is large, it is not efficient to use all possible negative cases in", + "type": "text" + }, + { + "bbox": [ + 441, + 94, + 453, + 104 + ], + "score": 0.86, + "content": "E ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 93, + 506, + 107 + ], + "score": 1.0, + "content": ". So, we use", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "negative sampling as in training word or graph embeddings (Mikolov et al., 2013; Tang et al., 2015;", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 343, + 128 + ], + "score": 1.0, + "content": "Grover & Leskovec, 2016), arbitrarily choosing a total of", + "type": "text" + }, + { + "bbox": [ + 343, + 115, + 376, + 127 + ], + "score": 0.92, + "content": "p _ { n } \\cdot | E |", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 114, + 450, + 128 + ], + "score": 1.0, + "content": "negative samples", + "type": "text" + }, + { + "bbox": [ + 450, + 115, + 466, + 126 + ], + "score": 0.89, + "content": "E ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 114, + 490, + 128 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 491, + 116, + 504, + 125 + ], + "score": 0.86, + "content": "E ^ { c }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 243, + 140 + ], + "score": 1.0, + "content": "where the negative sampling ratio", + "type": "text" + }, + { + "bbox": [ + 244, + 126, + 281, + 138 + ], + "score": 0.92, + "content": "p _ { n } \\in \\mathbb { R } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "is a hyperparameter. SuperGAT is capable of modeling", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 432, + 151 + ], + "score": 1.0, + "content": "graphs that are sparse with a sufficiently large number of negative samples (i.e.,", + "type": "text" + }, + { + "bbox": [ + 432, + 137, + 500, + 149 + ], + "score": 0.91, + "content": "\\mathsf { | } V \\times V | \\gg | E | )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 137, + 506, + 151 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 479, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 479, + 161 + ], + "score": 1.0, + "content": "but this is generally not a problem because most real-world graphs are sparse (Chung, 2010).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 82, + 506, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 433, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 433, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 288, + 180 + ], + "score": 1.0, + "content": "We define the optimization objective of layer", + "type": "text" + }, + { + "bbox": [ + 289, + 166, + 293, + 175 + ], + "score": 0.63, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 162, + 414, + 180 + ], + "score": 1.0, + "content": "as a binary cross-entropy loss", + "type": "text" + }, + { + "bbox": [ + 414, + 164, + 429, + 177 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { E } ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 162, + 433, + 180 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 162, + 433, + 180 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 184, + 455, + 202 + ], + "lines": [ + { + "bbox": [ + 155, + 184, + 455, + 202 + ], + "spans": [ + { + "bbox": [ + 155, + 184, + 455, + 202 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathcal { L } _ { E } ^ { l } = - \\frac { 1 } { \\vert E \\cup E ^ { - } \\vert } \\sum _ { ( j , i ) \\in E \\cup E ^ { - } } \\mathbf { 1 } _ { ( j , i ) = 1 } \\cdot \\log \\phi _ { i j } ^ { l } + \\mathbf { 1 } _ { ( j , i ) = 0 } \\cdot \\log \\left( 1 - \\phi _ { i j } ^ { l } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "f95d77b197178a44c18529809f1b5757ef98cf00db6518580a3452e935fd53b9.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 155, + 184, + 455, + 202 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 209, + 505, + 232 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 504, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 325, + 223 + ], + "score": 1.0, + "content": "where 1· is an indicator function. 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Note that SuperGAT has equivalent time and space", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 247, + 327 + ], + "score": 1.0, + "content": "complexity as GAT. To compute", + "type": "text" + }, + { + "bbox": [ + 247, + 313, + 262, + 326 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { E } ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "for one head, we need additional operations in terms of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 324, + 336, + 339 + ], + "spans": [ + { + "bbox": [ + 107, + 325, + 180, + 338 + ], + "score": 0.92, + "content": "O ( F ^ { l } \\cdot | E \\cup E ^ { - } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 324, + 336, + 339 + ], + "score": 1.0, + "content": ", and we do not need extra parameters.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 290, + 506, + 339 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 355, + 200, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 201, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 201, + 369 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "Our primary research objective is to design graph attentions that are effective with edge self-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "supervision. To do this, we pose four specific research questions. We first analyze what basic graph", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 404, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 504, + 415 + ], + "score": 1.0, + "content": "attentions (GO and DP) learn (RQ1 and 2) and how that can be improved with edge self-supervision", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 501, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 501, + 428 + ], + "score": 1.0, + "content": "(RQ3 and 4). We describe each research question and the corresponding experiment design below.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 381, + 505, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "RQ1. Does graph attention learn label-agreement? First, we evaluate what the graph attentions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 117, + 464 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 451, + 148, + 462 + ], + "score": 0.31, + "content": "\\mathrm { G A T _ { G O } }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 450, + 165, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 165, + 451, + 195, + 462 + ], + "score": 0.4, + "content": "\\mathrm { G A T _ { D P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "learn without edge supervision. For this, we present ground-truth of relational", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "importance and a metric to assess graph attention with ground-truth. Wang et al. (2019a) showed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "that node representations in the connected component converge to the same value in deep GATs.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "If there is an edge between nodes with different labels, then it will be hard to distinguish the two", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "corresponding labels with GAT of sufficiently many layers; that is, ideal attention should give all", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "weights to label-agreed neighbors. In that sense, we choose label-agreement between nodes as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 517, + 220, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 220, + 529 + ], + "score": 1.0, + "content": "ground-truth of importance.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 439, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "We compare label-agreement and graph attention based on Kullback–Leibler divergence of the nor-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 543, + 504, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 178, + 558 + ], + "score": 1.0, + "content": "malized attention", + "type": "text" + }, + { + "bbox": [ + 178, + 545, + 284, + 556 + ], + "score": 0.9, + "content": "\\pmb { \\alpha } _ { k } = [ \\alpha _ { k k } , \\alpha _ { k 1 } , . . . , \\alpha _ { k J } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 543, + 497, + 558 + ], + "score": 1.0, + "content": "with label agreement distribution for the center node", + "type": "text" + }, + { + "bbox": [ + 497, + 545, + 504, + 554 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 495, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 195, + 568 + ], + "score": 1.0, + "content": "and its neighbors 1 to", + "type": "text" + }, + { + "bbox": [ + 195, + 556, + 203, + 565 + ], + "score": 0.79, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 556, + 341, + 568 + ], + "score": 1.0, + "content": ". The label agreement distribution,", + "type": "text" + }, + { + "bbox": [ + 341, + 555, + 438, + 568 + ], + "score": 0.93, + "content": "\\ell _ { k } = [ \\ell _ { k k } , \\ell _ { k 1 } , \\dots , \\ell _ { k J } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 556, + 495, + 568 + ], + "score": 1.0, + "content": "is defined by,", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 533, + 505, + 568 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 574, + 468, + 590 + ], + "lines": [ + { + "bbox": [ + 144, + 574, + 468, + 590 + ], + "spans": [ + { + "bbox": [ + 144, + 574, + 468, + 590 + ], + "score": 0.51, + "content": "\\begin{array} { r } { \\ell _ { k j } = \\hat { \\ell } _ { k j } / \\sum _ { s } \\hat { \\ell } _ { k s } , \\quad \\hat { \\ell } _ { k j } = 1 \\ ( \\mathrm { i f } \\ k \\ \\mathrm { a n d } \\ j \\ \\mathrm { h a v e \\ t h e \\ s a m e \\ l a b e l ) \\ { o r \\ } 0 \\ ( o t h e r w i s e ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "0cfdb7c2638e9bd22eac6e1d84d990c6e5c48688aa247584c3a744d062f1859e.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 144, + 574, + 468, + 590 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 596, + 503, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "We employ KL divergence in Eq. 8, whose value becomes small when attention captures well the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 311, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 311, + 621 + ], + "score": 1.0, + "content": "label-agreement between a node and its neighbors.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 596, + 505, + 621 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 626, + 401, + 642 + ], + "lines": [ + { + "bbox": [ + 210, + 626, + 401, + 642 + ], + "spans": [ + { + "bbox": [ + 210, + 626, + 401, + 642 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathrm { K L D } ( \\alpha _ { k } , \\ell _ { k } ) = \\sum _ { j \\in \\mathbb { N } _ { k } \\cup \\{ k \\} } \\alpha _ { k j } \\log ( \\alpha _ { k j } / \\ell _ { k j } ) } \\end{array}", + "type": "interline_equation", + "image_path": "e5ec21531b26537c5cde7a98dfe304c1d14df930e8da819b7d309ea9e46a949f.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 210, + 626, + 401, + 642 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "RQ2. Is graph attention predictive of edge presence? To evaluate how well edge information is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "encoded in SuperGAT, we conduct link prediction experiments with SuperGATGO and SuperGATDP", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 131, + 689 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 131, + 677, + 145, + 689 + ], + "score": 0.89, + "content": "\\phi _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "of the last layer as a predictor. We measure the performance by AUC over multiple runs.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 380, + 700 + ], + "score": 1.0, + "content": "Since link prediction performance depends on the mixing coefficient", + "type": "text" + }, + { + "bbox": [ + 381, + 688, + 394, + 699 + ], + "score": 0.89, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "in Eq. 6, we adopt multiple", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 695, + 507, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 226, + 711 + ], + "score": 0.91, + "content": "\\lambda _ { E } \\in \\{ 1 0 ^ { \\overset { . } { - } 3 } , 1 0 ^ { - 2 } , \\overset { . } { \\cdot } \\overset { . } { \\cdot } , 1 0 ^ { 3 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 695, + 507, + 714 + ], + "score": 1.0, + "content": ". We train with an incomplete set of edges, and test with the missing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "edges and the same number of negative samples. At the same time, node classification performance", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 721, + 493, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 493, + 733 + ], + "score": 1.0, + "content": "is measured with the same settings to see how learning edge presence affects node classification.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 654, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "RQ3. Which graph attention should we use for given graphs? The above two research ques-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "tions explore what different graph attention learns with or without supervision of edge presence.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "Then, which graph attention is effective among them for given graphs? We hypothesize that differ-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "ent graph attention will have different abilities to model graphs under various homophily and average", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "degree. We choose these two properties among various graph statistics because they determine the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "quality and quantity of labels in our self-supervised task. From the perspective of supervised learn-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "ing of graph attention with edge labels, the learning result depends on how noisy labels are (i.e., how", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "low the homophily is) and how many labels exist (i.e., how high the average degree is). So, we gen-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 369, + 183 + ], + "score": 1.0, + "content": "erate 144 synthetic graphs (Section 4.1) controlling 9 homophily", + "type": "text" + }, + { + "bbox": [ + 369, + 171, + 411, + 182 + ], + "score": 0.31, + "content": "( 0 . 1 - 0 . 9 )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "and 16 average degree", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 107, + 181, + 145, + 192 + ], + "score": 0.62, + "content": "( 1 - 1 0 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "and perform the node classification task in the transductive setting with GCN, GATGO,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 240, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 181, + 206 + ], + "score": 1.0, + "content": "SuperGATSD, and", + "type": "text" + }, + { + "bbox": [ + 181, + 192, + 236, + 204 + ], + "score": 0.3, + "content": "\\mathrm { S u p e r G A T _ { M X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 191, + 240, + 206 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "In RQ3, there are also practical reasons to use the average degree and homophily, out of many graph", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "properties (e.g., diameter, degree sequence, degree distribution, average clustering coefficient). First,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "the graph property can be computed efficiently even for large graphs. Second, there should be an", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "algorithm that can generate graphs by controlling the property of interest only. Third, the property", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "should be a scalar value because if the synthetic graph space is too wide, it would be impossible to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "conduct an experiment with sufficient coverage. Average degree and homophily satisfy the above", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 464, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 464, + 287 + ], + "score": 1.0, + "content": "conditions and are suitable for our experiment, unlike some of the other graph properties.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "RQ4. Does design choice based on RQ3 generalize to real-world datasets? Experiments on", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "score": 1.0, + "content": "synthetic datasets provide an understanding of graph attention models’ performance, but they are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "oversimplified versions of real-world graphs. Can design choice from synthetic datasets be gener-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "alized to real-world datasets, considering more complex structures and rich features in real-world", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "graphs? To answer this question, we conduct experiments on 17 real-world datasets with the various", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "average degree (1.8 – 35.8) and homophily (0.16 – 0.91), and compare them with synthetic graph", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 370, + 192, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 192, + 381 + ], + "score": 1.0, + "content": "experiments in RQ3.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 398, + 176, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 177, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 177, + 411 + ], + "score": 1.0, + "content": "4.1 DATASETS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "Real-world datasets We use a total of 17 real-world datasets (Cora, CiteSeer, PubMed, Cora-ML,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 432, + 504, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 504, + 444 + ], + "score": 1.0, + "content": "Cora-Full, DBLP, ogbn-arxiv, CS, Physics, Photo, Computers, Wiki-CS, Four-Univ, Chameleon,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "Crocodile, Flickr, and PPI) in diverse domains (citation, co-authorship, co-purchase, web page, and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 188, + 466 + ], + "score": 1.0, + "content": "biology) and scales", + "type": "text" + }, + { + "bbox": [ + 188, + 454, + 230, + 465 + ], + "score": 0.47, + "content": "( 2 \\mathrm { k } - 1 6 9 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "nodes). We try to use their original settings as much as possible. To", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "verify research questions 1 and 2, we choose four classic benchmarks: Cora, CiteSeer, PubMed in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "the transductive setting, and PPI in the inductive setting. See appendix A.1 for detailed description,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 487, + 373, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 373, + 500 + ], + "score": 1.0, + "content": "splits, statistics (including degree and homophily), and references.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 515, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 355, + 527 + ], + "score": 1.0, + "content": "Synthetic datasets We generate random partition graphs of", + "type": "text" + }, + { + "bbox": [ + 356, + 518, + 363, + 525 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 515, + 445, + 527 + ], + "score": 1.0, + "content": "nodes per class and", + "type": "text" + }, + { + "bbox": [ + 445, + 518, + 451, + 525 + ], + "score": 0.54, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "classes (For-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "tunato, 2010), using NetworkX library (Hagberg et al., 2008). A random partition graph is a graph", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 294, + 550 + ], + "score": 1.0, + "content": "of communities controlled by two probabilities", + "type": "text" + }, + { + "bbox": [ + 294, + 538, + 309, + 549 + ], + "score": 0.87, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 537, + 326, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 538, + 344, + 549 + ], + "score": 0.88, + "content": "p _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 537, + 506, + 550 + ], + "score": 1.0, + "content": ". If the nodes have the same class labels,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 204, + 560 + ], + "score": 1.0, + "content": "they are connected with", + "type": "text" + }, + { + "bbox": [ + 204, + 549, + 218, + 559 + ], + "score": 0.86, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 548, + 380, + 560 + ], + "score": 1.0, + "content": ", and otherwise, they are connected with", + "type": "text" + }, + { + "bbox": [ + 380, + 549, + 399, + 559 + ], + "score": 0.86, + "content": "p _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 548, + 505, + 560 + ], + "score": 1.0, + "content": ". To generate a graph with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 192, + 572 + ], + "score": 1.0, + "content": "an average degree of", + "type": "text" + }, + { + "bbox": [ + 192, + 559, + 244, + 571 + ], + "score": 0.93, + "content": "d _ { a v g } = n \\cdot \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 559, + 292, + 572 + ], + "score": 1.0, + "content": ", we choose", + "type": "text" + }, + { + "bbox": [ + 293, + 560, + 307, + 570 + ], + "score": 0.87, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 559, + 325, + 572 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 560, + 344, + 570 + ], + "score": 0.88, + "content": "p _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 559, + 358, + 572 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 359, + 559, + 459, + 571 + ], + "score": 0.91, + "content": "p _ { i n } + ( c - 1 ) \\cdot p _ { o u t } = \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 559, + 506, + 572 + ], + "score": 1.0, + "content": ". The input", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "score": 1.0, + "content": "features of nodes are sampled from overlapping multi-Gaussian distributions (Abu-El-Haija et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 580, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 163, + 595 + ], + "score": 1.0, + "content": "2019). We set", + "type": "text" + }, + { + "bbox": [ + 163, + 582, + 171, + 591 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 580, + 200, + 595 + ], + "score": 1.0, + "content": "to 500,", + "type": "text" + }, + { + "bbox": [ + 200, + 582, + 207, + 591 + ], + "score": 0.62, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 580, + 277, + 595 + ], + "score": 1.0, + "content": "to 10, and choose", + "type": "text" + }, + { + "bbox": [ + 277, + 581, + 297, + 593 + ], + "score": 0.91, + "content": "d _ { a v g }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 580, + 375, + 595 + ], + "score": 1.0, + "content": "between 1 and 100,", + "type": "text" + }, + { + "bbox": [ + 375, + 582, + 390, + 592 + ], + "score": 0.81, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 580, + 412, + 595 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 413, + 581, + 501, + 593 + ], + "score": 0.93, + "content": "\\{ 0 . 1 \\delta , 0 . 2 \\delta , \\ldots , 0 . 9 \\delta \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 580, + 506, + 595 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 592, + 418, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 418, + 604 + ], + "score": 1.0, + "content": "We use 20 samples per class for training, 500 for validation and 1000 for test.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 108, + 621, + 233, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 234, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 234, + 633 + ], + "score": 1.0, + "content": "4.2 EXPERIMENTAL SET-UP", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "We follow the experimental set-up of GAT with minor adjustments. All parameters are initialized", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "by Glorot initialization (Glorot & Bengio, 2010) and optimized by Adam (Kingma & Ba, 2014).", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "We apply L2 regularization, dropout (Srivastava et al., 2014) to features and attention coefficients,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "and early stopping on validation loss and accuracy. We use ELU (Clevert et al., 2016) as a non-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 174, + 700 + ], + "score": 1.0, + "content": "linear activation", + "type": "text" + }, + { + "bbox": [ + 174, + 690, + 181, + 699 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 688, + 421, + 700 + ], + "score": 1.0, + "content": ". Unless specified, we employ a two-layer SuperGAT with", + "type": "text" + }, + { + "bbox": [ + 422, + 688, + 452, + 698 + ], + "score": 0.9, + "content": "F = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "features and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 137, + 709 + ], + "score": 0.89, + "content": "K = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "attention heads (total 64 features). All models are implemented in PyTorch (Paszke et al.,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "2019) and PyTorch Geometric (Fey & Lenssen, 2019). See appendix A.5 for detailed model and", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 233, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 233, + 733 + ], + "score": 1.0, + "content": "hyperparameter configurations.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5 + } + ], + "page_idx": 4, + "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, + 751, + 308, + 760 + ], + "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, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "RQ3. Which graph attention should we use for given graphs? The above two research ques-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "tions explore what different graph attention learns with or without supervision of edge presence.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "Then, which graph attention is effective among them for given graphs? We hypothesize that differ-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "ent graph attention will have different abilities to model graphs under various homophily and average", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "degree. We choose these two properties among various graph statistics because they determine the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "quality and quantity of labels in our self-supervised task. From the perspective of supervised learn-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "ing of graph attention with edge labels, the learning result depends on how noisy labels are (i.e., how", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "low the homophily is) and how many labels exist (i.e., how high the average degree is). So, we gen-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 369, + 183 + ], + "score": 1.0, + "content": "erate 144 synthetic graphs (Section 4.1) controlling 9 homophily", + "type": "text" + }, + { + "bbox": [ + 369, + 171, + 411, + 182 + ], + "score": 0.31, + "content": "( 0 . 1 - 0 . 9 )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "and 16 average degree", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 107, + 181, + 145, + 192 + ], + "score": 0.62, + "content": "( 1 - 1 0 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "and perform the node classification task in the transductive setting with GCN, GATGO,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 240, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 181, + 206 + ], + "score": 1.0, + "content": "SuperGATSD, and", + "type": "text" + }, + { + "bbox": [ + 181, + 192, + 236, + 204 + ], + "score": 0.3, + "content": "\\mathrm { S u p e r G A T _ { M X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 191, + 240, + 206 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 81, + 506, + 206 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "In RQ3, there are also practical reasons to use the average degree and homophily, out of many graph", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "properties (e.g., diameter, degree sequence, degree distribution, average clustering coefficient). First,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "the graph property can be computed efficiently even for large graphs. Second, there should be an", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "algorithm that can generate graphs by controlling the property of interest only. Third, the property", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "should be a scalar value because if the synthetic graph space is too wide, it would be impossible to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "conduct an experiment with sufficient coverage. Average degree and homophily satisfy the above", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 464, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 464, + 287 + ], + "score": 1.0, + "content": "conditions and are suitable for our experiment, unlike some of the other graph properties.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 209, + 506, + 287 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "RQ4. Does design choice based on RQ3 generalize to real-world datasets? Experiments on", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "score": 1.0, + "content": "synthetic datasets provide an understanding of graph attention models’ performance, but they are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "oversimplified versions of real-world graphs. Can design choice from synthetic datasets be gener-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "alized to real-world datasets, considering more complex structures and rich features in real-world", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "graphs? To answer this question, we conduct experiments on 17 real-world datasets with the various", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "average degree (1.8 – 35.8) and homophily (0.16 – 0.91), and compare them with synthetic graph", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 370, + 192, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 192, + 381 + ], + "score": 1.0, + "content": "experiments in RQ3.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 303, + 505, + 381 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 398, + 176, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 177, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 177, + 411 + ], + "score": 1.0, + "content": "4.1 DATASETS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "Real-world datasets We use a total of 17 real-world datasets (Cora, CiteSeer, PubMed, Cora-ML,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 432, + 504, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 504, + 444 + ], + "score": 1.0, + "content": "Cora-Full, DBLP, ogbn-arxiv, CS, Physics, Photo, Computers, Wiki-CS, Four-Univ, Chameleon,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "Crocodile, Flickr, and PPI) in diverse domains (citation, co-authorship, co-purchase, web page, and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 188, + 466 + ], + "score": 1.0, + "content": "biology) and scales", + "type": "text" + }, + { + "bbox": [ + 188, + 454, + 230, + 465 + ], + "score": 0.47, + "content": "( 2 \\mathrm { k } - 1 6 9 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "nodes). We try to use their original settings as much as possible. To", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "verify research questions 1 and 2, we choose four classic benchmarks: Cora, CiteSeer, PubMed in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "the transductive setting, and PPI in the inductive setting. See appendix A.1 for detailed description,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 487, + 373, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 373, + 500 + ], + "score": 1.0, + "content": "splits, statistics (including degree and homophily), and references.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 420, + 505, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 515, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 355, + 527 + ], + "score": 1.0, + "content": "Synthetic datasets We generate random partition graphs of", + "type": "text" + }, + { + "bbox": [ + 356, + 518, + 363, + 525 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 515, + 445, + 527 + ], + "score": 1.0, + "content": "nodes per class and", + "type": "text" + }, + { + "bbox": [ + 445, + 518, + 451, + 525 + ], + "score": 0.54, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "classes (For-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "tunato, 2010), using NetworkX library (Hagberg et al., 2008). A random partition graph is a graph", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 294, + 550 + ], + "score": 1.0, + "content": "of communities controlled by two probabilities", + "type": "text" + }, + { + "bbox": [ + 294, + 538, + 309, + 549 + ], + "score": 0.87, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 537, + 326, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 538, + 344, + 549 + ], + "score": 0.88, + "content": "p _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 537, + 506, + 550 + ], + "score": 1.0, + "content": ". If the nodes have the same class labels,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 204, + 560 + ], + "score": 1.0, + "content": "they are connected with", + "type": "text" + }, + { + "bbox": [ + 204, + 549, + 218, + 559 + ], + "score": 0.86, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 548, + 380, + 560 + ], + "score": 1.0, + "content": ", and otherwise, they are connected with", + "type": "text" + }, + { + "bbox": [ + 380, + 549, + 399, + 559 + ], + "score": 0.86, + "content": "p _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 548, + 505, + 560 + ], + "score": 1.0, + "content": ". To generate a graph with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 192, + 572 + ], + "score": 1.0, + "content": "an average degree of", + "type": "text" + }, + { + "bbox": [ + 192, + 559, + 244, + 571 + ], + "score": 0.93, + "content": "d _ { a v g } = n \\cdot \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 559, + 292, + 572 + ], + "score": 1.0, + "content": ", we choose", + "type": "text" + }, + { + "bbox": [ + 293, + 560, + 307, + 570 + ], + "score": 0.87, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 559, + 325, + 572 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 560, + 344, + 570 + ], + "score": 0.88, + "content": "p _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 559, + 358, + 572 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 359, + 559, + 459, + 571 + ], + "score": 0.91, + "content": "p _ { i n } + ( c - 1 ) \\cdot p _ { o u t } = \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 559, + 506, + 572 + ], + "score": 1.0, + "content": ". The input", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "score": 1.0, + "content": "features of nodes are sampled from overlapping multi-Gaussian distributions (Abu-El-Haija et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 580, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 163, + 595 + ], + "score": 1.0, + "content": "2019). We set", + "type": "text" + }, + { + "bbox": [ + 163, + 582, + 171, + 591 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 580, + 200, + 595 + ], + "score": 1.0, + "content": "to 500,", + "type": "text" + }, + { + "bbox": [ + 200, + 582, + 207, + 591 + ], + "score": 0.62, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 580, + 277, + 595 + ], + "score": 1.0, + "content": "to 10, and choose", + "type": "text" + }, + { + "bbox": [ + 277, + 581, + 297, + 593 + ], + "score": 0.91, + "content": "d _ { a v g }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 580, + 375, + 595 + ], + "score": 1.0, + "content": "between 1 and 100,", + "type": "text" + }, + { + "bbox": [ + 375, + 582, + 390, + 592 + ], + "score": 0.81, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 580, + 412, + 595 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 413, + 581, + 501, + 593 + ], + "score": 0.93, + "content": "\\{ 0 . 1 \\delta , 0 . 2 \\delta , \\ldots , 0 . 9 \\delta \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 580, + 506, + 595 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 592, + 418, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 418, + 604 + ], + "score": 1.0, + "content": "We use 20 samples per class for training, 500 for validation and 1000 for test.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 515, + 506, + 604 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 621, + 233, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 234, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 234, + 633 + ], + "score": 1.0, + "content": "4.2 EXPERIMENTAL SET-UP", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "We follow the experimental set-up of GAT with minor adjustments. All parameters are initialized", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "by Glorot initialization (Glorot & Bengio, 2010) and optimized by Adam (Kingma & Ba, 2014).", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "We apply L2 regularization, dropout (Srivastava et al., 2014) to features and attention coefficients,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "and early stopping on validation loss and accuracy. We use ELU (Clevert et al., 2016) as a non-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 174, + 700 + ], + "score": 1.0, + "content": "linear activation", + "type": "text" + }, + { + "bbox": [ + 174, + 690, + 181, + 699 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 688, + 421, + 700 + ], + "score": 1.0, + "content": ". Unless specified, we employ a two-layer SuperGAT with", + "type": "text" + }, + { + "bbox": [ + 422, + 688, + 452, + 698 + ], + "score": 0.9, + "content": "F = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "features and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 137, + 709 + ], + "score": 0.89, + "content": "K = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "attention heads (total 64 features). All models are implemented in PyTorch (Paszke et al.,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "2019) and PyTorch Geometric (Fey & Lenssen, 2019). See appendix A.5 for detailed model and", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 233, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 233, + 733 + ], + "score": 1.0, + "content": "hyperparameter configurations.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 80, + 496, + 154 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 80, + 496, + 154 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 80, + 496, + 154 + ], + "spans": [ + { + "bbox": [ + 112, + 80, + 496, + 154 + ], + "score": 0.961, + "type": "image", + "image_path": "5655dcd1d4856266ece3055fb81f4694564136af3508e8f265ea50b58ed87e85.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 80, + 496, + 104.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 104.66666666666667, + 496, + 129.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 129.33333333333334, + 496, + 154.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 160, + 504, + 182 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "Figure 2: Distribution of KL divergence between normalized attention and label-agreement on all", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 171, + 429, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 429, + 183 + ], + "score": 1.0, + "content": "nodes and layers for Cora dataset (Left: two-layer GAT, Right: four-layer GAT).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 195, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "Baselines For all datasets, we compare our model against representative graph neural models:", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "score": 1.0, + "content": "graph convolutional network (GCN) (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "and graph attention network (GAT) (Velickovi ˇ c et al., 2018). Furthermore, for Cora, CiteSeer, and ´", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "PubMed, we choose recent graph neural architectures that learn aggregation coefficients (or dis-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 238, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 252 + ], + "score": 1.0, + "content": "crete structures) over edges: constrained graph attention network (CGAT1) (Wang et al., 2019a),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "graph learning-convolutional network (GLCN) (Jiang et al., 2019), learning discrete structure", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "(LDS) (Franceschi et al., 2019), graph agreement model (GAM) (Stretcu et al., 2019), and Neu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 272, + 497, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 497, + 284 + ], + "score": 1.0, + "content": "ralSparse (NS in short) (Zheng et al., 2020). For the PPI, we use CGAT as an additional baseline.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 108, + 299, + 172, + 311 + ], + "lines": [ + { + "bbox": [ + 104, + 297, + 174, + 314 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 174, + 314 + ], + "score": 1.0, + "content": "5 RESULTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "This section describes the experimental results which answer the research questions in Section 4. We", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "include a qualitative analysis of attention, quantitative comparisons on node classification and link", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "prediction, and recipes of graph attention design. The results for sensitivity analysis of important", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 357, + 276, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 276, + 368 + ], + "score": 1.0, + "content": "hyper-parameters are in the appendix B.5.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 106, + 379, + 479, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 480, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 480, + 392 + ], + "score": 1.0, + "content": "Does graph attention learn label-agreement? GO learns label-agreement better than DP.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "We draw box plots of KL divergence between attention and label agreement distributions of two-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "layer and four-layer GAT with GO and DP attention for Cora dataset in Figure 2. We see similar", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "patterns in other datasets and place their plots in the appendix B.3. At the rightmost of each sub-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "figure, we draw the KLD distribution when uniform attention is given to all neighborhoods. Note", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 504, + 451 + ], + "score": 1.0, + "content": "that the maximum value of KLD of each node is different since the degree of nodes is different.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 499, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 499, + 463 + ], + "score": 1.0, + "content": "Also, the KLD distribution shows a long-tail shape like a degree distribution of real-world graphs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "There are three observations regarding distributions of KLD. First, we observe that the KLD dis-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "tribution of GO attention shows a pattern similar to the uniform attention for all citation datasets.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "score": 1.0, + "content": "This implies that trained GO attention is similar to the uniform distribution, which is in line with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "previously reported results in the case of entropy2 (Wang et al., 2019b). Second, KLD values of DP", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "attention tend to be larger than those of GO attention for the last layer, resulting in bigger long-tails.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "This mismatch between the learned distribution of DP attention and the label agreement distribution", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 104, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "suggests that DP attention does not learn label-agreement in the neighborhood. Third, the deeper the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "model (more than two), the larger the KLD value of DP attention in the last layer. This is because", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "score": 1.0, + "content": "the variance of DP attention increases as the layer gets deeper, as explained below in Proposition 1.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 504, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 189, + 582 + ], + "score": 1.0, + "content": "Proposition 1. For", + "type": "text" + }, + { + "bbox": [ + 190, + 569, + 244, + 580 + ], + "score": 0.44, + "content": "l + 1 t h \\ G A T", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 568, + 270, + 582 + ], + "score": 1.0, + "content": "layer,", + "type": "text" + }, + { + "bbox": [ + 270, + 569, + 279, + 581 + ], + "score": 0.28, + "content": "i f", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 568, + 313, + 582 + ], + "score": 1.0, + "content": "W and", + "type": "text" + }, + { + "bbox": [ + 313, + 571, + 321, + 579 + ], + "score": 0.44, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "are independent and identically drawn from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 289, + 592 + ], + "score": 1.0, + "content": "zero-mean uniform distribution with variance", + "type": "text" + }, + { + "bbox": [ + 289, + 579, + 302, + 591 + ], + "score": 0.9, + "content": "\\sigma _ { w } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 579, + 320, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 321, + 579, + 333, + 591 + ], + "score": 0.89, + "content": "\\sigma _ { a } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "respectively, assuming that parameters are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 590, + 438, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 225, + 604 + ], + "score": 1.0, + "content": "independent to input features", + "type": "text" + }, + { + "bbox": [ + 225, + 590, + 236, + 601 + ], + "score": 0.86, + "content": "h ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 590, + 302, + 604 + ], + "score": 1.0, + "content": "and elements of", + "type": "text" + }, + { + "bbox": [ + 303, + 591, + 313, + 601 + ], + "score": 0.85, + "content": "h ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 590, + 438, + 604 + ], + "score": 1.0, + "content": "are independent to each other,", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 603, + 484, + 619 + ], + "lines": [ + { + "bbox": [ + 114, + 603, + 484, + 619 + ], + "spans": [ + { + "bbox": [ + 114, + 603, + 484, + 619 + ], + "score": 0.76, + "content": "\\begin{array} { r } { V a r [ e _ { i j , G o } ^ { l + 1 } ] = 2 F ^ { l + 1 } \\sigma _ { w } ^ { 2 } \\sigma _ { a } ^ { 2 } \\mathbb { E } ( \\Vert h ^ { l } \\Vert _ { 2 } ^ { 2 } ) ~ a n d ~ V a r [ e _ { i j , D P } ^ { l + 1 } ] \\geq F ^ { l + 1 } \\sigma _ { w } ^ { 4 } \\left( \\frac { 4 } { 5 } \\mathbb { E } \\left( ( h _ { i } ^ { l } ) ^ { \\top } h _ { j } ^ { l } ) ^ { 2 } \\right) + V a r ( ( h _ { i } ^ { l } ) ^ { \\top } h _ { j } ^ { l } ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "84b7db261b5ef5a176d392c4a621ee3a65e4a98f5366a3c40c84e36bc980a895.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 114, + 603, + 484, + 619 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "The proof is given in the appendix B.1. While the variance of GO depends on the norm of features", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "only, the variance of DP depends on the expectation of the square of input’s dot-product and variance", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 370, + 662 + ], + "score": 1.0, + "content": "of input’s dot-product. 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Furthermore, for Cora, CiteSeer, and ´", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "PubMed, we choose recent graph neural architectures that learn aggregation coefficients (or dis-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 238, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 252 + ], + "score": 1.0, + "content": "crete structures) over edges: constrained graph attention network (CGAT1) (Wang et al., 2019a),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "graph learning-convolutional network (GLCN) (Jiang et al., 2019), learning discrete structure", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "(LDS) (Franceschi et al., 2019), graph agreement model (GAM) (Stretcu et al., 2019), and Neu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 272, + 497, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 497, + 284 + ], + "score": 1.0, + "content": "ralSparse (NS in short) (Zheng et al., 2020). For the PPI, we use CGAT as an additional baseline.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 194, + 506, + 284 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 299, + 172, + 311 + ], + "lines": [ + { + "bbox": [ + 104, + 297, + 174, + 314 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 174, + 314 + ], + "score": 1.0, + "content": "5 RESULTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "This section describes the experimental results which answer the research questions in Section 4. We", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "include a qualitative analysis of attention, quantitative comparisons on node classification and link", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "prediction, and recipes of graph attention design. The results for sensitivity analysis of important", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 357, + 276, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 276, + 368 + ], + "score": 1.0, + "content": "hyper-parameters are in the appendix B.5.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 322, + 506, + 368 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 379, + 479, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 480, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 480, + 392 + ], + "score": 1.0, + "content": "Does graph attention learn label-agreement? GO learns label-agreement better than DP.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "We draw box plots of KL divergence between attention and label agreement distributions of two-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "layer and four-layer GAT with GO and DP attention for Cora dataset in Figure 2. We see similar", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "patterns in other datasets and place their plots in the appendix B.3. At the rightmost of each sub-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "figure, we draw the KLD distribution when uniform attention is given to all neighborhoods. Note", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 504, + 451 + ], + "score": 1.0, + "content": "that the maximum value of KLD of each node is different since the degree of nodes is different.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 499, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 499, + 463 + ], + "score": 1.0, + "content": "Also, the KLD distribution shows a long-tail shape like a degree distribution of real-world graphs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 396, + 505, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "There are three observations regarding distributions of KLD. First, we observe that the KLD dis-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "tribution of GO attention shows a pattern similar to the uniform attention for all citation datasets.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "score": 1.0, + "content": "This implies that trained GO attention is similar to the uniform distribution, which is in line with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "previously reported results in the case of entropy2 (Wang et al., 2019b). 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We report accuracy (Cora, CiteSeer, PubMed) and micro f1-score", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 175, + 340, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 340, + 188 + ], + "score": 1.0, + "content": "(PPI) for node classification, and AUC for link prediction.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 118, + 191, + 488, + 280 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 118, + 191, + 488, + 280 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 118, + 191, + 488, + 280 + ], + "spans": [ + { + "bbox": [ + 118, + 191, + 488, + 280 + ], + "score": 0.963, + "type": "image", + "image_path": "b6414dea89a6faa66f22beaef6a90fd6cf1b9ed48b5f745505be88d742a3e778.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 118, + 191, + 488, + 220.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 118, + 220.66666666666666, + 488, + 250.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 118, + 250.33333333333331, + 488, + 280.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 283, + 503, + 307 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 282, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 332, + 297 + ], + "score": 1.0, + "content": "Figure 4: Mean test accuracy gains (of 5 runs) against", + "type": "text" + }, + { + "bbox": [ + 333, + 284, + 364, + 295 + ], + "score": 0.9, + "content": "\\mathrm { G A T _ { G O } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 282, + 505, + 297 + ], + "score": 1.0, + "content": "on synthetic datasets, varying ho-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 294, + 297, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 297, + 307 + ], + "score": 1.0, + "content": "mophily and average degree of the input graph.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 8.25 + }, + { + "type": "title", + "bbox": [ + 105, + 315, + 491, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 489, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 473, + 329 + ], + "score": 1.0, + "content": "Is graph attention predictive for edge presence? DP predicts edge presence better than", + "type": "text" + }, + { + "bbox": [ + 474, + 315, + 489, + 325 + ], + "score": 0.32, + "content": "G O", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "In Figure 3, we report the mean AUC over multiple runs (5 for PPI and 10 for others) for link", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 342, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 447, + 356 + ], + "score": 1.0, + "content": "prediction (red lines) and node classification (gray lines). As the mixing coefficient", + "type": "text" + }, + { + "bbox": [ + 448, + 343, + 462, + 354 + ], + "score": 0.89, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 342, + 506, + 356 + ], + "score": 1.0, + "content": "increases,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 352, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 506, + 368 + ], + "score": 1.0, + "content": "the link prediction score increases in all datasets and attentions. This is a natural result considering", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 124, + 377 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 365, + 138, + 376 + ], + "score": 0.89, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 364, + 383, + 377 + ], + "score": 1.0, + "content": "is the weight factor of self-supervised graph attention loss (", + "type": "text" + }, + { + "bbox": [ + 383, + 365, + 398, + 376 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "in Equation 6). For three", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 461, + 388 + ], + "score": 1.0, + "content": "out of four datasets, DP attention outperforms GO for link prediction for all range of", + "type": "text" + }, + { + "bbox": [ + 462, + 376, + 476, + 387 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "in our", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 268, + 399 + ], + "score": 1.0, + "content": "experiment. Surprisingly, even for small", + "type": "text" + }, + { + "bbox": [ + 269, + 387, + 282, + 397 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 386, + 505, + 399 + ], + "score": 1.0, + "content": ", DP attention shows around 80 AUC, much higher than", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "the performance of GO attention. PPI is an exception where GO attention shows higher performance", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 145, + 420 + ], + "score": 1.0, + "content": "for small", + "type": "text" + }, + { + "bbox": [ + 146, + 408, + 159, + 419 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "than DP, but the difference is slight. The results of this experiment demonstrate that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 419, + 375, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 375, + 432 + ], + "score": 1.0, + "content": "DP attention is more suitable than GO attention in encoding edges.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 436, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "This figure also includes node classification performance. For all datasets except PubMed, we ob-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 447, + 504, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 504, + 459 + ], + "score": 1.0, + "content": "serve a trade-off between node classification and link prediction; that is, node classification perfor-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 324, + 470 + ], + "score": 1.0, + "content": "mance decreases in SuperGATGO and SuperGATDP as", + "type": "text" + }, + { + "bbox": [ + 324, + 459, + 338, + 469 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "increases and thus link prediction perfor-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 425, + 482 + ], + "score": 1.0, + "content": "mance increases. PubMed also shows a decrease in performance at the largest", + "type": "text" + }, + { + "bbox": [ + 425, + 470, + 439, + 480 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "we have tested.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "This implies that it is hard to learn the relational importance from edges by simply optimizing graph", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 492, + 221, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 221, + 503 + ], + "score": 1.0, + "content": "attention for link prediction.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 514, + 503, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 529 + ], + "score": 1.0, + "content": "Which graph attention should we use for given graphs? It depends on homophily and average", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 526, + 187, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 187, + 538 + ], + "score": 1.0, + "content": "degree of the graph.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 410, + 556 + ], + "score": 1.0, + "content": "In Figure 4, we draw the mean test accuracy gains (over 5 runs) against", + "type": "text" + }, + { + "bbox": [ + 410, + 543, + 442, + 554 + ], + "score": 0.83, + "content": "\\mathrm { G A T _ { G O } }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 541, + 505, + 556 + ], + "score": 1.0, + "content": "as the average", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "degree increases from 1 to 100, for different values of homophily, on 64 synthetic graphs with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 564, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 191, + 577 + ], + "score": 1.0, + "content": "GCN, SuperGATSD,", + "type": "text" + }, + { + "bbox": [ + 191, + 565, + 247, + 576 + ], + "score": 0.28, + "content": "\\mathrm { S u p e r G A T _ { M X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 564, + 496, + 577 + ], + "score": 1.0, + "content": ". See full results from appendix B.2. We define homophily", + "type": "text" + }, + { + "bbox": [ + 497, + 565, + 504, + 574 + ], + "score": 0.72, + "content": "h", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 575, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 587 + ], + "score": 1.0, + "content": "as the average ratio of neighbors with the same label as the center node (Pei et al., 2020). That", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 586, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 117, + 605 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 586, + 282, + 606 + ], + "score": 0.93, + "content": "\\begin{array} { r } { h = \\frac { 1 } { | V | } \\bar { \\sum } _ { i \\in V } \\left( \\sum _ { j \\in \\mathbb { N } _ { i } } \\mathbf { \\bar { 1 } } _ { l ( i ) = l ( j ) } / | \\mathbb { N } _ { i } | \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 587, + 315, + 605 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 315, + 590, + 330, + 602 + ], + "score": 0.87, + "content": "l ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 587, + 414, + 605 + ], + "score": 1.0, + "content": "is the label of node", + "type": "text" + }, + { + "bbox": [ + 415, + 591, + 420, + 600 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 587, + 507, + 605 + ], + "score": 1.0, + "content": ". The expectation of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 326, + 619 + ], + "score": 1.0, + "content": "homophily for random partition graphs is analytically", + "type": "text" + }, + { + "bbox": [ + 327, + 605, + 351, + 617 + ], + "score": 0.92, + "content": "p _ { i n } / \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 603, + 506, + 619 + ], + "score": 1.0, + "content": ", and we just adopt this value to label", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 615, + 256, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 256, + 629 + ], + "score": 1.0, + "content": "the homophily of graphs in Figure 4.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 467, + 646 + ], + "score": 1.0, + "content": "We make the following observations from this figure. First, if the homophily is low", + "type": "text" + }, + { + "bbox": [ + 468, + 633, + 501, + 644 + ], + "score": 0.86, + "content": "( \\le ~ 0 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 632, + 505, + 646 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "SuperGATSD performs best among models because DP attention tends to focus on a small number", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "of neighbors. This result empirically confirms what we analytically found in Proposition 1. Second,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "even when homophily is low, the performance gain of SuperGAT against GAT increases as the av-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "erage degree increases to a certain level (around 10), meaning relation modeling can benefit from", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "self-supervision if there are sufficiently many edges providing supervision. This is in agreement", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "with prior study of label noise for deep neural networks where they find that the absolute amount of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "data with correct labels affects the learning quality more than the ratio between data with noisy and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "correct labels (Rolnick et al., 2017). Third, if the average degree and homophily are high enough,", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + } + ], + "page_idx": 6, + "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, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 119, + 79, + 487, + 150 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 79, + 487, + 150 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 79, + 487, + 150 + ], + "spans": [ + { + "bbox": [ + 119, + 79, + 487, + 150 + ], + "score": 0.953, + "type": "image", + "image_path": "8879eb0d78cf1cb5bd50b0a8f96799c8a3680def49289c4b6ddce0786606c356.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 119, + 79, + 487, + 102.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 119, + 102.66666666666667, + 487, + 126.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 119, + 126.33333333333334, + 487, + 150.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 154, + 503, + 187 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "Figure 3: Test performance on node classification and link prediction for GO and DP attentions", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 225, + 177 + ], + "score": 1.0, + "content": "against the mixing coefficient", + "type": "text" + }, + { + "bbox": [ + 226, + 165, + 239, + 176 + ], + "score": 0.9, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 164, + 505, + 177 + ], + "score": 1.0, + "content": ". We report accuracy (Cora, CiteSeer, PubMed) and micro f1-score", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 175, + 340, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 340, + 188 + ], + "score": 1.0, + "content": "(PPI) for node classification, and AUC for link prediction.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 118, + 191, + 488, + 280 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 118, + 191, + 488, + 280 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 118, + 191, + 488, + 280 + ], + "spans": [ + { + "bbox": [ + 118, + 191, + 488, + 280 + ], + "score": 0.963, + "type": "image", + "image_path": "b6414dea89a6faa66f22beaef6a90fd6cf1b9ed48b5f745505be88d742a3e778.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 118, + 191, + 488, + 220.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 118, + 220.66666666666666, + 488, + 250.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 118, + 250.33333333333331, + 488, + 280.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 283, + 503, + 307 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 282, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 332, + 297 + ], + "score": 1.0, + "content": "Figure 4: Mean test accuracy gains (of 5 runs) against", + "type": "text" + }, + { + "bbox": [ + 333, + 284, + 364, + 295 + ], + "score": 0.9, + "content": "\\mathrm { G A T _ { G O } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 282, + 505, + 297 + ], + "score": 1.0, + "content": "on synthetic datasets, varying ho-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 294, + 297, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 297, + 307 + ], + "score": 1.0, + "content": "mophily and average degree of the input graph.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 8.25 + }, + { + "type": "title", + "bbox": [ + 105, + 315, + 491, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 489, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 473, + 329 + ], + "score": 1.0, + "content": "Is graph attention predictive for edge presence? DP predicts edge presence better than", + "type": "text" + }, + { + "bbox": [ + 474, + 315, + 489, + 325 + ], + "score": 0.32, + "content": "G O", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "In Figure 3, we report the mean AUC over multiple runs (5 for PPI and 10 for others) for link", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 342, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 447, + 356 + ], + "score": 1.0, + "content": "prediction (red lines) and node classification (gray lines). As the mixing coefficient", + "type": "text" + }, + { + "bbox": [ + 448, + 343, + 462, + 354 + ], + "score": 0.89, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 342, + 506, + 356 + ], + "score": 1.0, + "content": "increases,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 352, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 506, + 368 + ], + "score": 1.0, + "content": "the link prediction score increases in all datasets and attentions. This is a natural result considering", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 124, + 377 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 365, + 138, + 376 + ], + "score": 0.89, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 364, + 383, + 377 + ], + "score": 1.0, + "content": "is the weight factor of self-supervised graph attention loss (", + "type": "text" + }, + { + "bbox": [ + 383, + 365, + 398, + 376 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "in Equation 6). For three", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 461, + 388 + ], + "score": 1.0, + "content": "out of four datasets, DP attention outperforms GO for link prediction for all range of", + "type": "text" + }, + { + "bbox": [ + 462, + 376, + 476, + 387 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "in our", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 268, + 399 + ], + "score": 1.0, + "content": "experiment. Surprisingly, even for small", + "type": "text" + }, + { + "bbox": [ + 269, + 387, + 282, + 397 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 386, + 505, + 399 + ], + "score": 1.0, + "content": ", DP attention shows around 80 AUC, much higher than", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "the performance of GO attention. PPI is an exception where GO attention shows higher performance", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 145, + 420 + ], + "score": 1.0, + "content": "for small", + "type": "text" + }, + { + "bbox": [ + 146, + 408, + 159, + 419 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "than DP, but the difference is slight. The results of this experiment demonstrate that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 419, + 375, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 375, + 432 + ], + "score": 1.0, + "content": "DP attention is more suitable than GO attention in encoding edges.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 331, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 436, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "This figure also includes node classification performance. For all datasets except PubMed, we ob-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 447, + 504, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 504, + 459 + ], + "score": 1.0, + "content": "serve a trade-off between node classification and link prediction; that is, node classification perfor-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 324, + 470 + ], + "score": 1.0, + "content": "mance decreases in SuperGATGO and SuperGATDP as", + "type": "text" + }, + { + "bbox": [ + 324, + 459, + 338, + 469 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "increases and thus link prediction perfor-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 425, + 482 + ], + "score": 1.0, + "content": "mance increases. PubMed also shows a decrease in performance at the largest", + "type": "text" + }, + { + "bbox": [ + 425, + 470, + 439, + 480 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "we have tested.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "This implies that it is hard to learn the relational importance from edges by simply optimizing graph", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 492, + 221, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 221, + 503 + ], + "score": 1.0, + "content": "attention for link prediction.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 437, + 505, + 503 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 514, + 503, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 529 + ], + "score": 1.0, + "content": "Which graph attention should we use for given graphs? It depends on homophily and average", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 526, + 187, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 187, + 538 + ], + "score": 1.0, + "content": "degree of the graph.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 410, + 556 + ], + "score": 1.0, + "content": "In Figure 4, we draw the mean test accuracy gains (over 5 runs) against", + "type": "text" + }, + { + "bbox": [ + 410, + 543, + 442, + 554 + ], + "score": 0.83, + "content": "\\mathrm { G A T _ { G O } }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 541, + 505, + 556 + ], + "score": 1.0, + "content": "as the average", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "degree increases from 1 to 100, for different values of homophily, on 64 synthetic graphs with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 564, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 191, + 577 + ], + "score": 1.0, + "content": "GCN, SuperGATSD,", + "type": "text" + }, + { + "bbox": [ + 191, + 565, + 247, + 576 + ], + "score": 0.28, + "content": "\\mathrm { S u p e r G A T _ { M X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 564, + 496, + 577 + ], + "score": 1.0, + "content": ". See full results from appendix B.2. We define homophily", + "type": "text" + }, + { + "bbox": [ + 497, + 565, + 504, + 574 + ], + "score": 0.72, + "content": "h", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 575, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 587 + ], + "score": 1.0, + "content": "as the average ratio of neighbors with the same label as the center node (Pei et al., 2020). That", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 586, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 117, + 605 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 586, + 282, + 606 + ], + "score": 0.93, + "content": "\\begin{array} { r } { h = \\frac { 1 } { | V | } \\bar { \\sum } _ { i \\in V } \\left( \\sum _ { j \\in \\mathbb { N } _ { i } } \\mathbf { \\bar { 1 } } _ { l ( i ) = l ( j ) } / | \\mathbb { N } _ { i } | \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 587, + 315, + 605 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 315, + 590, + 330, + 602 + ], + "score": 0.87, + "content": "l ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 587, + 414, + 605 + ], + "score": 1.0, + "content": "is the label of node", + "type": "text" + }, + { + "bbox": [ + 415, + 591, + 420, + 600 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 587, + 507, + 605 + ], + "score": 1.0, + "content": ". The expectation of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 326, + 619 + ], + "score": 1.0, + "content": "homophily for random partition graphs is analytically", + "type": "text" + }, + { + "bbox": [ + 327, + 605, + 351, + 617 + ], + "score": 0.92, + "content": "p _ { i n } / \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 603, + 506, + 619 + ], + "score": 1.0, + "content": ", and we just adopt this value to label", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 615, + 256, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 256, + 629 + ], + "score": 1.0, + "content": "the homophily of graphs in Figure 4.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 541, + 507, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 467, + 646 + ], + "score": 1.0, + "content": "We make the following observations from this figure. First, if the homophily is low", + "type": "text" + }, + { + "bbox": [ + 468, + 633, + 501, + 644 + ], + "score": 0.86, + "content": "( \\le ~ 0 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 632, + 505, + 646 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "SuperGATSD performs best among models because DP attention tends to focus on a small number", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "of neighbors. This result empirically confirms what we analytically found in Proposition 1. Second,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "even when homophily is low, the performance gain of SuperGAT against GAT increases as the av-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "erage degree increases to a certain level (around 10), meaning relation modeling can benefit from", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "self-supervision if there are sufficiently many edges providing supervision. This is in agreement", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "with prior study of label noise for deep neural networks where they find that the absolute amount of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "data with correct labels affects the learning quality more than the ratio between data with noisy and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "correct labels (Rolnick et al., 2017). Third, if the average degree and homophily are high enough,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "there is no difference between all models, including GCNs. If there are more correct edges beyond", + "type": "text", + "cross_page": true + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "a certain amount, we can learn fine representation without self-supervision. 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Modelogbn-arxiv CSPhysicsCora-ML Cora-Full DBLPCham.Four-Univ Wiki-CSPhotoComp.FlickrCroco.
GCN33.3±1.291.5±0.292.5±0.285.0±0.459.5±0.277.8±0.533.1±0.974.8±0.674.0±1.091.6±0.684.5±1.451.2±0.432.6±0.4
GraphSAGE54.6±0.390.0±0.192.2±0.183.7±0.459.2±0.278.7±0.641.0±0.974.4±0.677.5±0.590.4±1.183.0±1.450.7±0.253.0±1.0
GAT54.1±0.589.5±0.291.2±0.683.2±0.658.7±0.378.2±1.540.8±0.774.2±0.777.6±0.691.8±0.685.7±0.950.9±0.253.3±1.0
SuperGATsD54.50.388.80.491.6±0.584.50.455.80.679.40.841.60.776.20.877.9±0.786.8±2.582.2±0.945.1±1.253.3±0.9
SuperGATMX55.1±0.290.20.291.90.584.70.459.60280.70.742.00.875.30.677.9±0.591.8±0.9 85.7±1.1 50.8±0.253.3±0.9
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ModelCoraCiteSeerPubMed
GCN+81.570.379.0
GraphSAGE82.1 ± 0.671.9 ± 0.978.0 ± 0.7
CGAT81.4 ± 1.170.1 ± 0.978.1 ±1.0
GLCN+85.572.078.3
LDSt84.175-
GCN + GAM†86.273.586.0
GCN + NSt83.7 ±1.474.1 ± 1.41
GATt83.0±0.772.5 ± 0.779.0± 0.4
SuperGATsD82.70.672.5±0.881.30.5
SuperGATMX84.3*0.672.6±0.881.70.5
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ModelPPI
GCN GraphSAGE61.5 ± 0.4 59.0 ±1.2
CGAT GAT68.3 ± 1.7 72.2 ± 0.6
SuperGATsD74.40.4
SuperGATMX67.21.2
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The size is the performance gain of SuperGAT against GAT,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 445, + 535 + ], + "score": 1.0, + "content": "and the color indicates the best model. If the difference is not statistically significant (", + "type": "text" + }, + { + "bbox": [ + 445, + 524, + 451, + 534 + ], + "score": 0.7, + "content": "\\dot { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 522, + 476, + 535 + ], + "score": 1.0, + "content": "-value", + "type": "text" + }, + { + "bbox": [ + 476, + 523, + 502, + 534 + ], + "score": 0.64, + "content": "\\geq . 0 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 522, + 505, + 535 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 357, + 547 + ], + "score": 1.0, + "content": "between GAT and SuperGAT, and between SuperGATMX and", + "type": "text" + }, + { + "bbox": [ + 357, + 534, + 410, + 545 + ], + "score": 0.26, + "content": "\\mathrm { S u p e r G A T _ { S D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 532, + 505, + 547 + ], + "score": 1.0, + "content": ", we mark as GAT-Any", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "and SuperGAT-Any, respectively. 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Average degree and homophily can be found in the appendix A.1, and experimental", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "results of graph attention models are summarized in Tables 1, 2 and 3. 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Wiki-CS (N), located in the SuperGAT-Any region", + "type": "text" + }, + { + "bbox": [ + 375, + 655, + 389, + 666 + ], + "score": 0.38, + "content": "( \\_ )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 654, + 505, + 668 + ], + "score": 1.0, + "content": ", also show no difference in", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "performance between SuperGATs. Nine datasets (N), which SuperGATMX shows the highest per-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "formance, are located in the MX regions (\u0004) or within the margin of two squares. 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Modelogbn-arxiv CSPhysicsCora-ML Cora-Full DBLPCham.Four-Univ Wiki-CSPhotoComp.FlickrCroco.
GCN33.3±1.291.5±0.292.5±0.285.0±0.459.5±0.277.8±0.533.1±0.974.8±0.674.0±1.091.6±0.684.5±1.451.2±0.432.6±0.4
GraphSAGE54.6±0.390.0±0.192.2±0.183.7±0.459.2±0.278.7±0.641.0±0.974.4±0.677.5±0.590.4±1.183.0±1.450.7±0.253.0±1.0
GAT54.1±0.589.5±0.291.2±0.683.2±0.658.7±0.378.2±1.540.8±0.774.2±0.777.6±0.691.8±0.685.7±0.950.9±0.253.3±1.0
SuperGATsD54.50.388.80.491.6±0.584.50.455.80.679.40.841.60.776.20.877.9±0.786.8±2.582.2±0.945.1±1.253.3±0.9
SuperGATMX55.1±0.290.20.291.90.584.70.459.60280.70.742.00.875.30.677.9±0.591.8±0.9 85.7±1.1 50.8±0.253.3±0.9
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ModelCoraCiteSeerPubMed
GCN+81.570.379.0
GraphSAGE82.1 ± 0.671.9 ± 0.978.0 ± 0.7
CGAT81.4 ± 1.170.1 ± 0.978.1 ±1.0
GLCN+85.572.078.3
LDSt84.175-
GCN + GAM†86.273.586.0
GCN + NSt83.7 ±1.474.1 ± 1.41
GATt83.0±0.772.5 ± 0.779.0± 0.4
SuperGATsD82.70.672.5±0.881.30.5
SuperGATMX84.3*0.672.6±0.881.70.5
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ModelPPI
GCN GraphSAGE61.5 ± 0.4 59.0 ±1.2
CGAT GAT68.3 ± 1.7 72.2 ± 0.6
SuperGATsD74.40.4
SuperGATMX67.21.2
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The size is the performance gain of SuperGAT against GAT,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 445, + 535 + ], + "score": 1.0, + "content": "and the color indicates the best model. 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We call this plot a recipe since it introduces the optimal attention", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 555, + 219, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 219, + 568 + ], + "score": 1.0, + "content": "to a specific region’s graph.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 500, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "Now we map the results of 17 real-world datasets in Figure 5 according to their average degree and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "homophily. 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We put unpaired t-test results of SuperGAT with", + "type": "text" + }, + { + "bbox": [ + 366, + 616, + 398, + 627 + ], + "score": 0.79, + "content": "\\mathrm { G A T _ { G O } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 616, + 459, + 629 + ], + "score": 1.0, + "content": "with asterisks.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 572, + 505, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "score": 1.0, + "content": "We find that the graph attention recipe based on synthetic experiments can generalize across real-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 432, + 656 + ], + "score": 1.0, + "content": "world graphs. PPI and Four-Univ (N) are surrounded by squares of SuperGATSD", + "type": "text" + }, + { + "bbox": [ + 433, + 644, + 448, + 655 + ], + "score": 0.28, + "content": "( \\_ )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "at the bottom", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 374, + 668 + ], + "score": 1.0, + "content": "of the plane. Wiki-CS (N), located in the SuperGAT-Any region", + "type": "text" + }, + { + "bbox": [ + 375, + 655, + 389, + 666 + ], + "score": 0.38, + "content": "( \\_ )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 654, + 505, + 668 + ], + "score": 1.0, + "content": ", also show no difference in", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "performance between SuperGATs. Nine datasets (N), which SuperGATMX shows the highest per-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "formance, are located in the MX regions (\u0004) or within the margin of two squares. Note that there are", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "two MX regions: lower-middle average degree (2.5 - 7.5) and high homophily (0.8 - 0.9), and upper-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 201, + 711 + ], + "score": 1.0, + "content": "middle average degree", + "type": "text" + }, + { + "bbox": [ + 201, + 699, + 240, + 710 + ], + "score": 0.26, + "content": "( 7 . 5 - 5 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 699, + 363, + 711 + ], + "score": 1.0, + "content": "and lower-middle homophily", + "type": "text" + }, + { + "bbox": [ + 363, + 699, + 405, + 710 + ], + "score": 0.49, + "content": "( 0 . 3 \\textrm { - } 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". There are five datasets", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 356, + 723 + ], + "score": 1.0, + "content": "with no significant performance change across graph attention", + "type": "text" + }, + { + "bbox": [ + 356, + 711, + 370, + 721 + ], + "score": 0.54, + "content": "\\mathbf { \\Psi } ( \\mathbf { \\blacktriangle } )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 709, + 505, + 723 + ], + "score": 1.0, + "content": ". CiteSeer, Photo, and Computers", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 348, + 733 + ], + "score": 1.0, + "content": "are within a margin of one square from the GAT-Any region", + "type": "text" + }, + { + "bbox": [ + 348, + 721, + 363, + 732 + ], + "score": 0.35, + "content": "( \\sqcup )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "; however, Flickr and Crocodile are", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "in the SuperGAT-Any region. To find out the cause of this irregularity, we examine the distribution", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "of degree and per-node homophily (appendix A.4). We observe a more complex mixture distribution", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "of homophily and average degree in Flickr and Crocodile, and this seems to be equivalent to mixing", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 340, + 504, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 504, + 353 + ], + "score": 1.0, + "content": "graphs of different characteristics, resulting in inconsistent results with our attention design recipe.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 631, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 171, + 87, + 435, + 262 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 171, + 87, + 435, + 262 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 171, + 87, + 435, + 262 + ], + "spans": [ + { + "bbox": [ + 171, + 87, + 435, + 262 + ], + "score": 0.971, + "type": "image", + "image_path": "c2ba813c72d3a1b9241f24df38a8a48755c3ca74c3c2ce39298546d3f985bb01.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 171, + 87, + 435, + 145.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 171, + 145.33333333333334, + 435, + 203.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 171, + 203.66666666666669, + 435, + 262.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 270, + 504, + 292 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 283 + ], + "score": 1.0, + "content": "Figure 5: The best-performed graph attention design for synthetic and real-world graphs with various", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 280, + 233, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 233, + 294 + ], + "score": 1.0, + "content": "average degree and homophily.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "in the SuperGAT-Any region. To find out the cause of this irregularity, we examine the distribution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "of degree and per-node homophily (appendix A.4). We observe a more complex mixture distribution", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "of homophily and average degree in Flickr and Crocodile, and this seems to be equivalent to mixing", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 340, + 504, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 504, + 353 + ], + "score": 1.0, + "content": "graphs of different characteristics, resulting in inconsistent results with our attention design recipe.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "Comparison with baselines For a total of 17 datasets, SuperGAT outperforms GCN for 13", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "datasets, GAT for 12 datasets, GraphSAGE for 16 datasets. Interestingly, for CS, Physics, Cora-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "ML, and Flickr, in which our model performs worse than GCN, GAT also cannot surpass GCN.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "It is not yet known when the degree-normalized aggregation of GCN outperforms the attention-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "based aggregation, and more research is needed to figure out how to embed the degree information", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "into graph attention. Tables 2 and 3 show performance comparisons between SuperGAT and re-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 470, + 441 + ], + "score": 1.0, + "content": "cent GNNs for Cora, CiteSeer, PubMed, and PPI. Our model performs better for CiteSeer", + "type": "text" + }, + { + "bbox": [ + 471, + 429, + 503, + 440 + ], + "score": 0.84, + "content": "( 0 . 6 \\% { \\mathfrak { p } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 429, + 505, + 441 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 162, + 452 + ], + "score": 1.0, + "content": "and PubMed", + "type": "text" + }, + { + "bbox": [ + 162, + 440, + 195, + 451 + ], + "score": 0.85, + "content": "( 3 . 4 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 440, + 472, + 452 + ], + "score": 1.0, + "content": "than GLCN, which gives regularization to all relations in a graph.", + "type": "text" + }, + { + "bbox": [ + 472, + 441, + 505, + 450 + ], + "score": 0.6, + "content": "\\mathrm { G C N + }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 365, + 463 + ], + "score": 1.0, + "content": "NS (NeuralSparse) performs better than our model for CiteSeer", + "type": "text" + }, + { + "bbox": [ + 366, + 451, + 398, + 462 + ], + "score": 0.88, + "content": "( 1 . 5 \\% { \\mathfrak { p } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 451, + 468, + 463 + ], + "score": 1.0, + "content": "but not for Cora", + "type": "text" + }, + { + "bbox": [ + 468, + 451, + 500, + 462 + ], + "score": 0.85, + "content": "( 0 . 6 \\% \\mathfrak { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 451, + 505, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "CGAT modified for semi-supervised learning shows lower performance than GAT. Although LDS", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "and GAM which use iterative training show better performance except for LDS on Cora, these mod-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 484, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 445, + 495 + ], + "score": 1.0, + "content": "els require significantly more computation for the iterative training. For example,", + "type": "text" + }, + { + "bbox": [ + 446, + 484, + 505, + 494 + ], + "score": 0.81, + "content": "\\mathrm { G C N } + \\mathrm { G A M }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 495, + 504, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 253, + 507 + ], + "score": 1.0, + "content": "compared to our model needs more", + "type": "text" + }, + { + "bbox": [ + 253, + 495, + 273, + 505 + ], + "score": 0.86, + "content": "\\times 3 4", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 495, + 392, + 507 + ], + "score": 1.0, + "content": "more training time for Cora,", + "type": "text" + }, + { + "bbox": [ + 392, + 495, + 412, + 506 + ], + "score": 0.87, + "content": "\\times 7 2", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 495, + 484, + 507 + ], + "score": 1.0, + "content": "for CiteSeer, and", + "type": "text" + }, + { + "bbox": [ + 485, + 495, + 504, + 505 + ], + "score": 0.85, + "content": "\\times 8 2", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "for PubMed. See appendix A.7 and B.4 for the experimental set-up and the result of wall-clock time", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 516, + 144, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 144, + 529 + ], + "score": 1.0, + "content": "analysis.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 543, + 195, + 556 + ], + "lines": [ + { + "bbox": [ + 104, + 541, + 197, + 559 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 197, + 559 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "We proposed novel graph neural architecture designs to self-supervise graph attention following the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "input graph’s characteristics. 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Nodes are documents, and edges are citations. The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 231, + 382, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 382, + 245 + ], + "score": 1.0, + "content": "task for all citation network datasets is to classify each paper’s topic.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "score": 1.0, + "content": "Cora, CiteSeer, PubMed We use three benchmark datasets for semi-supervised node classifica-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "tion tasks in the transductive setting (Sen et al., 2008; Yang et al., 2016). The features of the nodes", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "are bag-of-words representations of documents. We follow the train/validation/test split of previ-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "score": 1.0, + "content": "ous work (Kipf & Welling, 2017). We use 20 samples per class for training, 500 samples for the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 300, + 272, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 272, + 312 + ], + "score": 1.0, + "content": "validation, and 1000 samples for the test.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "Cora-ML, Cora-Full, DBLP These are other citation network datasets from Bojchevski &", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "Gunnemann (2018). Node features are bag-of-words representations of documents. For CoraFull ¨", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "with features more than 5000, we reduce the dimension to 500 by performing PCA. We use the split", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 355, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 370 + ], + "score": 1.0, + "content": "setting in Shchur et al. (2018): 20 samples per class for training, 30 samples per class for validation,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 367, + 186, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 186, + 380 + ], + "score": 1.0, + "content": "the rest for the test.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "ogbn-arxiv The ogbn-arxiv is a recently proposed large-scale dataset of citation networks (Hu", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 402, + 504, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 504, + 414 + ], + "score": 1.0, + "content": "et al., 2020a; Wang et al., 2020). Nodes represent arXiv papers, and edges indicate citations be-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "tween papers, and node features are mean vectors of skip-gram word embeddings of their titles and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 424, + 448, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 448, + 438 + ], + "score": 1.0, + "content": "abstracts. We use the public split by publication dates provided by the original paper.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 108, + 448, + 243, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 244, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 244, + 461 + ], + "score": 1.0, + "content": "A.1.2 CO-AUTHOR NETWORK", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "CS, Physics The CS and Physics are co-author networks in each domain (Shchur et al., 2018).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "Nodes are authors, and edges mean whether two authors co-authored a paper. Node features are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "paper keywords from the author’s papers, and we reduce the original dimension (6805 and 8415) to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "500 using PCA. The split is the 20-per-class/30-per-class/rest from Shchur et al. (2018). The goal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 360, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 360, + 526 + ], + "score": 1.0, + "content": "of this task is to classify each author’s respective field of study.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 535, + 247, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 248, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 248, + 548 + ], + "score": 1.0, + "content": "A.1.3 AMAZON CO-PURCHASE", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Photo, Computers The Photo and Computers are parts of the Amazon co-purchase", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "graph (McAuley et al., 2015; Shchur et al., 2018). Nodes are goods, and edges indicate whether", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "two goods are frequently purchased together, and node features are a bag-of-words representation of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "product reviews. The split is the 20-per-class/30-per-class/rest from Shchur et al. (2018). The task", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 256, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 256, + 612 + ], + "score": 1.0, + "content": "is to classify the categories of goods.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 622, + 236, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 237, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 237, + 635 + ], + "score": 1.0, + "content": "A.1.4 WEB PAGE NETWORK", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 504, + 697 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 655 + ], + "score": 1.0, + "content": "Wiki-CS The Wiki-CS dataset is computer science related page networks in Wikipedia (Mernyei", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "& Cangea, 2020). Nodes represent articles about computer science, and edges represent hyperlinks", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "between articles. The features of nodes are mean vectors of GloVe word embeddings of articles.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 673, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 687 + ], + "score": 1.0, + "content": "There are 20 standard splits, and we experiment with five random seeds for each split (total 100", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 686, + 339, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 339, + 698 + ], + "score": 1.0, + "content": "runs). The task is to classify the main category of articles.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "Chameleon, Crocodile These datasets are Wikipedia page networks about specific topics,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "Chameleon and Crocodile (Rozemberczki et al., 2019). Nodes are articles, and edges are mutual", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + } + ], + "page_idx": 14, + "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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 250, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 252, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 252, + 95 + ], + "score": 1.0, + "content": "A EXPERIMENTAL SET-UP", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 106, + 235, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 236, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 236, + 118 + ], + "score": 1.0, + "content": "A.1 REAL-WORLD DATASET", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 127, + 505, + 189 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "In this section, we describe details (including nodes, edges, features, labels, and splits) of real-world", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "datasets. We report statistics of real-world datasets in Tables 4 and 5. For multi-label graphs (PPI),", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "we extend the homophily to the average of the ratio of shared labels on neighbors over nodes, i.e.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 159, + 507, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 298, + 180 + ], + "score": 0.91, + "content": "\\begin{array} { r } { h = \\frac { 1 } { | V | } \\sum _ { i \\in V } \\left( \\sum _ { j \\in \\mathbb { N } _ { i } } \\left| \\mathbb { C } _ { i } \\cap \\mathbb { C } _ { j } \\right| / ( | \\tilde { \\mathbb { N } _ { i } } | \\cdot | \\mathbb { C } | ) \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 161, + 329, + 179 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 329, + 164, + 341, + 174 + ], + "score": 0.88, + "content": "\\mathbb { C } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 161, + 443, + 179 + ], + "score": 1.0, + "content": "is a set of labels for node", + "type": "text" + }, + { + "bbox": [ + 443, + 164, + 448, + 173 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 161, + 466, + 179 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 466, + 164, + 475, + 173 + ], + "score": 0.77, + "content": "\\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 161, + 507, + 179 + ], + "score": 1.0, + "content": "is a set", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 177, + 158, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 158, + 190 + ], + "score": 1.0, + "content": "of all labels.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 127, + 507, + 190 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 201, + 231, + 213 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 232, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 232, + 214 + ], + "score": 1.0, + "content": "A.1.1 CITATION NETWORK", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 221, + 504, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "We use a total of 7 citation network datasets. Nodes are documents, and edges are citations. The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 231, + 382, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 382, + 245 + ], + "score": 1.0, + "content": "task for all citation network datasets is to classify each paper’s topic.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 221, + 505, + 245 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "score": 1.0, + "content": "Cora, CiteSeer, PubMed We use three benchmark datasets for semi-supervised node classifica-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "tion tasks in the transductive setting (Sen et al., 2008; Yang et al., 2016). The features of the nodes", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "are bag-of-words representations of documents. We follow the train/validation/test split of previ-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "score": 1.0, + "content": "ous work (Kipf & Welling, 2017). We use 20 samples per class for training, 500 samples for the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 300, + 272, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 272, + 312 + ], + "score": 1.0, + "content": "validation, and 1000 samples for the test.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 255, + 506, + 312 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "Cora-ML, Cora-Full, DBLP These are other citation network datasets from Bojchevski &", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "Gunnemann (2018). Node features are bag-of-words representations of documents. For CoraFull ¨", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "with features more than 5000, we reduce the dimension to 500 by performing PCA. We use the split", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 355, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 370 + ], + "score": 1.0, + "content": "setting in Shchur et al. (2018): 20 samples per class for training, 30 samples per class for validation,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 367, + 186, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 186, + 380 + ], + "score": 1.0, + "content": "the rest for the test.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 323, + 505, + 380 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "ogbn-arxiv The ogbn-arxiv is a recently proposed large-scale dataset of citation networks (Hu", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 402, + 504, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 504, + 414 + ], + "score": 1.0, + "content": "et al., 2020a; Wang et al., 2020). Nodes represent arXiv papers, and edges indicate citations be-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "tween papers, and node features are mean vectors of skip-gram word embeddings of their titles and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 424, + 448, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 448, + 438 + ], + "score": 1.0, + "content": "abstracts. We use the public split by publication dates provided by the original paper.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 391, + 505, + 438 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 448, + 243, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 244, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 244, + 461 + ], + "score": 1.0, + "content": "A.1.2 CO-AUTHOR NETWORK", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "CS, Physics The CS and Physics are co-author networks in each domain (Shchur et al., 2018).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "Nodes are authors, and edges mean whether two authors co-authored a paper. Node features are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "paper keywords from the author’s papers, and we reduce the original dimension (6805 and 8415) to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "500 using PCA. The split is the 20-per-class/30-per-class/rest from Shchur et al. (2018). The goal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 360, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 360, + 526 + ], + "score": 1.0, + "content": "of this task is to classify each author’s respective field of study.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 468, + 506, + 526 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 535, + 247, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 248, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 248, + 548 + ], + "score": 1.0, + "content": "A.1.3 AMAZON CO-PURCHASE", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Photo, Computers The Photo and Computers are parts of the Amazon co-purchase", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "graph (McAuley et al., 2015; Shchur et al., 2018). Nodes are goods, and edges indicate whether", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "two goods are frequently purchased together, and node features are a bag-of-words representation of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "product reviews. The split is the 20-per-class/30-per-class/rest from Shchur et al. (2018). The task", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 256, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 256, + 612 + ], + "score": 1.0, + "content": "is to classify the categories of goods.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 554, + 506, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 622, + 236, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 237, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 237, + 635 + ], + "score": 1.0, + "content": "A.1.4 WEB PAGE NETWORK", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 504, + 697 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 655 + ], + "score": 1.0, + "content": "Wiki-CS The Wiki-CS dataset is computer science related page networks in Wikipedia (Mernyei", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "& Cangea, 2020). Nodes represent articles about computer science, and edges represent hyperlinks", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "between articles. The features of nodes are mean vectors of GloVe word embeddings of articles.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 673, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 687 + ], + "score": 1.0, + "content": "There are 20 standard splits, and we experiment with five random seeds for each split (total 100", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 686, + 339, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 339, + 698 + ], + "score": 1.0, + "content": "runs). The task is to classify the main category of articles.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 641, + 506, + 698 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "Chameleon, Crocodile These datasets are Wikipedia page networks about specific topics,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "Chameleon and Crocodile (Rozemberczki et al., 2019). Nodes are articles, and edges are mutual", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "score": 1.0, + "content": "links between them. Node features are a bag-of-words representation with informative nouns in the", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "article. The number of features is 13183, but we use a reduced dimension of 500 by PCA. The split", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "is 20-per-class/30-per-class/rest from Shchur et al. (2018). The original dataset is for the regression", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 357, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 504, + 368 + ], + "score": 1.0, + "content": "task to predict monthly traffic, but we group values into six bins and make it a classification problem.", + "type": "text", + "cross_page": true + } + ], + "index": 19 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 708, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 210, + 101, + 398, + 302 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 181, + 80, + 429, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 181, + 79, + 429, + 94 + ], + "spans": [ + { + "bbox": [ + 181, + 79, + 429, + 94 + ], + "score": 1.0, + "content": "Table 4: Average degree and homophily of real-world graphs.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 210, + 101, + 398, + 302 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 210, + 101, + 398, + 302 + ], + "spans": [ + { + "bbox": [ + 210, + 101, + 398, + 302 + ], + "score": 0.979, + "html": "
DatasetDegreeHomophily
Four-Univ1.83 ± 1.710.16
PPI28.0 ± 39.260.17
Chameleon15.85 ± 18.200.21
Crocodile15.48 ± 15.970.26
Flickr10.08 ± 31.750.32
Cora-Full6.41 ± 8.790.59
ogbn-arxiv7.68 ± 9.050.63
Wiki-CS26.40 ± 36.040.68
CiteSeer2.78 ± 3.390.72
PubMed4.50 ± 7.430.79
Cora-ML5.45 ± 8.240.81
DBLP5.97 ± 9.350.81
Computers35.76 ± 70.310.81
Cora3.90 ± 5.230.83
CS8.93 ± 9.110.83
Photo31.13 ± 47.270.85
Physics14.38 ± 15.570.91
", + "type": "table", + "image_path": "b9159c1c04784a741ef8d2cffa3058a7116e2fd1ad941983f278094e78f11a0c.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 210, + 101, + 398, + 114.4 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 210, + 114.4, + 398, + 127.80000000000001 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 210, + 127.80000000000001, + 398, + 141.20000000000002 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 210, + 141.20000000000002, + 398, + 154.60000000000002 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 210, + 154.60000000000002, + 398, + 168.00000000000003 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 210, + 168.00000000000003, + 398, + 181.40000000000003 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 210, + 181.40000000000003, + 398, + 194.80000000000004 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 210, + 194.80000000000004, + 398, + 208.20000000000005 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 210, + 208.20000000000005, + 398, + 221.60000000000005 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 210, + 221.60000000000005, + 398, + 235.00000000000006 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 210, + 235.00000000000006, + 398, + 248.40000000000006 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 210, + 248.40000000000006, + 398, + 261.80000000000007 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 210, + 261.80000000000007, + 398, + 275.20000000000005 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 210, + 275.20000000000005, + 398, + 288.6 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 210, + 288.6, + 398, + 302.0 + ], + "spans": [], + "index": 15 + } + ] + } + ], + "index": 4.0 + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 505, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "score": 1.0, + "content": "links between them. Node features are a bag-of-words representation with informative nouns in the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "article. The number of features is 13183, but we use a reduced dimension of 500 by PCA. The split", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "is 20-per-class/30-per-class/rest from Shchur et al. (2018). The original dataset is for the regression", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 357, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 504, + 368 + ], + "score": 1.0, + "content": "task to predict monthly traffic, but we group values into six bins and make it a classification problem.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 382, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 395 + ], + "score": 1.0, + "content": "Four-Univ The Four-Univ dataset is a web page networks from computer science departments", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "of diverse universities (Craven et al., 1998). Nodes are web pages, edges are hyperlinks between", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "them, and node features are TF-IDF vectors of web page’s contents. There are five graphs consists", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "of four universities (Cornell, Texas, Washington, and Wisconsin) and a miscellaneous graph from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "other universities. As the original authors suggested3, we use three graphs of universities and a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 437, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 504, + 449 + ], + "score": 1.0, + "content": "miscellaneous graph for training, another one graph for validation (Cornell), and the other one graph", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "for the test (Texas). Classification labels are types of web pages (student, faculty, staff, department,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 460, + 190, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 190, + 471 + ], + "score": 1.0, + "content": "course, and project).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 485, + 175, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 176, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 176, + 497 + ], + "score": 1.0, + "content": "A.1.5 FLICKR", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "The Flickr dataset is a graph of images from Flickr (McAuley & Leskovec, 2012; Zeng et al., 2020).", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "Nodes are images, and edges indicate whether two images share common properties such as geo-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "graphic location, gallery, and users commented. Node features are a bag-of-words representation of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "score": 1.0, + "content": "images. We use labels and split in in Zeng et al. (2020). For labels, they construct seven classes by", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 548, + 242, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 242, + 562 + ], + "score": 1.0, + "content": "manually merging 81 image tags.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 108, + 574, + 283, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 284, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 284, + 587 + ], + "score": 1.0, + "content": "A.1.6 PROTEIN-PROTEIN INTERACTION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "The protein-protein interaction (PPI) dataset (Zitnik & Leskovec, 2017; Hamilton et al., 2017; Sub-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "ramanian et al., 2005) is a well-known benchmark in the inductive setting. A graph is given for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "score": 1.0, + "content": "human tissue, the nodes are proteins, the node’s features are biological signatures like genes, and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "the edges illustrate proteins’ interactions. The dataset consists of 20 training graphs, two validation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 427, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 427, + 652 + ], + "score": 1.0, + "content": "graphs, and two test graphs. This dataset has multi-labels of gene ontology sets.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 665, + 212, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 214, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 214, + 677 + ], + "score": 1.0, + "content": "A.2 LINK PREDICTION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 222, + 699 + ], + "score": 1.0, + "content": "For link prediction, we split", + "type": "text" + }, + { + "bbox": [ + 222, + 687, + 237, + 697 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 686, + 256, + 699 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 256, + 687, + 276, + 697 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "of edges for validation and test set, respectively. We fix", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 698, + 493, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 493, + 711 + ], + "score": 1.0, + "content": "the negative edges for the test set and sample negative edges for the training set at each iteration.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 115, + 722, + 458, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 720, + 459, + 733 + ], + "spans": [ + { + "bbox": [ + 118, + 720, + 459, + 733 + ], + "score": 1.0, + "content": "3http://www.cs.cmu.edu/afs/cs.cmu.edu/project/theo-20/www/data/", + "type": "text" + } + ] + } + ] + }, + { + "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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 210, + 101, + 398, + 302 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 181, + 80, + 429, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 181, + 79, + 429, + 94 + ], + "spans": [ + { + "bbox": [ + 181, + 79, + 429, + 94 + ], + "score": 1.0, + "content": "Table 4: Average degree and homophily of real-world graphs.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 210, + 101, + 398, + 302 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 210, + 101, + 398, + 302 + ], + "spans": [ + { + "bbox": [ + 210, + 101, + 398, + 302 + ], + "score": 0.979, + "html": "
DatasetDegreeHomophily
Four-Univ1.83 ± 1.710.16
PPI28.0 ± 39.260.17
Chameleon15.85 ± 18.200.21
Crocodile15.48 ± 15.970.26
Flickr10.08 ± 31.750.32
Cora-Full6.41 ± 8.790.59
ogbn-arxiv7.68 ± 9.050.63
Wiki-CS26.40 ± 36.040.68
CiteSeer2.78 ± 3.390.72
PubMed4.50 ± 7.430.79
Cora-ML5.45 ± 8.240.81
DBLP5.97 ± 9.350.81
Computers35.76 ± 70.310.81
Cora3.90 ± 5.230.83
CS8.93 ± 9.110.83
Photo31.13 ± 47.270.85
Physics14.38 ± 15.570.91
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Nodes are web pages, edges are hyperlinks between", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "them, and node features are TF-IDF vectors of web page’s contents. There are five graphs consists", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "of four universities (Cornell, Texas, Washington, and Wisconsin) and a miscellaneous graph from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "other universities. As the original authors suggested3, we use three graphs of universities and a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 437, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 504, + 449 + ], + "score": 1.0, + "content": "miscellaneous graph for training, another one graph for validation (Cornell), and the other one graph", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "for the test (Texas). Classification labels are types of web pages (student, faculty, staff, department,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 460, + 190, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 190, + 471 + ], + "score": 1.0, + "content": "course, and project).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 381, + 506, + 471 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 485, + 175, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 176, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 176, + 497 + ], + "score": 1.0, + "content": "A.1.5 FLICKR", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "The Flickr dataset is a graph of images from Flickr (McAuley & Leskovec, 2012; Zeng et al., 2020).", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "Nodes are images, and edges indicate whether two images share common properties such as geo-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "graphic location, gallery, and users commented. Node features are a bag-of-words representation of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "score": 1.0, + "content": "images. We use labels and split in in Zeng et al. (2020). For labels, they construct seven classes by", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 548, + 242, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 242, + 562 + ], + "score": 1.0, + "content": "manually merging 81 image tags.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 504, + 506, + 562 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 574, + 283, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 284, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 284, + 587 + ], + "score": 1.0, + "content": "A.1.6 PROTEIN-PROTEIN INTERACTION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "The protein-protein interaction (PPI) dataset (Zitnik & Leskovec, 2017; Hamilton et al., 2017; Sub-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "ramanian et al., 2005) is a well-known benchmark in the inductive setting. A graph is given for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "score": 1.0, + "content": "human tissue, the nodes are proteins, the node’s features are biological signatures like genes, and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "the edges illustrate proteins’ interactions. The dataset consists of 20 training graphs, two validation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 427, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 427, + 652 + ], + "score": 1.0, + "content": "graphs, and two test graphs. This dataset has multi-labels of gene ontology sets.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 595, + 505, + 652 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 665, + 212, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 214, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 214, + 677 + ], + "score": 1.0, + "content": "A.2 LINK PREDICTION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 222, + 699 + ], + "score": 1.0, + "content": "For link prediction, we split", + "type": "text" + }, + { + "bbox": [ + 222, + 687, + 237, + 697 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 686, + 256, + 699 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 256, + 687, + 276, + 697 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "of edges for validation and test set, respectively. We fix", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 698, + 493, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 493, + 711 + ], + "score": 1.0, + "content": "the negative edges for the test set and sample negative edges for the training set at each iteration.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 686, + 505, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 100, + 505, + 259 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 217, + 80, + 394, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 215, + 79, + 395, + 93 + ], + "spans": [ + { + "bbox": [ + 215, + 79, + 395, + 93 + ], + "score": 1.0, + "content": "Table 5: Statistics of the real-world datasets.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 100, + 505, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 100, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 505, + 259 + ], + "score": 0.983, + "html": "
Dataset#Nodes#Edges#Features#ClassesSplit#Training Nodes# Val.Nodes#TestNodes
Four-Univ4518342620006fixed4014 (3 Gs)248 (1 G)256 (1G)
PPI5694481871650121fixed44906 (20 Gs)6514 (2 Gs)5524 (2 Gs)
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Crocodile116311800205006random12018011331
Flickr892504498785007fixed446252231222313
Cora-Full197936342150070random1395204916349
ogbn-arxiv169343116624312840fixed909412979948603
Wiki-CS1170129711030010fixed58017695847
CiteSeer3327473237036fixed1205001000
PubMed19717443385003fixed605001000
Cora-ML2995815828797random1402102645
DBLP177165286716394random8012017516
Computers1375224586176710random20030013252
Cora2708542914337fixed1405001000
CS183338189450015random30045017583
Photo76501190817458random1602407250
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Therefore,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "we give some small examples of synthetic datasets to see qualitatively how the average degree and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 225, + 333 + ], + "score": 1.0, + "content": "homophily vary according to", + "type": "text" + }, + { + "bbox": [ + 226, + 321, + 232, + 330 + ], + "score": 0.71, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 320, + 250, + 333 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 250, + 321, + 264, + 332 + ], + "score": 0.88, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 320, + 506, + 333 + ], + "score": 1.0, + "content": ". Specifically, we draw 2D t-SNE plot of node features and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 398, + 344 + ], + "score": 1.0, + "content": "edges in Figure 6. In this figure, we can observe that the average degree", + "type": "text" + }, + { + "bbox": [ + 398, + 331, + 452, + 343 + ], + "score": 0.93, + "content": "( d _ { a v g } = n \\cdot \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 331, + 506, + 344 + ], + "score": 1.0, + "content": ") increases as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 113, + 352 + ], + "score": 0.61, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 342, + 221, + 354 + ], + "score": 1.0, + "content": "increases, and homophily", + "type": "text" + }, + { + "bbox": [ + 222, + 342, + 271, + 354 + ], + "score": 0.91, + "content": "( h = p _ { i n } / \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 342, + 324, + 354 + ], + "score": 1.0, + "content": "increases as", + "type": "text" + }, + { + "bbox": [ + 325, + 344, + 339, + 354 + ], + "score": 0.85, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "increases. Note that these are raw input", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "features sampled from the 2D Gaussian distribution, not learned node representation. We use the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 362, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 506, + 377 + ], + "score": 1.0, + "content": "code from the prior work4 (Abu-El-Haija et al., 2019) and apply normalization by standard score.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 374, + 460, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 152, + 387 + ], + "score": 1.0, + "content": "We choose", + "type": "text" + }, + { + "bbox": [ + 152, + 375, + 159, + 385 + ], + "score": 0.62, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 374, + 182, + 387 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 182, + 375, + 232, + 387 + ], + "score": 0.85, + "content": "\\{ 0 . 0 2 5 , 0 . 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 374, + 236, + 387 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 236, + 375, + 250, + 387 + ], + "score": 0.8, + "content": "p _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 374, + 274, + 387 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 274, + 375, + 345, + 387 + ], + "score": 0.92, + "content": "\\{ 0 . 1 \\delta , 0 . 5 \\delta , 0 . 9 \\delta \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 374, + 379, + 387 + ], + "score": 1.0, + "content": ", and fix", + "type": "text" + }, + { + "bbox": [ + 379, + 375, + 415, + 385 + ], + "score": 0.9, + "content": "n = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 374, + 432, + 387 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 433, + 375, + 456, + 385 + ], + "score": 0.9, + "content": "c = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 374, + 460, + 387 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "image", + "bbox": [ + 124, + 400, + 487, + 578 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 124, + 400, + 487, + 578 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 400, + 487, + 578 + ], + "spans": [ + { + "bbox": [ + 124, + 400, + 487, + 578 + ], + "score": 0.976, + "type": "image", + "image_path": "e3b9079245f2af111484609e3b99820050dc228cc53143bc13cefd45bf64735d.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 124, + 400, + 487, + 459.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 124, + 459.3333333333333, + 487, + 518.6666666666666 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 124, + 518.6666666666666, + 487, + 578.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 592, + 504, + 615 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "score": 1.0, + "content": "Figure 6: t-SNE plots of node features and edges for synthetic graph examples. 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DatasetModel入2EpePn
CoraSuperGATsD0.00782993594510.882669370.80.5
SuperGATMX0.00822886497311.346574530.80.5
CiteSeerSuperGATsD0.048238086570.090739928280.80.3
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SuperGATMX2.19E-0410.45205180.60.5
PPISuperGATsD3.39E-070.00103435184210.5
SuperGATmX1.00E-071.79E-0610.5
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ModelCoraCiteSeerPubMed
GAT*87.2 ± 0.377.3 ± 0.387.0 ± 0.3
CGAT*88.2 ± 0.378.9 ± 0.287.4 ± 0.3
CGAT (Our Impl.: 2-layer w/ 64-features)88.9 ± 0.378.9 ± 0.286.9 ± 0.2
SuperGATMx (2-layer w/ 64-features)88.7 ± 0.279.1 ± 0.287.0 ± 0.1
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To clearly see the difference between the various", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 345, + 414, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 414, + 358 + ], + "score": 1.0, + "content": "graph attention designs, we intentionally keep a simple model configuration.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 504, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 492, + 388 + ], + "score": 1.0, + "content": "Hyperparameter For real-world datasets, we tune two hyperparameters (mixing coefficients", + "type": "text" + }, + { + "bbox": [ + 492, + 375, + 504, + 386 + ], + "score": 0.85, + "content": "\\lambda _ { 2 }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 386, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 123, + 398 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 386, + 137, + 397 + ], + "score": 0.85, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 386, + 506, + 398 + ], + "score": 1.0, + "content": ") by Bayesian optimization for the mean performance of 3 random seeds. We choose negative", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 166, + 410 + ], + "score": 1.0, + "content": "sampling ratio", + "type": "text" + }, + { + "bbox": [ + 166, + 398, + 178, + 408 + ], + "score": 0.85, + "content": "p _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 396, + 201, + 410 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 202, + 396, + 277, + 408 + ], + "score": 0.89, + "content": "\\{ 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 396, + 379, + 410 + ], + "score": 1.0, + "content": ", and edge sampling ratio", + "type": "text" + }, + { + "bbox": [ + 379, + 398, + 390, + 408 + ], + "score": 0.85, + "content": "p _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 396, + 414, + 410 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 414, + 396, + 471, + 408 + ], + "score": 0.89, + "content": "\\{ 0 . 6 , 0 . 8 , 1 . 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 396, + 506, + 410 + ], + "score": 1.0, + "content": ". We fix", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "dropout probability to 0.0 for PPI, 0.2 for ogbn-arxiv, 0.6 for others. We set learning rate to 0.05", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "(ogbn-arxiv), 0.01 (PubMed, PPI, Wiki-CS, Photo, Computers, CS, Physics, Crocodile, Cora-Full,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "DBLP), 0.005 (Cora, CiteSeer, Cora-ML, Chameleon), 0.001 (Four-Univ). For ogbn-arxiv, we set", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "the number of features per head to 16 and the number of heads in the last layer to one; otherwise,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 452, + 359, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 359, + 464 + ], + "score": 1.0, + "content": "we use eight features per head and eight heads in the last layer.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 108, + 467, + 504, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 241, + 481 + ], + "score": 1.0, + "content": "For synthetic datasets, we choose", + "type": "text" + }, + { + "bbox": [ + 241, + 469, + 255, + 479 + ], + "score": 0.89, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 465, + 277, + 481 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 277, + 467, + 453, + 480 + ], + "score": 0.87, + "content": "\\{ 1 0 ^ { - 5 } , 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 1 0 ^ { - 2 } , 1 0 ^ { - 1 } , 1 , 1 0 , 1 0 ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 465, + 470, + 481 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 469, + 482, + 479 + ], + "score": 0.87, + "content": "\\lambda _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 465, + 505, + 481 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 477, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 107, + 478, + 188, + 491 + ], + "score": 0.91, + "content": "\\{ 1 0 ^ { - 7 } , 1 0 ^ { - 5 } , 1 0 ^ { - 3 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 477, + 420, + 493 + ], + "score": 1.0, + "content": ". We fix learning rate to 0.01, dropout probability to 0.2,", + "type": "text" + }, + { + "bbox": [ + 420, + 481, + 432, + 491 + ], + "score": 0.77, + "content": "p _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 477, + 482, + 493 + ], + "score": 1.0, + "content": "to 0.5, and,", + "type": "text" + }, + { + "bbox": [ + 482, + 481, + 493, + 491 + ], + "score": 0.86, + "content": "p _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 477, + 506, + 493 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 490, + 218, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 218, + 503 + ], + "score": 1.0, + "content": "0.8 for all synthetic graphs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "Table 6 describes hyperparameters for SuperGAT on four real-world datasets. For other datasets and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 518, + 370, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 370, + 531 + ], + "score": 1.0, + "content": "experiments, please see the code (./SuperGAT/args.yaml).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 549, + 242, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 243, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 243, + 561 + ], + "score": 1.0, + "content": "A.6 CGAT IMPLEMENTATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 452, + 583 + ], + "score": 1.0, + "content": "CGAT (Wang et al., 2019a) has two auxiliary losses: graph structure based constraint", + "type": "text" + }, + { + "bbox": [ + 452, + 572, + 464, + 583 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 572, + 505, + 583 + ], + "score": 1.0, + "content": "and class", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 583, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 188, + 594 + ], + "score": 1.0, + "content": "boundary constraint", + "type": "text" + }, + { + "bbox": [ + 189, + 583, + 200, + 593 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 583, + 371, + 594 + ], + "score": 1.0, + "content": ". 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In fact, in the CGAT paper, they conduct experiments in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 482, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 482, + 733 + ], + "score": 1.0, + "content": "full-supervised settings; that is, they use all nodes in training except validation and test nodes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + } + ], + "page_idx": 18, + "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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 142, + 101, + 469, + 205 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 173, + 80, + 437, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 173, + 79, + 438, + 93 + ], + "spans": [ + { + "bbox": [ + 173, + 79, + 438, + 93 + ], + "score": 1.0, + "content": "Table 6: Hyperparameters for experiments on real-world datasets.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 142, + 101, + 469, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 142, + 101, + 469, + 205 + ], + "spans": [ + { + "bbox": [ + 142, + 101, + 469, + 205 + ], + "score": 0.983, + "html": "
DatasetModel入2EpePn
CoraSuperGATsD0.00782993594510.882669370.80.5
SuperGATMX0.00822886497311.346574530.80.5
CiteSeerSuperGATsD0.048238086570.090739928280.80.3
SuperGATMX0.041613218320.013081692730.80.5
PubMedSuperGATsD0.000203092756318.825603330.60.7
SuperGATMX2.19E-0410.45205180.60.5
PPISuperGATsD3.39E-070.00103435184210.5
SuperGATmX1.00E-071.79E-0610.5
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ModelCoraCiteSeerPubMed
GAT*87.2 ± 0.377.3 ± 0.387.0 ± 0.3
CGAT*88.2 ± 0.378.9 ± 0.287.4 ± 0.3
CGAT (Our Impl.: 2-layer w/ 64-features)88.9 ± 0.378.9 ± 0.286.9 ± 0.2
SuperGATMx (2-layer w/ 64-features)88.7 ± 0.279.1 ± 0.287.0 ± 0.1
", + "type": "table", + "image_path": "a77d48fd59f9ce3b5cf8313d2e0f38c5fd22084a45cf3b6831ec7df4a40415da.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 132, + 250, + 479, + 269.6666666666667 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 132, + 269.6666666666667, + 479, + 289.33333333333337 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 132, + 289.33333333333337, + 479, + 309.00000000000006 + ], + "spans": [], + "index": 8 + } + ] + } + ], + "index": 5.75 + }, + { + "type": "text", + "bbox": [ + 108, + 334, + 503, + 357 + ], + "lines": [], + "index": 9.5, + "bbox_fs": [ + 105, + 333, + 505, + 358 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 504, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 492, + 388 + ], + "score": 1.0, + "content": "Hyperparameter For real-world datasets, we tune two hyperparameters (mixing coefficients", + "type": "text" + }, + { + "bbox": [ + 492, + 375, + 504, + 386 + ], + "score": 0.85, + "content": "\\lambda _ { 2 }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 386, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 123, + 398 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 386, + 137, + 397 + ], + "score": 0.85, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 386, + 506, + 398 + ], + "score": 1.0, + "content": ") by Bayesian optimization for the mean performance of 3 random seeds. We choose negative", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 166, + 410 + ], + "score": 1.0, + "content": "sampling ratio", + "type": "text" + }, + { + "bbox": [ + 166, + 398, + 178, + 408 + ], + "score": 0.85, + "content": "p _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 396, + 201, + 410 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 202, + 396, + 277, + 408 + ], + "score": 0.89, + "content": "\\{ 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 396, + 379, + 410 + ], + "score": 1.0, + "content": ", and edge sampling ratio", + "type": "text" + }, + { + "bbox": [ + 379, + 398, + 390, + 408 + ], + "score": 0.85, + "content": "p _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 396, + 414, + 410 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 414, + 396, + 471, + 408 + ], + "score": 0.89, + "content": "\\{ 0 . 6 , 0 . 8 , 1 . 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 396, + 506, + 410 + ], + "score": 1.0, + "content": ". We fix", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "dropout probability to 0.0 for PPI, 0.2 for ogbn-arxiv, 0.6 for others. We set learning rate to 0.05", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "(ogbn-arxiv), 0.01 (PubMed, PPI, Wiki-CS, Photo, Computers, CS, Physics, Crocodile, Cora-Full,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "DBLP), 0.005 (Cora, CiteSeer, Cora-ML, Chameleon), 0.001 (Four-Univ). For ogbn-arxiv, we set", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "the number of features per head to 16 and the number of heads in the last layer to one; otherwise,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 452, + 359, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 359, + 464 + ], + "score": 1.0, + "content": "we use eight features per head and eight heads in the last layer.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 375, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 467, + 504, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 241, + 481 + ], + "score": 1.0, + "content": "For synthetic datasets, we choose", + "type": "text" + }, + { + "bbox": [ + 241, + 469, + 255, + 479 + ], + "score": 0.89, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 465, + 277, + 481 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 277, + 467, + 453, + 480 + ], + "score": 0.87, + "content": "\\{ 1 0 ^ { - 5 } , 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 1 0 ^ { - 2 } , 1 0 ^ { - 1 } , 1 , 1 0 , 1 0 ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 465, + 470, + 481 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 469, + 482, + 479 + ], + "score": 0.87, + "content": "\\lambda _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 465, + 505, + 481 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 477, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 107, + 478, + 188, + 491 + ], + "score": 0.91, + "content": "\\{ 1 0 ^ { - 7 } , 1 0 ^ { - 5 } , 1 0 ^ { - 3 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 477, + 420, + 493 + ], + "score": 1.0, + "content": ". We fix learning rate to 0.01, dropout probability to 0.2,", + "type": "text" + }, + { + "bbox": [ + 420, + 481, + 432, + 491 + ], + "score": 0.77, + "content": "p _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 477, + 482, + 493 + ], + "score": 1.0, + "content": "to 0.5, and,", + "type": "text" + }, + { + "bbox": [ + 482, + 481, + 493, + 491 + ], + "score": 0.86, + "content": "p _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 477, + 506, + 493 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 490, + 218, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 218, + 503 + ], + "score": 1.0, + "content": "0.8 for all synthetic graphs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 465, + 506, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "Table 6 describes hyperparameters for SuperGAT on four real-world datasets. For other datasets and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 518, + 370, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 370, + 531 + ], + "score": 1.0, + "content": "experiments, please see the code (./SuperGAT/args.yaml).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 507, + 505, + 531 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 549, + 242, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 243, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 243, + 561 + ], + "score": 1.0, + "content": "A.6 CGAT IMPLEMENTATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 452, + 583 + ], + "score": 1.0, + "content": "CGAT (Wang et al., 2019a) has two auxiliary losses: graph structure based constraint", + "type": "text" + }, + { + "bbox": [ + 452, + 572, + 464, + 583 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 572, + 505, + 583 + ], + "score": 1.0, + "content": "and class", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 583, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 188, + 594 + ], + "score": 1.0, + "content": "boundary constraint", + "type": "text" + }, + { + "bbox": [ + 189, + 583, + 200, + 593 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 583, + 371, + 594 + ], + "score": 1.0, + "content": ". We borrow their notation for this section:", + "type": "text" + }, + { + "bbox": [ + 371, + 583, + 380, + 593 + ], + "score": 0.76, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 583, + 450, + 594 + ], + "score": 1.0, + "content": "as a set of nodes,", + "type": "text" + }, + { + "bbox": [ + 450, + 583, + 462, + 594 + ], + "score": 0.87, + "content": "\\mathbb { N } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 583, + 505, + 594 + ], + "score": 1.0, + "content": "as a set of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 186, + 606 + ], + "score": 1.0, + "content": "one-hop neighbors,", + "type": "text" + }, + { + "bbox": [ + 186, + 594, + 201, + 606 + ], + "score": 0.89, + "content": "\\mathbb { N } _ { i } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 592, + 357, + 606 + ], + "score": 1.0, + "content": "as a set of neighbors that share labels,", + "type": "text" + }, + { + "bbox": [ + 358, + 594, + 373, + 605 + ], + "score": 0.87, + "content": "\\mathbb { N } _ { i } ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "as a set of neighbors that do not", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 158, + 617 + ], + "score": 1.0, + "content": "share labels,", + "type": "text" + }, + { + "bbox": [ + 158, + 605, + 164, + 616 + ], + "score": 0.83, + "content": "\\zeta", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 604, + 334, + 617 + ], + "score": 1.0, + "content": "· as a margin between attention values, and", + "type": "text" + }, + { + "bbox": [ + 334, + 605, + 373, + 617 + ], + "score": 0.92, + "content": "\\phi \\left( v _ { i } , v _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "as unnormalized attention value.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 572, + 506, + 617 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 626, + 435, + 690 + ], + "lines": [ + { + "bbox": [ + 176, + 626, + 435, + 690 + ], + "spans": [ + { + "bbox": [ + 176, + 626, + 435, + 690 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathcal { L } _ { g } = \\displaystyle \\sum _ { i \\in \\mathcal { V } } \\displaystyle \\sum _ { j \\in \\mathbb { N } _ { i } \\setminus \\mathbb { N } _ { i } ^ { - } } \\sum _ { k \\in ( \\mathcal { V } \\setminus \\mathbb { N } _ { i } ) } \\operatorname* { m a x } \\left( 0 , \\phi \\left( v _ { i } , v _ { k } \\right) + \\zeta _ { g } - \\phi \\left( v _ { i } , v _ { j } \\right) \\right) } \\\\ & { \\mathcal { L } _ { b } = \\displaystyle \\sum _ { i \\in \\mathcal { V } } \\sum _ { j \\in \\mathbb { N } _ { i } ^ { + } } \\sum _ { k \\in \\mathbb { N } _ { i } ^ { - } } \\operatorname* { m a x } \\left( 0 , \\phi \\left( v _ { i } , v _ { k } \\right) + \\zeta _ { b } - \\phi \\left( v _ { i } , v _ { j } \\right) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "bdc18dee3eebd9aa6f7b0c94514e28757d07a625598ea87c807cbce8887b6f3c.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 176, + 626, + 435, + 647.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 176, + 647.3333333333334, + 435, + 668.6666666666667 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 176, + 668.6666666666667, + 435, + 690.0000000000001 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Since label information is included in these two losses, they are difficult to use in semi-supervised", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "settings that provide few labeled samples. In fact, in the CGAT paper, they conduct experiments in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 482, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 482, + 733 + ], + "score": 1.0, + "content": "full-supervised settings; that is, they use all nodes in training except validation and test nodes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 699, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 343, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 344, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 172, + 98 + ], + "score": 1.0, + "content": "So, we only use", + "type": "text" + }, + { + "bbox": [ + 172, + 83, + 185, + 95 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 79, + 344, + 98 + ], + "score": 1.0, + "content": "modified for semi-supervised learning.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 99, + 430, + 129 + ], + "lines": [ + { + "bbox": [ + 179, + 99, + 430, + 129 + ], + "spans": [ + { + "bbox": [ + 179, + 99, + 430, + 129 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { g } ^ { \\mathrm { S S L } } = \\sum _ { i \\in \\mathcal { V } } \\sum _ { j \\in \\mathbb { N } _ { i } } \\sum _ { k \\in ( \\mathcal { V } \\backslash \\mathbb { N } _ { i } ) } \\operatorname* { m a x } \\left( 0 , \\phi \\left( v _ { i } , v _ { k } \\right) + \\zeta _ { g } - \\phi \\left( v _ { i } , v _ { j } \\right) \\right)", + "type": "interline_equation", + "image_path": "e81f2265a8a62b545c154da359d998c8e5dc2fe7fbc90b21a0b78edd55af149f.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 179, + 99, + 430, + 109.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 179, + 109.0, + 430, + 119.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 179, + 119.0, + 430, + 129.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 139, + 460, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 139, + 460, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 128, + 153 + ], + "score": 1.0, + "content": "With", + "type": "text" + }, + { + "bbox": [ + 129, + 140, + 140, + 151 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 139, + 460, + 153 + ], + "score": 1.0, + "content": ", the multi-class cross-entropy on node labels, CGAT’s optimization objective is", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 157, + 357, + 171 + ], + "lines": [ + { + "bbox": [ + 253, + 157, + 357, + 171 + ], + "spans": [ + { + "bbox": [ + 253, + 157, + 357, + 171 + ], + "score": 0.93, + "content": "\\mathcal { L } = \\mathcal { L } _ { c } + \\lambda _ { g } \\mathcal { L } _ { g } + \\lambda _ { b } \\mathcal { L } _ { b } ,", + "type": "interline_equation", + "image_path": "263deb5b254c25fb593a2f7fc3e2ce853d11c6b1c430c5f06fe94863042c6ce3.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 253, + 157, + 357, + 171 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 175, + 239, + 187 + ], + "lines": [ + { + "bbox": [ + 106, + 175, + 239, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 239, + 187 + ], + "score": 1.0, + "content": "and our modified CGAT’s loss is", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 266, + 191, + 345, + 207 + ], + "lines": [ + { + "bbox": [ + 266, + 191, + 345, + 207 + ], + "spans": [ + { + "bbox": [ + 266, + 191, + 345, + 207 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { L } = \\mathcal { L } _ { c } + \\lambda _ { g } \\mathcal { L } _ { g } ^ { \\mathrm { S S L } } . } \\end{array}", + "type": "interline_equation", + "image_path": "fc919f51eb7cf286a4398f0fdc0ce69ab31acd658c7020440037c5a575b645e1.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 266, + 191, + 345, + 207 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 504, + 251 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 275, + 231 + ], + "score": 1.0, + "content": "In addition to losses, CGAT proposes top-", + "type": "text" + }, + { + "bbox": [ + 275, + 218, + 282, + 228 + ], + "score": 0.29, + "content": "\\mathbf { \\nabla } \\cdot \\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "softmax and node importance based negative sampling", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 227, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 160, + 241 + ], + "score": 1.0, + "content": "(NINS). Top-", + "type": "text" + }, + { + "bbox": [ + 160, + 229, + 167, + 239 + ], + "score": 0.42, + "content": "\\mathbf { \\nabla } \\cdot \\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 227, + 301, + 241 + ], + "score": 1.0, + "content": "softmax picks up nodes with top-", + "type": "text" + }, + { + "bbox": [ + 302, + 229, + 308, + 238 + ], + "score": 0.63, + "content": "\\mathbf { \\nabla } \\cdot \\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 227, + 506, + 241 + ], + "score": 1.0, + "content": "attention values among neighbors. 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h _ { j , t } ^ { l } ) \\right) } \\\\ & { = \\mathbb { E } ( W _ { k , \\cdot } ^ { 4 } ) \\mathbb { E } \\left( \\displaystyle \\sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right) + \\mathbb { E } ( W _ { k , \\cdot } ^ { 2 } ) ^ { 2 } \\mathbb { E } \\left( \\displaystyle \\sum _ { r } ^ { r } ( h _ { i , r } ^ { l } ) ^ { 2 } \\displaystyle \\sum _ { r } ^ { r } ( h _ { j , r } ^ { l } ) ^ { 2 } - \\displaystyle \\sum _ { r } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "78c23b06e35ecc114379a54f70b71effdabf1355d7b03027cfa3fd2c0b2e3d0a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 78, + 481, + 124.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 124.0, + 481, + 170.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 170.0, + 481, + 216.0 + ], + "spans": [], + "index": 2 + } + ] + }, + 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"score": 0.93, + "content": "e _ { i j , \\mathrm { D P } } ^ { l + 1 }", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 534, + 471, + 716 + ], + "lines": [ + { + "bbox": [ + 138, + 534, + 471, + 716 + ], + "spans": [ + { + "bbox": [ + 138, + 534, + 471, + 716 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathrm { ~ \\forall i a r ~ } ( c _ { \\delta , ( 1 , 1 ) } ^ { i ( 1 , 1 ) } ) } \\\\ & { = \\mathrm { ~ \\operatorname* { W i a r } ~ } ( h _ { i } ^ { i } , h _ { j } ^ { i } ) } \\\\ & { = \\mathrm { ~ \\operatorname* { W i a r } ~ } ( \\sum _ { r ^ { \\prime \\prime } = 1 } ^ { r ^ { \\prime \\prime } + 1 } h _ { i , r } ^ { i } h _ { j , x } ^ { i } ) } \\\\ & { = E ^ { i + 1 } \\mathrm { ~ \\operatorname* { W a r } ~ } ( h _ { i } ^ { j } , h _ { j } ^ { i } ) } \\\\ & { = F ^ { i + 1 } \\mathrm { ~ { \\mathbb { E } } ~ } [ ( h _ { i } ^ { i } h _ { j } ^ { i } ) ^ { 2 } 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{ { \\mathbb { S } } } \\{ ( h _ { i } ^ { i } ) ^ { 2 } h _ { j } ^ { i } \\} ) } \\end{array}", + "type": "interline_equation", + "image_path": "57632c9b644a8152b5789930512298cb6268805641b8d47466acb5879bbca639.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 138, + 534, + 471, + 594.6666666666666 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 138, + 594.6666666666666, + 471, + 655.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 138, + 655.3333333333333, + 471, + 715.9999999999999 + ], + "spans": [], + "index": 13 + } + ] + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 293, + 38 + ], + "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": 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"
Avg.degree Homophily
GCN 0.10.2 0.30.4 0.50.6 0.7 0.8 0.9
1 32.7 ± 1.135.1±0.434.7 ±0.440.8±0.443.4±0.346.2± 0.5 49.7±0.554.8±0.9 56.2±0.3
1.5 28.7± 0.530.0±0.634.6±0.440.8± 0.441.7 ± 0.446.9 ± 0.5 52.1±0.654.5± 0.563.6± 0.5
2.5 24.6±0.227.9 ±0.931.0±0.339.7±0.644.0± 0.651.4± 0.8 56.6±0.565.2 ±0.674.5± 0.4
3.5 25.8±0.830.5±0.831.9± 0.738.4± 0.445.0 ±1.049.8 ± 0.560.8±0.770.3±0.379.5± 0.3
5 25.0±0.326.3± 0.533.0 ± 1.139.8±0.750.2 ±0.558.6 ± 1.168.8±0.378.0 ±1.988.5±0.2
7.5 26.1± 0.529.8±0.634.0±0.643.8±0.952.0±0.470.7±3.274.6± 2.488.6±0.895.0±0.2
10 25.4± 0.629.2 ± 0.537.1± 0.547.6 ± 1.058.1±0.973.1 ± 2.885.3± 2.692.7 ± 1.197.9± 0.5
12.5 24.0±0.427.6±0.539.6±0.848.9±0.465.8 ± 3.177.8 ± 2.488.5±1.595.0 ±0.999.2± 0.0
15 22.1 ± 0.528.2±0.644.0± 0.753.0±0.569.9 ± 4.779.1 ± 2.092.2 ±1.898.0±0.499.6±0.3
20 24.3±0.630.9±0.741.9 ± 0.958.1 ± 1.275.0 ± 2.986.7 ± 1.196.1±0.398.9±0.199.8 ±0.1
25 26.6± 1.031.1± 0.344.9±0.262.4± 0.780.1± 2.691.0 ± 1.597.5± 0.599.5± 0.2100.0 ± 0.0
32.5 26.3± 0.733.8±0.751.8±0.867.5 ± 2.983.5 ± 1.995.7 ± 0.398.4±0.2100.0±0.0100.0±0.0
40 23.7±0.534.2± 0.553.4± 0.472.8 ± 1.587.6± 0.996.1 ± 0.699.7 ± 0.199.9 ± 0.0100.0 ± 0.0
50 25.0 ± 1.136.4± 1.055.1 ± 0.582.7 ± 3.091.2 ± 0.798.6±0.499.8±0.099.9 ±0.0100.0±0.0
75 25.9±0.640.6± 0.668.0 ± 1.087.9± 2.697.1± 0.699.6±0.1100.0 ±0.0100.0 ± 0.0100.0 ±0.0
100 25.1 ± 0.542.0±0.972.4 ± 1.492.6± 0.498.6± 0.2100.0 ±0.0100.0±0.0100.0±0.0100.0±0.0
GATGO 0.10.20.30.40.50.60.70.80.9
1 33.7±0.936.7 ±0.835.2± 1.240.1± 0.743.0± 0.846.3± 0.649.4± 0.553.8 ±1.456.1 ± 0.5
1.5 28.8 ±1.033.0±0.534.8±0.940.7± 0.442.2 ± 0.946.7± 1.052.2 ±0.853.6 ± 1.462.8 ± 0.5
2.5 28.8±1.230.6±0.632.3± 0.540.1 ± 0.844.0 ± 1.352.2± 0.655.5±1.464.6 ± 1.373.1± 0.6
3.5 28.0±0.433.4± 1.633.5± 0.640.1 ± 1.046.2 ± 0.650.8± 0.662.5±0.671.7± 0.678.0 ± 0.5
5 28.0±1.330.0±0.836.9± 0.642.1± 0.453.6± 0.761.0 ± 1.170.4±0.578.7 ±1.287.5± 0.6
7.5 30.0±0.931.6 ± 2.040.4± 0.545.0 ±1.059.5 ± 0.672.3 ± 1.079.4±0.690.4± 0.595.0 ± 0.5
10 31.0 ± 1.434.0 ±1.243.1 ± 0.654.6±0.566.1 ± 0.677.4 ± 1.590.6± 1.494.7 ±0.798.4± 0.5
12.5 29.8 ± 1.435.5± 1.747.3 ± 0.958.6±1.772.1 ± 1.486.3± 0.590.7±0.796.2 ±0.398.9±0.2
15 31.9 ±1.634.6 ± 1.649.5± 0.962.0±0.777.2 ± 1.585.9 ± 0.394.4± 0.998.5± 0.599.4± 0.1
2034.4 ±1.838.3± 1.654.1 ± 1.970.0 ± 1.183.9 ±1.093.7± 0.197.5±0.399.0 ±0.399.7 ± 0.0
2535.8±2.042.0± 2.457.7 ± 0.677.7 ± 0.987.9 ±1.295.0±0.798.7±0.699.6±0.399.9 ± 0.1
32.537.4 ±1.244.7 ± 1.166.5 ± 1.979.9 ± 1.291.4 ± 1.498.0± 0.599.0±0.399.8 ± 0.1100.0 ± 0.0
4037.5± 2.045.1± 0.966.5 ± 1.185.7 ± 1.593.5 ± 1.097.6± 0.699.5± 0.199.9 ± 0.199.9± 0.1
5038.7± 1.849.5± 2.368.9± 2.589.5± 0.896.0± 0.899.0± 0.499.7 ± 0.299.8 ± 0.1100.0 ±0.0
7539.6±3.053.5 ±1.777.6 ± 2.692.8± 2.398.0 ±1.299.5 ± 0.399.8±0.2100.0±0.0100.0 ± 0.0
10041.3 ± 1.256.6 ± 1.481.1 ± 3.595.9 ± 1.299.0 ± 0.499.8 ± 0.199.9 ± 0.1100.0 ± 0.1100.0 ± 0.0
SuperGATsD0.10.20.30.40.50.60.70.80.9
138.5±1.040.5±1.539.5 ±1.642.3±0.944.0± 0.547.1 ± 0.550.1 ± 1.053.0±0.355.1± 1.0
1.536.7 ± 1.237.8 ± 1.842.0± 0.442.9±0.845.0± 0.747.8 ± 0.552.9 ±0.754.7 ± 1.263.0±0.4
2.535.7±1.837.3 ± 1.639.2 ± 1.447.8 ±1.349.0± 0.356.1 ± 0.559.4±0.465.9 ±0.870.6 ± 1.2
3.535.7± 1.039.2 ± 1.941.4 ± 0.544.0±0.751.0± 0.355.0 ± 1.265.1 ± 1.173.5 ± 1.078.1±0.6
537.0± 1.539.0 ± 1.545.0 ±1.748.1 ± 1.057.5± 0.666.1 ± 0.974.4 ± 0.281.0 ± 1.088.3 ± 0.8
7.536.8 ± 1.538.7±0.747.3± 0.551.1 ±0.861.2 ± 0.675.5± 0.582.3±0.991.0±0.295.2± 0.3
1039.4 ± 1.142.3 ±0.750.5± 1.258.6±0.370.0± 0.580.1 ±0.590.8±0.395.5± 0.698.7±0.3
12.538.5± 1.042.0±0.750.5± 0.662.4± 0.573.6± 0.486.8± 0.492.9± 0.498.0± 0.699.3± 0.3
1540.4 ± 1.142.8± 0.753.6± 0.367.4± 0.479.7± 0.389.0 ± 1.096.3±0.399.1 ± 0.299.6± 0.3
2037.8±0.842.1 ± 0.956.9 ±0.970.0 ± 0.586.1 ±0.895.6± 0.399.1± 0.199.6± 0.299.7± 0.1
2540.0± 1.048.8±1.259.5±0.578.7±0.290.7 ±0.297.5± 0.199.4 ± 0.1100.0±0.099.9 ± 0.1
32.539.7 ± 1.148.5±0.769.4± 0.482.7 ± 0.594.3 ±0.299.3± 0.199.7 ± 0.199.9 ± 0.199.9 ±0.0
4044.2 ± 1.148.7± 1.369.2 ± 0.788.3± 0.497.1 ± 0.199.7± 0.199.9±0.099.8± 0.2100.0 ±0.0
5044.3 ± 0.753.2 ± 0.673.4 ±1.291.2 ± 0.497.7 ±0.2100.0 ±0.0100.0 ±0.0100.0±0.0100.0 ±0.0
7544.8± 0.756.1± 0.982.7±0.695.7 ±0.299.8±0.299.9 ± 0.1100.0±0.0100.0±0.0100.0±0.0
10043.1 ± 1.360.7±0.787.4± 0.398.3± 0.199.9 ± 0.0100.0 ±0.0100.0 ±0.0100.0 ±0.0100.0 ± 0.0
SuperGATMX0.10.20.30.40.50.60.70.80.9
133.4±0.736.3±0.835.4±0.740.5±0.343.4±0.647.0±0.850.0±0.554.2 ± 1.156.4±0.4
1.528.8±0.731.5 ± 1.036.0±0.841.1 ± 1.141.3 ±0.647.5± 0.852.3±0.754.8±0.563.4± 0.3
2.526.6±0.930.5±1.333.6±0.842.7 ± 1.544.7 ± 0.552.2 ± 0.556.5 ± 0.866.7 ±0.874.5 ± 0.7
3.527.7± 0.933.9 ± 1.737.9± 0.742.4± 0.649.7 ±0.854.2 ± 1.165.3 ± 1.173.2± 0.779.4 ± 1.5
528.4 ± 2.133.4±0.840.9±0.846.2± 0.456.4±0.264.1 ± 1.874.6± 1.483.1 ± 1.090.6±0.9
7.531.1 ± 0.735.0±0.946.5 ± 1.151.9 ± 0.764.3 ± 1.575.2 ± 1.682.1 ± 1.392.6 ± 1.895.8±0.2
1032.6± 1.741.5 ± 1.448.9 ± 1.261.7 ± 1.170.4 ± 1.381.3 ± 1.191.3±0.795.1 ± 0.899.2± 0.3
12.534.2 ± 3.141.1 ± 0.853.3 ± 1.065.9 ± 1.177.2 ± 1.488.1 ±0.792.6± 2.697.5 ± 0.999.3±0.2
1536.4 ± 1.339.9 ± 2.153.8±1.568.5±0.881.0±0.490.2 ± 1.096.1 ±0.999.2 ± 0.299.6± 0.1
2035.2 ± 3.141.0 ± 2.162.3 ±1.273.9 ± 0.986.3±0.695.2 ± 1.398.6±0.799.6± 0.199.9± 0.2
2536.8 ± 1.946.6± 1.762.7 ± 1.181.9 ± 1.790.7±0.896.8 ± 1.599.3±0.399.8±0.2100.0 ± 0.0
32.536.7± 0.846.8± 0.568.9± 0.683.8±0.793.7 ± 1.098.4±0.799.8± 0.1100.0±0.0100.0 ± 0.0
4039.8± 2.448.8 ±1.772.1 ±0.887.5 ±1.796.6±0.898.9±0.399.9 ± 0.1100.0±0.0100.0±0.0
5042.1 ± 1.353.6± 2.375.1 ± 1.992.7 ± 1.097.1 ± 0.999.6± 0.2100.0±0.0100.0±0.0100.0±0.0
7541.9 ± 1.155.9± 2.480.8±1.295.0 ± 1.199.5± 0.299.9 ±0.1100.0 ±0.0100.0 ±0.0100.0 ±0.0
10039.6± 2.059.2 ± 1.884.2 ± 3.196.9 ± 0.799.7 ± 0.1100.0± 0.0100.0 ±0.0100.0± 0.0100.0±0.0
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Avg.degree Homophily
GCN 0.10.2 0.30.4 0.50.6 0.7 0.8 0.9
1 32.7 ± 1.135.1±0.434.7 ±0.440.8±0.443.4±0.346.2± 0.5 49.7±0.554.8±0.9 56.2±0.3
1.5 28.7± 0.530.0±0.634.6±0.440.8± 0.441.7 ± 0.446.9 ± 0.5 52.1±0.654.5± 0.563.6± 0.5
2.5 24.6±0.227.9 ±0.931.0±0.339.7±0.644.0± 0.651.4± 0.8 56.6±0.565.2 ±0.674.5± 0.4
3.5 25.8±0.830.5±0.831.9± 0.738.4± 0.445.0 ±1.049.8 ± 0.560.8±0.770.3±0.379.5± 0.3
5 25.0±0.326.3± 0.533.0 ± 1.139.8±0.750.2 ±0.558.6 ± 1.168.8±0.378.0 ±1.988.5±0.2
7.5 26.1± 0.529.8±0.634.0±0.643.8±0.952.0±0.470.7±3.274.6± 2.488.6±0.895.0±0.2
10 25.4± 0.629.2 ± 0.537.1± 0.547.6 ± 1.058.1±0.973.1 ± 2.885.3± 2.692.7 ± 1.197.9± 0.5
12.5 24.0±0.427.6±0.539.6±0.848.9±0.465.8 ± 3.177.8 ± 2.488.5±1.595.0 ±0.999.2± 0.0
15 22.1 ± 0.528.2±0.644.0± 0.753.0±0.569.9 ± 4.779.1 ± 2.092.2 ±1.898.0±0.499.6±0.3
20 24.3±0.630.9±0.741.9 ± 0.958.1 ± 1.275.0 ± 2.986.7 ± 1.196.1±0.398.9±0.199.8 ±0.1
25 26.6± 1.031.1± 0.344.9±0.262.4± 0.780.1± 2.691.0 ± 1.597.5± 0.599.5± 0.2100.0 ± 0.0
32.5 26.3± 0.733.8±0.751.8±0.867.5 ± 2.983.5 ± 1.995.7 ± 0.398.4±0.2100.0±0.0100.0±0.0
40 23.7±0.534.2± 0.553.4± 0.472.8 ± 1.587.6± 0.996.1 ± 0.699.7 ± 0.199.9 ± 0.0100.0 ± 0.0
50 25.0 ± 1.136.4± 1.055.1 ± 0.582.7 ± 3.091.2 ± 0.798.6±0.499.8±0.099.9 ±0.0100.0±0.0
75 25.9±0.640.6± 0.668.0 ± 1.087.9± 2.697.1± 0.699.6±0.1100.0 ±0.0100.0 ± 0.0100.0 ±0.0
100 25.1 ± 0.542.0±0.972.4 ± 1.492.6± 0.498.6± 0.2100.0 ±0.0100.0±0.0100.0±0.0100.0±0.0
GATGO 0.10.20.30.40.50.60.70.80.9
1 33.7±0.936.7 ±0.835.2± 1.240.1± 0.743.0± 0.846.3± 0.649.4± 0.553.8 ±1.456.1 ± 0.5
1.5 28.8 ±1.033.0±0.534.8±0.940.7± 0.442.2 ± 0.946.7± 1.052.2 ±0.853.6 ± 1.462.8 ± 0.5
2.5 28.8±1.230.6±0.632.3± 0.540.1 ± 0.844.0 ± 1.352.2± 0.655.5±1.464.6 ± 1.373.1± 0.6
3.5 28.0±0.433.4± 1.633.5± 0.640.1 ± 1.046.2 ± 0.650.8± 0.662.5±0.671.7± 0.678.0 ± 0.5
5 28.0±1.330.0±0.836.9± 0.642.1± 0.453.6± 0.761.0 ± 1.170.4±0.578.7 ±1.287.5± 0.6
7.5 30.0±0.931.6 ± 2.040.4± 0.545.0 ±1.059.5 ± 0.672.3 ± 1.079.4±0.690.4± 0.595.0 ± 0.5
10 31.0 ± 1.434.0 ±1.243.1 ± 0.654.6±0.566.1 ± 0.677.4 ± 1.590.6± 1.494.7 ±0.798.4± 0.5
12.5 29.8 ± 1.435.5± 1.747.3 ± 0.958.6±1.772.1 ± 1.486.3± 0.590.7±0.796.2 ±0.398.9±0.2
15 31.9 ±1.634.6 ± 1.649.5± 0.962.0±0.777.2 ± 1.585.9 ± 0.394.4± 0.998.5± 0.599.4± 0.1
2034.4 ±1.838.3± 1.654.1 ± 1.970.0 ± 1.183.9 ±1.093.7± 0.197.5±0.399.0 ±0.399.7 ± 0.0
2535.8±2.042.0± 2.457.7 ± 0.677.7 ± 0.987.9 ±1.295.0±0.798.7±0.699.6±0.399.9 ± 0.1
32.537.4 ±1.244.7 ± 1.166.5 ± 1.979.9 ± 1.291.4 ± 1.498.0± 0.599.0±0.399.8 ± 0.1100.0 ± 0.0
4037.5± 2.045.1± 0.966.5 ± 1.185.7 ± 1.593.5 ± 1.097.6± 0.699.5± 0.199.9 ± 0.199.9± 0.1
5038.7± 1.849.5± 2.368.9± 2.589.5± 0.896.0± 0.899.0± 0.499.7 ± 0.299.8 ± 0.1100.0 ±0.0
7539.6±3.053.5 ±1.777.6 ± 2.692.8± 2.398.0 ±1.299.5 ± 0.399.8±0.2100.0±0.0100.0 ± 0.0
10041.3 ± 1.256.6 ± 1.481.1 ± 3.595.9 ± 1.299.0 ± 0.499.8 ± 0.199.9 ± 0.1100.0 ± 0.1100.0 ± 0.0
SuperGATsD0.10.20.30.40.50.60.70.80.9
138.5±1.040.5±1.539.5 ±1.642.3±0.944.0± 0.547.1 ± 0.550.1 ± 1.053.0±0.355.1± 1.0
1.536.7 ± 1.237.8 ± 1.842.0± 0.442.9±0.845.0± 0.747.8 ± 0.552.9 ±0.754.7 ± 1.263.0±0.4
2.535.7±1.837.3 ± 1.639.2 ± 1.447.8 ±1.349.0± 0.356.1 ± 0.559.4±0.465.9 ±0.870.6 ± 1.2
3.535.7± 1.039.2 ± 1.941.4 ± 0.544.0±0.751.0± 0.355.0 ± 1.265.1 ± 1.173.5 ± 1.078.1±0.6
537.0± 1.539.0 ± 1.545.0 ±1.748.1 ± 1.057.5± 0.666.1 ± 0.974.4 ± 0.281.0 ± 1.088.3 ± 0.8
7.536.8 ± 1.538.7±0.747.3± 0.551.1 ±0.861.2 ± 0.675.5± 0.582.3±0.991.0±0.295.2± 0.3
1039.4 ± 1.142.3 ±0.750.5± 1.258.6±0.370.0± 0.580.1 ±0.590.8±0.395.5± 0.698.7±0.3
12.538.5± 1.042.0±0.750.5± 0.662.4± 0.573.6± 0.486.8± 0.492.9± 0.498.0± 0.699.3± 0.3
1540.4 ± 1.142.8± 0.753.6± 0.367.4± 0.479.7± 0.389.0 ± 1.096.3±0.399.1 ± 0.299.6± 0.3
2037.8±0.842.1 ± 0.956.9 ±0.970.0 ± 0.586.1 ±0.895.6± 0.399.1± 0.199.6± 0.299.7± 0.1
2540.0± 1.048.8±1.259.5±0.578.7±0.290.7 ±0.297.5± 0.199.4 ± 0.1100.0±0.099.9 ± 0.1
32.539.7 ± 1.148.5±0.769.4± 0.482.7 ± 0.594.3 ±0.299.3± 0.199.7 ± 0.199.9 ± 0.199.9 ±0.0
4044.2 ± 1.148.7± 1.369.2 ± 0.788.3± 0.497.1 ± 0.199.7± 0.199.9±0.099.8± 0.2100.0 ±0.0
5044.3 ± 0.753.2 ± 0.673.4 ±1.291.2 ± 0.497.7 ±0.2100.0 ±0.0100.0 ±0.0100.0±0.0100.0 ±0.0
7544.8± 0.756.1± 0.982.7±0.695.7 ±0.299.8±0.299.9 ± 0.1100.0±0.0100.0±0.0100.0±0.0
10043.1 ± 1.360.7±0.787.4± 0.398.3± 0.199.9 ± 0.0100.0 ±0.0100.0 ±0.0100.0 ±0.0100.0 ± 0.0
SuperGATMX0.10.20.30.40.50.60.70.80.9
133.4±0.736.3±0.835.4±0.740.5±0.343.4±0.647.0±0.850.0±0.554.2 ± 1.156.4±0.4
1.528.8±0.731.5 ± 1.036.0±0.841.1 ± 1.141.3 ±0.647.5± 0.852.3±0.754.8±0.563.4± 0.3
2.526.6±0.930.5±1.333.6±0.842.7 ± 1.544.7 ± 0.552.2 ± 0.556.5 ± 0.866.7 ±0.874.5 ± 0.7
3.527.7± 0.933.9 ± 1.737.9± 0.742.4± 0.649.7 ±0.854.2 ± 1.165.3 ± 1.173.2± 0.779.4 ± 1.5
528.4 ± 2.133.4±0.840.9±0.846.2± 0.456.4±0.264.1 ± 1.874.6± 1.483.1 ± 1.090.6±0.9
7.531.1 ± 0.735.0±0.946.5 ± 1.151.9 ± 0.764.3 ± 1.575.2 ± 1.682.1 ± 1.392.6 ± 1.895.8±0.2
1032.6± 1.741.5 ± 1.448.9 ± 1.261.7 ± 1.170.4 ± 1.381.3 ± 1.191.3±0.795.1 ± 0.899.2± 0.3
12.534.2 ± 3.141.1 ± 0.853.3 ± 1.065.9 ± 1.177.2 ± 1.488.1 ±0.792.6± 2.697.5 ± 0.999.3±0.2
1536.4 ± 1.339.9 ± 2.153.8±1.568.5±0.881.0±0.490.2 ± 1.096.1 ±0.999.2 ± 0.299.6± 0.1
2035.2 ± 3.141.0 ± 2.162.3 ±1.273.9 ± 0.986.3±0.695.2 ± 1.398.6±0.799.6± 0.199.9± 0.2
2536.8 ± 1.946.6± 1.762.7 ± 1.181.9 ± 1.790.7±0.896.8 ± 1.599.3±0.399.8±0.2100.0 ± 0.0
32.536.7± 0.846.8± 0.568.9± 0.683.8±0.793.7 ± 1.098.4±0.799.8± 0.1100.0±0.0100.0 ± 0.0
4039.8± 2.448.8 ±1.772.1 ±0.887.5 ±1.796.6±0.898.9±0.399.9 ± 0.1100.0±0.0100.0±0.0
5042.1 ± 1.353.6± 2.375.1 ± 1.992.7 ± 1.097.1 ± 0.999.6± 0.2100.0±0.0100.0±0.0100.0±0.0
7541.9 ± 1.155.9± 2.480.8±1.295.0 ± 1.199.5± 0.299.9 ±0.1100.0 ±0.0100.0 ±0.0100.0 ±0.0
10039.6± 2.059.2 ± 1.884.2 ± 3.196.9 ± 0.799.7 ± 0.1100.0± 0.0100.0 ±0.0100.0± 0.0100.0±0.0
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ModelCoraCiteSeerPubMed
GAT11.3 ± 2.720.4 ± 6.721.1 ± 2.0
GCN+GAM709.3 ± 235.91099.3 ± 812.56923.3 ± 7042.0
SuperGATMX30.8 ± 0.519.3 ± 1.1151.4 ± 11.4
SuperGATMx+ MPNS20.8 ± 0.215.2 ± 0.184.6 ± 0.9
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As shown in the paper,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "we can see that DP attention does not capture label-agreement rather than GO attention. Also, the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 585, + 385, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 385, + 599 + ], + "score": 1.0, + "content": "degree of this phenomenon becomes stronger as the layer goes down.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 611, + 254, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 255, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 255, + 624 + ], + "score": 1.0, + "content": "B.4 WALL-CLOCK TIME RESULT", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "In Table 9, we report the mean wall-clock time (over three runs) of the training of GAT, GAM,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 644, + 504, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 504, + 656 + ], + "score": 1.0, + "content": "and SuperGATMX. 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There are three observations in this experiment.", + "type": "text" + }, + { + "bbox": [ + 409, + 666, + 466, + 677 + ], + "score": 0.7, + "content": "\\mathrm { G C N + G A M }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "is highly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 252, + 690 + ], + "score": 1.0, + "content": "time-intensive in the training stage", + "type": "text" + }, + { + "bbox": [ + 253, + 677, + 324, + 688 + ], + "score": 0.89, + "content": "( \\times 5 3 . 9 - \\times 3 2 8 . 1 ", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "versus GAT) for all datasets. Compared to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 199, + 701 + ], + "score": 1.0, + "content": "GAT, our model needs", + "type": "text" + }, + { + "bbox": [ + 199, + 688, + 221, + 699 + ], + "score": 0.87, + "content": "\\times 2 . 7", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 686, + 353, + 701 + ], + "score": 1.0, + "content": "more training time for Cora and", + "type": "text" + }, + { + "bbox": [ + 353, + 688, + 375, + 698 + ], + "score": 0.87, + "content": "\\times 7 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "for PubMed, and we reduce the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 332, + 712 + ], + "score": 1.0, + "content": "time by applying multi-processing to negative sampling", + "type": "text" + }, + { + "bbox": [ + 332, + 699, + 356, + 710 + ], + "score": 0.84, + "content": "( \\times 1 . 8", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 698, + 409, + 712 + ], + "score": 1.0, + "content": "for Cora and", + "type": "text" + }, + { + "bbox": [ + 409, + 699, + 431, + 709 + ], + "score": 0.87, + "content": "\\times 4 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "for PubMed). For", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "CiteSeer, we can see that SuperGATMX ends faster than GAT because of faster convergence and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 720, + 164, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 164, + 734 + ], + "score": 1.0, + "content": "fewer epochs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + } + ], + "page_idx": 23, + "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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 81, + 502, + 376 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 81, + 502, + 376 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 81, + 502, + 376 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 502, + 376 + ], + "score": 0.975, + "type": "image", + "image_path": "762ab1e4ccc2a8a192f78d09ad595597707a91b0e7af9118f15205550b082f32.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 81, + 502, + 179.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 179.33333333333331, + 502, + 277.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 277.66666666666663, + 502, + 375.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 385, + 505, + 418 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "Figure 8: Distribution of KL divergence between normalized attention and label-agreement on all", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "nodes and layers for Cora, CiteSeer, PubMed, and PPI (Left: two-layer GAT, Right: four-layer", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 405, + 136, + 421 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 136, + 421 + ], + "score": 1.0, + "content": "GAT).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 141, + 451, + 469, + 510 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 431, + 504, + 443 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "score": 1.0, + "content": "Table 9: Mean wall-clock time (seconds) of three runs of the training process on real-world datasets.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 141, + 451, + 469, + 510 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 141, + 451, + 469, + 510 + ], + "spans": [ + { + "bbox": [ + 141, + 451, + 469, + 510 + ], + "score": 0.977, + "html": "
ModelCoraCiteSeerPubMed
GAT11.3 ± 2.720.4 ± 6.721.1 ± 2.0
GCN+GAM709.3 ± 235.91099.3 ± 812.56923.3 ± 7042.0
SuperGATMX30.8 ± 0.519.3 ± 1.1151.4 ± 11.4
SuperGATMx+ MPNS20.8 ± 0.215.2 ± 0.184.6 ± 0.9
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Performance on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 202, + 481 + ], + "score": 1.0, + "content": "PPI is the largest when", + "type": "text" + }, + { + "bbox": [ + 202, + 469, + 216, + 479 + ], + "score": 0.89, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 468, + 227, + 481 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 227, + 468, + 249, + 479 + ], + "score": 0.9, + "content": "\\bar { 1 } 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 468, + 506, + 481 + ], + "score": 1.0, + "content": ", but the difference is relatively small comparing to others. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "observe that there is an optimal level of the edge supervision for each dataset, and using too large", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 490, + 291, + 502 + ], + "spans": [ + { + "bbox": [ + 107, + 490, + 120, + 501 + ], + "score": 0.88, + "content": "\\lambda _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 491, + 291, + 502 + ], + "score": 1.0, + "content": "degrades node classification performance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 123, + 519 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 509, + 135, + 519 + ], + "score": 0.86, + "content": "p _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 507, + 505, + 519 + ], + "score": 1.0, + "content": ", using too many negative samples has been shown to decrease performance. 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ModelCoraCiteSeerPubMed
GCN+81.570.379.0
GraphSAGE82.1 ± 0.671.9 ± 0.978.0 ± 0.7
CGAT81.4 ± 1.170.1 ± 0.978.1 ±1.0
GLCN+85.572.078.3
LDSt84.175-
GCN + GAM†86.273.586.0
GCN + NSt83.7 ±1.474.1 ± 1.41
GATt83.0±0.772.5 ± 0.779.0± 0.4
SuperGATsD82.70.672.5±0.881.30.5
SuperGATMX84.3*0.672.6±0.881.70.5
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Modelogbn-arxiv CSPhysicsCora-ML Cora-Full DBLPCham.Four-Univ Wiki-CSPhotoComp.FlickrCroco.
GCN33.3±1.291.5±0.292.5±0.285.0±0.459.5±0.277.8±0.533.1±0.974.8±0.674.0±1.091.6±0.684.5±1.451.2±0.432.6±0.4
GraphSAGE54.6±0.390.0±0.192.2±0.183.7±0.459.2±0.278.7±0.641.0±0.974.4±0.677.5±0.590.4±1.183.0±1.450.7±0.253.0±1.0
GAT54.1±0.589.5±0.291.2±0.683.2±0.658.7±0.378.2±1.540.8±0.774.2±0.777.6±0.691.8±0.685.7±0.950.9±0.253.3±1.0
SuperGATsD54.50.388.80.491.6±0.584.50.455.80.679.40.841.60.776.20.877.9±0.786.8±2.582.2±0.945.1±1.253.3±0.9
SuperGATMX55.1±0.290.20.291.90.584.70.459.60280.70.742.00.875.30.677.9±0.591.8±0.9 85.7±1.1 50.8±0.253.3±0.9
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ModelPPI
GCN GraphSAGE61.5 ± 0.4 59.0 ±1.2
CGAT GAT68.3 ± 1.7 72.2 ± 0.6
SuperGATsD74.40.4
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DatasetDegreeHomophily
Four-Univ1.83 ± 1.710.16
PPI28.0 ± 39.260.17
Chameleon15.85 ± 18.200.21
Crocodile15.48 ± 15.970.26
Flickr10.08 ± 31.750.32
Cora-Full6.41 ± 8.790.59
ogbn-arxiv7.68 ± 9.050.63
Wiki-CS26.40 ± 36.040.68
CiteSeer2.78 ± 3.390.72
PubMed4.50 ± 7.430.79
Cora-ML5.45 ± 8.240.81
DBLP5.97 ± 9.350.81
Computers35.76 ± 70.310.81
Cora3.90 ± 5.230.83
CS8.93 ± 9.110.83
Photo31.13 ± 47.270.85
Physics14.38 ± 15.570.91
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Wiki-CS1170129711030010fixed58017695847
CiteSeer3327473237036fixed1205001000
PubMed19717443385003fixed605001000
Cora-ML2995815828797random1402102645
DBLP177165286716394random8012017516
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DatasetModel入2EpePn
CoraSuperGATsD0.00782993594510.882669370.80.5
SuperGATMX0.00822886497311.346574530.80.5
CiteSeerSuperGATsD0.048238086570.090739928280.80.3
SuperGATMX0.041613218320.013081692730.80.5
PubMedSuperGATsD0.000203092756318.825603330.60.7
SuperGATMX2.19E-0410.45205180.60.5
PPISuperGATsD3.39E-070.00103435184210.5
SuperGATmX1.00E-071.79E-0610.5
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ModelCoraCiteSeerPubMed
GAT*87.2 ± 0.377.3 ± 0.387.0 ± 0.3
CGAT*88.2 ± 0.378.9 ± 0.287.4 ± 0.3
CGAT (Our Impl.: 2-layer w/ 64-features)88.9 ± 0.378.9 ± 0.286.9 ± 0.2
SuperGATMx (2-layer w/ 64-features)88.7 ± 0.279.1 ± 0.287.0 ± 0.1
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) } \\\\ & { = E ^ { i + 1 } \\mathrm { ~ \\operatorname* { W a r } ~ } ( h _ { i } ^ { j } , h _ { j } ^ { i } ) } \\\\ & { = F ^ { i + 1 } \\mathrm { ~ { \\mathbb { E } } ~ } [ ( h _ { i } ^ { i } h _ { j } ^ { i } ) ^ { 2 } ) - \\mathrm { ~ { \\mathbb { E } } } ( h _ { i } ^ { i } h _ { j } ^ { j } ) ^ { 2 } ] } \\\\ & { = F ^ { i + 1 } \\mathrm { ~ { \\mathbb { C } } ~ } \\mathrm { e } ^ { i } \\mathrm { ~ { W } ~ } ( \\frac { 4 } { 5 } ( h _ { i } ^ { i } h _ { j } ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } + \\frac { 1 } { 1 0 } \\sum _ { \\phi \\in \\mathbb { R } _ { j } } ( h _ { i , \\phi } ^ { i } h _ { j , + } ^ { i } + h _ { i , } ^ { i } h _ { j , \\phi } ^ { i } ) ^ { 2 } + \\mathrm { { \\mathbb { S } } } ( h _ { i , \\phi } ^ { i } h _ { j , + } ^ { i } ) ^ { 2 } ) } \\\\ & { \\quad + F ^ { i + 1 } \\sigma _ { \\phi } ^ { i } \\mathbb { E } ( \\mathbb { E } ( ( h _ { i } ^ { i } h _ { j } ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } - \\mathbb { E } [ ( h _ { i } ^ { i } h _ { j } ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } ] ) } \\\\ & { \\geq F ^ { i + 1 } \\sigma _ { \\phi } ^ { i } ( \\frac { 4 } { 5 } \\mathbb { E } ( ( h _ { i } ^ { i } h _ { j } ^ { \\top } ) ^ { 2 } h _ { j } ^ { i } ) + \\mathrm { { \\mathbb { S } } } \\{ ( h _ { i } ^ { i } ) ^ { 2 } h _ { j } ^ { i } \\} ) } \\end{array}" + }, + { + "category_id": 13, + "poly": [ + 840, + 1166, + 950, + 1166, + 950, + 1200, + 840, + 1200 + ], + "score": 0.93, + "latex": "\\boldsymbol { \\mathcal { U } } ( - \\boldsymbol { u } , \\boldsymbol { u } )" + }, + { + "category_id": 13, + "poly": [ + 514, + 1443, + 578, + 1443, + 578, + 1485, + 514, + 1485 + ], + "score": 0.93, + "latex": "e _ { i j , \\mathrm { D P } } ^ { l + 1 }" + }, + { + "category_id": 13, + "poly": [ + 1110, + 1165, + 1146, + 1165, + 1146, + 1200, + 1110, + 1200 + ], + "score": 0.9, + "latex": "\\sigma _ { w } ^ { 2 }" + }, + { + "category_id": 14, + "poly": [ + 597, + 1256, + 1101, + 1256, + 1101, + 1423, + 597, + 1423 + ], + "score": 0.84, + "latex": "\\begin{array} { c } { { \\displaystyle \\sigma _ { w } ^ { 2 } = \\frac 1 { 1 2 } ( u - ( - u ) ) ^ { 2 } = \\frac 1 3 u ^ { 2 } } } \\\\ { { \\displaystyle \\mathbb { E } ( W _ { \\cdot , \\cdot } ^ { 4 } ) = \\frac 1 5 \\sum _ { r = 0 } ^ { 4 } ( - u ) ^ { r } u ^ { 4 - r } = \\frac 1 5 u ^ { 4 } = \\frac 9 5 \\sigma _ { w } ^ { 4 } } } \\end{array}" + }, + { + "category_id": 14, + "poly": [ + 298, + 643, + 1391, + 643, + 1391, + 1112, + 298, + 1112 + ], + "score": 0.8, + "latex": "\\begin{array} { r l r } { { + 2 \\mathbb { R } ( W _ { k , * } ^ { 2 } ) ^ { 2 } \\mathbb { R } ( ( \\sum _ { r } h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } - \\sum _ { r } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } ) } } \\\\ & { } & { = \\frac { 9 } { 5 } \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( \\sum _ { r = 1 } ^ { r + 1 } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } ) + \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( \\| h _ { i } ^ { i } \\| _ { 2 } ^ { 2 } \\| h _ { j } ^ { i } \\| _ { 2 } ^ { 2 } - 3 \\sum _ { r = 1 } ^ { r + 1 } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } ) + 2 \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( ( ( h _ { i } ^ { i } ) ^ { \\top } h _ { j , r } ^ { i } ) ^ { 2 } } \\\\ & { } & { ( 3 7 ) } \\\\ & { } & { = \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { R } ( \\| h _ { i } ^ { i } \\| _ { 2 } ^ { 2 } \\| h _ { j } ^ { i } \\| _ { 2 } ^ { 2 } - \\frac { 6 } { 5 } \\sum _ { r = 1 } ^ { r + 1 } ( h _ { i , r } ^ { i } h _ { j , r } ^ { i } ) ^ { 2 } + 2 ( ( h _ { i } ^ { i } ) ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } ) } \\\\ & { } & { ( 3 8 ) } \\\\ & { } & = \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( \\frac { 4 } { 5 } ( h _ { i } ^ { i } ) ^ { \\top } h _ { j } ^ { i } ) ^ { 2 } + \\frac { 1 } { 1 0 } \\sum _ { s \\not = 1 } ^ { r } ( ( h _ { i , s } ^ { i } h _ { j , s } ^ { i } + h _ { i , r } ^ { i } h _ { j , s } ^ { i } ) ^ { 2 } + 8 ( h _ { i , r } ^ { i } h _ { j , s } ^ { i } ) ^ { 2 } ) + \\sigma _ { \\mathrm { w } } ^ { 4 } \\mathbb { E } ( ( h \\end{array}" + }, + { + "category_id": 14, + "poly": [ + 620, + 1204, + 1077, + 1204, + 1077, + 1247, + 620, + 1247 + ], + "score": 0.7, + "latex": "\\mathbb { E } ( W _ { \\cdot , \\cdot } ^ { 2 } ) = \\operatorname { V a r } ( W _ { \\cdot , \\cdot } ) + \\mathbb { E } ( W _ { \\cdot , \\cdot } ) ^ { 2 } = \\sigma _ { w } ^ { 2 }" + }, + { + "category_id": 14, + "poly": [ + 297, + 495, + 1333, + 495, + 1333, + 601, + 297, + 601 + ], + "score": 0.58, + "latex": "= \\mathbb { E } ( W _ { k , \\cdot } ^ { 4 } ) \\mathbb { E } \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right) + \\mathbb { E } ( W _ { k , \\cdot } ^ { 2 } ) ^ { 2 } \\mathbb { E } \\left( \\sum _ { r } ( h _ { i , r } ^ { l } ) ^ { 2 } \\sum _ { r } ( h _ { j , r } ^ { l } ) ^ { 2 } - \\sum _ { r } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right)" + }, + { + "category_id": 14, + "poly": [ + 295, + 219, + 1338, + 219, + 1338, + 599, + 295, + 599 + ], + "score": 0.58, + "latex": "\\begin{array} { r l } & { \\mathbb { E } ( h _ { i , k } ^ { \\ell } h _ { j , k } ^ { \\ell } ) } \\\\ & { = \\mathbb { E } \\left( \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { i , r } ^ { \\ell } \\right) ^ { 2 } \\left( \\sum _ { r = 1 } ^ { F ^ { l + 1 } } W _ { k r } h _ { j , r } ^ { l } \\right) ^ { 2 } \\right) } \\\\ & { = \\mathbb { E } ( W _ { k , \\cdot } ^ { 4 } ) \\mathbb { E } \\left( \\displaystyle \\sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right) + \\mathbb { E } ( W _ { k , \\cdot } ^ { 2 } ) ^ { 2 } \\mathbb { E } \\left( \\displaystyle \\sum _ { s \\neq t } ^ { F ^ { l + 1 } } ( h _ { i , s } ^ { l } h _ { j , t } ^ { l } ) ^ { 2 } + 2 ( h _ { i , s } ^ { l } h _ { j , s } ^ { l } ) ( h _ { i , t } ^ { l } h _ { j , t } ^ { l } ) \\right) } \\\\ & { = \\mathbb { E } ( W _ { k , \\cdot } ^ { 4 } ) \\mathbb { E } \\left( \\displaystyle \\sum _ { r = 1 } ^ { F ^ { l + 1 } } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right) + \\mathbb { E } ( W _ { k , \\cdot } ^ { 2 } ) ^ { 2 } \\mathbb { E } \\left( \\displaystyle \\sum _ { r } ^ { r } ( h _ { i , r } ^ { l } ) ^ { 2 } \\displaystyle \\sum _ { r } ^ { r } ( h _ { j , r } ^ { l } ) ^ { 2 } - \\displaystyle \\sum _ { r } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } \\right) } \\end{array}" + }, + { + "category_id": 14, + "poly": [ + 311, + 638, + 1407, + 638, + 1407, + 860, + 311, + 860 + ], + "score": 0.55, + "latex": "\\begin{array} { r l r } { { + 2 \\mathbb { E } ( W _ { k , \\cdot } ^ { 2 } ) ^ { 2 } \\mathbb { E } ( ( \\sum _ { r } h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } - \\sum _ { r } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } ) } } \\\\ & { } & { = \\frac { 9 } { 5 } \\sigma _ { w } ^ { 4 } \\mathbb { E } ( \\sum _ { r = 1 } ^ { l + 1 } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } ) + \\sigma _ { w } ^ { 4 } \\mathbb { E } ( \\| h _ { i } ^ { l } \\| _ { 2 } ^ { 2 } \\| h _ { j } ^ { l } \\| _ { 2 } ^ { 2 } - 3 \\sum _ { r = 1 } ^ { l + 1 } ( h _ { i , r } ^ { l } h _ { j , r } ^ { l } ) ^ { 2 } ) + 2 \\sigma _ { w } ^ { 4 } \\mathbb { E } ( ( ( h _ { i } ^ { l } ) ^ { \\top } h _ { j } ^ { l } ) ^ { 2 } ) } \\end{array}" + }, + { + "category_id": 15, + "poly": [ + 295.0, + 71.0, + 815.0, + 71.0, + 815.0, + 108.0, + 295.0, + 108.0 + ], + "score": 1.0, + "text": "" + }, + { + "category_id": 15, + "poly": [ + 1378.0, + 2007.0, + 1403.0, + 2007.0, + 1403.0, + 2033.0, + 1378.0, + 2033.0 + ], + "score": 1.0, + "text": "" + }, + { + "category_id": 15, + "poly": [ + 830.0, + 2085.0, + 870.0, + 2085.0, + 870.0, + 2124.0, + 830.0, + 2124.0 + ], + "score": 1.0, + "text": "" + }, + { + "category_id": 15, + "poly": [ + 830.0, + 2085.0, + 870.0, + 2085.0, + 870.0, + 2124.0, + 830.0, + 2124.0 + ], + "score": 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Avg.degree Homophily
GCN 0.10.2 0.30.4 0.50.6 0.7 0.8 0.9
1 32.7 ± 1.135.1±0.434.7 ±0.440.8±0.443.4±0.346.2± 0.5 49.7±0.554.8±0.9 56.2±0.3
1.5 28.7± 0.530.0±0.634.6±0.440.8± 0.441.7 ± 0.446.9 ± 0.5 52.1±0.654.5± 0.563.6± 0.5
2.5 24.6±0.227.9 ±0.931.0±0.339.7±0.644.0± 0.651.4± 0.8 56.6±0.565.2 ±0.674.5± 0.4
3.5 25.8±0.830.5±0.831.9± 0.738.4± 0.445.0 ±1.049.8 ± 0.560.8±0.770.3±0.379.5± 0.3
5 25.0±0.326.3± 0.533.0 ± 1.139.8±0.750.2 ±0.558.6 ± 1.168.8±0.378.0 ±1.988.5±0.2
7.5 26.1± 0.529.8±0.634.0±0.643.8±0.952.0±0.470.7±3.274.6± 2.488.6±0.895.0±0.2
10 25.4± 0.629.2 ± 0.537.1± 0.547.6 ± 1.058.1±0.973.1 ± 2.885.3± 2.692.7 ± 1.197.9± 0.5
12.5 24.0±0.427.6±0.539.6±0.848.9±0.465.8 ± 3.177.8 ± 2.488.5±1.595.0 ±0.999.2± 0.0
15 22.1 ± 0.528.2±0.644.0± 0.753.0±0.569.9 ± 4.779.1 ± 2.092.2 ±1.898.0±0.499.6±0.3
20 24.3±0.630.9±0.741.9 ± 0.958.1 ± 1.275.0 ± 2.986.7 ± 1.196.1±0.398.9±0.199.8 ±0.1
25 26.6± 1.031.1± 0.344.9±0.262.4± 0.780.1± 2.691.0 ± 1.597.5± 0.599.5± 0.2100.0 ± 0.0
32.5 26.3± 0.733.8±0.751.8±0.867.5 ± 2.983.5 ± 1.995.7 ± 0.398.4±0.2100.0±0.0100.0±0.0
40 23.7±0.534.2± 0.553.4± 0.472.8 ± 1.587.6± 0.996.1 ± 0.699.7 ± 0.199.9 ± 0.0100.0 ± 0.0
50 25.0 ± 1.136.4± 1.055.1 ± 0.582.7 ± 3.091.2 ± 0.798.6±0.499.8±0.099.9 ±0.0100.0±0.0
75 25.9±0.640.6± 0.668.0 ± 1.087.9± 2.697.1± 0.699.6±0.1100.0 ±0.0100.0 ± 0.0100.0 ±0.0
100 25.1 ± 0.542.0±0.972.4 ± 1.492.6± 0.498.6± 0.2100.0 ±0.0100.0±0.0100.0±0.0100.0±0.0
GATGO 0.10.20.30.40.50.60.70.80.9
1 33.7±0.936.7 ±0.835.2± 1.240.1± 0.743.0± 0.846.3± 0.649.4± 0.553.8 ±1.456.1 ± 0.5
1.5 28.8 ±1.033.0±0.534.8±0.940.7± 0.442.2 ± 0.946.7± 1.052.2 ±0.853.6 ± 1.462.8 ± 0.5
2.5 28.8±1.230.6±0.632.3± 0.540.1 ± 0.844.0 ± 1.352.2± 0.655.5±1.464.6 ± 1.373.1± 0.6
3.5 28.0±0.433.4± 1.633.5± 0.640.1 ± 1.046.2 ± 0.650.8± 0.662.5±0.671.7± 0.678.0 ± 0.5
5 28.0±1.330.0±0.836.9± 0.642.1± 0.453.6± 0.761.0 ± 1.170.4±0.578.7 ±1.287.5± 0.6
7.5 30.0±0.931.6 ± 2.040.4± 0.545.0 ±1.059.5 ± 0.672.3 ± 1.079.4±0.690.4± 0.595.0 ± 0.5
10 31.0 ± 1.434.0 ±1.243.1 ± 0.654.6±0.566.1 ± 0.677.4 ± 1.590.6± 1.494.7 ±0.798.4± 0.5
12.5 29.8 ± 1.435.5± 1.747.3 ± 0.958.6±1.772.1 ± 1.486.3± 0.590.7±0.796.2 ±0.398.9±0.2
15 31.9 ±1.634.6 ± 1.649.5± 0.962.0±0.777.2 ± 1.585.9 ± 0.394.4± 0.998.5± 0.599.4± 0.1
2034.4 ±1.838.3± 1.654.1 ± 1.970.0 ± 1.183.9 ±1.093.7± 0.197.5±0.399.0 ±0.399.7 ± 0.0
2535.8±2.042.0± 2.457.7 ± 0.677.7 ± 0.987.9 ±1.295.0±0.798.7±0.699.6±0.399.9 ± 0.1
32.537.4 ±1.244.7 ± 1.166.5 ± 1.979.9 ± 1.291.4 ± 1.498.0± 0.599.0±0.399.8 ± 0.1100.0 ± 0.0
4037.5± 2.045.1± 0.966.5 ± 1.185.7 ± 1.593.5 ± 1.097.6± 0.699.5± 0.199.9 ± 0.199.9± 0.1
5038.7± 1.849.5± 2.368.9± 2.589.5± 0.896.0± 0.899.0± 0.499.7 ± 0.299.8 ± 0.1100.0 ±0.0
7539.6±3.053.5 ±1.777.6 ± 2.692.8± 2.398.0 ±1.299.5 ± 0.399.8±0.2100.0±0.0100.0 ± 0.0
10041.3 ± 1.256.6 ± 1.481.1 ± 3.595.9 ± 1.299.0 ± 0.499.8 ± 0.199.9 ± 0.1100.0 ± 0.1100.0 ± 0.0
SuperGATsD0.10.20.30.40.50.60.70.80.9
138.5±1.040.5±1.539.5 ±1.642.3±0.944.0± 0.547.1 ± 0.550.1 ± 1.053.0±0.355.1± 1.0
1.536.7 ± 1.237.8 ± 1.842.0± 0.442.9±0.845.0± 0.747.8 ± 0.552.9 ±0.754.7 ± 1.263.0±0.4
2.535.7±1.837.3 ± 1.639.2 ± 1.447.8 ±1.349.0± 0.356.1 ± 0.559.4±0.465.9 ±0.870.6 ± 1.2
3.535.7± 1.039.2 ± 1.941.4 ± 0.544.0±0.751.0± 0.355.0 ± 1.265.1 ± 1.173.5 ± 1.078.1±0.6
537.0± 1.539.0 ± 1.545.0 ±1.748.1 ± 1.057.5± 0.666.1 ± 0.974.4 ± 0.281.0 ± 1.088.3 ± 0.8
7.536.8 ± 1.538.7±0.747.3± 0.551.1 ±0.861.2 ± 0.675.5± 0.582.3±0.991.0±0.295.2± 0.3
1039.4 ± 1.142.3 ±0.750.5± 1.258.6±0.370.0± 0.580.1 ±0.590.8±0.395.5± 0.698.7±0.3
12.538.5± 1.042.0±0.750.5± 0.662.4± 0.573.6± 0.486.8± 0.492.9± 0.498.0± 0.699.3± 0.3
1540.4 ± 1.142.8± 0.753.6± 0.367.4± 0.479.7± 0.389.0 ± 1.096.3±0.399.1 ± 0.299.6± 0.3
2037.8±0.842.1 ± 0.956.9 ±0.970.0 ± 0.586.1 ±0.895.6± 0.399.1± 0.199.6± 0.299.7± 0.1
2540.0± 1.048.8±1.259.5±0.578.7±0.290.7 ±0.297.5± 0.199.4 ± 0.1100.0±0.099.9 ± 0.1
32.539.7 ± 1.148.5±0.769.4± 0.482.7 ± 0.594.3 ±0.299.3± 0.199.7 ± 0.199.9 ± 0.199.9 ±0.0
4044.2 ± 1.148.7± 1.369.2 ± 0.788.3± 0.497.1 ± 0.199.7± 0.199.9±0.099.8± 0.2100.0 ±0.0
5044.3 ± 0.753.2 ± 0.673.4 ±1.291.2 ± 0.497.7 ±0.2100.0 ±0.0100.0 ±0.0100.0±0.0100.0 ±0.0
7544.8± 0.756.1± 0.982.7±0.695.7 ±0.299.8±0.299.9 ± 0.1100.0±0.0100.0±0.0100.0±0.0
10043.1 ± 1.360.7±0.787.4± 0.398.3± 0.199.9 ± 0.0100.0 ±0.0100.0 ±0.0100.0 ±0.0100.0 ± 0.0
SuperGATMX0.10.20.30.40.50.60.70.80.9
133.4±0.736.3±0.835.4±0.740.5±0.343.4±0.647.0±0.850.0±0.554.2 ± 1.156.4±0.4
1.528.8±0.731.5 ± 1.036.0±0.841.1 ± 1.141.3 ±0.647.5± 0.852.3±0.754.8±0.563.4± 0.3
2.526.6±0.930.5±1.333.6±0.842.7 ± 1.544.7 ± 0.552.2 ± 0.556.5 ± 0.866.7 ±0.874.5 ± 0.7
3.527.7± 0.933.9 ± 1.737.9± 0.742.4± 0.649.7 ±0.854.2 ± 1.165.3 ± 1.173.2± 0.779.4 ± 1.5
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1709.0, + 1038.0, + 1709.0 + ], + "score": 1.0, + "text": "" + } + ], + "page_info": { + "page_no": 24, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/hhU9TEvB6AF/hhU9TEvB6AF.md b/parse/train/hhU9TEvB6AF/hhU9TEvB6AF.md new file mode 100644 index 0000000000000000000000000000000000000000..9b54f1745b5cf8e6de440350ecf8dce3241239d2 --- /dev/null +++ b/parse/train/hhU9TEvB6AF/hhU9TEvB6AF.md @@ -0,0 +1,273 @@ +# Meta Learning Backpropagation And Improving It + +Louis Kirsch1, Jürgen Schmidhuber1,2 +1The Swiss AI Lab IDSIA, University of Lugano (USI) & SUPSI, Lugano, Switzerland +2King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia {louis, juergen}@idsia.ch + +# Abstract + +Many concepts have been proposed for meta learning with neural networks (NNs), e.g., NNs that learn to reprogram fast weights, Hebbian plasticity, learned learning rules, and meta recurrent NNs. Our Variable Shared Meta Learning (VSML) unifies the above and demonstrates that simple weight-sharing and sparsity in an NN is sufficient to express powerful learning algorithms (LAs) in a reusable fashion. A simple implementation of VSML where the weights of a neural network are replaced by tiny LSTMs allows for implementing the backpropagation LA solely by running in forward-mode. It can even meta learn new LAs that differ from online backpropagation and generalize to datasets outside of the meta training distribution without explicit gradient calculation. Introspection reveals that our meta learned LAs learn through fast association in a way that is qualitatively different from gradient descent. + +# 1 Introduction + +The shift from standard machine learning to meta learning involves learning the learning algorithm (LA) itself, reducing the burden on the human designer to craft useful learning algorithms [43]. Recent meta learning has primarily focused on generalization from training tasks to similar test tasks, e.g., few-shot learning [11], or from training environments to similar test environments [17]. This contrasts human-engineered LAs that generalize across a wide range of datasets or environments. Without such generalization, meta learned LAs can not entirely replace human-engineered variants. Recent work demonstrated that meta learning can also successfully generate more general LAs that generalize across wide spectra of environments [20, 1, 31], e.g., from toy environments to Mujoco and Atari. Unfortunately, however, many recent approaches still rely on a large number of human-designed and unmodifiable inner-loop components such as backpropagation. + +Is it possible to implement modifiable versions of backpropagation or related algorithms as part of the end-to-end differentiable activation dynamics of a neural net (NN), instead of inserting them as separate fixed routines? Here we propose the Variable Shared Meta Learning (VSML) principle for this purpose. It introduces a novel way of using sparsity and weight-sharing in NNs for meta learning. We build on the arguably simplest neural meta learner, the meta recurrent neural network (Meta RNN) [16, 10, 56], by replacing the weights of a neural network with tiny LSTMs. The resulting system can be viewed as many RNNs passing messages to each other, or as one big RNN with a sparse shared weight matrix, or as a system learning each neuron’s functionality and its LA. VSML generalizes the principle behind end-to-end differentiable fast weight programmers [45, 46, 3, 41], hyper networks [14], learned learning rules [4, 13, 33], and hebbian-like synaptic plasticity [44, 46, 25, 26, 30]. Our mechanism, VSML, can implement backpropagation solely in the forward-dynamics of an RNN. Consequently, it enables meta-optimization of backproplike algorithms. We envision a future where novel methods of credit assignment can be meta learned while still generalizing across vastly different tasks. This may lead to improvements in sample efficiency, memory efficiency, continual learning, and others. As a first step, our system meta learns online LAs from scratch that frequently learn faster than gradient descent and generalize to datasets outside of the meta training distribution (e.g., from MNIST to Fashion MNIST). Our VSML RNN is the first neural meta learner without hard-coded backpropagation that shows such strong generalization. + +# 2 Background + +Deep learning-based meta learning that does not rely on fixed gradient descent in the inner loop has historically fallen into two categories, 1) Learnable weight update mechanisms that allow for changing the parameters of an NN to implement a learning rule (FWPs / LLRs), and 2) Learning algorithms implemented in black-box models such as recurrent neural networks (Meta RNNs). + +Fast weight programmers & Learned learning rules (FWPs / LLRs) In a standard NN, the weights are updated by a fixed LA. This framework can be extended to meta learning by defining an explicit architecture that allows for modifying these weights. This weight-update architecture augments a standard NN architecture. NNs that generate or change the weights of another or the same NN are known as fast weight programmers (FWPs) [44, 45, 46, 3, 41], hypernetworks [14], NNs with synaptic plasticity [25, 26, 30] or learned learning rules [4, 13, 33]. Often these architectures make use of local Hebbian-like update rules or outer-products, and we summarize this category as FWPs / LLRs. In FWPs / LLRs the variables $V _ { L }$ that are subject to learning are the weights of the network, whereas the meta-variables $V _ { M }$ that implement the LA are defined by the weight-update architecture. Note that the dimensionality of $V _ { L }$ and $V _ { M }$ can be defined independently of each other and often $V _ { M }$ are reused in a coordinate-wise fashion for $V _ { L }$ resulting in $| V _ { L } | \gg | V _ { M } |$ , where $| \cdot |$ is the number of elements. + +Black-box learning in activations (Meta RNNs) It was shown that an RNN such as LSTM can learn to implement an LA [16] when the reward or error is given as an input [47]. After meta training, the LA is encoded in the weights of this RNN and determines learning during meta testing. The activations serve as the memory used for the LA solution. We refer to this as Meta RNNs [16, 10, 56] (Also sometimes referred to as memory-based meta learning.). They are conceptually simpler than FWPs / LLRs as no additional weight-update rules with many degrees of freedom need to be defined. In Meta RNNs $V _ { L }$ are the RNN activations and $V _ { M }$ are the parameters for the RNN. Note that an RNN with $N$ neurons will yield $| V _ { L } | = O ( N )$ and $| V _ { M } | = \stackrel { \cdot } { O } ( N ^ { 2 } )$ [46]. This means that the LA is largely overparameterized whereas the available memory for learning is very small, making this approach prone to overfitting [20]. As a result, the RNN parameters often encode task-specific solutions instead of generic LAs. Meta learning a simple and generalizing LA would benefit from $| V _ { L } | \gg | V _ { M } |$ . Previous approaches have tried to mend this issue by adding architectural complexity through additional memory mechanisms [53, 29, 40, 27, 42]. + +# 3 Variable Shared Meta Learning (VSML) + +In VSML we build on the simplicity of Meta RNNs while retaining $| V _ { L } | \gg | V _ { M } |$ from FWPs / LLRs. We do this by reusing the same few parameters $V _ { M }$ many times in an RNN (via variable sharing) and introducing sparsity in the connectivity. This yields several interpretations for VSML: + +$\textcircled{8}$ VSML as a single Meta RNN with a sparse shared weight matrix (Figure 1a). The most general description. +$\textcircled{8}$ VSML as message passing between RNNs (Figure 1b). We choose a simple sharing and sparsity scheme for the weight matrix such that it corresponds to multiple RNNs with shared parameters that exchange information. +$©$ VSML as complex neurons with learned updates (Figure 1c). When choosing a specific connectivity between RNNs, states / activations $V _ { L }$ of these RNNs can be interpreted as the weights of a conventional NN, consequently blurring the distinction between a weight and an activation. + +Introducing variable sharing to Meta RNNs We begin by formalizing Meta RNNs which often use multiplicative gates such as the LSTM [12, 15] or its variant GRU [6]. For notational simplicity, we consider a vanilla RNN. Let $s \in \mathbb { R } ^ { N }$ be the hidden state of an RNN. The update for an element $j \in \{ 1 , \ldots , N \}$ is given by + +![](images/b1ae57af27d23bff08bcfc6f114db04ece9a7f8b38923249b2e9e1b42ad6a50f.jpg) +Figure 1: Different perspectives on VSML: (a) A single Meta RNN [16] where entries in the weight matrix are shared or zero. (a) VSML consists of many sub-RNNs with shared parameters $V _ { M }$ passing messages between each other. (c) VSML implements an NN with complex neurons (here 2 neurons). $V _ { M }$ determines the nature of weights, how these are used in the neural computation, and the LA by which those are updated. Each weight $w _ { a b } \in \mathbb { R }$ is represented by the multi-dimensional RNN state ${ s _ { a b } } \in \mathbb { R } ^ { N }$ . Neuron activations correspond to messages $\overrightarrow { m }$ passed between sub-RNNs. + +$$ +s _ { j } \gets f _ { \mathrm { R N N } } ( s ) _ { j } = \sigma ( \sum _ { i } s _ { i } W _ { i j } ) , +$$ + +where $\sigma$ is a non-linear activation function, $W \in \mathbb { R } ^ { N \times N }$ , and the bias is omitted for simplicity. We also omit inputs by assuming a subset of $s$ to be externally provided. Each application of Equation 1 reflects a single time tick in the RNN. + +We now introduce variable sharing (reusing $W$ ) into the RNN by duplicating the computation along two axes of size $A , B$ (here $A = B$ , which will later be relaxed) giving $\bar { s } \in \mathbb { R } ^ { A \times B \times N }$ . For $a \in \{ 1 , \dotsc , A \} , b \in \{ 1 , \dotsc , B \}$ we have + +$$ +s _ { a b j } \gets f _ { \mathrm { R N N } } ( s _ { a b } ) _ { j } = \sigma ( \sum _ { i } s _ { a b i } W _ { i j } ) . +$$ + +This can be viewed as multiple RNNs arranged on a 2-dimensional grid, with shared parameters that update independent states. Here, we chose a particular arrangement (two axes) that will facilitate the interpretation $©$ of RNNs as weights. + +VSML as message passing between RNNs The computation so far describes $A \cdot B$ independent RNNs. We connect those by passing messages (interpretation $\textcircled{8}$ ) + +$$ +s _ { a b } \gets f _ { \mathrm { R N N } } ( s _ { a b } , \overrightarrow { m } _ { a } ) , +$$ + +where the message $\begin{array} { r } { \vec { m } _ { b } = \sum _ { a ^ { \prime } } f _ { \vec { m } } ( s _ { a ^ { \prime } b } ) } \end{array}$ with $b \in \{ 1 , \dots , A = B \}$ , $f _ { \overrightarrow { m } } : \mathbb { R } ^ { N } \mathbb { R } ^ { N ^ { \prime } }$ is fed as an additional input to each RNN. This is related to Graph Neural Networks [51, 58]. Summing over the axis $A$ (elements $a ^ { \prime }$ ) corresponds to an RNN connectivity mimicking those of weights in an NN (to facilitate interpretation $©$ . We emphasise that other schemes based on different kinds of message passing and graph connectivity are possible. For a simple $f _ { \overrightarrow { m } }$ defined by the matrix $C \in \mathbb { R } ^ { N \times N }$ , we may equivalently write + +$$ +s _ { a b j } \gets \sigma ( \sum _ { i } s _ { a b i } W _ { i j } + \sum _ { a ^ { \prime } } f _ { \overrightarrow { m } } ( s _ { a ^ { \prime } a } ) ) = \sigma ( \sum _ { i } s _ { a b i } W _ { i j } + \sum _ { a ^ { \prime } , i } s _ { a ^ { \prime } a i } C _ { i j } ) . +$$ + +This constitutes the minimal version of VSML with $V _ { M } : = ( W , C )$ and is visualized in Figure 1b. + +VSML as a Meta RNN with a sparse shared weight matrix It is trivial to see that with $A = 1$ and $B = 1$ we obtain a single RNN and Equation 4 recovers the original Meta RNN Equation 1. In the general case, we can derive an equivalent formulation that corresponds to a single standard RNN with a single matrix $\tilde { W }$ that has entries of zero and shared entries + +$$ +s _ { a b j } \gets \sigma ( \sum _ { c , d , i } s _ { c d i } \tilde { W } _ { c d i a b j } ) , +$$ + +where the six axes can be flattened to obtain the two axes. For Equation 4 and Equation 5 to be equivalent, $\tilde { W }$ must satisfy (derivation in Appendix A) + +$$ +\tilde { W } _ { c d i a b j } = \left\{ \begin{array} { l l } { C _ { i j } , } & { \mathrm { i f ~ } d = a \wedge ( d \neq b \vee c \neq a ) . } \\ { W _ { i j } , } & { \mathrm { i f ~ } d \neq a \wedge d = b \wedge c = a . } \\ { C _ { i j } + W _ { i j } , } & { \mathrm { i f ~ } d = a \wedge d = b \wedge c = a . } \\ { 0 , } & { \mathrm { o t h e r w i s e . } } \end{array} \right. +$$ + +This corresponds to interpretation $\textcircled{8}$ with the weight matrix visualized in Figure 1a. To distinguish between the single sparse shared RNN and the connected RNNs, we now call the latter sub-RNNs. + +VSML as complex neurons with learned updates The arrangement and connectivity of the subRNNs as described in the previous paragraphs corresponds to that of weights in a standard NN. Thus, in interpretation $@$ , VSML can be viewed as defining complex neurons where each sub-RNN corresponds to a weight in a standard NN as visualized in Figure 1c. All these sub-RNNs share the same parameters but have distinct states. The current formulation corresponds to a single NN layer that is run recurrently. We will generalize this to other architectures in the next section. A corresponds to the dimensionality of the inputs and $B$ to that of the outputs in that layer. + +The role of weights in a standard neural network is now assigned to the states of RNNs. This allows these RNNs to define both the neural forward computation as well as the learning algorithm that determines how the network is updated (where the mechanism is shared across the network). In the case of backpropagation, this would correspond to the forward and backward passes being implemented purely in the recurrent dynamics. We will demonstrate the practical feasibility of this in Section 3.2. The emergence of + +![](images/0ccd0fdb8442f57b99da7d0266c5c9a62738c64d643b90443c68ed18becb66cf.jpg) +Figure 2: The neural interpretation of VSML replaces all weights of a standard NN with tiny LSTMs using shared parameters (resembling complex neurons). This allows these LSTMs to define both the neural forward computation as well as the learning algorithm that determines how the network is updated. Information flows forward and backward in the network through multi-dimensional messages $\overrightarrow { m }$ and $\scriptstyle { \overleftarrow { m } }$ , generalizing the dynamics of an NN trained using backpropagation. + +RNN states as weights quickly leads to confusing terminology when RNNs have ‘meta weights’. +Instead, we simply refer to meta variables $V _ { M }$ and learned variables $V _ { L }$ . + +With this interpretation, VSML can be seen as a generalization of learned learning rules [4, 13, 33] and Hebbian-like differentiable mechanisms or fast weights more generally [44, 46, 25, 26] where RNNs replace explicit weight updates. In standard NNs, weights and activations have multiplicative interactions. For VSML RNNs to mimic such computation we require multiplicative interactions between parts of the state $s$ . Fortunately, LSTMs already incorporate this through gating and can be directly used in place of RNNs. + +Stacking VSML RNNs and feeding inputs To get a structure similar to one of the non-recurrent deep feed-forward architectures (FNNs), we stack multiple VSML RNNs where their states are untied and their parameters are tied.1 This is visualized with two layers in Figure 2 where the states $s ^ { ( 2 ) }$ of the second column of sub-RNNs are distinct from the first column $\mathbf { \boldsymbol { s } } ^ { ( 1 ) }$ . The parameters $A ^ { ( k ) }$ and $B ^ { ( k ) }$ describing layer sizes can then be varied for each layer $k \in \{ 1 , \ldots , K \}$ constrained by $A ^ { ( k ) } ~ = ~ B ^ { ( k - 1 ) }$ . The updated Equation 3 with distinct layers $k$ is given by $s _ { a b } ^ { ( k ) } f _ { \mathrm { R N N } } ( s _ { a b } ^ { ( k ) } , \overrightarrow { m } _ { a } ^ { ( k ) } )$ where $\begin{array} { r } { \vec { m } _ { b } ^ { ( k + 1 ) } : = \sum _ { a ^ { \prime } } \bar { f } _ { \vec { m } } ( s _ { a ^ { \prime } b } ^ { ( k ) } ) } \end{array}$ with $b \in \{ 1 , \ldots , B ^ { ( k ) } = A ^ { ( k + 1 ) } \}$ + +To prevent information from flowing only forward in the network, we use an additional backward message + +$$ +s _ { a b } ^ { ( k ) } \gets f _ { R N N } \big ( s _ { a b } ^ { ( k ) } , \overrightarrow { m } _ { a } ^ { ( k ) } , \overleftarrow { m } _ { b } ^ { ( k ) } \big ) , +$$ + +where $\begin{array} { r } { \overleftarrow { m } _ { a } ^ { ( k - 1 ) } : = \sum _ { b ^ { \prime } } f _ { \overleftarrow { m } } ( s _ { a b ^ { \prime } } ^ { ( k ) } ) } \end{array}$ with $a \in$ $\{ 1 , . . . , A ^ { ( k ) } = B ^ { ( k - 1 ) } \}$ (visualized in Figure 3). The backward transformation is given by f←−m : RN → RN 00 . + +Similarly, other neural architectures can be explicitly constructed (e.g. convolutional NNs, Section B.2). Some architectures may be learned implicitly if positional information is fed into each sub-RNN (Appendix C). We then update all states $s ^ { ( k ) }$ in sequence $1 , \ldots , K$ to mimic sequential layer execution. We may also apply multiple RNN ticks for each layer $k$ . + +![](images/a5613155d304964c2e8b7d7e753eadbc7c640e13696034612df8983dd1ae3ff5.jpg) +Figure 3: VSML with forward messages $\overrightarrow { m }$ and backward messages $\scriptstyle { \overleftarrow { m } }$ to form a two-layer NN with shared LSTM parameters but distinct states. + +To provide the VSML RNN with data, each time we execute the operations of the first layer, a single new datum $x \in \mathbb { R } ^ { A ( 1 ) }$ (e.g. one flattened image) is distributed across all sub-RNNs. In our present experiments, we match the axis $A ( 1 )$ to the input datum dimensionality such that each dimension (e.g., pixel) is fed to different RNNs. This corresponds to initializing the forward message $\vec { m } _ { a 1 } ^ { ( 1 ) } : = \boldsymbol { x } _ { a }$ dding . Fina $\overrightarrow { m }$ with zeros if nece we feed the error larly, we read the outpuat the output such that $\boldsymbol { \hat { y } } \in \mathbb { R } ^ { B ( K ) }$ from. See $\hat { y } _ { a } : = \overrightarrow { m } _ { a 1 } ^ { ( K + 1 ) }$ $e \in \mathbb { R } ^ { B ( K ) }$ $ _ { b 1 } : = e _ { b }$ Figure 2 for a visualization. Alternatively, multiple input or output dimensions could be patched together and fed into fewer sub-RNNs. + +# 3.1 Meta learning general-purpose learning algorithms from scratch + +Having formalized VSML, we can now use end-to-end meta learning to create LAs from scratch in Algorithm 1. We simply optimize the LSTM parameters $V _ { M }$ to minimize the sum of prediction losses over many time steps starting with random states $V _ { L } : = \{ s _ { a b } ^ { ( k ) } \}$ . We focus on meta learning online LAs where one example is fed at a time as done in Meta RNNs [16, 56, 10]. Meta training may be performed using end-to-end gradient descent or gradient-free optimization such as evolutionary strategies [57, 38]. The latter is significantly more efficient on VSML compared to standard NNs due to the small parameter space $V _ { M }$ . Crucially, during meta testing, no explicit gradient descent is used. + +# Algorithm 1 VSML: Meta Training + +Require: Dataset(s) $D = \{ ( x _ { i } , y _ { i } ) \}$ $V _ { M } \gets$ initialize LSTM parameters while meta loss has not converged do . Outer loop in parallel over datasets $D$ ${ \cal V } _ { L } = \{ s _ { a b } ^ { ( k ) } \} $ initialize LSTM states $\forall a , b , k$ for $( x , y ) ^ { \top } \in \{ ( x _ { 1 } , y _ { 1 } ) , \dotsc , ( x _ { T } , y _ { T } ) \} \subset D$ do $\triangleright$ Inner loop over $T$ examples $ { \vec { m } } _ { a 1 } ^ { ( 1 ) } : = x _ { a }$ ∀a $\triangleright$ Initialize from input image $\mathbf { X }$ for $\bar { k } \in \{ 1 , \ldots , K \}$ do $\triangleright$ Iterating over $K$ layers $s _ { a b } ^ { ( k ) } \gets f _ { R N N } \big ( s _ { a b } ^ { ( k ) } , \overrightarrow { m } _ { a } ^ { ( k ) } , \overleftarrow { m } _ { b } ^ { ( k ) } \big ) \quad \forall a , b$ . Equation 7 ←−m m (k+1)b := Pa0 f−→m (s(k)a0b) (k−1)a := Pb0 f←−m (s(k)ab0 ) ∀a ∀b . Create backward message . Create forward message yˆa := −→m(K+1)a1 ∀a $\triangleright$ Read output e := ∇yˆL(ˆy, y) . Compute error at outputs using loss $L$ ←−m(K)b1 := eb ∀b $\triangleright$ Input errors $\begin{array} { r } { V _ { M } \gets V _ { M } - \alpha \nabla _ { V _ { M } } \sum _ { t = 1 } ^ { T } L ( \hat { y } ( t ) , y ( t ) ) } \end{array}$ , obtaining $\nabla _ { V _ { M } }$ either by • back-propagation through the inner loop • evolution strategies, using a search distribution $p ( V _ { M } )$ + +# 3.2 Learning to implement backpropagation in RNNs + +![](images/c33778dd74e4643cf3ee884dd15c7cd4f5d7da2efc2ed6fb055153758c4435ca.jpg) +Figure 4: To implement backpropagation we optimize the VSML RNN to use and update weights $w$ and biases $b$ as part of the state $s _ { a b }$ in each subRNN. Inputs are pre-synaptic $x$ and error $e$ . Outputs are post-synaptic $\hat { y }$ and error $\cdot$ . + +An alternative to end-to-end meta learning is to first investigate whether the VSML RNN can implement backpropagation. Due to the algorithm’s ubiquitous use, it seems desirable to be able to meta learn backpropagation-like algorithms. Here we investigate how VSML RNNs can learn to implement backpropagation purely in their recurrent dynamics. We do this by optimizing $V _ { M }$ to (1) store a weight $w$ and bias $b$ as a subset of each state $s _ { a b }$ , (2) compute $y = \operatorname { t a n h } ( x ) w + b$ to implement neural forward computation, and (3) update $w$ and $b$ according to the backpropagation algorithm [23]. We call this process learning algorithm cloning and it is visualized in Figure 4. + +We designate an element of each message $\vec { m } _ { a } ^ { ( k ) }$ , $\{ \overline { { m } } _ { b } ^ { ( k ) }$ , $f _ { \vec { m } } ( s _ { a b } ^ { ( k ) } )$ $f _ { \overleftarrow { m } } ( s _ { a b } ^ { ( k ) } )$ as the input $x$ , error $e$ , and output $\hat { y }$ and error $\hat { e } ^ { \prime }$ . Similarly, we set $w : = s _ { a b 1 }$ and $b : = s _ { a b 2 }$ . We then optimize $V _ { M }$ via gradient descent to regress $\hat { y }$ , $\Delta w$ , $\Delta b$ , and $\hat { e } ^ { \prime }$ toward their respective targets. We can either generate the training dataset $D : = \{ ( x , \bar { w } , b , y , e , \bar { e ^ { \prime } } ) _ { i } \}$ randomly or run a ‘shadow’ NN on some supervised problem and fit the VSML RNN to its activations and parameter updates. Multiple iterations in the VSML RNN would then correspond to evaluating the network and updating it via backpropagation. The activations from the forward pass necessary for credit assignment could be memorized as part of the state $s$ or be explicitly stored and fed back. For simplicity, we chose the latter to clone backpropagation. We continuously run the VSML RNN forward, alternately running the layers in order $1 , \ldots , K$ and in reverse order $K , \ldots , 1$ .2 + +# 4 Experiments + +First, we demonstrate the capabilities of the VSML RNN by showing that it can implement neural forward computation and backpropagation in its recurrent dynamics on the MNIST [21] and Fashion MNIST [59] dataset. Then, we show how we can meta learn an LA from scratch on one set of datasets and then successfully apply it to another (out of distribution). Such generalization is enabled by extensive variable sharing where we have very few meta variables $| V _ { M } | ~ \approx ~ 2 , 4 0 0$ and many learned variables $| V _ { L } | \approx 2 5 7 , 2 0 0$ . We also investigate the robustness of the discovered LA. Finally, we introspect the meta learned LA and compare it to gradient descent. + +![](images/dbb2f1c0cc8cade0dc1f709d779c51876fa020bd13a4a622b6679683d6100ede.jpg) +Figure 5: The VSML RNN is optimized to implement backpropagation in its recurrent dynamics on MNIST, then tested both on MNIST and Fashion MNIST where it performs learning purely by unrolling the LSTM. We test on shallow and deep architectures (single hidden layer of 32 units). Standard deviations are over 6 seeds. + +Our implementation uses LSTMs and the message interpretation from Equation 7. Hyperparameters, training details, and additional experiments can be found in the appendix. + +# 4.1 VSML RNNs can implement backpropagation + +As described in Section 3.2, we optimize the VSML RNN to implement backpropagation. We structure the sub-RNNs to mimic a feed-forward NN with either one hidden layer or no hidden layers. To obtain training targets, we instantiate a standard NN, the shadow network, and feed it MNIST data. After cloning, we then run the LA encoded in the VSML RNN on the MNIST and Fashion MNIST dataset and observe that it performs learning purely in its recurrent dynamics, making explicit gradient calculations unnecessary. Figure 5 shows the learning curve on these two datasets. Notably, learning works both on MNIST (within distribution) and on Fashion MNIST (out of distribution). We observe that the loss is decently minimized, albeit regular gradient descent still outperforms our cloned backpropagation. This may be due to non-zero errors during learning algorithm cloning, in particular when these errors accumulate in the deeper architecture. It is also possible that the VSML states (‘weights’) deviate too far from ranges seen during cloning, in particular in the deep case when the loss starts increasing. We obtain $87 \%$ (deep) and $90 \%$ (shallow) test accuracy on MNIST and $76 \%$ (deep) and $80 \%$ (shallow) on Fashion MNIST (focusing on successful cloning over performance). + +# 4.2 Meta learning supervised learning from scratch + +In the previous experiments, we have established that VSML is expressive enough to metaoptimize backpropagation-like algorithms. Instead of cloning an LA, we now meta learn from scratch as described in Section 3.1. Here, we use a single layer ( $K = 1$ ) from input to output dimension and run it for two ticks per image with $N = 1 6$ and $N ^ { \prime } = N ^ { \prime \prime } = 8$ . First, the VSML RNN is meta trained end-to-end using evolutionary strategies (ES) [38] on MNIST to minimize the sum of cross-entropies over 500 data points starting from random state initializations. As each image is unique and $V _ { M }$ can not memorize the data, we are implicitly optimizing the VSML RNN to generalize to future inputs given all inputs it has seen so far. We do not pre-train $V _ { M }$ with a human-engineered LA. + +![](images/7b033cbe1d2c5cc6f6e774fb5334f6d5b86759ee6986d192914ef27be40db278.jpg) +Figure 6: The VSML RNN can be directly meta trained on MNIST to minimize the sum of errors when classifying online starting from a random state initialization. This allows for faster learning during meta testing compared to online gradient descent with Adam on the same dataset and even generalizes to a different dataset (Fashion MNIST). In comparison, a standard Meta RNN [16] strongly overfits in the same setting. Standard deviations are over 128 seeds. + +During meta testing on MNIST (Figure 6) we plot the cumulativputs on the y axis $\textstyle { \bigl ( } { \frac { 1 } { T } } \sum _ { t = 1 } ^ { T } c _ { t }$ n all previous after example $T$ + +indicating prediction correctness). For each example, the prediction when this example was fed to the RNN is used, thus measuring sample efficient learning. This evaluation protocol is similar to the one used in Meta RNNs [56, 10]. We observe that learning is considerably faster compared to the baseline of online gradient descent (no mini batching, the learning rate appropriately tuned). One possibility is that VSML simply overfits to the training distribution. We reject this possibility by meta testing the same unmodified RNN on a different dataset, here Fashion MNIST. Learning still works well, meaning we have meta learned a fairly general LA (although performance at convergence still lacks behind a little). This generalization is achieved without using any hardcoded gradients during meta testing purely by running the RNN forward. In comparison to VSML, a Meta RNN heavily overfits. + +# 4.3 Robustness to varying inputs and outputs + +A defining property of VSML is that the same parameters $V _ { M }$ can be used to learn on varying input and output sizes. Further, the architecture and thus the meta learned LA is invariant to the order of inputs and outputs. In this experiment, we investigate how robust we are to such changes. We meta train across MNIST with 3, 4, 6, and 7 classes. Likewise, we train across rescaled versions with 14x14, $2 8 \mathbf { x } 2 8$ , and $3 2 \mathrm { x } 3 2$ pixels. We also randomly project all inputs using a linear transformation, with the transforma + +![](images/c628948506067fe04ffbb9c085630d6448553be811bf1db3b99f5a94303b235f.jpg) +Figure 7: The meta learned learning algorithm is robust to permutations and size changes in the inputs and outputs. All six configurations have not been seen during training and perform comparable to the unchanged reference. Standard deviations are over 32 seeds. + +tion fixed for all inner learning steps. In Figure 7 we meta test on 6 configurations that were not seen during meta training. Performance on all of these configurations is comparable to the unchanged reference from the previous section. In particular, the invariance to random projections suggests that we have meta learned a learning algorithm beyond transferring learned representations [compare 11, 54, 55]. + +![](images/319cf9db2760a541d350ef7249d327ed4957d5292c4dc26af98cc095f61ec93b.jpg) +Figure 8: Online learning on various datasets. Cumulative accuracy in $\%$ after having seen $2 \mathrm { k }$ training examples evaluated after each prediction starting with random states (VSML, Meta RNN, HebbianFW, FWMemory) or random parameters (SGD). Standard deviations are over 32 meta test training runs. Meta testing is done on the official test set of each dataset. Meta training is on subsets of datasets excluding the Sum Sign dataset. Unseen tasks, most relevant from a general-purpose LA perspective, are opaque. + +# 4.4 Varying datasets + +To better understand how different meta training distributions and meta test datasets affect VSML RNNs and our baselines, we present several different combinations in Figure 8. The opaque bars represent tasks that were not seen during meta training and are thus most relevant for this analysis. This includes four additional datasets: (1) Kuzushiji MNIST [7] with 10 classes, (2) EMNIST [9] with 62 classes, (3) A randomly generated classification dataset (Random) with 20 data points that changes with each step in the outer loop, and (4) Sum Sign which generates random inputs and requires classifying the sign of the sum of all inputs. Meta training is done over 500 randomly drawn samples per outer iteration. Each algorithm is meta trained for $1 0 \mathrm { k }$ outer iterations. Inputs are randomly projected as in Section 4.3 (for VSML; the baselines did not benefit from these augmentations). We again report the cumulative accuracy on all data seen since the beginning of meta test training. We compare to SGD with a single layer, matching the architecture of VSML, and a hidden layer, matching the number of weights to the size of $V _ { L }$ in VSML. We also have included a Hebbian fast weight baseline [25] and an external (fast weight) memory approach [42]. + +We observe that VSML generalizes much better than Meta RNNs, Hebbian fast weights, and the external memory. These baselines overfit to the training environments. Notably, VSML even generalizes to the unseen tasks Random and Sum Sign which have no shared structure with the other datasets. In many cases VSML’s performance is similar to SGD but a little more sample efficient in the beginning of training (learning curves in Appendix B). This suggests that our meta learned LAs are good at quickly associating new inputs with their labels. We further investigate this in the next Section 5. + +# 5 Analysis + +Given that VSML seems to learn faster than online gradient descent in many cases we would like to qualitatively investigate how learning differs. We first meta train on the full MNIST dataset as before. During meta testing, we plot the output probabilities for each digit against the number of samples seen in Figure 9. We highlight the ground truth input class $\boxed { \begin{array} { r l } \end{array} }$ as well as the predicted class $\bigcirc$ . In this case, our meta test dataset consists of MNIST digits with two examples of each type. The same digit is always repeated twice. This allows us to observe and visualize the effect with only a few examples. We have done the same introspection with the full dataset in Appendix B. + +We observe that in VSML almost all failed predictions are followed by the correct prediction with high certainty. In contrast, SGD + +![](images/eaac26941486bc755c21019531e0870ec146a660bf3bfc809cb2fca51d5e7b20.jpg) +Figure 9: Introspection of how output probabilities change after observing an input and the prediction error when meta testing on MNIST with two examples for each type. We highlight the ground truth class $\boxed { \begin{array} { r l } \end{array} }$ as well as the predicted class $\bigcirc$ . The top plot shows VSML quickly associating the input images with the right label, almost always making the right prediction the second time with high confidence. The bottom plot shows the same dataset processed by SGD with Adam which fails to learn quickly. + +makes many incorrect predictions and fails to adapt correctly in only 20 steps. It seems that SGD learns qualitatively different from VSML. The VSML RNN meta learns to quickly associate new inputs with their class whereas SGD fails to do so. We tried several different SGD learning rates and considered multiple steps on the same input. In both cases, SGD does not behave similar to VSML, either learning much slower or forgetting previous examples. As evident from high accuracies in Figure 8, VSML does not only memorize inputs using this strategy of fast association, but the associations generalize to future unseen inputs. + +# 6 Related Work + +Memory based meta learning (Meta RNNs) Memory-based meta learning in RNNs [16, 10, 56] is a simple neural meta learner (see Section 2). Unfortunately, the LA encoded in the RNN parameters is largely overparameterized which leads to overfitting. VSML demonstrates that weight sharing can address this issue resulting in more general-purpose LAs. + +Learned Learning Rules / Fast Weights NNs that generate or change the weights of another or the same NN are known as fast weight programmers [44], hypernetworks [14], NNs with synaptic plasticity [25] or learned learning rules [4] (see Section 2). In VSML we do not require explicit architectures for weight updates as weights are emergent from RNN state updates. In addition to the learning rule, we implicitly learn how the neural forward computation is defined. Concurrent to this work, fast weights have also been used to meta learn more general LAs [39]. + +Learned gradient-based optimizers Meta learning has been used to find optimizers that update the parameters of a model by taking the loss and gradient with respect to these parameters as an input [34, 2, 22, 24]. In this work, we are interested in meta learning that does not rely on fixed gradient calculation in the inner loop. + +Discrete program search An interesting alternative to distributed variable updates in VSML is meta learning via discrete program search [48, 35]. In this paradigm, a separate programming language needs to be defined that gives rise to neural computation when its instructions are combined. This led to the automated rediscovery of backpropagation [35]. In VSML we demonstrate that a symbolic programming language is not required and general-purpose LAs can be discovered and encoded in variable-shared RNNs. Search over neural network parameters is usually easier compared to symbolic program search due to smoothness in the loss landscape. + +Multi-agent systems In the reinforcement learning setting multiple agents can be modeled with shared parameters [50, 32, 18], also in the context of meta learning [36]. This is related to the variable sharing in VSML depending on how the agent-environment boundary is drawn. Unlike these works, we demonstrate the advantage of variable sharing in meta learning more general-purpose LAs and present a weight update interpretation. + +# 7 Discussion and Limitations + +The research community has perfected the art of leveraging backpropagation for learning for a long time. At the same time, there are open questions such as how to minimize memory requirements, effectively learn online and continually, learn sample efficiently, learn without separate backward phases, and others. VSML suggests that instead of building on top of backpropagation as a fixed routine, meta learning offers an alternative to discover general-purpose LAs. Nevertheless, this paper is only a proof of concept—until now we have only investigated small-scale problems and performance does not yet quite match the mini-batched setting with large quantities of data. In particular, we observed premature convergence of the solution at meta test time which calls for further investigations. Scaling our system to harder problems and larger meta task distributions will be important future work. + +The computational cost of the current VSML variant is also larger than the one of standard backpropagation. If we run a sub-RNN for each weight in a standard NN with $W$ weights, the cost is in $O ( W N ^ { 2 } )$ , where $N$ is the state size of a sub-RNN. If $N$ is small enough, and our experiments suggest small $N$ may be feasible, this may be an acceptable cost. However, VSML is not bound to the interpretation of a sub-RNN as one weight. Future work may relax this particular choice. + +Meta optimization is also prone to local minima. In particular, when the number of ticks between input and feedback increases (e.g. deeper architectures), credit assignment becomes harder. Early experiments suggest that diverse meta task distributions can help mitigate these issues. Additionally, learning horizons are limited when using backprop-based meta optimization. Using ES allowed for training across longer horizons and more stable optimization. + +VSML can also be viewed as regularizing the NN weights that encode the LA through a representational bottleneck. It is conceivable that LA generalization as obtained by VSML can also be achieved through other regularization techniques. Unlike most regularizers, VSML also introduces substantial reuse of the same learning principle and permutation invariance through variable sharing. + +# 8 Conclusion + +We introduced Variable Shared Meta Learning (VSML), a simple principle of weight sharing and sparsity for meta learning powerful learning algorithms (LAs). Our implementation replaces the weights of a neural network with tiny LSTMs that share parameters. We discuss connections to meta recurrent neural networks, fast weight generators (hyper networks), and learned learning rules. + +Using learning algorithm cloning, VSML RNNs can learn to implement the backpropagation algorithm and its parameter updates encoded implicitly in the recurrent dynamics. On MNIST it learns to predict well without any human-designed explicit computational graph for gradient calculation. + +VSML can meta learn from scratch supervised LAs that do not explicitly rely on gradient computation and that generalize to unseen datasets. Introspection reveals that VSML LAs learn by fast association in a way that is qualitatively different from stochastic gradient descent. This leads to gains in sample efficiency. Future work will focus on reinforcement learning settings, improvements of meta learning, larger task distributions, and learning over longer horizons. + +# Acknowledgements + +We thank Sjoerd van Steenkiste, Imanol Schlag, Kazuki Irie, and the anonymous reviewers for their comments and feedback. This work was supported by the ERC Advanced Grant (no: 742870) and computational resources by the Swiss National Supercomputing Centre (CSCS, projects $s 9 7 8$ and s1041). We also thank NVIDIA Corporation for donating several DGX machines as part of the Pioneers of AI Research Award, IBM for donating a Minsky machine, and weights & biases [5] for their great experiment tracking software and support. + +# References + +[1] F. Alet, M. F. Schneider, T. Lozano-Perez, and L. P. Kaelbling. Meta-learning curiosity algorithms. In International Conference on Learning Representations, 2020. +[2] M. Andrychowicz, M. Denil, S. G. Colmenarejo, M. W. Hoffman, D. Pfau, T. Schaul, B. Shillingford, and N. 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CoRR, abs/1708.0, 2017. \ No newline at end of file diff --git a/parse/train/hhU9TEvB6AF/hhU9TEvB6AF_content_list.json b/parse/train/hhU9TEvB6AF/hhU9TEvB6AF_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..129143fd18f51ea8c4feaaa08b0a38287afc6fe2 --- /dev/null +++ b/parse/train/hhU9TEvB6AF/hhU9TEvB6AF_content_list.json @@ -0,0 +1,1193 @@ +[ + { + "type": "text", + "text": "Meta Learning Backpropagation And Improving It ", + "text_level": 1, + "bbox": [ + 187, + 122, + 808, + 148 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Louis Kirsch1, Jürgen Schmidhuber1,2 \n1The Swiss AI Lab IDSIA, University of Lugano (USI) & SUPSI, Lugano, Switzerland \n2King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia {louis, juergen}@idsia.ch ", + "bbox": [ + 210, + 199, + 789, + 258 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 292, + 535, + 310 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many concepts have been proposed for meta learning with neural networks (NNs), e.g., NNs that learn to reprogram fast weights, Hebbian plasticity, learned learning rules, and meta recurrent NNs. Our Variable Shared Meta Learning (VSML) unifies the above and demonstrates that simple weight-sharing and sparsity in an NN is sufficient to express powerful learning algorithms (LAs) in a reusable fashion. A simple implementation of VSML where the weights of a neural network are replaced by tiny LSTMs allows for implementing the backpropagation LA solely by running in forward-mode. It can even meta learn new LAs that differ from online backpropagation and generalize to datasets outside of the meta training distribution without explicit gradient calculation. Introspection reveals that our meta learned LAs learn through fast association in a way that is qualitatively different from gradient descent. ", + "bbox": [ + 233, + 325, + 766, + 491 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 517, + 310, + 535 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The shift from standard machine learning to meta learning involves learning the learning algorithm (LA) itself, reducing the burden on the human designer to craft useful learning algorithms [43]. Recent meta learning has primarily focused on generalization from training tasks to similar test tasks, e.g., few-shot learning [11], or from training environments to similar test environments [17]. This contrasts human-engineered LAs that generalize across a wide range of datasets or environments. Without such generalization, meta learned LAs can not entirely replace human-engineered variants. Recent work demonstrated that meta learning can also successfully generate more general LAs that generalize across wide spectra of environments [20, 1, 31], e.g., from toy environments to Mujoco and Atari. Unfortunately, however, many recent approaches still rely on a large number of human-designed and unmodifiable inner-loop components such as backpropagation. ", + "bbox": [ + 174, + 547, + 825, + 688 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Is it possible to implement modifiable versions of backpropagation or related algorithms as part of the end-to-end differentiable activation dynamics of a neural net (NN), instead of inserting them as separate fixed routines? Here we propose the Variable Shared Meta Learning (VSML) principle for this purpose. It introduces a novel way of using sparsity and weight-sharing in NNs for meta learning. We build on the arguably simplest neural meta learner, the meta recurrent neural network (Meta RNN) [16, 10, 56], by replacing the weights of a neural network with tiny LSTMs. The resulting system can be viewed as many RNNs passing messages to each other, or as one big RNN with a sparse shared weight matrix, or as a system learning each neuron’s functionality and its LA. VSML generalizes the principle behind end-to-end differentiable fast weight programmers [45, 46, 3, 41], hyper networks [14], learned learning rules [4, 13, 33], and hebbian-like synaptic plasticity [44, 46, 25, 26, 30]. Our mechanism, VSML, can implement backpropagation solely in the forward-dynamics of an RNN. Consequently, it enables meta-optimization of backproplike algorithms. We envision a future where novel methods of credit assignment can be meta learned while still generalizing across vastly different tasks. This may lead to improvements in sample efficiency, memory efficiency, continual learning, and others. As a first step, our system meta learns online LAs from scratch that frequently learn faster than gradient descent and generalize to datasets outside of the meta training distribution (e.g., from MNIST to Fashion MNIST). Our VSML RNN is the first neural meta learner without hard-coded backpropagation that shows such strong generalization. ", + "bbox": [ + 174, + 693, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Background ", + "text_level": 1, + "bbox": [ + 174, + 165, + 308, + 183 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Deep learning-based meta learning that does not rely on fixed gradient descent in the inner loop has historically fallen into two categories, 1) Learnable weight update mechanisms that allow for changing the parameters of an NN to implement a learning rule (FWPs / LLRs), and 2) Learning algorithms implemented in black-box models such as recurrent neural networks (Meta RNNs). ", + "bbox": [ + 174, + 196, + 825, + 252 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Fast weight programmers & Learned learning rules (FWPs / LLRs) In a standard NN, the weights are updated by a fixed LA. This framework can be extended to meta learning by defining an explicit architecture that allows for modifying these weights. This weight-update architecture augments a standard NN architecture. NNs that generate or change the weights of another or the same NN are known as fast weight programmers (FWPs) [44, 45, 46, 3, 41], hypernetworks [14], NNs with synaptic plasticity [25, 26, 30] or learned learning rules [4, 13, 33]. Often these architectures make use of local Hebbian-like update rules or outer-products, and we summarize this category as FWPs / LLRs. In FWPs / LLRs the variables $V _ { L }$ that are subject to learning are the weights of the network, whereas the meta-variables $V _ { M }$ that implement the LA are defined by the weight-update architecture. Note that the dimensionality of $V _ { L }$ and $V _ { M }$ can be defined independently of each other and often $V _ { M }$ are reused in a coordinate-wise fashion for $V _ { L }$ resulting in $| V _ { L } | \\gg | V _ { M } |$ , where $| \\cdot |$ is the number of elements. ", + "bbox": [ + 173, + 266, + 825, + 431 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Black-box learning in activations (Meta RNNs) It was shown that an RNN such as LSTM can learn to implement an LA [16] when the reward or error is given as an input [47]. After meta training, the LA is encoded in the weights of this RNN and determines learning during meta testing. The activations serve as the memory used for the LA solution. We refer to this as Meta RNNs [16, 10, 56] (Also sometimes referred to as memory-based meta learning.). They are conceptually simpler than FWPs / LLRs as no additional weight-update rules with many degrees of freedom need to be defined. In Meta RNNs $V _ { L }$ are the RNN activations and $V _ { M }$ are the parameters for the RNN. Note that an RNN with $N$ neurons will yield $| V _ { L } | = O ( N )$ and $| V _ { M } | = \\stackrel { \\cdot } { O } ( N ^ { 2 } )$ [46]. This means that the LA is largely overparameterized whereas the available memory for learning is very small, making this approach prone to overfitting [20]. As a result, the RNN parameters often encode task-specific solutions instead of generic LAs. Meta learning a simple and generalizing LA would benefit from $| V _ { L } | \\gg | V _ { M } |$ . Previous approaches have tried to mend this issue by adding architectural complexity through additional memory mechanisms [53, 29, 40, 27, 42]. ", + "bbox": [ + 173, + 446, + 826, + 626 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 Variable Shared Meta Learning (VSML) ", + "text_level": 1, + "bbox": [ + 173, + 643, + 545, + 662 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In VSML we build on the simplicity of Meta RNNs while retaining $| V _ { L } | \\gg | V _ { M } |$ from FWPs / LLRs. We do this by reusing the same few parameters $V _ { M }$ many times in an RNN (via variable sharing) and introducing sparsity in the connectivity. This yields several interpretations for VSML: ", + "bbox": [ + 174, + 675, + 826, + 717 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "$\\textcircled{8}$ VSML as a single Meta RNN with a sparse shared weight matrix (Figure 1a). The most general description. \n$\\textcircled{8}$ VSML as message passing between RNNs (Figure 1b). We choose a simple sharing and sparsity scheme for the weight matrix such that it corresponds to multiple RNNs with shared parameters that exchange information. \n$©$ VSML as complex neurons with learned updates (Figure 1c). When choosing a specific connectivity between RNNs, states / activations $V _ { L }$ of these RNNs can be interpreted as the weights of a conventional NN, consequently blurring the distinction between a weight and an activation. ", + "bbox": [ + 205, + 728, + 825, + 866 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Introducing variable sharing to Meta RNNs We begin by formalizing Meta RNNs which often use multiplicative gates such as the LSTM [12, 15] or its variant GRU [6]. For notational simplicity, we consider a vanilla RNN. Let $s \\in \\mathbb { R } ^ { N }$ be the hidden state of an RNN. The update for an element $j \\in \\{ 1 , \\ldots , N \\}$ is given by ", + "bbox": [ + 173, + 882, + 821, + 911 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/b1ae57af27d23bff08bcfc6f114db04ece9a7f8b38923249b2e9e1b42ad6a50f.jpg", + "image_caption": [ + "Figure 1: Different perspectives on VSML: (a) A single Meta RNN [16] where entries in the weight matrix are shared or zero. (a) VSML consists of many sub-RNNs with shared parameters $V _ { M }$ passing messages between each other. (c) VSML implements an NN with complex neurons (here 2 neurons). $V _ { M }$ determines the nature of weights, how these are used in the neural computation, and the LA by which those are updated. Each weight $w _ { a b } \\in \\mathbb { R }$ is represented by the multi-dimensional RNN state ${ s _ { a b } } \\in \\mathbb { R } ^ { N }$ . Neuron activations correspond to messages $\\overrightarrow { m }$ passed between sub-RNNs. " + ], + "image_footnote": [], + "bbox": [ + 191, + 87, + 818, + 223 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 339, + 823, + 369 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/aec312823b19fbd3996491d88091d87c5725063323dbba06de0ca3464831a762.jpg", + "text": "$$\ns _ { j } \\gets f _ { \\mathrm { R N N } } ( s ) _ { j } = \\sigma ( \\sum _ { i } s _ { i } W _ { i j } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 388, + 367, + 606, + 401 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\sigma$ is a non-linear activation function, $W \\in \\mathbb { R } ^ { N \\times N }$ , and the bias is omitted for simplicity. We also omit inputs by assuming a subset of $s$ to be externally provided. Each application of Equation 1 reflects a single time tick in the RNN. ", + "bbox": [ + 173, + 402, + 825, + 448 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We now introduce variable sharing (reusing $W$ ) into the RNN by duplicating the computation along two axes of size $A , B$ (here $A = B$ , which will later be relaxed) giving $\\bar { s } \\in \\mathbb { R } ^ { A \\times B \\times N }$ . For $a \\in \\{ 1 , \\dotsc , A \\} , b \\in \\{ 1 , \\dotsc , B \\}$ we have ", + "bbox": [ + 173, + 452, + 825, + 497 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/bd446ad5b95522b305babefe9b762f10f82ec8bf83cb2ee5e6827d94874df36a.jpg", + "text": "$$\ns _ { a b j } \\gets f _ { \\mathrm { R N N } } ( s _ { a b } ) _ { j } = \\sigma ( \\sum _ { i } s _ { a b i } W _ { i j } ) .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 502, + 629, + 535 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This can be viewed as multiple RNNs arranged on a 2-dimensional grid, with shared parameters that update independent states. Here, we chose a particular arrangement (two axes) that will facilitate the interpretation $©$ of RNNs as weights. ", + "bbox": [ + 173, + 539, + 825, + 584 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "VSML as message passing between RNNs The computation so far describes $A \\cdot B$ independent RNNs. We connect those by passing messages (interpretation $\\textcircled{8}$ ) ", + "bbox": [ + 173, + 597, + 823, + 627 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dbe020d5af66f40b67afaa6858cc17392b6229b77c392d47d8f437e8f795275d.jpg", + "text": "$$\ns _ { a b } \\gets f _ { \\mathrm { R N N } } ( s _ { a b } , \\overrightarrow { m } _ { a } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 421, + 631, + 575, + 648 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the message $\\begin{array} { r } { \\vec { m } _ { b } = \\sum _ { a ^ { \\prime } } f _ { \\vec { m } } ( s _ { a ^ { \\prime } b } ) } \\end{array}$ with $b \\in \\{ 1 , \\dots , A = B \\}$ , $f _ { \\overrightarrow { m } } : \\mathbb { R } ^ { N } \\mathbb { R } ^ { N ^ { \\prime } }$ is fed as an additional input to each RNN. This is related to Graph Neural Networks [51, 58]. Summing over the axis $A$ (elements $a ^ { \\prime }$ ) corresponds to an RNN connectivity mimicking those of weights in an NN (to facilitate interpretation $©$ . We emphasise that other schemes based on different kinds of message passing and graph connectivity are possible. For a simple $f _ { \\overrightarrow { m } }$ defined by the matrix $C \\in \\mathbb { R } ^ { N \\times N }$ , we may equivalently write ", + "bbox": [ + 173, + 655, + 826, + 744 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a4fa3ae13d4eb053bd2e1b570e336ff4662b2e762e505e889dc05c1a0c6ebc4a.jpg", + "text": "$$\ns _ { a b j } \\gets \\sigma ( \\sum _ { i } s _ { a b i } W _ { i j } + \\sum _ { a ^ { \\prime } } f _ { \\overrightarrow { m } } ( s _ { a ^ { \\prime } a } ) ) = \\sigma ( \\sum _ { i } s _ { a b i } W _ { i j } + \\sum _ { a ^ { \\prime } , i } s _ { a ^ { \\prime } a i } C _ { i j } ) .\n$$", + "text_format": "latex", + "bbox": [ + 251, + 748, + 745, + 782 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This constitutes the minimal version of VSML with $V _ { M } : = ( W , C )$ and is visualized in Figure 1b. ", + "bbox": [ + 174, + 787, + 813, + 805 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "VSML as a Meta RNN with a sparse shared weight matrix It is trivial to see that with $A = 1$ and $B = 1$ we obtain a single RNN and Equation 4 recovers the original Meta RNN Equation 1. In the general case, we can derive an equivalent formulation that corresponds to a single standard RNN with a single matrix $\\tilde { W }$ that has entries of zero and shared entries ", + "bbox": [ + 173, + 816, + 825, + 876 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/eeb3ae8d9676dd584d8b30e1f2e7cd87e1831ed2eae316f1470b3cdab7e9006f.jpg", + "text": "$$\ns _ { a b j } \\gets \\sigma ( \\sum _ { c , d , i } s _ { c d i } \\tilde { W } _ { c d i a b j } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 405, + 881, + 593, + 915 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the six axes can be flattened to obtain the two axes. For Equation 4 and Equation 5 to be equivalent, $\\tilde { W }$ must satisfy (derivation in Appendix A) ", + "bbox": [ + 169, + 90, + 823, + 122 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/aba9ee62d2cb9e4441538f13c5f01b18fd44f40ab8e22078076d3cef95f3da60.jpg", + "text": "$$\n\\tilde { W } _ { c d i a b j } = \\left\\{ \\begin{array} { l l } { C _ { i j } , } & { \\mathrm { i f ~ } d = a \\wedge ( d \\neq b \\vee c \\neq a ) . } \\\\ { W _ { i j } , } & { \\mathrm { i f ~ } d \\neq a \\wedge d = b \\wedge c = a . } \\\\ { C _ { i j } + W _ { i j } , } & { \\mathrm { i f ~ } d = a \\wedge d = b \\wedge c = a . } \\\\ { 0 , } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 315, + 131, + 678, + 199 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This corresponds to interpretation $\\textcircled{8}$ with the weight matrix visualized in Figure 1a. To distinguish between the single sparse shared RNN and the connected RNNs, we now call the latter sub-RNNs. ", + "bbox": [ + 173, + 210, + 825, + 239 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "VSML as complex neurons with learned updates The arrangement and connectivity of the subRNNs as described in the previous paragraphs corresponds to that of weights in a standard NN. Thus, in interpretation $@$ , VSML can be viewed as defining complex neurons where each sub-RNN corresponds to a weight in a standard NN as visualized in Figure 1c. All these sub-RNNs share the same parameters but have distinct states. The current formulation corresponds to a single NN layer that is run recurrently. We will generalize this to other architectures in the next section. A corresponds to the dimensionality of the inputs and $B$ to that of the outputs in that layer. ", + "bbox": [ + 173, + 256, + 826, + 356 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The role of weights in a standard neural network is now assigned to the states of RNNs. This allows these RNNs to define both the neural forward computation as well as the learning algorithm that determines how the network is updated (where the mechanism is shared across the network). In the case of backpropagation, this would correspond to the forward and backward passes being implemented purely in the recurrent dynamics. We will demonstrate the practical feasibility of this in Section 3.2. The emergence of ", + "bbox": [ + 174, + 361, + 356, + 637 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/0ccd0fdb8442f57b99da7d0266c5c9a62738c64d643b90443c68ed18becb66cf.jpg", + "image_caption": [ + "Figure 2: The neural interpretation of VSML replaces all weights of a standard NN with tiny LSTMs using shared parameters (resembling complex neurons). This allows these LSTMs to define both the neural forward computation as well as the learning algorithm that determines how the network is updated. Information flows forward and backward in the network through multi-dimensional messages $\\overrightarrow { m }$ and $\\scriptstyle { \\overleftarrow { m } }$ , generalizing the dynamics of an NN trained using backpropagation. " + ], + "image_footnote": [], + "bbox": [ + 369, + 366, + 823, + 518 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "RNN states as weights quickly leads to confusing terminology when RNNs have ‘meta weights’. \nInstead, we simply refer to meta variables $V _ { M }$ and learned variables $V _ { L }$ . ", + "bbox": [ + 173, + 637, + 821, + 665 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "With this interpretation, VSML can be seen as a generalization of learned learning rules [4, 13, 33] and Hebbian-like differentiable mechanisms or fast weights more generally [44, 46, 25, 26] where RNNs replace explicit weight updates. In standard NNs, weights and activations have multiplicative interactions. For VSML RNNs to mimic such computation we require multiplicative interactions between parts of the state $s$ . Fortunately, LSTMs already incorporate this through gating and can be directly used in place of RNNs. ", + "bbox": [ + 173, + 671, + 825, + 755 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Stacking VSML RNNs and feeding inputs To get a structure similar to one of the non-recurrent deep feed-forward architectures (FNNs), we stack multiple VSML RNNs where their states are untied and their parameters are tied.1 This is visualized with two layers in Figure 2 where the states $s ^ { ( 2 ) }$ of the second column of sub-RNNs are distinct from the first column $\\mathbf { \\boldsymbol { s } } ^ { ( 1 ) }$ . The parameters $A ^ { ( k ) }$ and $B ^ { ( k ) }$ describing layer sizes can then be varied for each layer $k \\in \\{ 1 , \\ldots , K \\}$ constrained by $A ^ { ( k ) } ~ = ~ B ^ { ( k - 1 ) }$ . The updated Equation 3 with distinct layers $k$ is given by $s _ { a b } ^ { ( k ) } f _ { \\mathrm { R N N } } ( s _ { a b } ^ { ( k ) } , \\overrightarrow { m } _ { a } ^ { ( k ) } )$ where $\\begin{array} { r } { \\vec { m } _ { b } ^ { ( k + 1 ) } : = \\sum _ { a ^ { \\prime } } \\bar { f } _ { \\vec { m } } ( s _ { a ^ { \\prime } b } ^ { ( k ) } ) } \\end{array}$ with $b \\in \\{ 1 , \\ldots , B ^ { ( k ) } = A ^ { ( k + 1 ) } \\}$ ", + "bbox": [ + 173, + 771, + 826, + 880 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To prevent information from flowing only forward in the network, we use an additional backward message ", + "bbox": [ + 173, + 92, + 486, + 133 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/92c1ed6925a96948824b0b65811fe3cd0faa25865ff4b0c03f5daa7040659ab8.jpg", + "text": "$$\ns _ { a b } ^ { ( k ) } \\gets f _ { R N N } \\big ( s _ { a b } ^ { ( k ) } , \\overrightarrow { m } _ { a } ^ { ( k ) } , \\overleftarrow { m } _ { b } ^ { ( k ) } \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 217, + 140, + 439, + 162 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\overleftarrow { m } _ { a } ^ { ( k - 1 ) } : = \\sum _ { b ^ { \\prime } } f _ { \\overleftarrow { m } } ( s _ { a b ^ { \\prime } } ^ { ( k ) } ) } \\end{array}$ with $a \\in$ $\\{ 1 , . . . , A ^ { ( k ) } = B ^ { ( k - 1 ) } \\}$ (visualized in Figure 3). The backward transformation is given by f←−m : RN → RN 00 . ", + "bbox": [ + 173, + 167, + 485, + 234 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Similarly, other neural architectures can be explicitly constructed (e.g. convolutional NNs, Section B.2). Some architectures may be learned implicitly if positional information is fed into each sub-RNN (Appendix C). We then update all states $s ^ { ( k ) }$ in sequence $1 , \\ldots , K$ to mimic sequential layer execution. We may also apply multiple RNN ticks for each layer $k$ . ", + "bbox": [ + 174, + 239, + 485, + 325 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/a5613155d304964c2e8b7d7e753eadbc7c640e13696034612df8983dd1ae3ff5.jpg", + "image_caption": [ + "Figure 3: VSML with forward messages $\\overrightarrow { m }$ and backward messages $\\scriptstyle { \\overleftarrow { m } }$ to form a two-layer NN with shared LSTM parameters but distinct states. " + ], + "image_footnote": [], + "bbox": [ + 500, + 92, + 821, + 252 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 324, + 766, + 338 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To provide the VSML RNN with data, each time we execute the operations of the first layer, a single new datum $x \\in \\mathbb { R } ^ { A ( 1 ) }$ (e.g. one flattened image) is distributed across all sub-RNNs. In our present experiments, we match the axis $A ( 1 )$ to the input datum dimensionality such that each dimension (e.g., pixel) is fed to different RNNs. This corresponds to initializing the forward message $\\vec { m } _ { a 1 } ^ { ( 1 ) } : = \\boldsymbol { x } _ { a }$ dding . Fina $\\overrightarrow { m }$ with zeros if nece we feed the error larly, we read the outpuat the output such that $\\boldsymbol { \\hat { y } } \\in \\mathbb { R } ^ { B ( K ) }$ from. See $\\hat { y } _ { a } : = \\overrightarrow { m } _ { a 1 } ^ { ( K + 1 ) }$ $e \\in \\mathbb { R } ^ { B ( K ) }$ $ _ { b 1 } : = e _ { b }$ Figure 2 for a visualization. Alternatively, multiple input or output dimensions could be patched together and fed into fewer sub-RNNs. ", + "bbox": [ + 173, + 344, + 826, + 465 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 Meta learning general-purpose learning algorithms from scratch ", + "text_level": 1, + "bbox": [ + 176, + 482, + 661, + 497 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Having formalized VSML, we can now use end-to-end meta learning to create LAs from scratch in Algorithm 1. We simply optimize the LSTM parameters $V _ { M }$ to minimize the sum of prediction losses over many time steps starting with random states $V _ { L } : = \\{ s _ { a b } ^ { ( k ) } \\}$ . We focus on meta learning online LAs where one example is fed at a time as done in Meta RNNs [16, 56, 10]. Meta training may be performed using end-to-end gradient descent or gradient-free optimization such as evolutionary strategies [57, 38]. The latter is significantly more efficient on VSML compared to standard NNs due to the small parameter space $V _ { M }$ . Crucially, during meta testing, no explicit gradient descent is used. ", + "bbox": [ + 173, + 507, + 825, + 608 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 VSML: Meta Training ", + "text_level": 1, + "bbox": [ + 174, + 625, + 411, + 640 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Require: Dataset(s) $D = \\{ ( x _ { i } , y _ { i } ) \\}$ $V _ { M } \\gets$ initialize LSTM parameters while meta loss has not converged do . Outer loop in parallel over datasets $D$ ${ \\cal V } _ { L } = \\{ s _ { a b } ^ { ( k ) } \\} $ initialize LSTM states $\\forall a , b , k$ for $( x , y ) ^ { \\top } \\in \\{ ( x _ { 1 } , y _ { 1 } ) , \\dotsc , ( x _ { T } , y _ { T } ) \\} \\subset D$ do $\\triangleright$ Inner loop over $T$ examples $ { \\vec { m } } _ { a 1 } ^ { ( 1 ) } : = x _ { a }$ ∀a $\\triangleright$ Initialize from input image $\\mathbf { X }$ for $\\bar { k } \\in \\{ 1 , \\ldots , K \\}$ do $\\triangleright$ Iterating over $K$ layers $s _ { a b } ^ { ( k ) } \\gets f _ { R N N } \\big ( s _ { a b } ^ { ( k ) } , \\overrightarrow { m } _ { a } ^ { ( k ) } , \\overleftarrow { m } _ { b } ^ { ( k ) } \\big ) \\quad \\forall a , b$ . Equation 7 ←−m m (k+1)b := Pa0 f−→m (s(k)a0b) (k−1)a := Pb0 f←−m (s(k)ab0 ) ∀a ∀b . Create backward message . Create forward message yˆa := −→m(K+1)a1 ∀a $\\triangleright$ Read output e := ∇yˆL(ˆy, y) . Compute error at outputs using loss $L$ ←−m(K)b1 := eb ∀b $\\triangleright$ Input errors $\\begin{array} { r } { V _ { M } \\gets V _ { M } - \\alpha \\nabla _ { V _ { M } } \\sum _ { t = 1 } ^ { T } L ( \\hat { y } ( t ) , y ( t ) ) } \\end{array}$ , obtaining $\\nabla _ { V _ { M } }$ either by • back-propagation through the inner loop • evolution strategies, using a search distribution $p ( V _ { M } )$ ", + "bbox": [ + 165, + 640, + 826, + 902 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 Learning to implement backpropagation in RNNs ", + "text_level": 1, + "bbox": [ + 176, + 92, + 555, + 106 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/c33778dd74e4643cf3ee884dd15c7cd4f5d7da2efc2ed6fb055153758c4435ca.jpg", + "image_caption": [ + "Figure 4: To implement backpropagation we optimize the VSML RNN to use and update weights $w$ and biases $b$ as part of the state $s _ { a b }$ in each subRNN. Inputs are pre-synaptic $x$ and error $e$ . Outputs are post-synaptic $\\hat { y }$ and error $\\cdot$ . " + ], + "image_footnote": [], + "bbox": [ + 200, + 121, + 343, + 242 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "An alternative to end-to-end meta learning is to first investigate whether the VSML RNN can implement backpropagation. Due to the algorithm’s ubiquitous use, it seems desirable to be able to meta learn backpropagation-like algorithms. Here we investigate how VSML RNNs can learn to implement backpropagation purely in their recurrent dynamics. We do this by optimizing $V _ { M }$ to (1) store a weight $w$ and bias $b$ as a subset of each state $s _ { a b }$ , (2) compute $y = \\operatorname { t a n h } ( x ) w + b$ to implement neural forward computation, and (3) update $w$ and $b$ according to the backpropagation algorithm [23]. We call this process learning algorithm cloning and it is visualized in Figure 4. ", + "bbox": [ + 383, + 117, + 825, + 268 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We designate an element of each message $\\vec { m } _ { a } ^ { ( k ) }$ , $\\{ \\overline { { m } } _ { b } ^ { ( k ) }$ , $f _ { \\vec { m } } ( s _ { a b } ^ { ( k ) } )$ $f _ { \\overleftarrow { m } } ( s _ { a b } ^ { ( k ) } )$ as the input $x$ , error $e$ , and output $\\hat { y }$ and error $\\hat { e } ^ { \\prime }$ . Similarly, we set $w : = s _ { a b 1 }$ and $b : = s _ { a b 2 }$ . We then optimize $V _ { M }$ via gradient descent to regress $\\hat { y }$ , $\\Delta w$ , $\\Delta b$ , and $\\hat { e } ^ { \\prime }$ toward their respective targets. We can either generate the training dataset $D : = \\{ ( x , \\bar { w } , b , y , e , \\bar { e ^ { \\prime } } ) _ { i } \\}$ randomly or run a ‘shadow’ NN on some supervised problem and fit the VSML RNN to its activations and parameter updates. Multiple iterations in the VSML RNN would then correspond to evaluating the network and updating it via backpropagation. The activations from the forward pass necessary for credit assignment could be memorized as part of the state $s$ or be explicitly stored and fed back. For simplicity, we chose the latter to clone backpropagation. We continuously run the VSML RNN forward, alternately running the layers in order $1 , \\ldots , K$ and in reverse order $K , \\ldots , 1$ .2 ", + "bbox": [ + 385, + 275, + 825, + 380 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 381, + 825, + 450 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 469, + 312, + 487 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "First, we demonstrate the capabilities of the VSML RNN by showing that it can implement neural forward computation and backpropagation in its recurrent dynamics on the MNIST [21] and Fashion MNIST [59] dataset. Then, we show how we can meta learn an LA from scratch on one set of datasets and then successfully apply it to another (out of distribution). Such generalization is enabled by extensive variable sharing where we have very few meta variables $| V _ { M } | ~ \\approx ~ 2 , 4 0 0$ and many learned variables $| V _ { L } | \\approx 2 5 7 , 2 0 0$ . We also investigate the robustness of the discovered LA. Finally, we introspect the meta learned LA and compare it to gradient descent. ", + "bbox": [ + 174, + 502, + 485, + 709 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/dbb2f1c0cc8cade0dc1f709d779c51876fa020bd13a4a622b6679683d6100ede.jpg", + "image_caption": [ + "Figure 5: The VSML RNN is optimized to implement backpropagation in its recurrent dynamics on MNIST, then tested both on MNIST and Fashion MNIST where it performs learning purely by unrolling the LSTM. We test on shallow and deep architectures (single hidden layer of 32 units). Standard deviations are over 6 seeds. " + ], + "image_footnote": [], + "bbox": [ + 496, + 482, + 823, + 602 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our implementation uses LSTMs and the message interpretation from Equation 7. Hyperparameters, training details, and additional experiments can be found in the appendix. ", + "bbox": [ + 173, + 715, + 825, + 744 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 VSML RNNs can implement backpropagation ", + "text_level": 1, + "bbox": [ + 174, + 762, + 534, + 776 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As described in Section 3.2, we optimize the VSML RNN to implement backpropagation. We structure the sub-RNNs to mimic a feed-forward NN with either one hidden layer or no hidden layers. To obtain training targets, we instantiate a standard NN, the shadow network, and feed it MNIST data. After cloning, we then run the LA encoded in the VSML RNN on the MNIST and Fashion MNIST dataset and observe that it performs learning purely in its recurrent dynamics, making explicit gradient calculations unnecessary. Figure 5 shows the learning curve on these two datasets. Notably, learning works both on MNIST (within distribution) and on Fashion MNIST (out of distribution). We observe that the loss is decently minimized, albeit regular gradient descent still outperforms our cloned backpropagation. This may be due to non-zero errors during learning algorithm cloning, in particular when these errors accumulate in the deeper architecture. It is also possible that the VSML states (‘weights’) deviate too far from ranges seen during cloning, in particular in the deep case when the loss starts increasing. We obtain $87 \\%$ (deep) and $90 \\%$ (shallow) test accuracy on MNIST and $76 \\%$ (deep) and $80 \\%$ (shallow) on Fashion MNIST (focusing on successful cloning over performance). ", + "bbox": [ + 174, + 787, + 825, + 872 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 92, + 826, + 189 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 Meta learning supervised learning from scratch ", + "text_level": 1, + "bbox": [ + 173, + 208, + 544, + 223 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the previous experiments, we have established that VSML is expressive enough to metaoptimize backpropagation-like algorithms. Instead of cloning an LA, we now meta learn from scratch as described in Section 3.1. Here, we use a single layer ( $K = 1$ ) from input to output dimension and run it for two ticks per image with $N = 1 6$ and $N ^ { \\prime } = N ^ { \\prime \\prime } = 8$ . First, the VSML RNN is meta trained end-to-end using evolutionary strategies (ES) [38] on MNIST to minimize the sum of cross-entropies over 500 data points starting from random state initializations. As each image is unique and $V _ { M }$ can not memorize the data, we are implicitly optimizing the VSML RNN to generalize to future inputs given all inputs it has seen so far. We do not pre-train $V _ { M }$ with a human-engineered LA. ", + "bbox": [ + 174, + 234, + 485, + 469 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/7b033cbe1d2c5cc6f6e774fb5334f6d5b86759ee6986d192914ef27be40db278.jpg", + "image_caption": [ + "Figure 6: The VSML RNN can be directly meta trained on MNIST to minimize the sum of errors when classifying online starting from a random state initialization. This allows for faster learning during meta testing compared to online gradient descent with Adam on the same dataset and even generalizes to a different dataset (Fashion MNIST). In comparison, a standard Meta RNN [16] strongly overfits in the same setting. Standard deviations are over 128 seeds. " + ], + "image_footnote": [], + "bbox": [ + 496, + 232, + 821, + 359 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "During meta testing on MNIST (Figure 6) we plot the cumulativputs on the y axis $\\textstyle { \\bigl ( } { \\frac { 1 } { T } } \\sum _ { t = 1 } ^ { T } c _ { t }$ n all previous after example $T$ ", + "bbox": [ + 174, + 476, + 483, + 521 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "indicating prediction correctness). For each example, the prediction when this example was fed to the RNN is used, thus measuring sample efficient learning. This evaluation protocol is similar to the one used in Meta RNNs [56, 10]. We observe that learning is considerably faster compared to the baseline of online gradient descent (no mini batching, the learning rate appropriately tuned). One possibility is that VSML simply overfits to the training distribution. We reject this possibility by meta testing the same unmodified RNN on a different dataset, here Fashion MNIST. Learning still works well, meaning we have meta learned a fairly general LA (although performance at convergence still lacks behind a little). This generalization is achieved without using any hardcoded gradients during meta testing purely by running the RNN forward. In comparison to VSML, a Meta RNN heavily overfits. ", + "bbox": [ + 173, + 520, + 826, + 657 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 Robustness to varying inputs and outputs ", + "text_level": 1, + "bbox": [ + 174, + 678, + 501, + 691 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "A defining property of VSML is that the same parameters $V _ { M }$ can be used to learn on varying input and output sizes. Further, the architecture and thus the meta learned LA is invariant to the order of inputs and outputs. In this experiment, we investigate how robust we are to such changes. We meta train across MNIST with 3, 4, 6, and 7 classes. Likewise, we train across rescaled versions with 14x14, $2 8 \\mathbf { x } 2 8$ , and $3 2 \\mathrm { x } 3 2$ pixels. We also randomly project all inputs using a linear transformation, with the transforma", + "bbox": [ + 173, + 704, + 419, + 897 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/c628948506067fe04ffbb9c085630d6448553be811bf1db3b99f5a94303b235f.jpg", + "image_caption": [ + "Figure 7: The meta learned learning algorithm is robust to permutations and size changes in the inputs and outputs. All six configurations have not been seen during training and perform comparable to the unchanged reference. Standard deviations are over 32 seeds. " + ], + "image_footnote": [], + "bbox": [ + 433, + 693, + 825, + 815 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "tion fixed for all inner learning steps. In Figure 7 we meta test on 6 configurations that were not seen during meta training. Performance on all of these configurations is comparable to the unchanged reference from the previous section. In particular, the invariance to random projections suggests that we have meta learned a learning algorithm beyond transferring learned representations [compare 11, 54, 55]. ", + "bbox": [ + 174, + 897, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/319cf9db2760a541d350ef7249d327ed4957d5292c4dc26af98cc095f61ec93b.jpg", + "image_caption": [ + "Figure 8: Online learning on various datasets. Cumulative accuracy in $\\%$ after having seen $2 \\mathrm { k }$ training examples evaluated after each prediction starting with random states (VSML, Meta RNN, HebbianFW, FWMemory) or random parameters (SGD). Standard deviations are over 32 meta test training runs. Meta testing is done on the official test set of each dataset. Meta training is on subsets of datasets excluding the Sum Sign dataset. Unseen tasks, most relevant from a general-purpose LA perspective, are opaque. " + ], + "image_footnote": [], + "bbox": [ + 173, + 87, + 826, + 424 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 526, + 825, + 582 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 Varying datasets ", + "text_level": 1, + "bbox": [ + 174, + 602, + 328, + 617 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To better understand how different meta training distributions and meta test datasets affect VSML RNNs and our baselines, we present several different combinations in Figure 8. The opaque bars represent tasks that were not seen during meta training and are thus most relevant for this analysis. This includes four additional datasets: (1) Kuzushiji MNIST [7] with 10 classes, (2) EMNIST [9] with 62 classes, (3) A randomly generated classification dataset (Random) with 20 data points that changes with each step in the outer loop, and (4) Sum Sign which generates random inputs and requires classifying the sign of the sum of all inputs. Meta training is done over 500 randomly drawn samples per outer iteration. Each algorithm is meta trained for $1 0 \\mathrm { k }$ outer iterations. Inputs are randomly projected as in Section 4.3 (for VSML; the baselines did not benefit from these augmentations). We again report the cumulative accuracy on all data seen since the beginning of meta test training. We compare to SGD with a single layer, matching the architecture of VSML, and a hidden layer, matching the number of weights to the size of $V _ { L }$ in VSML. We also have included a Hebbian fast weight baseline [25] and an external (fast weight) memory approach [42]. ", + "bbox": [ + 174, + 628, + 825, + 808 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We observe that VSML generalizes much better than Meta RNNs, Hebbian fast weights, and the external memory. These baselines overfit to the training environments. Notably, VSML even generalizes to the unseen tasks Random and Sum Sign which have no shared structure with the other datasets. In many cases VSML’s performance is similar to SGD but a little more sample efficient in the beginning of training (learning curves in Appendix B). This suggests that our meta learned LAs are good at quickly associating new inputs with their labels. We further investigate this in the next Section 5. ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 Analysis ", + "text_level": 1, + "bbox": [ + 174, + 89, + 276, + 106 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Given that VSML seems to learn faster than online gradient descent in many cases we would like to qualitatively investigate how learning differs. We first meta train on the full MNIST dataset as before. During meta testing, we plot the output probabilities for each digit against the number of samples seen in Figure 9. We highlight the ground truth input class $\\boxed { \\begin{array} { r l } \\end{array} }$ as well as the predicted class $\\bigcirc$ . In this case, our meta test dataset consists of MNIST digits with two examples of each type. The same digit is always repeated twice. This allows us to observe and visualize the effect with only a few examples. We have done the same introspection with the full dataset in Appendix B. ", + "bbox": [ + 174, + 121, + 356, + 481 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We observe that in VSML almost all failed predictions are followed by the correct prediction with high certainty. In contrast, SGD ", + "bbox": [ + 173, + 486, + 354, + 555 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/eaac26941486bc755c21019531e0870ec146a660bf3bfc809cb2fca51d5e7b20.jpg", + "image_caption": [ + "Figure 9: Introspection of how output probabilities change after observing an input and the prediction error when meta testing on MNIST with two examples for each type. We highlight the ground truth class $\\boxed { \\begin{array} { r l } \\end{array} }$ as well as the predicted class $\\bigcirc$ . The top plot shows VSML quickly associating the input images with the right label, almost always making the right prediction the second time with high confidence. The bottom plot shows the same dataset processed by SGD with Adam which fails to learn quickly. " + ], + "image_footnote": [], + "bbox": [ + 367, + 116, + 826, + 431 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "makes many incorrect predictions and fails to adapt correctly in only 20 steps. It seems that SGD learns qualitatively different from VSML. The VSML RNN meta learns to quickly associate new inputs with their class whereas SGD fails to do so. We tried several different SGD learning rates and considered multiple steps on the same input. In both cases, SGD does not behave similar to VSML, either learning much slower or forgetting previous examples. As evident from high accuracies in Figure 8, VSML does not only memorize inputs using this strategy of fast association, but the associations generalize to future unseen inputs. ", + "bbox": [ + 174, + 555, + 825, + 651 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 671, + 321, + 689 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Memory based meta learning (Meta RNNs) Memory-based meta learning in RNNs [16, 10, 56] is a simple neural meta learner (see Section 2). Unfortunately, the LA encoded in the RNN parameters is largely overparameterized which leads to overfitting. VSML demonstrates that weight sharing can address this issue resulting in more general-purpose LAs. ", + "bbox": [ + 174, + 704, + 825, + 760 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Learned Learning Rules / Fast Weights NNs that generate or change the weights of another or the same NN are known as fast weight programmers [44], hypernetworks [14], NNs with synaptic plasticity [25] or learned learning rules [4] (see Section 2). In VSML we do not require explicit architectures for weight updates as weights are emergent from RNN state updates. In addition to the learning rule, we implicitly learn how the neural forward computation is defined. Concurrent to this work, fast weights have also been used to meta learn more general LAs [39]. ", + "bbox": [ + 173, + 766, + 825, + 849 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Learned gradient-based optimizers Meta learning has been used to find optimizers that update the parameters of a model by taking the loss and gradient with respect to these parameters as an input [34, 2, 22, 24]. In this work, we are interested in meta learning that does not rely on fixed gradient calculation in the inner loop. ", + "bbox": [ + 176, + 856, + 823, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Discrete program search An interesting alternative to distributed variable updates in VSML is meta learning via discrete program search [48, 35]. In this paradigm, a separate programming language needs to be defined that gives rise to neural computation when its instructions are combined. This led to the automated rediscovery of backpropagation [35]. In VSML we demonstrate that a symbolic programming language is not required and general-purpose LAs can be discovered and encoded in variable-shared RNNs. Search over neural network parameters is usually easier compared to symbolic program search due to smoothness in the loss landscape. ", + "bbox": [ + 174, + 92, + 825, + 188 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Multi-agent systems In the reinforcement learning setting multiple agents can be modeled with shared parameters [50, 32, 18], also in the context of meta learning [36]. This is related to the variable sharing in VSML depending on how the agent-environment boundary is drawn. Unlike these works, we demonstrate the advantage of variable sharing in meta learning more general-purpose LAs and present a weight update interpretation. ", + "bbox": [ + 174, + 194, + 825, + 263 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 Discussion and Limitations ", + "text_level": 1, + "bbox": [ + 176, + 282, + 431, + 300 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The research community has perfected the art of leveraging backpropagation for learning for a long time. At the same time, there are open questions such as how to minimize memory requirements, effectively learn online and continually, learn sample efficiently, learn without separate backward phases, and others. VSML suggests that instead of building on top of backpropagation as a fixed routine, meta learning offers an alternative to discover general-purpose LAs. Nevertheless, this paper is only a proof of concept—until now we have only investigated small-scale problems and performance does not yet quite match the mini-batched setting with large quantities of data. In particular, we observed premature convergence of the solution at meta test time which calls for further investigations. Scaling our system to harder problems and larger meta task distributions will be important future work. ", + "bbox": [ + 174, + 314, + 825, + 453 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The computational cost of the current VSML variant is also larger than the one of standard backpropagation. If we run a sub-RNN for each weight in a standard NN with $W$ weights, the cost is in $O ( W N ^ { 2 } )$ , where $N$ is the state size of a sub-RNN. If $N$ is small enough, and our experiments suggest small $N$ may be feasible, this may be an acceptable cost. However, VSML is not bound to the interpretation of a sub-RNN as one weight. Future work may relax this particular choice. ", + "bbox": [ + 174, + 459, + 825, + 529 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Meta optimization is also prone to local minima. In particular, when the number of ticks between input and feedback increases (e.g. deeper architectures), credit assignment becomes harder. Early experiments suggest that diverse meta task distributions can help mitigate these issues. Additionally, learning horizons are limited when using backprop-based meta optimization. Using ES allowed for training across longer horizons and more stable optimization. ", + "bbox": [ + 174, + 535, + 825, + 604 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "VSML can also be viewed as regularizing the NN weights that encode the LA through a representational bottleneck. It is conceivable that LA generalization as obtained by VSML can also be achieved through other regularization techniques. Unlike most regularizers, VSML also introduces substantial reuse of the same learning principle and permutation invariance through variable sharing. ", + "bbox": [ + 174, + 611, + 825, + 666 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 685, + 299, + 703 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We introduced Variable Shared Meta Learning (VSML), a simple principle of weight sharing and sparsity for meta learning powerful learning algorithms (LAs). Our implementation replaces the weights of a neural network with tiny LSTMs that share parameters. We discuss connections to meta recurrent neural networks, fast weight generators (hyper networks), and learned learning rules. ", + "bbox": [ + 174, + 717, + 825, + 773 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Using learning algorithm cloning, VSML RNNs can learn to implement the backpropagation algorithm and its parameter updates encoded implicitly in the recurrent dynamics. On MNIST it learns to predict well without any human-designed explicit computational graph for gradient calculation. ", + "bbox": [ + 176, + 779, + 821, + 821 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "VSML can meta learn from scratch supervised LAs that do not explicitly rely on gradient computation and that generalize to unseen datasets. Introspection reveals that VSML LAs learn by fast association in a way that is qualitatively different from stochastic gradient descent. This leads to gains in sample efficiency. Future work will focus on reinforcement learning settings, improvements of meta learning, larger task distributions, and learning over longer horizons. ", + "bbox": [ + 174, + 827, + 825, + 897 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements ", + "text_level": 1, + "bbox": [ + 176, + 89, + 338, + 106 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We thank Sjoerd van Steenkiste, Imanol Schlag, Kazuki Irie, and the anonymous reviewers for their comments and feedback. This work was supported by the ERC Advanced Grant (no: 742870) and computational resources by the Swiss National Supercomputing Centre (CSCS, projects $s 9 7 8$ and s1041). We also thank NVIDIA Corporation for donating several DGX machines as part of the Pioneers of AI Research Award, IBM for donating a Minsky machine, and weights & biases [5] for their great experiment tracking software and support. 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It can even meta learn new LAs that differ from online", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 346, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 142, + 346, + 469, + 357 + ], + "score": 1.0, + "content": "backpropagation and generalize to datasets outside of the meta training distribution", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 357, + 469, + 368 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 469, + 368 + ], + "score": 1.0, + "content": "without explicit gradient calculation. Introspection reveals that our meta learned", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 367, + 469, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 469, + 379 + ], + "score": 1.0, + "content": "LAs learn through fast association in a way that is qualitatively different from", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 379, + 212, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 379, + 212, + 389 + ], + "score": 1.0, + "content": "gradient descent.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5, + "bbox_fs": [ + 140, + 258, + 471, + 389 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 410, + 190, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 192, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 192, + 426 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "The shift from standard machine learning to meta learning involves learning the learning algorithm", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "(LA) itself, reducing the burden on the human designer to craft useful learning algorithms [43].", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 457, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 506, + 469 + ], + "score": 1.0, + "content": "Recent meta learning has primarily focused on generalization from training tasks to similar test tasks,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "e.g., few-shot learning [11], or from training environments to similar test environments [17]. This", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 479, + 507, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 507, + 491 + ], + "score": 1.0, + "content": "contrasts human-engineered LAs that generalize across a wide range of datasets or environments.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 489, + 507, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 507, + 503 + ], + "score": 1.0, + "content": "Without such generalization, meta learned LAs can not entirely replace human-engineered variants.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Recent work demonstrated that meta learning can also successfully generate more general LAs", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "that generalize across wide spectra of environments [20, 1, 31], e.g., from toy environments to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "Mujoco and Atari. Unfortunately, however, many recent approaches still rely on a large number of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 533, + 444, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 444, + 546 + ], + "score": 1.0, + "content": "human-designed and unmodifiable inner-loop components such as backpropagation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 434, + 507, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "Is it possible to implement modifiable versions of backpropagation or related algorithms as part", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "score": 1.0, + "content": "of the end-to-end differentiable activation dynamics of a neural net (NN), instead of inserting", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "them as separate fixed routines? Here we propose the Variable Shared Meta Learning (VSML)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "principle for this purpose. It introduces a novel way of using sparsity and weight-sharing in NNs", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "for meta learning. We build on the arguably simplest neural meta learner, the meta recurrent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 506, + 618 + ], + "score": 1.0, + "content": "neural network (Meta RNN) [16, 10, 56], by replacing the weights of a neural network with tiny", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "LSTMs. The resulting system can be viewed as many RNNs passing messages to each other,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "or as one big RNN with a sparse shared weight matrix, or as a system learning each neuron’s", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "functionality and its LA. VSML generalizes the principle behind end-to-end differentiable fast weight", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "programmers [45, 46, 3, 41], hyper networks [14], learned learning rules [4, 13, 33], and hebbian-like", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "score": 1.0, + "content": "synaptic plasticity [44, 46, 25, 26, 30]. Our mechanism, VSML, can implement backpropagation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "score": 1.0, + "content": "solely in the forward-dynamics of an RNN. Consequently, it enables meta-optimization of backprop-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "like algorithms. We envision a future where novel methods of credit assignment can be meta learned", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 690, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 705 + ], + "score": 1.0, + "content": "while still generalizing across vastly different tasks. This may lead to improvements in sample", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "score": 1.0, + "content": "efficiency, memory efficiency, continual learning, and others. As a first step, our system meta", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "learns online LAs from scratch that frequently learn faster than gradient descent and generalize to", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 507, + 96 + ], + "score": 1.0, + "content": "datasets outside of the meta training distribution (e.g., from MNIST to Fashion MNIST). Our VSML", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "score": 1.0, + "content": "RNN is the first neural meta learner without hard-coded backpropagation that shows such strong", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 168, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 168, + 118 + ], + "score": 1.0, + "content": "generalization.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 550, + 506, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "learns online LAs from scratch that frequently learn faster than gradient descent and generalize to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 507, + 96 + ], + "score": 1.0, + "content": "datasets outside of the meta training distribution (e.g., from MNIST to Fashion MNIST). Our VSML", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "score": 1.0, + "content": "RNN is the first neural meta learner without hard-coded backpropagation that shows such strong", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 168, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 168, + 118 + ], + "score": 1.0, + "content": "generalization.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 131, + 189, + 145 + ], + "lines": [ + { + "bbox": [ + 103, + 129, + 191, + 149 + ], + "spans": [ + { + "bbox": [ + 103, + 129, + 191, + 149 + ], + "score": 1.0, + "content": "2 Background", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 156, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "score": 1.0, + "content": "Deep learning-based meta learning that does not rely on fixed gradient descent in the inner loop", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 166, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 180 + ], + "score": 1.0, + "content": "has historically fallen into two categories, 1) Learnable weight update mechanisms that allow for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "changing the parameters of an NN to implement a learning rule (FWPs / LLRs), and 2) Learning", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 486, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 486, + 201 + ], + "score": 1.0, + "content": "algorithms implemented in black-box models such as recurrent neural networks (Meta RNNs).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "Fast weight programmers & Learned learning rules (FWPs / LLRs) In a standard NN, the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "weights are updated by a fixed LA. This framework can be extended to meta learning by defining", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "an explicit architecture that allows for modifying these weights. This weight-update architecture", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "score": 1.0, + "content": "augments a standard NN architecture. NNs that generate or change the weights of another or the same", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "NN are known as fast weight programmers (FWPs) [44, 45, 46, 3, 41], hypernetworks [14], NNs", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "with synaptic plasticity [25, 26, 30] or learned learning rules [4, 13, 33]. Often these architectures", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 275, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 290 + ], + "score": 1.0, + "content": "make use of local Hebbian-like update rules or outer-products, and we summarize this category as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 290, + 300 + ], + "score": 1.0, + "content": "FWPs / LLRs. In FWPs / LLRs the variables", + "type": "text" + }, + { + "bbox": [ + 291, + 288, + 304, + 299 + ], + "score": 0.88, + "content": "V _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "that are subject to learning are the weights of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 257, + 311 + ], + "score": 1.0, + "content": "network, whereas the meta-variables", + "type": "text" + }, + { + "bbox": [ + 258, + 299, + 273, + 309 + ], + "score": 0.9, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "that implement the LA are defined by the weight-update", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 284, + 322 + ], + "score": 1.0, + "content": "architecture. Note that the dimensionality of", + "type": "text" + }, + { + "bbox": [ + 285, + 310, + 297, + 321 + ], + "score": 0.89, + "content": "V _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 309, + 315, + 322 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 316, + 309, + 331, + 321 + ], + "score": 0.91, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "can be defined independently of each other", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 146, + 334 + ], + "score": 1.0, + "content": "and often", + "type": "text" + }, + { + "bbox": [ + 146, + 321, + 162, + 331 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 320, + 334, + 334 + ], + "score": 1.0, + "content": "are reused in a coordinate-wise fashion for", + "type": "text" + }, + { + "bbox": [ + 335, + 321, + 347, + 332 + ], + "score": 0.88, + "content": "V _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 320, + 396, + 334 + ], + "score": 1.0, + "content": "resulting in", + "type": "text" + }, + { + "bbox": [ + 396, + 320, + 450, + 333 + ], + "score": 0.93, + "content": "| V _ { L } | \\gg | V _ { M } |", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 320, + 480, + 334 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 481, + 320, + 495, + 333 + ], + "score": 0.84, + "content": "| \\cdot |", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 331, + 205, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 205, + 343 + ], + "score": 1.0, + "content": "the number of elements.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 506, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "Black-box learning in activations (Meta RNNs) It was shown that an RNN such as LSTM can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 363, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 506, + 379 + ], + "score": 1.0, + "content": "learn to implement an LA [16] when the reward or error is given as an input [47]. After meta training,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "the LA is encoded in the weights of this RNN and determines learning during meta testing. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "activations serve as the memory used for the LA solution. We refer to this as Meta RNNs [16, 10, 56]", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "(Also sometimes referred to as memory-based meta learning.). They are conceptually simpler than", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "score": 1.0, + "content": "FWPs / LLRs as no additional weight-update rules with many degrees of freedom need to be defined.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 420, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 169, + 432 + ], + "score": 1.0, + "content": "In Meta RNNs", + "type": "text" + }, + { + "bbox": [ + 169, + 420, + 182, + 430 + ], + "score": 0.88, + "content": "V _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 420, + 301, + 432 + ], + "score": 1.0, + "content": "are the RNN activations and", + "type": "text" + }, + { + "bbox": [ + 302, + 420, + 317, + 430 + ], + "score": 0.9, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 420, + 506, + 432 + ], + "score": 1.0, + "content": "are the parameters for the RNN. Note that an", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 151, + 443 + ], + "score": 1.0, + "content": "RNN with", + "type": "text" + }, + { + "bbox": [ + 151, + 431, + 162, + 440 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 429, + 240, + 443 + ], + "score": 1.0, + "content": "neurons will yield", + "type": "text" + }, + { + "bbox": [ + 241, + 430, + 298, + 442 + ], + "score": 0.93, + "content": "| V _ { L } | = O ( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 429, + 317, + 443 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 430, + 381, + 442 + ], + "score": 0.95, + "content": "| V _ { M } | = \\stackrel { \\cdot } { O } ( N ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "[46]. This means that the LA", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "is largely overparameterized whereas the available memory for learning is very small, making this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "approach prone to overfitting [20]. As a result, the RNN parameters often encode task-specific", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "solutions instead of generic LAs. Meta learning a simple and generalizing LA would benefit from", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 473, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 107, + 474, + 160, + 486 + ], + "score": 0.92, + "content": "| V _ { L } | \\gg | V _ { M } |", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 473, + 505, + 488 + ], + "score": 1.0, + "content": ". Previous approaches have tried to mend this issue by adding architectural complexity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 485, + 351, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 351, + 497 + ], + "score": 1.0, + "content": "through additional memory mechanisms [53, 29, 40, 27, 42].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 106, + 510, + 334, + 525 + ], + "lines": [ + { + "bbox": [ + 103, + 509, + 335, + 528 + ], + "spans": [ + { + "bbox": [ + 103, + 509, + 335, + 528 + ], + "score": 1.0, + "content": "3 Variable Shared Meta Learning (VSML)", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 535, + 506, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 371, + 548 + ], + "score": 1.0, + "content": "In VSML we build on the simplicity of Meta RNNs while retaining", + "type": "text" + }, + { + "bbox": [ + 372, + 535, + 426, + 547 + ], + "score": 0.93, + "content": "| V _ { L } | \\gg | V _ { M } |", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "from FWPs / LLRs.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 293, + 559 + ], + "score": 1.0, + "content": "We do this by reusing the same few parameters", + "type": "text" + }, + { + "bbox": [ + 293, + 547, + 308, + 557 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "many times in an RNN (via variable sharing) and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 557, + 451, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 451, + 569 + ], + "score": 1.0, + "content": "introducing sparsity in the connectivity. This yields several interpretations for VSML:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 126, + 577, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 125, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 125, + 576, + 140, + 590 + ], + "score": 0.82, + "content": "\\textcircled{8}", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "VSML as a single Meta RNN with a sparse shared weight matrix (Figure 1a). The", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 142, + 589, + 244, + 600 + ], + "spans": [ + { + "bbox": [ + 142, + 589, + 244, + 600 + ], + "score": 1.0, + "content": "most general description.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 126, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 126, + 603, + 140, + 616 + ], + "score": 0.82, + "content": "\\textcircled{8}", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 603, + 506, + 618 + ], + "score": 1.0, + "content": "VSML as message passing between RNNs (Figure 1b). We choose a simple sharing and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 142, + 615, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 142, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "sparsity scheme for the weight matrix such that it corresponds to multiple RNNs with shared", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 142, + 627, + 298, + 639 + ], + "spans": [ + { + "bbox": [ + 142, + 627, + 298, + 639 + ], + "score": 1.0, + "content": "parameters that exchange information.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 126, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 126, + 640, + 140, + 653 + ], + "score": 0.8, + "content": "©", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "VSML as complex neurons with learned updates (Figure 1c). When choosing a specific", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 141, + 653, + 332, + 665 + ], + "score": 1.0, + "content": "connectivity between RNNs, states / activations", + "type": "text" + }, + { + "bbox": [ + 333, + 653, + 345, + 663 + ], + "score": 0.89, + "content": "V _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "of these RNNs can be interpreted as the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 664, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 141, + 664, + 506, + 676 + ], + "score": 1.0, + "content": "weights of a conventional NN, consequently blurring the distinction between a weight and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 142, + 675, + 198, + 686 + ], + "spans": [ + { + "bbox": [ + 142, + 675, + 198, + 686 + ], + "score": 1.0, + "content": "an activation.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Introducing variable sharing to Meta RNNs We begin by formalizing Meta RNNs which often", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "use multiplicative gates such as the LSTM [12, 15] or its variant GRU [6]. For notational simplicity,", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 72, + 507, + 118 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 131, + 189, + 145 + ], + "lines": [ + { + "bbox": [ + 103, + 129, + 191, + 149 + ], + "spans": [ + { + "bbox": [ + 103, + 129, + 191, + 149 + ], + "score": 1.0, + "content": "2 Background", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 156, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "score": 1.0, + "content": "Deep learning-based meta learning that does not rely on fixed gradient descent in the inner loop", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 166, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 180 + ], + "score": 1.0, + "content": "has historically fallen into two categories, 1) Learnable weight update mechanisms that allow for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "changing the parameters of an NN to implement a learning rule (FWPs / LLRs), and 2) Learning", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 486, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 486, + 201 + ], + "score": 1.0, + "content": "algorithms implemented in black-box models such as recurrent neural networks (Meta RNNs).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 156, + 506, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "Fast weight programmers & Learned learning rules (FWPs / LLRs) In a standard NN, the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "weights are updated by a fixed LA. This framework can be extended to meta learning by defining", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "an explicit architecture that allows for modifying these weights. This weight-update architecture", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "score": 1.0, + "content": "augments a standard NN architecture. NNs that generate or change the weights of another or the same", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "NN are known as fast weight programmers (FWPs) [44, 45, 46, 3, 41], hypernetworks [14], NNs", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "with synaptic plasticity [25, 26, 30] or learned learning rules [4, 13, 33]. Often these architectures", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 275, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 290 + ], + "score": 1.0, + "content": "make use of local Hebbian-like update rules or outer-products, and we summarize this category as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 290, + 300 + ], + "score": 1.0, + "content": "FWPs / LLRs. In FWPs / LLRs the variables", + "type": "text" + }, + { + "bbox": [ + 291, + 288, + 304, + 299 + ], + "score": 0.88, + "content": "V _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "that are subject to learning are the weights of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 257, + 311 + ], + "score": 1.0, + "content": "network, whereas the meta-variables", + "type": "text" + }, + { + "bbox": [ + 258, + 299, + 273, + 309 + ], + "score": 0.9, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "that implement the LA are defined by the weight-update", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 284, + 322 + ], + "score": 1.0, + "content": "architecture. Note that the dimensionality of", + "type": "text" + }, + { + "bbox": [ + 285, + 310, + 297, + 321 + ], + "score": 0.89, + "content": "V _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 309, + 315, + 322 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 316, + 309, + 331, + 321 + ], + "score": 0.91, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "can be defined independently of each other", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 146, + 334 + ], + "score": 1.0, + "content": "and often", + "type": "text" + }, + { + "bbox": [ + 146, + 321, + 162, + 331 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 320, + 334, + 334 + ], + "score": 1.0, + "content": "are reused in a coordinate-wise fashion for", + "type": "text" + }, + { + "bbox": [ + 335, + 321, + 347, + 332 + ], + "score": 0.88, + "content": "V _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 320, + 396, + 334 + ], + "score": 1.0, + "content": "resulting in", + "type": "text" + }, + { + "bbox": [ + 396, + 320, + 450, + 333 + ], + "score": 0.93, + "content": "| V _ { L } | \\gg | V _ { M } |", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 320, + 480, + 334 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 481, + 320, + 495, + 333 + ], + "score": 0.84, + "content": "| \\cdot |", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 331, + 205, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 205, + 343 + ], + "score": 1.0, + "content": "the number of elements.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 212, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 506, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "Black-box learning in activations (Meta RNNs) It was shown that an RNN such as LSTM can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 363, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 506, + 379 + ], + "score": 1.0, + "content": "learn to implement an LA [16] when the reward or error is given as an input [47]. After meta training,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "the LA is encoded in the weights of this RNN and determines learning during meta testing. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "activations serve as the memory used for the LA solution. We refer to this as Meta RNNs [16, 10, 56]", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "(Also sometimes referred to as memory-based meta learning.). They are conceptually simpler than", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "score": 1.0, + "content": "FWPs / LLRs as no additional weight-update rules with many degrees of freedom need to be defined.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 420, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 169, + 432 + ], + "score": 1.0, + "content": "In Meta RNNs", + "type": "text" + }, + { + "bbox": [ + 169, + 420, + 182, + 430 + ], + "score": 0.88, + "content": "V _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 420, + 301, + 432 + ], + "score": 1.0, + "content": "are the RNN activations and", + "type": "text" + }, + { + "bbox": [ + 302, + 420, + 317, + 430 + ], + "score": 0.9, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 420, + 506, + 432 + ], + "score": 1.0, + "content": "are the parameters for the RNN. Note that an", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 151, + 443 + ], + "score": 1.0, + "content": "RNN with", + "type": "text" + }, + { + "bbox": [ + 151, + 431, + 162, + 440 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 429, + 240, + 443 + ], + "score": 1.0, + "content": "neurons will yield", + "type": "text" + }, + { + "bbox": [ + 241, + 430, + 298, + 442 + ], + "score": 0.93, + "content": "| V _ { L } | = O ( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 429, + 317, + 443 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 430, + 381, + 442 + ], + "score": 0.95, + "content": "| V _ { M } | = \\stackrel { \\cdot } { O } ( N ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "[46]. This means that the LA", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "is largely overparameterized whereas the available memory for learning is very small, making this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "approach prone to overfitting [20]. As a result, the RNN parameters often encode task-specific", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "solutions instead of generic LAs. Meta learning a simple and generalizing LA would benefit from", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 473, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 107, + 474, + 160, + 486 + ], + "score": 0.92, + "content": "| V _ { L } | \\gg | V _ { M } |", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 473, + 505, + 488 + ], + "score": 1.0, + "content": ". Previous approaches have tried to mend this issue by adding architectural complexity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 485, + 351, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 351, + 497 + ], + "score": 1.0, + "content": "through additional memory mechanisms [53, 29, 40, 27, 42].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 353, + 506, + 497 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 510, + 334, + 525 + ], + "lines": [ + { + "bbox": [ + 103, + 509, + 335, + 528 + ], + "spans": [ + { + "bbox": [ + 103, + 509, + 335, + 528 + ], + "score": 1.0, + "content": "3 Variable Shared Meta Learning (VSML)", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 535, + 506, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 371, + 548 + ], + "score": 1.0, + "content": "In VSML we build on the simplicity of Meta RNNs while retaining", + "type": "text" + }, + { + "bbox": [ + 372, + 535, + 426, + 547 + ], + "score": 0.93, + "content": "| V _ { L } | \\gg | V _ { M } |", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "from FWPs / LLRs.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 293, + 559 + ], + "score": 1.0, + "content": "We do this by reusing the same few parameters", + "type": "text" + }, + { + "bbox": [ + 293, + 547, + 308, + 557 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "many times in an RNN (via variable sharing) and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 557, + 451, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 451, + 569 + ], + "score": 1.0, + "content": "introducing sparsity in the connectivity. This yields several interpretations for VSML:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 534, + 506, + 569 + ] + }, + { + "type": "list", + "bbox": [ + 126, + 577, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 125, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 125, + 576, + 140, + 590 + ], + "score": 0.82, + "content": "\\textcircled{8}", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "VSML as a single Meta RNN with a sparse shared weight matrix (Figure 1a). The", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 589, + 244, + 600 + ], + "spans": [ + { + "bbox": [ + 142, + 589, + 244, + 600 + ], + "score": 1.0, + "content": "most general description.", + "type": "text" + } + ], + "index": 39, + "is_list_end_line": true + }, + { + "bbox": [ + 126, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 126, + 603, + 140, + 616 + ], + "score": 0.82, + "content": "\\textcircled{8}", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 603, + 506, + 618 + ], + "score": 1.0, + "content": "VSML as message passing between RNNs (Figure 1b). We choose a simple sharing and", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 615, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 142, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "sparsity scheme for the weight matrix such that it corresponds to multiple RNNs with shared", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 142, + 627, + 298, + 639 + ], + "spans": [ + { + "bbox": [ + 142, + 627, + 298, + 639 + ], + "score": 1.0, + "content": "parameters that exchange information.", + "type": "text" + } + ], + "index": 42, + "is_list_end_line": true + }, + { + "bbox": [ + 126, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 126, + 640, + 140, + 653 + ], + "score": 0.8, + "content": "©", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "VSML as complex neurons with learned updates (Figure 1c). 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(a) VSML consists of many sub-RNNs with shared parameters", + "type": "text" + }, + { + "bbox": [ + 456, + 194, + 472, + 204 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 192, + 506, + 207 + ], + "score": 1.0, + "content": "passing", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "messages between each other. (c) VSML implements an NN with complex neurons (here 2 neurons).", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 107, + 216, + 122, + 226 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "determines the nature of weights, how these are used in the neural computation, and the LA by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 261, + 239 + ], + "score": 1.0, + "content": "which those are updated. Each weight", + "type": "text" + }, + { + "bbox": [ + 262, + 226, + 298, + 237 + ], + "score": 0.93, + "content": "w _ { a b } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "is represented by the multi-dimensional RNN state", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 236, + 452, + 252 + ], + "spans": [ + { + "bbox": [ + 107, + 237, + 147, + 250 + ], + "score": 0.93, + "content": "{ s _ { a b } } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 236, + 327, + 252 + ], + "score": 1.0, + "content": ". Neuron activations correspond to messages", + "type": "text" + }, + { + "bbox": [ + 327, + 237, + 338, + 249 + ], + "score": 0.83, + "content": "\\overrightarrow { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 236, + 452, + 252 + ], + "score": 1.0, + "content": "passed between sub-RNNs.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 108, + 269, + 504, + 293 + ], + "lines": [ + { + "bbox": [ + 104, + 267, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 236, + 284 + ], + "score": 1.0, + "content": "we consider a vanilla RNN. Let", + "type": "text" + }, + { + "bbox": [ + 236, + 269, + 268, + 281 + ], + "score": 0.92, + "content": "s \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 267, + 506, + 284 + ], + "score": 1.0, + "content": "be the hidden state of an RNN. 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We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 269, + 344 + ], + "score": 1.0, + "content": "also omit inputs by assuming a subset of", + "type": "text" + }, + { + "bbox": [ + 270, + 334, + 275, + 342 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "to be externally provided. Each application of Equation 1", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 343, + 259, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 259, + 354 + ], + "score": 1.0, + "content": "reflects a single time tick in the RNN.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 358, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 293, + 372 + ], + "score": 1.0, + "content": "We now introduce variable sharing (reusing", + "type": "text" + }, + { + "bbox": [ + 293, + 360, + 305, + 370 + ], + "score": 0.73, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 358, + 506, + 372 + ], + "score": 1.0, + "content": ") into the RNN by duplicating the computation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 369, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 200, + 385 + ], + "score": 1.0, + "content": "along two axes of size", + "type": "text" + }, + { + "bbox": [ + 200, + 372, + 221, + 383 + ], + "score": 0.89, + "content": "A , B", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 369, + 246, + 385 + ], + "score": 1.0, + "content": "(here", + "type": "text" + }, + { + "bbox": [ + 246, + 372, + 277, + 382 + ], + "score": 0.91, + "content": "A = B", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 369, + 425, + 385 + ], + "score": 1.0, + "content": ", which will later be relaxed) giving", + "type": "text" + }, + { + "bbox": [ + 425, + 370, + 483, + 382 + ], + "score": 0.92, + "content": "\\bar { s } \\in \\mathbb { R } ^ { A \\times B \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 369, + 506, + 385 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 383, + 272, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 236, + 395 + ], + "score": 0.89, + "content": "a \\in \\{ 1 , \\dotsc , A \\} , b \\in \\{ 1 , \\dotsc , B \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 383, + 272, + 395 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 398, + 385, + 424 + ], + "lines": [ + { + "bbox": [ + 225, + 398, + 385, + 424 + ], + "spans": [ + { + "bbox": [ + 225, + 398, + 385, + 424 + ], + "score": 0.95, + "content": "s _ { a b j } \\gets f _ { \\mathrm { R N N } } ( s _ { a b } ) _ { j } = \\sigma ( \\sum _ { i } s _ { a b i } W _ { i j } ) .", + "type": "interline_equation", + "image_path": "bd446ad5b95522b305babefe9b762f10f82ec8bf83cb2ee5e6827d94874df36a.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 225, + 398, + 385, + 424 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "This can be viewed as multiple RNNs arranged on a 2-dimensional grid, with shared parameters that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "update independent states. 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We connect those by passing messages (interpretation", + "type": "text" + }, + { + "bbox": [ + 354, + 484, + 366, + 497 + ], + "score": 0.85, + "content": "\\textcircled{8}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 484, + 371, + 499 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 500, + 352, + 514 + ], + "lines": [ + { + "bbox": [ + 258, + 500, + 352, + 514 + ], + "spans": [ + { + "bbox": [ + 258, + 500, + 352, + 514 + ], + "score": 0.91, + "content": "s _ { a b } \\gets f _ { \\mathrm { R N N } } ( s _ { a b } , \\overrightarrow { m } _ { a } ) ,", + "type": "interline_equation", + "image_path": "dbe020d5af66f40b67afaa6858cc17392b6229b77c392d47d8f437e8f795275d.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 258, + 500, + 352, + 514 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 506, + 590 + ], + "lines": [ + { + "bbox": [ + 104, + 518, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 518, + 186, + 534 + ], + "score": 1.0, + "content": "where the message", + "type": "text" + }, + { + "bbox": [ + 187, + 520, + 271, + 533 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\vec { m } _ { b } = \\sum _ { a ^ { \\prime } } f _ { \\vec { m } } ( s _ { a ^ { \\prime } b } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 518, + 294, + 534 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 294, + 520, + 379, + 533 + ], + "score": 0.87, + "content": "b \\in \\{ 1 , \\dots , A = B \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 518, + 383, + 534 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 383, + 518, + 454, + 533 + ], + "score": 0.89, + "content": "f _ { \\overrightarrow { m } } : \\mathbb { R } ^ { N } \\mathbb { R } ^ { N ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 518, + 506, + 534 + ], + "score": 1.0, + "content": "is fed as an", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "score": 1.0, + "content": "additional input to each RNN. This is related to Graph Neural Networks [51, 58]. Summing over the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 125, + 556 + ], + "score": 1.0, + "content": "axis", + "type": "text" + }, + { + "bbox": [ + 125, + 543, + 134, + 553 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 542, + 176, + 556 + ], + "score": 1.0, + "content": "(elements", + "type": "text" + }, + { + "bbox": [ + 177, + 543, + 186, + 553 + ], + "score": 0.74, + "content": "a ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 542, + 506, + 556 + ], + "score": 1.0, + "content": ") corresponds to an RNN connectivity mimicking those of weights in an NN (to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 552, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 200, + 569 + ], + "score": 1.0, + "content": "facilitate interpretation", + "type": "text" + }, + { + "bbox": [ + 200, + 554, + 216, + 567 + ], + "score": 0.88, + "content": "©", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 552, + 506, + 569 + ], + "score": 1.0, + "content": ". We emphasise that other schemes based on different kinds of message", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 565, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 335, + 581 + ], + "score": 1.0, + "content": "passing and graph connectivity are possible. For a simple", + "type": "text" + }, + { + "bbox": [ + 336, + 568, + 350, + 579 + ], + "score": 0.9, + "content": "f _ { \\overrightarrow { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 565, + 438, + 581 + ], + "score": 1.0, + "content": "defined by the matrix", + "type": "text" + }, + { + "bbox": [ + 438, + 566, + 487, + 578 + ], + "score": 0.91, + "content": "C \\in \\mathbb { R } ^ { N \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 565, + 505, + 581 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 578, + 200, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 200, + 591 + ], + "score": 1.0, + "content": "may equivalently write", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 593, + 456, + 620 + ], + "lines": [ + { + "bbox": [ + 154, + 593, + 456, + 620 + ], + "spans": [ + { + "bbox": [ + 154, + 593, + 456, + 620 + ], + "score": 0.93, + "content": "s _ { a b j } \\gets \\sigma ( \\sum _ { i } s _ { a b i } W _ { i j } + \\sum _ { a ^ { \\prime } } f _ { \\overrightarrow { m } } ( s _ { a ^ { \\prime } a } ) ) = \\sigma ( \\sum _ { i } s _ { a b i } W _ { i j } + \\sum _ { a ^ { \\prime } , i } s _ { a ^ { \\prime } a i } C _ { i j } ) .", + "type": "interline_equation", + "image_path": "a4fa3ae13d4eb053bd2e1b570e336ff4662b2e762e505e889dc05c1a0c6ebc4a.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 154, + 593, + 456, + 602.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 154, + 602.0, + 456, + 611.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 154, + 611.0, + 456, + 620.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 624, + 498, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 499, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 315, + 638 + ], + "score": 1.0, + "content": "This constitutes the minimal version of VSML with", + "type": "text" + }, + { + "bbox": [ + 315, + 625, + 375, + 637 + ], + "score": 0.93, + "content": "V _ { M } : = ( W , C )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 624, + 499, + 638 + ], + "score": 1.0, + "content": "and is visualized in Figure 1b.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 647, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 477, + 660 + ], + "score": 1.0, + "content": "VSML as a Meta RNN with a sparse shared weight matrix It is trivial to see that with", + "type": "text" + }, + { + "bbox": [ + 477, + 648, + 505, + 658 + ], + "score": 0.87, + "content": "A = 1", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 123, + 671 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 659, + 151, + 669 + ], + "score": 0.89, + "content": "B = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "we obtain a single RNN and Equation 4 recovers the original Meta RNN Equation 1. In", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "the general case, we can derive an equivalent formulation that corresponds to a single standard RNN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 681, + 369, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 188, + 694 + ], + "score": 1.0, + "content": "with a single matrix", + "type": "text" + }, + { + "bbox": [ + 188, + 681, + 200, + 692 + ], + "score": 0.86, + "content": "\\tilde { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 682, + 369, + 694 + ], + "score": 1.0, + "content": "that has entries of zero and shared entries", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 698, + 363, + 725 + ], + "lines": [ + { + "bbox": [ + 248, + 698, + 363, + 725 + ], + "spans": [ + { + "bbox": [ + 248, + 698, + 363, + 725 + ], + "score": 0.94, + "content": "s _ { a b j } \\gets \\sigma ( \\sum _ { c , d , i } s _ { c d i } \\tilde { W } _ { c d i a b j } ) ,", + "type": "interline_equation", + "image_path": "eeb3ae8d9676dd584d8b30e1f2e7cd87e1831ed2eae316f1470b3cdab7e9006f.jpg" + } + ] + } + ], + "index": 40.5, + "virtual_lines": [ + { + "bbox": [ + 248, + 698, + 363, + 711.5 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 248, + 711.5, + 363, + 725.0 + ], + "spans": [], + "index": 41 + } + ] + } + ], + "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": "image", + "bbox": [ + 117, + 69, + 501, + 177 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 69, + 501, + 177 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 69, + 501, + 177 + ], + "spans": [ + { + "bbox": [ + 117, + 69, + 501, + 177 + ], + "score": 0.806, + "type": "image", + "image_path": "b1ae57af27d23bff08bcfc6f114db04ece9a7f8b38923249b2e9e1b42ad6a50f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 69, + 501, + 105.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 105.0, + 501, + 141.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 141.0, + 501, + 177.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 182, + 506, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "Figure 1: Different perspectives on VSML: (a) A single Meta RNN [16] where entries in the weight", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 192, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 456, + 207 + ], + "score": 1.0, + "content": "matrix are shared or zero. (a) VSML consists of many sub-RNNs with shared parameters", + "type": "text" + }, + { + "bbox": [ + 456, + 194, + 472, + 204 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 192, + 506, + 207 + ], + "score": 1.0, + "content": "passing", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "messages between each other. (c) VSML implements an NN with complex neurons (here 2 neurons).", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 107, + 216, + 122, + 226 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "determines the nature of weights, how these are used in the neural computation, and the LA by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 261, + 239 + ], + "score": 1.0, + "content": "which those are updated. Each weight", + "type": "text" + }, + { + "bbox": [ + 262, + 226, + 298, + 237 + ], + "score": 0.93, + "content": "w _ { a b } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "is represented by the multi-dimensional RNN state", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 236, + 452, + 252 + ], + "spans": [ + { + "bbox": [ + 107, + 237, + 147, + 250 + ], + "score": 0.93, + "content": "{ s _ { a b } } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 236, + 327, + 252 + ], + "score": 1.0, + "content": ". 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We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 269, + 344 + ], + "score": 1.0, + "content": "also omit inputs by assuming a subset of", + "type": "text" + }, + { + "bbox": [ + 270, + 334, + 275, + 342 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "to be externally provided. Each application of Equation 1", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 343, + 259, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 259, + 354 + ], + "score": 1.0, + "content": "reflects a single time tick in the RNN.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 317, + 506, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 358, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 293, + 372 + ], + "score": 1.0, + "content": "We now introduce variable sharing (reusing", + "type": "text" + }, + { + "bbox": [ + 293, + 360, + 305, + 370 + ], + "score": 0.73, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 358, + 506, + 372 + ], + "score": 1.0, + "content": ") into the RNN by duplicating the computation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 369, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 200, + 385 + ], + "score": 1.0, + "content": "along two axes of size", + "type": "text" + }, + { + "bbox": [ + 200, + 372, + 221, + 383 + ], + "score": 0.89, + "content": "A , B", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 369, + 246, + 385 + ], + "score": 1.0, + "content": "(here", + "type": "text" + }, + { + "bbox": [ + 246, + 372, + 277, + 382 + ], + "score": 0.91, + "content": "A = B", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 369, + 425, + 385 + ], + "score": 1.0, + "content": ", which will later be relaxed) giving", + "type": "text" + }, + { + "bbox": [ + 425, + 370, + 483, + 382 + ], + "score": 0.92, + "content": "\\bar { s } \\in \\mathbb { R } ^ { A \\times B \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 369, + 506, + 385 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 383, + 272, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 236, + 395 + ], + "score": 0.89, + "content": "a \\in \\{ 1 , \\dotsc , A \\} , b \\in \\{ 1 , \\dotsc , B \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 383, + 272, + 395 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 358, + 506, + 395 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 398, + 385, + 424 + ], + "lines": [ + { + "bbox": [ + 225, + 398, + 385, + 424 + ], + "spans": [ + { + "bbox": [ + 225, + 398, + 385, + 424 + ], + "score": 0.95, + "content": "s _ { a b j } \\gets f _ { \\mathrm { R N N } } ( s _ { a b } ) _ { j } = \\sigma ( \\sum _ { i } s _ { a b i } W _ { i j } ) .", + "type": "interline_equation", + "image_path": "bd446ad5b95522b305babefe9b762f10f82ec8bf83cb2ee5e6827d94874df36a.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 225, + 398, + 385, + 424 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "This can be viewed as multiple RNNs arranged on a 2-dimensional grid, with shared parameters that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "update independent states. Here, we chose a particular arrangement (two axes) that will facilitate the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 449, + 261, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 162, + 464 + ], + "score": 1.0, + "content": "interpretation", + "type": "text" + }, + { + "bbox": [ + 162, + 449, + 176, + 462 + ], + "score": 0.89, + "content": "©", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 450, + 261, + 464 + ], + "score": 1.0, + "content": "of RNNs as weights.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 427, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 504, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 428, + 486 + ], + "score": 1.0, + "content": "VSML as message passing between RNNs The computation so far describes", + "type": "text" + }, + { + "bbox": [ + 429, + 474, + 452, + 484 + ], + "score": 0.9, + "content": "A \\cdot B", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "independent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 484, + 371, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 353, + 499 + ], + "score": 1.0, + "content": "RNNs. We connect those by passing messages (interpretation", + "type": "text" + }, + { + "bbox": [ + 354, + 484, + 366, + 497 + ], + "score": 0.85, + "content": "\\textcircled{8}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 484, + 371, + 499 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 473, + 505, + 499 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 500, + 352, + 514 + ], + "lines": [ + { + "bbox": [ + 258, + 500, + 352, + 514 + ], + "spans": [ + { + "bbox": [ + 258, + 500, + 352, + 514 + ], + "score": 0.91, + "content": "s _ { a b } \\gets f _ { \\mathrm { R N N } } ( s _ { a b } , \\overrightarrow { m } _ { a } ) ,", + "type": "interline_equation", + "image_path": "dbe020d5af66f40b67afaa6858cc17392b6229b77c392d47d8f437e8f795275d.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 258, + 500, + 352, + 514 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 506, + 590 + ], + "lines": [ + { + "bbox": [ + 104, + 518, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 518, + 186, + 534 + ], + "score": 1.0, + "content": "where the message", + "type": "text" + }, + { + "bbox": [ + 187, + 520, + 271, + 533 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\vec { m } _ { b } = \\sum _ { a ^ { \\prime } } f _ { \\vec { m } } ( s _ { a ^ { \\prime } b } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 518, + 294, + 534 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 294, + 520, + 379, + 533 + ], + "score": 0.87, + "content": "b \\in \\{ 1 , \\dots , A = B \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 518, + 383, + 534 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 383, + 518, + 454, + 533 + ], + "score": 0.89, + "content": "f _ { \\overrightarrow { m } } : \\mathbb { R } ^ { N } \\mathbb { R } ^ { N ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 518, + 506, + 534 + ], + "score": 1.0, + "content": "is fed as an", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "score": 1.0, + "content": "additional input to each RNN. This is related to Graph Neural Networks [51, 58]. Summing over the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 125, + 556 + ], + "score": 1.0, + "content": "axis", + "type": "text" + }, + { + "bbox": [ + 125, + 543, + 134, + 553 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 542, + 176, + 556 + ], + "score": 1.0, + "content": "(elements", + "type": "text" + }, + { + "bbox": [ + 177, + 543, + 186, + 553 + ], + "score": 0.74, + "content": "a ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 542, + 506, + 556 + ], + "score": 1.0, + "content": ") corresponds to an RNN connectivity mimicking those of weights in an NN (to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 552, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 200, + 569 + ], + "score": 1.0, + "content": "facilitate interpretation", + "type": "text" + }, + { + "bbox": [ + 200, + 554, + 216, + 567 + ], + "score": 0.88, + "content": "©", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 552, + 506, + 569 + ], + "score": 1.0, + "content": ". We emphasise that other schemes based on different kinds of message", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 565, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 335, + 581 + ], + "score": 1.0, + "content": "passing and graph connectivity are possible. For a simple", + "type": "text" + }, + { + "bbox": [ + 336, + 568, + 350, + 579 + ], + "score": 0.9, + "content": "f _ { \\overrightarrow { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 565, + 438, + 581 + ], + "score": 1.0, + "content": "defined by the matrix", + "type": "text" + }, + { + "bbox": [ + 438, + 566, + 487, + 578 + ], + "score": 0.91, + "content": "C \\in \\mathbb { R } ^ { N \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 565, + 505, + 581 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 578, + 200, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 200, + 591 + ], + "score": 1.0, + "content": "may equivalently write", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 518, + 506, + 591 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 593, + 456, + 620 + ], + "lines": [ + { + "bbox": [ + 154, + 593, + 456, + 620 + ], + "spans": [ + { + "bbox": [ + 154, + 593, + 456, + 620 + ], + "score": 0.93, + "content": "s _ { a b j } \\gets \\sigma ( \\sum _ { i } s _ { a b i } W _ { i j } + \\sum _ { a ^ { \\prime } } f _ { \\overrightarrow { m } } ( s _ { a ^ { \\prime } a } ) ) = \\sigma ( \\sum _ { i } s _ { a b i } W _ { i j } + \\sum _ { a ^ { \\prime } , i } s _ { a ^ { \\prime } a i } C _ { i j } ) .", + "type": "interline_equation", + "image_path": "a4fa3ae13d4eb053bd2e1b570e336ff4662b2e762e505e889dc05c1a0c6ebc4a.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 154, + 593, + 456, + 602.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 154, + 602.0, + 456, + 611.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 154, + 611.0, + 456, + 620.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 624, + 498, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 499, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 315, + 638 + ], + "score": 1.0, + "content": "This constitutes the minimal version of VSML with", + "type": "text" + }, + { + "bbox": [ + 315, + 625, + 375, + 637 + ], + "score": 0.93, + "content": "V _ { M } : = ( W , C )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 624, + 499, + 638 + ], + "score": 1.0, + "content": "and is visualized in Figure 1b.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 624, + 499, + 638 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 647, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 477, + 660 + ], + "score": 1.0, + "content": "VSML as a Meta RNN with a sparse shared weight matrix It is trivial to see that with", + "type": "text" + }, + { + "bbox": [ + 477, + 648, + 505, + 658 + ], + "score": 0.87, + "content": "A = 1", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 123, + 671 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 659, + 151, + 669 + ], + "score": 0.89, + "content": "B = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "we obtain a single RNN and Equation 4 recovers the original Meta RNN Equation 1. In", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "the general case, we can derive an equivalent formulation that corresponds to a single standard RNN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 681, + 369, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 188, + 694 + ], + "score": 1.0, + "content": "with a single matrix", + "type": "text" + }, + { + "bbox": [ + 188, + 681, + 200, + 692 + ], + "score": 0.86, + "content": "\\tilde { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 682, + 369, + 694 + ], + "score": 1.0, + "content": "that has entries of zero and shared entries", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 647, + 505, + 694 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 698, + 363, + 725 + ], + "lines": [ + { + "bbox": [ + 248, + 698, + 363, + 725 + ], + "spans": [ + { + "bbox": [ + 248, + 698, + 363, + 725 + ], + "score": 0.94, + "content": "s _ { a b j } \\gets \\sigma ( \\sum _ { c , d , i } s _ { c d i } \\tilde { W } _ { c d i a b j } ) ,", + "type": "interline_equation", + "image_path": "eeb3ae8d9676dd584d8b30e1f2e7cd87e1831ed2eae316f1470b3cdab7e9006f.jpg" + } + ] + } + ], + "index": 40.5, + "virtual_lines": [ + { + "bbox": [ + 248, + 698, + 363, + 711.5 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 248, + 711.5, + 363, + 725.0 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 72, + 504, + 97 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "where the six axes can be flattened to obtain the two axes. For Equation 4 and Equation 5 to be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 327, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 85, + 153, + 98 + ], + "score": 1.0, + "content": "equivalent,", + "type": "text" + }, + { + "bbox": [ + 153, + 84, + 164, + 95 + ], + "score": 0.86, + "content": "\\tilde { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 85, + 327, + 98 + ], + "score": 1.0, + "content": "must satisfy (derivation in Appendix A)", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 104, + 415, + 158 + ], + "lines": [ + { + "bbox": [ + 193, + 104, + 415, + 158 + ], + "spans": [ + { + "bbox": [ + 193, + 104, + 415, + 158 + ], + "score": 0.94, + "content": "\\tilde { W } _ { c d i a b j } = \\left\\{ \\begin{array} { l l } { C _ { i j } , } & { \\mathrm { i f ~ } d = a \\wedge ( d \\neq b \\vee c \\neq a ) . } \\\\ { W _ { i j } , } & { \\mathrm { i f ~ } d \\neq a \\wedge d = b \\wedge c = a . } \\\\ { C _ { i j } + W _ { i j } , } & { \\mathrm { i f ~ } d = a \\wedge d = b \\wedge c = a . } \\\\ { 0 , } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "aba9ee62d2cb9e4441538f13c5f01b18fd44f40ab8e22078076d3cef95f3da60.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 193, + 104, + 415, + 117.5 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 193, + 117.5, + 415, + 131.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 193, + 131.0, + 415, + 144.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 193, + 144.5, + 415, + 158.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 505, + 190 + ], + "lines": [ + { + "bbox": [ + 106, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 243, + 180 + ], + "score": 1.0, + "content": "This corresponds to interpretation", + "type": "text" + }, + { + "bbox": [ + 243, + 166, + 257, + 179 + ], + "score": 0.88, + "content": "\\textcircled{8}", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "with the weight matrix visualized in Figure 1a. To distinguish", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 500, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 500, + 191 + ], + "score": 1.0, + "content": "between the single sparse shared RNN and the connected RNNs, we now call the latter sub-RNNs.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 506, + 282 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "VSML as complex neurons with learned updates The arrangement and connectivity of the sub-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 507, + 228 + ], + "score": 1.0, + "content": "RNNs as described in the previous paragraphs corresponds to that of weights in a standard NN.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 196, + 240 + ], + "score": 1.0, + "content": "Thus, in interpretation", + "type": "text" + }, + { + "bbox": [ + 196, + 225, + 210, + 238 + ], + "score": 0.85, + "content": "@", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 225, + 506, + 240 + ], + "score": 1.0, + "content": ", VSML can be viewed as defining complex neurons where each sub-RNN", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 238, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 250 + ], + "score": 1.0, + "content": "corresponds to a weight in a standard NN as visualized in Figure 1c. All these sub-RNNs share", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "the same parameters but have distinct states. The current formulation corresponds to a single NN", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "layer that is run recurrently. We will generalize this to other architectures in the next section. A", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 460, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 313, + 284 + ], + "score": 1.0, + "content": "corresponds to the dimensionality of the inputs and", + "type": "text" + }, + { + "bbox": [ + 313, + 271, + 322, + 280 + ], + "score": 0.8, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 270, + 460, + 284 + ], + "score": 1.0, + "content": "to that of the outputs in that layer.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 218, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 219, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 219, + 299 + ], + "score": 1.0, + "content": "The role of weights in a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 297, + 218, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 218, + 308 + ], + "score": 1.0, + "content": "standard neural network is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 308, + 218, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 218, + 320 + ], + "score": 1.0, + "content": "now assigned to the states", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 218, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 218, + 331 + ], + "score": 1.0, + "content": "of RNNs. 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To distinguish", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 500, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 500, + 191 + ], + "score": 1.0, + "content": "between the single sparse shared RNN and the connected RNNs, we now call the latter sub-RNNs.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 166, + 505, + 191 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 506, + 282 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "VSML as complex neurons with learned updates The arrangement and connectivity of the sub-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 507, + 228 + ], + "score": 1.0, + "content": "RNNs as described in the previous paragraphs corresponds to that of weights in a standard NN.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 196, + 240 + ], + "score": 1.0, + "content": "Thus, in interpretation", + "type": "text" + }, + { + "bbox": [ + 196, + 225, + 210, + 238 + ], + "score": 0.85, + "content": "@", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 225, + 506, + 240 + ], + "score": 1.0, + "content": ", VSML can be viewed as defining complex neurons where each sub-RNN", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 238, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 250 + ], + "score": 1.0, + "content": "corresponds to a weight in a standard NN as visualized in Figure 1c. All these sub-RNNs share", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "the same parameters but have distinct states. The current formulation corresponds to a single NN", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "layer that is run recurrently. We will generalize this to other architectures in the next section. 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The param-", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 104, + 654, + 504, + 671 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 129, + 671 + ], + "score": 1.0, + "content": "eters", + "type": "text" + }, + { + "bbox": [ + 130, + 656, + 149, + 668 + ], + "score": 0.89, + "content": "A ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 654, + 171, + 671 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 171, + 656, + 191, + 667 + ], + "score": 0.9, + "content": "B ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 654, + 433, + 671 + ], + "score": 1.0, + "content": "describing layer sizes can then be varied for each layer", + "type": "text" + }, + { + "bbox": [ + 433, + 657, + 504, + 669 + ], + "score": 0.93, + "content": "k \\in \\{ 1 , \\ldots , K \\}", + "type": "inline_equation" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 666, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 172, + 684 + ], + "score": 1.0, + "content": "constrained by", + "type": "text" + }, + { + "bbox": [ + 172, + 668, + 241, + 680 + ], + "score": 0.9, + "content": "A ^ { ( k ) } ~ = ~ B ^ { ( k - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 666, + 444, + 684 + ], + "score": 1.0, + "content": ". The updated Equation 3 with distinct layers", + "type": "text" + }, + { + "bbox": [ + 445, + 670, + 452, + 680 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 666, + 506, + 684 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 675, + 503, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 212, + 696 + ], + "score": 0.92, + "content": "s _ { a b } ^ { ( k ) } f _ { \\mathrm { R N N } } ( s _ { a b } ^ { ( k ) } , \\overrightarrow { m } _ { a } ^ { ( k ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 675, + 244, + 703 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 245, + 680, + 353, + 696 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\vec { m } _ { b } ^ { ( k + 1 ) } : = \\sum _ { a ^ { \\prime } } \\bar { f } _ { \\vec { m } } ( s _ { a ^ { \\prime } b } ^ { ( k ) } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 675, + 378, + 703 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 378, + 682, + 503, + 695 + ], + "score": 0.92, + "content": "b \\in \\{ 1 , \\ldots , B ^ { ( k ) } = A ^ { ( k + 1 ) } \\}", + "type": "inline_equation" + } + ], + "index": 59 + } + ], + "index": 56, + "bbox_fs": [ + 104, + 611, + 508, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 298, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 299, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 299, + 85 + ], + "score": 1.0, + "content": "To prevent information from flowing only for-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 298, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 298, + 95 + ], + "score": 1.0, + "content": "ward in the network, we use an additional back-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 165, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 165, + 107 + ], + "score": 1.0, + "content": "ward message", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 111, + 269, + 129 + ], + "lines": [ + { + "bbox": [ + 133, + 111, + 269, + 129 + ], + "spans": [ + { + "bbox": [ + 133, + 111, + 269, + 129 + ], + "score": 0.91, + "content": "s _ { a b } ^ { ( k ) } \\gets f _ { R N N } \\big ( s _ { a b } ^ { ( k ) } , \\overrightarrow { m } _ { a } ^ { ( k ) } , \\overleftarrow { m } _ { b } ^ { ( k ) } \\big ) ,", + "type": "interline_equation", + "image_path": "92c1ed6925a96948824b0b65811fe3cd0faa25865ff4b0c03f5daa7040659ab8.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 133, + 111, + 269, + 129 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 297, + 186 + ], + "lines": [ + { + "bbox": [ + 101, + 128, + 297, + 155 + ], + "spans": [ + { + "bbox": [ + 101, + 128, + 136, + 155 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 136, + 133, + 249, + 149 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\overleftarrow { m } _ { a } ^ { ( k - 1 ) } : = \\sum _ { b ^ { \\prime } } f _ { \\overleftarrow { m } } ( s _ { a b ^ { \\prime } } ^ { ( k ) } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 128, + 275, + 155 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 275, + 137, + 297, + 148 + ], + "score": 0.82, + "content": "a \\in", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 145, + 300, + 164 + ], + "spans": [ + { + "bbox": [ + 107, + 149, + 214, + 162 + ], + "score": 0.87, + "content": "\\{ 1 , . . . , A ^ { ( k ) } = B ^ { ( k - 1 ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 145, + 300, + 164 + ], + "score": 1.0, + "content": "(visualized in Fig-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 297, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 297, + 173 + ], + "score": 1.0, + "content": "ure 3). 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Some architectures may be", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 298, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 298, + 235 + ], + "score": 1.0, + "content": "learned implicitly if positional information is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 234, + 297, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 297, + 246 + ], + "score": 1.0, + "content": "fed into each sub-RNN (Appendix C). 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We may also apply multiple RNN ticks for each layer", + "type": "text" + }, + { + "bbox": [ + 460, + 257, + 466, + 267 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 256, + 470, + 270 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 506, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "score": 1.0, + "content": "To provide the VSML RNN with data, each time we execute the operations of the first layer, a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 282, + 507, + 299 + ], + "spans": [ + { + "bbox": [ + 104, + 282, + 183, + 299 + ], + "score": 1.0, + "content": "single new datum", + "type": "text" + }, + { + "bbox": [ + 183, + 284, + 229, + 297 + ], + "score": 0.92, + "content": "x \\in \\mathbb { R } ^ { A ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 282, + 507, + 299 + ], + "score": 1.0, + "content": "(e.g. one flattened image) is distributed across all sub-RNNs. In", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 284, + 309 + ], + "score": 1.0, + "content": "our present experiments, we match the axis", + "type": "text" + }, + { + "bbox": [ + 284, + 297, + 305, + 309 + ], + "score": 0.86, + "content": "A ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "to the input datum dimensionality such that each", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 306, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 506, + 322 + ], + "score": 1.0, + "content": "dimension (e.g., pixel) is fed to different RNNs. This corresponds to initializing the forward message", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 315, + 507, + 355 + ], + "spans": [ + { + "bbox": [ + 107, + 318, + 156, + 334 + ], + "score": 0.9, + "content": "\\vec { m } _ { a 1 } ^ { ( 1 ) } : = \\boldsymbol { x } _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 315, + 196, + 355 + ], + "score": 1.0, + "content": "dding . Fina", + "type": "text" + }, + { + "bbox": [ + 197, + 320, + 208, + 332 + ], + "score": 0.82, + "content": "\\overrightarrow { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 315, + 284, + 355 + ], + "score": 1.0, + "content": "with zeros if nece we feed the error", + "type": "text" + }, + { + "bbox": [ + 332, + 315, + 429, + 355 + ], + "score": 1.0, + "content": "larly, we read the outpuat the output such that", + "type": "text" + }, + { + "bbox": [ + 434, + 319, + 481, + 333 + ], + "score": 0.92, + "content": "\\boldsymbol { \\hat { y } } \\in \\mathbb { R } ^ { B ( K ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 315, + 507, + 355 + ], + "score": 1.0, + "content": "from. See", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 333, + 481, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 169, + 348 + ], + "score": 0.91, + "content": "\\hat { y } _ { a } : = \\overrightarrow { m } _ { a 1 } ^ { ( K + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 334, + 332, + 346 + ], + "score": 0.92, + "content": "e \\in \\mathbb { R } ^ { B ( K ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 333, + 481, + 348 + ], + "score": 0.91, + "content": " _ { b 1 } : = e _ { b }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "Figure 2 for a visualization. Alternatively, multiple input or output dimensions could be patched", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 357, + 264, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 264, + 369 + ], + "score": 1.0, + "content": "together and fed into fewer sub-RNNs.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 382, + 405, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 406, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 406, + 396 + ], + "score": 1.0, + "content": "3.1 Meta learning general-purpose learning algorithms from scratch", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Having formalized VSML, we can now use end-to-end meta learning to create LAs from scratch", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 348, + 425 + ], + "score": 1.0, + "content": "in Algorithm 1. We simply optimize the LSTM parameters", + "type": "text" + }, + { + "bbox": [ + 348, + 413, + 363, + 424 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "to minimize the sum of prediction", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 103, + 424, + 507, + 443 + ], + "spans": [ + { + "bbox": [ + 103, + 425, + 334, + 443 + ], + "score": 1.0, + "content": "losses over many time steps starting with random states", + "type": "text" + }, + { + "bbox": [ + 335, + 424, + 389, + 439 + ], + "score": 0.93, + "content": "V _ { L } : = \\{ s _ { a b } ^ { ( k ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 425, + 507, + 443 + ], + "score": 1.0, + "content": ". We focus on meta learning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 435, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 506, + 451 + ], + "score": 1.0, + "content": "online LAs where one example is fed at a time as done in Meta RNNs [16, 56, 10]. Meta training may", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "be performed using end-to-end gradient descent or gradient-free optimization such as evolutionary", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "strategies [57, 38]. The latter is significantly more efficient on VSML compared to standard NNs due", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 221, + 483 + ], + "score": 1.0, + "content": "to the small parameter space", + "type": "text" + }, + { + "bbox": [ + 221, + 471, + 237, + 481 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 470, + 506, + 483 + ], + "score": 1.0, + "content": ". Crucially, during meta testing, no explicit gradient descent is used.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 107, + 495, + 252, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 254, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 254, + 510 + ], + "score": 1.0, + "content": "Algorithm 1 VSML: Meta Training", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 101, + 507, + 506, + 715 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 254, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 192, + 522 + ], + "score": 1.0, + "content": "Require: Dataset(s)", + "type": "text" + }, + { + "bbox": [ + 192, + 509, + 254, + 521 + ], + "score": 0.88, + "content": "D = \\{ ( x _ { i } , y _ { i } ) \\}", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 117, + 520, + 259, + 532 + ], + "spans": [ + { + "bbox": [ + 117, + 520, + 146, + 531 + ], + "score": 0.89, + "content": "V _ { M } \\gets", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 520, + 259, + 532 + ], + "score": 1.0, + "content": "initialize LSTM parameters", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 117, + 530, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 117, + 530, + 269, + 543 + ], + "score": 1.0, + "content": "while meta loss has not converged do", + "type": "text" + }, + { + "bbox": [ + 344, + 530, + 494, + 543 + ], + "score": 1.0, + "content": ". Outer loop in parallel over datasets", + "type": "text" + }, + { + "bbox": [ + 494, + 532, + 504, + 541 + ], + "score": 0.78, + "content": "D", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 132, + 541, + 329, + 561 + ], + "spans": [ + { + "bbox": [ + 132, + 542, + 197, + 556 + ], + "score": 0.86, + "content": "{ \\cal V } _ { L } = \\{ s _ { a b } ^ { ( k ) } \\} ", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 541, + 297, + 561 + ], + "score": 1.0, + "content": "initialize LSTM states", + "type": "text" + }, + { + "bbox": [ + 298, + 545, + 329, + 556 + ], + "score": 0.74, + "content": "\\forall a , b , k", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 130, + 554, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 130, + 554, + 146, + 569 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 147, + 556, + 304, + 568 + ], + "score": 0.83, + "content": "( x , y ) ^ { \\top } \\in \\{ ( x _ { 1 } , y _ { 1 } ) , \\dotsc , ( x _ { T } , y _ { T } ) \\} \\subset D", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 554, + 319, + 569 + ], + "score": 1.0, + "content": "do", + "type": "text" + }, + { + "bbox": [ + 383, + 557, + 391, + 566 + ], + "score": 0.33, + "content": "\\triangleright", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 554, + 455, + 567 + ], + "score": 1.0, + "content": "Inner loop over", + "type": "text" + }, + { + "bbox": [ + 455, + 556, + 464, + 565 + ], + "score": 0.73, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "examples", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 142, + 562, + 504, + 588 + ], + "spans": [ + { + "bbox": [ + 142, + 562, + 146, + 588 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 147, + 567, + 195, + 581 + ], + "score": 0.55, + "content": " { \\vec { m } } _ { a 1 } ^ { ( 1 ) } : = x _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 562, + 218, + 588 + ], + "score": 1.0, + "content": "∀a", + "type": "text" + }, + { + "bbox": [ + 380, + 572, + 388, + 579 + ], + "score": 0.3, + "content": "\\triangleright", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 568, + 497, + 582 + ], + "score": 1.0, + "content": "Initialize from input image", + "type": "text" + }, + { + "bbox": [ + 497, + 572, + 504, + 579 + ], + "score": 0.54, + "content": "\\mathbf { X }", + "type": "inline_equation" + } + ], + "index": 49 + }, + { + "bbox": [ + 145, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 145, + 579, + 162, + 592 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 162, + 581, + 226, + 592 + ], + "score": 0.75, + "content": "\\bar { k } \\in \\{ 1 , \\ldots , K \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 579, + 241, + 592 + ], + "score": 1.0, + "content": "do", + "type": "text" + }, + { + "bbox": [ + 402, + 582, + 410, + 590 + ], + "score": 0.51, + "content": "\\triangleright", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 579, + 467, + 593 + ], + "score": 1.0, + "content": "Iterating over", + "type": "text" + }, + { + "bbox": [ + 467, + 581, + 478, + 590 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "layers", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 161, + 591, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 161, + 591, + 325, + 606 + ], + "score": 0.51, + "content": "s _ { a b } ^ { ( k ) } \\gets f _ { R N N } \\big ( s _ { a b } ^ { ( k ) } , \\overrightarrow { m } _ { a } ^ { ( k ) } , \\overleftarrow { m } _ { b } ^ { ( k ) } \\big ) \\quad \\forall a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 594, + 506, + 606 + ], + "score": 1.0, + "content": ". 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The activations from the forward pass necessary for credit assignment could be", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 231, + 335 + ], + "score": 1.0, + "content": "memorized as part of the state", + "type": "text" + }, + { + "bbox": [ + 231, + 325, + 237, + 333 + ], + "score": 0.65, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "or be explicitly stored and fed back. For simplicity, we chose the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 332, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 506, + 348 + ], + "score": 1.0, + "content": "latter to clone backpropagation. We continuously run the VSML RNN forward, alternately running", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 344, + 347, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 181, + 357 + ], + "score": 1.0, + "content": "the layers in order", + "type": "text" + }, + { + "bbox": [ + 182, + 345, + 219, + 356 + ], + "score": 0.91, + "content": "1 , \\ldots , K", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 344, + 301, + 357 + ], + "score": 1.0, + "content": "and in reverse order", + "type": "text" + }, + { + "bbox": [ + 302, + 345, + 339, + 356 + ], + "score": 0.86, + "content": "K , \\ldots , 1", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 344, + 347, + 357 + ], + "score": 1.0, + "content": ".2", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 107, + 372, + 191, + 386 + ], + "lines": [ + { + "bbox": [ + 104, + 370, + 193, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 370, + 193, + 389 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 297, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 296, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 296, + 410 + ], + "score": 1.0, + "content": "First, we demonstrate the capabilities of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 409, + 297, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 297, + 422 + ], + "score": 1.0, + "content": "VSML RNN by showing that it can implement", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 420, + 298, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 298, + 433 + ], + "score": 1.0, + "content": "neural forward computation and backpropaga-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 431, + 298, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 298, + 443 + ], + "score": 1.0, + "content": "tion in its recurrent dynamics on the MNIST [21]", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 442, + 297, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 297, + 454 + ], + "score": 1.0, + "content": "and Fashion MNIST [59] dataset. Then, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 453, + 297, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 297, + 465 + ], + "score": 1.0, + "content": "show how we can meta learn an LA from scratch", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 464, + 298, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 298, + 476 + ], + "score": 1.0, + "content": "on one set of datasets and then successfully ap-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 475, + 297, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 297, + 486 + ], + "score": 1.0, + "content": "ply it to another (out of distribution). Such", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 486, + 297, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 297, + 497 + ], + "score": 1.0, + "content": "generalization is enabled by extensive variable", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 496, + 297, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 297, + 509 + ], + "score": 1.0, + "content": "sharing where we have very few meta variables", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 507, + 297, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 507, + 174, + 519 + ], + "score": 0.87, + "content": "| V _ { M } | ~ \\approx ~ 2 , 4 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 507, + 297, + 520 + ], + "score": 1.0, + "content": "and many learned variables", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 518, + 298, + 531 + ], + "spans": [ + { + "bbox": [ + 107, + 519, + 173, + 530 + ], + "score": 0.75, + "content": "| V _ { L } | \\approx 2 5 7 , 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 518, + 298, + 531 + ], + "score": 1.0, + "content": ". We also investigate the robust-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 528, + 297, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 297, + 542 + ], + "score": 1.0, + "content": "ness of the discovered LA. Finally, we introspect", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 540, + 297, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 297, + 553 + ], + "score": 1.0, + "content": "the meta learned LA and compare it to gradient", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 551, + 141, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 141, + 563 + ], + "score": 1.0, + "content": "descent.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42 + }, + { + "type": "image", + "bbox": [ + 304, + 382, + 504, + 477 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 304, + 382, + 504, + 477 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 304, + 382, + 504, + 477 + ], + "spans": [ + { + "bbox": [ + 304, + 382, + 504, + 477 + ], + "score": 0.969, + "type": "image", + "image_path": "dbb2f1c0cc8cade0dc1f709d779c51876fa020bd13a4a622b6679683d6100ede.jpg" + } + ] + } + ], + "index": 53, + "virtual_lines": [ + { + "bbox": [ + 304, + 382, + 504, + 395.57142857142856 + ], + "spans": [], + "index": 50 + }, + { + "bbox": [ + 304, + 395.57142857142856, + 504, + 409.1428571428571 + ], + "spans": [], + "index": 51 + }, + { + "bbox": [ + 304, + 409.1428571428571, + 504, + 422.71428571428567 + ], + "spans": [], + "index": 52 + }, + { + "bbox": [ + 304, + 422.71428571428567, + 504, + 436.2857142857142 + ], + "spans": [], + "index": 53 + }, + { + "bbox": [ + 304, + 436.2857142857142, + 504, + 449.8571428571428 + ], + "spans": [], + "index": 54 + }, + { + "bbox": [ + 304, + 449.8571428571428, + 504, + 463.42857142857133 + ], + "spans": [], + "index": 55 + }, + { + "bbox": [ + 304, + 463.42857142857133, + 504, + 476.9999999999999 + ], + "spans": [], + "index": 56 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 482, + 505, + 558 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 303, + 481, + 507, + 494 + ], + "spans": [ + { + "bbox": [ + 303, + 481, + 507, + 494 + ], + "score": 1.0, + "content": "Figure 5: The VSML RNN is optimized to im-", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 303, + 493, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 303, + 493, + 506, + 505 + ], + "score": 1.0, + "content": "plement backpropagation in its recurrent dynam-", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 303, + 504, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 303, + 504, + 505, + 514 + ], + "score": 1.0, + "content": "ics on MNIST, then tested both on MNIST and", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 303, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 303, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "Fashion MNIST where it performs learning purely", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 303, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 303, + 526, + 506, + 538 + ], + "score": 1.0, + "content": "by unrolling the LSTM. We test on shallow and", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 303, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 303, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "deep architectures (single hidden layer of 32 units).", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 304, + 547, + 454, + 558 + ], + "spans": [ + { + "bbox": [ + 304, + 547, + 454, + 558 + ], + "score": 1.0, + "content": "Standard deviations are over 6 seeds.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 60 + } + ], + "index": 56.5 + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 506, + 581 + ], + "score": 1.0, + "content": "Our implementation uses LSTMs and the message interpretation from Equation 7. 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We", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "structure the sub-RNNs to mimic a feed-forward NN with either one hidden layer or no hidden layers.", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "score": 1.0, + "content": "To obtain training targets, we instantiate a standard NN, the shadow network, and feed it MNIST", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "data. 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Due to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 235, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 235, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "the algorithm’s ubiquitous use, it seems desirable to be able to meta", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 234, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 234, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "learn backpropagation-like algorithms. Here we investigate how", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 234, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 234, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "VSML RNNs can learn to implement backpropagation purely in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 234, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 234, + 147, + 441, + 160 + ], + "score": 1.0, + "content": "their recurrent dynamics. 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Similarly,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 234, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 234, + 247, + 262, + 259 + ], + "score": 1.0, + "content": "we set", + "type": "text" + }, + { + "bbox": [ + 263, + 249, + 304, + 258 + ], + "score": 0.88, + "content": "w : = s _ { a b 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 247, + 322, + 259 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 322, + 247, + 361, + 258 + ], + "score": 0.89, + "content": "b : = s _ { a b 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 247, + 438, + 259 + ], + "score": 1.0, + "content": ". We then optimize", + "type": "text" + }, + { + "bbox": [ + 438, + 247, + 454, + 258 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "via gradient", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 235, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 235, + 257, + 308, + 270 + ], + "score": 1.0, + "content": "descent to regress", + "type": "text" + }, + { + "bbox": [ + 308, + 258, + 315, + 269 + ], + "score": 0.71, + "content": "\\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 257, + 318, + 270 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 318, + 258, + 335, + 268 + ], + "score": 0.41, + "content": "\\Delta w", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 257, + 339, + 270 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 339, + 258, + 353, + 268 + ], + "score": 0.62, + "content": "\\Delta b", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 257, + 373, + 270 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 373, + 258, + 382, + 267 + ], + "score": 0.85, + "content": "\\hat { e } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "toward their respective targets.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 234, + 268, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 234, + 268, + 400, + 281 + ], + "score": 1.0, + "content": "We can either generate the training dataset", + "type": "text" + }, + { + "bbox": [ + 400, + 268, + 504, + 281 + ], + "score": 0.91, + "content": "D : = \\{ ( x , \\bar { w } , b , y , e , \\bar { e ^ { \\prime } } ) _ { i } \\}", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 234, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 234, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "randomly or run a ‘shadow’ NN on some supervised problem and fit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 234, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 234, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "the VSML RNN to its activations and parameter updates. Multiple", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "iterations in the VSML RNN would then correspond to evaluating the network and updating it via", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 312, + 504, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 504, + 324 + ], + "score": 1.0, + "content": "backpropagation. The activations from the forward pass necessary for credit assignment could be", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 231, + 335 + ], + "score": 1.0, + "content": "memorized as part of the state", + "type": "text" + }, + { + "bbox": [ + 231, + 325, + 237, + 333 + ], + "score": 0.65, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "or be explicitly stored and fed back. For simplicity, we chose the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 332, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 506, + 348 + ], + "score": 1.0, + "content": "latter to clone backpropagation. We continuously run the VSML RNN forward, alternately running", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 344, + 347, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 181, + 357 + ], + "score": 1.0, + "content": "the layers in order", + "type": "text" + }, + { + "bbox": [ + 182, + 345, + 219, + 356 + ], + "score": 0.91, + "content": "1 , \\ldots , K", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 344, + 301, + 357 + ], + "score": 1.0, + "content": "and in reverse order", + "type": "text" + }, + { + "bbox": [ + 302, + 345, + 339, + 356 + ], + "score": 0.86, + "content": "K , \\ldots , 1", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 344, + 347, + 357 + ], + "score": 1.0, + "content": ".2", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 25, + "bbox_fs": [ + 230, + 212, + 507, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 357 + ], + "lines": [], + "index": 31, + "bbox_fs": [ + 104, + 300, + 506, + 357 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 372, + 191, + 386 + ], + "lines": [ + { + "bbox": [ + 104, + 370, + 193, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 370, + 193, + 389 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 297, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 296, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 296, + 410 + ], + "score": 1.0, + "content": "First, we demonstrate the capabilities of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 409, + 297, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 297, + 422 + ], + "score": 1.0, + "content": "VSML RNN by showing that it can implement", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 420, + 298, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 298, + 433 + ], + "score": 1.0, + "content": "neural forward computation and backpropaga-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 431, + 298, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 298, + 443 + ], + "score": 1.0, + "content": "tion in its recurrent dynamics on the MNIST [21]", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 442, + 297, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 297, + 454 + ], + "score": 1.0, + "content": "and Fashion MNIST [59] dataset. Then, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 453, + 297, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 297, + 465 + ], + "score": 1.0, + "content": "show how we can meta learn an LA from scratch", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 464, + 298, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 298, + 476 + ], + "score": 1.0, + "content": "on one set of datasets and then successfully ap-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 475, + 297, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 297, + 486 + ], + "score": 1.0, + "content": "ply it to another (out of distribution). Such", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 486, + 297, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 297, + 497 + ], + "score": 1.0, + "content": "generalization is enabled by extensive variable", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 496, + 297, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 297, + 509 + ], + "score": 1.0, + "content": "sharing where we have very few meta variables", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 507, + 297, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 507, + 174, + 519 + ], + "score": 0.87, + "content": "| V _ { M } | ~ \\approx ~ 2 , 4 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 507, + 297, + 520 + ], + "score": 1.0, + "content": "and many learned variables", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 518, + 298, + 531 + ], + "spans": [ + { + "bbox": [ + 107, + 519, + 173, + 530 + ], + "score": 0.75, + "content": "| V _ { L } | \\approx 2 5 7 , 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 518, + 298, + 531 + ], + "score": 1.0, + "content": ". We also investigate the robust-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 528, + 297, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 297, + 542 + ], + "score": 1.0, + "content": "ness of the discovered LA. 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We test on shallow and", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 303, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 303, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "deep architectures (single hidden layer of 32 units).", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 304, + 547, + 454, + 558 + ], + "spans": [ + { + "bbox": [ + 304, + 547, + 454, + 558 + ], + "score": 1.0, + "content": "Standard deviations are over 6 seeds.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 60 + } + ], + "index": 56.5 + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 506, + 581 + ], + "score": 1.0, + "content": "Our implementation uses LSTMs and the message interpretation from Equation 7. Hyperparameters,", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 578, + 401, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 401, + 591 + ], + "score": 1.0, + "content": "training details, and additional experiments can be found in the appendix.", + "type": "text" + } + ], + "index": 65 + } + ], + "index": 64.5, + "bbox_fs": [ + 106, + 566, + 506, + 591 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 604, + 327, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 329, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 329, + 619 + ], + "score": 1.0, + "content": "4.1 VSML RNNs can implement backpropagation", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 66 + }, + { + "type": "text", + "bbox": [ + 107, + 624, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 625, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 505, + 636 + ], + "score": 1.0, + "content": "As described in Section 3.2, we optimize the VSML RNN to implement backpropagation. We", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "structure the sub-RNNs to mimic a feed-forward NN with either one hidden layer or no hidden layers.", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "score": 1.0, + "content": "To obtain training targets, we instantiate a standard NN, the shadow network, and feed it MNIST", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "data. After cloning, we then run the LA encoded in the VSML RNN on the MNIST and Fashion", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 106, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "MNIST dataset and observe that it performs learning purely in its recurrent dynamics, making explicit", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 105, + 680, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 692 + ], + "score": 1.0, + "content": "gradient calculations unnecessary. Figure 5 shows the learning curve on these two datasets. Notably,", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "learning works both on MNIST (within distribution) and on Fashion MNIST (out of distribution).", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "We observe that the loss is decently minimized, albeit regular gradient descent still outperforms our", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "cloned backpropagation. This may be due to non-zero errors during learning algorithm cloning, in", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 117 + ], + "score": 1.0, + "content": "particular when these errors accumulate in the deeper architecture. It is also possible that the VSML", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "states (‘weights’) deviate too far from ranges seen during cloning, in particular in the deep case when", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 248, + 140 + ], + "score": 1.0, + "content": "the loss starts increasing. We obtain", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 249, + 127, + 268, + 138 + ], + "score": 0.86, + "content": "87 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 268, + 127, + 312, + 140 + ], + "score": 1.0, + "content": "(deep) and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 312, + 127, + 332, + 138 + ], + "score": 0.87, + "content": "90 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 332, + 127, + 485, + 140 + ], + "score": 1.0, + "content": "(shallow) test accuracy on MNIST and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 485, + 127, + 505, + 138 + ], + "score": 0.85, + "content": "76 \\%", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 498, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 151, + 150 + ], + "score": 1.0, + "content": "(deep) and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 151, + 138, + 171, + 149 + ], + "score": 0.86, + "content": "80 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 171, + 138, + 498, + 150 + ], + "score": 1.0, + "content": "(shallow) on Fashion MNIST (focusing on successful cloning over performance).", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 69.5, + "bbox_fs": [ + 105, + 625, + 506, + 692 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 506, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "learning works both on MNIST (within distribution) and on Fashion MNIST (out of distribution).", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "We observe that the loss is decently minimized, albeit regular gradient descent still outperforms our", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "cloned backpropagation. This may be due to non-zero errors during learning algorithm cloning, in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 117 + ], + "score": 1.0, + "content": "particular when these errors accumulate in the deeper architecture. It is also possible that the VSML", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "states (‘weights’) deviate too far from ranges seen during cloning, in particular in the deep case when", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 248, + 140 + ], + "score": 1.0, + "content": "the loss starts increasing. We obtain", + "type": "text" + }, + { + "bbox": [ + 249, + 127, + 268, + 138 + ], + "score": 0.86, + "content": "87 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 127, + 312, + 140 + ], + "score": 1.0, + "content": "(deep) and", + "type": "text" + }, + { + "bbox": [ + 312, + 127, + 332, + 138 + ], + "score": 0.87, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 127, + 485, + 140 + ], + "score": 1.0, + "content": "(shallow) test accuracy on MNIST and", + "type": "text" + }, + { + "bbox": [ + 485, + 127, + 505, + 138 + ], + "score": 0.85, + "content": "76 \\%", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 498, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 151, + 150 + ], + "score": 1.0, + "content": "(deep) and", + "type": "text" + }, + { + "bbox": [ + 151, + 138, + 171, + 149 + ], + "score": 0.86, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 138, + 498, + 150 + ], + "score": 1.0, + "content": "(shallow) on Fashion MNIST (focusing on successful cloning over performance).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 106, + 165, + 333, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 334, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 334, + 179 + ], + "score": 1.0, + "content": "4.2 Meta learning supervised learning from scratch", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 297, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 298, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 298, + 198 + ], + "score": 1.0, + "content": "In the previous experiments, we have estab-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 196, + 298, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 298, + 209 + ], + "score": 1.0, + "content": "lished that VSML is expressive enough to meta-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 208, + 298, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 298, + 220 + ], + "score": 1.0, + "content": "optimize backpropagation-like algorithms. In-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 219, + 297, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 297, + 231 + ], + "score": 1.0, + "content": "stead of cloning an LA, we now meta learn from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 230, + 297, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 297, + 242 + ], + "score": 1.0, + "content": "scratch as described in Section 3.1. Here, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 241, + 297, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 181, + 253 + ], + "score": 1.0, + "content": "use a single layer (", + "type": "text" + }, + { + "bbox": [ + 181, + 241, + 210, + 251 + ], + "score": 0.87, + "content": "K = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 241, + 297, + 253 + ], + "score": 1.0, + "content": ") from input to output", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 252, + 297, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 297, + 264 + ], + "score": 1.0, + "content": "dimension and run it for two ticks per image", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 297, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 127, + 274 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 128, + 263, + 165, + 273 + ], + "score": 0.9, + "content": "N = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 262, + 184, + 274 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 185, + 262, + 250, + 272 + ], + "score": 0.92, + "content": "N ^ { \\prime } = N ^ { \\prime \\prime } = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 262, + 297, + 274 + ], + "score": 1.0, + "content": ". First, the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 273, + 297, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 297, + 285 + ], + "score": 1.0, + "content": "VSML RNN is meta trained end-to-end using", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 297, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 297, + 296 + ], + "score": 1.0, + "content": "evolutionary strategies (ES) [38] on MNIST to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 295, + 297, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 297, + 307 + ], + "score": 1.0, + "content": "minimize the sum of cross-entropies over 500", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 306, + 298, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 298, + 318 + ], + "score": 1.0, + "content": "data points starting from random state initializa-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 317, + 297, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 248, + 329 + ], + "score": 1.0, + "content": "tions. As each image is unique and", + "type": "text" + }, + { + "bbox": [ + 248, + 317, + 264, + 328 + ], + "score": 0.9, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 317, + 297, + 329 + ], + "score": 1.0, + "content": "can not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 327, + 297, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 297, + 341 + ], + "score": 1.0, + "content": "memorize the data, we are implicitly optimizing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 338, + 297, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 297, + 352 + ], + "score": 1.0, + "content": "the VSML RNN to generalize to future inputs", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 350, + 297, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 297, + 361 + ], + "score": 1.0, + "content": "given all inputs it has seen so far. We do not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 360, + 283, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 143, + 374 + ], + "score": 1.0, + "content": "pre-train", + "type": "text" + }, + { + "bbox": [ + 144, + 361, + 159, + 372 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 360, + 283, + 374 + ], + "score": 1.0, + "content": "with a human-engineered LA.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 16 + }, + { + "type": "image", + "bbox": [ + 304, + 184, + 503, + 285 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 304, + 184, + 503, + 285 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 184, + 503, + 285 + ], + "spans": [ + { + "bbox": [ + 304, + 184, + 503, + 285 + ], + "score": 0.969, + "type": "image", + "image_path": "7b033cbe1d2c5cc6f6e774fb5334f6d5b86759ee6986d192914ef27be40db278.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 304, + 184, + 503, + 196.625 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 304, + 196.625, + 503, + 209.25 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 304, + 209.25, + 503, + 221.875 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 304, + 221.875, + 503, + 234.5 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 304, + 234.5, + 503, + 247.125 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 304, + 247.125, + 503, + 259.75 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 304, + 259.75, + 503, + 272.375 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 304, + 272.375, + 503, + 285.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 290, + 505, + 399 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 303, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 303, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "Figure 6: The VSML RNN can be directly meta", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 304, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 304, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "trained on MNIST to minimize the sum of errors", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 304, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 304, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "when classifying online starting from a random", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 303, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 303, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "state initialization. This allows for faster learning", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 304, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 304, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "during meta testing compared to online gradient", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 304, + 344, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 304, + 344, + 505, + 355 + ], + "score": 1.0, + "content": "descent with Adam on the same dataset and even", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 303, + 356, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 303, + 356, + 506, + 367 + ], + "score": 1.0, + "content": "generalizes to a different dataset (Fashion MNIST).", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 303, + 366, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 303, + 366, + 506, + 378 + ], + "score": 1.0, + "content": "In comparison, a standard Meta RNN [16] strongly", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 303, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 303, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "overfits in the same setting. Standard deviations", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 303, + 388, + 383, + 399 + ], + "spans": [ + { + "bbox": [ + 303, + 388, + 383, + 399 + ], + "score": 1.0, + "content": "are over 128 seeds.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38.5 + } + ], + "index": 34.0 + }, + { + "type": "text", + "bbox": [ + 107, + 377, + 296, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 297, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 297, + 389 + ], + "score": 1.0, + "content": "During meta testing on MNIST (Figure 6) we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 101, + 388, + 296, + 420 + ], + "spans": [ + { + "bbox": [ + 101, + 388, + 181, + 420 + ], + "score": 1.0, + "content": "plot the cumulativputs on the y axis", + "type": "text" + }, + { + "bbox": [ + 181, + 398, + 228, + 414 + ], + "score": 0.92, + "content": "\\textstyle { \\bigl ( } { \\frac { 1 } { T } } \\sum _ { t = 1 } ^ { T } c _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 388, + 287, + 420 + ], + "score": 1.0, + "content": "n all previous after example", + "type": "text" + }, + { + "bbox": [ + 287, + 401, + 296, + 411 + ], + "score": 0.7, + "content": "T", + "type": "inline_equation" + } + ], + "index": 44 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 506, + 521 + ], + "lines": [ + { + "bbox": [ + 163, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 163, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "indicating prediction correctness). For each example, the prediction when this example", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "score": 1.0, + "content": "was fed to the RNN is used, thus measuring sample efficient learning. This evaluation protocol", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "is similar to the one used in Meta RNNs [56, 10]. We observe that learning is considerably faster", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "compared to the baseline of online gradient descent (no mini batching, the learning rate appropriately", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "tuned). One possibility is that VSML simply overfits to the training distribution. We reject this", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 467, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 478 + ], + "score": 1.0, + "content": "possibility by meta testing the same unmodified RNN on a different dataset, here Fashion MNIST.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "score": 1.0, + "content": "Learning still works well, meaning we have meta learned a fairly general LA (although performance", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "at convergence still lacks behind a little). This generalization is achieved without using any hardcoded", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "gradients during meta testing purely by running the RNN forward. In comparison to VSML, a Meta", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 509, + 196, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 196, + 522 + ], + "score": 1.0, + "content": "RNN heavily overfits.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 49.5 + }, + { + "type": "title", + "bbox": [ + 107, + 537, + 307, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 307, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 307, + 551 + ], + "score": 1.0, + "content": "4.3 Robustness to varying inputs and outputs", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 55 + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 257, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 258, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 258, + 570 + ], + "score": 1.0, + "content": "A defining property of VSML is that", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 569, + 258, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 191, + 581 + ], + "score": 1.0, + "content": "the same parameters", + "type": "text" + }, + { + "bbox": [ + 191, + 569, + 207, + 580 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 569, + 258, + 581 + ], + "score": 1.0, + "content": "can be used", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 579, + 257, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 257, + 592 + ], + "score": 1.0, + "content": "to learn on varying input and output", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 590, + 258, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 258, + 602 + ], + "score": 1.0, + "content": "sizes. Further, the architecture and", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 601, + 257, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 257, + 613 + ], + "score": 1.0, + "content": "thus the meta learned LA is invariant", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 613, + 257, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 257, + 624 + ], + "score": 1.0, + "content": "to the order of inputs and outputs. In", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 624, + 258, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 258, + 635 + ], + "score": 1.0, + "content": "this experiment, we investigate how", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 634, + 258, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 258, + 646 + ], + "score": 1.0, + "content": "robust we are to such changes. We", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 645, + 258, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 258, + 657 + ], + "score": 1.0, + "content": "meta train across MNIST with 3, 4,", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 657, + 257, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 257, + 667 + ], + "score": 1.0, + "content": "6, and 7 classes. Likewise, we train", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 106, + 667, + 258, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 258, + 678 + ], + "score": 1.0, + "content": "across rescaled versions with 14x14,", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 677, + 259, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 132, + 689 + ], + "score": 0.63, + "content": "2 8 \\mathbf { x } 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 677, + 152, + 690 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 152, + 678, + 178, + 689 + ], + "score": 0.83, + "content": "3 2 \\mathrm { x } 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 677, + 259, + 690 + ], + "score": 1.0, + "content": "pixels. 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All", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 264, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 264, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "six configurations have not been seen during training and", + "type": "text" + } + ], + "index": 80 + }, + { + "bbox": [ + 263, + 684, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 263, + 684, + 505, + 695 + ], + "score": 1.0, + "content": "perform comparable to the unchanged reference. 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In-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 219, + 297, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 297, + 231 + ], + "score": 1.0, + "content": "stead of cloning an LA, we now meta learn from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 230, + 297, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 297, + 242 + ], + "score": 1.0, + "content": "scratch as described in Section 3.1. 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As each image is unique and", + "type": "text" + }, + { + "bbox": [ + 248, + 317, + 264, + 328 + ], + "score": 0.9, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 317, + 297, + 329 + ], + "score": 1.0, + "content": "can not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 327, + 297, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 297, + 341 + ], + "score": 1.0, + "content": "memorize the data, we are implicitly optimizing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 338, + 297, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 297, + 352 + ], + "score": 1.0, + "content": "the VSML RNN to generalize to future inputs", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 350, + 297, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 297, + 361 + ], + "score": 1.0, + "content": "given all inputs it has seen so far. 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Standard deviations", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 303, + 388, + 383, + 399 + ], + "spans": [ + { + "bbox": [ + 303, + 388, + 383, + 399 + ], + "score": 1.0, + "content": "are over 128 seeds.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38.5 + } + ], + "index": 34.0 + }, + { + "type": "text", + "bbox": [ + 107, + 377, + 296, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 297, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 297, + 389 + ], + "score": 1.0, + "content": "During meta testing on MNIST (Figure 6) we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 101, + 388, + 296, + 420 + ], + "spans": [ + { + "bbox": [ + 101, + 388, + 181, + 420 + ], + "score": 1.0, + "content": "plot the cumulativputs on the y axis", + "type": "text" + }, + { + "bbox": [ + 181, + 398, + 228, + 414 + ], + "score": 0.92, + "content": "\\textstyle { \\bigl ( } { \\frac { 1 } { T } } \\sum _ { t = 1 } ^ { T } c _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 388, + 287, + 420 + ], + "score": 1.0, + "content": "n all previous after example", + "type": "text" + }, + { + "bbox": [ + 287, + 401, + 296, + 411 + ], + "score": 0.7, + "content": "T", + "type": "inline_equation" + } + ], + "index": 44 + } + ], + "index": 34.5, + "bbox_fs": [ + 101, + 376, + 297, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 506, + 521 + ], + "lines": [ + { + "bbox": [ + 163, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 163, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "indicating prediction correctness). For each example, the prediction when this example", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "score": 1.0, + "content": "was fed to the RNN is used, thus measuring sample efficient learning. This evaluation protocol", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "is similar to the one used in Meta RNNs [56, 10]. We observe that learning is considerably faster", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "compared to the baseline of online gradient descent (no mini batching, the learning rate appropriately", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "tuned). One possibility is that VSML simply overfits to the training distribution. We reject this", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 467, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 478 + ], + "score": 1.0, + "content": "possibility by meta testing the same unmodified RNN on a different dataset, here Fashion MNIST.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "score": 1.0, + "content": "Learning still works well, meaning we have meta learned a fairly general LA (although performance", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "at convergence still lacks behind a little). This generalization is achieved without using any hardcoded", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "gradients during meta testing purely by running the RNN forward. In comparison to VSML, a Meta", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 509, + 196, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 196, + 522 + ], + "score": 1.0, + "content": "RNN heavily overfits.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 412, + 506, + 522 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 537, + 307, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 307, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 307, + 551 + ], + "score": 1.0, + "content": "4.3 Robustness to varying inputs and outputs", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 55 + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 257, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 258, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 258, + 570 + ], + "score": 1.0, + "content": "A defining property of VSML is that", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 569, + 258, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 191, + 581 + ], + "score": 1.0, + "content": "the same parameters", + "type": "text" + }, + { + "bbox": [ + 191, + 569, + 207, + 580 + ], + "score": 0.89, + "content": "V _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 569, + 258, + 581 + ], + "score": 1.0, + "content": "can be used", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 579, + 257, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 257, + 592 + ], + "score": 1.0, + "content": "to learn on varying input and output", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 590, + 258, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 258, + 602 + ], + "score": 1.0, + "content": "sizes. 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We", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 645, + 258, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 258, + 657 + ], + "score": 1.0, + "content": "meta train across MNIST with 3, 4,", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 657, + 257, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 257, + 667 + ], + "score": 1.0, + "content": "6, and 7 classes. Likewise, we train", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 106, + 667, + 258, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 258, + 678 + ], + "score": 1.0, + "content": "across rescaled versions with 14x14,", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 677, + 259, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 132, + 689 + ], + "score": 0.63, + "content": "2 8 \\mathbf { x } 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 677, + 152, + 690 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 152, + 678, + 178, + 689 + ], + "score": 0.83, + "content": "3 2 \\mathrm { x } 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 677, + 259, + 690 + ], + "score": 1.0, + "content": "pixels. 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All", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 264, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 264, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "six configurations have not been seen during training and", + "type": "text" + } + ], + "index": 80 + }, + { + "bbox": [ + 263, + 684, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 263, + 684, + 505, + 695 + ], + "score": 1.0, + "content": "perform comparable to the unchanged reference. Standard", + "type": "text" + } + ], + "index": 81 + }, + { + "bbox": [ + 264, + 694, + 381, + 705 + ], + "spans": [ + { + "bbox": [ + 264, + 694, + 381, + 705 + ], + "score": 1.0, + "content": "deviations are over 32 seeds.", + "type": "text" + } + ], + "index": 82 + } + ], + "index": 80 + } + ], + "index": 76.75 + }, + { + "type": "text", + "bbox": [ + 107, + 711, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "tion fixed for all inner learning steps. In Figure 7 we meta test on 6 configurations that were not seen", + "type": "text" + } + ], + "index": 83 + }, + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "during meta training. Performance on all of these configurations is comparable to the unchanged", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 428, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 440 + ], + "score": 1.0, + "content": "reference from the previous section. In particular, the invariance to random projections suggests that", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "we have meta learned a learning algorithm beyond transferring learned representations [compare", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 449, + 155, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 155, + 462 + ], + "score": 1.0, + "content": "11, 54, 55].", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 83, + "bbox_fs": [ + 106, + 710, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 69, + 506, + 336 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 69, + 506, + 336 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 69, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 69, + 506, + 336 + ], + "score": 0.975, + "type": "image", + "image_path": "319cf9db2760a541d350ef7249d327ed4957d5292c4dc26af98cc095f61ec93b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 69, + 506, + 158.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 158.0, + 506, + 247.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 247.0, + 506, + 336.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 341, + 505, + 407 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 406, + 354 + ], + "score": 1.0, + "content": "Figure 8: Online learning on various datasets. Cumulative accuracy in", + "type": "text" + }, + { + "bbox": [ + 406, + 342, + 416, + 352 + ], + "score": 0.81, + "content": "\\%", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 341, + 492, + 354 + ], + "score": 1.0, + "content": "after having seen", + "type": "text" + }, + { + "bbox": [ + 492, + 342, + 505, + 352 + ], + "score": 0.54, + "content": "2 \\mathrm { k }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "training examples evaluated after each prediction starting with random states (VSML, Meta RNN,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "HebbianFW, FWMemory) or random parameters (SGD). 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During", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 183, + 218, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 218, + 195 + ], + "score": 1.0, + "content": "meta testing, we plot the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 195, + 218, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 218, + 205 + ], + "score": 1.0, + "content": "output probabilities for each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 205, + 219, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 219, + 217 + ], + "score": 1.0, + "content": "digit against the number of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 216, + 219, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 219, + 228 + ], + "score": 1.0, + "content": "samples seen in Figure 9.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 226, + 219, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 219, + 240 + ], + "score": 1.0, + "content": "We highlight the ground", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 238, + 218, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 175, + 249 + ], + "score": 1.0, + "content": "truth input class", + "type": "text" + }, + { + "bbox": [ + 175, + 238, + 185, + 248 + ], + "score": 0.6, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 238, + 218, + 249 + ], + "score": 1.0, + "content": "as well", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 249, + 218, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 204, + 261 + ], + "score": 1.0, + "content": "as the predicted class", + "type": "text" + }, + { + "bbox": [ + 204, + 249, + 216, + 261 + ], + "score": 0.79, + "content": "\\bigcirc", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 249, + 218, + 261 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 259, + 218, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 218, + 272 + ], + "score": 1.0, + "content": "In this case, our meta test", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 270, + 219, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 219, + 281 + ], + "score": 1.0, + "content": "dataset consists of MNIST", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 281, + 219, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 219, + 293 + ], + "score": 1.0, + "content": "digits with two examples of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 293, + 218, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 218, + 304 + ], + "score": 1.0, + "content": "each type. The same digit is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 304, + 218, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 218, + 315 + ], + "score": 1.0, + "content": "always repeated twice. This", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 314, + 219, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 219, + 325 + ], + "score": 1.0, + "content": "allows us to observe and vi-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 218, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 218, + 337 + ], + "score": 1.0, + "content": "sualize the effect with only", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 336, + 218, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 218, + 348 + ], + "score": 1.0, + "content": "a few examples. We have", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 347, + 218, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 218, + 359 + ], + "score": 1.0, + "content": "done the same introspection", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 357, + 219, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 219, + 371 + ], + "score": 1.0, + "content": "with the full dataset in Ap-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 369, + 147, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 147, + 380 + ], + "score": 1.0, + "content": "pendix B.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 217, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 218, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 218, + 396 + ], + "score": 1.0, + "content": "We observe that in VSML", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 396, + 218, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 218, + 407 + ], + "score": 1.0, + "content": "almost all failed predictions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 407, + 219, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 219, + 419 + ], + "score": 1.0, + "content": "are followed by the correct", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 418, + 219, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 219, + 429 + ], + "score": 1.0, + "content": "prediction with high cer-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 429, + 218, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 218, + 441 + ], + "score": 1.0, + "content": "tainty. 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The top plot shows VSML quickly", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 224, + 390, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 224, + 390, + 506, + 404 + ], + "score": 1.0, + "content": "associating the input images with the right label, almost always making", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 225, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 225, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "the right prediction the second time with high confidence. The bottom", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 225, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 225, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "plot shows the same dataset processed by SGD with Adam which fails", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 224, + 423, + 292, + 437 + ], + "spans": [ + { + "bbox": [ + 224, + 423, + 292, + 437 + ], + "score": 1.0, + "content": "to learn quickly.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 55.5 + } + ], + "index": 48.5 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 504, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 504, + 452 + ], + "score": 1.0, + "content": "makes many incorrect predictions and fails to adapt correctly in only 20 steps. It seems that SGD", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "learns qualitatively different from VSML. The VSML RNN meta learns to quickly associate new", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "inputs with their class whereas SGD fails to do so. We tried several different SGD learning rates", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "and considered multiple steps on the same input. In both cases, SGD does not behave similar to", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "VSML, either learning much slower or forgetting previous examples. As evident from high accuracies", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "in Figure 8, VSML does not only memorize inputs using this strategy of fast association, but the", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 506, + 297, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 297, + 518 + ], + "score": 1.0, + "content": "associations generalize to future unseen inputs.", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 63 + }, + { + "type": "title", + "bbox": [ + 107, + 532, + 197, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 198, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 198, + 547 + ], + "score": 1.0, + "content": "6 Related Work", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 67 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "Memory based meta learning (Meta RNNs) Memory-based meta learning in RNNs [16, 10, 56] is", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 104, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "a simple neural meta learner (see Section 2). Unfortunately, the LA encoded in the RNN parameters", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "is largely overparameterized which leads to overfitting. VSML demonstrates that weight sharing can", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 105, + 591, + 337, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 337, + 603 + ], + "score": 1.0, + "content": "address this issue resulting in more general-purpose LAs.", + "type": "text" + } + ], + "index": 71 + } + ], + "index": 69.5 + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "Learned Learning Rules / Fast Weights NNs that generate or change the weights of another or", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "the same NN are known as fast weight programmers [44], hypernetworks [14], NNs with synaptic", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "plasticity [25] or learned learning rules [4] (see Section 2). In VSML we do not require explicit", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "architectures for weight updates as weights are emergent from RNN state updates. In addition to the", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "learning rule, we implicitly learn how the neural forward computation is defined. Concurrent to this", + "type": "text" + } + ], + "index": 76 + }, + { + "bbox": [ + 106, + 662, + 414, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 414, + 674 + ], + "score": 1.0, + "content": "work, fast weights have also been used to meta learn more general LAs [39].", + "type": "text" + } + ], + "index": 77 + } + ], + "index": 74.5 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 104, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "Learned gradient-based optimizers Meta learning has been used to find optimizers that update", + "type": "text" + } + ], + "index": 78 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "the parameters of a model by taking the loss and gradient with respect to these parameters as an", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "input [34, 2, 22, 24]. In this work, we are interested in meta learning that does not rely on fixed", + "type": "text" + } + ], + "index": 80 + }, + { + "bbox": [ + 106, + 711, + 259, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 259, + 724 + ], + "score": 1.0, + "content": "gradient calculation in the inner loop.", + "type": "text" + } + ], + "index": 81 + } + ], + "index": 79.5 + } + ], + "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": "title", + "bbox": [ + 107, + 71, + 169, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 171, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 171, + 88 + ], + "score": 1.0, + "content": "5 Analysis", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 218, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 95, + 218, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 218, + 108 + ], + "score": 1.0, + "content": "Given that VSML seems to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 219, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 219, + 119 + ], + "score": 1.0, + "content": "learn faster than online gra-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 117, + 218, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 218, + 130 + ], + "score": 1.0, + "content": "dient descent in many cases", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 219, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 219, + 141 + ], + "score": 1.0, + "content": "we would like to qualita-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 140, + 219, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 219, + 151 + ], + "score": 1.0, + "content": "tively investigate how learn-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 219, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 219, + 163 + ], + "score": 1.0, + "content": "ing differs. We first meta", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 218, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 218, + 173 + ], + "score": 1.0, + "content": "train on the full MNIST", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 218, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 218, + 185 + ], + "score": 1.0, + "content": "dataset as before. During", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 183, + 218, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 218, + 195 + ], + "score": 1.0, + "content": "meta testing, we plot the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 195, + 218, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 218, + 205 + ], + "score": 1.0, + "content": "output probabilities for each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 205, + 219, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 219, + 217 + ], + "score": 1.0, + "content": "digit against the number of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 216, + 219, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 219, + 228 + ], + "score": 1.0, + "content": "samples seen in Figure 9.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 226, + 219, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 219, + 240 + ], + "score": 1.0, + "content": "We highlight the ground", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 238, + 218, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 175, + 249 + ], + "score": 1.0, + "content": "truth input class", + "type": "text" + }, + { + "bbox": [ + 175, + 238, + 185, + 248 + ], + "score": 0.6, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 238, + 218, + 249 + ], + "score": 1.0, + "content": "as well", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 249, + 218, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 204, + 261 + ], + "score": 1.0, + "content": "as the predicted class", + "type": "text" + }, + { + "bbox": [ + 204, + 249, + 216, + 261 + ], + "score": 0.79, + "content": "\\bigcirc", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 249, + 218, + 261 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 259, + 218, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 218, + 272 + ], + "score": 1.0, + "content": "In this case, our meta test", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 270, + 219, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 219, + 281 + ], + "score": 1.0, + "content": "dataset consists of MNIST", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 281, + 219, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 219, + 293 + ], + "score": 1.0, + "content": "digits with two examples of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 293, + 218, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 218, + 304 + ], + "score": 1.0, + "content": "each type. The same digit is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 304, + 218, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 218, + 315 + ], + "score": 1.0, + "content": "always repeated twice. This", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 314, + 219, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 219, + 325 + ], + "score": 1.0, + "content": "allows us to observe and vi-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 218, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 218, + 337 + ], + "score": 1.0, + "content": "sualize the effect with only", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 336, + 218, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 218, + 348 + ], + "score": 1.0, + "content": "a few examples. We have", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 347, + 218, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 218, + 359 + ], + "score": 1.0, + "content": "done the same introspection", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 357, + 219, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 219, + 371 + ], + "score": 1.0, + "content": "with the full dataset in Ap-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 369, + 147, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 147, + 380 + ], + "score": 1.0, + "content": "pendix B.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 95, + 219, + 380 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 217, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 218, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 218, + 396 + ], + "score": 1.0, + "content": "We observe that in VSML", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 396, + 218, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 218, + 407 + ], + "score": 1.0, + "content": "almost all failed predictions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 407, + 219, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 219, + 419 + ], + "score": 1.0, + "content": "are followed by the correct", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 418, + 219, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 219, + 429 + ], + "score": 1.0, + "content": "prediction with high cer-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 429, + 218, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 218, + 441 + ], + "score": 1.0, + "content": "tainty. 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We highlight the ground truth class", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 225, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 225, + 380, + 235, + 390 + ], + "score": 0.41, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 379, + 351, + 392 + ], + "score": 1.0, + "content": "as well as the predicted class", + "type": "text" + }, + { + "bbox": [ + 351, + 380, + 362, + 391 + ], + "score": 0.83, + "content": "\\bigcirc", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 379, + 505, + 392 + ], + "score": 1.0, + "content": ". The top plot shows VSML quickly", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 224, + 390, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 224, + 390, + 506, + 404 + ], + "score": 1.0, + "content": "associating the input images with the right label, almost always making", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 225, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 225, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "the right prediction the second time with high confidence. The bottom", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 225, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 225, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "plot shows the same dataset processed by SGD with Adam which fails", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 224, + 423, + 292, + 437 + ], + "spans": [ + { + "bbox": [ + 224, + 423, + 292, + 437 + ], + "score": 1.0, + "content": "to learn quickly.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 55.5 + } + ], + "index": 48.5 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 504, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 504, + 452 + ], + "score": 1.0, + "content": "makes many incorrect predictions and fails to adapt correctly in only 20 steps. It seems that SGD", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "learns qualitatively different from VSML. The VSML RNN meta learns to quickly associate new", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "inputs with their class whereas SGD fails to do so. We tried several different SGD learning rates", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "and considered multiple steps on the same input. In both cases, SGD does not behave similar to", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "VSML, either learning much slower or forgetting previous examples. As evident from high accuracies", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "in Figure 8, VSML does not only memorize inputs using this strategy of fast association, but the", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 506, + 297, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 297, + 518 + ], + "score": 1.0, + "content": "associations generalize to future unseen inputs.", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 63, + "bbox_fs": [ + 105, + 441, + 506, + 518 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 532, + 197, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 198, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 198, + 547 + ], + "score": 1.0, + "content": "6 Related Work", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 67 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "Memory based meta learning (Meta RNNs) Memory-based meta learning in RNNs [16, 10, 56] is", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 104, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "a simple neural meta learner (see Section 2). Unfortunately, the LA encoded in the RNN parameters", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "is largely overparameterized which leads to overfitting. VSML demonstrates that weight sharing can", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 105, + 591, + 337, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 337, + 603 + ], + "score": 1.0, + "content": "address this issue resulting in more general-purpose LAs.", + "type": "text" + } + ], + "index": 71 + } + ], + "index": 69.5, + "bbox_fs": [ + 104, + 558, + 506, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "Learned Learning Rules / Fast Weights NNs that generate or change the weights of another or", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "the same NN are known as fast weight programmers [44], hypernetworks [14], NNs with synaptic", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "plasticity [25] or learned learning rules [4] (see Section 2). In VSML we do not require explicit", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "architectures for weight updates as weights are emergent from RNN state updates. In addition to the", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "learning rule, we implicitly learn how the neural forward computation is defined. Concurrent to this", + "type": "text" + } + ], + "index": 76 + }, + { + "bbox": [ + 106, + 662, + 414, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 414, + 674 + ], + "score": 1.0, + "content": "work, fast weights have also been used to meta learn more general LAs [39].", + "type": "text" + } + ], + "index": 77 + } + ], + "index": 74.5, + "bbox_fs": [ + 105, + 606, + 506, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 104, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "Learned gradient-based optimizers Meta learning has been used to find optimizers that update", + "type": "text" + } + ], + "index": 78 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "the parameters of a model by taking the loss and gradient with respect to these parameters as an", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "input [34, 2, 22, 24]. In this work, we are interested in meta learning that does not rely on fixed", + "type": "text" + } + ], + "index": 80 + }, + { + "bbox": [ + 106, + 711, + 259, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 259, + 724 + ], + "score": 1.0, + "content": "gradient calculation in the inner loop.", + "type": "text" + } + ], + "index": 81 + } + ], + "index": 79.5, + "bbox_fs": [ + 104, + 677, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "Discrete program search An interesting alternative to distributed variable updates in VSML is meta", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 97 + ], + "score": 1.0, + "content": "learning via discrete program search [48, 35]. In this paradigm, a separate programming language", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "needs to be defined that gives rise to neural computation when its instructions are combined. This led", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "to the automated rediscovery of backpropagation [35]. In VSML we demonstrate that a symbolic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "programming language is not required and general-purpose LAs can be discovered and encoded in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "variable-shared RNNs. Search over neural network parameters is usually easier compared to symbolic", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 334, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 334, + 151 + ], + "score": 1.0, + "content": "program search due to smoothness in the loss landscape.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Multi-agent systems In the reinforcement learning setting multiple agents can be modeled with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "score": 1.0, + "content": "shared parameters [50, 32, 18], also in the context of meta learning [36]. This is related to the variable", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "sharing in VSML depending on how the agent-environment boundary is drawn. Unlike these works,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "we demonstrate the advantage of variable sharing in meta learning more general-purpose LAs and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 199, + 262, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 262, + 211 + ], + "score": 1.0, + "content": "present a weight update interpretation.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 108, + 224, + 264, + 238 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 265, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 265, + 240 + ], + "score": 1.0, + "content": "7 Discussion and Limitations", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "score": 1.0, + "content": "The research community has perfected the art of leveraging backpropagation for learning for a long", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 507, + 274 + ], + "score": 1.0, + "content": "time. At the same time, there are open questions such as how to minimize memory requirements,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "effectively learn online and continually, learn sample efficiently, learn without separate backward", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "score": 1.0, + "content": "phases, and others. VSML suggests that instead of building on top of backpropagation as a fixed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "routine, meta learning offers an alternative to discover general-purpose LAs. Nevertheless, this", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "paper is only a proof of concept—until now we have only investigated small-scale problems and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "performance does not yet quite match the mini-batched setting with large quantities of data. In", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "score": 1.0, + "content": "particular, we observed premature convergence of the solution at meta test time which calls for further", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "investigations. Scaling our system to harder problems and larger meta task distributions will be", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 348, + 199, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 199, + 360 + ], + "score": 1.0, + "content": "important future work.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "The computational cost of the current VSML variant is also larger than the one of standard back-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 411, + 388 + ], + "score": 1.0, + "content": "propagation. If we run a sub-RNN for each weight in a standard NN with", + "type": "text" + }, + { + "bbox": [ + 411, + 375, + 423, + 385 + ], + "score": 0.44, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "weights, the cost is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 117, + 399 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 386, + 158, + 398 + ], + "score": 0.92, + "content": "O ( W N ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 385, + 190, + 399 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 190, + 387, + 200, + 396 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 385, + 338, + 399 + ], + "score": 1.0, + "content": "is the state size of a sub-RNN. If", + "type": "text" + }, + { + "bbox": [ + 338, + 387, + 349, + 396 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "is small enough, and our experiments", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 163, + 410 + ], + "score": 1.0, + "content": "suggest small", + "type": "text" + }, + { + "bbox": [ + 164, + 397, + 174, + 407 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "may be feasible, this may be an acceptable cost. However, VSML is not bound to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 407, + 478, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 478, + 421 + ], + "score": 1.0, + "content": "the interpretation of a sub-RNN as one weight. Future work may relax this particular choice.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "Meta optimization is also prone to local minima. In particular, when the number of ticks between", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "input and feedback increases (e.g. deeper architectures), credit assignment becomes harder. Early", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "experiments suggest that diverse meta task distributions can help mitigate these issues. Additionally,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "learning horizons are limited when using backprop-based meta optimization. Using ES allowed for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 468, + 353, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 353, + 481 + ], + "score": 1.0, + "content": "training across longer horizons and more stable optimization.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "VSML can also be viewed as regularizing the NN weights that encode the LA through a representa-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "tional bottleneck. It is conceivable that LA generalization as obtained by VSML can also be achieved", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "through other regularization techniques. Unlike most regularizers, VSML also introduces substantial", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 516, + 464, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 464, + 531 + ], + "score": 1.0, + "content": "reuse of the same learning principle and permutation invariance through variable sharing.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 107, + 543, + 183, + 557 + ], + "lines": [ + { + "bbox": [ + 104, + 541, + 185, + 560 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 185, + 560 + ], + "score": 1.0, + "content": "8 Conclusion", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "We introduced Variable Shared Meta Learning (VSML), a simple principle of weight sharing and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "sparsity for meta learning powerful learning algorithms (LAs). Our implementation replaces the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "weights of a neural network with tiny LSTMs that share parameters. We discuss connections to meta", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 603, + 484, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 484, + 613 + ], + "score": 1.0, + "content": "recurrent neural networks, fast weight generators (hyper networks), and learned learning rules.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 108, + 617, + 503, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 631 + ], + "score": 1.0, + "content": "Using learning algorithm cloning, VSML RNNs can learn to implement the backpropagation algo-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "rithm and its parameter updates encoded implicitly in the recurrent dynamics. On MNIST it learns to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 639, + 489, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 489, + 653 + ], + "score": 1.0, + "content": "predict well without any human-designed explicit computational graph for gradient calculation.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "VSML can meta learn from scratch supervised LAs that do not explicitly rely on gradient computation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "and that generalize to unseen datasets. Introspection reveals that VSML LAs learn by fast association", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "in a way that is qualitatively different from stochastic gradient descent. This leads to gains in sample", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 686, + 507, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 507, + 703 + ], + "score": 1.0, + "content": "efficiency. Future work will focus on reinforcement learning settings, improvements of meta learning,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 699, + 344, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 344, + 713 + ], + "score": 1.0, + "content": "larger task distributions, and learning over longer horizons.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 742, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "Discrete program search An interesting alternative to distributed variable updates in VSML is meta", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 97 + ], + "score": 1.0, + "content": "learning via discrete program search [48, 35]. In this paradigm, a separate programming language", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "needs to be defined that gives rise to neural computation when its instructions are combined. This led", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "to the automated rediscovery of backpropagation [35]. In VSML we demonstrate that a symbolic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "programming language is not required and general-purpose LAs can be discovered and encoded in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "variable-shared RNNs. Search over neural network parameters is usually easier compared to symbolic", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 334, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 334, + 151 + ], + "score": 1.0, + "content": "program search due to smoothness in the loss landscape.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 72, + 506, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Multi-agent systems In the reinforcement learning setting multiple agents can be modeled with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "score": 1.0, + "content": "shared parameters [50, 32, 18], also in the context of meta learning [36]. This is related to the variable", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "sharing in VSML depending on how the agent-environment boundary is drawn. Unlike these works,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "we demonstrate the advantage of variable sharing in meta learning more general-purpose LAs and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 199, + 262, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 262, + 211 + ], + "score": 1.0, + "content": "present a weight update interpretation.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 154, + 506, + 211 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 224, + 264, + 238 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 265, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 265, + 240 + ], + "score": 1.0, + "content": "7 Discussion and Limitations", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "score": 1.0, + "content": "The research community has perfected the art of leveraging backpropagation for learning for a long", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 507, + 274 + ], + "score": 1.0, + "content": "time. At the same time, there are open questions such as how to minimize memory requirements,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "effectively learn online and continually, learn sample efficiently, learn without separate backward", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "score": 1.0, + "content": "phases, and others. VSML suggests that instead of building on top of backpropagation as a fixed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "routine, meta learning offers an alternative to discover general-purpose LAs. Nevertheless, this", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "paper is only a proof of concept—until now we have only investigated small-scale problems and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "performance does not yet quite match the mini-batched setting with large quantities of data. In", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 338 + ], + "score": 1.0, + "content": "particular, we observed premature convergence of the solution at meta test time which calls for further", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "investigations. Scaling our system to harder problems and larger meta task distributions will be", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 348, + 199, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 199, + 360 + ], + "score": 1.0, + "content": "important future work.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 248, + 507, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "The computational cost of the current VSML variant is also larger than the one of standard back-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 411, + 388 + ], + "score": 1.0, + "content": "propagation. If we run a sub-RNN for each weight in a standard NN with", + "type": "text" + }, + { + "bbox": [ + 411, + 375, + 423, + 385 + ], + "score": 0.44, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "weights, the cost is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 117, + 399 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 386, + 158, + 398 + ], + "score": 0.92, + "content": "O ( W N ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 385, + 190, + 399 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 190, + 387, + 200, + 396 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 385, + 338, + 399 + ], + "score": 1.0, + "content": "is the state size of a sub-RNN. If", + "type": "text" + }, + { + "bbox": [ + 338, + 387, + 349, + 396 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "is small enough, and our experiments", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 163, + 410 + ], + "score": 1.0, + "content": "suggest small", + "type": "text" + }, + { + "bbox": [ + 164, + 397, + 174, + 407 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "may be feasible, this may be an acceptable cost. However, VSML is not bound to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 407, + 478, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 478, + 421 + ], + "score": 1.0, + "content": "the interpretation of a sub-RNN as one weight. Future work may relax this particular choice.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 364, + 506, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "Meta optimization is also prone to local minima. In particular, when the number of ticks between", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "input and feedback increases (e.g. deeper architectures), credit assignment becomes harder. Early", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "experiments suggest that diverse meta task distributions can help mitigate these issues. Additionally,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "learning horizons are limited when using backprop-based meta optimization. Using ES allowed for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 468, + 353, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 353, + 481 + ], + "score": 1.0, + "content": "training across longer horizons and more stable optimization.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 425, + 506, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "VSML can also be viewed as regularizing the NN weights that encode the LA through a representa-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "tional bottleneck. It is conceivable that LA generalization as obtained by VSML can also be achieved", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "through other regularization techniques. Unlike most regularizers, VSML also introduces substantial", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 516, + 464, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 464, + 531 + ], + "score": 1.0, + "content": "reuse of the same learning principle and permutation invariance through variable sharing.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 484, + 506, + 531 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 543, + 183, + 557 + ], + "lines": [ + { + "bbox": [ + 104, + 541, + 185, + 560 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 185, + 560 + ], + "score": 1.0, + "content": "8 Conclusion", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "We introduced Variable Shared Meta Learning (VSML), a simple principle of weight sharing and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "sparsity for meta learning powerful learning algorithms (LAs). Our implementation replaces the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "weights of a neural network with tiny LSTMs that share parameters. We discuss connections to meta", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 603, + 484, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 484, + 613 + ], + "score": 1.0, + "content": "recurrent neural networks, fast weight generators (hyper networks), and learned learning rules.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 568, + 505, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 617, + 503, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 631 + ], + "score": 1.0, + "content": "Using learning algorithm cloning, VSML RNNs can learn to implement the backpropagation algo-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "rithm and its parameter updates encoded implicitly in the recurrent dynamics. On MNIST it learns to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 639, + 489, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 489, + 653 + ], + "score": 1.0, + "content": "predict well without any human-designed explicit computational graph for gradient calculation.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 616, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "VSML can meta learn from scratch supervised LAs that do not explicitly rely on gradient computation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "and that generalize to unseen datasets. Introspection reveals that VSML LAs learn by fast association", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "in a way that is qualitatively different from stochastic gradient descent. This leads to gains in sample", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 686, + 507, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 507, + 703 + ], + "score": 1.0, + "content": "efficiency. Future work will focus on reinforcement learning settings, improvements of meta learning,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 699, + 344, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 344, + 713 + ], + "score": 1.0, + "content": "larger task distributions, and learning over longer horizons.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 655, + 507, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 71, + 207, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 69, + 208, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 69, + 208, + 87 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 96, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 506, + 109 + ], + "score": 1.0, + "content": "We thank Sjoerd van Steenkiste, Imanol Schlag, Kazuki Irie, and the anonymous reviewers for their", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "score": 1.0, + "content": "comments and feedback. This work was supported by the ERC Advanced Grant (no: 742870) and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 466, + 130 + ], + "score": 1.0, + "content": "computational resources by the Swiss National Supercomputing Centre (CSCS, projects", + "type": "text" + }, + { + "bbox": [ + 466, + 118, + 487, + 128 + ], + "score": 0.58, + "content": "s 9 7 8", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 142 + ], + "score": 1.0, + "content": "s1041). We also thank NVIDIA Corporation for donating several DGX machines as part of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "score": 1.0, + "content": "Pioneers of AI Research Award, IBM for donating a Minsky machine, and weights & biases [5] for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 151, + 320, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 320, + 164 + ], + "score": 1.0, + "content": "their great experiment tracking software and support.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 178, + 163, + 191 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 165, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 165, + 193 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 197, + 507, + 725 + ], + "lines": [ + { + "bbox": [ + 110, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 110, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "[1] F. 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Require: Dataset(s) D = {(xi,yi)} VM ← initialize LSTM parameters while meta loss has not converged do (k)Outer loop in parallel over datasets D
for (x,y)∈{(x1,y1)...,(xT,yr)} CDdo> Inner loop over T examples
fork ∈{1,...,K} doInitialize from input image X Iterating over K layers
(k) ←fRNN(s， (k) mb (k)) Sab Aa,bEquation 7
mb (k+1) £afm(s) (k）) := 6Create forward message
ma (k-1) ∑f(s) := (k） Aa Create backward message
Aa a1Read output
e:=VgL(y,y)Compute error at outputs using loss L
凯 :=eb∀b>Input errors
· back-propagation through the inner loop · evolution strategies, using a search distribution p(VM)
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In attempting to explain the origin of adversarial examples, previous studies have typically focused on the fact that neural networks operate on high dimensional data, they overfit, or they are too linear. Here we show that distributions of logit differences have a universal functional form. This functional form is independent of architecture, dataset, and training protocol; nor does it change during training. This leads to adversarial error having a universal scaling, as a power-law, with respect to the size of the adversarial perturbation. We show that this universality holds for a broad range of datasets (MNIST, CIFAR10, ImageNet, and random data), models (including state-of-the-art deep networks, linear models, adversarially trained networks, and networks trained on randomly shuffled labels), and attacks (FGSM, step l.l., PGD). Motivated by these results, we study the effects of reducing prediction entropy on adversarial robustness. Finally, we study the effect of network architectures on adversarial sensitivity. To do this, we use neural architecture search with reinforcement learning to find adversarially robust architectures on CIFAR10. Our resulting architecture is more robust to white and black box attacks compared to previous attempts. + +# 1 INTRODUCTION + +An intriguing aspect of deep learning models in computer vision is that while they can classify images with high accuracy, they fail catastrophically when those same images are perturbed slightly in an adversarial fashion (Szegedy et al., 2013; Goodfellow et al., 2014). The prevalence of adversarial examples presents challenges to our understanding of how deep networks generalize and pose security risks in real world applications (Papernot et al., 2016a; Kurakin et al., 2016a). Several techniques have been proposed to defend against adversarial examples. Adversarial training (Goodfellow et al., 2014) augments the training data with adversarial examples. It has been shown that using stronger adversarial attacks in adversarial training can increase the robustness to stronger attacks, but at the cost of a decrease in clean accuracy (i.e. accuracy on samples that have not been adversarially perturbed) (Madry et al., 2017). Defensive distillation (Papernot et al., 2016b), feature squeezing (Xu et al., 2017), and Parseval training (Cisse et al., 2017) have also been shown to make models more robust against adversarial attacks. + +The goal of this work is to study the common properties of adversarial examples. We calculate the adversarial error, defined as the difference between clean accuracy and adversarial accuracy at a given size of adversarial perturbation (). Surprisingly, adversarial error has a similar dependence on small values of $\epsilon$ for all network models and datasets we studied, including linear, fully-connected, simple convolutional networks, Inception v3 (Szegedy et al., 2016), Inception-ResNet v2, Inception v4 (Szegedy et al., 2017), ResNet v1, ResNet v2 (He et al., 2016), NasNet-A (Zoph & Le, 2016; Zoph et al., 2017), adversarially trained Inception v3 (Kurakin et al., 2016b) and Inception-ResNet v2 (Tramer et al., 2017), and networks trained on randomly shuffled labels of MNIST. Adversarial \` error due to the Fast Gradient Sign Method (FGSM), its L2-norm variant, and Projected Gradient Descent (PGD) attack grows as a power-law like $A \epsilon ^ { B }$ with $B$ between 0.9 and 1.3. By contrast, we find that adversarial error caused by one-step least likely class method (step l.l.) also scales as a power-law where $B$ is between 1.8 and 2.5 for small $\epsilon$ . This observed universality points to a mysterious commonality between these models and datasets, despite the different number of channels, pixels, and classes present. Adversarial error caused by FGSM on the training set of randomly shuffled labels of MNIST (LeCun & Cortes) also has the power-law form where $B = 1 . 2$ , which implies that the universality is not a result of the specific content of these datasets nor the ability of the model to generalize. + +To discover the mechanism behind this universality we show how, at small $\epsilon$ , the success of an adversarial attack depends on the input-logit Jacobian of the model and on the logits of the network. We demonstrate that the susceptibility of a model to FGSM and PGD attacks is in large part dictated by the cumulative distribution of the difference between the most likely logit and the second most likely logit. We observe that this cumulative distribution has a universal form among all datasets and models studied, including randomly produced data. Together, we believe these results provide a compelling story regarding the susceptibility of machine learning models to adversarial examples at small $\epsilon$ . + +We show that training with single-step adversarial examples offers protection against large $\epsilon$ attacks (between 0.2 and 32), but does not help appreciably at defending against small $\epsilon$ attacks (below 0.2). At $\epsilon = 0 . 2$ , all ImageNet models we studied incur 10 to $2 5 \%$ adversarial error, and surprisingly, vanilla NASNet-A (best clean accuracy in our study) has a lower adversarial error than adversarially trained Inception-ResNet v2 or Inception v3 (Kurakin et al., 2016b) (Fig. 1(a)). In light of these results, we explore a different avenue to adversarial robustness through architecture selection. We perform neural architecture search (NAS) using reinforcement learning (Zoph & Le, 2016; Zoph et al., 2017). These techniques allow us to find several architectures that are especially robust to adversarial perturbations. In addition, by analyzing the adversarial robustness of the tens-of-thousands of architectures constructed by NAS, we gain insights into the relationship between size of a model, its clean accuracy, and its adversarial robustness. In summary, the key contributions of our work are: + +• We study the functional form of adversarial error and logit differences across several models and datasets, which turn out to be universal. We analytically derive the commonality in the power-law tails of the logit differences, and show how it leads to the commonality in the form of adversarial error. +• We observe that although the qualitative form of logit differences and adversarial error is universal, it can be quantitatively improved with entropy regularization and better network architectures. +• We study the dependence of adversarial robustness on the network architecture via NAS. We show that while adversarial accuracy is strongly correlated with clean accuracy, it is only weakly correlated with model size. Our work leads to architectures that are more robust to white-box and black-box attacks on CIFAR10 (Krizhevsky & Hinton, 2009) than previous studies. + +# 2 SURPRISING UNIVERSALITY OF ADVERSARIAL ERROR AT SMALL  + +FGSM computes adversarial examples as: + +$$ +\boldsymbol { x } ^ { \mathrm { a d v } } = \boldsymbol { x } + \epsilon \mathrm { s i g n } \left( \nabla _ { \boldsymbol { x } } L ( \boldsymbol { x } , \boldsymbol { y } ) \right) , +$$ + +where $x$ is the clean image, $y$ is the correct label for that image, $x ^ { \mathrm { a d v } }$ is the adversarial image, $\epsilon$ is the size of the adversarial perturbation, and $L ( x , y )$ is the loss function. $\epsilon$ values are specified in range [0,255]. We only study white-box attacks in this section. + +We begin with a preliminary examination of the architectural dependence of adversarial robustness. To that end, in Fig. 1 (a) we plot the test set adversarial error due to an FGSM attack as a function of $\epsilon$ for several models on ImageNet (Russakovsky et al., 2015). We note that for $\epsilon < 0 . 2$ , the adversarial error follows a power law form with an exponent between 0.9 and 1.1 for all models studied. Even adversarially trained models Kurakin et al. (2016b), while adversarially much more robust for larger values of $\epsilon$ , follow a similar form and reach as large as $20 \%$ adversarial error at smaller $\epsilon$ . + +In light of the surprising commonality in adversarial error at small- $\epsilon$ , we investigate whether there is any way to get a different form for the adversarial error. To do this, we evaluate the adversarial error due to step l.l. attack, which is computed as: + +$$ +x ^ { \mathrm { a d v } } = x - \epsilon \mathrm { s i g n } \left( \nabla _ { x } L ( x , y _ { l . l . } ) \right) , +$$ + +where $y _ { l . l . }$ is least likely class predicted by the network on clean image $x$ (Kurakin et al., 2016b). The adversarial error also follows a power law, however with a larger exponent. The exponents range from 1.8 to 2.2 for ImageNet models (Fig. 1(b)), and 1.8 to 2.5 for models trained on MNIST (Fig. 2(c)) and CIFAR10. Thus, we see that attack protocol can change the exponent of the observed power-law. + +![](images/96e44a0d4a3fd367b209184ba9b25c3986431a5b883e6cfe8c25726059b999c8.jpg) +Figure 1: Test set adversarial error as a function of $\epsilon$ for models trained on ImageNet due to FGSM and step l.l. attack in (a) and (b), respectively. adv. tr. denotes models that are adversarially trained (Kurakin et al., 2016b). In (b), we also show two of the power law fits with straight lines. + +To test the limits of the universality observed in Fig. 1, we perform a number of more extensive tests. First, we investigate the effect of architecture by stochastically sampling thousands of different neural networks and train them on MNIST. We then measure their adversarial error due to FGSM on the test set. The architectures we sample are either fully-connected networks with 1-4 hidden layers and 30-2000 hidden nodes in each layer, or simple convolutional networks with dropout rates between 0-0.5. The adversarial error of representative linear, fully-connected, and convolutional networks are shown in Fig. 2 (a). As above, these models all have the same form of adversarial error with a powerlaw dependence on $\epsilon$ with exponents between 0.9 and 1.2. + +See Fig. 9 in the Appendix for a plot with all of the generated networks. We perform the same analysis on a 32-layer ResNet trained on CIFAR10 (He et al., 2016), which achieves a $9 2 . 6 \%$ clean accuracy on the test set. The result is shown in Appendix Fig. 10, where the adversarial error follows a power law with an exponent of 0.99 up to an $\epsilon$ of 1. + +Next, we probe the relationship between generalization and adversarial robustness following a similar approach to Zhang et al. (2016). In particular, we train a fully connected network on MNIST with shuffled labels until it reaches perfect accuracy on the training set. The adversarial error on the training set is shown in Fig. 2(b). Once again we see that the adversarial error follows an almost identical power-law form at small $\epsilon$ with an exponent of 1.2. + +Finally, we further investigate the dependence of adversarial robustness on attack protocol. We plot in Fig. 2 (c) the adversarial error on MNIST with an $L _ { \infty }$ -normalized FGSM attack, an $L _ { 2 }$ - normalized FGSM attack, and a 20-step projected gradient descent (PGD) attack (Madry et al., 2017). We see that despite the anomalous exponent observed for step-l.l. attacks, the other attack methods display the same universality with exponents of 1.1, 1.2, and 1.3 for L2-norm, FGSM, and PGD attacks, respectively. step-l.l. attack on MNIST has an exponent of 2.3. + +# 3 A MEAN-FIELD THEORY OF ADVERSARIAL PERTURBATIONS + +# 3.1 LINEAR RESPONSE + +We now offer a theoretical explanation for the observed universal behavior of adversarial error. The breadth of these observations shows that adversarial error for small adversarial perturbations does not depend on the specifics of the neural network, which implies that we can understand the small $\epsilon$ regime by making simplifying approximations. We begin by considering the linear response of a neural network to adversarial perturbations. Another approach to adversarial examples that considers the linear response of the network can be found in Nayebi & Ganguli (2017). The effect of margins on adversarial robustness has been brought up in ?. + +![](images/71a2d3a2071be278b30c68ebe087ca022a4b5a62472b42ec80cadf2f1fe43425.jpg) +Figure 2: Points represent adversarial error as a function of $\epsilon$ for models trained on MNIST. Straight lines are power-law fits. (a) FGSM attack on fully-connected 3-layer network (FC), a linear model, and a convolutional network (b) FGSM attack on FC trained on randomly shuffled labels (evaluated on the training set). (c) Different attacks on FC. + +We will study an $L _ { 2 }$ -variant of the FGSM attack. Here, the adversarial perturbation is given by, + +$$ +x ^ { \mathrm { a d v } } = x + \epsilon \frac { \nabla _ { x } L } { | | \nabla _ { x } L | | _ { 2 } } +$$ + +instead of the more commonly used $\ell _ { \infty }$ variant. As shown above, the form and exponent of adversarial error is qualitatively insensitive to this choice (see Fig. 2(c)). We will now attempt to compute the minimum $\epsilon$ , that we call $\hat { \epsilon } ( x )$ , required before the class assigned to an input, $x$ , changes. Assuming the network was able to perfectly classify clean images, the adversarial error rate will then be $P ( \bar { \hat { \epsilon } } < \epsilon )$ . While perfect classification will not be achieved in practice, the insensitivity of the form of adversarial error to clean accuracy demonstrated above for many systems suggests that this approximation is sound. + +Notationally, we will refer to the output of the network as ${ \hat { y } } _ { i } ( x )$ and the corresponding logits as $h _ { i } ( x )$ . The class prediction of the network will then be arg $\operatorname* { m a x } ( \hat { y } ( x ) ) = \operatorname { a r g m a x } ( h ( x ) )$ . For simplicity we will choose an ordering of the logits such that $h _ { 1 } ( x ) \geq h _ { 2 } ( x ) \geq \cdot \cdot \cdot \geq h _ { N } ( x )$ . We can then enumerate a set of logit-differences, $\Delta _ { i j } ( x ) = h _ { i } ( x ) - \dot { h _ { j } } ( x )$ . If an adversarial perturbation is to successfully cause the network to make an erroneous prediction, then it must be true that $h _ { 1 } ( x ^ { \mathrm { a d v } } ) < h _ { j } ( \dot { x } ^ { \mathrm { a d v } } )$ for at least one $j$ . + +We calculate the response of the logits to adversarial perturbation. We consider the linearized response of the network and find that in the limit of small $\epsilon$ (see Appendix 6.2.1), + +$$ +h ( x ^ { \mathrm { a d v } } ) = h ( x ) + \epsilon \frac { J ^ { T } J \delta } { | | J \delta | | _ { 2 } } + \mathcal { O } ( \epsilon ^ { 2 } ) +$$ + +where $J _ { i j } = \partial h _ { j } / \partial x _ { i }$ is the input-logit Jacobian of the network and $\delta _ { i } = \partial L / \partial h _ { i }$ is the error of the outputs of the network. For notational convenience we will define $\Gamma ( x ) = J ^ { T } J \delta / | J \delta | | _ { 2 }$ . In this linear model we therefore predict that the logit-differences will scale as follows, + +$$ +\Delta _ { i j } ( x ^ { \mathrm { a d v } } ) \approx \Delta _ { i j } ( x ) + \epsilon ( \Gamma _ { i } ( x ) - \Gamma _ { j } ( x ) ) + \mathcal { O } ( \epsilon ^ { 2 } ) . +$$ + +Recall that the adversarial perturbation will successfully cause the network to just barely misclassify an input precisely when $\Delta _ { 1 j } ( x ^ { \mathrm { a d v } } ) = 0$ for at least one $j$ . We can predict per-class $\epsilon$ -thresholds beyond which the network will misclassify a given point, + +$$ +\hat { \epsilon } _ { j } ( x ) = \frac { \Delta _ { 1 j } ( x ) } { \Gamma _ { j } ( x ) - \Gamma _ { 1 } ( x ) } . +$$ + +Together this allows us to compute a linear approximation to $\hat { \epsilon }$ given by $\hat { \epsilon } _ { \mathrm { l i n e a r } } ( x ) = \operatorname* { m i n } _ { j } ( \epsilon _ { j } ( x ) )$ + +We can confirm that the change in the logits for small changes in the inputs is well-described by this linear model. This is shown in fig. 3 (a) where we see the logits for a single example upon perturbation over a range of $\epsilon$ . In particular, for small $\epsilon$ we see an excellent agreement between the linear approximation and the true logit dynamics. We also see that the $\epsilon$ where the first and second logit cross is well-approximated by the linear prediction. In Fig. 3 (b) we plot $\hat { \epsilon } _ { \mathrm { l i n e a r } } ( x )$ against $\hat { \epsilon } ( x )$ evaluated on every MNIST example in the test set. The white dashed line is the line $\hat { \epsilon } ( x ) = \hat { \epsilon } _ { \mathrm { l i n e a r } } ( x )$ . We see that when $\hat { \epsilon } ( x )$ is small the $\hat { \epsilon } _ { \mathrm { l i n e a r } } ( x )$ concentrate increasingly around the $\hat { \epsilon } ( x )$ . Together these results show that the linear response predictions are valid for small adversarial perturbations. + +![](images/85f4e581191d1aa2d9b2fce3b1c7c6baf8327e01427ec09910818f8323319cd2.jpg) +Figure 3: Linear approximation for the response of the logits to adversarial perturbation. (a) The dynamics of logits to an adversarial perturbation as a function of size $\epsilon$ for a single training example. Dashed lines show the linear approximation. Colors indicate the ranking of the logit from largest (red) to smallest (blue). (b) The smallest $\epsilon$ needed to fool the network (ˆ) for individual test examples compared with the prediction from the linear theory. (c) Mean field prediction of $\hat { \epsilon }$ . + +While the linear model outlined above gives excellent agreement in the $\epsilon 0$ limit, the $\Gamma _ { i } ( x )$ are themselves complicated objects (being functions of the Jacobian). This makes the analytic evaluation of Eq. (6) challenging. We therefore introduce a “mean-field” approximation to Eq. (6) by replacing $\Gamma _ { i } ( x )$ by its average over the dataset, $\langle \Gamma _ { i } ( x ) \rangle$ . Similar independence approximations have previously been successful in analyzing the expressivity and trainability of neural networks (Schoenholz et al., 2016; Poole et al., 2016). Finally, we observe that the vast majority of the time (for example, more than $9 5 \%$ of successful FGSM attacks for $\epsilon < 5 0$ ), it is $\Delta _ { 1 2 } ( x ^ { \mathrm { a d v } } )$ that goes to zero before any of the other $\Delta _ { 1 j }$ . We therefore assume that this will be the dominant failure mode for neural networks and write down a mean-field estimate for $\hat { \epsilon }$ , + +$$ +\hat { \epsilon } _ { \mathrm { M . F . } } = \frac { \Delta _ { 1 2 } ( x ) } { \langle \Gamma _ { 2 } \rangle - \langle \Gamma _ { 1 } \rangle } . +$$ + +We show in Fig. 3 (c) that this approximation continues to be strongly correlated with ˆ. Together these results suggest that the adversarial error rate for perturbations of size $\epsilon$ will be + +$$ +P ( \hat { \epsilon } \le \epsilon ) \approx P \left[ \Delta _ { 1 2 } \le \epsilon ( \langle \Gamma _ { 2 } \rangle - \langle \Gamma _ { 1 } \rangle ) \right] = P ( \Delta _ { 1 2 } \le \tilde { \epsilon } ) +$$ + +where we have defined $\tilde { \epsilon } = \epsilon ( \left. \Gamma _ { 2 } \right. - \left. \Gamma _ { 1 } \right. )$ to be a network-specific rescaling of $\epsilon$ . We therefore expect the adversarial error rate at small $\epsilon$ to be dictated by the cumulative distribution of $\Delta _ { 1 2 }$ for attacks that effectively target the second most likely class (e.g. FGSM, PGD ...etc.). + +# 3.2 UNIVERSAL PROPERTIES OF THE LOGIT DIFFERENCE DISTRIBUTION + +With the results from the preceding section in hand, we now investigate the distribution of logit differences, $P ( \Delta _ { 1 j } )$ . Since we are particularly interested in the small $\epsilon$ regime, we seek to compute $P ( \Delta _ { 1 j } )$ for small $\Delta _ { 1 j }$ . To make progress we will again make a mean field approximation and assume that each of the logits are i.i.d. with arbitrary distribution. With this approximation we find that for small $\Delta _ { 1 j }$ (see Appendix 6.2.2), + +$$ +P ( \Delta _ { 1 j } ) = C \Delta _ { 1 j } ^ { j - 2 } + \mathcal { O } ( \Delta _ { 1 j } ^ { j - 1 } ) +$$ + +where $C$ is a network specific constant. An interesting consequence of this result is that the difference that we are particularly interested in, $P ( \Delta _ { 1 2 } )$ , scales as $\mathcal { O } ( 1 )$ as $\Delta _ { 1 2 } 0$ . This implies that, generically, we expect the most likely logit and second most likely logit to have a finite probability of being arbitrarily close together. We interpret this as an inherent uncertainty in the predictions of neural networks. + +While it is not obvious that the assumption of a factorial logit distribution is valid here, we will see that Eq. (9) captures the universal features of the distribution at small values of the logit difference. Indeed, from the previous section we see that the form of Eq. (9) implies that the adversarial error rate should scale as follows + +$$ +P ( \hat { \epsilon } < \epsilon ) \approx P ( \Delta _ { 1 2 } < \tilde { \epsilon } ) \approx C \tilde { \epsilon } + \mathcal { O } ( \tilde { \epsilon } ^ { 2 } ) +$$ + +as was broadly observed in the previous section. + +To further test whether or not the mean field approximation is valid, we evaluate $\Delta _ { 1 j }$ for a number of different neural network architectures and datasets. We find that on all datasets and models studied, the $\Delta _ { 1 j }$ distributions have power-law tails. As predicted, $\Delta _ { 1 2 }$ has a power-law tail with an exponent of about 0, and $\Delta _ { 1 j }$ for $j > 2$ have power-law tails with positive exponents increasing with $j$ . We note, however, that the powers are typically not integral for large $j$ . It seems likely that this breakdown is the result of correlations between the logits. In Fig. 4, we compare distribution of $\Delta _ { 1 j }$ with $j = 2 , 3 , 4$ for ImageNet and logits that are independently sampled from a uniform distribution for 5 million samples with 10 classes. In Fig. 6 we see similar results for MNIST. Together these results verify our predictions over a vast set of networks and datasets. + +The empirical results above show that adversarial error has a power-law form for well-studied and random datasets, simple full connected networks as well as complicated state-of-the-art models. It follows that the prevalence and commonality of adversarial examples is not due to the depth of the model (for example, see the linear model in Fig. 2 (a)), or the high-dimensionality of the datasets. Rather, they are due to the fact that lots of examples have small $\Delta _ { 1 2 }$ values. This makes it easy to find examples to fool the model at test-time. + +It is interesting that the distribution of $\Delta _ { 1 j }$ , at small $\Delta _ { 1 j }$ , for trained models is essentially identical to that of i.i.d. random logits (especially for $j = 2$ ). This suggests that while our training procedures are good at modifying the largest logit in a way that leads to good clean accuracies, these procedures do not induce strong enough correlations between the logits to disrupt the essential scaling uncovered above. This problem is reminiscent of the problem distillation (Hinton et al., 2015) attempts to solve, by incorporating information about ratios of incorrect classes. This might be one of the reasons defensive distillation improves adversarial robustness. It would be interesting to study the distributions of logit differences during training of distillation networks. + +![](images/dfac972faeb31b0e791b25975de81e491597615df9ab61798b0539d59233dbbe.jpg) +Figure 4: (a) Distribution of $\Delta _ { 1 j }$ of NASNet-A trained on ImageNet. (b) Distribution of $\Delta _ { 1 j }$ for logits that are sampled independently from a uniform random distribution for 5 million samples with 10 classes. $\Delta _ { 1 j }$ of other models are in Appendix Fig. 14 + +Given the large density of small more robust. Our proposed los $\Delta _ { 1 2 }$ values, we study an entronction can be written as: $\begin{array} { r } { \mathrm { \bar { l o s s } } = \mathrm { o l d } \mathrm { \bar { l o s s } } - \lambda \sum _ { i = 1 } ^ { n } p _ { i } \log p _ { i } } \end{array}$ where is a hyperparameter, $n$ is the number of classes, and $p _ { i }$ are the outputs of the neural network. This regularization term has been used by Miyato et al. (2017) for semi-supervised learning tasks. It aims to increase the confidence of the network on each sample, which is the opposite of previous regularization attempts that penalized confidence to increase generalization accuracy (Pereyra et al., + +2017; Szegedy et al., 2015). By penalizing the entropy of the softmax outputs, we aim to increase the logit differences. + +In Fig. 5, we show that entropy regularization with a $\lambda = 4 . 5$ increases the adversarial robustness both for regularly trained networks and step l.l. adversarially trained networks, and both for permutation invariant and regular MNIST. We note that the same qualitative results hold for other values of $\lambda$ we tried. Despite the increase in adversarial accuracy, the permutation invariant MNIST model has $0 . 8 \%$ lower clean accuracy when trained with the entropy penalty. In Appendix Fig. 11, we show that a wide ResNet trained with the entropy regularizer has improved robustness with no loss in clean accuracy. + +![](images/c1869225e39aa09f2326fc23af8d98fa31a28439eb1d8da0aed3cd53b95f57fc.jpg) +Figure 5: Step l.l. attack adversarial accuracy as a function of $\epsilon$ for CNN and permutation invariant MNIST in (a) and (b), respectively. Regular training (purple), entropy regularization (red), adversarial training (green), and adversarial training with entropy regularization (blue) have been implemented. Adversarial training was done using the step l.l. method. In (c), we show the PGD attack adversarial accuracy on permutation invariant MNIST trained with and without step l.l. adversarial training. + +We investigate whether the increased adversarial robustness is due to increased logit differences. In Fig. 6, we plot the distribution of $\Delta _ { 1 j }$ for $j$ up to 4, for two networks trained on permutation invariant MNIST, with and without entropy regularization. As expected, margins are shifted to larger values and density of samples with small $\Delta _ { 1 j }$ are reduced. The tails still follow a power-law form with the same exponents, however there are fewer samples with small margins compared to a regularly trained network. Although entropy regularization made our networks white-box attacks, it did not lead to a significant improvement against black-box attacks. For this reason, we focus on the influence of network architectures on adversarial sensitivity below. + +![](images/77dfdc0bb5f0bde75a3a7b36d8a17c63ceb148055a493141d5e6090904603350.jpg) +Figure 6: Distribution of $\Delta _ { 1 j }$ up to $j = 5$ for permutation invariant MNIST trained with and without entropy regularization in red and purple, respectively. + +# 4 ROLE OF NETWORK ARCHITECTURES IN ADVERSARIAL ROBUSTNESS + +Several recent papers observe that larger networks are more robust against adversarial examples, regardless if they are adversarially trained or not (Kurakin et al., 2016b; Madry et al., 2017). However, it is not clear if network architectures play an important role in adversarial robustness. Are larger models more robust because they have more trainable parameters, or simply because they have higher clean accuracy? Is it possible to find more robust network architectures that do not necessarily have more parameters? We run several experiments to answer these questions, as well as to find an adversarially more robust model on CIFAR10. We perform neural architecture search (NAS) with reinforcement learning. Our search space and procedure are almost exactly the same as in Zoph et al. (2017). One difference is that we restrict the search space so that the normal cell must be the same as the reduction cell. This reduces the complexity of the search space as now we have only half as many predictions. Finally, we increase the number of prediction steps from 5 to 7 to slightly gain back the complexity that was lost when we restricted the normal cell to be equal to the reduction cell. + +We carry out two experiments: + +• Experiment 1: NAS where child models are trained with clean and step l.l. adversarial examples and the reward is computed on the validation set with FGSM adversarial accuracy at $\epsilon = 8$ . +• Experiment 2: NAS where child models are trained with clean and PGD adversarial examples and the reward is computed on the validation set with FGSM adversarial accuracy at $\epsilon = 8$ . + +In both experiments, child models are trained for 10 epochs on a training set of 25 thousand samples. Child models are trained on mini-batches where half of the samples are adversarially perturbed, following the procedure in (Kurakin et al., 2016b). At the end of each experiment, we pick the child model with the highest FGSM adversarial accuracy at $\epsilon = 8$ on the validation set of 5000 samples, and scale up the number of filters. We train the enlarged models for 100 epochs on the full training set of 45000 samples for 12 different hyperparameter sets, and pick the one with the highest adversarial accuracy on the validation set. Finally, we report below the performance of these models on a held-out test set of 10 thousand samples. To provide a comparison with the results of our two experiments, we also run a vanilla NAS where the reward is clean validation accuracy. We will refer to the best architecture from vanilla NAS as NAS Baseline. When trained using the setup above only on clean examples, NAS Baseline reaches a test set accuracy of $9 5 . 3 \%$ . + +![](images/45725418240ff962864ee69771205d42909fa7e90b2d0a5a3159452fdd5e53aa.jpg) +Figure 7: (a) step l.l. adversarial accuracy of NAS Baseline trained with and without step l.l. adversarial examples in green and red, respectively. Best model from Experiment 1 is shown in blue. (b) PGD adversarial accuracy of NAS Baseline trained with and without PGD adversarial examples in green and red, respectively. Best model from Experiment 2 is shown in blue. + +We present the results of Experiment 1 in Fig 7(a). Here the green curve is the adversarial accuracy NAS Baseline. The blue curve is the adversarial accuracy of a network architecture that was found by Experiment 1. Both of these architectures are trained with the same adversarial training procedure. We try the same sets of hyperparameters and report here the models with best adversarial accuracy at $\epsilon = 8$ on the validation set. Adversarial training reduced the clean accuracy by $0 . 2 \%$ . Adversarially trained models both have clean accuracy of $9 5 . 1 \%$ on the test set, whereas the model that was trained without adversarial training reached $9 5 . 3 \%$ accuracy. + +We next use PGD adversarial examples in the training of child models, to find architectures that are more robust to any adversarial attack within an $\epsilon$ ball (Madry et al., 2017). Following the training procedure by Madry et al. (2017), we use 7 steps of size 2, for a total $\epsilon = 8$ . We present the results of Experiment 2 in Fig. 7(b). As was the case in Experiment 1, the architecture found by adversarial NAS leads to a more robust model. At $\epsilon = 8$ , the architecture from Experiment 2 reaches a $17 \%$ higher adversarial accuracy on PGD examples. We compare our results to the results by Madry et al. (2017). Madry et al. (2017) trained only on PGD examples, whereas half of our minibatches are clean examples. Despite this, we match their accuracy on white-box PGD attacks. Against other white- and black-box attacks our model is more robust, and our clean accuracy is $5 . 9 \%$ higher. We also note that NAS Baseline model has 4.9 million trainable parameters, whereas the model from Experiments 1 and 2 have 2.3 million and 3.5 million parameters, respectively. NAS found an adversarially more robust architecture with many fewer parameters. Best architecture from Experiment 2 and NAS Baseline are presented in Appendix Fig. 16. Finally, we study the performance statistics of child models during NAS. In Fig. 8, we report the results for 9360 child models that were trained during Experiment 1. As explained above, these models are only trained for 10 epochs. In Fig. 8(a), we see that the correlation between adversarialy accuracy and the number of trainable parameters of the model is not very strong. On the other hand, adversarial accuracy is strongly correlated with clean accuracy (Fig. 8(b)). We hypothesize that this is the reason both Madry et al. (2017) and Kurakin et al. (2016b) found that making networks larger increased adversarial robustness, because it also increased the clean accuracy. This implies that commonly used architectures, like Inception v3 and ResNet, benefit from having more parameters. This however was not the case for most child models during NAS. On the other hand, having a high clean accuracy is not sufficient for adversarial robustness. As seen in Fig. 8(c), there is a large variance in the adversarial accuracy of models with good clean accuracy. The range of adversarial accuracies in the histogram of models with larger than $85 \%$ clean accuracy is $22 \%$ and the standard deviation is $2 . 6 \%$ . For this reason, our experiments led to more robust architectures than NAS Baseline. + +Table 1: Performance of our best architecture from Experiment 2 at $\epsilon = 8$ . Black-box attacks are sourced from a copy of the network independently initialized and trained. + +
White-boxBlack-box
CleanFGSMstep 1.1.PGDFGSMstep 1.1.PGD
Madry et al. (2017)87.3%56.1%150.0%67.0 %164.2%
This work93.2%63.6%77.9%50.1%78.1 %84.9%75.0%
+ +![](images/0207438d01698c7a498a79f722099d8aa9424e42d55368ce98b83ff84f3350d9.jpg) +Figure 8: Performance of child models on the validation set. FGSM adversarial accuracy at $\epsilon = 8$ vs. number of trainable parameters and clean accuracy in (a) and (b), respectively. Black dots represent each child model, purple line is a running average. Figure (c) is the histogram of the adversarial accuracy for models with clean accuracy larger than $85 \%$ . + +# 5 CONCLUSION + +In this paper we studied common properties of adversarial examples across different models and datasets. We theoretically derived a universality in logit differences and adversarial error of machine learning models. We showed that architecture plays an important role in adversarial robustness, which correlates strongly with clean accuracy. + +# REFERENCES + +Moustapha Cisse, Piotr Bojanowski, Edouard Grave, Yann Dauphin, and Nicolas Usunier. Parseval networks: Improving robustness to adversarial examples. 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Learning transferable architectures for scalable image recognition. arXiv preprint arXiv:1707.07012, 2017. + +# 6 APPENDIX + +# 6.1 FURTHER EXPERIMENTS + +![](images/b670571accf1ec3148d8641417d4a6b43fdf7c353720d8907853cc342b1807d3.jpg) +Figure 9: Adversarial error for hundreds of models trained on MNIST, including fully-connected and convolutional models. We only show models with clean accuracy larger than $80 \%$ . + +![](images/0cda1546173e20803b367c7e26e3db58e2dabb30fc58a20a86e6556b80a83117.jpg) +Figure 10: Points represent the adversarial error due to FGSM as a function of $\epsilon$ for a 32-layer ResNet trained on CIFAR10. Straight line is a power law fit with an exponent of 0.99. + +![](images/19c453a2f89cb39d463cccfb8014f80f6dddd6dcf92298ac8ab322959b8de828.jpg) +Figure 11: Effect of adding an entropy regularization $( \lambda = 3 . 0 )$ ): step l.l. adversarial accuracy of wide ResNet on CIFAR10, with and without entropy regularization. Both models have a clean accuracy of $9 4 \%$ . They were both trained for 100 epochs with the same hyperparameters. + +# 6.2 DERIVATIONS + +Here we derive several of the results found in the main text. + +# 6.2.1 LINEAR RESPONSE + +First, we compute the linear response of the network to the $L _ { 2 }$ FGSM attack. Let, $f : \mathbb { R } ^ { N } \to \mathbb { R } ^ { M }$ be a neural network (or other model) structured such that $f = \operatorname { s o f t m a x } \circ h$ where $h : \mathbb { R } ^ { N } \to \mathbb { R } ^ { M }$ maps inputs to logits. Additionally define a loss $L : \mathbb { R } ^ { M } \to \mathbb { R }$ which can be cross-entropy, $L ^ { 2 }$ , etc.. To generate adversarial examples we start with an input $\boldsymbol { x } \in \mathbb { R } ^ { N }$ and a corresponding target $t \in \mathbb { R } ^ { M }$ such that $t _ { \beta } = 1$ if $\beta = \gamma$ for some $\gamma$ and $t _ { \beta } = 0$ otherwise. We assume our network gets the answer correct so that $h _ { \gamma } > h _ { \beta }$ for all $\beta \neq \gamma$ . Then we apply the adversarial perturbation, + +$$ +x _ { \alpha } ^ { \prime } = x _ { \alpha } + \epsilon \frac { \nabla _ { x } L } { | | \nabla _ { x } L | | _ { 2 } } . +$$ + +Note that we can write + +$$ +\nabla _ { x } L = { \frac { \partial L } { \partial x _ { \alpha } } } = \sum _ { \beta } { \frac { \partial h _ { \beta } } { \partial x _ { \alpha } } } { \frac { \partial L } { \partial h _ { \beta } } } = \sum _ { \beta } J _ { \alpha \beta } { \frac { \partial L } { \partial h _ { \beta } } } = J \delta . +$$ + +Where we associate $J _ { \alpha \beta } = \partial h _ { \beta } / \partial x _ { \alpha }$ with the input-to-logit Jacobian linking the inputs to the logits and $\delta = \partial L / \partial h _ { \beta }$ the error of the outputs of the network. + +![](images/1d995c793653d9be88818c8a1da8189e47e3b0c071c1153919643df49d15ff93.jpg) +Figure 12: Fully connected network trained on MNIST with hinge loss. (a) Distributions of logit differences. (b) Red dots represent the adversarial error when FGSM attack uses the same hinge loss from training. Blue dots represent the adversarial error when FGSM attack uses a cross-entropy loss to create the adversarial examples.The line is a power-law fit with an exponent of 0.98 + +![](images/917b69d3b380562fee50ab581eb866d83a8e46674d25a3848452158d9ecbe7a4.jpg) +Figure 13: Fully connected network trained on MNIST with L2-norm loss. (a) Distributions of logit differences. (b) Black dots represent the adversarial error due to FGSM. The line is a power-law fit with an exponent of 1.02 + +We can compute the change to the logits of the network due to this perturbation. We find, + +$$ +\begin{array} { l } { \displaystyle h ( \boldsymbol { x } ^ { \prime } ) = h ( \boldsymbol { x } + \epsilon \nabla _ { \boldsymbol { x } } L / | | \nabla _ { \boldsymbol { x } } L | | _ { 2 } ) } \\ { \displaystyle h _ { \beta } ^ { \prime } \approx h _ { \beta } + \frac { \epsilon } { | | \nabla _ { \boldsymbol { x } } L | | _ { 2 } } \sum _ { \alpha } \frac { \partial h _ { \beta } } { \partial x _ { \alpha } } \frac { \partial L } { \partial x _ { \alpha } } + \mathcal { O } ( \epsilon ^ { 2 } ) } \\ { \displaystyle ~ = h _ { \beta } + \frac { \epsilon } { | | \nabla _ { \boldsymbol { x } } L | | _ { 2 } } \sum _ { \alpha \delta } \frac { \partial h _ { \beta } } { \partial x _ { \alpha } } \frac { \partial h _ { \delta } } { \partial x _ { \alpha } } \frac { \partial L } { \partial h _ { \delta } } } \end{array} +$$ + +where we have plugged in for eq. (11). Expressing the above equation in terms of the Jacobian, it follows that we can write the effect of the adversarial perturbation on the logits by, + +$$ +h ^ { \prime } = h + \epsilon \frac { J ^ { T } J \delta } { | | J \delta | | _ { 2 } } +$$ + +as postulated. + +![](images/c88352a66455538e3ceb30d420cf36b2a3ba9a8e1ccaa6907bbe9c8b2c3f27af.jpg) +Figure 14: (a) Distribution of $\Delta _ { 1 N }$ for other ImageNet models. + +![](images/0969b6740619fa492073f02b297f60c5b900679948f3b95b0f698427fc978a31.jpg) +Figure 15: (a) FGSM adversarial accuracy of NAS Baseline trained with and without step l.l. adversarial examples in green and red, respectively. Best model from Experiment 1 is shown in blue. (b) FGSM adversarial accuracy of NAS Baseline trained with and without PGD adversarial examples in green and red, respectively. Best model from Experiment 2 is shown in blue. + +6.2.2 UNIVERSAL PROPERTIES OF THE LOGIT DIFFERENCE DISTRIBUTION + +We now show that + +$$ +P ( \Delta _ { 1 j } ) = C \Delta _ { 1 j } ^ { j - 2 } + \mathcal { O } ( \Delta _ { 1 j } ^ { j - 1 } ) . +$$ + +To make progress we will again make a mean field approximation and assume that each of the logits are i.i.d. with arbitrary distribution $P ( h )$ . We denote the cumulative distribution $F ( h )$ . While it is not obvious that the factorial approximation is valid here, we will see that the resulting distribution of $P ( \Delta _ { 1 j } )$ shares many qualitative similarities with the distribution observed in real networks. + +We first change variables from the logits to a sorted version of the logits, $r _ { i }$ . The ranked logits are defined such that $r _ { 1 } = \operatorname* { m a x } ( \{ h _ { i } \} )$ , $r _ { 2 } = \mathrm { m a x } ( \{ h _ { i } \} \backslash \{ r _ { 1 } \} ) , \cdot \cdot \cdot$ . Our first result is to compute the resulting joint distribution between $r _ { 1 }$ and $r _ { j }$ , + +$$ +P _ { j } ( r _ { 1 } , r _ { j } ) = A ( N , j ) F ^ { N - j } ( r _ { j } ) \left[ F ( r _ { 1 } ) - F ( r _ { j } ) \right] ^ { j - 2 } P ( r _ { j } ) P ( r _ { 1 } ) +$$ + +where $A ( N , j ) = N ( N - 1 ) { \binom { N - 2 } { j - 2 } }$ is a combinatorial factor. Eq. (18) has a simple interpretation. $F ^ { N - j } ( r _ { j } )$ is the probability that there are $N - j$ variables less than $r _ { j }$ ; $[ F ( r _ { 1 } ) - F ( r _ { j } ) ] ^ { j - 2 }$ is the probability that $j - 2$ variables are between $r _ { j }$ and $r _ { 1 }$ ; $P ( r _ { j } ) P ( r _ { 1 } )$ is the probability that there is one variable equal to each of $r _ { 1 }$ and $r _ { j }$ . The combinatorial factor can be understood since there are $N$ ways of selecting $r _ { 1 }$ , $N - 1$ ways of selecting $r _ { j }$ , and $\binom { N - 2 } { j - 2 }$ ways of choosing $j - 2$ variables out of the remaining $N - 2$ to be between $r _ { j }$ and $r _ { 1 }$ . + +In terms of eq. (18) we can compute the distribution over $\Delta _ { 1 j }$ to be given by, + +$$ +\begin{array} { l } { { P ( \Delta _ { 1 j } ) = \displaystyle \int d r P _ { j } ( r + \Delta _ { 1 j } , r ) } } \\ { { \displaystyle \qquad = A ( N , j ) \int d r F ^ { N - j } ( r ) \left[ F ( r + \Delta _ { 1 j } ) - F ( r ) \right] ^ { j - 2 } P ( r ) P ( r + \Delta _ { 1 j } ) . } } \end{array} +$$ + +We can analyze this equation for small $\Delta _ { 1 j }$ . Expanding to lowest order in $\Delta _ { 1 j }$ , + +$$ +\begin{array} { l } { P ( \Delta _ { 1 j } ) \approx A ( N , j ) \displaystyle \int d r { \cal F } ^ { N - j } ( r ) \left[ { \cal F } ( r ) + \Delta _ { 1 j } P ( r ) - { \cal F } ( r ) \right] ^ { j - 2 } P ( r ) \left[ P ( r ) + \Delta _ { 1 j } \frac { d P ( r ) } { d r } \right] } \\ { \displaystyle ~ ( \Delta ( N , j ) \Delta _ { 1 j } ^ { j - 2 } \int d r { \cal F } ^ { N - j } ( r ) P ^ { j } ( r ) + \mathcal { O } ( \Delta _ { 1 j } ^ { j - 1 } ) . ~ } \end{array} +$$ + +Since the term in the integral does not depend on $\Delta _ { 1 j }$ the result follows with, + +$$ +{ \cal C } = { \cal N } ( N - 1 ) { \binom { N - 2 } { j - 2 } } \int d r F ^ { N - j } ( r ) P ^ { j } ( r ) . +$$ + +# 6.2.3 ARCHITECTURES + +![](images/156b53fcd92de96aae41c52a10d1da280a0afb23260f06022ef070bf60dba302.jpg) +Figure 16: Left: Best architecture from Experiment 1. Right: Architecture of NAS Baseline. 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